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POLITECNICO OF MILAN
SPECIALISTIC THESIS
Performance analysis of finite element computer
programs in the linear viscoelastic domain.
Author: Supervisor:
Giulia CIRILLI Dr. Andrea PISANI
A thesis submitted in fulfillment of the requirements
for the degree of Engineer of Structure
in the
Architecture, Built Environment and Construction Engineering
Department of the Politecnico of Milan
December 21, 2017
II
Table of contents
- Notation IV
- Introduction 1
- Basic equation of elasticity 2
- The Structural Problem for a linear elastic material 7
o Principle of Virtual Displacement 7
o The Rayleigh-Ritz method 9
o Matrix formulation of the displacement analysis 11
Example of application a frame 23
Example of application a cantilever beam 25
- The time-dependent behavior of the concrete 26
o The creep strain 27
o The creep function 28
o The relaxation function 30
o The link between creep function and relaxation function 32
o Volterra Integral 33
o The basic theorem of the linear viscoelasticity 34
First principle of the linear elasticity 34
Second principle of the linear elasticity 35
Principle of acquisition of the modified system stress distribution 35
- General model for concrete 39
o CEB 1990 model 39
o CEB 2010 model 43
o ACI model 46
o Bazant-Baweja Model (B3) 50
- Computation of the Volterra’s integral 53
- General method 54
III
o Trapezoidal rule 54
o Gauss 58
- Algebraic method 62
o M.S.M. 65
o E.M.M. 66
o A.A.E.M.M. 67
- Simplified method 69
o The hereditary model 69
o The Kelvin-Voigt model 70
o Aging model 72
o Dirichlet 75
- Structural analysis for viscoelastic material 77
o The link between the creep function and the relaxation function
for a three-dimensional problem 79
o Principle of virtual displacements for a three-dimensional problem 81
o Transformation of the coordinates 84
o Assembling phase 85
o Solution of the problem with the algebraic method 87
Example of application for a frame 95
Example of application for a cantilever beam 99
- Examination of the scientific literature 101
- MIDAS/Gen solution 108
• Numerical comparison 110
First example 110
Second example 119
• Building in construction 124
- Conclusion 147
- Bibliography 148
IV
Notation
Symbols used throughout most of the thesis are listed. Symbols less frequently used, or that have differ-
ent meanings in different contexts, are defined where they are used.
Mathematical symbols
Rectangular matrix or square matrix, diagonal matrix
Column vector, row vector
T Matrix transpose
1, T
Matrix inverse, transpose of inverse ( inverse od transpose)
Norm of a matrix or vector
Time differentiation; for example,
2 2/ , /du dt d u dtu u
, Partial differentiation if the following subscript is a letter, for example
2, / , , /x xyw w x w w x y
Latin symbols
A Area or cross-sectional area
a Generalized d.o.f. (also known as generalized coordinates)
B Spatial derivatives of field variables are [B]{d}
Cm Field continuity of degree m
C Dumping matrix; constrain matrix
D Displacement; flexural rigidity of a plate or a shell
V
D, d Nodal d.o.f. of structure and element, respectively
d.o.f. Degree(s) of freedom
E Modulus of elasticity
E Matrix of elastic stiffness; [E]=E in one dimension
F Body forces per unit volume
G Shear modulus
h Characteristic length; convective heat transfer coefficient
I Moment of inertia of cross-sectional area
I Unit matrix, also called identity matrix
J Creep function
J Determinant of [J]
K Spring stiffness, or bar stiffness AE/L, or thermal conductivity
K, k Conventional stiffness matrix of structure, element
Kσ, kσ Stress stiffness matrix of structure, element
L, LT Length of element, length of structure
L, m, n Direction Cosines
M, m Mass matrix of structure, element
Nels Number of elements
N Shape (or basis, or interpolation) functions
0, 0 Null matrix, null vector
P Externally applied concentrated loads on structure nodes
p Pressure; degree of a complete polynomial
q Distributed load, per unit length or per unit area
R Total load on structure nodes; eR P r
re Loads applied to nodes by an element
S Surface or surface area
T Temperature
t Thickness; time
T Transformation matrix
VI
U, U0 Strain energy, strain energy per unit volume
u, v, w, Displacement components in coordinate directions
u Vector of displacements, T
u u v w
V Volume
x, y, z Cartesian coordinates
Greek symbols
α Coefficient of thermal expansion; penalty number
Γ Jacobian matrix inverse, [Γ]=[J]-1
ε, ε0 Vector of strains, vector od initial strains
η Generalized axial stress
θx, θy, θz Rotation components about coordinate axes
κ, κ Matrix of thermal conductivities, vector of curvatures
λ Eigenvalue; Lagrange multiplier
ν Poisson’s ratio
ξ Damping ratio (ratio of actual damping to critical damping)
ξ, η, ζ Reference coordinates of isoparametric elements
Π A functional; for example, Πp=potential energy functional
χ Generalized curvature
ρ Mass density
σ, σ0 Vector of stresses, vector of initial stresses
Φ Modal matrix
Φ Surface tractions
ϕ Creep coefficient
ω, ω2 Circular frequency in radians per second, spectral matrix
1
Introduction
The concrete has a time-dependent development of mechanical and thermal properties. To compute
the stress or the strain on early age concrete, accurate material models are available. The description of
mechanical short-term properties is possible precisely with existing linear elastic models. The viscoelas-
tic behavior of early age concrete is still an object of research. In the last decades, many different mod-
els were made for creep and relaxation function of early age concrete. These models describe the creep
strain, as a function of the loading stress and the age of the material. For very simple model, such as
with constant loads, the creep strain can be computed with good accuracy, but for variable stress histo-
ries the creep strain must be calculated stepwise, based on the validity of the principle of superposition.
In the first part of the thesis, the Structural Problem and the dynamic response of the material are stud-
ied, with some hypothesis to simplify the model: a constant Poisson’s ratio and a linear elastic constitu-
tive law. Then, the time-dependent behavior of a viscoelastic material is analyzed, the meaning of creep
and shrinkage are explained, with their problems in computation. The solution of these problems is face
up with different method, such as the formulas are provided by law, the general method, the algebraic
method and the simplified models. The finite element method for a viscoelastic material is the main
topic of this elaborate, making the assumption that the material has a constant Poisson’s coefficient in
time.
Firstly, the scientific literature is analyzed, to have a better view of the problem. In most of the articles,
the propose solutions are always incomplete to define a full exact method in viscoelastic domain, for
different reasons, that we explain in the homonymous chapter. The literature shows to be enough poor
on the topic that we examine. Indeed, it is possible to develop a solution based on the algebraic
method, that can simplify the computation of the viscoelastic finite element method. This way brings to
considerably reduce the time of calculation, because the solution of the algebraized method, that is
start being set, a solution that is comparable to a double elastic law. The problem is reduces from a con-
volution integral to a linear problem that , surely, has fewer problems in the computation.
Then it was possible to compare the method of solution by the Gauss approximation, with a computer
program called MIDAS, giving a couple of examples on the application.
Time domain finite element analysis models are often considered to assess the dynamic behavior of
solid concrete structure, especially for tall building, that has a long constructional period. In the period
of these research, we analyze the problem on a building, that is going to be made the next year. It per-
fectly fits the characteristic of a model where it is possible to underline the problem.
2
Basic equations of Elasticity
The Computational Mechanics is the science that create methods and tools to represent in mathemati-
cal form all the physical phenomena. The calculation softwires greatly simplified the solution and the
development of the mathematical models. The mechanical representation of the structural engineering
can be the Linear Elastic Problem, which is the simplest theorist way. It is composed by linear equation
and it gives a good approximation for problems with a simple geometry and not considering the viscos-
ity of the material which is going to be analyzed in the next chapter.
The linear elastic problem considers the materials as continua, which means the matter fills the entire
region off the space it occupies, despite the fact the matter is made of atoms and is discrete. For a con-
tinuous ad regular substance, the derivatives are definable. More precisely it is going to be considered a
Cauchy’s element which is characterized by the infinitesimal measure of stresses and of strains. The ma-
terial is linear elastic. The external forces are supposed to be quasi-static and the strains of the body,
considerable small. Thanks to this last hypothesis, the equilibrium configuration of the body can be
taken as the initial configuration.
Figure 1. Generic deformable body
Defining the known values:
- the volume forces F that act on the volume (self-weight, inertia forces, …);
- the surface forces f, acting on S, are known on the unconstrained surface Sf;
- imposed displacement s0 on the constrained surface Su.
To define the problem in an easy way we are going to assemble the known values in column matrixes:
3
1 1 01
2 2 0 02
3 3 03
; ;
F f s
F F f f s s
F f s
(0.1)
The incognita, that allows to define the stress field of the body, are:
- the displacement tensor s(x) defined by 3 scalar components sj, j=1,2,3;
- the strain tensor ε(x) or Cauchy’s strain tensor, which is a double symmetric tensor and it is
characterized by 6 independent scalar components εij(=εij), j,i=1,2,3;
- the stress tensor σ(x) or Cauchy’s stress tensor, which is a double symmetric tensor and it is
characterized by 6 independent scalar components σij(=σij), j,i=1,2,3.
Figure 2. Behavior of the stresses acting on an element of infinitesimal volume
These incognita can be expected in three column matrices in the following way:
11 11
22 22
1
33 33
2
12 12
3
23 23
31 31
; ;2
2
2
s
s s
s
(0.2)
The triplet of Cartesian axes is (x, y, z), so all the variables can be defined as:
4
0
0 0
0
; ; ; ; ;
x x
y y
x x x x
z z
y y y y
xy xy
z z z z
yz yz
zx zx
F f s s
F F f f s s s s
F f s s
(0.3)
The governing equation, at the base of the linear elastic boundary value problem, are based on tensor
partial differential equations for the equilibrium and infinitesimal strain displacement relation, to ensure
the compatibility. The system of differential equation is completed by the constitutive law, that is repre-
sented by a set of linear algebraic relation for a linear elastic material. If the material cannot be simpli-
fied as elastic, a more complex viscous-elastic constitutive law, which is characterized by significate inte-
grals, replace the linear elastic equations.
The equilibrium equations are represented by three partial differential equation (PDE) of the first order,
linear, defined on the volume V of the body and by three boundary conditions on the free surface of the
body (Sf):
0 in VF (0.4)
in S fn f (0.5)
where 𝜕 is a differential operator of equilibrium and n is the normal unit vector always going out of the
boundary of the body, defined by the following matrixes:
/ 0 0 / 0 /
0 / 0 / / 0
0 0 / 0 / /
x y z
y x z
z y x
(0.6)
0 0 0
0 0 0
0 0 0
x y z
y x z
z y x
n n n
n n n n
n n n
(0.7)
5
Figure 3. Stresses acting on the volume of the element posed on the discontinuity surface of the stress field
The compatibility equations are defined by six partial differential equations of the first order, linear and
defined on the volume:
Ts (0.8)
They explicit that the small displacement tensor represents the symmetric part of the displacement gra-
dient. These equations are completed by the boundary conditions on the constrain part of the body (Su):
0s s (0.9)
The constitutive law depends on the nature of the material. The linear elastic constitutive law is repre-
sented by six linear equation that connect the stresses to the strains. The link is made by the elastic ten-
sor, a quadruple tensor. It can be written in a direct or indirect form respectively by the stiffness E or the
yielding E-1:
E (0.10)
1
E
(0.11)
Considering an isotropous elastic material, with a constant elastic modulus E and Poisson’s ratio ν, the
yielding matrix is defined as follow:
1 1
1 0 0 0
1 0 0 0
1 0 0 01 1
0 0 0 2(1 ) 0 0
0 0 0 0 2(1 ) 0
0 0 0 0 0 2(1 )
EE E
(0.12)
6
Defining the constant term 𝜆:
𝜆 =𝜈
(1 + 𝜈)(1 − 2𝜈)
equal to the Lamè constant divided by the elastic modulus, it is possible to define the stiffness matrix as
follow:
1 0 0 0
1 0 0 0
1 0 0 0
10 0 0 0 0
2 1
10 0 0 0 0
2 1
10 0 0 0 0
2 1
E E E
(0.13)
The expressions (0.10) and (0.11) become:
E (0.14)
11
E
(0.15)
7
The Structural Problem for a linear elastic material
The goal of this chapter is to describe the Finite Element Method, for problem with a linear elastic con-
stitutive law, in small stresses and small displacement.
In the chapter, it is shown the elastic problem and the difficulties of the final equations. The numerical
methods permit to find a good approximation of the solution, also for structures with complex geome-
tries and particular dispositions of the loads. The Finite Element method is the greatest of these numeri-
cal methods and it is based on the principle of virtual displacement, that is described below, or the sta-
tionarity of the potential energy.
The principle of virtual displacement
The principle of virtual work describe that the work of the active forces is zero for every virtual displace-
ment, which is infinitesimal and allowed by the constrains. If the body is in equilibrium, the work of the
acting forces is zero for every virtual displacement. For deformable bodies, the principle of virtual work
is still valid considering the internal work associated with the virtual deformation of the bodies.
The principle of virtual displacement says that, the principle of virtual work is a necessary and sufficient
condition for the equilibrium of a rigid body (or of a system of rigid bodies), for every small displacement
cinematically admissible that is imposed on the initial configuration. An infinitesimal displacement field s
imposed to the body in the initial configuration can be considered as the equilibrium configuration, if
the hypothesize of small displacements is made. If the displacements are possible with the constrain
whom assigned displacement are s0 and the associated small displacement field ε. It needs to be verified
the following conditions:
1
in V2
Ts s (1.1)
0 in Sus s (1.2)
The cinematically admissible displacement field with the assigned subsidence are infinite. Among these
there is the effective displacement produced by the load acting on the body.
The principle of virtual displacement is demonstrable showing that the stress field σ* is statically admis-
sible. A group of force of volume F*, a group of force of surface f* and of stresses σ* are definable stati-
cally admissible, if they verified the following conditions:
0 in VF (1.3)
in S fn f (1.4)
For hypothesis:
00 * * ( * ) *
f u
T T T T
V S S VF s dV f s dS n s dS dV (1.5)
8
∀ s cinematically admissible, where * is a generic double symmetric tensor and n is the unit vector
always perpendicular to the surface. It’s possible to define that:
1
* *2
T T Ts s (1.6)
Using the symmetry of the tensor *:
1
* *2
T T Ts s s (1.7)
That can also be write as:
* * *T T Ts s s (1.8)
Using the theorem of the divergence on the first term as follow:
* * *T T T
V S Vs dV s n dS s dV (1.9)
The equation (1.5) becomes:
00 * * ( * ) * ( * )
f u
T T T T T
V S S V SF s dV f s dS n s dS s dV n s dS (1.10)
That become:
0 * * * *f
T T
V SF s dV f n s dS (1.11)
It needs to be valid for all the displacement field s , so it must be zero in all the volume V and the free
surface Sf:
* * in VF (1.12)
* * in S ff n (1.13)
Which means that the stress field * is statically admissible with forces *F in V and *f on Sf.
Considering the static of σ, F, f coincident with the reality, the hypothesis of small displacement, on the
body at the initial configuration, is made, with an infinitesimal variation, compatible with the constrain
and respecting the compatibility with the real displacement δs, δε. The kinematic variables need to sat-
isfy the following compatibility equations:
1
in V2
Ts s (1.14)
0 in Sus (1.15)
In conclusion, the principle of virtual displacement permits to say that a sufficient condition for the equi-
librium of a deformable body is that:
9
f
T T T
V V SdV F s dV f s dS (1.16)
∀ δs, δε real displacement and strain field.
The main points of the Finite Element Method are two. The first is the description of the deformation of
the body with a finite number of variables or degrees of freedom, associable to the Rayleigh-Ritz
method. The second is the division of the continuous body in finite elements properly connected be-
tween each other’s.
The Rayleigh-Ritz method
In the following chapter is explained the application of the Rayleigh-Ritz method for a generic linear
elastic problem. It is an important approximation for the solution for a hypothetic elastic problem, that
is solved by the principle of virtual displacement.
Among all the solution kinematically admissible, the real one, that is also in equilibrium, respect the fol-
lowing condition:
f
T T T
V V SdV F s dV f s dS (2.1)
𝛿휀̂, 𝛿�̂� 𝜖 𝑌 ; 𝑌 ≡ {𝛿휀̂, 𝛿�̂� ∶ 𝛿휀̂ = 𝐶 𝛿�̂� 𝑖𝑛 𝑉; 𝛿�̂� = 𝛿�̂�0 𝑖𝑛 𝑆𝑢}.
The class Y of all the kinematically admissible solution contains the exact one of the elastic problem. The
Rayleigh-Ritz method searches the best solution in a subclass �̃� of the class Y minimizing the value of the
equation to get closer to the real solution represented by the satisfaction of the equation.
The definition of the subclass �̃� is represented by an algebraic expression for the displacements and
strains. This expression must respect the compatibility equation on the domain and on the boundary.
The subclass �̃� is represented by a finite number of parameter n, contained in the vector u. The dis-
placement is expressed as a linear combination of known functions:
( )s N x u (2.2)
N(x) are the shape functions that are polynomial and trigonometric for the Rayleigh-Ritz method.
The model of the strains is derivate from the model of the displacement through the linear compatibility
previously shown that affirms 1
2
Ts s . The strains is defined as follow:
( )B x u (2.3)
B(x) is a matrix composed by the derivatives of the shape functions.
10
The displacements s and the strains respect the compatibility condition in an exact form. Substituting
these parameters, the equation of the principle of virtual displacement (2.1) and substituting the first
stress with the constitutive law (0.10), it can be written as:
f
T T T T
V V Su B EB udV F N udV f N udS (2.4)
or
f
T T T T
V V Su B E B udV F N udV f N udS (2.5)
Once stated that vector u is independent of the local cartesian coordinates, eq.(2.5) can be written as
f
T T T T
V V Su B E B dV u F N dV u f N dS u (2.6)
Eq. (2.6) implies that:
0T T T
u k P (2.7)
where:
T T T
V V Vk B EBdV B E BdV E B BdV (2.8)
f
T T T
V SP F NdV f NdS (2.9)
Knowing that the stiffness matrix k is symmetricT
k k , the expression (2.7) becomes:
0ku P (2.10)
The main linear system (2.10) has n incognita in the vector u that govern the elastic problem with the
Rayleigh-Ritz approximation and it can be obtained also by the stationarity of the potential energy. Once
obtained the vector u , it is possible to compute the displacement, the strains and the stresses associ-
ated to it:
( )s N x u (2.11)
( )B x u (2.12)
( ( ) )E B x u (2.13)
The solution of the problem satisfies precisely the compatibility and the linear elastic constitutive law;
the equilibrium conditions are partially respected. The principal difficulty of the Rayleigh-Ritz method is
to fine the right displacement model that satisfies the compatibility condition on the boundary for every
11
shape. The FEM applies the condition of the compatibility on each part of the full body and the problem
of the Rayleigh-Ritz method is reduced.
Matrix formulation of the displacement analysis
Since the sixties the Finite Element Method had become the most general and used method for the so-
lution of partial differential equations. The develop of the calculator permitted to advance this method
and to solve the big calculation that are at the base. The FEM has no limit to the solution of solid me-
chanic problems.
The hypotheses of the FEM are:
- small strains and stresses
- linear elastic material
- no dynamic effects
The fundamental ides at the bases of the method are the subdivision of the element in a lot of small
parts, called Finite Element, and the introduction of a displacement model on each FE. Each part is de-
fined by a finite number of parameters that are the degrees of freedom of the system. The displacement
model is the same used by the Rayleigh-Ritz model for a generic body. The phases of the method are the
followed:
1. idealization
2. Discretization
3. Modelling
4. Transformation of coordinates
5. Assembling
6. Writing of the Principle of virtual displacement
7. Solution of the linear system
8. Reconstruction of the full solution
The idealization is the main part that permit to simplify the problem in an easily treatable scheme with
the equation of the continuous or structural mechanic. The stresses are representable by the static
loads, the dynamic loads, the thermic parameters or others. For the correct schematization of the prob-
lem it is important to analyze all the stress acting on the structure. The behavior of the material needs to
be chosen as close as possible to the real answer of the material with the applied loads. Most of the
cases are acceptable with the elastic linear scheme for low level of loads. The geometry must be simpli-
fied giving particular attention to those parts where the stresses and the strain is going to be computed.
12
Figure 4. Construction and relative structural scheme
The discretization is the subdivision of the continuous in Finite Elements. It is constructer a grid of FE,
called mesh. The thickness of the mesh is smaller where more precise information are requested or
where the stress model changes faster. On the contrary the mesh thickness is large in all the rest of the
model to reduce the computation costs. The ratio between the maximum dimension and the minimum
dimension, called shape ratio, needs to be not too far from 1 to have good numerical results. It is also
important to overlap the boundary of the elements and the discontinuity surfaces of the material. This
part is easily automatable, but it is still fundamental to choose the shape of the finite elements, the den-
sity and sometimes the total number. It is choose a real example with a 2D beam element. The structure
is characterized by a simple geometry easy to analyses and discretize, to arrive in conclusion to approxi-
mate structural theory that greatly simplify the study. These kinematic theories are based on some hy-
pothesis on the displacement field. It is assumed to be an Eulero-Bernoulli beam model that has at the
basis the main hypothesis: after deformation the cross-section, that firstly is orthogonal to the main
axis, continues to be plane, orthogonal to the main axis and it conserve the same shape. It is possible to
notice the changing in the figure below, that in in the next phase is going to explain the displacement
field.
Figure 5. Kinematic of the Eulero-Bernoulli model of the beam
13
The conditions that the cross sections section continue to be plane after deformation and they conserve
the orthogonality with the main axis, mean that the strain εx varies linearly on the height of the beam.
Figure 6. Example of subdivision of a structure of finite elements, with a thinner mesh
The modelling phase is composed by choosing the model for each displacement field on each finite ele-
ment of the mesh. The introduction of this model has the same consequence of the Rayleigh-Ritz prob-
lem: the deformed shape is reduced compare to the real possibility of the system.
Figure 7. Finite Element for plane and tridimensional problem
14
The degrees of freedom are chosen on the significative point of the element, called nodes. Using a soft-
ware that solve this type of problem, the only choice is about the best type of element to solve the
problem.
Figure 8. Continuous body discretized in finite elements
Now, it is given attention to a general finite element i, that is part of the continuous discretized in ne ele-
ments. To describe a good model, the references are made on a local reference system, generally differ-
ent from the global one and the coordinate in the new system are represented in the vector xL. The ele-
ment i is composed by n nodes that are characterized by the nodal displacement components UiL. The
model of the displacement for the Rayleigh-Ritz method views in the previous chapter is:
( ) ( )L L L
i iis x N x u (3.1)
Figure 9. Singular finite element and relative nodal displacement
15
The matrix ( )L
iN x contains the interpolation functions that defines the dependence between the dis-
placement in the finite element and the displacement parameters of the nodes. These functions are
called shape functions and the matrix is the shape function matrix. The definition of this matrix is the
main problem to formulate the displacement model. At this point it is easy to derive the strain model on
the element i by the linear compatibility contained in the operator C:
( ) ( ) ( )L L L L L
i i iiix C N x u B x u (3.2)
Based on the hypothesis of a linear elastic material, governed by the stiffness matrix di ,the stress model
can be computed:
1 ei n
In this model, there are also the inelastic strain i
p that are supposed to be known. The material is con-
sidered homogeneous with a matrix i
d constant in the element i.
The displacement fields for the Eulero-Bernoulli beam, that is previously introduce, is the following, as
the figure 5 explain clearly:
( , ) ( ) sin ( )
( , ) ( ) (1 cos ( ))
x
y
s x y u x y x
s x y v x y x
(3.3)
The components u(x) and v(x) are the displacement component of the main axis in direction x and y, and
ϕ(x) is the rotation of the generic section of the beams. For infinitesimal angles of rotation, the displace-
ment field becomes:
( )( , ) ( )
( , ) ( )
x
y
dv xs x y u x y
dx
s x y v x
(3.4)
The displacement field is completely defined by the terms u(x) and v(x) that are the generalized dis-
placement of the Eulero-Bernoulli beam model, in matrix form:
( , ) ( )
; ( , ) ( )
x
y
s x y u xs u
s x y v x
(3.5)
The kinematic model is expressed by the following relation:
( , ) ( )s x y nu x (3.6)
Where the matrix n is the differential operator:
1
0 1
dy
n dx
(3.7)
16
The condition on the boundary needs to be define by a constrain for the displacement in x, y and the
rotation or a fixed value. The strain field is characterized by the only component not equal to zero:
2
2
( , ) ( ) ( )( , ) x
x
s x y du x d v xx y y
x dx dx
(3.8)
It can also be written the following form:
( , ) ( ) ( )x x y x y x (3.9)
Where ( )x is the axial deformation of the centroid fiber and ( )x is the curvature of the deformed
axis (the elastic line). Both are the generalized strain of the Eulero-Bernoulli beam model. The strain
field is now completed, the vector of the generalized strain is:
( )
( )
xq
x
(3.10)
The strain field can be expressed in the matrix forma as follow:
( , ) ( ) ( )x y b y q x (3.11)
where the vector ε, in this case, contain the only component εx and b is a matrix where is contained the
kinematic constrain:
1,b y (3.12)
The generalized variables of deformation and displacement are connected by the following relation:
q u (3.13)
Where the differential operator of linear compatibility is:
2
2
0
0
d
dx
d
dx
(3.14)
The connection is the same formally given by the internal compatibility equation for a generic body.
It is possible now to write the principle of virtual displacement as in the Rayleigh-Ritz method [(2.1) -
(2.9)]:
i i fi
T T T
ii i i iiV V SdV F s dV f s dS (3.15)
The same passage of the previous chapter is made to arrive at the final form:
i i fi
LT T L T L T L
ii i i iii i i iiV V Si
u B E B dV u F N dV u f N dS u (3.16)
17
All the shape matrix and the derivative of the shape matrix depends on the singular finite element i. The
elastic modulus is also dependent on it because the structure is not homogeneous and the properties
varies for each finite element i. The considered element has an external surface that can be divided in
three zones: one that is free Sf, one that is constrained Su and one that is link to the other elements of
the mesh, called interface Si. It is possible to write the following linear system adding the new parts of
the external work due to the reacting forces r on the constrained surface Su and to the interaction forces
rI that the element exchange with the other elements on the interface SI:
T T T
LT L L L L L L L L
i i Iii i i iiu k u P u R u R u (3.17)
where
i
L T
i i ii Vk B E B dV (3.18)
i Ii
TL T T
i i iV SP F N dV f N dS (3.19)
ui
TL T
i iSR r N dS (3.20)
Ii
TL T
Ii I iSR r N dS (3.21)
The linear system represents the equilibrium condition at the nodes of the finite element i for the forces
applied on the loaded surface, the reactive forces on the constrain surface, the interaction forces at the
connection boundary and the elastic forces due to the deformation of the element. The stiffness matrix
of the singular finite element is symmetric, so the transpose is equal to the normal matrix T
L L
i ik k .
The transformation of the coordinate is due to the difference of local referment system chosen in the
previous phase with the global one.
Figure 10. Global and Local reference system for a generic EF
18
To refer everything to the same reference system, a matrix of rotation Ti is introduced. It connects the
displacement vector of the finite element i in the local reference system, to the one expressed in the
global coordinates:
L
i iiu T u (3.22)
Substituting in the previous models we obtain:
( ) ( ) ( )
( ) ( ) ( )
L L L L
i i iii i
L L L L
i i ii i i
s x N x u N x T u
x B x u B x T u
(3.23)
The Assembling phase is important to arrive to an only model referred to the continuous or to the struc-
ture. Starting from the kinematic model of each finite element it is possible to find the degrees of free-
dom that are in common for different finite elements mutually linked. It is necessary to impose condi-
tions between the nodal displacement of finite elements close each other. These are the compatibility
conditions which are expressed by the connectivity matrix i
L where the parameter is 0 or 1. A vector of
displacement U is defined. It contains all the displacement of the nodes of the mesh without repetition
and considering also the nodes of the constrain boundary:
i=1 ni eiu L u (3.24)
Figure 11. Assembling phase to come back to the continuous body composed by different FE
Substituting in the models it is obtained:
( ) ( ) ( )
( ) ( ) ( )
L L L
i ii i ii i
L L L
i ii i i i i
s x N x T u N x T L u
x B x T u B x T L u
(3.25)
The constitutive law for a linear elastic multidimensional material is expressed in the formula (0.10), it is
possible to write:
( ) ( ) ( ) ( )L LL L
i i i i i i i ix E x E B x T L u E B x T L u (3.26)
19
The single finite element I is taken into account, isolated from the total mesh. It is developed now the
principle of virtual displacement for the full model composed by summing all the components of each ne
elements:
Ku P R (3.27)
where:
1
enT L
i iii
K L k L
(3.28)
1
enT L
iii
P L P
(3.29)
1
( )en
T L L
i Iiii
R L R R
(3.30)
In the last expression obtain by the Principle of Virtual Displacement it is missing the component due to
the interaction forces RI on the interfaces of the elements because it is zero for the action-reaction prin-
ciple. The meaning of the stiffness matrix is: the element Kij represent the force on the degree of free-
dom i when it is assigned a unit value to the degree of freedom j and zero to all the rest. The hat on the
displacement vector u means that the compatibility equations are respected.
Before to find the solution of the linear system, the external constrains need to be imposed. This is pos-
sible dividing the degrees of freedom in two parts: the nodal displacement of the free nodes LU and the
known nodal displacement of the constrain nodes VU . In consequence also the stiffness matrix K and
the two vectors P and R are divided, the vector P contains only known values, the part of the vector
R linked to the free nodes is zero and the other is incognita:
0LL LV LL
V VVVL VV
K K u P
u P RK K
(3.31)
The equations of the linear system are the equilibrium condition at the nodes of the mesh of finite ele-
ments. On every free node acts forces equivalent to the surface and volume loads and elastic forces due
to the elastic deformation of the body. The incognita must satisfy the equilibrium on each node of the
mesh. The system is composed by two subsystems of equation:
LL VLL LV
V VL VVL VV
K u K u P
K u K u P R
(3.32)
In the first system, the incognita are in the vector Lu and in the second one, they are in the vector VR .
The matrixLL
K is not singular so it is invertible, it is possible to solve the first system for the vector
Lu and the second for VR once it is known the other incognita vector:
20
1( )LL VLL LV
V VL VVL VV
u K P K u
R K u K u P
(3.33)
In the calculation software, the longest phase is inversion of the stiffness matrix to compute the dis-
placement of the free nodes. For linear elastic problems, the stiffness matrix is symmetric and positive
defined. Another property of the stiffness matrix is to be a band matrix, which is a sparse matrix whose
non-zero entry are confined to a diagonal band, comprising the main diagonal and zero or more diago-
nals on either side. This effect is justified by the meaning of the stiffness matrix components, the entries
out of the diagonal are forces on the nodes when are acting degrees of freedom of other elements.
To simplify the model, the free nodes are considered and the system becomes:
LLLL
K u P (3.34)
which is:
K u P (3.35)
In this way the vector of the reaction forces in the constrain VR disappear and the only incognita is the
free node displacement vector u .
For the Eulero-Bernoulli beam model the generalized stresses can be computed identifying the multi-
plied coefficient of the virtual strain in the expression of the internal work. The same operation can be
done to compute the generalized displacement in the external virtual work. The matrix Q, that contain
all the generalized stresses, is:
x
T A
x
A x
A
dAN
Q b dAMydA
(3.36)
The generalized stresses N, M are the resultants of the axial stresses acting on the section and the re-
sultant moment of these stresses on the main inertial axis z. The virtual internal work for a piece of
beam of finite length L can be written as follow:
0 0 0
( ) ( ' '')
L L LT
iL Q q dx N M dx Nu Mv dx (3.37)
Where the apex means the derivation in x.
For the beam element of the Eulero-Bernoulli model, in the same way, the equality of the internal and
external virtual work is imposed to obtain the equilibrium equation at the basis of the problem. The vir-
tual displacement equation is considered as a sufficient condition for the equilibrium. A piece of beam of
length L, on which are acting external distributed forces axial n(x) and transversal p(x) and moment at
the edge, is shown in the figure.
21
Figure 12. Piece of beam with the external and internal forces in evidence
The external virtual work can be written as follow:
0 0
0
( ( ) ( ) ( ) ( )) (0) ( ) ( )
L
e LL n x u x p x v x dx H u V v x W L (3.38)
Imposing the principle of virtual work, so Li=Le for every virtual kinematic of the beam, it is possible to
arrive at the equilibrium equations at the basis of the problem. The equality is expressed by:
0
0
0 0
0 ' '' 0 0 '
' 0 0 ' ' ' 0 ' 0
L
L L
L
N n u M p v dx N L H u L N H u m L V v L
M V v M L W v L M W v
(3.39)
The equilibrium equation for each piece of beam is derived:
( )dN
n xdx
(3.40)
2
2( )
d Mp x
dx (3.41)
And the relative boundary conditions:
0(0)N H , 0(0)M W
,0
0
dMV
dx
,
( ) LN L H,
( ) LM L W ,
L
L
dMV
dx
.
22
Supposing that the material is linear elastic, the model of the Eulero-Bernoulli beam can be completed.
It is possible to define the constitutive law in the definition of Q in a direct form. Then it is possible to
use the kinematic model ( . ) ( ) ( )x x y b y q x to arrive at the definition of Q=Q(q). The beam is com-
posed by a homogeneous linear elastic and isotropic material, characterized by a constant elastic modu-
lus E and a constant Poisson coefficient ν. The hypothesis at the basis of the model is that the only com-
ponent not equal to zero is εx, so the connection between the stresses and the strain can be expressed
in the direct form by:
( 2 ) , , 0.x x y z x xy yz xzG (3.42)
In terms of generalized variables is expressed by:
( 2 ) ( 2 )T T T
x x
A A A
Q b dA b G dA b G bdA q Eq
(3.43)
In a matrix form can be explicit by:
( 2 ) 0
0 ( 2 )
N G A
M G J
(3.44)
Where A is the section area and J is the moment of inertia of the section referring to z. this connection
makes the beam too much stiff for the initial kinematic hypothesis that all the transverse section stays
plane after defamation. In the reality, they are free to contract, except for the zone close to the con-
strains. For this, to renounce a perfect kinematic formulation, the edometric modulus ( 2 )G can be
substitute to the elastic modulus of the material:
0
0
EAE
EJ
(3.45)
This underline the result of the Saint Venant’s pressoflession. The reconstruction of the full solution is the final part of the Finite element method. This phase com-
putes the displacements, the strains and the stresses for each finite element. The equations are those of
the previous phases:
( ) ( )
( ) ( ) i=1 n
( ) ( )
L L L
i ii
L L L
i i ei
L L L
i ii i
s x N x u
x B x u
x E B x u
(3.46)
The final solution satisfies exactly the constitutive law and the compatibility equation, but it is approxi-
mate for the respect of the equilibrium conditions.
23
Example for a frame
The frame in the following figure is consideres.
Figure 13. Example of an iperstatic frame
The axial and the bending effects are uncoupled and the deformations are small. The frame is character-
ized by 6 degrees of freedom, three for each node:
i
i i
i
u
u v
(3.47)
Figure 14. Unknown displacements of the frame
The element displacement coordinates are referred to a local axis ,x y . As in the explanation of the
finite element method, the local reference system can be transformed to the global (x, y) one through
the transformation matrix T :
24
i iu T u (3.48)
11
11
1 1
2 2
22
22
cos sin 0 0 0 0
sin cos 0 0 0 0
0 0 1 0 0 0
0 0 0 cos sin 0
0 0 0 sin cos 0
0 0 0 0 0 1
u u
v v
uu
vv
(3.49)
Figure 15. Global and local coordinates in evidence
It is now possible to write the element matrix equations for the computation of the virtual displacement
vector being everything known:
13 2 3 2
1
2 21
2
2
23 2 3 2
2 2
0 0 0 0
12 6 12 60 0
6 4 6 20 0
0 0 0 0
12 6 12 60 0
6 2 6 40 0
i
EA EA
l l
EI EI EI EIu
l l l lv
EI EI EI EI
l l l lP
EA EA u
l lv
EI EI EI EI
l l l l
EI EI EI EI
l l l l
(3.50)
The problem is computed since the displacement are calculated.
25
Example of a cantilever beam
For a cantilever beam subjected to a concentrated load P [39], we solve the problem by replacing the
force with equivalent nodal forces acting at each end of the beam as it is shown in the figure.
Figure 16. Cantilever beam subjected to concentrated load and relative simplification
The right-end vertical displacement and rotation need to be computed. In the discretization of the
beam, here, only one element is considered, with nodes at each end of the beam. The concentrated
load are substitute by the appropriate loading case. The equation (3.35) can now be written, substitut-
ing the right beam element stiffness matrix as follows:
2
23
2
12 6 2
6 4
8
P
vLEI
PLL Ll
(3.51)
26
The time- dependent behavior of the concrete
The viscosity for the concrete is a strain added to the elastic and plastic ones. It is very important be-
cause it arrives to considerable value compared to the elastic strain. In the constitutive law, it has a di-
rect dependence with the time even without changes in the tensional state. It is composed by a stress-
dependent strain and a stress-independent strain.
The concrete strain components are simplified in the following formula by the MC 2010:
0 0 0( , ) ( ) ( , ) ( ) ( )c ci cc cT cst t t t t t t (4.1)
Or
0 0( , ) ( , ) ( )c c cnt t t t t (4.2)
휀𝑐𝑖(𝑡0) is the initial strain at loading (usually the elastic strain);
휀𝑐𝑐(𝑡, 𝑡0) is the creep strain at time 𝑡 > 𝑡0;
휀𝑐𝑇(𝑡) is the thermal strain;
휀𝑐𝑠(𝑡) is the shrinkage strain;
휀𝑐𝜎(𝑡, 𝑡0) is the stress-dependent strain: 휀𝑐𝜎(𝑡, 𝑡0) = 휀𝑐𝑖(𝑡0) + 휀𝑐𝑐(𝑡, 𝑡0);
휀𝑐𝑛(𝑡) is the stress-independent strain: 휀𝑐𝑛(𝑡) = 휀𝑐𝑇(𝑡) + 휀𝑐𝑠(𝑡).
Shrinkage is a time-dependent deformation associated with a volumetric change of the cement paste
which involves a decrease in the volume of unrestrained concrete members during hardening and dry-
ing. The drying shrinkage is the first shown type of shrinkage due to changes in surface energy of desic-
cate gel particle. The deformation of the concrete increase with time, even the load is kept constant.
The asymptotic value is dependent on the water-cement ratio, the aggregate-cement ratio, the ratio be-
tween the section area and the perimeter in contact with the air and on the relative humidity. Con-
straining the shrinkage, it’s possible to notice a variation of the stresses to maintain the self-equilibrium.
Creep is a phenomenon due to the effect of the stress on the microstructure of the cement paste: the
adsorbed water layers tend to become thinner between gel particles transmitting compressive stress,
firstly it occurs rapidly and then it slows with time. The cement changes his property with time, the
strength becomes higher and the elastic modulus grows. When the concrete is loaded with a condition
of relative humidity equal to 1 and of constant temperature, the deformation variable in time is called
basic creep. If the temperature is variable in time, it is called transitional thermal creep. Creep in con-
crete so can vary with the following factors:
- Increase in water/cement ration increases creep;
- Creep decrease with decrease in the age and strength of concrete when the concrete is sub-
jected to stress;
- Creep deformations increase in ambient temperature and decrease in humidity;
27
- Creep depends on many other factors related to the quality of concrete and condition of expo-
sure such as the type, amount and maximum size of aggregate; type of cement; amount of ce-
ment paste; size and shape of the concrete mass; amount of steel reinforcement; and curing
conditions.
The creep strain
If concrete is subjected to sustained loads, it continues to deform further with time. This phenomenon,
discover in 1907 by Hatt, is now commonly referred to as creep.
The reinforced concrete structures are rather sensitive to creep effects that have a significant im-
portance. The principal hypothesis is to consider the concrete as a time-variable linearly visco-elastic
material with a Poisson ratio constant in time. It is also guarantee the validity of the principle of super-
position. This principle only yields accurate results if some conditions are met. First, the stresses in the
structure must stay within service limits, i.e. less than 40% of the compressive strength of the concrete.
Second, there cannot be significant chances in the environmental conditions.
For a constant stress 𝜎𝑐(𝑡0) applied at time 𝑡0, with a stress value lower than 0.4 fck (where fck maximum
compressive strength of the concrete), the stress-dependent strain at time t can be expressed as:
00 0 0 0 0 0
0 0
( , )1( , ) ( ) ( , ) ( ) ( ) ( , )
( ) ( )c ci cc c c
ci ci
t tt t t t t t t J t t
E t E t
(4.3)
𝜑(𝑡, 𝑡0) is the creep coefficient;
𝐸𝑐𝑖 is the modulus of elasticity at age of 28 days;
𝐽(𝑡, 𝑡0) is the creep function or compliance function, representing the total stress-dependent strain per
unit stress;
𝐸𝑐𝑖(𝑡0) is the modulus of elasticity at the time of loading 𝑡0
Figure 17. Loaded specimen with constant stress and relative graph of the deformation in time
28
Concrete is considered as an aging linear viscoelastic material and the diagram of figure shows the de-
pendence of the strain with the time.
Figure 18. Diagram of the strain changing in time of a specimen loaded and unloaded
The first deformation 휀𝑐𝑖 is an elastic deformation and the second 휀𝑐𝑐 , which dependent by the time, is
the viscous part. The more the time it passes, the more the viscous strain goes to the stability, after
some years it doesn’t change anymore.
As shown in the diagram, removing the load, there is an elastic recovery, which is smaller than the load-
ing one for the aging properties of the concrete, a creep one and a third residual strain will remain per-
manently because the gel particles are bounded in the deformed position.
The creep function
The creep function, or compliance function, reflects the time evolution of strain, that is sum of the elas-
tic strain and the creep one, in test of unit stress level:
0
( ) ( , ') ( ') 't
t J t t d t dt (4.4)
It is characterized by some properties:
- It grows constantly at time t: 𝜕𝐽(𝑡,𝑡0)
𝜕𝑡≥ 0;
- For 𝑡 → ∞ there is an asymptote, that is hard to define in experimental ways;
29
- The structure response changes respect to the time of loading, because the elastic modulus
grows with time and so the elastic strain becomes lower. This happens for the aging properties
of the concrete. The following figure shows the reduction of the creep response due to load ap-
plications in different time, the later the smaller are the elastic and creep strain.
-
- Figure 19. Relationship of time and age of loading to creep function
If there is more than one load 𝜎𝑐1 and 𝜎𝑐2 at the same starting time𝑡0it is possible to compute the total
strain 휀𝑐𝑡𝑜𝑡:
휀𝑐1(𝑡, 𝑡0) = 𝜎𝑐1(𝑡0) 𝐽(𝑡, 𝑡0)
휀𝑐2(𝑡, 𝑡0) = 𝜎𝑐2(𝑡0) 𝐽(𝑡, 𝑡0)
휀𝑐𝑡𝑜𝑡(𝑡, 𝑡0) = 휀𝑐1(𝑡, 𝑡0) + 휀𝑐12(𝑡, 𝑡0) = (𝜎𝑐1(𝑡0) + 𝜎𝑐1(𝑡0)) 𝐽(𝑡, 𝑡0) = 𝜎𝑐3(𝑡0) 𝐽(𝑡, 𝑡0)
It is evident the validity of the principle of superposition of the effects. This principle is unviable when
the hypothesis of the same time of application of the loads is missing. For aging phenomena, some stud-
ies of McHenry and Maslov resolved the problem. McHenry defined a constitutive law, where the be-
havior of the concrete can be computed with the summation of small increment of stresses in various
times as shown in the figure:
30
Figure 20. Diagram of the stress changing in time and relative approximation
The total deformation can be expressed by the following integral:
휀𝑐1(𝑡, 𝑡0) = ∫ⅆ𝜎(𝑡′)
ⅆ𝑡′
𝑡
0
∙ 𝐽(𝑡, 𝑡′) ⅆ𝑡′ + 휀𝑐𝑛(𝑡)
This law is valid if in his stress history the material is not affected by big variation of load. For variable
stresses and the principle of superposition consistent with respect to the assumption of linearity, the
constitutive equation can be written in terms of Riemann integral as follows:
0
0 0 0
( ')( , ) ( ) ( , ) ( , ') ' ( )
'
t
c cnt
d tt t t J t t J t t dt t
dt
(4.5)
For realistic forms of 𝐽(𝑡, 𝑡0), this integral cannot be solved analytically but approximate numerical solu-
tions are easier, where the integral is approximated by a summation.
The relaxation function
The relaxation function 𝑅(𝑡, 𝑡0), which represent the uniaxial stress history caused by a constant strain
휀 = 1 enforced at any age 𝑡0 ≤ 𝑡. If the relaxation function is known, the superposition principle can be
written as:
0
( ) ( , ') ( ') ( ')t
c c cnt R t t d t t (4.6)
31
Figure 21. Loaded specimen with constant strain and relative graph of the stress in time
For a constant deformation ( )c ot at time 𝑡0, the stress component at time t is:
𝜎𝑐(𝑡, 𝑡0) = [𝜎𝑐𝑒(𝑡0) − 𝜎𝑐𝑐(𝑡, 𝑡0)] = 𝜎𝑐𝑒(𝑡0) [1 −𝜎𝑐𝑐(𝑡, 𝑡0)
𝜎𝑐𝑒(𝑡0)] = 휀𝑐(𝑡0)𝐸𝑐𝑖(𝑡0)[1 − 𝜌(𝑡, 𝑡0)] =
= 휀𝑐(𝑡0)𝑅(𝑡, 𝑡0)
𝜎𝑐𝑒(𝑡0) is the initial tension after deformation,
𝜎𝑐𝑐(𝑡, 𝑡0) is the stress recovered by the concrete after time,
𝜌(𝑡, 𝑡0) is the relaxation coefficient.
The relaxation function is strictly dependent by the creep function. The two functions are equivalent in-
tegrals which represent the mutual connection between the integral and the solution.
The properties of the relaxation function are slightly different than the creep ones:
- For the thermodynamic 𝑅(𝑡, 𝑡0) ≥ 0
- It decreases constantly at time t: 𝜕𝑅(𝑡,𝑡0)
𝜕𝑡≥ 0;
- For 𝑡 → ∞ there is an asymptote;
- The behavior of the concrete is different based on the loading time for the dependence of the
elastic modulus with the time: 𝑅(𝑡, 𝑡0) = 𝐸𝑐𝑖(𝑡0);
- The creep results depend on the aging phenomenon.
32
The link between creep function and relaxation function
The creep function J and the relaxation function R are dependent each other. If the stress fields and the
strain field are not constant in time, calling t’ a generic instant between the time of loading t0 and the
time t, in the computation of the strain and the stress, the following two functions are valid [31]:
0
0
0 0 0
0 0
( ')( , ) ( ) ( , ) ( , ') '
'
( ')( ) ( ) ( , ) ( , ') '
'
tc
c ct
tc
c ct
d tt t t J t t J t t dt
dt
d tt t R t t R t t dt
dt
(4.7)
The two are dependent between each other. The two functions are equivalent integral, and they repre-
sent the mutual connection between the nucleus of the integral and the nucleus of the solution. This
integral is also called convolution integral. These relations show the validity of the principle of superpo-
sition. The deformation is assumed as the summation of the resultant effects of a constant stress 0( )c t
and the effect produces by incremental stresses ( ')
'
cd t
dt
applied at time t’ constant too.
Imposing 1 c t , the two functions (4.7) become:
0
0
0 0
0
( ) ( , )
( ', ) ( , )( , ') ' 1
' ( )
c
t
tc
t J t t
dJ t t R t tR t t dt
dt E t
(4.8)
Knowing the definition of 0 0
0
1( , )
( )c
J t tE t
. This equation shows the dependence between the creep
function and the relaxation function.
In case of constant deformation 1c in time, the second equation of the (4.7) becomes:
0( ) ( , t )t R t (4.9)
A differential of the previous equation for t=t’ is made:
0( , t )( ) '
'
R td t dt
t
(4.10)
Taking into account the solution of the function in the viscous constitutive law described in (4.7), t fol-
lowing relation is obtained:
0
00 0 0
( ', )( , ) ( , ) ( , ') ' 1
'
t
t
R t tR t t J t t J t t dt
t
(4.11)
If the elastic part is separated from the viscous one, by the introduction of the creep coefficient, the re-
lation can be written as:
33
0
00 0
( ', t )( , ') ' 1 E ( ) ( , )
'
t
ct
R tJ t t dt t J t t
t
(4.12)
Volterra Integral
It is common that, when a structure is analyzed, the deformation ε are known and not the stresses σ. In
this case the following equation is a Volterra integral equation, which is a variable limits integral where
the core function of the integral is directly dependent to the upper bound of the integral:
0
0( ) [ ( ) ( )] ( , ')t
tt t t R t t (4.13)
The Volterra integral has he following properties:
- If the core of the integral, that in this case is J(t,t’), is monotonic growing, continuous and as-
ymptotic, the solution of the homogeneous integral equation 0
( ') ( , ')t
d t J t t is ( ) 0t ;
- Considering the properties of the creep function J(t,t’), if the function ε is defined, the equation
admits only one solution;
- If 0( ) ( ) 1t t , the integral must be 0
( ') ( , ') 1t
d t J t t and the equation, that solve the
integral, is called solving nucleus of the Volterra integral. This function is called Relaxation func-
tion R (t,t’).
Until now it is described a way to simplify the model and not to solve the Volterra integral that is at the
basis of the problem. Considering more particular complex stress history, such as structure with an im-
portant hyperstaticity or structure made in different phases, the theoretical models taken into account
aren’t completed to solve the problem.
The problem can easily be solved in analytical ways if the creep coefficient is the core of the integral, be-
cause this function is known. If in the core of the integral it is contained the relaxation function, the so-
lution of the problem is more complicated because the convolution integral is hard to solve:
휀𝑐(𝑡) = 𝜎𝑐(𝑡0) 𝐽(𝑡, 𝑡0) + ∫ 𝐽(t, t′)𝑡
𝑡0
𝜕𝜎𝑐(t′)
𝜕t′ⅆt′ + 휀𝑐𝑛(𝑡)
𝜎𝑐(𝑡) = [휀𝑐(𝑡) − 휀𝑐𝑛(𝑡)] 𝑅(𝑡, 𝑡0) + ∫ ⅆ[휀𝑐(𝑡) − 휀𝑐𝑛(𝑡)] ∙ 𝑅(𝑡, 𝑡0)𝑡
𝑡0
ⅆ𝑡
The algebraic methods are based on the approximate connection between the stresses and the viscous
strains; the law is always expressed by algebraic equation that solve the complex problem.
34
The basic theorem of the linear viscoelasticity
In the real life, the homogeneous structures correspond to a unique concrete casting. This type of struc-
tures in linear visco-elastic constitutive law are characterized by three theorems that are called theo-
rems of the linear viscoelasticity. Obviously, no reinforced concrete structure can be considered as a ho-
mogeneous one, for the presence of the reinforcement, but it does not critically modify the behavior of
the structure. For this reason, it is possible to analyze these structures based on the same theorems.
All the theorems are based on some hypothesis:
- Rigid restrains
- Only statical actions, 0
- Rheological homogeneous histotrophic material
The first principle of the linear viscoelasticity
The first theorem of the linear viscoelasticity [23] says that:
“In a homogeneous structure with linear visco-elastic behavior, constant Poisson coefficient in time and
rigid restrains, for constant surfaces and/or volume forces applied at t=t0 (creep type problem), the ini-
tial elastic state of stress is not modified by creep, while the initial elastic state of deformation is modi-
fied through an affinity corresponding to the total creep factor”, i.e.:
00
( ) ( )
( ) ( ) ( , ') ( ')
el
ij ij
tel
i co i
t t
u t E t t t du t
(4.14)
for 0c icu .
The application of this principle, when possible, is really useful. By this the iperstatic actions acting on
the structure can be evaluated in a pure elastic phase. The displacements follow the second expression
where el
iu is the elastic answer of the structure at t’ time of loading, el
idu is the increment of the elas-
tic displacement between the instants τ and t’+dt’ due to the increment of load dpj (t’).
The deformations are defined by a law which says that ‘in case of constant static action in time, the total
deformation grows proportionally with the elastic ones’. By this, respecting all the hypothesis the vis-
cous problem can be neglected for the resistance verifies, because it is enough the simple elastic com-
putation. The deformation verifies, that grows in time, needs more verifies, but it is not a problem be-
cause the creep function J(t,t’) is known.
35
The second principle of the linear viscoelasticity
The statement of the principle is:
“In a homogeneous structure with linear visco-elastic behavior, constant Poisson coefficient in time and
rigid restrains, the elastic state of deformation due to a system of prescribed variable boundary dis-
placements and/or inelastic strains is not modified by creep, while the elastic state of stress is computed
by the sum of the elastic strain and the relaxation effects”, i.e.:
0
00
0 0
( ) ( )
( , ) 1( ) ( ) ( , ') ( ')
( ) ( )
el
i i
tel el
ij ij ij
co co
t t
R t tt t R t t d t
E t E t
(4.15)
For 0i ip f .
The first expression shows the hypothesis that the deformation is constantly the same of the elastic one.
For this reason, it is possible to consider in the second expression the auto-equilibrate tensional state
0( )ed t . The two formulas represent an equilibrate and compatible problem. Moreover, it satisfies the
Volterra integral which means that it is the only possible solution.
As for the first principle, there is a law of the isomorphism based on the stresses and it says that ‘if the
hypothesis are verified and the imposed deformation are constant in time, the stresses decrease pro-
portionally with the elastic ones’. For this second law, the creep problem dose not influence the stains
and the displacements and the elastic computation is sufficient to know them. For iperstatic structure,
the internal action needs to be computed, because there can be auto-stresses state in equilibrium with
the constrain action. Moreover, the relaxing of the concrete create a reduction in time of the constrain
actions that provoke an internal redistribution of the stress field. The computation of this variation per-
mit to compute the relaxation effects due by the R (t, τ) that is not known as the creep functions. The
relaxation function can be computed as already saw by the following expression:
0
00 0
( ', t )( , ') ' E ( ) ( , ) 1
'
t
ct
R tJ t t dt t J t t
t
(4.16)
or by tables of the national legislation in graphic way.
Principle of the Principle of acquisition of the modified system stress distribution
This principle interests all the structures that have been subjected to a change of the static scheme. This
happens in all the prefabrications, in bridge built by distinct phases and in all the structure with post-
poned constrains. The loads stay unchanged, but the variation of the static scheme provokes a variation
in time of the internal stresses, that are going to equilibrate for the new static scheme. To better under-
stand the process, an example is made.
36
A continuous beam n time iperstatic, with fixed constrains and made by a viscous-elastic isotropic mate-
rial with a constant Poisson coefficient is loaded by a constant load p in time. For the first principle of
the linear viscoelasticity the stress state stays unchanged in time but the deformation state changes as
the elastic initial strain. The rotation in correspondence of the simple support of the beam is considered:
0 0 0( , ) (1 ( , )) ( )p pet t t t t (4.17)
where
0( )pe t is the elastic rotation for 0t t due to the load p,
0( , )p t t is the viscoelastic rotation for 0t t due to the load p.
In correspondence of the support in analysis at time 0*t t , a loose constrain is introduced to block the
rotation and the angular distortion of the value of the viscoelastic rotation:
0 0 0( *, ) (1 ( *, )) ( )p pet t t t t (4.18)
In an interval of time ( *)t t t , the constrain, called participated constrain, block the additional an-
gular rotation:
0 0 0 0 0( , ) ( *, ) ( ( , ) ( *, )) ( )p p pet t t t t t t t t (4.19)
At the same moment, it stats to be an iperstatic reaction ( , *)X t t of the constrain. This change the mo-
ment diagram in time and all the reaction forces already presents on the structure before the addition
of the constrain. The viscoelastic rotation is ( , *)X t t , due only to the iperstatic ( , *)X t t applied to the
principal structure. The viscoelastic rotation , ( )p X t in *t t needs to respect the compatibility for the
structure subjected to the postponed constrain in the following way:
, 0 0( ) ( , ) ( , *) ( *, ) cost for t t*p X p X pt t t t t t t (4.20)
For the first principle of the linear viscoelasticity, in case of a structure subjected only to the static load
( , *)X X t t the equation can be written as followed:
0 0
* *( , *) ( ) ( , ') ( ') ( ) ( , ') ( ') for t t*
t t
X Xe let t
t t E t J t t d t E t J t t dX t (4.21)
where
le is the elastic rotation for X=1;
( ')Xed t is the elastic rotation compatible with a variation ( )dX t of the iperstatic reaction of the post-
poned constrain.
Substituting this last expression in the main expression of the total viscoelastic rotation , ( )p X t be-
comes:
0 0 0 0 0 0
*( ) ( , ') ( ') ( ( , ) ( *, )) ( ( , ) ( *, )) ( )
t
le p p pet
E t J t t dX t t t t t t t t t t (4.22)
37
where 0( )eX t is the elastic relation of the postponed iperstatic constrain.
Both the iperstatic reaction and the elastic rotation are applied on the principal structure at time t=t0,
the compatibility condition is:
, 0 0 0 0 0 0( ) ( ) ( ) ( ) ( ) ( ) 0 for t = tp X pe Xe pe le et t t t t X t (4.23)
Form this last equation (4.23) is easy to derivate the following expression:
0 0 0( ) ( ) ( )pe le et t X t (4.24)
Substituting this expression in the formula(4.22) it
0 0 0 0
*( , ') ( ') ( ) ( ( , ) ( *, )) / ( ) ( )
t
et
J t t dX t X t t t t t E t Y t (4.25)
The same procedure can be applied adding n postponed constrains. In this case there would be n iper-
static reaction 1 2 3, , , , nX X X X . Moreover, there would be an expression as the last one, all decou-
pled each other and each one dependent of the incognita iX .
This last expression is the Volterra integral that has an integral nucleus 0( , )J t t and a resultant nucleus
0( , )R t t . The solution of this formula can be written as:
0 0
* *0
( ) ( ', )( ) ( , ') ( ') ( , ') '
( ) '
t te
t t
X t t tX t R t t dY t R t t dt
E t t
(4.26)
Remembering that:
( , ')1 1 1
( , ') ( ') ( , ') 1 ( , ')( ') ( ') ( ') ( ')
ve v
e
t tJ t t t t t t t
E t E t t E t
(4.27)
It is deducted that:
0 0
0
( ', ) ( ', ) 1
' ' ( )
t t J t t
t t E t
(4.28)
Substituting in the solution of the Volterra integral in (4.26), it is obtained:
00
* *
( ', )( ) ( , ') ( ') ( ) ( , ') '
'
t t
et t
J t tX t R t t dY t X t R t t dt
t
(4.29)
In this last expression, all the components in the integral are positive, this means that also the solution
( , *)X t t is positive and it grows in time. This condition causes an increase of the stresses of the post-
poned constrain and his iperstatic reaction goes to a starting value of 0 in t=t* to a maximum value in
t . The main difficult to solve the Volterra integral, to arrive to a value of X for a generic t, is that
the relaxation function ( , ')R t t is not known but it must be computed. For this reason, the legislation
gives tabulate value for the following function:
38
00 0
*
( ', )( , , ) ( , ') '
'
t
t
t tt t t R t t dt
t
(4.30)
This can also be given in dimensionless form as 28/ E , which in this case is
0/ ( )E t . In this
case the computation of the iperstatic action ( )X t is immediate by the expression:
28
0( )e
EX X
E t or
eX X
The maximum value of ( , *)X t t is in correspondence of the maximum of the function, so for
0( , , *)t t , changing t with a constant t* or a changing t* with a constant t. The postponed constrain,
as already anticipate, gets bigger in time and the closer it is to the loading time 0t . The maximum value
of the function is reached in 0*t t and t , it is equal to 1. In
0*t t , it is:
0 0
0 00 0
( ', ) ( ', )( , , ) ( , ') ' ( , ') '
' '
t t
t t
t t J t tt t t R t t dt R t t dt
t t
(4.31)
This last expression is the convolution integral. Remembering that:
0
0 0
0
( ', ) ( , )( , ') ' 1 ( , ') 0
' ( )
t
tc
J t t R t tR t t dt t t
t E t
(4.32)
It is possible to derivate:
0
0 0
0
( , )( , , ) 1
( )c
R t tt t t
E t (4.33)
00 0
0
min{ ( , )}max{ ( , , )} 1
( )c
R t tt t t
E t
This impose that the relation X (t, t*) has an upper bound Xe, the asymptotic value for t and
0*t t .
To increase the effect of the reacquisition of the principal state, it is possible to act in t0 and change it as
close as possible to the origin of the time. In this way, it is possible to load at the maximum the concrete
since the starting when the material is young. The statement of the principle of acquisition of the modi-
fied system stress distribution is:
“If the restraint conditions in a structure n times redundant, subjected to forces constant in time,
change immediately after loading (when concrete is still young), its stress distribution tends to evolve in
time to get a final shape like the one corresponding to the application of the loads to the structure in his
final restraint condition.”
39
General models for concrete
The creep properties of concrete are usually determined by measuring the creep coefficient. This result
represents the most accurate way of determining the creep strains, but creep measurements typically
cost time, which is usually unjustifiable for common engineering projects.
The models previously explained are not capable to describe entirely all the aspect of the deformation
behavior of the concrete. The Kelvin-Voigt model can be a good approximation for the concrete already
old and the Dischinger model represent valid simplification for a young concrete. In absence of test-re-
sults many authors suggested mathematical models to predict the creep coefficient in function of time
to be used in structural calculations. These models permit to correct the missing point of the approxima-
tion problem and to have a single model to take into account the hereditary and the aging of the con-
crete.
The models that are analyzed in this chapter is:
- CEB 1990 model
- CEB 2010 model
- ACI model
- Bazant-Baweja Model (B3)
The first two model are proposed by the European institute and the last two are the American for the
analysis of the same problem.
CEB 1990 Model
The creep function of the model is given by the following expression [2]:
0
0
0 28
( , )1( , )
( )c c
t tJ t t
E t E
(5.1)
0
1
( )cE t is the instantaneous elastic strain;
0
28
( , )
c
t t
E
is the creep strain.
The formula to compute the elastic modulus of the concrete at 28 days is:
40
1
34 4
28
82,15 10 2,15 10
10 10
ck cmc
f fE
(5.2)
ckf is the characteristic compressive strength of a concrete at 28 days in [N/mm2], evaluate on cylindric
spaceman at temperature of 20°C;
cmf is the mean value of compressive strength of concrete associated to the ckf , 8cm ckf f MPa .
The elastic modulus changes in time for the aging property of the material and it can be compute by the
following law for times different than 28 days:
28( ) ( )c E cE t t E (5.3)
0.51 28/
( )s t
E t e (5.4)
s is function of the rapidity of the concrete to dry.
The creep coefficient is computed by two factors:
0 0 0( , ) ( )ct t t t (5.5)
0 is the theoretical creep coefficient;
0( )c t t is the coefficient that consider the creep effect in time.
The theoretical creep coefficient can be computed by the following formula:
0 0( ) ( )RH cmf t (5.6)
where:
3
11001
0,46100
RH
RH
h
(5.7)
5,3( )
10
cm
cm
ff
;
0 0,2
0
1( )
0,1t
t
.
The coefficient, that takes into account the creep effect in time, can be computed as follow:
41
0,3
0
0
0
( )c
H
t tt t
t t
(5.8)
18
0 0
150 1 1,2 250 1500H
RH h
RH h
;
2 cAh
u is the notational size of member (mm), where cA is the cross-section and u is the perimeter of
the member in contact with the atmosphere;
0h is a standard value equal to 100 mm;
RH is the relative humidity of the ambient environment (%);
RH0 is a standard value always equal to 100%;
t is the age of the concrete (days) at the considered moment;
t0 is the age of concrete at loading (days), it considers the effect of type of cement on the creep coeffi-
cient by the equation
0 0, 1,2
0, 1,
91 0.5
2 /T
T T
t tt t
days;
1,Tt is a constant value equal to 1 day;
Α is a power which depends on the type of cement;
0,Tt is the age of concrete at loading (days) 0
400013.65
273 ( )/
0,
1
i
nT t T
T i
t
t t e
;
it is the number of days where a temperature T prevails;
( )iT t is the temperature (°C) during the time period it ;
0T is a constant value equal to 1°C.
Considering a time t of total time when the concrete is under stresses, the age at the time of meas-
urement is:
0,Tt t t
42
Figure 22. Dimensionless creep function CEB90, for h=200mm, RH=80%, fcm=28n/mm
Figure 23. Dimensionless Relaxation function CEB90, for h=200mm, RH80%, fcm=28n/mm
The total shrinkage or swelling strains can be calculated as follow:
( , ) ( )cs s cso s st t t t (5.9)
where
cso is the notional shrinkage coefficient;
s is the coefficient to describe the development of shrinkage with time;
t is the age of concrete(days);
ts is the age of concrete(days) at the beginning of shrinkage of swelling.
The notional shrinkage coefficient may be computed by:
43
( )cso s cm RHf (5.10)
with
6( ) 160 10 (9 / ) 10s cm sc cm cmof f f
where
fcm is the mean compressive strength of concrete at the age of 28 days (MPa)
fcmo is a standard value equal to 10 MPa
βsc is a coefficient that depends on the type of cement (4 for slowly hardening cements SL, 5 for normal
or rapid hardening cements N and R, 8 for rapid hardening high strength cements RS)
1.55 for 40% RH < 99%RH sRH
0.25 for RH 99%RH
3
1sRH
o
RH
RH
The development of the shrinkage may be obtained from:
0.5
1
2
1
( ) /( )
350 / ( ) /
ss s
o s
t t tt t
h h t t t
(5.11)
With the same parameter defined for the computation of the creep function.
CEB 2010 Model
This model [1] follows the same path of the CEB 1990 for the definition of the creep function and elastic
modulus associated to the material. The definition of the creep coefficient varies, and it is considered as
a sum of two main factors, with t the age of concrete in days at the considered moment and to the age
of the concrete ad loading in days:
( , ) ( , ) ( , )o bc o dc ot t t t t t (5.12)
The first component ( , )bc ot t is the basic creep coefficient and it is computable by the formula:
( , ) ( ) ( , )bc o bc cm bc ot t f t t (5.13)
with
0.7
1.8( )
( )bc cm
cm
ff
44
2
0
0,
30( , ) ln 0.035 1bc o
adj
t t t tt
where the fcm is the mean compressive strength at an age of 28 days in MPa.
The second term of the main formula (5.12) is the drying creep coefficient that can be estimated from:
0 0( , ) ( ) ( ) ( ) ( , )dc o dc cm dc dc dct t f RH t t t (5.14)
with
1.4
412( )
( )dc cm
cm
ff
3
1100( )
0.1100
dc
RH
RHh
0 0.2
0,
1( )
0.1dc
adj
tt
Where RH is the relative humidity of the ambient environment in % and h =2Ac/u is the notional size of
member in mm, where Ac is the cross section in mm2 and u is the perimeter of the member in contact
with the atmosphere.
The development of the drying creep with time is shown in the following formula:
0( )
0
0
0
( , )( )
t
dc
h
t tt t
t t
(5.15)
with
0
0,
1( )
3.52.3
adj
t
t
1.5 250 1500cm cmh f fh
with
0.5
35cmf
cmf
Changing the age of loading of the concrete t0 with the time adjusted t0,adj, it is possible to take into ac-
count the effect of type of cement in the creep coefficient. The adjusted age of loading of the concrete
can be computed as follow:
45
0, 0, 1.2
0,
91 0.5 days
2adj T
T
t tt
(5.16)
where t0,T is the age of concrete at loading in days adjusted taking into account the elevated or reduced
temperature on the maturity of concrete computable as follow:
400013.65
270 ( )
1
i
nT t
T i
i
t t e
Δti is the number of days where a temperature T prevails and T(Δti) is the temperature in °C during the
time period Δti.
Consider elevated level of stresses, that are in the range 0 00.4 ( ) 0.6 ( )cm c cmf t f t , the non-linear-
ity of creep needs to be taken into account. It is expressed by the formula:
1.5 0.4
0 0( , ) ( , ) for 0.4<k 0.6k
t t t t e
(5.17)
where
0( , )t t is the non-linear notional creep coefficient;
0( , )t t is the creep coefficient previously computed;
0( )
c
cm
kf t
is the stress strength ratio.
The computation of the shrinkage in the CEB2010 varies compared the CEB1990. The swelling strains is
given by the formula:
( , ) ( ) ( , )cs s cas cds st t t t t (5.18)
The first component represent the autogenous shrinkage that occurs without loss of moisture. This is
due to hydration reactions in cement paste which subtract water molecules. It can be computed by:
0( ) ( ) ( )cas cas cm ast f t (5.19)
where
2.5
6
0
/10( ) 10
6 /10
cmcas cm as
cm
ff
f
0.2( ) 1 t
as t e is a time function
αas is a coefficient dependent on the type of cement as shown in the table below:
46
Figure 24. Table of the coefficient αi (Table 5.1-12 CEB2010)
The second term of the main equation (5.18) is the drying shrinkage that is the primary type of shrink-
age. It Is caused by changes on surface energy of gel particle due to desiccation. It is given by:
0( , ) ( ) ( ) ( )cds s cds cm RH ds st t f RH t t (5.20)
where:
t is the age (in days);
ts id the concrete age at the beginning of drying in days;
2 6
0 1( ) 220 110 10ds cmf
cds cm dsf e is the notional drying shrinkage coefficient;
3
1
1
1.55 1 for 40 RH<99%( ) 100
0.25 for RH 99%
s
RH
s
RH
RH
0.5
2( )
0.035 ( )
s
ds s
s
t tt t
h t t
0.1
1
351.0s
cmf
1 2,ds ds are coefficient dependent on the type of cement.
ACI COMMITTEE 209 Model
The American model [3] has at the base the Branson and Christianson model of the 1971. It was review
in the ’80 and in the ’90. It has a validity in standard condition and in case of different situations, it gives
the opportunity to modify some parameters. The compliance function is given by:
0
0
1 ( , )( , )
cmto
t tJ t t
E
(5.21)
47
where:
cmtoE is the modulus of elasticity at the time of loading at the age t0 (MPa or psi);
0( , )t t is the creep coefficient s the ratio of the creep strain to the elastic strain at the start of loading
at the age t0 (days).
The secant modulus of elasticity od concrete Ectmo at any time t0 of loading is given by:
1.50.043mcto c cmtoE f (MPA) in SI units;
1.533mcto c cmtoE f (psi) in in.-lb units;
c is the unit weight of concrete (kg/m3 or lb/ft3);
fcmto is the mean concrete compressive strength at time of loading (MPa or psi).
The general equation to predict the compressive strength at any time t is given by:
28ctm cm
tf f
a bt
(5.22)
where 28cmf is the concrete mean compressive strength at 28 days in MPa or psi, a (in days) and b are
constant and t is the age of the concrete. The ratio a/b is the age of the concrete in days at which one
half of the ultimate (in time) compressive strength of concrete is reached.
Figure 25. Table of the CEB for the computation of the coefficient a and b
The two constants a and b are function of both the type of cement used and the type of curing em-
ployed. The ranges of a and b for the normal weight, sand lightweight, and all lightweight concretes (us-
ing both moist and steam curing, and Types I and III cement) are: a = 0.05 to 9.25, and b = 0.67 to 0.98.
Typical recommended values are given in Table
The creep model proposed by ACI 209R-92 has two components that determine the asymptotic value
and the time development of creep. The predicted parameter is the creep coefficient 0( , )t t at con-
crete age t due to a load applied at the age t0 and it is given by the following formula:
0
0
0
( , )( )
u
t tt t
d t t
(5.23)
where and d are constants for a given member shape and size that define the time-ratio part;
u is the ultimate creep coefficient.
48
For standard condition, in absence of specific creep data for local aggregates and conditions, the aver-
age value proposed for the ultimate creep coefficient u is 2.35. For conditions other than the standard
conditions, the value of the ultimate creep coefficient u needs to be modified by a correction factor γc,
computable by the following formula:
, , , , , ,c c to c RH c vs c s c sh (5.24)
It is a sum of correction factors defined as follow. For ages at application of load greater than 7 days
form moist cured concrete or greater than l to 3 days for steam-cured concrete, the age of loading fac-
tor for creep γc,to is estimated from:
0.118
, 1.25c to ot for moist curing;
0.1118
, 1.25c to ot for steam curing.
The factor dependent by the ambient relative humidity h (in decimals) can be computed by:
, 1.27 0.67c RH h for 0.40h ;
for a 0.40h , values higher than 1.0 should be used for creep γh.
The third coefficient of the formula allows for the size of the member in term of volume-surface ratio,
for members with a volume-surface ratio other than 38 mm (1.5 in.), or an average thickness other than
150 mm (6 in.):
0.0213( / )
,
2(1 1.13 )
3
V S
c vs e
in SI units
0.54( / )
,
2(1 1.13 )
3
V S
c vs e
in in.-lb units
where:
V is the specimen volume in mm3 or in3;
S id the specimen surface area in mm2 or in2.
It is possible to use the average-thickness methods to account for the effects of member size on ϕu. It
tends to compute correction factor bigger compared with the method of the volume-surface. The stand-
ard condition is signed by the average thickness of the member equal to 150mm (6 in.) or the volume
surface ratio 37.5mm (1.5 in.). For value lower than the standard one it is possible to use the following
tables.
Figure 26. Table for the computation of the creep factor for cases different than the standard
49
For a member that exceed the standard conditions it is possible to compute the creep factor as follow:
during the first year of loading ( ) 1ot t year :
,
,
1.10 0.00092
1.10 0.00363( / )
c d
c d
d
V S
in SI units;
,
,
1.10 0.023
1.10 0.092( / )
c d
c d
d
V S
in in.-lb units;
for the ultimate values, ( ) 1ot t year :
,
,
1.10 0.00067
1.10 0.00268( / )
c d
c d
d
V S
in SI units;
,
,
1.10 0.017
1.10 0.068( / )
c d
c d
d
V S
in in.-lb units.
Where d=4(V/S) is the average thickness on mm or inches of the part of the member under considera-
tion.
The other factors depend on the composition of the concrete and are:
- the slump factor ,c s , where s is the slump of fresh concrete:
, 0.82 0.00264c s s in SI units;
, 0.82 0.067c s s in in.-lb units;
- the fine aggregate factor ,c , where ψ is the ratio of fine aggregate to total aggregate by
weight expressed as percentage: , 0.88 0.0024c ;
- air content factor ,c , where α is the air content in percent:
, 0.46 0.09 1c .
Figure 27. Comparison between the creep function computed by the CEB 90 and by the ACI 209
50
BAZANT-BAWEJA B3 Model
The Bazant-Baweja model [4] was made in the 2000 and it is the last version of the first Bazant model
published in the ’70 in the Northwestern University. It is an evolution of the model of the 1995 that is
already a development of many shrinkage and creep prediction methods. The last version of 2000 is sim-
pler and better theoretically justified than the previous methods. The effect of concrete composition
and design strength on the model parameters is the main source of error.
The constitutive law the define the model is the following:
The elastic modulus of the concrete for a generic time t is computed by the elastic modulus at 28 days
multiplied for a corrective function depending on the time as shown by the following formula:
284 0.85
cmt cm
tE E
t
(5.25)
The creep function is computed by the sum of three coefficients:
1( , ) ( , ) ( , , )o o o d o cJ t t q C t t C t t t (5.26)
The first coefficient represents the instantaneous strain and it is obtained by the elastic modulus at 28
days:
1
28
1 0.6
o cm
qE E
28 284734cm cmE f
Where:
28cmf is the compressive strength of the concrete at 28 days;
Eo is the elastic modulus asymptotic, it is different from the normal one because it is independent by the
time.
The second coefficient of the main formula to compute the creep coefficient Co represents the basic
creep of the element. It is composed by three terms: an aging viscoelastic term, a nonaging viscoelastic
term and an aging flow term:
2 3 4( , ) ( , ) ln 1 lnn
o o o o
o
tC t t q Q t t q t t q
t
(5.27)
The aging viscoelastic compliance term is composed by the two terms that are:
6 0.5 0.9
2 28185.4 10 cmq c f in SI units
6 0.5 0.9
2 2886.814 10 cmq c f in in.-lb units;
51
1/ ( )( )
( )( , ) ( ) 1
( , )
oo
r tr t
f o
o f o
o
Q tQ t t Q t
Z t t
;
12/9 4/9
0( ) 0.086( ) 1.21( )f o oQ t t t
;
( , ) ( ) ln 1 ( )m n
o o oZ t t t t t ;
0.12
( ) 1.7 8o or t t .
where m and n are empirical parameters whose value can be taken the same for all normal concretes
(m=0.5 and n =0.1).
The second term of (5.27) which represent the non-aging viscoelastic term is define by the following pa-
rameter:
4
3 20.29( / )q w c q
where the ratio w/c represent the water cement ratio.
The last parameter of the main equation (5.27) can be defined by the following parameter:
6 0.7
4 20.3 10 ( / )q a c in SI units,
6 0.7
4 0.14 10 ( / )q a c in in.-lb units.
The compliance function for drying creep it is defined by the following formula that consider the drying
before loading:
0.5
8 ( )8 ( )
5( , , ) oH tH t
d o cC t t t q e e (5.28)
Where the parameter q5 is the drying creep compliance parameter that is dependent by the strength at
28 days fcm28 and by the ultimate shrinkage strain εsh∞ can be computed by the following formula:
0.61 6
5 280.757 10cm shq f
where H(t) and H(to) are spatial average of pore relative humidity computable by:
( ) 1 (1 ) ( )cH t h S t t
( ) 1 (1 ) ( )o o cH t h S t t
Where ( )cS t t and ( )o cS t t are the time function for shrinkage calculated at the age of concrete t
and at the age of concrete t0 in days, respectively, with τsh the shrinkage half-time:
0.5
( ) tanh cc
sh
t tS t t
,
52
0.5
( ) tanh o co c
sh
t tS t t
.
53
Computation of the Volterra’s integral
The analysis of the structure and the verification of the cross sections are based on the satisfaction of
the following equations, as it is already saw for the structural problem of a linear elastic material:
- the equilibrium equations
- the compatibility equations
- the constitutive law of the material
Compared to a linear elastic material, the viscoelastic material has ha different constitutive law, that is
characterized by the Volterra integral as shown below:
0 0 0
0
( ')( , ) ( ) ( , ) ( , ') ' ( )
'
t
c cn
d tt t t J t t J t t dt t
dt
(5.29)
The Volterra integral is an integral variable in time and it does not appear in the Hooke constitutive law.
The experimental solutions have high cost and different difficulties. The solution of this problem is done
by different methods that can be defined exact if the error to solve the integral is small. In the following
chapters, the general methods, the algebraic methods and the simplified models are analyzed. These
three groups have different approaches to face the problem.
54
General method
The general methods are the most correct way to solve the Volterra integral. They use a mathematical
procedure based on the discretization of the structural history in its time evolution. The new model can
be solved by a pseudo-elastic incremental problem. The equations with the Volterra integral in it be-
come algebraic equations where the integrals are substitute by typically numerical equation of squaring.
By this method it is possible to approximate step-by-step the solution of the viscous problem for the
concrete. There are many ways to simplify the main model, the most used are the trapeze method and
the Gauss method.
Trapezoidal Rule
The trapezoidal rule is the most used exact method for the resolution of the Volterra integral by a nu-
merical procedure. The technique of the squaring trapeze approximates the creep function, expressed
function of the stresses, to a broken polygonal. The area under the curve of the broken line is composed
by a summation of the areas of the i-elements with a trapeze shape that compose the approximation of
the full Volterra integral.
Figure 28. Graphic representation of the Volterra integral with the trapeze method
The t0-t interval (where t is the age of the concrete and t0 is the time of loading of the concrete) is subdi-
vided in different interval ti (1 < i < t). The variation of the stresses in the i-interval can be expressed by a
difference of the stresses acting at the starting and at the end of the interval:
1( ) ( )i i it t
55
The computation of the integral is possible by the principle of superposition, it is make the summation
of each area of the trapezes:
0
1
1
( ) ( )( , ) ( , ) ( , )
2
kt
k i k it
i
dJ t d J t t J t t
d
(5.30)
The procedure starts from the deformation history of the element which needs to be known consider
the description of the increasing of the stresses starting from the time t=ti:
1 1 1 1 1 0 1 1 1
1( ) ( ) ( , ) ( , ) ( )
2t t J t t J t t t t
2 2 2 2 2 0 1 2 2 2 1 2 2 1
1 1( ) ( ) ( , ) ( , ) ( , ) ( , ) ( )
2 2t t J t t J t t J t t J t t t t
( )k kt t
The resultant stress at time tk is given by the summation of the all differences of stresses previously
computed:
1 1
( )k
c k it
(5.31)
The bigger part of the deformation is developed in the first period after the application of the load and it
becomes stabile in the longer period. It is useful to have smaller intervals in the first period after the
loading to have a better analysis and longer one with the growing of the time. To define in an effective
way the sampling points it is possible to use a good scale of reference with this properties that charac-
terize the deformation phenomena. This scale is the logarithmic one given by the CEB bulletin
n°142/142bis:
1
1 0 0( ) 10mi it t t t
with 16m , 0 it t t .
The examples in Figure 29 represent:
a) a viscous-elastic beam c with a spring e in the middle of the length and a distributed uniform
load p of a beam shown
b) A column loaded by the axial force p, made by elastic reinforcement and visco-elastic concrete.
56
Figure 29. Iperstatic and isostatic structure of a beam and a column.
They are both solve by the force method. The iperstatic constrain, that is put in evidence in the figure,
connects the elastic part to the viscous-elastic one. The service structure without the iperstatic constrain
becomes isostatic. The principle of superposition permit to consider the visco-elastic strain in corre-
spondence of the constrain as the sum of the strain due to the external load and the strain due to the
iperstatic action. The strain related to the elastic and visco-elastic part, that is the difference of the two
parts, to satisfy the compatibility needs to the zero. For this structure, the equation function of the time
t to solve the problem is:
0
11 11 1 0 11 0 0 0 11 0 0
( )( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( , )
t
e c c c ct
dXX t X t t E t t t t E t t d
d
10 0 0 0( ) ( ) ( , ) 0c ct E t t t
where:
1 11( ) eX t is the strain (positive to the bottom) of the elastic part of the structure acting in reaction to
the iperstatic action 1( )X t and linearly dependent by the flexibility (Hooke law);
10 0 0 0( ) ( ) ( , )c ct E t t t is the positive strain (positive to the top) due to the homogeneous visco-elastic
part for the effect of the external loading;
0
11 0 11 0 0 0 11 0 0
( )( ) ( ) ( ) ( , ) ( ) ( ) ( , )
t
c c c ct
dXX t t E t t t t E t t d
d
is the strain (positive to the top) of
the homogeneous visco-elastic part due to the iperstatic action 1( )X t .
The first two components in the expression of the strain are due to the first theorem of the linear visco-
elasticity. The terms are the elastic coefficients for t=t0, for a concrete elastic modulus Ec(t0).
57
By the trapeze method the Volterra integral that is in the main equation is substitute by a summation.
This operation is synthetized in the following picture where on the left the functions
11 0 0( , ) ( ) ( )c ct t E t are drawn in variation of loading time 1t .
Figure 30. Graphic representation of the solution of the Volterra integral by the trapeze method
The dark area is represented by the term:
1 0 11 0 0 0,c c kX t t E t t t
If the intervals of time 1i it t are define in an effective way, the real curve can be computed with small
errors with the broken polygonal. The dark area of the Figure 30 at time kt t is computed as a sum-
mation of the trapeze areas as follow:
11 11 1 0 11 0 0 0 1 11 0 0
1
( , ) ( , )( ) ( ) ( ) ( ) ( , ) ( ) ( )
2
kk i k i
k e c c k i c c
i
t t t tX t X t t E t t t X t E t
10 0 0 0( ) ( ) ( , ) 0c c kt E t t t
where 1 1 1 1( ) ( )i i iX X t X t
If we subtract the equation of the area under the curve, that is computed by the trapeze method at time
t=tk, is subtracted to the area under the curve at time t=tk-1:
111 1 1 0 11 0 0 0 1 0 1 11 0 0
( , ) ( , )( ) ( ) ( ) ( , ) ( , ) ( ) ( )
2
k k k ke k c c k k k c c
t t t tX X t t E t t t t t X t E t
58
11 1 1 1
1 11 0 0
1
( , ) ( , ) ( , ) ( , )( ) ( )
2
kk i k i k i k i
i c c
i
t t t t t t t tX t E t
10 0 0 0 1 0( ) ( ) ( , ) ( , ) 0c c k kt E t t t t t
Substituting the following parameters:
1 1 1 111 0 0
( , ) ( , ) ( , ) ( , )( ) ( )
2
k i k i k i k iki c c
t t t t t t t td t E t
,
111 0 0 11
( , ) ( , )( ) ( )
2
k k k kkk c c e
t t t td t E t
,
0 1 0 11 0 10 0 0 0 1 0( ) ( ) ( ) ( ) ( , ) ( , )k c c c k kd X t t t E t t t t t .
The equation of the difference of time becomes:
1
1 0 1
1
1 k
k k i ki
ikk
X d X dd
.
The elastic solution can be computed as:
10 01 0
11 11 0
( )( )
( )
c
e c
tX t
t
.
Imposing k=1 and k=2 it is possible to calculate:
0111
11
dX
d , 12 20 11 21
22
1X d X d
d .
In the same way, it is possible to compute all the other coefficient until t=tk:
1 1 0 1
1
k
k i
i
X t X t X
.
This example is referred to a problem with one iperstatic action. It can be applied for all the visco-elastic
linear problem and with some correction also to cracked structure or to visco-elastic non-linear prob-
lems. These situations need a good software for the computation of the solution that result complex.
Gauss method
The previous method of the trapeze is based on a squaring Newton-Cotes system of n=2, but there are
other ways to simplify the Volterra integral that use squaring system with an order bigger than the sec-
ond or, Radau, Lobotto, Chebeysev or Gauss formulas. The squaring formulas of Gauss are the more
used in the engineering goal because with the same number of sampling points they give a more precise
59
result and they give an exact solution of the polynomial is of 2n-1 order (compared to the n-1 order of
the squaring formula of Newton-Cotes. The integral with the squaring Gauss formulas becomes [5]:
0 1 '
( ')( ') ( , ') ( , )
'j
nt
j jt
j t t
d td t J t t c A J t t
dt
(5.32)
where:
iA are the weight factors or quadrature weight and function of the sampling points;
jt are the abscissas of the sampling points, or integration nodes.
In this case 2 points, with a weight of 1, are considered, they are a good average for the precision of the
result.
For a better result is better to reduce the integration interval [5]. The subdivision of the interval of time
is better in a logarithmic scale because the creep phenomena is greater just after the loading time and it
goes to a constant value in time. The interval in time (t0, t) is subdivide in k-1 intervals, with the ex-
tremes ti and distant of tk each other:
1
1 0 010mk kt t t t
The Volterra integral expressed by the Gauss integration previously descripted is:
0
1
1 1 '
( ')( ') ( , ') ( , )
'ij
k nt
i j k ijt
i j t t
d td t J t t c A J t t
dt
(5.33)
with:
1 1( )
2
j i i i i
ij
a t t t tt
1( )
2
i ii
t tc
where aj is the abscissa in the j-point in the dimensionless interval [-1, 1]. To approximate the term
( )d
d
it is possible to use the first derivative of the interpolation polynomial of Lagrange of grade n.
The derivative can be computed if ( )pt is known on the n+1 abscissa (with p=0, 1, 2, 3, … n):
0
( ')' ( ') ( )
n
p p
p
d tL t t
d
(5.34)
with:
0
( )' ( ')
' ' ' ( ')
nn
p
s p s ns p
L tt t t t t
0n p nt t t
60
This form is used when the abscissa of the sampling points pt are divided by the same distance.
The previous term ( )d
d
is computed in
ijt , one of the sampling points of Gauss posed in the in-
terval with the endpoints ti and ti+1. The instants on the abscissa tp, where the function is computed,
are the endpoint of the n intervals in which there is the i-interval. For the Gauss method, the n intervals
are the summation of the q intervals bigger than ti and the r intervals that are after, n=q+r. The last
equation (5.34) computed in ijt is:
( )
' ( ) ( )
ij
i r
p ij p
p i qt
dL t t
d
(5.35)
with
0
( )' ( )
' ( )
i rq r ij
p ij
s ij p ij s q r ijs p
tL t
t t t t t
Substituting in the equation that represent the main Volterra integral (5.33), it is obtained:
0
1
1 1 0
( )( ') ( , ') ( ) ( , )
' ( )
k n n i rt q r ij
i j p k ijt
i j s i q s ij p ij s q r ijs p
td t J t t c A t J t t
t t t t t
(5.36)
The only value known is the linear elastic solution 0( )pt t and to obtain in all the instants tk>t0
the last equation needs to be solved k times, changing the q and r values. By this criterion and knowing
that the squaring function is more precise in the center than the tk endpoints of the intervals, the func-
tion ( )t is approximate with a broken polygonal, whom derivative is a steps functions where the dis-
continuities are the point of abscissa ti:
1
1
( )( ')
'
i i
i i
t td t
dt t t
with 1i it t .
The Volterra integral can be finally computed as:
0
11
1 11
( )( ') ( , ') ( , )
k nti i
j k ijt
i ji i
t td t J t t A J t t
t t
(5.37)
The Gauss method is more precise in the numerical derivation procedure than the trapeze method. The
interpolation gaussian formula considers the sampling points in the temporal interval (ti+1, ti), symmetric
to its midpoint, instead of the endpoint ti and ti+1 considered in the trapezoidal rule [5].
Taking into account the document [22] it is possible to know the abscissas and the weight factors that
compute the Volterra integral for a Gaussian integration:
Abscissas = ix
61
The value of ix is 0.577, if the weight is 1 and like in this case n=2, the number of gaussian sampling
point is 2.
62
Algebraic methods
These methods allowed to compute trough a pretty hard computational procedure a good approxima-
tion of the integral problem. The methods are based on purely algebraic linear equations that substitute
the complex equation composed by the Volterra integral. As the figure shows, these methods approxi-
mate the creep function as a constant value EF, compared to the real value of the curve CG [21].
Figure 31. Comparison between the real integral CG and the algebraic integral of the creep function in variation of the stress
The principal equations at the basis of the viscoelastic constitutive law, that define the creep function
and the relaxation one, with a concrete age t e and an application of load t’, are:
0
0
0 0 0
0 0
( ')( , ) ( ) ( , ) ( , ') '
'
( ')( ) ( ) ( , ) ( , ') '
'
tc
c ct
tc
c ct
d tt t t J t t J t t dt
dt
d tt t R t t R t t dt
dt
(5.38)
The strain of the concrete dependent on the stresses only can be expressed by a linear function of the
creep coefficient 0( , )t t , it is the sum of the elastic strain and the viscous one that is the incognita:
0
0 0 1 0 1 0
0
( )( , ) ( , ) ( , )
( )
cc c c c
ci
tt t t t t t
E t
(5.39)
In the second equation of the (5.38) it can be substitute the expression on the strain as follow:
0
0 00 1
0 0
( ) ( )( ) ( , ) ( , ') ( , ') '
( ) ' ( )
t
cc c
ci cit
t tt R t t t t R t t dt
E t t E t
(5.40)
where 0
0
( )0
' ( )ci
t
t E t
.
The last equation (5.40) becomes:
63
0
00 1
0
( ) ( , ')( ) ( , ) ( , ') '
( ) '
t
c c
ci t
t t tt R t t R t t dt
E t t
(5.41)
The equation that defines the creep function can be derived in time:
0
( , ') 11 ( , ')
' ' ( )ci
J t tt t
t t E t
(5.42)
Knowing that the elastic modulus stays constant in time E(t’) =cost, the equation (5.42) becomes:
0
( , ') ( , ')( )
' 'ci
J t t t tE t
t t
(5.43)
It is possible to substitute (5.43) in the equation that define the stress (5.41):
0
00 1 0
0
( ) ( , ')( ) ( , ) ( ) ( , ') '
( ) '
t
c c ci
ci t
t J t tt R t t E t R t t dt
E t t
(5.44)
The integral can be expressed by:
0
0
0
( , )( , ')( , ') ' 1
' ( )
t
t
R t tJ t tR t t dt
t E t
The main equation (5.44) becomes:
0 00 1 0
0 0
( ) ( , )( ) ( , ) ( ) 1
( ) ( )c c ci
ci
t R t tt R t t E t
E t E t
(5.45)
It is possible to isolate the strain that is the incognita of the equation:
00
01
00
0
( )( ) ( , )
( )
( , )( ) 1
( )
c
cic
ci
ci
tt R t t
E t
R t tE t
E t
(5.46)
The total viscoelastic strain is computable as:
00
0 00 0
0 00
0
( )( ) ( , )
( ) ( )( , ) ( , )
( ) ( , )( ) 1
( )
c
c cic
ci
ci
ci
tt R t t
t E tt t t t
E t R t tE t
E t
0 00 0 0
0 0 0
0
( ) ( ) ( ) 1( , ) 1 ( , ) ( , )
( ) ( ) ( , )1
( )
cc
ci ci
ci
t t tt t t t t t
E t E t R t t
E t
(5.47)
The aging coefficient 0( , )t t is given by the following formula:
64
0
00
0
1 1( , )
( , )( , )1
( )ci
t tt tR t t
E t
(5.48)
Substituting in the main function (5.47):
0 00 0 0 0
0 0
( ) ( ) ( )( , ) 1 ( , ) 1 ( , ) ( , )
( ) ( )
cc
ci ci
t t tt t t t t t t t
E t E t
(5.49)
This method can be applicable if the aging coefficient is known. This coefficient is function of the relaxa-
tion function 0( , )R t t that also needs to be known. The main factor is the creep coefficient 0( , )t t that
permits to compute the creep function 0( , )J t t and the relaxation function 0( , )R t t .
The computation of the creep coefficient is at the basis of the algebraic methods. It can be computed by
tables, by graph or by the resolution of equations as the following:
0
0 0
( , ')' 1 ( ) ( , )
'
t
ci
t
J t tdt E t J t t
t
(5.50)
The method just descripted is acceptable if the strain of the concrete varies linearly. It cannot be taken
into account in case of big stresses, thermal strain or cyclic loads. This model does not consider the elas-
tic modulus at 28 days because it is too much big for concrete loaded in early age.
The aging coefficient 0( , )t t is a growing function that is dependent by t and t’. it is always positive be-
cause it is function of the relaxation function 0( , )R t t that is always positive for 0 0( , ) ( )ciR t t E t . The
maximum value of the relaxation function is reached at time equal to infinite, for the model in which the
creep coefficient 0( , )t t maximize the value of the relaxation function.
The minimum value of the aging coefficient was discovered by Mola [21] that says that it cannot over-
pass the following limits:
'
'
'
1lim ( , ') 1 ( , ') 1
( , ')2
t t
t t
t t
dE
dtt t t t
dR t t
dt
(5.51)
This is due to the fact that '
0t t
dE
dt
; '
( , ')0
t t
dR t t
dt
and the term '
'
1 1( , ')
t t
t t
dE
dt
dR t t
dt
.
65
The aging coefficient for Mola is limited by the two values 1
2 and 1, but in case of constant elastic mod-
ulus '
0t t
dE
dt
.
The CNR 10025 and the Eurocode 2 computed the aging modulus 0( , )t t . They show that considering
the climatic condition present in Europe and an infinite time which can be consider as 43 10 days, the
aging coefficient can be approximated as 0( , ) 0.8t t . This last value is an important approximation in
the computation of the creep coefficient.
M.S.M.
The M.S.M. [21], which means “Mean Stress Method”, is the method of the mean stress as the name
explains. It can be applied when the aging coefficient is the minimum admissible value:
0
1( , ) 0.5
2t t (5.52)
The elastic modulus is considered constant 0( ) ( )ci ciE t E t .
The superposition integral is approximated rounded down, it brings to an overestimation of the stress
difference 0( ) ( )c ct t . The computation of the strain previously computed becomes the following:
0 00 0 0
0 0
( ) ( ) ( )( , ') 1 ( , ) 1 ( , ) ( , )
( ) ( )
cc
ci
t t tt t t t t t t t
E t E t
(5.53)
Remembering that the elastic modulus is considered constant. This method gives good results only
when the computation of the aging coefficient gives back a really small error. The mistake grows with
the creep coefficient as the following relation shows:
0 0
0
0
1( , ) ( , )
( , )1
( )
t t t tR t t
E t
(5.54)
The lower value of the creep coefficient corresponds to the greater value of the relaxation function and
small differential stresses. In the following figure, it is compared the real result and the value of the area
computed by the lower value of the aging coefficient.
66
Figure 32. Approximation of the creep function with an aging coefficient equal to 0.5
E.E.M.
The E.E.M. [21], which means “Effective Modulus Method”, is an algebraic method that impose the ag-
ing coefficient 0( , )t t equal to 1, which means that it rounds it up. The method gives a good approxi-
mation of the result if the stress is constant or if the material is only characterized by hereditary viscous
properties, for example in the concrete loaded in old age. The equation is correct for t and it is
approximated for a generic t. For a generic strain 0( , )c t t , a round up approximation of the aging coeffi-
cient brings to an underestimation of the differential stress 0( ) ( )c ct t . This method is good for con-
crete that are loaded in old age, and as already said for concrete that has hereditary viscous properties.
In the figure, it is shown the approximation of the area that compute the creep function. Compared to
the M.S.M., this area is bigger than the area under the real curve.
67
Figure 33. Approximation of the creep function with an aging coefficient equal to 1
A.A.E.M.M.
The A.A.E.M.M. [21], that means “Age-Adjusted Effective Modulus Method”, is an average of the other
two method previously descripted. In this case the values of the aging coefficient 0( , )t t are given by
the CEB in a graphic form or for values greater than 43 10t days it is possible to compute it by the
approximated Chiorino formula:
0.5
0.5
'( , )
'
tt
n t
(5.55)
where:
3
4
/10000
0.131 1 0.772 2.917 10 0.772 0.0114 8
50cmh
h RHn h f
e
2 cAh
u is the notational size of the member (mm), where cA is the cross-section and u is the perime-
ter of the member in contact with the atmosphere;
68
RH is the relative humidity of the ambient environment (%);
t is the age of the concrete (days) at the considered moment;
t’ is the age of loading of the concrete (days);
fcm is the compression strength at 28 days.
In the figure is evident that the approximated area is like the area under the real curve.
Figure 34. Approximation of the creep function with the A.A.E.M.M.
The approximation of the Chiornio formula are acceptable in the following limits: 50 600mm h mm 50% 80%RH The error, respecting these limits, is really law. The loading time t0 is never lower than 14 days, so
0( , )t has the value of 00.7 ( , ) 0.85t as Trost [36] said.
69
Simplified models
The hereditary model
The computational hereditary model takes into account the Boltzmannian materials which are charac-
terized by the same properties irrespective of the time. The answer of the material, to a load or to a
fixed deformation, is the same independently by the time t’ of loading. This model is obtained connect-
ing schemes of the elastic law in series and schemes of the viscous one in parallel. These two models re-
spectively made by Hook and by Newton respect the following constitutive law:
- Elastic model: 휀(𝑡) =𝜎(𝑡)
𝐸 (Hooke)
- Viscous model: d𝜀(𝑡)
𝑑𝑡=
𝜎(𝑡)
𝜈 (Newton)
Figure 35. Graphic representation of the elastic and the viscous models
The two parameters 𝐸 and 𝜈, that have significant importance in the models, are constant in the time.
This allowed to write a creep function for 𝐽 and a relaxation function 𝑅 neglecting the aging. The model
is loaded at time t’ and the creep function at time t is:
𝐽(𝑡, 𝑡′) =1
𝐸+ 𝐶(𝑡 − 𝑡′)
This law shows that changing the time of loading, the creep function shifts on the t axis with the same
shape and the same asymptote like in the figure:
70
Figure 36. Representation of the Creep function with different loading times
The kelvin-Voigt model
Figure 37. Graphical representation of the hereditary Kelvin-Voigt model
Among the hereditary models it is important the Kelvin-Voigt model. It gets the creep and the relaxation
functions with an easy practical use. The connection between the stresses and strains is obtained start-
ing from the following equations:
*
1R( , ') 1 exp ( ')
1
Et t t t
(Hooke’s model)
11
1
( )( )
tt
(Newton’s model)
33
3
( )( )
tt
E
(Hooke’s model)
71
1 2( ) ( )t t
* 0.4
1.4
1 2 3( ) ( ) ( ) ( )t t t t (6.1)
Deleting the variable with the index in the (6.1), the differential equation that links the stress ( )t to
the strain ( )t becomes:
1 1
3 2 3 2
( ) ( )1 ( ) ( )
E Et tt t
E E
(6.2)
3E E is the instantaneous elastic modulus;
3
1
E
E is the final creep coefficient;
* 2
1E
is the retardation time.
For an assigned constant stress history 𝜎(𝑡) = 𝑐𝑜𝑛𝑠𝑡 = 1 applied at time t’ < t, if the previous equation
is integrated, the creep function of the model can be derived:
28
( ) ( ) 1 ( ) 1 ( )( )
( )
d t d t d t d tt
dt dt E t dt E dt
(6.3)
If a constant strain history 휀(𝑡) = 𝑐𝑜𝑛𝑠𝑡 = 1, applied at time t’ < t, is assigned, it is possible to obtain
the relaxation function of the model:
*
1R( , ') 1 exp ( ')
1
Et t t t
(6.4)
By this last two equations it is possible to solve the creep problem (with an assigned 𝜎(𝑡)) or the relaxa-
tion problem (with an assigned 휀(𝑡)), integrating on known variables.
In the case of non-aging viscoelasticity, its compliance function is given by:
*( ')/1
( , ') (1 ) ( ')t tJ t t e H t tE
(6.5)
Where H is the Heaviside step function.
The right-hand of the equation depends on the rime lag t-t’ only, rather than on the times t and t’ sepa-
rately. This is typical of non-aging materials. By coupling several time Kelvin units in series, a Kelvin chain
is obtained by a compliance function as follow:
72
( ')/
10
1 1( , ') ( ') (1 ) ( ')
Mt t
J t t t t e H t tE E
(6.6)
Where φ if the compliance function considered as a function of a single variable, the time lag t-t’.
The hereditary models are not capable to describe the aging of the material, that has a significant role to
describe the creep of the material, especially for non-Boltzmannian material as the concrete.
Aging model
To solve the problem of the aging there are a lot of model that are based on this phenomenon. They
consider that the creep strain is dependent just by the aging, without any hereditary characteristics, so
they are completely irreversible. For this the answer of the material to the unloading is totally elastic
and equal to the initial instantaneous elastic strain, the creep is completely neglected.
The following method was based on the experimental results of Glanville and Whitney, it was formu-
lated by Dischinger and it is called rate-of-creep method. It corresponds to a differential stress-strain re-
lation based on an aging Maxwell model. Aside from providing various instructive simple solutions, the
method is still useful in that it provides an approximate bound on the exact solution. When the stress
varies monotonically in time, which is true of most of problems, the rate-of-creep method generally
gives a bound on the exact solution which is opposite to the bound obtained from the classical effective
modulus. It is based on the hypothesis of parallelism of the creep functions, so for time far enough from
the loading time, all the curves have the same tangent. For a constant stress history 𝜎(𝑡) = 𝑐𝑜𝑛𝑠𝑡 = 1
applied at time, for a generic loading time t’ > 0t , the hypothesis is expressed by the following equation:
28 28
( ) 1 ( ) ( )d t d t t
d E dt E
(6.7)
where 0( , )C t t is the creep strain at time t for a unit stress applied at time 0t .
Two different time of loading t’ and 0t are condidered, as the figure shows the two creep functions
( , ')J t t and 0( , )J t t have the same shape but different starting point
73
Figure 38. representation of the deformation for an aging model
Dischinger writes the same hypothesis in this other way:
t
It introduces the creep coefficient referred to a modulus at 28 days. Substituting this equation in the
Volterra integral and deriving in t, the differential equation that links the stress and the strain is:
28
( ) ( ) 1 ( ) 1 ( )( )
( )
d t d t d t d tt
dt dt E t dt E dt
(6.8)
This equation (6.8) is a differential equation of I order with variable coefficient. This equation substitutes
the principle of superposition equation hypotheses by McHenry. It is easily solvable knowing the creep
coefficient referred to a modulus at 28 days. To compute this coefficient, Dischinger uses the expo-
nential form:
0 0 0( , ) (1 exp( ( )) ( , )t t t t g t t (6.9)
That is:
0 0 0 0( , ') ( , ) ( ', ) (exp( ( ' ) exp( ( ))t t t t t t t t t t (6.10)
At infinite time, the creep coefficient can be considered constant for the asymptote of the creep func-
tion. Substituting the analytical form of ( , ')t t and integrating with ( ) 1t and ( ) 0t , it is possi-
ble to obtain the relaxation function:
0'
28
( )R( , ') ( ') exp 1 exp( ( ))
t
t
E dt t E t t t
E d
(6.11)
74
the result of experimental testing on concrete specimens shows the limited variation of the elastic mod-
ulus in time, 281.13E E . This difference is neglected and the elastic modulus is considered con-
stant in time. The creep and the relaxation functions becomes:
0 0
28 28
1 1( , ') 1 exp( ( ' )) exp( ( )) 1 ( , ')J t t t t t t t t
E E (6.12)
28 0 0 28R( , ') exp exp( ( ' )) exp( ( )) exp( ( , '))t t E t t t t E t t (6.13)
The creep function has a horizontal asymptote for t . The Volterra’s integral equation can be ex-
pressed also in this way:
28 28
( ) 1 ( ) ( )d t d t t
d E dt E
(6.14)
This is a differential equation of the first order with constant coefficient.
In the following figure, it is shown the difference between the Kelvin-Voigt’s model and the Dischinger
model. The second has values of the relaxation function bigger than the first one. Moreover, as it is pre-
viously said, for Dischinger model at time of unloading 0t , there is not the component of the creep
strain, so the behavior of the residual strain at the unloading time is irreversible.
Figure 39. Differences between the relaxing function for the Kelvin-Voigt and the Dischinger models
75
Figure 40. Example of the deformation for the Dischinger model with a complete unloading at to
The problem of the Dischinger model is that it does not consider the reversible part of the creep strain.
For a better approximation of the solution, Rüsch and Jungwirth modify the model on the basis of exper-
imental tests. The experiments show that the reversible part develop faster than the irreversible one,
more precisely the reversible part, which doesn’t depend form the application time of the stress, is 0.4
the elastic one:
* 28
1.4
EE
* 0.4
1.4
Dirichlet
To a deeper understanding of the behavior of a viscoelastic material, especially with regards to the time
scale at which the viscous processes take place, the concept of retardation spectrum turns out to be
useful. The viscoelastic behavior of the concrete is considered trough an incremental process in time by
a numerical method based on a creep approximation with a finite number of Dirichlet series terms [11].
A beam is loaded at time t0 , without any thermal variation strain. The concrete strain of the beam is:
0 0( , ) ( , ) ( )c c cnt t t t t (6.15)
휀𝑐𝜎(𝑡, 𝑡0) is the stress-dependent strain;
휀𝑐𝑛(𝑡) is the stress-independent strain which in this case is equal to the shrinkage because the thermal
component is equal to zero by hypothesis: 휀𝑐𝑛(𝑡) = 휀𝑐𝑇(𝑡) + 휀𝑐𝑠(𝑡) = 휀𝑐𝑠(𝑡).
The creep strain can be expressed considered approximately linear in relation to the stress whenever
the stress values remain inferior than half of the characteristic value of concrete strength:
0 0 0( , ) ( ) ( , )c ct t t J t t (6.16)
76
Since the concrete stresses will be variable in time, the stress dependent strain for a generic instant t
shall be written, using the superposition principle, by the following constitutive relation:
0
0 0 0( , ) ( ) ( , ) ( , )t
cc c
tt t t J t t J t d
(6.17)
This is the Volterra integral equation, the implementation of this relation is the adopted numerical
model requires specified algorithm since the concrete stresses are unknown. To solve this problem a
system of linear differential equation can be obtain approximating the creep function with a finite num-
ber of Dirichlet series terms as follows:
0
0 0
1
( , ) ( ) 1 i
t tn
j
j
J t t a t e
(6.18)
where the coefficient 0( )ja t is related to the initial shapes of specific creep curves at the loading appli-
cation time t0 and the values i are related to the shapes of specific creep curves over a period of time.
Another approximation with the Dirichlet series terms can be done on the Relaxation function, in the
same way as for the creep function. In this way it would be possible to express the inverse form of equa-
tion (6.17)that define the stresses (4.6).
77
Structural analysis for viscoelastic material
In the finite element analysis made in the previous chapter, deformations remain small, so the linear re-
lations could be used to represent the strain in a body. Now, there is the possibility of bigger defor-
mation during a loading process. For a viscouselastic material, it is important to distinguish the refer-
ence configuration, where the position is known and the current, or deformed configuration after load-
ing is applied. The relationship, describing the finite deformation behavior of solids, involve equations
related to both, the reference and the deformed configurations. The creep behavior, for reinforced con-
crete members, is analyzed in the chapter before. The long-term effects of creep and shrinkage lead to a
progressive increase in the deformation and a change of the cracking pattern over time. If the structural
members are subjected to high levels of sustained loads, the creep and the shrinkage effects may put
the structure out of service, may reduce the residual load-carrying capacity of the member over time, or
may even lead to premature failures. The time-dependent variation of the internal stresses in time, re-
sults from the linear brittle behavior of concrete in tension, from the nonlinear behavior in compression,
and, also, from the restraint of the long-term effects by the steel reinforcement. This stress redistribu-
tion is combined with the shifting of the neutral axis with time. In this chapter, it is expanded an inte-
gral-type computational approach, for the analysis of time-dependent effects of concrete structures. In
this approach, the finite element method is coupled with a numerical solution of the Volterra integral
equation used to describe an aging linear viscoelastic material. This method is justified based on the
principle of superposition and this is evident in the phase of assembling. As already said various time this
principle is suitable for simple and complex structure only if some conditions are met. First, the stresses
of the structure need to be lower than the 40% of the compressive strength of the concrete and second,
there cannot be significant changes in the environmental conditions [1,2,8,9]. Assuming these condi-
tions valid and using the same reference matrix and vectors used for the structural analysis for an elastic
material defined in the first chapter [(0.1) - (0.3)], it is possible to define the finite element method for a
viscous elastic material. In the previous chapter is exposed the differences of the strain due to creep ef-
fects. These change the constitutive law at the basis of the problem.
It is consider a structure composed by a viscoelastic material, with a Poisson coefficient constant in time,
the dynamic stress equilibrium at a material point inside the finite element can be represented as a sum
of the elastic part and the viscous one, as it is already saw in the explanation of the creep for a one-di-
mensional problem. The structure is continuous but not homogenous for the hypothesis of distinct
phases of construction that provoke different properties of the material. The structure is subdivided in
different finite elements i, that are continuous and homogeneous. To extend the problem to a three-
dimension material, the expression(4.4) can be reported in matrix form, which represent the constitu-
tive law for a finite element i:
0
( ) ( , ') ( ') 't
i i it J t t t dt (6.19)
where the creep function can be express as follow by the hypothesis of Poisson’s coefficient constant in time [30]:
78
1
0 0 0
0 1 0 0
0 1 0 0
0 1 0 0( , ) ( , ) ( , )
00 0 0 2(1 ) 0
2(1 )0 0 0 0 0
0 0 0 0 2(1 )0
i i
i i
i i
i ii ii
i
i
J t t J t t J t t
(6.20)
With this hypothesis, the equation (6.19) becomes:
1
0
( ')( ) ( , ') '
'
ti
i ii
d tt J t t dt
dt
(6.21)
where the matrix 1
i
can go outside of the integral because it is independent on the time. Being valid
the principle of superposition it is possible to divide the integral in two terms as follow:
0
1 1
0 0
( ')( ) ( , ) ( ) ( , ') '
'
ti
i ii ii i t
tt J t t t J t t dt
t
(6.22)
The first term represents an immediate response of the material to the loading at time t0 and the second
term shows all the creep deformation in the successive instant since t0 until t. The term 0( , )iJ t t charac-
terizes each finite element because of the hypothesis of a heterogeneous material.
The inverse equation that connect the stress field to the strain field expressed in the equation (4.6) can
be written in matrix form for a three dimensional problem as below:
0
( ')( ) ( , ') '
'
ti
i i
tt R t t dt
t
(6.23)
For the hypothesis of Poisson’s coefficient constant in time, the relaxation function can be expressed as
follow:
79
( , ') ( , ')
1 0 0 0(1 )(1 2 ) (1 )(1 2 )
1 0 0 0(1 )(1 2 ) (1 )(1 2 )
1 0 0 0(1 )(1 2 ) (1 )(1 2 )
( , ')1
0 0 0 0 02 1
10 0 0 0 0
2 1
10 0 0 0 0
2 1
ii i
i i
i i i i
i i
i i i i
i i
i i i i
i
i
i
i
R t t R t t
R t t
(6.24)
Substituting this expression (6.24), the equation (6.23) becomes:
0
( ')( ) ( , ') '
'
ti
i ii
tt R t t dt
t
(6.25)
For the superposition principle, it becomes:
0
0
0 0
0 0
( ')( ) ( , ) ( ) ( , ') '
'
( ')( , ) ( ) ( , ') '
'
ti
i ii it
ti
ii ii i t
tt R t t t R t t dt
t
tR t t t R t t dt
t
(6.26)
The link between the creep function and the relaxation function for a three-dimensional problem
The link between the creep and the relaxation function is expressed in the equation (4.12) for a one-di-
mensional problem. To arrive at the same formula for a three-dimensional problem, the constitutive law
for a viscoelastic material with a constant Poisson’s coefficient expressed in(6.26) and (6.22) is:
0
0
1 1
0 0
0 0
( ')( ) ( , ) ( ) ( , ') '
'
( ')( ) ( , ) ( ) ( , ') '
'
t
t
t
t
d tt J t t t J t t dt
dt
d tt R t t t R t t dt
dt
(6.27)
A constant stress I t is imposed and the two functions (6.27) become:
80
0
1
0
0 0
( ) ( , )
( ')( , ) ( ) ( , ') '
'
t
t
t J t t I
d tI R t t t R t t dt
dt
(6.28)
Substituting the value of the strain obtain in the first expression, in the second one:
0
1
0
1
1 0
0 0 0
( ) ( , )
( ', )( , ) ( , ) ( , ') '
'
t
t
t J t t I
dJ t t II R t t J t t I R t t dt
dt
(6.29)
Knowing the definition of 0 0
0
1( , )
( )c
J t tE t
and that the Poisson’s coefficient is constant in time, so it
can go outside of the integral, the final expression of the dependence between the creep function and
the relaxation function is:
0
0 0
0
( ', ) ( , )( , ') '
' ( )
t
tc
dJ t t R t tI R t t dt I I
dt E t (6.30)
that is:
0
0 0
0
( ', ) ( , )( , ') ' 1
' ( )
t
tc
dJ t t R t tR t t dt
dt E t (6.31)
or
0
0 0
0
( , ) ( ', )1 ( , ') '
( ) '
t
tc
R t t dJ t tR t t dt
E t dt (6.32)
In case of constant deformation I in time, the equations of the (6.27) become:
0
1 1
0 0
0
( ')( , ) ( ) ( , ') '
'
( ) ( , )
t
t
d tI J t t t J t t dt
dt
t R t t I
(6.33)
The relation that can be obtain by the first equation is the following:
0
1 1 0
0 0 0
( ', )( , ) ( , ) ( , ') '
'
t
t
dR t t II J t t R t t I J t t dt
dt
(6.34)
By definition 0 0( , )R t t = 0( )cE t so the last equation(6.34) becomes:
0
00 0
( ', )( , ) ( ) ( , ') '
'
t
ct
dR t tI J t t E t I J t t dt I
dt (6.35)
81
that is:
0
00 0
( ', )1 ( , ) ( ) ( , ') '
'
t
ct
dR t tJ t t E t J t t dt
dt (6.36)
These two equations (6.36) and (6.32) are the same obtained for a one dimensional problem in the pre-
vious chapters (4.12) and(4.8).
Principle of Virtual Displacement for a three-dimensional problem
By the definition of the constitutive law(6.26), now it is possible to arrive at the definition of the princi-
ple of virtual displacement. Looking at the formula of the principle of virtual displacement for an elastic
material(1.16), everything that is on the right of the equation stay the same, because it does not contain
terms on the constitution of the material, but the first part of the equation evolves in a different way for
a viscous elastic material than for an elastic one that is analyzed in the chapters before. The first integral
of the principle of virtual work for a viscouselastic material changes by the constitutive law (6.26) as be-
low:
0
0
0 0
0 0
( ')( ) ( , ) ( , ') ' ( )
'
( ')( , ) ( ) ( , ') ' ( )
'
TtT T T Ti
i i ii iV V ti
TtT T Ti
i ii ii iV t
tdV t R t t R t t dt t dV
t
tR t t t R t t dt t dV
t
(6.37)
Since the coefficient ( , ')iR t t does not depend on the coordinates (x,y,z), it can be put outside of the in-
tegral in the first term. The matrix i
is symmetric as it is shown in eq.(6.24), so the transposed matrix is
the same as the normal one T
i i The equation (6.37) becomes:
0
0 0
( ')( , ) ( ) ( ) ( , ') ' ( )
'
TtT T i
i i i ii ii iV V V ti
tdV R t t t t dV R t t dt t dV
t
(6.38)
The Rayleigh-Ritz’s approximations are the same that are used for the linear elastic material (2.2) and (2.3):
( , , , ) ( , , ) ( )
( , , , ) ( , , ) ( )
LL L L L L L
i ii
LL L L L L L
i ii
s x y z t N x y z u t
s x y z t N x y z u t
(6.39)
( , , , ) ( , , ) ( )
( , , , ) ( , , ) ( )
LL L L L L L
i ii
LL L L L L L
i ii
x y z t B x y z u t
x y z t B x y z u t
(6.40)
82
where the shape matrix i
N and the derivatives of the shape matrix i
B are defined and they depend on
the space only, but the deformation, the strain and the displacement are dependent on the time. Com-
pared to the linear elastic solution (3.16), this case has a dependence on the time due to the non-linear
behavior of the creep effects.
By the definitions (6.39) and (6.40) the eq. (6.38) becomes:
0
0 0
( ')( , ) ( ) ( ) ( , ') ' ( )
'
LT T
t iT LT T L Li
i i i ii ii i ii iV V V ti
u t BdV R t t u t B B u t dV R t t dt B u t dV
t
(6.41)
where the displacement vectors are dependent on the time only and are not dependent on the space.
For this reason, they can be brought outside of the integral in the volume:
0
0 0
( ')( , ) ( ) ( ) ( , ') ' ( )
'
LT T
t iT LT T L Li
i i i ii ii i ii iV V V ti
u t BdV R t t u t B B dV u t R t t dt B dV u t
t
(6.42)
The matrix of the derivate of the shape functions i
B depends on the space only ( , , )L L L
i iB B x y z ,
where ( , , )L L Lx y z are the local cartesian coordinates, and it is independent on the time. The matrix
iB can be put outside of the time integral:
00 0
( ')( , ) ( ) ( ) ( , ') ' ( )
'
LTtT LT T L T Li
i i i ii ii i i ii iV V V ti
u tdV R t t u t B B dV u t R t t dt B B dV u t
t
(6.43)
The integral in time in the brackets does not depends on the volume and can go outside of the integral:
00 0
( ')( , ) ( ) ( ) ( , ') ' ( )
'
LTtT LT T L T Li
i i i ii ii i i ii iV V t Vi
u tdV R t t u t B B dV u t R t t dt B B dV u t
t
(6.44)
It is now possible to put in evidence the integral in the volume that is present in both terms:
0
0 0
( ')( , ) ( ) ( , ') ' ( )
'
LTtT LT T Li
i i ii i i iiV t Vi
u tdV R t t u t R t t dt B B dV u t
t
(6.45)
The integral of the volume represents the dimensionless matrix:
83
T
i ii iVk B B dV (6.46)
related to the stiffness matrix of the elastic solution at time t0 as shown below:
0 0
0 0
1 1( ) ( )
( ) ( )
T
i ii iV
T
i i i iVi i
k B B dV
B E t B dV k tE t E t
(6.47)
This matrix is symmetric and the transposed form is the same as the normal one i
k = T
ik . The stiffness
matrix of the linear elastic material is dependent only on the time of loading, as the dimensionless stiff-
ness matrix that is defined in (6.47).
The equation (6.45) becomes:
0
0 0
( ')( , ) ( ) ( , ') ' ( )
'
LTtT LT Li
i i ii i iV ti
u tdV R t t u t R t t dt k u t
t
(6.48)
For the principle of virtual work, this last expression is equal to the external work that is the same ex-
pression of equation (2.1) :
00 0
( ')( , ) ( ) ( , ') ' ( ) ( ) ( )
' f
LTtLT L T L T Li
i i i ii i i i it V S
u tR t t u t R t t dt k u t F N dV u t f N dS u t
t
(6.49)
The vector of the external forces is:
f
T T T
i i iV SP F N dV f N dS (6.50)
Substituting in the equation (6.49), it becomes:
0
0 0
( ')( , ) ( ) ( , ') ' ( ) ( )
'
LTtLT L T Li
ii i ii i it
u tR t t u t R t t dt k u t P u t
t
(6.51)
This final solution can be compared to the linear elastic one (3.17). For a viscous elastic material, there is
a dependence of all the displacement on the time. The stiffness matrix for a viscous elastic material can-
not be assembled as in the linear elastic one because the dependence is not only on the time of loading
but also on all the successive instant until the time of computation. For this reason, the dimensionless
stiffness matrix i
k and the terms that contains ( , ')iR t t characterize the stiffness of the element and they
cannot be assembled in one matrix only as for the elastic material. The problems to solve the governing
equation for a viscous elastic material (6.51) are two:
84
- The reference to the relaxation function and not on the creep function that is the known one,
this problem can be solved analyzing the link between the creep and the relaxation function for
a three-dimensional problem
- The integral in time that is in the brackets, a Volterra integral, that needs one of the methods
previously analyzed in the chapters before to be solved
Transformation of the coordinates
This last equation (6.51) refers to a finite element i that is still being analyzed in singularly. In the follow-
ing phase, the referment system is changed from local to general. The matrix of shape i
N and its deriv-
ative i
B are referred both to a local system of cartesian coordinates:
( , , )
( , , )
L L L
i i
L L L
i i
N N x y z
B B x y z
(6.52)
To change the referent system is necessary to use a rotation matrix i
T that connects the displacement
in the local system to the displacement in the global one as follows:
L
i ii
L
i ii
u T u
u T u
(6.53)
where the displacement vector iu is referred to the global reference system and not anymore to the
local one. By these equations (6.53), the displacement and the strain fields become the following:
( , , , ) ( , , ) ( ) ( , , ) ( )
( , , , ) ( , , ) ( ) ( , , ) ( )
LL L L L L L L L L
i i iii i
LL L L L L L L L L
i i iii i
s x y z t N x y z u t N x y z T u t
s x y z t N x y z u t N x y z T u t
(6.54)
( , , , ) ( , , ) ( ) ( , , ) ( )
( , , , ) ( , , ) ( ) ( , , ) ( )
LL L L L L L L L L
i i ii i i
LL L L L L L L L L
i i ii i i
x y z t B x y z u t B x y z T u t
x y z t B x y z u t B x y z T u t
(6.55)
85
Assembling phase
It is now possible to pass to the assembling phase to arrive to an only model referred to the continuous
structure. It is necessary to define a matrix of connectivity i
L that links the displacement of the total
structure u to the one of each finite element iu as shown below:
i i
i i
u L u
u L u
(6.56)
The expression of the displacement and the strain fields can be written as follow:
( , , , ) ( , , ) ( ) ( , , ) ( )
( , , , ) ( , , ) ( ) ( , , ) ( )
L L L L L L L L L
i ii i ii i
L L L L L L L L L
i ii i ii i
s x y z t N x y z T u t N x y z T L u t
s x y z t N x y z T u t N x y z T L u t
(6.57)
( , , , ) ( , , ) ( ) ( , , ) ( )
( , , , ) ( , , ) ( ) ( , , ) ( )
L L L L L L L L L
i ii i i i i
L L L L L L L L L
i ii i i i i
x y z t B x y z T u t B x y z T L u t
x y z t B x y z T u t B x y z T L u t
(6.58)
Now, the expression of the of the principle of virtual work (6.51) can be completed without substantial
variations. Specifically, the displacement vector of the specific finite element, that is multiplied by the
matrix of transformation of the coordinates and the matrix of connectivity multiplied of equation (6.51),
is substituted and it is obtained the following equation:
00
( ')( , ) ( , ') '
'
T T T
tT T T Ti i
ii ii i i i i iit
u t L TR t t u L T R t t dt k T L u P T L u
t
(6.59)
The connectivity matrix i
L is independent on the time and it can go outside of the integral:
00
( ')( , ) ( , ') '
'
T
tT T T T
ii i i i i i i iit
u tR t t u R t t dt L T k T L u P T L u
t
(6.60)
The vector of small displacement can be remove on both side of the equation as follow:
00
( ')( , ) ( , ') '
'
T
tT T T T
ii i i i i i i iit
u tR t t u R t t dt L T k T L P T L
t
(6.61)
86
This equation refers on the singular finite element and all the coefficients depends on it, in exception of
the vector of the displacement. To put in evidence the displacement vector in the big brackets is neces-
sary to solve the time integral inside of those. The expression (6.61) can be also expressed in the follow-
ing way:
0
( ')( , ') '
'
T
t T T T
ii i i i i i ii
u tR t t dt L T k T L P T L
t
(6.62)
The problem of the assembling phase is the next passage that for the linear elastic material is made by
the summation of all the element stiffness matrixes to obtain the global stiffness matrix (3.28). This pro-
cedure is not possible for a viscous elastic material because the first term in the brackets is dependent
on each finite element for the relaxation function ( , ')iR t t that characterize them. The solution of the
Volterra integral needs to be done before this passage, then it is possible to divide the terms that are
dependent on the finite element and the global one.
For the computation of the stress field is not simple as for the linear elastic material since the depend-
ence of the strain with the stresses is not anymore linear. The stresses and the strain are connected by
the constitutive law previously defined (6.26) which is dependent by the time. To directly connect the
stress fields to the displacement one that is the output of the analysis it is possible to modify the consti-
tutive law (6.26) with the last expression of the Rayleigh-Ritz approximation (6.58):
00 0
( ')( ) ( , ) ( ) ( , ') '
'
ti i i
i i i t
d B T L u tt R t t B T L u t R t t dt
dt
(6.63)
All the terms that are independent by the time can put outside of the integral and the following relation
is obtained:
0
0
0 0
0 0
( ')( ) ( , ) ( ) ( , ') '
'
( ')( , ) ( ) ( , ') '
'
t
i i i i i i t
t
i i i t
d u tt R t t B T L u t B T L R t t dt
dt
d u tB T L R t t u t R t t dt
dt
(6.64)
For the computation of the stress field it is necessary to solve the Volterra integral. For the solution, it is
possible to use one of the methods explained in the previous chapters.
It is possible to add now to the solution just obtained, the shrinkage deformation which is not depend-
ent on the stresses. This can be computed following the same path with a thermic variation.
87
Solution of the problem with the algebraic method
To solve the problem by the Algebraic method previously explained, the constitutive law (5.38) for a
three-dimensional problem of a material with a constant Poisson’s coefficient is the starting point:
0
0
1 1
0 0
0 0
( ')( ) ( , ) ( ) ( , ') '
'
( ')( ) ( , ) ( ) ( , ') '
'
t
t
t
t
d tt J t t t J t t dt
dt
d tt R t t t R t t dt
dt
(6.65)
The strain can be expressed also as a function of the creep coefficient 0( , )t t . The strain is composed
by an elastic part and a viscous one that is dependent on the creep coefficient as it is shown below:
1
0 1 10 0 0 0
0
1( , ) ( , ) ( ) ( , )
( )c
t t t t t t tE t
(6.66)
The second equation of the (6.65) can be rewritten by using the (6.66) as follows (remember that
0 0( , ) 0t t ):
0
1 1
10 0 0 0
0 0
1 1( ) ( , ) ( ) ( , ') ( ) ( ', ) '
( ) ' ( )
t
tc c
dt R t t t R t t t t t dt
E t dt E t
(6.67)
Knowing that 1
0
0
1( ) 0
' ( )c
dt
dt E t
, because the term in the brackets is a constant, the last
equation (6.67) becomes:
0
1 0
10 0
0
( ', )1( ) ( , ) ( ) ( , ') '
( ) '
t
tc
d t tt R t t t R t t dt
E t dt
(6.68)
The definition of the creep function (5.1)can be derived in time as shown below:
00
0
( ', ) 11 ( ', )
' ' ( )c
J t tt t
t t E t
(6.69)
the elastic modulus at time of loading is constant in time:
0
10
' ( )ct E t
(6.70)
The equation (6.69) connects the creep function to the creep coefficient:
88
00 0
( ', )( ) ( ', )
' 'c
J t tE t t t
t t
(6.71)
This term can be substitute in the expression of the stress (6.68):
0
1 010 0 0
0
( ', )1( ) ( , ) ( ) ( , ') ( ) '
( ) '
t
ct
c
J t tt R t t t R t t E t dt
E t t
(6.72)
where it is possible to delete all the matrix containing the Poisson’s coefficient since it is constant in
time:
0
010 0 0
0
( ', )1( ) ( , ) ( ) ( , ') ( ) '
( ) '
t
ct
c
J t tt R t t t R t t E t dt
E t t
(6.73)
Substituting in this expression the link between the creep function and the relaxation function previ-
ously computed (6.31), last equation(6.73) becomes:
010 0 0
0 0
( , )1( ) ( , ) ( ) ( ) 1
( ) ( )c
c c
R t tt R t t t E t
E t E t
(6.74)
The viscous strain can be put in evidence from this last equation:
1
1 0 000 0
0
1 1 1( ) ( , ) ( )
( , )( ) ( )1
( )c c
c
t R t t tR t tE t E t
E t
(6.75)
The computation of the whole strain of the equation (6.66) will becomes:
1 1
0 0 0 0 000 0 0
0
1 1 1 1( , ) ( ) ( , ) ( ) ( , ) ( )
( , )( ) ( ) ( )1
( )c c c
c
t t t t t t R t t tR t tE t E t E t
E t
(6.76)
that can be rewrite as follows:
1 10 0 00 0
0 00 0 0
0 0
( , ) ( ) ( , )( ) 1( , ) 1 ( , )
( , ) ( , )( ) ( ) ( )1 1
( ) ( )c c c
c c
t t t R t ttt t t t
R t t R t tE t E t E t
E t E t
(6.77)
In the constant of the last and in the first term, it is summed and subtracted 1:
89
1 10 0 0
0 00 00 0 0
0 0
( , ) 1 1 ( ) ( , )( ) 1( , ) 1 ( , ) 1 1
( , ) ( , )( ) ( ) ( )1 1
( ) ( )c c c
c c
t t t R t ttt t t t
R t t R t tE t E t E t
E t E t
(6.78)
The last equation can be rewritten in the following way:
1 1 00 0 0 0
0 00 0 0
0 0
( )( ) 1 1 1( , ) 1 ( , ) 1 ( , ) ( , )
( , ) ( , )( ) ( , ) ( )1 1
( ) ( )c c
c c
ttt t t t t t t t
R t t R t tE t t t E t
E t E t
(6.79)
1 1 00 0 0
0 00 0 0 0
0 0
( )( ) 1 1 1 1( , ) 1 ( , ) ( , ) 1
( , ) ( , )( ) ( , ) ( ) ( , )1 1
( ) ( )c c
c c
ttt t t t t t
R t t R t tE t t t E t t t
E t E t
(6.80)
The following term can be defined:
0
0 0
0
1 1( , )
( , ) ( , )1
( )ci
t tR t t t t
E t
(6.81)
The equation (6.80) becomes:
1 1 0
0 0 0 0 0
0 0
( )( )( , ) 1 ( , ) ( , ) ( , ) 1 ( , )
( ) ( )c c
ttt t t t t t t t t t
E t E t
(6.82)
or
1 10 0
0 0 0 0
0 0
( ) ( ) ( )( , ) 1 ( , ) 1 ( , ) ( , )
( ) ( )c c
t t tt t t t t t t t
E t E t
(6.83)
The equation (6.83) can also be written with the displacements for the Rayleigh-Ritz approximation (2.2)
and (2.3):
1 1 0
0 0 0 0
0 0
( )( )( ) 1 ( , ) ( , ) ( , ) 1 ( , )
( ) ( )ii
c c
ttB u t t t t t t t t t
E t E t
(6.84)
where the only incognita is ( )t . The incognita is put in evidence:
90
1 1 0
0 0 0 0
0 0
( )( )1 ( , ) ( , ) ( ) ( , ) 1 ( , )
( ) ( )ii
c c
ttt t t t B u t t t t t
E t E t
(6.85)
10 00 0
0 0 0
( ) ( )( ) ( ) ( , ) 1 ( , )
1 ( , ) ( , ) ( )
c
iic
E t tt B u t t t t t
t t t t E t
(6.86)
The multiplier in front of the big square brackets is a constant and also the multiplier of the stress at
loading:
01 0
0 0
( )( , )
1 ( , ) ( , )
cE tC t t
t t t t
(6.87)
02 0 0
0
( , )( , ) 1 ( , )
( )c
t tC t t t t
E t
(6.88)
By these definitions the equation (6.86) can be rewritten as follows:
1
1 0 1 0 2 0 0( ) ( , ) ( ) ( , ) ( , ) ( )c iit C t t B u t C t t C t t t
(6.89)
that is:
1 0 1 0 2 0 0( ) ( , ) ( ) ( , ) ( , ) ( )c iit C t t B u t C t t C t t t (6.90)
Now, the principle of virtual work (1.16) can be expressed with this definition of the stresses(6.90), the
first part of the integral becomes:
1 0 0 1 0 2 0( ) ( , ) ( ) ( , ) ( , )
T LT T T
i i iiV Vi idV u t B C t t t C t t C t t dV
(6.91)
1 0 0 1 0 2 0( ) ( , ) ( ) ( , ) ( , ) ( )
T LT T T L
i i i ii iV VidV u t B C t t t C t t C t t B u t dV
(6.92)
The second part of the equation of the principle of virtual work stay the same [(6.49)-(6.51)]. The princi-
ple of virtual work becomes the following:
1 0 0 1 0 2 0( ) ( , ) ( ) ( , ) ( , ) ( ) ( )LT T T L T L
ii i i ii iVu t B C t t t C t t C t t B dV u t P u t (6.93)
The problems of the Volterra integral and of the reference to the relaxation function are avoid. The dis-
placement vector, which is the incognita, can be put in evidence as follows:
1 0 0 1 0 2 0
1 0 0 1 0 2 0
( ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( )
( ) ( , ) ( ) ( , ) ( , )
LT T L T L T L
ii i i i ii i iV V
LT T T T
ii ii i iV V
u t B C t t B dV u t P u t t C t t C t t B dV u t
u t B C t t B dV P t C t t C t t B dV
(6.94)
91
The problem is solved. The value of 0( , )t t can be assumed as 0.8 by Trost approximation [21] which
respects the limits value of the coefficient computed by Mola [21]1
2 and 1, the value of the creep coef-
ficient 0( , )t t and the value of the elastic modulus 0( )ciE t are computable by the CEB90 and the stress
at time of loading is known. At this point it is easy to compute the stresses at time t with equation (6.86)
because the displacement vector ( )L
iu t is known.
In the last equation(6.94), the constant 1 0( , )C t t is substitute as follows:
0
0 0 2 0
0 0 0 0
( )1( ) ( ) ( ) ( , )
1 ( , ) ( , ) 1 ( , ) ( , )
LT T T T cii ici i iV V
E tu t B E t B dV P t C t t B dV
t t t t t t t t
(6.95)
or
00 2 0
0 0 0 0
( )1( ) ( ) ( , )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T cii i ii V
E tu t k P t C t t B dV
t t t t t t t t
(6.96)
In this equation it is possible to recognize the elastic stiffness matrix already defined for the linear elastic
problem (3.18). Substituting, the equation (6.95) becomes ( 0 0 2 0, 0, ( , ) 0t t C t t ) :
( )LT L T
ii iu t k P (6.97)
System (6.96) has the same form of the elastic problem (6.97) but it contains more known terms. There
is a fictitious force, that is represented by the last term (the integral) of the equation, and an additional
known term in front of the displacements.
It can then be stated that the algebraic method is a fast way to compute the displacement for an as-
signed time t. The dependence of the displacement is only on the time t and on the time of loading t0,
but does not require to take into account small time steps between these two. For this reason, the costs
of computation are considerably reduced. Nevertheless
The computation is not that different compared to the linear elastic problem. The additional terms are
the two constants and an integral in the volume that correspond to a fictitious force applied on the
structure. This integral is easy to compute because everything is known and the dependence on the vol-
ume is of the stress vector and the derivate shape matrix.
Another advantage is that once that the problem is solved and the displacement are known, the compu-
tation of the stresses is easier. The formula (6.90) does not contain any more a Volterra integral for the
computation of the stresses and it is simple to solve.
92
Obviously, this is an approximated solution, as it was described above. Term 0( , )t t can be computed
exactly by means of equations (6.81) and (6.31), or, remembering its limits (see Mola [21] and (5.51))
and according to Trost [36] its value can be fixed at 0.8. Anyway this approximation demonstrated to
give very good solutions in many practical cases so that it is for instance adopted by the European code
EUROCODE 2 to determine the stress changes in prestressed concrete sections and the value of the pre-
strain to be adopted in ultimate limit state computation.
Consequently, the approximation of the creep function by means of a Dirichlet serie is useless and the
creep function can be use directly in the computation. The substitution of the creep function with the
product of the creep coefficient and the coefficient 0( , )t t allows to get a solution without other ap-
proximation in a simple way.
This method represents an approximated approach to the general analysis of the structure under long-
term loading, and therefore tests are needed to verify the extent of the error gathered, but this is a big
problem itself because it implies the knowledge of the exact solution that generally is unknown. There-
fore, this method will be tested by means of comparison with general solutions computed for specific
heterogeneous structure and by trying to verify the effectiveness of the theorems of viscoelasticity.
A piece of beam, as it is done in the Finite Element Method for an elastic material, is considered, in the
following figure it is represented the kinematic model that is considered.
Figure 41. Kinematic of Bernoulli-Euler model of the beam
The beam is loaded as shown in the following figure and it is simply supported on both edges.
93
Figure 42. piece of beam with external and internal forces in evidence
The material of the beam is a viscous-elastic one with a Poisson’s coefficient constant in time. It is now
possible to write the formulation obtain by the algebraic method (6.96). The material of the beam is
supposed to be homogeneous, indeed the coefficient 0( , )t t is constant and easily computable by the
CEB 90 (5.6). The coefficient 0( , )t t is consider equal to 0.8 for the Trost approximation [21]. The inte-
gral in the equation (6.96) contain all the variables that changes for the beam formulation. The deriva-
tives of the shape matrix i
B is the same as the one computed for an elastic material (3.12) because it
depends on the geometric properties of the beam. The stress vector 0( )i t is the same of the elastic
beam because in the first instant of loading the beam behave as elastic as stated in the first and the sec-
ond principle of linear viscoelasticity (4.14) and (4.15):
0 0( ) ( )el
i i i i ii it t E E b u E b u (6.98)
The validity of these principles is inside the definition of the creep function itself [2]:
0
0
0 28
( , )1( , )
( )c c
t tJ t t
E t E
(6.99)
The creep coefficient at time t0 is equal to 0, 0 0( , ) 0t t , so the constitutive law (4.3) at time of load-
ing t0 is equal to the elastic one:
1 1
0 0 0 0 0 0
0
1( , ) ( ) ( ) ( ) ( )
( )
el
ci i i cicici
t t E t t t tE t
(6.100)
The computation of the incognita displacement ( )LT
iu t is possible by the following formulation:
94
00 2 0
0 0 0 0
( )1( ) ( ) ( , )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T cii i ii V
E tu t k P t C t t B dV
t t t t t t t t
(6.101)
Everything that is a constant for the element and it is contained in the integral can go outside:
02 0 0
0 0 0 0
( )1( ) ( , ) ( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T Tcii i ii V
E tu t k P C t t t B dV
t t t t t t t t
(6.102)
The coefficient 2 0( , )C t t is also constant for the element, it can be substituted with the full terms ac-
cording to equation (6.88):
0 00 0
0 0 0 0 0
( ) ( , )1( ) 1 ( , ) ( )
1 ( , ) ( , ) 1 ( , ) ( , ) ( )
LT L T Tcii i ii V
c
E t t tu t k P t t t B dV
t t t t t t t t E t
(6.103)
that is:
0 0
0
0 0 0 0
( , ) 1 ( , )1( ) ( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T
ii i ii V
t t t tu t k P t B dV
t t t t t t t t
(6.104)
The stress vector that is inside the integral is referred to the first instant. At the initial moment the stress
vector is the same as the elastic stress vector for the first principle of viscoelasticity (4.14). The vector
0( )T
i t can be replaced in equation (6.104) with the expression given in the equation (2.13) (matrix
is symmetric) :
0 0
0 0
0 0 0 0
( , ) 1 ( , )1( ) ( ) ( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T T
ii i ci ii V
t t t tu t k P u t B E t B dV
t t t t t t t t
(6.105)
The displacement vector is independent of the volume and can be put outside of the integral:
0 0
0 0
0 0 0 0
( , ) 1 ( , )1( ) ( ) ( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T T
ii i ci ii V
t t t tu t k P u t B E t B dV
t t t t t t t t
(6.106)
In this last equation the stiffness elastic matrix is represented by the integral, and, therefore, equation
(6.106) becomes:
0 0
0
0 0 0 0
( , ) 1 ( , )1( ) ( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T L
ii ii i
t t t tu t k P u t k
t t t t t t t t
(6.107)
The last two terms are the vector of the external forces T
iP (see equation (3.32)), the equation (6.107)
is:
95
0 0
0 0 0 0
( , ) 1 ( , )1( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T T
i ii i
t t t tu t k P P
t t t t t t t t
(6.108)
The vector of external forces is put in evidence:
0 0 0 0
0 0 0 0
1 ( , ) ( , ) ( , ) 1 ( , )1( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T
ii i
t t t t t t t tu t k P
t t t t t t t t
(6.109)
that is:
0
0 0 0 0
1 ( , )1( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T
ii i
t tu t k P
t t t t t t t t
(6.110)
This last equation permits to compute the displacement vector just knowing the constants 0( , )t t and
0( , )t t as an elastic problem.
Example of application for a frame
For the frame, that is considered in the application of the finite element method for an elastic material,
the procedure is slightly different in the case of viscoelastic material with the algebraic method.
The figure below represents the degrees of freedom of the frame, the displacement in x and y direction
and the rotation for the point 1 and 2.
Figure 43. Degrees of freedom of the frame
Indeed, the displacement vector is:
96
1
1
1
2
2
2
u
v
uu
v
(6.111)
The last equation computed in the previous chapter (6.110) shows that the problem can be considered
as an elastic one by the addition of the two constants 0( , )t t and 0( , )t t , that characterize the viscoe-
lasticity. The coefficient 0( , )t t is considered 0.8 for Trost approximation and the creep coefficient
0( , )t t can be computed by the correspondent formula of the code [CEB 90 (5.5); CEB 10 (5.12)]. The
other matrices are the same computed in the example of the elastic frame (3.50). The vector of the
nodal forces in this case, where concentrated forces are applied on the first node, is represented as:
0
0
0
0
F
p
P
(6.112)
The problem is solved with the computation of the displacement vector with the following equations:
0
0 0 0 0
1 ( , )1( )
1 ( , ) ( , ) 1 ( , ) ( , )
LT L T
ii i
t tu t k P
t t t t t t t t
(6.113)
For the computation of the stresses in the beam it is possible to use the formula:
1 0 1 0 2 0 0( ) ( , ) ( ) ( , ) ( , ) ( )c iit C t t B u t C t t C t t t (6.114)
where the constant 1 0( , )C t t and 2 0( , )C t t are defined by the two formulas (6.87) and (6.88) by the al-
ready computed coefficients. The derivative of the shape matrix i
B needs to be computed for the calcu-
lation of the stresses.
For the derivation of the shape matrix, displacement field that defines the incognita is defined.
Firstly, the axial deformations are neglected and the transverse displacements and rotations show, as in
the Figure 44.
97
Figure 44. Vertical and rotational displacement of the beam
The deformed shape of a frame element, v(x), subjected to end-forces F, is a cubic polynomial. The poly-
nomial can be written in the following form:
2 3
0 1 2 3( )v x a a x a x a x (6.115)
The rotation of the beam can be computed as the derivative of the vertical displacement:
2
1 2 3( ) '( ) 2 3x v x a a x a x (6.116)
The four polynomial coefficient a1, a2, a3 and a4 are computed by matching the v(0), v’(0), v(l) and v’(l) to
the specified end displacement and rotation of the beam. Assuming that the rotation are small and ne-
glecting the shear deformation effects, it is imposed that:
1 0 1(0)v v a v (6.117)
1 1 1'(0)v a (6.118)
2 1 1 22 3
2 0 1 2 3 2 2 2
3 2( )
v vv l v a a l a l a l v a
l l
(6.119)
2 12 1 2
2 1 2 3 2 3 3 2
2'( ) 2 3
v vv l a a l a l a
l l
(6.120)
Neglecting the axial deformation, the shape matrix N can be defined by the relation of it with the verti-
cal displacement and the rotation with the known coefficient of the end-beam:
1
1
2
2
v
vN
v
(6.121)
The shape matrix is defined as:
98
2 3 2 3 2 3 2 3
2 3 2 2 3 2
2 2 2 2
2 3 2 2 3 2
3 2 2 3 21
6 6 4 3 6 6 2 31
x x x x x x x xx
l l l l l l l lN
x x x x x x x x
l l l l l l l l
(6.122)
The derivative of the shape matrix B can be simply computed by deriving the shape matrix:
2 2 2 2
2 3 2 2 3 2
2 3 2 2 3 2
6 6 4 3 6 6 2 31
6 12 4 6 6 12 2 6
x x x x x x x x
l l l l l l l lBx x x x
l l l l l l l l
(6.123)
This matrix is going to complete the equation (6.96).
For beam with transverse displacement v(x) but no axial deformation, the displacement of the end of
the beam is along the direction of the axial load N(x), which in this case is represented by the force F, as
it is shown in the Figure 43.
Figure 45. Axial deformation of the beam in the x direction due to the transverse deformation
The deformation increases with the rotation of the beam. With the approximation of a small angle of
rotation, this small displacement can be expressed as followed:
dv
du dvdx
(6.124)
This displacement in the x direction is only due to the vertical displacement. Frame elements carrying axial loads and undergoing to large lateral displacement have nonlinear behavior arising for the internal moment that are the product of the axial load F and the transverse displacement.
For the axial displacement the shape function can be derive by the following expression:
1 2( ) 1x x
u x u ul l
(6.125)
99
where the nodal displacement in x direction of point 1 and 2 is already substituted. The axial displace-
ment is now added to the previous system of equations (6.121):
1
1
1
2
2
2
u
vu
v Nu
v
(6.126)
where the shape matrix N is expressed as follows:
2 3 2 3 2 3 2 3
2 3 2 2 3 2
2 2 2 2
2 3 2 2 3 2
1 0 0 0 0
3 2 2 3 20 1 0
6 6 4 3 6 6 2 30 1 0
x x
l l
x x x x x x x xN x
l l l l l l l l
x x x x x x x x
l l l l l l l l
(6.127)
The derivative of the shape matrix B is:
2 2 2 2
2 3 2 2 3 2
2 3 2 2 3 2
1 10 0 0 0
6 6 4 3 6 6 2 30 1 0
6 12 4 6 6 12 2 60 0
l l
x x x x x x x xB
l l l l l l l l
x x x x
l l l l l l l l
(6.128)
It is now possible to solve the equation (6.114) for the computation of the stress vector.
Example of application for a cantilever beam
A cantilever beam is subjected to a concentrated transverse load 0P [40] (p.327) applied at the point x=a
of a beam element of length l. The corresponding consistent load vector P is computable from the ex-
pression of the work done by the external load as follows:
0 0
0 0( ) ( ) ( )
i il l T T
i ie i iW p x s dx P a x N u dx P N x a u (6.129)
100
where p(x) is the distributed load applied along the beam, the vector s is the deflection of the beam
defined in (2.2) and 1 if x =
( )0 if x
ax a
a
is the Dirac function.
For equation (6.129), the consistent load vector can be identified as:
0( )i i
P N x a P (6.130)
In this last case where the shape matrix is computed, with a concentrated load 0P applied at l=l/4, the
consistent load vector is:
2 3
2 3
2 3
2
0 0 0
2 3
2 3
2 3
2
0 0
3 2 191
32
2 49
64
0 04
113 2
32
7
64
T
i i
x x
l l
x x lx
l l lP N x P P P
x x
l l
lx x
l l
(6.131)
101
Examination of the scientific literature
The numerical procedure for a time dependent modelling of concrete has been analyzed by many au-
thors in different scientific journals.
The analysis made by Criel [14] take into account the viscoelastic constitutive law expressed by the com-
pliance function as Sassone and Cassalegno [35] did in their document:
1
,0
( ')( ) ( , ') '
'
ti
i c ref
d P tu t k E J t t dt
dt
(6.132)
where ,c refE is an elastic modulus of reference of the concrete used to compose, as it is done before
(6.47), the dimensionless stiffness matrix, as follows:
,
1i i
c ref
k kE
(6.133)
The authors do not justify or prove expression (6.132), that therefore does not spring from the well-
known and well-established equations from (6.19) to (6.51). It seems to be an assumption without any
logical and motivated explanation. In this way the problem of the presence of the unknown relaxation
function is overcome and there is direct dependence on the creep function. The approximation is not
correct on the base of the theoretical bases because the creep coefficients, as it is previously saw(6.19),
connects the strains to the stresses and, once that the stress vector is substituted it in the principle of
virtual displacement, it is not possible to change the dependence of the relaxation function on the
creep function without taking into account their link (6.36).
The article continues by solving the second problem, which is the solution of the Volterra integral in
equation (6.132) by the formulation of the trapezoidal rule. This part follows the method shown in the
omniums chapter. A number of k time intervals is defined, comprehended between 0 and t, these steps
do not require to have the same length. Firstly, the variation of the creep function in the time interval i
can be expressed as the difference of the final and the initial value:
0 1 0 0( , t ) ( , t ) ( , t )i i iJ t J t J t (6.134)
The computation of the integral is possible for the validity of the principle of superposition, by making the sum of each area of the trapezes:
10
1
( )( ')( , ') ' ( , ) ( , )
' 2
kt i jii i k j i k j
j
P tP tJ t t dt J t t J t t
t
(6.135)
where the resultant force at time t is given by the summation of the all differences of stresses previously
computed:
1
( )k
jk
j
P t P
(6.136)
102
The integral in time of the equation (6.132) becomes:
10
1
( ') 1( , ') ' ( , ) ( , )
' 2
T kti
ji i k j i k ji
j
P tJ t t dt J t t J t t P
t
(6.137)
To return to the fundamental relation that define the force vector in function of the displacement it is
possible to divide the summation from the term corresponding to the last step (tk-1, tk). The computation
of the variation of the nodal displacement in this last step is possible as shown below:
1
, 1
11
, 1 1 1 1
2
1( , ) ( , )
2
1( , ) ( , ) ( , ) ( , )
2
ii c ref i k k i k kk k
k
ic ref i k j i k j i k j i k jj
j
u k E J t t J t t P
k E J t t J t t J t t J t t P
(6.138)
The relation between the load increments ik
P and the additional deformation ik
u is given. For
the author this is the viscoelastic extension of a linear elastic relation as equation (3.27). This approxi-
mation is really big because it is not take into account the real constitutive law of the viscoelastic ele-
ment but there is a substitution of the stiffness matrix in the governing equation for a linear elastic ma-
terial.
Note that the trapezoidal rule is not the best way to perform this integral [5]. As already stated, the
Gauss quadrature formula (5.37) gives a much better approximation with the same time discretization.
When adopting this last method, equation (6.138) becomes:
1
1
0 0
0 11
( ) ( )( , ) ( ) ( , )
T Tk n
T Ti ij j
iii h i k ih ij hj j
u t u tR t t u t A R t t k P
t t
(6.139)
where the components are defined in the chapter of the Gauss method for the same formula (5.37).
Another author that takes into account the problem of the time dependent behavior of a continuous
composite beam is Virtuoso [11]. He firstly describes all the equations that governs the elastic problem
as it is done in the first chapter, then he considers the dependence of the stress on the time by the fol-
lowing constitutive law:
0
1 1
0 0
( ')( ) ( , ) ( ) ( , ') '
'
t
t
d tt J t t t J t t dt
dt
(6.140)
To solve the Volterra integral equation he approximates the creep function, which is multiple times
called by him creep coefficient, as a Dirichlet series terms as follows:
103
0
0 0
1
( , ) ( ) 1 j
t tn
j
j
J t t a t e
(6.141)
This system of differential equation can be solved numerically and, defining the shrinkage as an imposed
strain in each time interval, the incremental constitutive law can be expressed as followed:
* * * *
c cc shk i k k k kE
(6.142)
where *
i kE represent the equivalent elastic modulus to be adopted in the k interval which depends on
the creep function approximation, *
cck
is the creep strain in the k interval due to the stress history
of the previous intervals and *
shk
represent the shrinkage strain. All these terms are not explained
and therefore their use is mysterious.
In the next step, Virtuoso, as Criel, repeats equation (6.142) replacing the strain and the stress field with
the displacement and the force fields, without developing any substitution that could justified the new
formula. In short, the constitutive relation that connects the internal forces and the deformation field of
the element section for the time interval tk is written on his article [11] as follows:
* * *
, ,i i i shk i i kk k ku k p u u (6.143)
where:
* * * ** * 2 *
01, 1 1 1 11,., ,.
Tk
ik kA Ak k k k
u E dA E z dA
* * * *
01, 1 ,.,.,.T
shi shi kAk k ku E dA
where *
i kk is the stiffness matrix computed with the *
i kE for the concrete; the creep and shrinkage
are referred to the concrete reference point: * * *
1 01, 1k k k
z and
* *
01,sh shk k
.
The composition of all these terms is not explained and their real meaning is vague.
One more author that wrote an article on the incremental analysis of time-dependent effects in compo-
site structures is Jurkiewiez [30]. He considers the right constitutive law in both direct and inverse form
as in equations (6.65) and the relation (6.31) between the creep function and the relaxation function as
belows:
104
0
00
( , ) ( , ')1 ( ', ) '
( ) '
t
tc
R t t dJ t tR t t dt
E t dt (6.144)
where the elastic modulus ( )cE t should obviously be replaced by 0( )cE t (see eq. (6.32)), probably just
a printing error. He then solves this last equation with the trapezoidal rule.
He proposes also to solve the problem of the Relaxation function considering it as a Dirichlet serie as fol-
lows:
0
0 0 0 0
1
( , ) ( ) ( ) j
t tn
j
j
R t t E t E t e
(6.145)
considering the effect of time on the 0( )jE t parameters with another Dirichlet serie:
0
3
0 ,
1
( ) st
j j s
s
E t e
(6.146)
Unfortunately, the author does not explain how the building process needed to determine the stiffness
matrix and the vector of externally applied loads follow from these assumptions.
Another approach to investigate time hardening material such as concrete proposed by Bazant and Wu [32], by analogy with a generalized Maxwell’s model, is to represent the constitutive law of a viscoelastic material by linear differential equations [11]:
( ) ( ) ( )( ( ) *( )) j=0, ,rj j j jt t E t t t (6.147)
where
0 0 (6.148)
and
0
( ) ( ) tr
j
j
t t
(6.149)
( )t represent a set of r+1 interval variables, each of them being attached to the corresponding -
branch of the Maxwell’s model; *( )t represent any given stress-independent strain such as shrinkage
or temperature induced strain. the solution of the system of differential equations (6.147) can be ex-
pressed as an incremental expression for any finite time interval k of time(t,t+Δt):
( ) ( ) *his
E t t (6.150)
where * ( ) *E t being the increase in *( )t during the time interval k; ( )E t is a fictitious modu-
lus whose value depend on t and Δt; ( )his t accounts for the whole stress history since the beginning of
105
the loading period and it depends on the cumulative variables of the Dirichlet series ( )t at the be-
ginning of the time step k:
1
( ) 1 j
nthis
j
j
t e
(6.151)
The definition of the fictitious elastic modulus matrix ( )E t is not defined in the document. It is an im-
portant missing part because it defined the viscous model of the material considered.
The incremental formulation (6.150) can be generalized for three-dimensional isotropic material with a
Poisson’s coefficient constant in time, so taking into account the constitutive law previously defined
(6.26) and the definition of the increase of stresses * during the time interval k, he writes the follow-
ing formula:
0( ( , ) ( ) *)his
R t t t (6.152)
This equation can be considered correct after the definition of the stress-independent strain
* ( ) *E t and the division of the elastic modulus * ( ) *E t . This approximation is not
correct since the shrinkage and thermal strain being independent on the stresses are also independent
on the elastic modulus.
The time analysis must be performed as a step-by-step procedure, in order to scan the life-span of the
structure. The set of cumulative variables attached to every finite element, that is shown in the equation
(6.151), represents the integral in the constitutive law expressed in the starting equation (6.26) and they
must be update at the end of every computation step. Such approach is really expensive in matter of
both time and memory consuming, especially for big structure. In order to decrease the number of de-
grees of freedom and make the computation faster, he expresses the time-dependent behavior of the
structural elements in terms of generalized strains iD for each time step i of the total number k versus
the generalized stresses iS for each time step i. This approach yields an incremental relationship for
the behavior of the cross-section of the specific structural element during the time interval (t,t+Δt)[30]:
* i=1, ,k
his
ii i iiS H D S S (6.153)
where the his
iS depends on the cumulative variables (6.151) and it takes into account the whole stress
history; * iS takes into account all the stresses due to shrinkage or to variation of temperature. By this
approach it is possible to include the behavior of the constituent materials as expressed in the (6.150)
and (6.152) in the classic theories of beams, plates or shells. The value of ( )his
t the creep stress de-
pends in a linear way on the previous values of the normal stress at any given location within the cross-
section of the element. The distribution of the *( )t and the *( )t , might be non-linear but it is as-
sumed linearly distributed in practice to make the calculation easier.
106
In the Table below, it is shown in the case of a composite beam and in the case of a thin layered iso-
tropic shell:
Figure 46. definition of the variables of the classic theories of beams and shells
The stiffness matrix i
H depends on the composition of the element, on the actual time t and on the
time interval i.
If the external load is kept constant over the time interval i (t,t+Δt), the increase internal virtual work Δw
is given by the work Δwint k produced by the internal stresses his and * . When expressed in terms
of generalized variables, it yield for any element (i)[30]:
int T
i ii ii
w w D S dV (6.154)
At this point he substitutes the expression (6.153) in the last one and he complete the principle of virtual
work directly showing the following expression:
* his
K u P P (6.155)
107
This expression is referred on the whole structure and the finite elements as been already assembled as
it can be understood by the definition of the singular vectors:
( )T
i i i iK t B H B (6.156)
* *T
ii iVF B S dV (6.157)
( ) ( )his T his
ii iVF t B S t dV (6.158)
In this formulation the elastic and viscoelastic dependence is divided. The first matrix ( )K t is similar to
the elastic one (3.28) but the elastic modulus is dependent on the actual time t. In all these global matri-
ces he did not consider the connectivity matrix which are important for the assembling phase. The creep
part, main topic of this analysis, is simplified with a force matrix dependent on the time which is derived
by the Dirichlet series of the stresses computed before(6.151). This analysis is clearer than the others
but it contains a lot of missing detail which complete the precision of the method.
Note that none of these authors make a comparison with an exact solution take from the scientific liter-
ature and therefore none of them is able to show the extent of the error gathered by their approaches.
108
MIDAS/Gen solution
The software MIDAS/Gen permits to take into account in the program a structure constructed floor by
floor or with distinct parts constructed at various stages. This result has significant difference as the
scale of the building increase to the re typical structural analysis entails applying the loads to the com-
pleted structure at once. The structure constantly changes and evolves at the construction progress with
varying material properties such as modulus of elasticity and compressive strength due to different ma-
turities among continuous member. As previously saw in the chapters before all these properties are
dependent by the time and they characterize a heterogenous structure. Since the structural configura-
tion continuously chances with different loadings and support conditions, and each construction stage
affects the subsequent stages, the design of certain structural components may be even governed dur-
ing the construction. The time dependent material properties that are going to be considered are:
- The creep in concrete members with different maturities
- The shrinkage in concrete members with different maturities
- The compressive strength expressed as a function of time in concrete members
The program solves these problems in two diverse ways. It is possible to directly enter the creep coeffi-
cient for each element at each stage and it can be applied to the accumulated element stress to the pre-
sent time. Another method is to numerically express specific function for creep by integrating the stress
time history using the creep coefficients specified in the built-in standards within the program.
If the creep coefficients are entered by the user, the calculation method to compute the creep loading is
by these formulas [34]:
0 0 0( , ) ( , ) ( )cc ct t t t t (6.159)
0( ) ( , )cA
P E t t t dA (6.160)
In the respect of the first principle of the linear viscoelasticity that is previously saw.
The method for the numerical computation of total creep when specific functions of creep are numeri-
cally expressed, and stresses are integrated over time uses the superposition of each previous stage. In
the Analysis Reference manual of MIDAS, the total creep from a specific time of loading t0 to an actual
time t can be expressed by the following integration:
0 0
00
0 0
( , ) ( )( )
( )
t
cc
t t tt dt
E t t
(6.161)
This last formula has evidently a misprint error, because the right formulation of the expression is:
0
( , ') ( ')( ) '
( ') '
t
cct
t t tt dt
E t t
(6.162)
109
Assuming a constant stress at each stage, the stress-dependent strain can be simplified as a sum of the
strain at each stage:
1
1
( , )
( )
nj n j
cn j
j j
t t
E t
(6.163)
By this definition, the incremental creep strain cn between the stages n-1 and n can be expressed as
follow:
1 2
1
1 1
( , ) ( , )
( ) ( )
n nj n j j n j
cn cn cn j j
j jj j
t t t t
E t E t
(6.164)
At this point the program approximate the creep function as a Dirichlet functional summation to calcu-
late the incremental creep strain without having to save the entire stress time history, as follow:
0( )
0 00
10
( , )( ) 1
( )i
t tm
i
i
t t ta t e
E t
(6.165)
where the coefficients in front of the equal are computable by the CEB formula (5.12) and it is possible
to compute the term 0( )ia t .
By the definition (6.165) the incremental strain of the equation (6.164) becomes:
0 0( ) ( )
2
1 1
1 1
( ) ( ) 1i i
t t t tm n
cn j j n i n
i j
a t e a t e
(6.166)
0( )
1
1 i
t tm
cn in
i
A e
(6.167)
where
0( )
2
1 1
1
( ) ( )i
t tn
in j j n i n
j
A a t e a t
(6.168)
In MIDAS the term m is taken equal to 5, so in the computation of the equation (6.166), the coefficient
i is used 0.1, 1, 10, 100, 1000 sequentially.
In this way, the increment strain for each element at each stage can be obtained from the resulting
stress accumulated at the previous stage. This method gives a good analysis reflecting the change in
stresses. Time intervals for construction stages in general cases are relatively short and if the time inter-
val specified for a stage is too long, it needs to be internally divided into sub-time intervals to closely re-
flect the creep effects. Computed the characteristic of the creep, the time intervals should be preferably
divided into a log scale.
110
Numerical comparison
The effectiveness of the MIDAS program can be compared with an exact solution. These solutions are
compared with those obtained with the general solution. This solution is the one obtained by the Gauss
quadrature formula [5]. The solution of equation (6.139) can be achieved by computing first the solution
at time t=t0, then progressively the one time t=t1, then the one at time t=t2, and so on. To achieve this
goal the long-term constitutive law suggested by the European code [1,2] will be adopted for the con-
crete.
First example
The example consists of a simply supported deck 20 m long with a composite cross section shown in the
following figure:
Figure 47. Composite section with the relative nodes reference
The section is composed by a first casting of the inferior girder with the following material properties:
- Cylindrical characteristic compressive strength of the concrete at the age of 28 days fck = 40 MPa
- Relative humidity of ambient environment RH = 50 %
- Notional size of the member h = 200 mm
- Age of concrete at the beginning of the shrinkage 3 days
111
- Mean compressive strength of the concrete at age of 28 days fcm = 48 MPa
- Elastic modulus of the concrete Ec = 34644,62 MPa
- Elastic modulus of the steel (tendon) Es = 195000 MPa
- Poisson’s ratio ν = 0.2
The geometrical properties of the first casting are:
X(node) Y(node)
[mm] [mm]
- node 1 -585,00000 0,00000
- node 2 -595,00000 10,00000
- node 3 -595,00000 80,00000
- node 4 -150,00000 140,00000
- node 5 -70,00000 190,00000
- node 6 -70,00000 630,00000
- node 7 -150,00000 680,00000
- node 8 -150,00000 1000,00000
- node 9 150,00000 1000,00000
- node 10 150,00000 680,00000
- node 11 70,00000 630,00000
- node 12 70,00000 190,00000
- node 13 150,00000 140,00000
- node 14 595,00000 80,00000
- node 15 595,00000 10,00000
- node 16 585,00000 0,00000
The geometrical properties of the tendons are:
X(cavo) Y(cavo) A(cavo)
[mm] [mm] [mm²]
- tendon 1 -540.00000 50.00000 93.00000
- tendon 2 540.00000 50.00000 93.00000
112
- tendon 3 -460.00000 50.00000 93.00000
- tendon 4 460.00000 50.00000 93.00000
- tendon 5 -380.00000 50.00000 93.00000
- tendon 6 380.00000 50.00000 93.00000
- tendon 7 -260.00000 50.00000 186.00000
- tendon 8 260.00000 50.00000 186.00000
- tendon 9 -160.00000 50.00000 186.00000
- tendon 10 160.00000 50.00000 186.00000
- tendon 11 -80.00000 50.00000 186.00000
- tendon 12 80.00000 50.00000 186.00000
- tendon 13 0.00000 50.00000 186.00000
- tendon 14 -160.00000 90.00000 186.00000
- tendon 15 160.00000 90.00000 186.00000
- tendon 16 -80.00000 90.00000 186.00000
- tendon 17 80.00000 90.00000 186.00000
- tendon 18 0.00000 90.00000 186.00000
- tendon 19 -110.00000 960.00000 93.00000
- tendon 20 110.00000 960.00000 93.00000
Material property of the second casting of the upper slab:
- Cylindrical characteristic compressive strength of the concrete at the age of 28 days fck = 25 MPa
- Relative humidity of ambient environment RH = 75 %
- Notional size of the member h = 200 mm
- Age of concrete at the beginning of the shrinkage 3 days
- Mean compressive strength of the concrete at age of 28 days fcm = 33 MPa
- Elastic modulus of the concrete Ec = 30576, 89 MPa
- Poisson’s ratio ν = 0.2
The geometrical properties of the slab are:
113
X(node) Y(node)
[mm] [mm]
- node 1 -600,00000 1000,00000
- node 2 -600,00000 1220,00000
- node 3 600,00000 1220,00000
- node 4 600,00000 1000,00000
The phases of the constructional stage are defined by the following times:
- Time of casting t0 = 0 days
- End of curing of the girder t1 = 3 days
- Prestressing of the girder t2 = 7 days
- Casting of the slab t3 = 27 days
- End of curing of the slab and activation of the connection t4 = 30 days
- Application of the permanent loads t5 = 55 days
The permanent loads applied after 55 days are equal to 300 kg/m2 on all the top surface of the slab.
They comprehend the asphalt weight and all the additional weight of finishing.
The prestressing force is applied as an initial deformation imposed to the prestressing strands. Assuming
that the stress in the prestressing strands is equal to 1375 MPa, and allowing for a 3% reduction of the
stress because of relaxation in the prestressing bed, the initial deformation is equal to p1 = 1375·(1 –
0.03)/195000 6.84‰.
In the software two different beams are modelled, as shown in Figure 48.
Figure 48. MIDAS model
114
It was possible to define the particular geometry of the bottom beam by the extension software ‘Sectional
Properties calculator’ present in the tools of the program. Firstly, the section of the beam is draw in Au-
toCAD software and then it is imported in the software where the properties are defined as it is shown in
the Figure 49.
Figure 49. MIDAS/Sectional Properties Calculator for the definition of the bottom section point out in yellow
Displacements in y and z directions are blocked on both edges. By modelling the bottom beam as a linear
element, it was possible to add the 20 tendons with the properties previously descripted. The self-weight
is introduced as a linear distributed load for both beam. In the previous step of the formation of the top
plate, a temporary load is added to represent its weight that start acting on the beam before the end of
its curing.
Figure 50. Material properties of the model
115
Figure 51. Section geometrical properties
The structure, the boundary and the loads groups are created and put in the 4 constructional stages that
define the time period as it is shown in the figure.
Figure 52. Constructional stage of the model
To take into account the creep parameters, the CEB 2010 formulas are chosen, already descripted in the
homonymous chapter. For each material. the characteristic compressive strength of the concrete at the
age of 28 days, the relative humidity and the notional size of the member, that for simplicity it is taken
as 200mm, even if in the MIDAS program there is the possibility to auto calculate it for the 2D elements,
are defined.
116
Figure 53. Time dependent Material Properties for the C40/50
Figure 54.Time dependent Material Properties for the C40/50
The mean compressive strength and a normal speed of hardening for both materials for both materials
are also defined and the relative properties are linked as it is shown in the picture.
117
Figure 55. summary of the time dependent material link for both the concretes
In each constructional stage are set the times previously defined. In the constructional stage analysis
control data, it is taken into account both creep and shrinkage, as shown in the Figure 55.
118
Figure 56. Constructional stage analysis control data of the model
The modelling phase is ended and it is possible, now, to run the model to compute, by the Finite Ele-
ment Method descripted in the previous chapter, all the output data (stress, displacement, forces, … )
The goal is to compare the the MIDAS software results and the ones obtain with the exact solution by
the system of Volterra integral equations, which represents the mathematical synthesis of the physics of
this problem. The problem is solved by a refined step-by-step time integration method. This last proce-
dure gives rise to an error that can be minimized through a suitable choice of the time discretization
procedure [37].
The MIDAS results are reported in addendum. In the table of Figure 57 are compared the results of
MIDAS with the exact solution computed by the Gauss method.
119
Figure 57. comparison between the MIDAS and the General method results
In this table, the first response of the two approaches, that corresponds to the elastic response, is the
same, with an error lower than 10%. Even if both method use the hypothesis of Bernoulli-Navier for the
planar transverse section, the computation is different since the division in finite element of the homon-
ymous method. The more is refined the mesh of the model, the more accurate is the results. The mesh,
as it is shown in the Figure 48, is made of finite elements with a length of 1 meter. Taking a smaller
length of the finite element, it is possible to have a better result.
Second example
For what concern the validity of the creep coefficient given by the program, it is possible to develop a
model of the singular column as a beam constrained at the base. This example is easily computable by
the General model of the concrete. The different parameters are computed by the CEB 1990 Model as it
is descripted in the homonymous chapter. In the following table, there are reported all the data of the
column.
120
It is loaded on the top by a concentrated load P1 equal to 100 kN. All the coefficients are defined in the
chapter of the CEB 1990, in the equations from (5.1) to (5.11). A time line of 6000 days is considered.
The results obtained by the MIDAS software are compared with the results of the CEB 1990 formulas
that is reported in the Figure 59.
R_ck 27 N/mm^2
f_ck 22.41 N/mm^2
f_cm 30.41 N/mm^2
E 31213.23677 N/mm^2
t_i 7 gg
s 0.25 β_cc 0.778800783 β_e 0.882496903 E_c 27545.58477 N/mm^2
RH 0.7 RH_0 1
L 500 mm
h 500 mm
A 250000 mm^2
P 2000 mm
h 250 mm
h_0 100 mm
f_cm0 10 N/mm^2
β(f_cm) 3.039258637 β(t_i) 0.634609108 β_h 641.2576742
β_c(t-t_i) 0.257042604
φ_RH 1.480525848 φ_0 2.855551217
φ(t,t_i) 0.73399832 J(t,t_i) 5.98191E-05 mm^2/N
P1 10000 N
σ/Δσ 0.04 N/mm^2
Figure 58. MIDAS model of the
column
121
Figure 59. CEB 1990 formulas results of the column’s displacement due to P1
The elastic answer, that in the graph is represented by the beam displacement in the first instant, in-
stantaneous response, can be compared to the elastic result of MIDAS. The beam displacement is of
0.66 mm by using the CEB 1990 and of 0.63 mm with MIDAS. The instantaneous answer of the beam is
the same in both analysis and it can be consider exact since it presents an error lower than 5%. The
curve in excel continue to grow until it stabilizes, in this part the creep displacements is considered and
not the shrinkage, that can be taken into account by others formula of the CEB in the relative section.
The total creep displacement in excel is computable by the difference of the last displacement, that can
be considered static at 6000 days, and the first one, that is the elastic one. The total value of the dis-
placement due to the elastic effect and the creep one in the graph is at time equal to infinite and it is
1.1. In MIDAS it is equal to 1.21. The error is lower than 10% and it is acceptable as a good result.
It was possible to see the validity of the superposition principle by adding another load P2 = P1 that rep-
resent a second floor. This time, in the first story, it is take into account the self-weight too. A load of
2400 kg is added to represent the self-weight of each piece of the column and the graph of Figure 60 is
obtained.
0 1000 2000 3000 4000 5000 6000 7000
0
0.2
0.4
0.6
0.8
1
1.2
Time (days)
Dis
pla
cem
ent
(mm
)
Displacement δ
122
Figure 60. CEB 1990 formulas results of the column’s displacement due to self-weight
The graph of the displacement obtained by the CEB 1990 due to the concentrated loads P1 and P2 is re-
ported bellow.
Figure 61. CEB 1990 formulas results of the column’s displacement due to P1 and P2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
0.5
1
1.5
2
2.5
Time (days)
Dis
pla
cem
ent
(mm
)
Displacement δ
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (days)
Dis
pla
cem
ent
(mm
)
Displacement δ
123
The sum of the creep and the elastic displacement due to the self-weight and
the concentrated loads P1 and P2 is obtained by the sum of the two curves
at time as infinite. The value obtained by the CEB 1990 formulas is equal to
2.1 + 1.7 = 3.8 mm.
In the model of the program, a second floor is added as it is shown in the fig-
ure on the side. The result of the program of the creep and the elastic dis-
placement is equal to 3.5 mm.
The comparison of the results is less than 10 %, indeed, it is a good approxi-
mation.
Figure 62. MIDAS model of
the column with 2 floors
124
Building in construction
In tall concrete buildings, the problem of the creep and the shrinkage affects the service life behavior of
the structural elements. For this reason, the construction method and phases must be taken into ac-
count as a key factor in defining the evolution of the stresses and strains. In this chapter, it is analyze the
time-dependent behavior of the case of study of a building, that is going to be made the next year.
Figure 63. Architectonical rendering of the tower
The structure is going to be called ‘Gioia 22’, for the reason why it will be in the Gioia neighborhood in
Milan, in substitution of an existent building ‘ex INPS’. It is a structure with a 125 m of maximum height
out of soil, it is composed by 25 floors out of the ground and 4 in the basement.
Starting by an already made model in MIDAS, the Constructional stage of the building is the focus of the
work. Firstly, the probable times of the constructional periods are planned. In the following schedule are
define the time of construction for each component:
- 75 days for the mat foundation;
- for the basement floors a total of:
o 30 days for the vertical elements;
o 30 days for the horizontal elements;
125
- for the first and second floor of the tower a total of:
o 16 days for the vertical elements;
o 16 days for the horizontal elements;
- for the floors from the 3rd to the 23th a total of:
o 42 days for the core and law rise (present in the first 13 floors);
o 120 days for the vertical elements;
o 120 days for the horizontal elements;
- for the 24th and 25th a total of:
o 28 days for the vertical elements;
o 33 days for the horizontal elements;
- 20 days for the finishing.
By this schedule, it was possible to define the various stages in the program, for a total of 136. In the fol-
lowing picture,there are reported the main phases of the construction.
Figure 64. Constructional stage of the floor -3rd of the basement
In Figure 64, the construction of the first floor of the basement is shown, that corresponds to the -3rd
floor. In the Figure 64, the slab of the -3rd is not completed, because the fulfillment of the horizontal and
the vertical elements is divided in different zones, for the big extension of the plant plan, about 6500
mq.
126
Figure 65. Constructional stage of the conclusion of the basement
In Figure 65, the basement is completed. It is composed by 4 floors, including the 0, which are planned
to be made in less than 6 months, after the starting of the construction. The upper part has a smaller
plan plant, as it is shown in the Figure 66.
Figure 66. Constructional stage of the completion of the 15th floor
127
The Figure 66 corresponds to the completion of the law rise, that is made until the 13th floor. The built of
the core and the law rise is made three by three floors, by climbing formworks, as shown in the Figure
66, and on each floor, the horizontal and vertical elements are constructed dividing the North from the
South zone, for the big extension of the plan plant. The Figure 66 corresponds to a construction phase in
which, in the North zone, the horizontal and vertical elements are completed, but in the South one,
there are the vertical one. The podium is composed by the 1st and the 2nd floors.
Figure 67. Constructional stage of the completion of the concrete part of the structure
In this stage of construction, corresponding to Figure 67, the concrete part of the building is completed,
and to do it, it is estimated a time of 1 year and 3 months and half. In the Figure 67, it is evident how the
building twists with the growing of the height. In the South part, the columns are inclined, and they have
a projection of the last floor that doesn’t fall in the base plan plant of the 3rd floor, one column has a
projection that is also outside of the podium plan plant.
128
Figure 68. MIDAS model complete
The completion of the building is made by the addition of the steel elements, in the upper part of the
structure, and of all the finishes, considered as an additional weight. The permanent loads are applied
after the curing of each part, with different periods differed in extension. The variable loads compre-
hend all the non-structural forces, such as: offices, bathrooms, aisles, packings, square and fire fighter
loads. The variable loads, the seismic loads and the wind ones, are added in the end of the construc-
tional analysis. The non-structural permanent loads are added floor by floor, with an interval of 30 days
for each floor, starting from the casting of the last piece of low rise.
Both creep and shrinkage of all concretes are considered and, for each of material, the relative proper-
ties are defined, such as the characteristic compressive strength of the concretes at age of 28 days, the
notional size of each the member and the mean compressive strength of the concretes at the age of 28
days. It is considered a Relative Humidity of 75% and an age of 3 days of all the concrete at the begin-
ning of the shrinkage. Interestingly enough, the Figure 69 reports the variation of the creep function, for
a random concrete used in the model, C50/60. The formulas of the software are referred to the CEB
2010, as it is shown in the Figure 69, and they are explained in the homonymous chapter.
129
Figure 69. Creep coefficient developing in time for a concrete C50/60, with a time of loading at 28 days by the formulas of CEB
2010
By graph represented in Figure 69, it is possible to understand at which time the deformation, due to
the loads, can be considered constant. The facades are a sensitive element of the structure. These can-
not be affected by big deformations, indeed, they need to be applied in the exact moment when the
long term vertical deformation is considerable constant to the one at time infinite. By the figure 69’s
graph, it is possible to compute these differential deformations, that are going to affect the facades. The
difference of the value of the creep coefficient φ at time infinite, that is the coefficient studied in the
CEB ’90 chapter, to the creep coefficient at the time of application of the facades, permits the computa-
tion of the displacement that will act on the facades. It is made a computation of the creep coefficient
following the CEB Model Code ’90 formulas, for the same type of concrete C56/60 and with a loading
time of 28 days, the graph obtained is reported in Figure 70.
130
Figure 70. Creep coefficient developing in time for a concrete C50/60 by the CEB ’90 formulas
The curve follows the same path and it reach the same level. The creep coefficient computed in the pro-
gram is obtained by the formulas of the CEB Model Code 2010, the graph of Figure 70, instead, are com-
puted by the formula of the CEB ’90. The values, that are computed by the formulas of the older model,
are more in favor of security. To confirm the reliability of the computation handmade, it is also possible
to show the creep coefficient computed by the program by the formula of the CEB ’90 for the same ma-
terial with the same loading time, shown in Figure 71.
Figure 71. Creep coefficient developing in time for a concrete C50/60, with a time of loading at 28 days by the formulas of CEB
2090
0 1000 2000 3000 4000 5000 6000 7000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time [days]
Cre
ep c
oef
fici
ent
φ
Creep coefficient
131
The graphs of Figure 69 and 71 are the same.
Another graph, that is representative of the creep effects, is reported in the Figure 72, for the same con-
crete C45/50. By the graph of Figure 72, the behavior of the concrete in the first age is shown. In the
first period, it behaves as a fragile material, but at its age of 28 days it can be considered resistant for
the standard.
Figure 72. compressive strength growing in time of a concrete C45/50
After the modelling phase, the program is run and the results of the erection sequence are compared
with the one obtained by the conventional analysis. In the comparison, the structural permanent loads,
the non-structural permanent loads and the facades loads are considered. The conventional analysis re-
sults are taken as elastic, indeed they are not multiplied for any coefficient. The erection sequence anal-
ysis, instead, takes automatically into account a first effect of creep and shrinkage. To consider these ef-
fects the less as possible, in the constructional stage analysis it is chosen the first instant after the appli-
cation of the loads. Exactly, this moment corresponds to the stage 136, at 1170 days (3 years and 2
months and half) since the casting of the first piece of the mat foundation.
132
Figure 73. Plan plant of the basement floor with a key pad of the considered column
A group of five column, that are indicate in the Figure 73 as C1, C2, C3, C4 and C5, are representative to
compare the axial forces that act on each one. These columns correspond to the most loaded of all.
Figure 74. Axial forces acting on the chosen column at the -3rd floor in the erection sequence
133
Figure 75. Axial forces acting on the chosen column at the -3rd floor in the conventional analysis
By the comparison of the two analysis, the axial loads can be consider the same. In the most loaded col-
umn, correspondent to the C2 the difference is of the 2,14%, all the other columns present the same or
lower percental differences. In the following figure it is possible to check the axial forces in the same col-
umns in in 0 floor.
134
Figure 76. Axial forces acting on the chosen column at the 0 floor in the erection sequence
Figure 77. Axial forces acting on the chosen column at the 0 floor in the conventional analysis
In the comparison of the axial forces of the Figure 76 and 77 there are more discrepancies. In the col-
umn C2, the difference of the forces between the two model is the 2.6%. The axial force of the column
C4 on the two model differs of the 2.3%. The values are similar, and they can be consider acceptable to
confirm the models.
Figure 78. Axial forces acting on the chosen column at the 8th floor in the erection sequence
135
Figure 79. Axial forces acting on the chosen column at the 8th floor in the conventional analysis
In the Figure 78 and 79, the axial forces of the same column at the 8th floor are shown. The difference on
the most loaded column C2 is of 2.3%, but on the column C1 it differs of 11.7%. This column is the one
where the force varies the most. To have a better comparison, it is possible to do an average of the dif-
ference between the two models, by considering the area of influence of the 5 columns. The difference
of the summation of the axial forces acting in the 5 columns is of 2.69%, always a low percental.
Figure 80. Axial forces acting on the chosen column at the 16th floor in the erection analysis
136
Figure 81. Axial forces acting on the chosen column at the 16th floor in the conventional analysis
On the 16th floor, the axial forces, that are reported in the Figure 80 and 81, are acting. The highest value
of difference is on the column C2 and it is of 13%. Taking into account the area of influence the differ-
ence is reduced to 5.3%.
Figure 82. Axial forces acting on the chosen column at the 20th floor in the erection analysis
Figure 83. Axial forces acting on the chosen column at the 20th floor in the conventional analysis
137
In the comparison on the 20th floor, the total difference between the two models becomes of 8.9%, for
the 5 columns.
Figure 84. Axial forces acting on the chosen column at the 24th floor in the erection analysis
Figure 85. Axial forces acting on the chosen column at the 24th floor in the conventional analysis
In the last floor, the 24th, the highest difference between the two models are found. The discrepancies
have a value of 17%, that being lower than the 20% is still acceptable. In this last comparison, the punc-
tual forces have also higher differences, in the columns C2 and C4 the force change sign, but they are
equilibrated by the higher value of the axial forces in the lateral columns C1, C3 and C5.
The summation of the axial forces of the 5 columns, that is referred to the erection sequence, is always
resulted bigger than the one computed by conventional analysis, in all the floors. This is due to the fact
that in the constructional stage the loads start to accumulate also the effect of the creep and the shrink-
age.
138
The discrepancies of the results are due to the different application of the loads. In the real construction
process, when the construction dead loads are applied to the structure in a preceding stage, the mem-
ber forces of the same stage are not affected by the loads on the next one, which is yet to be con-
structed. In the case of conventional analysis, however, the non-existing loads of the following stage re-
strain the rotational deformations at the previous stage. As a result, the completed structure of all the
stages together sustains the construction loads. This causes the discrepancies between the analysis, es-
pecially in the higher floors, that are constructed later. Therefore, the conventional analysis method
may produce considerable discrepancies in design results.
The values, however, presents always a total difference lower than 20% and this confirms the validity of
the model. In a comparison of the moment, resulting on the structure, it would be found the same ap-
proximation of the results.
Figure 86. Plan plant of the displacement contour of the mat foundation in vertical direction with the conventional analysis
139
Figure 87. Plan plant of the displacement contour of the mat foundation in vertical direction with the erection sequence
In the previous two pictures, there are shown the vertical displacements of the mat foundation in a
plant plan. Looking at the legend of Figure 87, the displacements are higher of 0.5 mm in the construc-
tional stage analysis, than in the conventional one. As expected, the higher deformations are due to the
time effects, but their results are almost the same, since not all creep and shrinkage effects show until
that age. Referring to the constructional stage analysis, the point, where there is the biggest vertical de-
formation, is in correspondence of the core, 0.027 m. In correspondence of the North-East column there
is a displacement of 0.019 m. This difference creates, on the mat foundation, an angle of deformation
of:
0.008 180 150.04
10.8 360h
It is possible to compute the same angle of rotation of the mat foundation also for the conventional
analysis:
0.008 180 150.04
10.8 360h
This last value is results the same. Even if the displacement are bigger, the deformation conserves al-
most the same shape.
Another cause for discrepancies results from the differential shortening columns. This effect is based on
different hypothesis:
- The axial stress in the column is not affected by the variation of the stress distribution
140
- There is no interaction between the variations in the stress distribution in the various slabs
- The creep behavior is described by the aging, rate of creep model already analyzed in the
homonymous chapter
- The analysis is carried out taking into account the construction phases along the height of the
building
Now, it is compared the displacement of the North-East column to the displacement of core, in the point
that is closer to the chosen column. In this way, it is possible to see the effect of the column shortening.
The results obtain by the erection sequence analysis are now compared with the ones obtain by a con-
ventional analysis that doesn’t consider the time of construction, but it assumes that everything was
made in the same time. In this last analysis the computation of the axial deformation of the column ar-
rives to a result that grows inversely proportional to the story. Considering the constructional phases,
the displacements of a column have the following diagram, in relation to the story.
Figure 88. MIDAS results of the displacement of a column in function of the stories due after the construction until a time in
which they can be consider stabile
141
Figure 89. MIDAS results of the displacement of a column in function of the stories due just after the application of the last load
In these graphs, it is shown the vertical displacement function of the story of the building, for the corner
column in the North-East corner. The different lines are referred to the displacement due to the load, to
the creep, and the total displacement that is a summation of the previous and the one due to the shrink-
age. It considers the erection sequence analysis where the displacement, due to the loads applied to the
lower floors and to the same floor of the element just casted, are compensated on each floor. Indeed,
all the vertical displacements are due to the contribute subsequent to casting. This is understandable
from both graphs because the curves decrease in the top part. At time of casting the displacement is
equal to 0, so part of the top floors’ displacements is not affected by the lower floors displacements, for
this reason they result lower. The graph in Figure 88 represents the displacement at time equal to infi-
nite and the graph in Figure 89, is taken just after the application of the last loads. The results are
slightly different, in the second images the curves present a peak at almost 2/3 of the height. In the first
graph all the total displacements of the structure are reached. The decrease of the displacements in the
top part gets smaller at time equal to infinite, because more displacements are accumulated in the
newer elements than in the ones already stabilized. The different inclination of the curves, before and
after the 0 floor, is due to different sections and materials in the columns of the basement floors from
those of the tower.
The software program permits to see the simulation of the total vertical displacement growing in time.
The points located in the same North-East column are taken at the -3rd, 0 and 1st floors.
142
Figure 90. Vertical displacement in function of the time of points of the North-East column at the -3rd, 0 and 1st floor
The points at the first and 0 floor, start deforming after their casting, for this reason in the first period
the displacement is equal to 0 in the graph of Figure 90. For differences of material and section, the
point at the first floor deforms more than the one at the 0 floor. The point at the 0 and at -3rd floor have
the same inclination, because they are made with the same section and material. There is a higher gap
at the 307th day, because in that moment the finishes’ loads are applied on the structure.
It is possible to see also the displacement due to creep. It is chosen just the point at the first floor on the
same column.
Figure 91. Function due to the creep effects after the casting of a point at the base of the North-East column at the first floor
As it is shown in the graph of Figure 91, in the first life time of the element the creep displacements are
higher due for the aging effects. In the last period the creep goes to stabilize until the 4169th days. In the
higher jump corresponding to the last phase, the curve continues smoothly as a parable, keeping the
concave part facing upwards, but it is not shown for the few number of existent steps. The jump in the
last step can be consider already reached, when the β, which is the CEB ’90 coefficient representative of
143
the creep function and computable by the formula (5.8), becomes a value close to infinite one. It is pos-
sible, for each material, to compute this moment by the creep coefficient diagram previously shown in
the picture.
The column shortening effect determines a different shortening between the core and the columns, a
distortion of the slab between it, a redistribution of the vertical loads, some additional moments in the
slab and effects on non-load bearing element such as piping, finishes and façades. The column shorten-
ing is evident in the Figure 92, where it is reported the displacement of the erection sequence analysis
for the column previously saw and the point of the core closer to the column in analysis.
Figure 92. MIDAS results of the displacement of the North-East column and the closest point of the core at time infinite
In the graph of Figure 92, in the growing of the structure, the column shortening effects shows. The core
has a bigger vertical displacement than the column, because of different materials. The column is made
by a C70/85 concrete, instead the core is made by a C45/50 concrete, surely less resistant than the col-
umn one. The vertical deformation of the mat foundation, in the point where the core acts, 0.027 m, is
bigger than the one in the point of application of the column, 0.187 m. This can be seen at level B4 of
the graph. This affects the transparency of the column shortening effect. If the curves are tranlated to
have the same starting point at the base (B4), the core presents a lower vertical deformation compared
to the column, especially in the top part of the tower. In this case, the column shortening effect is coun-
terbalanced by the varied materials of the elements, indeed, as the graph shows, the difference of the
vertical displacements, of the two points on the top tower, is small enough.
Last but not least, the torsion of the structure is analyzed. The South part of the building has a shape
that turns and gets bigger in the top part. This shape generates forces in the x and y direction that are
the cause for the torsion. The results of the displacement in the x and y direction are reported in Figure
93 and 94.
144
Figure 93. Displacements in x direction of all the beam of the structure
Figure 94. Displacements in y direction of all the beam of the structure
145
Because of the creep and the shrinkage, the displacements grow also after that the structure is finished.
The tower continues to twist in time. The torsion needs to compute the displacement that affect the fa-
çade. The displacements in the x direction are lower than the one in the y axis, as shown in the Figures
93 and 94. The most affected column by the torsion is the column in the South-West corner, at ¾ of the
height that is zoomed in Figure 95.
Figure 95. Displacements in y direction zoomed on the model on the highest values between the 24th and 25th floors
The Figure 95 shows that on the border of the slab, the displacements, in y direction, are bigger than the
one on the column, because there is a rigid displacement of all the floor. In this way, the displacement,
that is going to affect the façade, can be computed, more precisely, the angle of torsion that is going to
affect them. The differential angle of torsion is computed by dividing the differential displacement by
the distance of the correspondent floors. This operation is made on some reference floors:
- Between 6th and 7th floors: 0.005 180 25
0.074.1 360h
- Between 12th and 13th floors: 0.004 180 22
0.064.1 360h
- Between 20th and 21th floors: 0.003 180 15
0.044.1 360h
146
- Between24th and 25th floors: 0.004 180 22
0.064.1 360h
The displacements in the plan of the slab grows in the first part of the building until they reach the maxi-
mum value at ¾ and they reduce in the top part. The maximum differential angle of torsion, indeed, is in
the lower part of the building, as the computations confirm. The angle between the deformed shape
and the no-deformed one is higher in the first part, than it goes to stabilize, until the maximum displace-
ment, and it change sign in the last part, always with a soft inclination.
In conclusion, we understand the importance of the constructional stage analysis for tall building, such
has this one. The differences between this analysis and the conventional one, are various. Assuming a
viscoelastic rate of creep ageing behavior of concrete, the evaluation of the stress distribution in the
slab permits the definition of the upper and lower bounds of their values in time. All the problems that
we analyzed in the building are a main part in the structural project.
147
Conclusion
The finite element analysis for a viscoelastic material is the main topic of the work. Extensive research
has been conducted to investigate this problem. The computer programs try to solve it by different
paths. The time-dependent behavior of the material can create problems on the building envelope,
especially for tall building. Thus, it needs to be considered in the design, especially for medium to high-
rise buildings. The problem is faced in the exercise efficiency and not in the strength capability. As it is
shown in the model of ‘Gioia22’, the consequences induced by time-dependent effects in composite
structures are really effective. Changes in the stress and strain distribution of the structure are the main
difference with a conventional analysis.
The theory of linear viscoelasticity is a present-day problem for the engineer. There are few and poor documents and articles that deal with it. It is hard to find a good work on the topic, with a clean explanation of all terms and a clear solution of the complex phases of the finite element method. There are various methods to solve the Volterra integral which is the biggest problem of the computation. In particular, the MIDAS software solves the problem with a more refined method. The problem is solved by a summation of the differential strain in each time step and the the Volterra integral is simplified as a Dirichlet summation of 5 terms. In this way, the time of computation becomes much longer. The precision is really high, but, especially if the finite elements are really small, the calculation takes a lot of time. The comparison between the MIDAS solution and the exact solution computed by the Gauss method is really good and it has an error largely smaller than 20%, in most of the time period.
After analyzing the scientific literature and the commercial computer program, it is outlined a solution following the algebraic method. The general solution of the finite element method is possible for the approximation made by the algebraic method. Particular attention is given to the solution of the Volterra integral. By this way, it was possible to considerably simplify the problem, bringing it back to a double elastic analysis and, therefore, reducing the time of computation. This way of computation is highly faster that all the others that had been reviewed. The velocity of computation is due to the elimination of the convolution integral. This method is also used by the European code to determine the long-term losses in prestressed concrete beams.
The algebraic method analysis made is a step of the definition of a new approach to solve the problem
in the viscoelastic domain. The approach has to be refined by adding all the specific details that can
characterize a model, such as: concentrated load on the nodes, thermal effect, pre-compression effect,
springs and constructional stages. Moreover, the level of precision of this method needs to be tested by
comparing its outcomes, with exact solutions. Obviously, these solutions are available in the scientific
literature just for particular structural elements, such as the one used to verify the reliability of the
MIDAS program. Referring to these particular solutions, it would be interesting to compare the error
gathered by the algebraic method, with the one obtained by the commercial computer programs, like
MIDAS, especially by relating this error with the CPU time (time of computation).
148
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