University of the German Armed Forces Munich

A method for the development and control of stiffness matrices for the calculation of beam and shell structures using the symbolic programming language MAPLE

N. Gebbeken, E. Pfeiffer, I. Videkhina

University of the German Armed Forces Munich

Faculty of Civil and Environmental Engineering Institute of Engineering Mechanics and Structural Mechanics Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken

Relevance of the topic

In structural engineering the design and calculation of beam and shell structures is a daily practice. Beam and shell elements can also be combined in spatial structures like bridges, multi-story buildings, tunnels, impressive architectural buildings etc.

University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken

Truss structure, Railway bridge Firth of Forth (Scotland)

Folded plate structure, Church in Las Vegas

Calculation methods

In the field of engineering mechanics, structural mechanics and structural informatics the calculation methods are based in many cases on the discretisation of continua, i.e. the reduction of the manifold of state variables to a finite number at discrete points.

Type of discretisation e.g.:

- Finite Difference Method (FDM)


X

Y

x

i,j

i,j+1 i+1,j+1i-1,j+1

i-1,j

i-1,j-1 i,j-1 i+1,j-1

i+1,j

x

y

y

Inside pointsof grid

Outside pointsof grid

Boundary of continuum

Centerpoint

ΔΔ

ΔΔ

i 1 , j i,j

i,j

i , j 1 i,j

i,j

O( x)x x

O( y)y y

f ff

f ff

Differential quotients are substitutedthrough difference quotients

Continuum

u1

v1

u3

v3

u2

v2


Calculation methods - type of discretisation

- Finite Element Method (FEM)

Static calculation of a concrete panel

First calculation step: Degrees of freedom in nodes.Second calculation step: From the primary unknowns the state variables at the edges of the elements and inside are derived.

Calculation methods - type of discretisation

- Meshfree particle solvers (e.g. Smooth Particle Hydrodynamics (SPH)) for high velocity impacts, large deformations and fragmentation

Aluminiumplate Fragment cloud

Experimental und numeric presentation of a high velocity impact:a 5 [mm] bullet with 5.2 [km/s] at a 1.5 [mm] Al-plate.


PD Dr.-Ing. habil. Stefan Hiermaier


FEM-Advantages:

Continua can easily be approximated with different elementgeometries (e.g. triangles, rectangles, tetrahedrons, cuboids)

The strict formalisation of the method enables a simple implementation of new elements in an existing calculus

The convergence of the discretised model to the real systembehaviour can be influenced with well-known strategies,e.g. refinement of the mesh, higher degrees of elementformulations, automated mesh adaptivity depending on stress gradients or local errors


Aspects about FEM

Extensive fundamentals in mathematics (infinitesimal calculus, calculus of variations, numerical integration, error estimation, error propagation etc.) and mechanics (e.g. nonlinearities of material and the geometry) are needed. Unexperienced users tend to use FEM-programmes as a „black box“.

Teaching the FEM-theory is much more time consuming as other

numerical methods, e.g. FDM At this point it is helpful to use the symbolic programming language MAPLE as an eLearning tool: the mathematical background is imparted without undue effort and effects of modified calculation steps or extensions of the FEM-theory can be studied easier!


The Finite Element Method (FEM) is mostly used for the analysis of structures.

Basic concept of FEM is a stiffness matrix R which implicates the vector U of node displacements with vector F of forces.

R U F

Of interest are state variables like moments (M), shear (Q) and normal forces (N), from which stresses (, ) and resistance capacities (R) are derived. It is necessary to assess the strength of structures depending on stresses. [ ] [ ]cal allowable

F

A

R

F

l

l l

l


Static System

Actions

Reaction forces

Deformation of System

M

V

MH

V

H

F

M

V

MH

V

H

1

T

A S F

A U

B S

R U F

Vector S of forces results from the strength of construction.

Vector U of the node displacements depends on the system stiffness.

Structures should not only be resistant to loads, but also limit deformations and be stable against local or global collapse.


In the design process of structures we have to take into account not only static actions, but different types of dynamic influences.

Typical threat potentials for structures:

- The stability against earthquakes

- The aerodynamic stability of filigran structures

- Weak spot analysis, risk minimisation

Collapse of the Tacoma Bridge at a wind velocity of 67 [km/h]

Consequences of an earthquake Consequences of wind-inducedvibrations on a suspension bridge

Citicorp Tower NYC


( )M U C U R U F t

dynamic problem static problem

- mass (M)- damping (C)- stiffness (R)

The most static and dynamic influences are represented in thefollowing equation:

Mercedes-multistoreyin Munich

wind loading

FEM for the solution of structural problems


Research goals:

1. The basic purpose of this work is the creation of an universal method for the development of stiffness matrices which are necessary for the calculation of engineering constructions using the symbolic programming language MAPLE.

2. Assessment of correctness of the obtained stiffness matrices.

Short overview of the fundamental equations for the calculation of beam and shell structures

ui ujj

wj

i

wi

Beam structures Shell structures

Differential equation for a single beam4

4

d w q

dx EJ

with w- deflection, EJ- bending stiffness (E- modul of elasticity, J- moment of inertia), x- longitudinal axis, q- line load

Beams with arbitrary loads and complex boundary conditions

2. Theory of second order3 2

3 2,

d q dw d

dx EJ dx dx

44 4

44 , with

4

d w q kbn w n

dx EJ EJ

1. Beam on elastic foundation

with n- relative stiffness of foundation, k- coefficient of elastic foundation, b- broadness of bearing

with - shearing strain

3. Biaxial bending4 2

4 2

d w N d w q

dx EJ dx EJ

with N- axial force

Differential equations for a disc (expressed in displacements)

2 2 2

2 2

2 2 2

2 2

1 10

2 2

1 10

2 2

u u v

x y x y

v v u

y x x y

Differential equation for a plate

4 4 4

4 2 2 42

w w w p

x x y y D


Calculation of beam structures

For the elaboration of the stiffness matrix for beams the following approach will be suggested:

1. Based on the differential equation for a beam the stiffness matrix is developed in a local coordinate system.

2. Consideration of the stiff or hinge connection in the nodes at the end of the beam.

3. Extension of element matrix formulations for beams with different characteristics, e.g. tension/ compression.

4. Transforming the expressions from the local coordinate system into the global coordinate system.

5. The element matrices are assembled in the global stiffness matrix.

FEM equations Equations from the strength of materials

Type of the development

Beam structure Tension-

compression

Bending without consideration of the

transverse strain

Bending with consideration of the transverse strain

Equation of equlibrium

A S F A

N dA A

M ydA A

M ydA 1

Geometrical relations

TA U du

dx

2

2

du d wy

dx dx mit

du d dwy

dx dx dx

2

Material law 1B S E E E 3

(2) (3) 1 TB A U S du

Edx

2

2

d wE y

dx mit

d dwE y

dx dx

4

1 TA B A U F du

N EAdx

2

2

d wM EJ

dx mit

d dwM EJ

dx dx

5

(4) (1) 1 1( )TA B A F U

du N

dx EA

2

2

d w M

dx EJ mit

M d dw

EJ dx dx

6

(6) (4) 1 1 1( )T TB A A B A F S N

A

My

J

My

J 7


R

Development of differential equations of beams with or without consideration of the transverse strain

4

2 3

31' 2

2''

23'''

4

241

0 1 2 36

0 0 2 6

0 0 0 62

0 0 0 0

1

IV

x

w w x x xC xw x xC qw M xC EJ x

w QC

w qx


Basic equations:4

4

d w q

dx EJ

2

2

d wM EJ

dx

3

3

d wQ EJ

dx

Solution:4

2 31 2 3 4( )

24

qxw x C C C x C x

EJ

Solution and derivatives in matrix form:

Algorithm for the elaboration of a stiffness matrixfor an ordinary beam

D

homogeneous particular

i

j

u qu L C L

u EJ

4

2 3

2 3

1

2

3

4

01 0 0 0 00 1 0 0

1 240 1 2 3

6

i

i

j

j

w C

C qw C EJ

C

ll l l

l l l

L u

LC

4

2 3

31' 2

2''

23'''

4

241

0 1 2 36

0 0 2 6

0 0 0 62

0 0 0 0

1

IV

x


w QC

w qx


Unit displacements of nodes

Substituting in the first two rows of the matrix D the coordinates for the nodes with x = 0 and x = l we get expressions corresponding to unit displacements of the nodes:

D

or

L1

4

2 3

31' 2

2''

23'''

4

241

0 1 2 36

0 0 2 6

0 0 0 62

0 0 0 0

1

IV

x


w QC

w qx


Substituting in the second two rows of the matrix D the coordinates for the nodes with x = 0 and x = l follow the shear forces and moments at the ends of a beam corresponding with the reactions:

f i

l

fwi

f j

fwj

Mi

l

Qi

Mj

Qj

Reaction forces and internal forces

22

-6

1

2

3

4

00 0 0 0

00 0 2 0

0 0 0

0 0 2 62

w i i

i i

w j j

j j

f Q C

f M CEJ q

f Q C

f M C

l

ll

f

C

1 1i

j

ff EJ L C q L

f

1L

or


1 1qC L u L L

EJ

We express the integration constants by the displacements of the nodes:

i

j

u qu L C L

u EJ

Replacing with deliversC

1 1f EJ L C q L

1 1 1 11 1 1 1 1q

f EJ L L u L L q L EJ L L u q L L L LEJ

or in simplified form:

qf EJ r u q f

rqf


Within meansqf EJ r u q f

r the relative stiffness matrix with EJ = 1

the relative load column with q = 1 qr

The final stiffness matrix r and the load column for an ordinary beam:qf

wii wj

j


44 4

44 , with

4

d w q kbn w n

dx EJ EJ

Solution:

1 2 3 4 4( ) cos( ) sin( ) cos( ) sin( )

4nx nx nx nx q

w x C e nx C e nx C e nx C e nxn EJ

Elaboration of the stiffness matrix for a beam on an elastic foundation

In analogous steps the development of the stiffness matrix for a beam on an elastic foundation leads to more difficult differential equations:

Basic equations:

n relative stiffness of foundation k coefficient of elastic foundationb broadness of bearing


Elaboration of the stiffness matrix for a beam on an elastic foundation

The final stiffness matrix r and the load column :qf


Basic equations:

3

3

2

2

d q

dx EJ

dw d

dx dx

dM EJ

dx

2

2

dQ EJ

dx

Solution:

32

1 2 3

4 2

0 1 2 3

( )6

2 3( ) ( 2 )

2 3 24 2

qxx C C x C x

EJ

x x EJ qx qxw x C C x C C x

GF EJ GF

Algorithm for the elaboration of a stiffness matrix for abeam element following the theory of second order

Considering transverse strain the algorithm changes substantially. Instead of only one equation two equations are obtained with the two unknowns bending and nodal distortion:

withEJ

GF (shearing strain)


The final stiffness matrix r and the load column for a beam element following the theory of second order:

qr

wii wj

j

resultmatr_r :=

12 E J

l ( )12 l26 E J

12 l2

12 E J

l ( )12 l26 E J

12 l2q l2

6 E J

12 l24 E J ( )3 l2

l ( )12 l2

6 E J

12 l22 E J ( ) 6 l2

l ( )12 l2q l2

12

12 E J

l ( )12 l2

6 E J

12 l212 E J

l ( )12 l2

6 E J

12 l2q l2

6 E J

12 l22 E J ( ) 6 l2

l ( )12 l2

6 E J

12 l24 E J ( )3 l2

l ( )12 l2q l2

12

x

z

m

n

o

m1

o1

n1

x

xz

x

z

m

n

o

m1

o1

n1

Axis of beam(bended)

x

x

Axis of beam(unformed)

Theory of first order Theory of second order


Single beam Beam on elastic

foundation Harmonic oscillation Biaxial bending Theory of second order

4

4

d w q

dx EJ

4414

41

4 ,

mit4

d w qn w

dx EJ

kbn

EJ

g

4424

2

42

,

mit

d W qn W

dx EJ

Fn

EJ

4 2234 2

3mit

d w d w qn

dx dx EJ

Nn

EJ

3

3

2

2

d q

dx EJ

dw d

dx dx

The formulas of the moment (M) and the shear force (Q)

2

2

d wM EJ

dx

3

3

d wQ EJ

dx

4

4

d wq EJ

dx

2

2

d wM EJ

dx

3

3

d wQ EJ

dx

4414

4d w

q EJ n wdx

2

2

d WM EJ

dx

3

3

d WQ EJ

dx

4424

d Wq EJ n W

dx

2

2

d wM EJ

dx

3 2233 2

d w d wQ EJ n

dx dx

4 2234 2

d w d wq EJ n

dx dx

mitd dw

M EJdx dx

2

2

dQ EJ

dx

3

3

dq EJ

dx

Fundamental equations for the calculation of beam structures used in the development of the stiffness matrix


Assessment of correctness of the stiffness matrices

Derivations of stiffness matrices are sometimes extensive and sophisticated in mathematics. Therefore, the test of the correctness of the mathematical calculus for this object is an important step in the development process of numerical methods.

There are two types of assessment:

1. Compatibility condition

2. Duplication of the length of the element


1. Compatibility condition

i j

x

Element 1 i jElement 2O-x

x

x

0r z r r z r z Fox xji jj ii ij

Equation of equilibrium at point О:

The displacement vectors and can be expressed as Taylor rows: zx

z

x

'

wzo w

in the centre point O

' '' '''2 3 45

' '' 2! ''' 3! 4!

' '' '''2 3 45

' '' 2! ''' 3! 4!

IVw w w w wx x xz x o x

x IV vw w w w w

IVw w w w wx x xz x o x

x IV vw w w w w

' '' '''2 3 4

5 0' '' 2 ''' 6 24

IVw w w w wx x xr r r r x r r r r r r r r o x Fji ii jj ij ji ij ji ij ji ij ji ijIV vw w w w w

After transformation:

rij


2. Duplication of the length of the element

x x

i jElement 1 i jElement 2O-x x

Equation of equilibrium at point -x, О, x :

0( )

( ) 0( ) ( )

0( )

r z r z roxii ij iq x

r z r r z r z r rox xji jj ii ij jq o iq o

r z r z ro xji jj jq x

Or in matrix form:

r r rii ij iq x

r r r r r rji jj ii ij jq o iq o

r r rji jj jq x

Rearrangement of rows and columns

Application of Jordan’s method with - new value of element and - initial value of element.

*r r r rij ij i j *rij


Calculation of shell structures

Wall- like girder Loaded plate Hall roof- like folded plate structure

Panel Plate Folded plate structure

x

y

p – Boundary load in plane

A BA and B – Reaction force in plane

y

x

P

Reaction force Plane

Load Plane

x

zy

Boundary of panel

+ =+ =


Systematic approach for the development of differential equations for a disc

Type of the development

Equation of equilibrium 0 0yx xy yxxy xyx y x y

1

Geometrical relations x y

u v u v

x y x y

2

Material law

2

2

1

1

1

1

x x y x x y

y y x y y x

E

E

E

E

3

(2) (3)

2 21 1

2 1

x y

E u v E u v

x y x y

E u v

y x

4

(4) (1)

2 2 2

2 2

2 2 2

2 2

1 10

2 2

1 10

2 2

u u v

x y x y

v v u

y x x y

5


The system of partial differential equations for discs changes to a system of ordinary differential equations if the displacements are approximated by trigonometric rows:

Inserting the results of this table into equation (5) from the previous tablewe get a system of ordinary differential equations:

2 1 12 02 2 2

21 12 022 2

d V dUV

dydy

d U dVU

dydy

( ) cos withn

u U y xL

( ) sin with

nv V y x

L

( ) sinu

U y xx

( ) cos

vV y x

x

( )cos

u dU yx

y dy

( )

sinv dV y

xy dy

22 ( ) cos

uU y x

x

2

22

( ) sinv

V y xx

2 2

2 2

( )cos

u d U yx

y dy

2 2

2 2

( )sin

v d V yx

y dy

2 ( )sin

u dU yx

x y dy

2 ( )

cosv dV y

xx y dy

Type of the development Bending of plate

Equation of equilibrium 2

2

h

x xh

m zdz

2

2

h

y yh

m zdz

2

2

h

xy xyh

m zdz

1

Geometrical relations w w

u z v zx y

2 2 2

2 22x y xy

w w wz z z

x y x y

2

Material law 21x x y

E

21y y x

E

2 1xy xy

E

3

(2) (3) 2 2

2 2 21x

E w wz

x y

2 2

2 2 21y

E w wz

y x

2

1xy

E wz

x y

4

2 2

2 2x

w wm D

x y

2 2

2 2y

w wm D

y x

2

1xy

wm D

x y

5

(4) (1) 2 2

2 2xmw w

x y D

2 2

2 2

ymw w

y x D

2

1xymw

x y D

6

(6) (4) 3

12 xx

mz

h

3

12 xy

mz

h

3

12 xyxy

mz

h 7

3

2with

12 1

E hD


Systematic approach for the development of differential equations for a plate


Systematic approach for the development of differential equations for a plate

Stress and internal force in plate element

x

dxdy

xy

dxdy

mxmy

dxdy

yz

xz

dxdy

qx

qy

dxdy

xyyx

dxdy

mxy

myx

Shearing stress

Torsion with shear

Shear force

Torsional moment

dx

dy

x

z, w(x,y)

p(x,y)

h/2h/2

Equation of equilibriumBalanced forces in z-direction:

yxqq

px y

Balanced moments for x- and y-axis:

xy yxx xy x

m mm mq q

y x x y

Equation of equilibrium after transformations: 2 22

2 22 (1)xy yx

m mmp

x x y y


Partial differential equation for a plate:4 4 4

4 2 2 42

w w w p

x x y y D

This changes to an ordinary differential equation if the displacements are approximated by trigonometric rows.

Inserting the results of the table in the above equation we get the ordinary differential equation:

4 4 42 4

4 2 42

d W d W d W p

dx dy dy D

( ) sin withn

w W y xL

( ) cosw

W y xx

( )sin

w dW yx

y dy

22

2( ) sin

wW y x

x

2 2

2 2

( )sin

w d W yx

y dy

2 ( )cos

w dW yx

x y dy

2 ( )

cosw dW y

xx y dy

33

3( ) cos

wW y x

x

3 3

3 3

( )sin

w d W yx

x dy

32

2

( )sin

w dW yx

x y dy

3 2

2

( )cos

w d W yx

y x dy

44

4( ) sin

wW y x

x

4 4

44 4

( )sin

w d W yx

y dy

4 22

2 2 2

( )sin

w d W yx

x y dy

4 2

22 2 2

( )sin

w d W yx

y x dy


- MAPLE permits a fast calculation of stiffness matrices for different element

types in symbolic form

- Elaboration of stiffness matrices can be automated

- Export of the results in other computer languages (C, C++, VB, Fortran) can help to implement stiffness matrices in different environments

- For students‘ education an understanding of algorithms is essential to test different FE-formulations

- Students can develop their own programmes for the FEM

Conclusion:

Documents

University of the German Armed Forces Munich