Specialization and additions to Stahlbeton I 4.1 Basics...• Avoidance through careful curing (prevention of significant water losses on the fresh concrete surface caused by high

4 Long-term effects

Specialization and additions to Stahlbeton I

4.1 Basics

02.12.2019 ETH Zurich | Chair of Concrete Structures and Bridge Design | Advanced Structural Concrete 1

Time-dependent behaviour of concrete

Shrinkage

Volume contraction without load

(Figure for free, unrestrained deformations → no restraint forces)

Creep

Increase of deformations under constant stress

Relaxation

Decrease of stress under constant strain

no load effect

t

volume contraction due to shrinkage

t

t

stress constant

t

initial deformation

creep deformation

t

strain constant

stress decrease due to relaxation

initial stress

t


εcσc

σc εc

σc εc

Long-term effects

Shrinkage

Early/capillary (Plastic) shrinkage (up to 4‰ → avoid!) • Capillary stresses during the evaporation of water from the fresh concrete lead to denser structure of the cement matrix in

the first few hours until hardening.• Avoidance through careful curing (prevention of significant water losses on the fresh concrete surface caused by high

concrete or air temperatures, low humidity and wind).

Autogenous and chemical shrinkage (normal concrete up to 0.3‰, UHPC up to 1.2‰)• Volume contraction during hydration, initially caused by the chemical integration of the water molecules into the hydration

products (first days). Afterwards, as soon as the water in the capillary pores is used up, it is mainly caused by capillary tension due to the lower internal relative humidity, leading the hydration to consume water from the gel pores (first weeks).

• Primarily dependent on W/C ratio: The lower the W/C ratio, the greater the autogenous shrinkage (significant effect only for W/C < 0.45 high-performance concrete, UHPC).

Drying shrinkage (up to approx. 0.3‰ outside at RH=70%, up to approx. 0.5‰ inside at RH=50%)• Volume contraction in hardened concrete by releasing water into the environment. Begins with formwork stripping or the

end of curing and lasts for years.• Magnitude primarily dependent on cement paste volume (cement, admixtures, entrapped air and water). Faster for high

W/C ratios, low air humidity and thin components.

02/12/2019 ETH Zurich | Chair of Concrete Structures and Bridge Design | Advanced Structural Concrete 3

CH

insi

de

CH

out

side

NB: End value isindependent of theinitiation of drying


Drying shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐(according to SIA 262)

Drying shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐𝑐 [‰] Progression ⁄𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) 𝜀𝜀𝑐𝑐𝑐𝑐𝑐




Drying shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐

Final drying shrinkage value 𝜀𝜀𝑐𝑐𝑐𝑐𝑐 [‰]

Autogenous shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (according to SIA 262)

Progression of autogenous shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 [‰]

𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 = 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 + 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡

+CH

insid

e

CH o

utsid

e



Drying shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐

Final drying shrinkage value 𝜀𝜀𝑐𝑐𝑐𝑐𝑐 [‰]

Autogenous shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (according to SIA 262)

Progression of autogenous shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 [‰]

𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 = 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡 + 𝜀𝜀𝑐𝑐𝑐𝑐 𝑡𝑡

+CH

insid

e

CH o

utsid

e

Long-term effects

Creep and relaxation

Cause / Phenomena• Stress leads to rearrangement or evaporation of water in the cement paste. The associated sliding and compaction

processes lead to volume contraction.• The standards assume that creep deformations cease after some decades (asymptotically approaching the long-term

creep coefficient ϕ∞). This is controversial today, there are e.g. older cantilever bridges indicating that creep deformationsmight keep increasing continuously. Yet only few experiments are available.

Influences on the magnitude of creep deformations• Load level (creep deformations approximately proportional to the load)• Cement paste volume (high cement paste volume = larger creep deformations)• Concrete compressive strength (high compressive strength = smaller creep deformations)• Age of the concrete (loading at a young age causes larger creep deformations)

Influences on the course of time• Creep is faster in smaller elements (thin components)• Creep is faster at low relative humidity (dry environment)

Relaxation• Creep and relaxation are related phenomena• Relaxation behaviour is influenced by the same variables as creep


Long-term effects

Creep

t

t

stress constant

• Increase in deformation under constant stress

• 𝜀𝜀𝑐𝑐 𝑡𝑡 = 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 + 𝜀𝜀𝑐𝑐𝑐𝑐(𝑡𝑡)

• 𝜀𝜀𝑐𝑐𝑐𝑐 = 𝜑𝜑(𝑡𝑡, 𝑡𝑡0) � 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 resp.𝜀𝜀𝑐𝑐 𝑡𝑡 = 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 1 + 𝜑𝜑(𝑡𝑡, 𝑡𝑡0)

• 𝜑𝜑(𝑡𝑡, 𝑡𝑡0) creep index

• Normal case: 𝜑𝜑𝑡𝑡=𝑐 ≅ 1.5 … 2.5 i.e. increase of deformations by factor 2.5...3.5

• Analogous behaviour under tension (non-cracked concrete)


εc

εc,t=0

ϕ (t)·εc,t=0

σc

Long-term effects

Relaxation (≈ creep at ε = const.)

• Decrease in stress under constant strain• Approximation (fictitious modulus of elasticity):

𝜎𝜎𝑐𝑐 𝑡𝑡 = 𝜎𝜎𝑐𝑐,𝑡𝑡=0 �1

1 + 𝜑𝜑

• Better approximation (derived e.g. using Trost method)

𝜎𝜎𝑐𝑐 𝑡𝑡 = 𝜎𝜎𝑐𝑐,𝑡𝑡=0 1 −𝜑𝜑(𝑡𝑡)

1 + 𝜇𝜇𝜑𝜑

• Normal case: 𝜑𝜑𝑡𝑡=𝑐 ≅ 1.5 … 2.5, 𝜇𝜇 = 𝑐𝑐𝑐𝑐. 0.75, i.e. reduction of the initial stress to approx. 25%.

• Decrease is less pronounced (to approx. 40%) if deformation isimposed slowly

t

t

strain constant


σc

εc

Long-term effects

Creep Relaxation (= creep)

t

t

t

t

stress constant strain constant

𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 =𝜎𝜎𝑐𝑐𝐸𝐸𝑐𝑐

𝜀𝜀𝑐𝑐𝑐𝑐(𝑡𝑡) = 𝜑𝜑(𝑡𝑡, 𝑡𝑡0) � 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒


εc

σc

σc

εc

Long-term effects

Creep - reversible and plastic part• The deformations of the concrete under load are composed of the elastic deformations 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 and the time-dependent creep deformations 𝜀𝜀𝑐𝑐𝑐𝑐• The creep deformations 𝜀𝜀𝑐𝑐𝑐𝑐 consist of a reversible component 𝜀𝜀𝑐𝑐𝑐𝑐 ,𝑟𝑟 (reversal sets in relatively quickly, half-life approx. 30 days) and an

irreversible (plastic) component 𝜀𝜀𝑐𝑐𝑐𝑐,𝑝𝑝:

The irreversible part 𝜀𝜀𝑐𝑐𝑐𝑐 ,𝑝𝑝 depends on the time of loading = concrete age at load application (old concrete is less prone to creep) and occursmuch slower than the reversible component.

• Example: Loading and complete unloading after a longer period of time (permanent stretching):

For simplicity, no distinction is made between the two components in the following slides , and no unloading is considered.

( ) ( ) ( ) ( ), , , ,c c el cc r cc p c el cct t t t= + + = +ε ε ε ε ε ε

𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒

𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒𝜀𝜀𝑐𝑐𝑐𝑐,𝑟𝑟 + 𝜀𝜀𝑐𝑐𝑐𝑐,𝑝𝑝(𝑡𝑡0)

t𝑡𝑡0 𝑡𝑡1

𝜀𝜀𝑐𝑐𝑐𝑐,𝑟𝑟 + 𝜀𝜀𝑐𝑐𝑐𝑐,𝑝𝑝(𝑡𝑡1)

𝜀𝜀𝑐𝑐𝑐𝑐,𝑝𝑝 𝑡𝑡0 − 𝜀𝜀𝑐𝑐𝑐𝑐,𝑝𝑝(𝑡𝑡1)irreversible part


εc

Long-term effectsCreep – Final value and progression(see also SIA 262, 3.1.2.6)

• Increase in deformation under constant stress•

with

• Normal case: 𝜑𝜑𝑡𝑡=𝑐 ≅ 1.5 … 2.5, i.e. increase of deformations by a factor of 2.5...3.5

• Analogous behaviour under tension (in uncracked concrete) t

t

𝜀𝜀𝑐𝑐𝑐𝑐 = 𝜑𝜑(𝑡𝑡, 𝑡𝑡0) � 𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒

stress 𝜎𝜎𝑐𝑐 = constant

𝜀𝜀𝑐𝑐,𝑒𝑒𝑒𝑒 =𝜎𝜎𝑐𝑐𝐸𝐸𝑐𝑐

𝑡𝑡0

𝑡𝑡0

( ) ( )( )

, 0 ,

0 ,

,

(1 , )c c el c el

c el

t t t

t t

ε = ε + ϕ ⋅ ε

= + ϕ ⋅ ε

t0t

creep coefficient

time

age of the concrete at the start of exposure

0t t− load duration

( )0,t tϕ


εc

σc

Long-term effects

Creep – Final value and progression(see also SIA 262, 3.1.2.6)

- Relative humidity:

- Stress level:

- Concrete compressive strength:

- Concrete age at loading:(corrected for the influence of the temperature: t0,eff → kT t0)

- Load duration (→ progression):

1.5 0.45

0.45 1)(für , sonst c

ckfc c ck ce f

σ −

σ σβ = σ > β =

... 25 / 30 30 / 37 35 / 45 ...

... 2.9 2.7 2.6 ...fcC C Cβ =

0 0( ) 1.2 0.2 ( 28d) 0.5t tβ ≈ β = =

0(( ) ) 1t tβ = ∞ − ≈

1.25 1.5 ( 80 65%)RH RHϕ ≈ ≈ (CH)

𝑡𝑡: time at which the creepcoefficient ϕ is determined

𝑡𝑡0: concrete age at timeof loading

( ) 0 00 , ( 1.( ) 2. )) 5( 5fcRH c t tt t tσϕ = ⋅β ⋅ ⋅β ⋅ − ≈ϕ β β


Long-term effects


CH in

side

CH o

utsid

e

: Beiwert für relative Luftfeuchtigkeit (RH: normalerweise Jahresmittel)RHϕ

CH in

side

CH o

utsid

e


( ) 0 00, ( 1.( ) 2. )) 5( 5fcRH c t tt t tσϕ = ⋅β ⋅ ⋅β ⋅ − ≈ϕ β β

Coefficient for relative humidity (RH: usually the annual mean)

Long-term effects


0( ) : Betonalter bei Belastung tβ : Lastdauer ( Zeitverlauf) RHϕ →

normal case(t0 = 28d)



Concrete age at loading Load duration (→ progression)

Long-term effects


0( ) : Betonalter bei Belastung tβ : Lastdauer ( Zeitverlauf) RHϕ →

normal case(t0 = 28d)



Concrete age at loading Load duration (→ progression)

4 Long-term effects

4.2 Effect of creep on the load-bearing and deformation behaviour


Long-term effects

Effect of creep on deformations of a structure• The effect of creep must always be taken into account when determining deformations due to permanent loads. The increase in deformation

due to creep is considerably smaller in the cracked stage II than in the non-cracked stage I (see Stahlbeton I).• Deformations are often controlling the design, for example in the case of:

- passive reinforced, slender girders (above h/L≈ 1/12)- passive reinforced slabs (flat slabs, canopies, slabs near facades area, non-load-bearing walls)- prestressed bridge girders, whose stresses in construction and final state differ strongly(cantilever construction, continuous beams cast span by span)

Effect of creep on internal forces and stresses• Restraint and residual stresses are partially relieved due to creep over time (relaxation).• For statically determinate (= isostatic) systems and for statically indeterminate systems with uniform creep properties, creep has no effect on

the internal forces• Significant internal force redistributions occur in statically indeterminate (= hyperstatic) systems as a result of changes of the static system

and non-uniform creep properties. The calculation of creep effects is complicated by the nterdependence (creep depends on the level of stress and vice versa).

Approaches for the calculation of creep and shrinkage problems• Method with age adjusted modulus of elasticity• Unit creep curve method (Dischinger method)• Rüsch Method (improved method Dischinger) • Creep step method• Trost Method (sufficiently accurate and suitable for manual calculations)


Long-term effects

Creep – Boltzmann superposition principle

• The creep strain due to any stress development σ(t) can generally be expressed as follows:

• For discrete stress increments (steps) ∆σi , which are applied at time ti results:

( ) ( )0

0

1 ,t

ccc

t t dE

τ=

τ=

∂σε = ϕ τ τ

∂τ∫( ) ( )

00

1 ,n

cc i iic

t t tE =

ε = ∆σ ⋅ϕ∑

sc

𝑡𝑡

( ), it tϕ

( )0

0

,t tϕ→ ∆σ( )1

1

,t tϕ→ ∆σ

𝑡𝑡𝑡𝑡0 𝑡𝑡1

𝑡𝑡𝑗𝑗𝑡𝑡0 𝑡𝑡102.12.2019 ETH Zurich | Chair of Concrete Structures and Bridge Design | Advanced Structural Concrete 19

∆σ0

∆σ1

Long-term effects

Creep – Boltzmann superposition principle

Incorrect procedure for determining creep deformations (creep from the respective load level for the entire load with new creep coefficient):

• (*) Effective = correct portion of creep caused by σ0in time interval t1…tj

• (**) Incorrectly determined portion of creep caused by σ0in time interval t1…tj

sc

𝑡𝑡

( ), it tϕ

σ0σ1

( )0

0

,t tϕ→ σ( )1

1

,t tϕ→ σ

𝑡𝑡𝑡𝑡0 𝑡𝑡1

( )1 0, (richtig)jt t∆ϕ → σ

𝑡𝑡𝑗𝑗𝑡𝑡0 𝑡𝑡1

( )1 0, (falsch)jt t∆ϕ → σ

(*)

(**)


Long-term effects

Approaches for the calculation of creep and shrinkage problems

Method with age adjusted modulus of elasticity• Effect of concrete age at loading neglected

→ same creep curve for all loads, shifted along abscissa (horizontal)• Unrealistic (overestimates ability of old concrete to resist creep)

𝑡𝑡

( ), it tϕ ( ) ( )10, ,tt tt = ϕ = ∞ϕ = ∞

𝑡𝑡1𝑡𝑡0

𝑡𝑡1 −𝑡𝑡0( )0

0

,t tϕ→ ∆σ( )1

1

,t tϕ→ ∆σ

𝑡𝑡


∆σ0

∆σ1

σc

Long-term effects


Method with age adjusted modulus of elasticity• Effect of concrete age at loading neglected

→ same creep curve for all loads, shifted along abscissa (horizontal)• Unrealistic (overestimates ability of old concrete to resist creep)• Unrealistic: corresponds to assumption of viscoelastic, i.e. fully reversible behaviour

𝑡𝑡𝑡𝑡1𝑡𝑡0 𝑡𝑡0 +𝑡𝑡1

𝑡𝑡

real behaviour

age adjusted modulus of elasticity


∆σ0

εc

σc

Long-term effects


Unit creep curve method (Dischinger method)• Same creep curve for all loads, shifted along ordinate (i.e. vertically)• Advantage: Representation in recursion formulae possible• Unrealistic: underestimates creep of old concrete

𝑡𝑡

( ), it tϕ

𝑡𝑡1𝑡𝑡0

( )0

0

,t tϕ→ ∆σ( )1

1

,t tϕ→ ∆σ

𝑡𝑡

( )0,it tϕ


σc

∆σ0

∆σ1

Long-term effects


Unit creep curve method (Dischinger method)• Same creep curve for all loads, shifted along ordinate (i.e. vertically)• Advantage: Representation in recursion formulae possible• Unrealistic: underestimates creep of old concrete• unrealistic: neglects viscoelastic behaviour (no reversible part)

𝑡𝑡𝑡𝑡1𝑡𝑡0

𝑡𝑡

Dischinger method

real behaviour


σc∆σ0

εc

Long-term effects


Rüsch method (improved Dischinger method)• Basically the same assumptions as Dischinger method• Superposition of the entire reversible part of creep deformations (neglected in the Dischinger method) with the elastic

elongation• Reasonably realistic, since the reversible portion of creep deformations occurs relatively quickly

𝑡𝑡

𝑡𝑡

Dischinger method

Rüsch method

real behaviour

𝑡𝑡1𝑡𝑡0


σc∆σ0

εc

Long-term effects


Creep step method• The stress history is only known in advance in simple cases (this was assumed in the previous considerations). In general,

it depends on the creep behaviour. The solution therefore usually requires an iterative or step-by-step approach.• Based on the Dischinger method a differential equation for creep behaviour can be formulated (also possible with the Rüsch

method). For numerical solutions, the creep step method can be used, which is based on a subdivision of the load history into time intervals or into "creep steps" (subdivision of the creep coefficient φ(𝑡𝑡 = ∞, 𝑡𝑡0) in equal creep intervals ∆φ, usually more appropriate).

• Linearisation of the creep and stress function per interval results in the increase of creep deformation in the time interval. ∆𝑡𝑡𝑖𝑖 = 𝑡𝑡𝑖𝑖 − 𝑡𝑡𝑖𝑖−1:

Change of the creep function during ∆𝑡𝑡𝑖𝑖Change of the concrete stress during ∆𝑡𝑡𝑖𝑖

• Total strain increase in time interval ∆𝑡𝑡𝑖𝑖 = 𝑡𝑡𝑖𝑖 − 𝑡𝑡𝑖𝑖−1:

1,i 1

0 01

; :2

:

i i icc i i i i

c ci i i

E E−

−

−

σ ∆σ ∆ϕ∆ε = ∆ϕ + ∆ϕ = ϕ − ϕ

∆σ = σ − σ

1,i ,i ,i ,i

0 0 0 0

12

i i i ic cc cs i i cs

c c c cE E E E−∆σ ∆σ σ ∆σ

∆ε = + ∆ε + ∆ε = + ∆ϕ + ∆ϕ + ∆ε


4 Long-term effects

4.3 Simplified method for the investigation of long-term effects


Long-term effects


Trost methodIn practice, a relatively large proportion of the total stress is applied at a time 𝑡𝑡0 followed by smaller stress increments ∆𝜎𝜎𝑖𝑖(additional loads, but also internal force redistributions). The Trost method takes advantage of this to avoid an iterative or step-by-step approach.

The creep function for the stress increments (𝜎𝜎 𝑡𝑡 − ∆𝜎𝜎0 = ∑𝑖𝑖=1𝑛𝑛 ∆𝜎𝜎𝑖𝑖) occurring at the time period 𝑡𝑡𝑖𝑖 > 𝑡𝑡0 (resp. 𝑡𝑡0 < 𝑡𝑡𝑖𝑖 ≤ ∞)is reduced with an ageing coefficient μ 𝑡𝑡 (sometimes also called «relaxation factor»).

The creep deformation due to the total change in stress according to Boltzmann's superposition principle is:

( ) ( ) ( ) ( ) ( ) ( )0 00 0 0

0

0

001 0

( ), , , ,n

ii

c cc

ic cc

tt tt t t t t t

E Et

Et

E=

σ σ⋅ µε = + = +

σ − σ∆σ⋅ ϕϕ ⋅⋅ ϕ ϕ⋅∑

Ageing coefficient (simplifiedby Trost to be identical for all load levels, 𝑡𝑡𝑖𝑖 > 𝑡𝑡0)General derivation see Marti, Theory of Structures, Chapter 7.4.2

𝑡𝑡𝑡𝑡0 𝑡𝑡1 𝑡𝑡𝑖𝑖

( ) 0tσ − ∆σ( )tσ


σc

∆σ0

∆σ1

∆σi

Long-term effects


Trost methodThe ageing coefficient results from the equation on the previous slide:

The total deformations at time t thus amount to:

( ) ( )( ) ( ) ( )( ) ( )00

0 0

001 ,1 ( ),

ccs

cc t t t

tt

EEt ttε = + +

σϕ

σ − σ+⋅ εµ ϕ+

shrinkageelastic deformations

( ) ( ) ( )( )

( )( ) ( )0 1

01 0 0 0 0

,, ( ) , ( )

,

n

i ini i

ii c c

t ttt t t t t t

E E t t t=

=

∆σ ⋅ϕσ − σ∆σ

⋅ ϕ = ⋅µ ⋅ ϕ → µ =σ − σ ⋅ ϕ

∑∑


𝑡𝑡𝑡𝑡0 𝑡𝑡1 𝑡𝑡𝑖𝑖

( ) 0tσ − ∆σ( )tσ

σc

∆σ0

∆σ1

∆σi

Long-term effects


Trost method• The stress curve is generally not known → µ(t) cannot be calculated directly in the way outlined on the previous slides• If the relaxation function is determined from the creep function (solution of a linear, inhomogeneous Volterra integral

equation), the corresponding ageing coefficient can be determined numerically [see Seelhofer 2009 or Marti, Theory of Structures]:

• The evaluation sows that τ µ(t) varies only slightly

→ Ageing coefficient µ independent of timesufficiently accurate for practical applications

→ for usual conditions (ϕ = 1.5...4)approximately µ ≈ 0.80


Long-term effects


Trost method

• With this approximation the total deformation at time t is:

• Alternative formulation using fictitious («refined age adjusted») moduli of elasticity for long-term influences:

( ) ( ) ( ) ( )

( ) ( )

00

0 0 0 0 0 0,

1 1 1

( ), , , , 0.8mit

c csc

t tE

t t t t t t t

ε = σ + ϕ + ∆σ + ⋅ϕ + ε

σ = ∆σ = σ = ∆σ = σ − σ ϕ > µϕ = ≈

µ

initial stress stresses added over time

( )

( )

( )

( )

( ) ( ) ( ) ( ) ( )0 0 0 0

0 0 0 0

0 0

: ,1 , 1 ,

1 , 1 ,

c cc cs cs c

c c c c

t t E Et t t E EE E E E t t t tt t t t

∆σ ∆σσ σ ′ ′′ε = + + ε = + + ε = =′ ′′ + ϕ + ⋅ϕ

+ ϕ + ⋅ϕµ

µ

initial stress stresses added over time


with

Final values at t = ∞ (µ = 0.8):

0.330.230.14

Long-term effects

Calculation of relaxation function based on creep coefficient and ageing factor• Relaxation function = stress curve for constant (imposed) initial strain,

Method with age adjusted modulus of elasticity

Trost method

( ) ( ) ( )0 0

0 0 0

0

0 0 0

( )1 1

( )1

1( ) ( ) 11 1

cc c c

ttE E E

t

t t

σ ∆σ σε = + ϕ + + ϕ =

ϕ→ ∆σ = −σ

+ ϕ ϕ

→ σ = σ + ∆σ = σ − = σ + ϕ + ϕ

( ) ( ) ( )0 0

0 0 0

0

0 0

( )1 1

( )1

( ) ( ) 11

cc c c

ttE E E

t

t t

σ ∆σ σε = + ϕ + + ⋅ϕ =

ϕ→ ∆σ = −σ

+ ⋅ϕ ϕ

→ σ = σ + ∆σ = σ − + ⋅ϕ

µ

µ

µ

sc

t

00

0c

cEσ

ε =i.e. initial strains remain constant

( )0,( )t tϕ = ϕ

( )0,( )t tϕ = ϕ

age adjusted modulus

Trost

Dischinger

0σ

0σ0σ0σ

The Trost method is simple and agrees well with experiments (better than more complicated procedures) → only this procedure is used in the following!


Long-term effects

Terminology and generalisation of the Force Method → Time-dependent Force Method• In the following, Trost’s Method is used in combination with the Force Method, known from «Theory of Structures I/II• To account for long-term effects, compatibility conditions are expressed here at different moments in time• Further information on the Force Method: Peter Marti, «Theory of Structures» resp. «Baustatik», Chapter 16. The following

summaries are taken from this book (p. 254 and p. 257, respectively):


Long-term effects

Redistribution of internal forces in statically indeterminate systemsSystems with uniform creep propertiesExample 1: Two-span beam, solution with force method

BS (basic system): Intermediate bearing removedRV: (redundant variable): Reaction at intermediate support Displacements in the basic system (elastic, t = t0):

Compatibility condition at time t0 :

Time-dependent compatibility condition with the Trost method:

→ Generalization to general systems is possible → With uniformcreep properties, the redundant variables of stat. indeterminate systems do not change!

( ) ( )4 3

10 11

2 25 1384 48

kg l lEI EI

δ = δ =

10

10 11 11

1111

(1 ) (1 ) ( ) (1 ) 01( ) 0 ( ) 01

B

B

e

B Be

B

R

R R t

R t R t

δ = δ ⋅ + ϕ + ⋅δ ⋅ + ϕ + ∆ ⋅δ ⋅ + µϕ =+ µϕ

+ ∆ ⋅δ = → ∆ϕ

δ+

+ ⋅ =δ

= 0 (compatibilityat time t0)

10( 1)q =δ

11(R 1)B =δ

l l

permanent loads (effective from t0): gk

10 11 0 BeRδ = δ + ⋅δ =

( ) ( )B Be BR t R R t= + ∆


Long-term effects

Redistribution of internal forces in statically indeterminate systemsSystems with uniform creep propertiesGeneralization to general systems

If the creep properties are uniform, redundant variables do not change in statically indeterminate systems!

11 11 1 10

0

10 11 1

1

1

1

0

0

0

: (* )0

0*

i

e i

ie i i

e

i i ii ie

i i

Xt t

X

Xt t

X−

δ δ δ

δ δ δ → = = ⋅

= + ⋅ =

δ δ δ

δ δ δ

RV at :

Can

Com

ge

patibility at

of redundant

( ) ( ) ( )10 11 1

1

1

0

1

0

( ) ( )

01 1 1

0

i ie i

i

i e

i i ii ie

X

X

X t X X t

Xt t

X

= + ∆

∆ → > + ϕ + ⋅ + ϕ +

δ δ δ δ δ δ

+ µϕ =

∆

→

variables using the Trost method:

Compatibility for :

using (**) ( )( )

11 1 1 11 1

1 1

101

0

00

10

i i

i ii i ii ii

X

X

X

X

δ δ ∆ δ δ + µϕ ⋅ = ≠ + ϕ δ δ ∆ δ δ

∆ → = ∆

: where

All coefficients multiplied by the

same factor!


Long-term effects

Redistribution of internal forces in statically indeterminate systemsSystems with uniform creep propertiesExample 2: Prestressed two-span beam, solution with force method

BS: Intermediate bearing removedRV: Reaction intermediate supportDisplacements in the basic system (elastic, t = t0):


Time-dependent compatibility condition with the Trost method:

→ Support reactions change due to time-dependent prestressinglosses (RV proportional to prestressing force = deviation force)

10( 1)q =δ

11(R 1)B =δ0 10 11( ( )) 0k Beg u t R+ ⋅ δ + ⋅ δ =δ =

permanent loads (effective from t0): gk

Deviation forces u(t) u(t) = u(t0)+∆u(t)

( ) ( )4 3

10 112 25 1

384 48l l

EI EIδ = δ =

( ) ( ) ( ) ( ) ( )

( )

0 10

0 10 1

11 10 11

10 11

10

11

1

( ) 1 1 ( ) 1 ( ) 1 01 1( ) ( ) 01 1

( (

)

)

(

)

k B

B

Be

k Be B

B

g u t R u t R t

u t R t

R t u t

g u t R+ ⋅

δ = + ⋅ δ + ϕ + ⋅ δ + ϕ + ∆ ⋅ δ + µϕ + ∆ ⋅ δ + µϕ =

+ µϕ + µϕ+ ∆ ⋅ δ + ∆ ⋅ δ =

+ ϕ + ϕδ

→

δ +

= −∆

⋅

δ

δ

∆= 0 (compatibilityat time t0)

l l

( ) ( )B Be BR t R R t= + ∆


Long-term effects

Redistribution of internal forces in statically indeterminate systemsSystems with non-uniform creep propertiesExample 3: Hinged frame with concrete beam and steel columns, solution with force methodDisplacements in the basic system (elastic, t = t0):


Time-dependent compatibility condition with the Trost method(support does not creep), taking into account the compatibility at t0:

10 10 11 11

2 22 4 3 32 10 28 4 3 48 4 16 4 3 16

R S R S

R R S

gl l l gl l l l l h lEI EI EI EI EI EI

δ = − ⋅ ⋅ ⋅ = − δ = δ = − ⋅ = δ = ⋅ ⋅ ⋅ =

( ) 10 10

11 11

10 10 11 111 0 11 0( )6

SS R

Rk

eS

RR

Seg lt XX

δ + δδ = → = − =

δ + δδ + δ + δ + δ =

( ) ( ) ( )( ) ( )

( ) ( )

10 10 11 11 11 11

10 11 1110 11

1

10 11 1

0 11

10 11 11 11

11 1

1

1

1 1 1

1 1

11 1 1

1

( ) 1 1 1 0

1 0

1 0 ( )1

S R S Re

S R S R S Re

R R S Re

R ReR R S R

e S R

t X X

X

X

X X

XX X t

δ = δ + δ + ϕ + δ + δ + ϕ + ∆ δ + δ + µϕ = + δ ⋅ϕ + + ⋅ δ ⋅ϕ + ∆ δ + δ + µϕ =

δ + ⋅ δ δ ϕ + ⋅ δ ϕ + ∆ δ + δ + µϕ = → ∆ = −ϕ δ

δ + δ δ +

δ

δ

+ + µϕ

→ ∆ 1 1 1( ) , ( ) 16 2 2 6 2k k

eg l g lX t X X t ϕ ϕ ϕ

= = = + + µϕ + µϕ + µϕ

/ 6SEI EI=

0(t )R IEI E t= =

4lh =

l

kg

BS+RV:

System and loads:

0 :M

1 1(X 1) :M =

2 8gl

4l

−

1 1 1( ) ( )eX t X X t= + ∆

Formula applies only for this example(system-dependent)


→ In the case of non-uniform creep properties internal forces are redistributed due to creep

Redistribution of internal forces in statically indeterminate systemsSystems with non-uniform creep properties - Generalization to general systems

11 11 1 10

0

10 11 1

1

1

1

0

0

0

: (* )0

0*

i

e i

ie i i

e

i i ii ie

i i

Xt t

X

Xt t

X−

δ δ δ

δ δ δ → = = ⋅

= + ⋅ =

δ δ δ

δ δ δ

RV at :

Can

Com

ge

patibility at

of redundant

( ) ( ) ( )10 11 1

1

1

0

1

0

( ) ( )

01 1 1

0

i ie i

i

i e

i i ii ie

X

X

X t X X t

Xt t

X

= + ∆

∆ → > + ϕ + ⋅ + ϕ +

δ δ δ δ δ δ

+ µϕ =

∆

→

variables using the Trost method:

Compatibility for :

using (**) ( )( )

11 1 1 11 1

1 1

101

0

00

10

i i

i ii i ii ii

X

X

X

X

δ δ ∆ δ δ + µϕ ⋅ = ≠ + ϕ δ δ ∆ δ δ

∆ → = ∆

: where

Long-term effects

no constant factor for creep influence(each RV has a different factor in general)

→ redundant variables must change!


→ In the case of non-uniform creep properties, internal force redistributions occur as a result of creep

( )0 1

0,

1

( ) ( ) ( ) 1 0

( ) ( )1 1

B B B B B

BB B EG B

B

t t M t

M t M M t

∆θ = θ = θ ⋅ϕ + ∆ ⋅ θ + µϕ =

θ ϕ ϕ→ ∆ = − = ⋅ =

θ + µϕ + µϕ

Influence of creep for system changesExample 4 - Connection of two simple beams with the same creep behaviour

System, load relevant for creep:

Construction sequence: 1. Two single span girders are positioned (lifted in)2. t=t0: Monolithic connection at B

BS+RV:

Bending moment and girder end rotation at B (per side) at t0:

Comparison: Bending moment at B on a single two-span beam:(«OC»: one casting)

Compatibility condition (relative rotation of girder ends at B remains constant after = t0):

At the intermediate support, a moment of approx. 80% of the two-span beam built in one casting «OC» develops due to creep.

Long-term effects

3

0 1,24 3

kB B

g l lEI EI

θ = θ =

3

0 0 0( ) 0, ( )24

kB Be B B

g lM t M tEI

= = θ = θ =

A B C

3

0 0( )24

kB B

g ltEI

θ = θ =gk (permanent loads and possibly deviation forces)

0(t ), ( ) 0B B B tθ = θ ∆θ =

0

( ) ( ) ( )B Be B BM t M M t M t=

= + ∆ = ∆

Bθ

20

B,1 8

B kOC

B

g lM

θ= − = −

θ


Influence of creep in system changes

The ratio of the moment at B to the moment of thesystem built in one casting «OC» for various pointsin time and creep coefficients:

As a general rule, in system changes, creep largely builds up the stress state of the system built in one casting σOC. The higher the creep coefficient, the closer it approximates the state of the system built in one casting («Einguss-System» in German). As an approximation one may use:

Long-term effects

( )( )0.6 0.8t A OC A

A

OC

S S S SSS

=∞ ≈ + −

Internal forces before system change (initial state)Internal forces of system built in one casting "OC"

ϕ ≈ 1 ϕ ≈ 2

A B C

3

0 0( )24

kB B

g ltEI

θ = θ =gk (permanent loads and possibly deviation forces)

0(t ), ( ) 0B B B tθ = θ ∆θ =

0

( ) ( ) ( )B Be B BM t M M t M t=

= + ∆ = ∆

Bθ


56 days 180 days 1 year 5 years

ϕ(t) 1.00 1.75 2.00 2.50

MB(t)/MB,Mono 0.56 0.73 0.77 0.83

Long-term effects

Influence of creep on restraint internal forces (caused by imposed deformations)Example 5a - three-span beam, time-independent («fast») support displacements s1, s2

Compatibility condition at time t = t0 :

Time-dependent compatibility condition (Trost method):

...ditto, inversely:

1s2 1 2( , )s s sθ

1 1 1( ) ( )AX t X X t= + ∆ 2 2 2( ) ( )AX t X X t= + ∆

1l

2s1 1 2( , )s s sθ

2l 3l

1 1 11 1 11 2 12 2 12

2 1 21 1 21 2 22 2 22

( ) ( ) (1 ) ( ) (1 ) 0( ) ( ) (1 ) ( ) (1 ) 0

A A

A A

t X X t X X tt X X t X X t

∆θ = θ ⋅ϕ + ∆ θ ⋅ + µϕ + θ ⋅ϕ + ∆ θ ⋅ + µϕ =∆θ = θ ⋅ϕ + ∆ θ ⋅ + µϕ + θ ⋅ϕ + ∆ θ ⋅ + µϕ =

11 11 2 12 1 11 11 12

2 21 2 22 2 22 21 22

A A s sA

A A s sA

X X XX X X

−θ + θ = θ θθ θ → = θ + θ = θ θθ θ

[ ]

[ ]

1

11 11 2 12 1 11 2 1211 11 12

2 21 22 21 21 2 22 1 21 2 22

2

1 1

2 2

( ) ( )( )1( ) 1( ) ( )

1

( )( ) 1

re

s

A As

sA A

s

A

A

X t X t X XX tX t

X t X t X X

X t XX t X

−

θϕ

∆ θ + ∆ θ = − θ + θϕ θ∆ θ θ + µϕ → = − ∆ θ θ θϕ + µϕ ∆ θ + ∆ θ = − θ + θ

+ µϕθ

ϕ∆ → = − ∆ + µϕ

( ) 11

sp. i iAX t X ϕ= − + µϕ

(analogous to relaxation function)

→ Time-independent restraint forces("fast imposed deformation") are reduced by creep (or relaxation) to 1/3...1/4 of the initial value


56 days 5 years

ϕ(t) 1.00 2.00

Xi(t)/XiA(t) 0.44 0.23

Long-term effects

Influence of creep on restraint internal forces (caused by imposed deformations)Example 5a - three-span beam, time-dependent («slow») support displacements s1, s2

Assumption: Settlement process (s1, s2) proportional to creep function:

Time-dependent compatibility condition (Trost method):

...ditto, inversely:

1s2 1 2( , )s s sθ

1 1 10

( ) ( )AX t X X t=

= + ∆

2 2 20

( ) ( )AX t X X t=

= + ∆

1l

2s1 1 2( , )s s sθ

2l 3l

1 1 11 2 12 1 ,

2 1 21 2 22 2 ,

( ) ( ) (1 ) ( ) (1 )

( ) ( ) (1 ) ( ) (1 )

s

s

t X t X t

t X t X t

∞∞

∞∞

ϕ∆θ = ∆ θ ⋅ + µϕ + ∆ θ ⋅ + µϕ = θ

ϕϕ

∆θ = ∆ θ ⋅ + µϕ + ∆ θ ⋅ + µϕ = θϕ

0, 0

0

0( , )( ) ( ) :0( , )

ii i i

i

st ts t s t s t tXt t ∞

∞

=ϕ ϕ= = ∞ = =

=ϕ = ∞ ϕ

11 11 2 12 1 ,1 ,1 11 12

2 21 22 2 ,1 21 2 22 2 ,

1 ,1,

2 2 ,

( ) ( )(1 ) ( )

( ) (1 )( ) ( )(1 )

( ) ( )( ) (1 ) (1

ss

ss

E eli iE el

E el

X t X tX tX tX t X t

XX t X t XX t X

−∞∞∞

∞∞∞

∞

∞ ∞

ϕ∆ θ + ∆ θ = θ

θϕ ϕ + µϕ ∆ θ θ → = ϕ ∆ θ θ θϕ + µϕ ∆ θ + ∆ θ = θϕ + µϕ

ϕ ϕ → = = ϕ + µϕ ϕ resp.

,,

)

isiE elX ∞θ

+ µϕ

value in elastic system subjected to without creep

:

→ Due to creep (or relaxation) time-dependent restraint forces("slow imposed deformation") reach only approx. 40% of the elastic (short-term) value


56 days 180 days 5 years

ϕ(t) 1.00 1.75 2.00

Xi(t)/XiE,el(t) 0.44 0.36 0.38

Long-term effects

Aspects not covered in the lecture:

Composite cross-sections of concrete and steel or precast concrete components and in-situ concrete→ Residual stresses or force redistributions due to shrinkage and creep of the concrete

(steel does neither creep nor shrink, prefabricated components creep less than in-situ concrete)→ Determination of the force redistributions from the compatibility condition (plane cross-sections remain plane)→ Consideration of creep due to time-dependent residual stresses with the Trost method

Effect of crack formation on creep behaviour→ In all previous slides, uncracked behaviour was assumed (results valid e.g. for girders fully prestressed under permanent loads)→ Crack formation and long-term effects influence one another→ Approximate calculation analogous to the non-cracked state with ficticious creep coefficient ϕ’:

- Determination of cracked elastic stiffness EIIIt=0 with Ec0 resp. EIIIt=∞ with Ec0 /(1+ϕ) (see Stahlbeton I)- Calculation with EIIIt=0 using the ficticious creep coefficient ϕ’= EIIIt=0 / EIIIt=∞−1.

Effect of creep on prestressed systems→ Prestress losses due to shrinkage, creep and relaxation of the prestressing steel see Stahlbeton II.→ Internal forces due to prestressing are to be taken into account when determining the creep-generating stresses. Treatment as anchor,

deviation and friction forces (prestressing on the load side) is advisable → Creep caused by sum of permanent loads and anchor and deviation forces due to prestressing.

→ For highly prestressed, deformation-sensitive systems, such as cantilever bridges during the construction stage (*), the long-term effects must be carefully investigated and upper/lower limit values must be used.

(*) large deformations due to dead weight (+) and prestressing (-), resulting deformation = difference, sensitive to assumptions made (there is no "safe side" when determining camber = «Überhöhung» in German)


Long-term effects

Summary • The term "long-term effects" covers shrinkage, creep and relaxation. Creep and relaxation of concrete are related phenomena.

• Due to the large variability of the material properties, the shrinkage and creep behaviour can only be determined approximately, even with complex calculations.

• All permanent actions (dead weight, superimposed loads, prestressing) cause creep.

• The stress history usually depends on the creep behaviour. The solution of creep problems therefore requires an iterative / step-by-step approach. For manual calculations, the Trost method (with an ageing coefficient of µ≈0.8 for stresses that do not act from the beginning) is appropriate.

• In statically indeterminate systems with uniform creep properties, the restraint forces due to creep change exclusively as a result of time-dependent prestress losses (RV due to prestressing is proportional to the prestressing force resp. the deviation forces).

• In statically indeterminate systems with non-uniform creep properties, the redundant variables change as a result of creep.

• After system changes, creep largely builds up the stress state of the system built in one casting. The more prone to creep the system components are, the closer it approximates the system built in one casting. For normal conditions (ϕ ≈ 2) approx. 80% of the latter is reached.

• Time-independent restraint forces ("fast imposed deformation") are reduced by creep(resp. relaxation) to 1/3...1/4 of the initial value. The reduction of the initial restraintforces are the larger, the more prone to creep the system components are.

• Time-dependent restraint forces ("slow imposed deformation") achieve as a result of creep (resp. relaxation) only approx. 40% of the elastic (short-term) value. The restraint forcesnever act in full-size and the more prone to creep the system parts are, the less they build up.

• Relaxation reduces the restraint forces, but not the deformations!

( ) 11i iAX t X ϕ

= − + µϕ

,( )(1 )i iE elX t X

∞

ϕ=

ϕ + µϕ


Documents

Specialization and additions to Stahlbeton I 4.1 Basics...• Avoidance through careful curing (prevention of significant water losses on the fresh concrete surface caused by high