Structural Effects of Creep

where the parameter

t¼150 1h þ ð1:2hÞ¹⁸i

sþ250ðdaysÞ

can be assumed equal to 500 for the most common environmental and structural situations.

Design Nominal Values

For application purposes, nominal estimations of the creepfinal coefficient can be assumed at the design stage, conventionally referred to some standard situations.

Values ofu_∞(t_o) are reported hereafter for a nominal age of loadingt_o= 14 days and for a relative humidity UR = 60%, considering three representative classes, respectively, of low, medium and high strength, combined with equivalent thick- ness values between small and medium.

s= 1.0 s= 2.0 s= 3.0

C20/25 3.30 2.98 2.83

C35/43 2.66 2.41 2.28

C50/60 2.29 2.07 1.97

Fornon-homogeneoussections and structures, also remaining within afirst-order behaviour, statically determined and undetermined cases have to be distinguished.

Instatically determined casesthe static regime is not influenced by the deformation behaviour of the material, and therefore stresses do not change due to creep. Deformations have viscous increments locally proportional to the relative creep coefficients.

In statically undetermined cases on the contrary the non-homogeneity of the material causes the mutual influence of static and geometric regimes with respect to the effects of creep: both stresses and deformations, starting from the initial elastic configuration, have variations according to the global constitutive law of the structural problem.

The second-order behaviour, in which the stress regime is influenced by the displacements anyway, takes the problem again as for the statically undetermined cases, with the necessity of a global viscoelastic analysis for homogeneous and statically determined structures too. An important case of this category of problems is the instability of columns under combined compression and bending, discussed in Chap.7.

(b) (d)

(a) (c)

Fig. 1.21 Structural effects of creep

1.3.1 Resolution of the Integral Equation

In order to show the computational aspects of the problem, one canfirst consider the algorithm that gives the response r=r(e) along the time, following a given deformation historye= e(t), based on a known creep functionv(t, t_o). In order to obtain this, Volterra’s integral equation is to be solved

eðtÞ ¼rovðt;t_oÞ þ Z^t

t₀

vðt;sÞdrðsÞ;

wherer= r(t) is the unknown function.

The solution is elaborated with an approximated numerical procedure which expresses the integral as a summation offinite contributions. The time interval (t−t_o) is then subdivided inkincrements (see Fig. 1.22), evaluating on one side the creep function for the chosen times:

vðtk;tiÞwith i¼0;1;. . .;k;

Fig. 1.22 Graphical representation of the numerical procedure

where tkcorresponds to the reading time t. Defining now through points, on the basis of a similar scansion of the curver= r(t), the functionv=v(r) represented in Fig.1.22c, the relevant equation, written fort=tk, can be set as

ekrovðtk;toÞ þ1 2

X^k

i¼1

vðtk;tiÞ þvðtk;ti1Þ

½ Dir

withek= e(tk) and having set

Dir¼riri1:

The area under the curvev=v(r) has therefore been expressed as summation of ktrapezoids.

Such equation can be progressively re-written for increasing times and therefore withk= 0, 1, 2,…. One will therefore have, with reference for example to the four intervals assumed in Fig.1.22and setting for brevityv_ki= v(t_k,t_i),

e0¼v00r0

e1¼v10r0þv11þv10

2 D¹r e2¼v₂₀r0þv21þv20

2 D1rþv22þv21

2 D2r e3¼v₃₀r0þv₃₁þv₃₀

2 D1rþv₃₂þv₃₁

2 D2rþv₃₃þv₃₂ 2 D3r e4¼v₄₀r0þv₄₁þv₄₀

2 D1rþv₄₂þv₄₁

2 D2rþv₄₃þv₄₂

2 D3rþv₄₄þv₄₃ 2 D4r: All together the equations form an algebraic triangular linear system that can be solved with a simple forward substitution done in parallel to the generation of the coefficients. The unknownsr_o,D1r,D2r,…are therefore progressively calculated and cumulated to give the responser₁,r₂,…,r_i,…,r_k.

In practice this procedure, automatically elaborated by electronic computation, is used to obtain the relaxation function settinge_o= e₁= = 1 and extending it to infinite time. The accuracy of the elaborations depends on the time subdivision done in the integration interval. Optimum results are obtained with a constant subdivision in logarithmic scale:

logðtiÞ logðti1Þ ¼loga;

and assuminga = 1.15 andD1t= t₁− t_o= 0.05 days. This is proposed the C.E.B.

Model Code that further suggests to extend the integration interval up to 10,000 days (30 years). Beyond such limit creep contributions are negligible.

1.3.2 General Method

In a statically undetermined non-homogeneous problem, where both functions r(t) ande(t) are unknown, the integral equation relative to the viscoelastic beha- viour of the material has to be supplemented by the law that expresses the structural behaviour.

Let us consider the simple example of a reinforced concrete column subjected to an axial force N constant in time. Let A_c and A_s be the cross-sectional areas of concrete and reinforcement and let E_c and E_s be the elastic moduli of the two materials. Stated first the deformation compatibility with ec=es =e, the initial balanced elastic solution is obtained immediately from

eo¼ N

EcAcð1þaeq_sÞ; rco¼E_ceo; withqs =As/Acandae= Es/Ec(see point2.1.1).

Following on, the migration of stresses from concrete to reinforcement steel has to fulfil the equilibrium relationship:

AcdrcþAsdrs¼0;

from which one obtains, being drs=E_sde, the differential equation drc

de ¼ q_sEs;

that, in the problem under consideration, supplements the Volterra’s integral equation.

Transposing the equilibrium equation tofinite differences, one has

Die¼ Dirc

q_sEs

;

and the procedure of numerical integration has to be modified as follows:

D¹rc¼2ðe1v10rcoÞ v11þv10

e1¼e0

D¹e¼ D1rc

q_sE_s e2¼e1þD¹e

D2rc¼2ðe2v20rcoÞ ðv21þv20ÞD1rc

v22þv21

D2e¼ D2rc

q_sEs

e3¼e2þD2e D³rc¼. . .:

This corresponds to evaluate, in every single time intervalDit, the creep effects of relaxation as if they were due to a contraction ei of constant value and to elastically compensate at time ti the consequent disequilibrium of stresses in the section with the additional contractionDie.

In the example presented, given that stressesrcare constant in the section and along the axis of the column, structural equilibrium can be imposed with one simple formula. But in general the equilibrium is expressed with integrals extended to the section and the structure. Consequent discretized numerical procedures, in addition to one of time integrations, lead to very onerous elaborations. From this onerous- ness comes the benefit of simplifying calculation, with respect to the general method presented above, with the approximated procedures reported below.

1.3.3 Algebraic Methods

Algebraic methods aim at the substitution of the integral in time, contemplated in the constitutive equation of viscoelasticity, with an algebraic equation, to avoid the discretized numerical procedure which follows step by step the history of the phenomenon. According to such methods, the actual continuous history of stress incrementsDr(s) =r(s)−rofollowing thefirst instantaneous load application can be substituted, in order to evaluate the creep effects at time t, with only one instantaneous increment Dr(t) =r(t)− ro (see Fig.1.23) applied from a given timet₁conveniently chosen in an intermediate position betweent_oandt:

eðt;toÞ rovðt;toÞ þDrðtÞvðt;t1Þ:

The Ageing Coefficient Method

A first method that gives very accurate results is the one called AAEMM (age-adjusted effective modulus method) orageing coefficient method. According to this method it is set as

Fig. 1.23 Approximate representation of stress history

vðt;t1Þ ¼ 1 Eo

1þvðt;toÞuðt;toÞ

½ ;

where the functionv(t, to) is called ageing coefficient and is obtained from vðt;toÞ ¼ Eo

E_orðt;t_oÞ 1 uðt;t_oÞ:

The evaluation of the relaxation function is therefore required. Practical appli- cations, which are generally limited to the analysis of thefinal response fort= ∞, can rely on appropriate tables ofr(t, t_o). The solution is obtained elaboratingfirst an instantaneous analysis of the structure at timet_o, evaluating the stresses relaxation at constant deformation with integrations on sections and structure with

DrðtÞ ¼eorovðt;toÞ vðt;t1Þ

and eventually redistributing the resulting unbalanced forces with a further incre- mental analysis.

The necessary double-structural analysis and the integrations in between bring still too onerous computations. For this reason the ageing coefficient method, profitably used in simple analyses of single section, is not normally used in the analysis of complex frames.

The Effective Modulus Method

For a quicker analysis of frames, the bigger approximations of the method called EMM (effective modulus method) have to be accepted, which let timet1coincide with the instantt_o, as if the total stressr(t) =ro+Dr(t) was applied with only one initial load step (t₁=t_o):

eðtÞ ¼rðtÞvðt;toÞ:

This implies only one instantaneous analysis of the structure where the elastic modulus of concrete has been simply adjusted with

EðtÞ ¼ Eo

1þuðt;toÞ:

Adopting this effective modulus the effects of creep are underestimated. The results have non-uniform approximations: bigger in configurations where creep effects highly influence the regime of redundancies, more limited in the opposite case. For statically determined cases or homogeneous configurations under static loads, even the effective modulus method gives exact results.

Eventually, in the case of successive iteration of permanent and instantaneous loads, a standard solution can be given to the problem, still in an approximated way.

For this more rigorous methods contemplate to follow the loading history, adding an instantaneous incremental analysis under accidental loads, to be cumulated to the previous viscoelastic response under permanent loads. The standard procedure on the contrary is limited to one only instantaneous analysis carried out with the weighted effective modulus:

EðtÞ ¼ E_o 1þcuðt;toÞ;

wherecdepends on the ratio between permanent loads and total loads. Acceptable results on an application level are obtained assuming forcthe square of this ratio.

Technical Method

With an approximation that overestimates creep effects, certain technical solutions assumet1=t, evaluating the relaxation with

DrðtÞ ¼ rouðt;toÞ;

that is, on the basis on a constant stress equal to the initial one. In this way for example the tension losses due to creep in pre-stressing cables are evaluated (see Sect.10.1.3).

To conclude it is to be noted that, as it can be deduced from linear analyses of serviceability states and nonlinear analyses taken to the failure limit, in the domain of second-order behaviour with displacements that are no more negligible, creep plays a determining role with respect to the resistance of the structure. This is for example the case of the already mentioned instability of slender columns under combined compression and bending actions.