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75 7 Design 36 The delayed introduction of additional restraints is sometime conceived as an artifice to improve the behaviour of the structure under permanent loads, when the corresponding stress distribution in the original structural system is more favourable, if compared to the one that would be obtained, for the same loads, in the final system. In such cases, the introduction of the additional restraints, intended to improve the response to live loads and the final robustness of the structure by an increase of its statical redundancy, substantially reduces the original benefits, because of the significant creep induced stress redistributions altering the original response to permanent loads (e.g. fixed concrete arches provisionally built as three-hinged arches with a centre line corresponding to the funicular curve for dead loads). Different values of t 0 and t 1 must be considered and repeated solutions of Eq. (7.2-36) must be performed to obtain (t,t 0,t 1 )) from J(t,t'). The procedure for numerical solution indicated in subclause (a) shall be adopted. For further information reference is made to M. A. Chiorino A Rational Approach to the Analysis of Creep Structural Effects, ACI SP-227, 2005, pp Design aids. For a given creep prediction model, and for the corresponding compliance function J(t,t'), design aids can be provided for the evaluation of the related functions R(t,t') and(t,t 0,t 1 ) in terms of set of graphs of these three functions and of the aging coefficient (t,t') introduced in subclause (refer to CEB Bulletin 215 for the creep prediction model of CEB Model Code 1990; a few examples for the models indicated in subclause are given in fib Bulletin 52), or in terms of computational programs to be downloaded (see e.g. or inserted in computational softwares. For further information reference is made to M. Sassone, M. A. Chiorino Design Aids for the Evaluation of Creep Induced Structural Effects, ACI SP-227, 2005, pp (d) Multiple changes in the structural system In cases where the transition from the initial to the final structural system is obtained by means of several different restraint modifications applied at times t i t 0 + (i = 1,.,j), the redistribution effects consequent to every single change in the structural system may be superimposed in time. Therefore, the system of the stresses evolves for t > t 1 according to Eq. (7.2-38) (4 th theorem of aging linear viscoelasticity): S j j1 ( 0 i1 where: el,1 el, i t) S S ( t', t, t ) (7.2-37) i S el,i is correction to be applied, in the associated elastic problem, to the elastic solution S el,i in order to respect the geometrical conditions imposed by the additional restraints of structural system i+1, imagined as introduced before the loads Effective homogeneous concrete structures with additional steel structural elements If the main cause of heterogeneity is represented by the presence of steel structural elements that may be considered equivalent to redundant elastic restraints, while the concrete part of the structure may still be approximately regarded as an effective homogeneous structure of averaged rheological

117 7 Design 78 The forces within a stress field or strut-and-tie model can be calculated with help of equilibrium conditions. When developing a model it is advisable to roughly take compatibility of deformations into consideration. In a first approximation, directions and magnitudes of the forces of the model may be orientated at the corresponding linear elastic stress state. When applying stress fields or strut-and-tie models (Figure ) the following steps may be considered: the geometry of the D-region may be assumed and have a minimum length equal to the maximum width of spread; a free-body with a (first) strut-and-tie model may be sketched. In order to minimize the effects of redistribution of forces (with consequences for crack width in the SLS) the struts should as much as possible be oriented to the compressive stress trajectories in the uncracked state; forces of the model have to be calculated such that they represent an equilibrium system of internal forces and external loads; the cross-section of the struts (compressive fields) and ties (tensile fields) shall be determined or checked; geometry of nodes shall be checked and detailing of reinforcement developed; the model has to be refined if necessary; and nodes, struts and ties of the final model must comply with the detailing of the reinforcement. Figure : Exemplary basic elements for strut-and-tie modelling

129 7 Design Verification of structural safety (ULS) for non-static loading Fatigue design The given fatigue strength of concrete is valid for concrete tested under sealed conditions (see subclause ). The fatigue strength of steel is given both for a normal environment and for a marine environment Scope The following design rules apply for the entire service life of concrete structures. The rules for reinforcing and prestressing steel should be applied if more than 10 4 load repetitions are expected; low-cycle fatigue is not covered. The verification of the design principle (see subclause ) can be performed according to the three methods given in subclauses , and , with an increasing refinement. The models for the analysis of stresses in reinforced and prestressed concrete members under fatigue loading are treated in subclause as well as concrete stress gradients. Subclause deals with shear design and in a method for calculating the increased deflections under fatigue loading is given. The relevant combination of loads is treated in subclause Analysis of stresses in reinforced and prestressed members under fatigue loading Linear elastic models may generally be used, and reinforced concrete in tension is considered to be cracked. The ratio of moduli of elasticity for steel and concrete may be taken as = 10. In the case of prestressed members it should be verified if the relevant section is sensitive to cracking. This holds true if any combination of loads (see subclause ) causes tensile stresses at the concrete surface. In that case the stress ranges for reinforcing steel and prestressing steel should be calculated assuming the cracked state. The effect of differences in bond behaviour of prestressing and reinforcing steel has to be taken into account. Unless a more refined method is used, this can be done using a linear elastic model for stress calculation which fulfils the compatibility in strains and multiplying the stress in the reinforcing steel by the following factor:

196 In specific cases, e.g. in incremental launching with precast elements, a minimum compressive stress may be required. As a rule the limit state of decompression should be required, if cracking or reopening of cracks has to be avoided under a given load combination. In a beam the state of decompression is reached when the section under consideration is compressed and the extreme fibre concrete stress is equal to zero. The occurrence of longitudinal cracks may lead to a reduction in durability. In the absence of other measures (such as an increase of concrete cover) it is recommended to limit the compressive stress for exposure classes XD, XS, and XF (section subclause 4.7.2). However, no limitation in serviceability conditions is necessary for stresses under bearings and anchorages through mechanical devices (e.g. anchor plate of prestressing tendons). The limit of 0.6f ck (t) is not a sharply defined value. Consequently, in the corresponding verification the prestressing force may be represented by its mean value, and in transient situations where the magnitude of variable actions is small (especially at transfer of prestressing forces in beams) the quasi-permanent combination of actions may be replaced by the characteristic combination. On the other hand the prestressing force and concrete strength should be considered in the verification with their values at the time at which the maximum stresses are reached. Measures should be envisaged for deformations if the span/effective depth ratio exceeds 85% of the value given in Table If creep is likely to significantly affect the behaviour of the member considered (e.g. with regard to loss of prestress, deformation, validity of the structural analysis) an alternative measure would be to limit the stress to Tensile stresses in the concrete Depending on the limit state considered, various stress limitations may apply. The limit state of decompression is the most relevant. Stresses may be calculated on the basis of a homogeneous uncracked concrete section. The contribution of reinforcement to the area and the section modulus of the cross section may be taken into account Limit state of decompression The limit state of decompression is defined as the state where concrete stresses are below or equal to zero in all principal directions Compressive stresses in the concrete Excessive compressive stress in the concrete under service load may lead to longitudinal cracks and high and hardly predictable creep, with serious consequences to prestress losses. When such effects are likely to occur, measures should be taken to limit the stresses to an appropriate level. If the stress does not exceed 0.6f ck (t): under the characteristic combination, longitudinal cracking is unlikely to occur; under the quasi-permanent combination of actions, creep and the corresponding prestress losses can be predicted with adequate accuracy. If under the quasi-permanent combination of actions the stress exceeds 0.4f ck (t), non-linearity should be regarded for the assessment of creep deformation (see subclause (d)). fib Bulletin 66: Model Code 2010, Final draft Volume 2 157

216 if the concrete in the compression area is still in the elastic state. Alternatively, instead of interpolation, an equivalent stiffness deduced from (Eq ) can be used for direct simplified calculation of deflection: For calculating long-term deflections due to creep and shrinkage the following simplified procedure can be used: for for where ϕ is the creep coefficient (see subclause ); a g a ϕ is the instantaneous deflection due to quasi- permanent loads; is the creep deflection; a sh is the shrinkage deflection. The stresses σ s and σ sr for the interpolation coefficient ζ are calculated at the most unfavourable section which is usually the section subjected to the maximum bending moment. σ sr / σ s in Eq. (7.6-16) may be replaced by M r / M for flexure and N r / N for pure tension, where M r is the cracking moment and N r is the cracking force. M and N represent the moment and normal force for the load combination considered. For loads with a duration long enough to cause creep, the total deformation including creep is obtained by using an effective modulus of elasticity of concrete according to: where: (7.6-18) is the creep coefficient corresponding to the load and time interval. Shrinkage curvatures may be assessed by: where: 1/r cs is the curvature due to shrinkage; ε cs is the free shrinkage strain (see subclause ); S I (7.6-19) is the first moment of area of the reinforcement about the centroid of the section; second moment of area of the section; α e is the effective modular ratio =. S and I should be calculated for the uncracked and the fully cracked condition. The final curvature is assessed by Eq. (7.6-14). fib Bulletin 66: Model Code 2010, Final draft Volume 2 177 2b1af7f3a8