Appendix Nonlinear analysis A Notation See also and
(1/r)m Average curvature at the section considered
(1/r)cr Curvature calculated on the basis of a cracked section
Myd Moment which produces the stress fyd in the reinforcement
Myk Moment which produces the stress fyk in the reinforcement
Coefficient which takes account of the bond properties of the reinforcement
"2 Coefficient which takes account of the nature and duration of loading
(c Strain at the extreme compression fibre, calculated ignoring tension stiffening esm Average steel strain, calculated taking account of tension stiffening esmr Average steel strain calculated on the basis of an uncracked section under the cracking load
£sy Yield strain of the reinforcement
£sym Strain corresponding to Bs = fyk (with fyk = fym) allowing for tension stiffening
Bs Steel stress calculated on the basis of a cracked section under the loading considered osr Steel stress, calculated on the basis of a cracked section under the cracking load
A2.1 General
P(1) Nonlinear methods of analysis may be used for both the serviceability and ultimate limit states, provided that the method satisfies equilibrium and compatibility.
P(2) At the ultimate limit state, the ability of local critical sections to withstand any inelastic deformations implied by the analysis shall be checked, taking appropriate account of uncertainties.
P(3) The deformations, and hence the distribution of internal forces and moments within the structure, should be calculated on the basis of the mean values of the material properties (such as Ecm, fctm etc.). The design values of the properties shall, however, be assumed at the critical zones where the ultimate resistance shall be calculated on the basis of Section 4.3.1.
P(4) For structures dominantly subjected to static loads, the effects of previous applications of loading may generally be ignored, and a monotonic increase of the intensity of the actions may be assumed.
A2.2 Refined approach for linear members subjected to bending with or without axial force
(1) Linear elements may be analysed by numerical methods which take, as their starting point, a design momentcurvature relationship combined with the assumption that, on average, plane sections remain plane.
As a simplification, the curvature may be derived from the relationship:
(cm is the average curvature at the section considered.
is the average steel strain calculated taking account of tension stiffening is the strain at the most compressed fibre (negative for compression) calculated ignoring tension stiffening.
(2) The stressstrain relationships for concrete and steel should be as given in Sections 4.2.1, 4.2.2 and 4.2.3.
(3) The contribution of the concrete in tension between cracks (tension stiffening) may be taken into account by the use of an effective average stressstrain curve for steel in cracked concrete. This may be obtained from the equation below:
sm where:
"2
is the average steel strain allowing for tension stiffening is the steel strain calculated on the basis of an uncracked section under the cracking load is the steel stress calculated on the basis of a cracked section under the cracking load is the steel stress calculated on the basis of a cracked section under the loading considered.
is a coefficient which takes account of the bond properties of the reinforcement ("i = 1 for deformed bars and 0.5 for plain bars)
is a coefficient which takes account of the duration and nature of the loading. ("2 = 1 for short term loading and 0.5 for long term or frequently repeated loading).
The relationship is valid between the cracking load, under which the maximum tensile stress in the concrete reaches fctm (see 3.1.2.3) and the load under which the reinforcement reaches yield. Figure A2.1 illustrates the relationship.
(4) Beyond the point corresponding to attainment of the design ultimate yield of the reinforcement (point F' in Figure A2.1), the section may be assumed to act as a plastic hinge carrying a constant moment independent of curvature or rotation until a limiting plastic rotation, given in Figure A2.2, is reached. This approach applies when the increase of moment beyond F' is negligible. The effects of transverse steel are neglected. The allowable plastic rotations given in Figure A2.2 take account of model uncertainty.
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