Response Evaluation - Fundamentals of Earthquake Engineering - page 198

Failure criteria for concrete defi ned in stress space are used extensively in structural applications.
The non-linear constitutive relationships are only applicable within the failure envelope. Once the
failure criteria are satisfi ed, the material suffers from one of two modes of failure: crushing or cracking,
under compression or tension, respectively. In compression, the material loses its strength upon the
satisfaction of the given criterion. For tensile failure, several techniques may be adopted to model
cracking using FE methods. These fall under two categories: ' discrete ' and ' smeared ' cracking models.
The detailed description of these methods is available in Chen (1982) among others. It suffi ces here to
note that the discrete crack model is a better choice if there is prior information about the crack patterns
so that the mesh can be refi ned along crack paths. It is adequate for problems where aggregate interlock
and dowel action are signifi cant. Discrete crack models are the better choice if local behaviour, e.g.
local stresses and crack sizes, is more important than overall behaviour, as for example load-defl ection
curves. On the other hand, the smeared crack technique is better suited to the assessment of overall
structural response. The latter technique, when used with isoparametric elements, is versatile, effi cient
and economical. As a consequence, the smeared crack model is the most suitable choice for earthquake
engineering applications where the global behaviour is usually the focus of the analysis.
Multi-linear elastic-plastic models may be employed for RC and composite steel and concrete struc-
tures as further discussed in Section 4.5.3.1. These are generally phenomenological models, which also
take into account the presence of steel reinforcement. They may accommodate the stiffness degradation
caused by the onset of concrete cracking and steel yielding (e.g. Takeda et al ., 1970; Saiidi and Sozen,
1979 ; Ibarra et al ., 2005, among others). Such models employ generally unixial formulations.
Most of the models described above consider strain-stress relationships for concrete under compres-
sive loads with an envelope curve, which matches the material response obtained under increasing
loads. If the stress is decreased, an unloading curve will be traced. Increasing the stress forces the
material along a reloading branch of the material response curve. Figure 4.9 shows the most commonly
used uniaxial relationship for concrete under monotonic and cyclic loading, which is the model by
Mander et al . (1988). The latter can be utilized to simulate the behaviour of both confi ned or ' core '
and unconfi ned or 'cover' concrete in cross sections modelled by fi bres as illustrated in Section
4.5.2 .
A simplifi ed non-linear concrete model, which may simulate constant active confi nement, is imple-
mented in Zeus-NL (Elnashai et al ., 2003); this model is derived from Mander's formulation. A constant
confi ning pressure is assumed, taking into account the maximum transverse pressure generated by

30

f c

0.85 f'cc

25

Confined
Concrete

f cc

20

15

f co

10

Unconfined
Concrete

5

E c

ε cu

E sec

0

0 0.002 0.004 0.006 0.008 0.01

ε co ε cc ε c

Strain

Figure 4.9 Uniaxial models for concrete: Mander et al . (1988) ( left ) and constant confi nement Zeus-NL model
used in the assessment of the sample SPEAR frame ( right )