Page 103 - Handbook of Materials Failure Analysis

98 CHAPTER 5 Failure analysis of reinforced concrete structures

in which E and G are longitudinal and transversal elasticity modules, respectively,
and ε and γ are normal and distortional strains, respectively.

2.2 PLASTICITY MODEL FOR STEEL

Steel, as most part of ductile materials, presents elastic mechanical behavior until
reaching its yield stress. After this stress level, there are some movements in the
internal crystals of the material, which give it a new strength capacity. In this phase,
named hardening, there is loss of stiffness, but the material still has strength capacity
until its failure limit. The models based on elastoplasticity theory are appropriate to
describe such mechanical behavior. Thus, the model chosen to describe the mechan-
ical behavior of reinforcements’ steel is composed by an elastoplastic constitutive
law with positive isotropic hardening. The criterion used to verify the elastoplastic
steel behavior is given by:
f ¼ σ s σ sy + Kψ (5.8)

in which σ s indicates the steel reinforcements’ layer stress, σ sy is the steel yielding
stress, K is the hardening plastic modulus, and ψ is the equivalent plastic strain
measurement.
The stress over each reinforcement layer is determined as follows:
f 0 ! σ ¼ Eε
f > 0 ! σ ¼ E t ε (5.9)
EK
in which E t is the tangent elastoplastic modulus, which is given by: E t ¼ = .
ð E + KÞ
3 GEOMETRIC NONLINEARITY

Figure 5.2 illustrates the initial and final configurations for a given point P, which
belong to a solid, after loading action. The horizontal and vertical displacements
of such point are defined by:
u p x, yÞ ¼ uxðÞ y sin θðÞ
ð
(5.10)
v p x, yð Þ ¼ vxðÞ y + ycos θðÞ
0
Considering a second-order approximation for displacements, where sin θðÞ ¼ v xðÞ
0 2
x
v ðÞ
and cos θðÞ ¼ 1 , Equation 5.10 is rewritten as follows:
2
0
u p x, yÞ ¼ uxðÞ yv xðÞ
ð
0 2 !
v xðÞ (5.11)
v p x, yð Þ ¼ vxðÞ y
2
in which u and v correspond, respectively, to horizontal and vertical displacement
fields for any point along the body.
Considering that geometric nonlinearity second-order terms are given by Green
strain measurement, the normal and distortional strain fields, ε xx and γ xy , respec-
tively, are written as follows: