Page 110 - Handbook of Materials Failure Analysis

4 Shear Strength Model 105

by the maximum value of the rotated main strain damaged portion toward the rein-
forcement direction. Mathematically, this condition is expressed as follows:

½
ε sw ¼ max ε 1 D sin αðފ (5.25)
in which ε sw represents the stirrup strain, α is the main tensile direction, and max
indicates the maximum operator.
The resultant shear force on each stirrup is assessed by σ sw A sw , where A sw cor-
responds to a single stirrup cross-section area and σ sw the stirrup normal stress. σ sw
value is obtained using the elastoplastic model over total stirrup strain, ε sw . Accord-
ing to the Ritter-M€ orsch’s truss model, the shear force resisted by stirrups is calcu-
lated for a range of width equal to the effective depth of section, d. Therefore, the
shear reinforcement contribution is written as follows:
V sw ¼ σ sw ρ bd (5.26)
sw
in which ρ sw is the transversal reinforcement ratio defined by A sw /(sb), s is the spac-
ing between stirrups, and b is the cross-section width.

4.4 SHEAR RESISTANCE IMPROVEMENTS INTO THE NONLINEAR
SOLUTION TECHNIQUE
The numerical formulation proposed in this study is nonlinear since displacements,
internal efforts, and external forces have a nonlinear dependency due to the repre-
sentation of the mechanical degradation as a function of the loading process. This
nonlinear problem is solved using the Newton-Raphson technique, which involves
prevision and correction steps. The stiffness matrix is evaluated considering the
actual damage state at each integration point and the loading process is transformed
into an incremental-iterative procedure. The nonequilibrated force vector is obtained
by the difference between the forces due to the applied load and the resistant forces,
which reflect the admissible equilibrium state considering damage and elastoplasti-
city criteria.
The shear forces due to the external load are mechanically equilibrated by con-
crete and reinforcements strength components, as previously presented. The total
shear mechanical resistance is provided by concrete, either intact or aggregate inter-
lock, reinforcements through dowel action, and reinforcements resistance itself.
Therefore, the shear mechanical resistance is expressed as follows:

V ¼ V c + V d + V sw (5.27)
The improvements proposed on the representation of shear strength mechanisms
reflect the admissible shear force on the equilibrium configuration. As a result, all
improvements proposed are accounted during the incremental-iterative nonlinear
process and during the determination of the nonequilibrated force vector.
The stiffness matrix is updated considering the actual mechanical damage state
of concrete and reinforcements. Therefore, the update process concerns the values
of longitudinal elasticity modulus of concrete and elastoplastic modulus of