Page 115 - Handbook of Materials Failure Analysis

110 CHAPTER 5 Failure analysis of reinforced concrete structures

penetration and reinforcements corrosion. The results obtained by the proposed
formulation are compared with experimental and numerical responses available in
literature.

7.1 TWO-DIMENSIONAL REINFORCED CONCRETE FRAME
The structure studied in this application is a reinforced concrete frame experimen-
tally analyzed by Vecchio and Emara [47] and numerically tested by Guner [48]
and La Borderie et al. [49]. The frame is analyzed by the nonlinear FEM model
assuming that mechanical finite element is governed by pure Euler-Bernoulli
approach (B), without shear contributions, and pure full Timoshenko approach
(TSD), including shear contributions. The loads and frame geometry are presented
in Figure 5.8.
Two types of support conditions were considered: case I—frame including
the support beam and case II—simple clamped-clamped frame, as illustrated in
Figure 5.9. The responses determined by La Borderie et al. [49] were obtained
assuming only case II. Concerning the types of analyses, Guner [48] used the soft-
ware SAP 2000, in which the structure is modeled considering mixed mechanical
behavior, that is, elastic-linear along the one-dimensional finite elements and plastic
hinges at the appropriate member ends. Such hinges are positioned at the end nodes
of some special finite elements, such as the joint of a beam and column to simulate
the existence of rigid offsets. The numerical analysis performed by La Borderie
et al. [49] was based on one-dimensional finite elements considering their own dam-
age model in which the inelastic strains from the damage were accounted.
The following material parameters were adopted: Young modulus of concrete
23,674 MPa, concrete compression strength f ck ¼ 30MPa, coefficient of Poisson
0.2, yield strength of steel 418 MPa, Young modulus of steel 192,500 MPa, and plas-
tic modulus of steel 19,250 MPa. The parameters adopted for Mazars damage model
were the following: ε d0 ¼ 0:000085, A T ¼ 1:145, B T ¼ 10330, A C ¼ 1:117, and
B C ¼ 1189. The horizontal load on the frame top was applied into steps of 5 kN.
The equilibrium trajectories for cases I and II are presented in Figures 5.10 and
5.11, respectively. The use of clamped supports, as observed in case II, provided
higher stiffness to the structure, since there was no rotation in these support nodes.
The support beams adopted in case I did not introduce significant difference for
model B. However, for the TSD model, considerable changes were observed in terms
of displacements after concrete cracking and especially in terms of ultimate load.
The high capacity of internal forces redistribution may be the main reason for this
type of behavior.
Tables 5.2 and 5.3 present the values of the loading and horizontal displacement
for both reinforcement steel yielding in node 1 and frame failure. The column error
(%) was evaluated from a comparison between experimental and each numerical
result for both yielding and ultimate loads.
The TSD model, for case I, represented better the real structural behavior of the
frame in terms of ultimate load and reinforcement steel yielding. The differences