Simplified and Advanced Analysis of Membrane Action of Concrete Slabs

For concrete under tension, the multi-directional fixed smeared crack model with linear tension softening after cracking was adopted (refer to Fig. 13). The main characteristics of the smeared crack model include:

* The total strain was decomposed into elastic and crack strains;

* The material was isotropic before cracking and became orthotropic upon the crack formation;

* The initiation of cracks was governed by a tension cut-off criterion and a threshold angle between two consecutive cracks. For successive initiation of the cracks, the following criteria must be satisfied simultaneously;

* The principal tensile stress exceeds the maximum stress condition; and

* The angle between the principal tensile stress and the existing crack exceeds the threshold angle of 60 degrees.

* A constant shear retention factor β = 0.2 was considered for the reduction of shear stiffness of concrete due to cracking.

The tensile strength was defined as ft = 0.21(f^sub ck^)2/3, in accordance with the lower bound value (5% fractile) specified in Eurocode 2, Part 1.1,22 For a given fracture energy G^sub f^, which is treated as a material property, the ultimate crack strain ε^sub cr,nn^ was determined from the shaded area in Fig. 13 as follows

... (17)

where h is the crack bandwidth that is dependant on the size of the elements and the integration scheme.

The reinforcement was modeled using a bilinear curve. The measured yield strength, ultimate strength, and ductility, as shown in Table 1, were adopted for each mesh reinforcement.

The Newton-Raphson iteration method was adopted in the analyses, incorporating the iteration-based adaptive loading with arc-length control. The line search algorithm for automatically scaling the incremental displacements, within the iteration process, was also included to improve the convergence rate and the efficiency of analyses.

Due to double symmetry of support and loading conditions, only a quarter of the concrete slabs were analyzed. A 6 x 6 mesh was adopted for the square slabs that gave an individual element size of 3.610 x 3.610 in. (91.7 x 91.7 mm). For the rectangular slabs, a 9 x 6 mesh was adopted, resulting in an individual element size of 3.717 x 3.610 in. (94.4 x 91.7 mm). A constant fracture energy of 0.343 lbf/in. (0.06 N/mm) was assumed23 for all slabs.

COMPARISON OF PREDICTIONS AND EXPERIMENTAL RESULTS

Figures 14 to 19 show the comparison between the test results, the simple method, and the advanced method. It can be seen in all the figures that the advanced method attempts to predict the full load-displacement history. In contrast, the simple method is based on rigid, perfectly plastic behavior (with change of geometry) and thus does not predict the elastic and elastic-plastic displacements. Therefore, the loaddisplacement from the simple method forms a failure envelope with the test results merging gradually toward the predicted response, when a good correlation between the analysis and test results is obtained.

For Slabs M1 and M2 (Fig. 14 and 15), where the mesh size was 0.095 in. (2.42 mm) in diameter at 2.0 in. (50.8 mm) spacing, the test results merged toward the simple predicted response presented by the authors. For comparison, the simple membrane design methods presented by Hayes9 and Sawczuk4 are also shown. It can be seen from Fig. 14 and 15 that both Hayes' and Sawczuk's method overpredict the slab's capacity, resulting in unconservative results. The comparison presented in Fig. 14 and 15 conforms to the conclusions previously drawn by Bailey15 that the methods derived by Hayes and Sawczuk lead to unconservative predictions due to the incorrect failure mode being assumed by Hayes and the incorrect assumption of full tensile capacity at the center of the slab by Sawczuk. Comparison with the remaining 12 tests presented in this paper showed the similar trend of both Hayes' and Sawczuck's method overpredicting the load capacity.