In assessing damage and reconstruction at Pompeii, the primary structural issue in is the response of the masonry walls, particularly the out-of-plane response. Inspection of walls at the Forum reveals many examples of the U-shaped scooped pattern damage that characterizes out-of-plane failure, but few instances of the diagonal cracking characteristic of in plane failure.
In out-of-plane response of masonry walls, there are three key phenomena that contribute to failure: 1) tension cracking of the masonry; 2) compression crushing of the masonry; and 3) instability due to slenderness effects. Unless the walls have high compressive axial loading, crushing is rarely significant; the behavior is typically dominated by cracking and instability [Yokel 1971, p. 51].
Walls at Pompeii typically carry light axial loads; timber roof and floor systems are common and building heights rarely exceed two stories, so modeling methods for the walls must account for tension cracking and slenderness effects, but need not consider crushing of the masonry. Toward that objective, the study is exploring a modelling method based on elastic solid elements connected by surface gap-interface elements, using large displacement analysis; the approach is called the block-interface model. The interface elements account for the lack of tension strength, and the large displacement analysis accounts for slenderness effects. This approach neglects non-linear behavior of the material in compression, and neglects the tension strength of the material, but it can be shown that the compression stresses remain in the elastic range, and analytic studies have shown that the tension strength of masonry typically has little influence on the out-of-plane behavior of a wall [Yokel 1971, p. 29].
This page presents selected preliminary results of studies aimed at developing and verifying the block-interface model to the out-of-plane behavior of unreinforced masonry walls, using the ABAQUS analysis program [HKS 1996].
|Unmagnified deformed shape for a top-restrained unreinforced masonry column subjected to self weight and lateral concentrated loads at mid-height.|
|Horizontal load factor vs. horizontal displacement at mid-height.|
The Abaqus 5.5 Input file block-2.inp is available.
|Magnified deformed shape for a top-restrained unreinforced masonry column subjected to self weight, a 1 psi surcharge and earthquake ground motion. Displacement magnification factor = 100|
Note the much greater degree of cracking and slipping in the upper blocks, this is because the friction and prestress forces are much smaller at the top.
The figure below shows the deformed shape of the model, well into the stages of non-linear behavior resulting from cracking and second order effects. Note the high curvature in the base region compared with upper part of the pier.
|The deformed shape of the block-interface model of the Mendola example. Note the concentration of curvature at the base of the pier. Displacement magnification = 10|
The figure below shows a detail of the pier base, using a higher magnification for the displacements to highlight the patterns of cracking. Note how the cracks increase in depth and width toward the bottom of the pier.
|A detailed view of cracking at the base as modelled by the block-interface approach. Displacement magnification = 30|
The figure below shows a comparison of the load deflection curves, with the sampled curve for the Mendola model shown in red and the curve for the block-interface model shown in yellow. Note the very close agreement of the two models. The most significant difference occurs at the peak of maximum load. The block-interface model effectively "chops off" this peak, where the slope of the curve abruptly changes from positive to negative. This is probably due to the fact that cracks propagate in discrete increments equal to the depth of a single element. When a crack propagates through the full depth there is an abrupt loss of stiffness. The model could come closer to the theoretical result by using a mesh which incorporates more elements through the depth, however the results obtained here are sufficiently accurate for most purposes.
|A comparison of load-deflection curves for the Mendola model and the block-interface model. The mendola curve is based on a few selected points from the published curve. [Mendola 1995, p. 1587]|
The input file col-2.inp is available.
|Next: Comparison with Experiment: The Yokel Studies|