The response of masonry walls to out-of-plane seismic excitation is one of the most complex and ill-understood areas of seismic analysis. [Paulay 1992, p. 623]
The following discussion reviews and categorizes important studies in the literature. The categories divide into experimental and analytic studies, further divided into static and dynamic, and one-way and two-way span conditions.
Note that unlike other parts of the presentation which are aimed at readers with basic working knowledge of structures and physics, the following discussion is aimed at readers with significant knowledge of structural engineering and seismic design.
The most extensive dynamic studies were performed by as part of an extensive investigation aimed at developing standards for the renovation of unreinforced masonry buildings in Los Angeles, particularly with respect to acceptable height to thickness ratios for walls [ABK 1981, Kariotis 1986] . The study examined the one-way behavior of eight specimens of varying construction and geometry under a range of gravity loads and several dynamic motions at top and bottom, simulating the input motion from the ground or from a diaphragm anchorage. The study concluded that the primary concern is "dynamic stability"--the equilibrium of the cracked wall under the influence of applied loads, self weight, and inertial loads-- rather than material stress levels. The key parameters identified for predicting failure were the wall slenderness, the ratio of applied gravity load to wall self weight, and the peak input velocities at the base and top of the wall.
Another important dynamic experimental study of out-of-plane behavior was recently published by Lam [1995]. The study focused on a one-way cantilever condition with the objective of comparing various analytic approaches with experimental results. The subject of the experiment was a 1 m tall wall subjected to the El Centro ground motion and its behavior was compared with analytic results obtained from non-linear finite element analysis of a stick model, the response spectrum method, and the "equal energy method," which assumes that the kinetic energy developed in linear and non-linear systems is equal when subjected to the same ground motion [Lam 1995, p. 186]. The study is noteworthy in the literature for including a comparison of dynamic testing and analytic results. One of the important pieces of data to emerge from the Lam study was the damping ratio for the masonry wall, which fell consistently within the range of 2.6% to 3.4% [Lam 1995, p. 181]. The conclusions concerning the comparison of analytic methods are discussed in the following section concerning analytic studies.
One of the simplest methods for dynamic analysis of one-way spans was published by Priestley in 1985 [Priestley 1985], with corrections in a subsequently published discussion [Priestley 1986], and final publication in 1992 [Paulay 1992 pp. 623-636]. The method uses first principles and approximating assumptions to determine the non-linear load-deflection curve for a masonry wall subjected to out-of-plane loading; the area under the load deflection curve determines the maximum energy that the wall can absorb up to failure. Using the "equal energy" approach, the method uses the maximum energy of the non-linear model to determine an acceleration level in a linear model that corresponds to the same energy level. If analysis for a particular ground motion or response spectrum shows that this energy level is exceeded in the linear model, then it is assumed that the non-linear model would fail as well.
The development of Priestley's method is based on the free body diagram shown below.
A free body diagram of an unreinforced masonry wall supported top and bottom, subjected to out-of-plane loading. The partial free body diagram shows that the horizontal forces and reactions produce a counterclockwise disturbing moment, while the vertical forces produce a clockwise restoring moment. The system becomes unstable when the restoring moment goes to zero. |
The partial free body diagram shows that the horizontal forces and reactions create a counterclockwise disturbing moment, while the vertical forces produce a clockwise restoring moment. Combining this equilibrium condition with the moment curvature relation for a pre-compressed cross section composed of a no-tension material, the method derives the non-linear relation between the magnitude of the horizontal pressure and the horizontal displacement at mid-height.
There are two important assumptions in Priestley's analysis. First is the assumption that the top and bottom of the wall are simply connected so that the vertical forces at the end always act at the centerline of the section. This assumption is unrealistic, but conservative. For most common construction conditions, the vertical resultants at the ends of the wall will act closer to the unloaded face, as bottom portion of the wall rocks on its base and the top portion rotates; thus, the restoring moment will actually be larger than predicted by Priestley's method. Second, is the fundamental assumption of the equal energy approach. There has been little study to confirm the accuracy of this approach [Lam 1995, p. 183], and Priestley says only "It is suggested that an estimate of the equivalent linear elastic response acceleration ... can be found by the equal-energy approach", without further justification [Paulay 1992, p. 630].
Lam's 1995 study of a one-way cantilever wall compared the predictions of the equal-energy approach with experimental results, and found that the equal-energy overestimated the acceleration required to fail the wall by a factor of 3.5 when the frequency of the equivalent model was calculated using uncracked properties (as Priestley's example does), and by a factor of two when the frequency was based on the cracked properties [Lam 1995 , p. 189]. Lam's study is too limited to generalize its conclusions. If the equal-energy method is inherently unconservative, as Lam's results suggest, it is possible that this effect may be offset by the Priestley's conservative assumption concerning the location of the vertical resultants at the wall ends. Further study is required to assess the applicability of the equal-energy approach to this problem.
Other analytic studies have focussed on static loading. Mendola [1995] recently published an approach for analyzing the response of one-way cantilevers to lateral loading, using first principles to account for large displacements and material with no tension strength. Although the method was developed to investigate seismic behavior of masonry structures, the analysis was limited to static loading of a cantilever. The paper did not address issues involved in extending the approach to dynamic loading of more complex configurations.
Considering two-way behavior, a number of studies have applied the yield line method, originally developed for reinforced concrete slabs, to unreinforced wall panels. It has been observed that the crack patterns in experimental specimens are very similar to patterns of yielding predicted by yield line theory, and that strength values predicted by applying yield line theory to unreinforced panels give reasonable agreement with experiment for many span and load conditions [Hendry 1981, p. 126].
It is widely recognized, however, that there is no theoretical justification for applying yield line theory to a brittle material, and it has been shown that the failure loads predicted by yield line theory always exceed actual values [Sinha 1978, p. 81]. Sinha [1978] proposed a "fracture line" method, which gives closer correlation with experimental results than yield line theory. The method accounts for both the brittle nature of masonry and the orthotropic behavior of brickwork, which for conventional running bond tends to be stronger and stiffer for horizontal spanning action, due to the interlocking action of the bricks. Sinha's method does not account for the effects of axial compression in a wall.
Rots [1991, pp. 49-50] identifies three basic approaches to modelling the mechanical properties of masonry:
The methods are listed in order of increasing abstraction. Representing joints as continuum elements accounts for the localized stresses and bonding between bricks and mortar resulting from their different mechanical properties. The second approach represents masonry as continuum elements and joints as gap-interface elements; it models joint behavior less accurately, but can reasonably predict the behavior of an entire panel. The final approach does not make a distinction between bricks and joints, treating the masonry assembly an anisotropic continuum; it is most appropriate for looking at large scale structures.
Since the first method focusses on highly localized phenomena, it has not been widely applied to the study of structural systems or members. The second approach has been applied by several researchers in studying in-plane behavior of masonry wall panels [Ali 1988, Gambarotta 1994, Lotfi 1994, Page 1978, Rots 1991] and the smeared approach has been used to model small masonry buildings, accounting only for in-plane effects [Benedetti 1984, Tanrikulu 1992].
A detail at the north gate of the Macellum. Note the hole in the brick masonry, which reveals the concrete core of the wall. The variety of stone and blocks that can be seen on the surface of the wall serve as permanent formwork for the concrete core. |
In considering the three fundamental approaches identified by Rots, it is clear that the approach of modelling joints as continuum elements is too detailed for the Pompeii study, which is concerned with overall patterns of failure rather than local stresses at the interface of block and mortar. The smeared crack approach is appropriate from the perspective of scale, however, available implementations of the model, such as that available with the ABAQUS program, are valid only for monotonic loading, and so cannot be used in the dynamic time history analysis that will probably be necessary to assess seismic response of the non-linear masonry. Modelling joints as discontinuum elements is somewhat questionable, because the walls are constructed with a continuous core withing a shell of blocks, rather than discrete blocks.
Taking these factors into account, the discontinuum model seems most appropriate. As noted above, this approach has been used by several researchers to investigate in-plane effects; its application to study out-of-plane effects, particularly for a two-way span condition, will provide useful knowledge not only about Pompeii, but about the general analysis of out-of-plane behavior of unreinforced masonry.
The application of the discontinuum approach will be called the "block-interface model"; preliminary application and verification studies using the model are discussed in the following documents.
Next: Preliminary ABAQUS Studies |