Three methods for analyzing vaulted unreinforced masonry structures subjected to horizontal ground motion are presented and compared: real time thrust-line analysis, rigid body dynamics (RBD), and discrete element analysis. These methods are used to analyze three structures of increasing complexity: the single rocking block, the masonry arch, and the masonry arch on buttresses. Comparison of the results verifies the accuracy of the methods as well as the advantages of each method. The following methods of analysis will be applied to evaluate the safety of unreinforced masonry structures in seismic areas: (1) thrust line analysis using graphic statics, (2) rigid body dynamics, and (3) discrete element analysis. These methods will be introduced independently and then applied to different problems for evaluation and comparison. 1.1 Thrust line analysis using graphic statics Although the primary focus of this research is the dynamic analysis of unreinforced masonry, it is valuable to begin with an ‘equivalent static analysis’, meaning the analysis of structures subjected to constant horizontal acceleration (first order earthquake simulation) and constant vertical acceleration (gravity). Equivalent static analysis is essentially a stability analysis which is solely based on geometry and is independent of scale. Under the assumptions of limit analysis, individual blocks are not free to slide or crush, but are only free to separate, or hinge. Hinges form when the ‘thrust line’ can no longer be contained within the masonry and reaches the surface of the masonry. At this point, the masonry can no longer support the forces, and the structure is no longer in equilibrium without hinging. However, when hinging occurs, the collapse of the structure is not necessarily eminent. Take for example, the specific case of an arch on spreading supports studied by Ochsendorf (2006). Three hinges must form immediately upon support displacements, but the thrust line remains within the masonry and the structure therefore remains stable until four hinges are formed at the point of collapse. Although the static analysis of arched structures has been studied for several years, Block (2005) recently developed a useful tool (http://web.mit.edu/masonry/interactiveThrust/) which uses graphic statics to achieve a rapid first order assessment of the stability of various masonry structures. The real time graphic statics framework allows the effect of geometrical changes such as arch thickness, buttress width, etc., to be readily evaluated. For the current paper, this tool was extended to simulate structures on tilting surfaces, an effective way to conduct an ‘equivalent static analysis’ (http://web.mit.edu/masonry). 1.2 Rigid body dynamics While applying a constant horizontal acceleration to a structure is a valuable first step in simulating response to an earthquake, the resulting allowable horizontal ground accelerations are clearly conservative. If instead, the duration of the horizontal ground acceleration is reduced, then higher accelerations can be withstood. As a result, analytical rigid body dynamics solutions have been used to characterize this dynamic behavior of rigid body problems. The fundamental rigid body dynamics problem is that of the single rocking block under horizontal base motion, which was first investigated by Housner (1963). Housner uses the same assumption as Heyman, but must also assume a restitution coefficient to account for the dynamic behavior which causes impact during rocking. Spanos (1984) expanded on the research of Housner, and plots the stability of the rocking block subjected to harmonic accelerations of varying frequency and amplitude. The nature of the solution would logically be the same for multiple block ‘structures’ but the analytical solution becomes difficult to obtain. Additionally, several other researchers have noted the shortcomings of the limit analysis assumptions, and have found analytical solutions which allow for the possibility of more complex behavior such as sliding and bouncing (Augusti and Sinopoli 1992, Shenton and Jones 1991). Rigid body dynamics was also used by Clemente (1998) and Oppenheim (1992) to study the semi-circular arch. Both of these studies effectively extend the idea of the single rocking block to a more realistic structure which still only has one degree of freedom. Their analysis also does not include the possibility of sliding or more complicated arch hinging mechanisms. 1.3 Discrete Element Analysis Although the analytical solutions to the rigid body dynamics problems provide insight into the nature of the dynamics of masonry structures, their complexity demonstrates the need for computational tools which can correctly address the problem of rigid block dynamics. Initially, finite element programs were the computational tools of choice for most engineers, but they were 974 Structural Analysis of Historical Constructions M. DeJong and J.A. Ochsendorf optimal for problems of elasticity not stability. Advances in non-linear finite element modeling have made them more appropriate for masonry structures, but the more recent application of discrete element modeling inherently captures the discontinuous nature of masonry and allows for fully dynamic analysis with large displacements. Discrete element programs are particularly suitable for masonry structures because they allow the definition of individual blocks within the structure. Constitutive properties of blocks and contacts must be defined and input. These programs typically solve the equations of motion of each individual block in the system using a time stepping scheme. Contact forces on each block are assumed to be proportional to the inter-penetration between blocks, which is determined using the input contact stiffness. The out of balance force after each time step is applied to the equations of motion in the next time step, making the magnitude of the time step critical for accurate modeling. Bićanić et al. (2003) applies finite element and discrete element methods to model an actual masonry arch bridge which was subjected to vertical loading until failure. Although this is a problem involving static loading, the authors conclude that discontinuous modeling frameworks provide a viable alternative for evaluating the structural integrity of masonry, and for predicting the ultimate collapse load and failure mode. In this research, the Universal Discrete Element Code (UDEC) was used and evaluated in comparison with the previously defined analysis methods (Cundall 1980). 2 THE ROCKING BLOCK PROBLEM The first problem which will be addressed using the analysis methods already introduced is the single rocking block on a rigid half-space. As stated earlier, this problem has been extensively investigated by several researchers, but it serves as a good starting place for a comparison of the analysis methods in question. 2.1 Thrust-line Analysis Although a thrust line analysis using a real-time graphic statics tool could be performed to achieve the constant horizontal acceleration which would cause the collapse of a single rigid block, the solution is trivial: H B where g xg / = ∗ = λ λ & & (1) where B and H are defined in Fig. 1, and g is the acceleration of gravity. Any dynamic horizontal ground motion which does not exceed this minimum level of acceleration will cause no rocking response of the rigid block. Figure 1 : Definition of rocking block problem on a rigid half-space. 2.2 Rigid Body Dynamics The RBD solution of the rigid block subjected to harmonic horizontal ground motion is not quite as straightforward. Spanos and Koh (1984) present the full derivation and come to the following linearized equations of motion: H
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