Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 Application of response spectrum analysis in historical buildings M.E. Stavroulaki, B. Leftheris Institute of Applied Mechanics, Department of Engineering Greece Abstract The basic concepts and assumptions used in the response spectrum analysis method is reviewed in this paper with respect to its application in historical buildings. More specifically, the methodology of modeling and the applica- tion of the response spectrum analysis is described and discussed through our work with the Lighthouse at the Venetian Harbor of Chania. 1 Introduction The main cause of damage in building structures during an earthquake is usually their response to ground induced motions. In order to evaluate the behaviour of the structure for this type of loading condition, the princip- les of structural dynamics must be applied to determine the stresses and deflections generated in the structure. The dynamic characteristics of the building is established by its natu- ral frequencies, modes and damping: the analysis is based on linear-elastic behaviour of materials and the ground input motion is a smoothed de- sign spectrum in order to calculate the maximum values of the structural response. In this work we describe the application of response spectrum analysis to the Lighthouse at the Venetian Harbour of Chania. We use the example, however, to discuss the requirements of response spectrum analy- sis for historical buildings in general. The Lighthouse is a masonry structure built by the Egyptians in 1838. Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 94 Dynamics, Repairs & Restoration 2 Finite Element modeling of masonry structures The finite element method of analysis requires the selection of an appro- priate model that would sufficiently represent the real structure. We chose finite elements, with the discretization of the structure that permit as much structural and architectural details as possible. The geometry of the Lighthouse consists of two parts: a) the base and b) the cylindrical tower with height of about seven (7) meters and 18 meters respectively. The thick- ness of the wall is about 0.8 meters, with small differences from place to place, including two (2) windows and a door. Appropriate modifications were done in the initial geometric CAD mo- del in order to reduce its complexity and enhance the numerical stability of the subsequent steps in the finite element structural analysis task. In this step parts of the structure which do not contribute to its structural integrity (i.e. architectural details) and parts of the structure which are not critical for the strength of the structure have been deleted from the model. For this structure it was necessary to apply the general three-dimensional finite element procedure. Tetrahedral solid elements were selected for the finite element modeling with eight corners, eight nodes and three translational de- grees of freedom for each node. Using linear elastic material, the evaluation of damage, the residual strength and the vibrational characteristics of the structure was done by means of in situ tests. In order to estimate the cur- rent strength of the Lighthouse we included the existing initial slope of the third part, which was probably caused by an earthquake in the beginning of the century. We have analysed two models: one with and one without the base of the Lighthouse. The final model with the base consists of 4015 elements and a system of 19083 equations. In parallel the model without consist of 1604 elements and a system of 7692 equations. The two finite element models are shown in figure 1. 3 Response Spectrum Analysis for massive masonry buildings Response spectrum analysis is widely used by civil engineers to compute the maximum expected response of a structure to complicated time history excitations such as the ground motions which occur in earthquakes. The method is based on the calculation of the vibration modes of the structure. The eigenvectors and eigenvalues produced by the real eigenvalue analysis are used to generate modal coordinates for further dynamic analysis by the modal superposition method. For a given seismic excitation we calculate Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 Dynamics, Repairs & Restoration 95 the peak absolute response of each mode and the participation factor. The complete solution for the system is obtained by superimposing the indepen- dent modal solutions. In order to evaluate the dynamic characteristics of the structural model the eigenproblen was solved and the eigenvectors and the eigenvalues were calculated. A variety of eigensystem solution methods were tested. Their numerical efficiency depends on the matrix topology, the order of system matrices and the number of eigenvalues and eigenvectors to be extracted. In our application, the Lanczos method was the appropriate solution tech- nique. It is considered to be one of the best available methods for use with large, sparse, symmetric matrices. The calculated eigenmodes for the two models are shown in figure 2,3. A comparison with the in situ measurements indicate the accuracy of the method we chose. For the dynamic structural analysis of the Lighthouse a design earth- quake spectrum was used which corresponds to the statistically expected earthquake according to the Aseismic Design Specifications. The elastic response spectrum for a single component of a single earthquake record cannot be used because the expected maximum response for the structure is critical and second because it is necessary to include safety requirements. The design spectrum for a given region has been used since it is constructed from various earthquakes tests, various distances and spectral accelerations, and considers the ability of the structure to develop non- critical plastifi- cation effects. Note that the latter point is critical for masonry structures which have an enormous ability to absorb kinetic energy internally. Using the design response spectra the following factors have been included in the analysis: • The seismic zone coefficient which depends on the order of expectation of damages • The importance factor which depends on the usage and the value of the building (like historical value) • The smoothed response spectrum * The soil profile coefficient which takes into account the soil conditions at the site and the relationship between the soil period arid the period of the structure (soil- structure interaction) The damping coefficient, which depends on the structural system ( usually considerable for masonries) The seismic behaviour coefficient which decreases the loading due to the nonelastic behaviour of the structure. * The coefficient which depends on the type of the foundation ( with or without underground floors) Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 96 Dynamics, Repairs & Restoration The proper design spectrum was inserted in the analysis as a base excita- tion [1][2]. Following the completion of the run the calculated participation factors give the measure of significance of each eigenvector in the overall response of the structure. These coefficients show the importance of the eigenmodes in our system. This is the principal difference between the mas- sive monumental structures and the steel reinforced concrete ones. The participation factors for the first ten eigenmodes of the two models ( with and without the base) are given in tables 1,2. The results show different participation factors for the two basic direc- tions concluding that the final response depends on the seismic excitation direction. Thus modes with height participation factors have significant par- ticipation on the vibrational energy of the whole structure and they should be included in the response spectrum analysis. Another important finding for masonry stiff massive structures, is that the design response spectrum according to the Aseismic Design Specificati- ons overestimates the earthquake induced loading for small eigenvalues by considering a horizontal first branch in the design spectrum, instead of the ascending real one. In fact most of the significant eigenvalues lie in this part of the spectrum. 4 Superposition of modal responses Using the response spectrum method for multiple degrees of freedom (MDOF) systems, the maximum modal response is obtained for each mode included in the analysis. These maximum response values cannot possibly occur at the same time; therefore, a means must be found to combine the modal maximum in such a way as to approximate the maximum total response. One way to do this superposition is the use of the sum of the absolute values (SAY) of the modal responses. This combination can be expressed as I. (i) Since this combination assumes that the maxima occur at the same time and that they also have the same sign, it produces an upper-bound estimate for the responses, which is a conservative and not economical assumption. A more reasonable estimate, based on probability theory, can be obtained by using the square-root-of-the- sum-of-the-squares (SRSS) method, which is expressed as N This method has been shown to give a good approximation of the re- sponse for two-dimensional structural systems. For three-dimensional sy- Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 Dynamics, Repairs & Restoration 97 stems, it has been shown that the complete-quadratic combination (CQC) method may offer a significant improvement in estimating the response of certain structural systems. This combination is expressed as AT TV T where the constant modal damping LJ „• r (4) Using the SRSS method for two-dimensional systems and the CQC me- thod for either two or three dimensional systems, gives a good approxi- mation of the maximum earthquake response of an elastic system, without requiring a complete time-history analysis [3],[1]. This is particularly im- portant for design purposes.