Application of Response Spectrum Analysis in Historical Buildings

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 application 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 design 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 analysis for historical buildings in general. The Lighthouse is a masonry structure built by the Egyptians in 1838.

94 Dynamics, Repairs & Restoration

2 Finite Element modeling of masonry

structures

The finite element method of analysis requires the selection of an appropriate 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 model 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 degrees 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

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 infigur e 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 earthquake 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)

The proper design spectrum was inserted in the analysis as a base excitation [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 massive monumental structures and the steel reinforced concrete ones. The participation factors for thefirs tte n 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 horizontalfirs t 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 response for two-dimensional structural systems. For three-dimensional sy-

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 method for either two or three dimensional systems, gives a good approximation of the maximum earthquake response of an elastic system, without requiring a complete time-history analysis [3],[1]. This is particularly important for design purposes. These three methods are well known because most of the design codes like the Eurocode 6 recommend their use. One should also observe that the difference between the SAV and SRSS method indicate in some sense the presence of densely placed eigenmodes and the clustering between them. In our application for dynamic analysis of the lighthouse the SRSS method was used. The evaluation of the modal damping is estimated from information in the bibliography. In the future it will be estimated from in situ measurements.

5 Discussion of Results

The calculation of the maximum strains and stresses using the superposition of the modal responses and the combination of them with the results of the static analysis (dead loads) provide us with the final stress and strain fields on the surfaces of the structure. Determination of failure areas is carried out using a permissible stress failure criterion where for each point of the structure the permissible stresses is defined [4],[5]. From the final results we conclude that stresses lie outside the permissible (safe) region in only small parts of the structure (local failure effects). Roughly speaking the failures were located in the first part (near the base) and the third part of the Lighthouse (where damages from a previous earthquake occured)(figure 4). It was also observed that the initial inclination of the Lighthouse affects both the static and the dynamic response of the structure and that the inclusion of the base (lower part) which is made of a lower quality material than the structural model, influences the dynamic response.

98 Dynamics, Repairs & Restoration

Acknowledgements

The financial support through the RE.CIT.E ROC-NORD EEC Pro- gramm is gratefully acknowledged.

References

[l] Eurocode No6, Common unified rules for masonry structures, report EUR 9888 EN, 1989.

[2] Polyakov, S.V ., Design of earhtquake resistant structure s^/[\i Publishers-Moscow, English translation, 1985.

[3] Naeim, Y.,The seismic design handbook,Va.n Nostrand Reinhold Publis- hers, New York, 1989.

[4] Hey man, J., Leaning towers, Meccamca, 1992, No27, 153-159.

[5] Salvatore di Pasquale, New trends in the analysis of masonry structures, Meccamca, 1992, No27 173-184.

Figure 1: The finite element models of Lighthouse.

Dynamics, Repairs & Restoration 99

Figure 2: First(1.07Hz), secondfl.l3Hz) and fourth(5.79Hz) mode.

Figure 3: First (0.932Hz) and seventh (9.02Hz) mode.

100 Dynamics, Repairs & Restoration

EARTHQUAKE IN DIRECTION X EARTHQUAKE N DIRECTION Y NUMBER PERIOD EFFECTIVE MASS PARTICIPATION EFFECTIVE MASS PARTICIPATION OF MODE (SEC) FACTOR (%) FACTOR (%) 1 0.93872 0.10963E+06 46 2946.1 2946.1 2 0.88635 1847.1 0.77 0.10978E+06 0.10978E+06 3 0.18176 709.36 0.29 7382.6 7382.6 4 0.17271 59195 24.84 60264 60264 5 0.16668 13067 5.48 18155 18155 6 0.16356 15393 6.46 3358 3358 7 0.10257 29862 12.53 36957 36957 8 0.71582E-01 2006.3 0.84 3239.8 3239.8 9 0.67226E-01 5323.9 2.23 1078.4 1078.4 10 0.58934E-01 1306.9 0.55 74.414 74.414

Table 1: Participation of modes - Model without base. EARTHQUAKE IN DIRECTION X EARTHQUAKE IN DIRECTION Y NUMBER PERIOD EFFECTIVE MASS PARTICIPATION EFFECTIVE MASS PARTICIPATION OF MODE (SEC) FACTOR (%) FACTOR (%) 1 1.07280 0.10887E+06 16.56 15356 2.27 2 1.00500 16164 2.46 0.11118E+06 16.42 3 0.20272 16918 2.57 53920 7.96 4 0.18870 43174 6.57 34736 5.13 5 0.18049 37028 5.63 14368 2.12 6 0.11879 9252.0 1.41 L_ 36517 5.39 7 0.11092 0.19248E+06 29.27 0.14409E+06 21.28 8 0.10028 j 0.23108E+06 35.14 0.22268E+06 32.89 9 0.84078E-01 1327.3 0.20 6295.9 0.93 10 0.73829E-01 1274.9 0.20 37880 5.60

Table 2: Participation of modes - Model with base.

4.7e+05 - 6.26+05 3.36+05 - 4.76+05

Figure 4: Maximum stesses violation. Sz for model without base and Syz for model with base.