Feb 6, 1992 - response of the firehouse of Gilroy during the Loma Prieta Earthquake are available. The firehouse survived the earthquake with little damage.
EERf's Forty-Foutth
SEISMIC
San Francisco,
Annual Meeting
EVALUATION OF UNREINFORCED MASONRY '''ITH FLEXIBLE DIAPHRAGMS Arturo
Tena
California
STRUCTURES
Colunga'
ABSTRACf A discr ete linear-elastic, multi-degree-of-freedom (MDOF) dynamic model developed for the dynamic analysis of unreinforced masonry (URM) structures with flexible diaphragms is presented. The discrete MDOF dynamic model represents the dynamic response of the structure in a given direction by a reduced number of discrete masses associated to translational degrees of freedom acting in that direction. The discrete model considers the f1exibility of the diaphragms and the rotations of the waJls, which are included into the global translational degrees of freedom through static condensation. Soil-structure interaction effects can be easily incorporated in the method. The applicability of the method in the study of the dynamic response of a URM structure with flexible diaphragms is presented. The structure, an old firehouse located at Gilroy, California, has been instrumented by California Strong Motion Instrumentation Program (CSMIP). Instrumented record s of the dynamic response of the firehouse of Gilroy during the Loma Prieta Earthquake are available. The firehouse survived the earthquake with little damage. The discrete dynamic model presented in this work has been able to reproduce well the recorded dynamic response of the firehouse of Gilroy during the Loma Prieta Earthquake.
II\TTRODUCTION Some attempts have been made to study the applicability of conventional analytical methods to predict the behavior of unreinforced masonry (URM) structures during earthquakes. The use of conventional special-purpose structural analysis programs such as ETABS (Ref. 13) have been found adequate to model buildings with essentially solid structural walls and rigid floor systems. More specialized structural analysis programs (COMBAT, Ref. 3) have been recently used to study masonry structures. These programs are attractive because they permit one to model panel s with openings, different foundation support conditions, as well as in-plane f100r f1exibility. However, this type of approach is restricted to static and response spectra elastic seismic analyses since both ET ABS and COMBAT neither perform time-step integration, nor consider geometric and material nonlinearities. In addition, both ETABS and COMBAT are unable to modeI orthotropic materials. 1
Graduat e Research
Universit y o] fIIinois
February 6-8, 1992
Assisrant,
al Urbana=Cham
De part ment of Civil Engineering, paign,
Urbana,
fIIinois,
Thursday, page 17
61801
U.S.A.
Student Paper
San Francisco,
EER/'s Forty-Fourth Annual Meeting
California
Little attention has been paid to reproduce the dynamic response of unreinforced masonry structures with flexible diaphragms. Surprisingly as it may seem, the state-of'-the -art in analysis of unreinforced masonry buildings does not go beyond single story buildings. LPMJI program (Ref. 5), as developed in recent years, is perhaps one of the most advanced programs since it models diaphragm flexibility. However, because wall rotations are not included since walls are modeled as shear springs, deflected shapes of multi-story walls cannot be represented. Excessive constraints at each floor level will result in overestimates of natural frequency. This work discusses a different approach to predict, study and evaluate the seismic dynamic behavior of unreinforced masonry buildings with flexible diaphragms. A discrete linear-elastic, multi-degree-of-freedom dynamic model is proposed to study response of low-rise URM buildings where elastic response may be expected during an earthquake. The discrete model considers the f1exibility of the diaphragms and the rotations of the walls, which are included into the global lateral stiffness of the wall elements through static condensation. Soil-structure interaction effects can be easily incorporated in the method. The resulting discrete model constitutes a powerful tool in the dynamic analysis of structures with flexible diaphragms considering soil-structure interaction effects. The discrete dynamic model is easy to implement in cornputer programs and is fast lO run on computers. Preliminary studies suggest that the discrete dynamic model presented herein is capable to represent the dynamic response of URM structures with flexible diaphragms well. DISCRETE LINEAR-ELASTIC, MULTI-DEGREE-OF-FREEDOM DYNAMIC MODEL
(MDOF)
The ability to study dynamic response of complicated structures with simplified methods is desirable. The method proposed herein fits into this category. The key assumption is to assume that the URM structures remain elastic during an earthquake. This hyporhesis seems reasonable for low-rise URM structures but may not necessarily apply for others. The method considers both the effects of flexibility of the diaphragms and the rotations of the walls in the overall response of the system. The discrete MDOF dynamic model can be visualized as an equivalent system of condensed beams (representing the perforated cantilever URM walls) linked by elastic springs (representing the flexible f100r systems). Response is measured by the transJationaJ degrees of freedom of these elements. Masses are lumped at the dynamic degrees of freedom, as illustrated in Fig. 1. Mass Discretization Masses are lumped at the intersection of the centroidal axes of walls and diaphragms, and at the center of each diaphragm. If required, rhe diaphragm's response can be represented by more than one intermediate mass. However, the use of only one intermediate mass to modeJ the diaphragm's response is appropriate for practical purposes. The lumped masses considered in the direction of interest are obtained by distributing the total mass of the structure according to a tributary strip area .criterion. Flexible
Diaphra~m
Modelin~
Flexible diaphragms in the direction ofnterest are represented by elastic shear springs as shown in Fig. 1. The stiffness of the elastic springs that represent the diaphragm
February 6-8, 1992
Thursday, page 18
Student Paper
EER/'s Fotty-Fourth Annual Meeting
San Francisco,
California
action can be roughly estimated with the in-plan e shear and bending stiffness of the floor system. The estimated stiffness value of the elastic springs obtained by this procedure should constitute a lower bound of the real stiffness of the diaphragm, since the contributions of the supporting beams and joists are neglected, particularly in wood sheathing floor systems. The discrete MDOF dynamic model has the capability to represent the stiffness of each diaphragm independently. The model can also represent local variations on the stiffness of each diaphragm, for example, when diaphragms do not have a uniform thickness across the span. Wall Modelin.: : Finite Element FIexibility Method Walls resisting lateral forces in the direction under study are represented by equivalent condensed beams with translational degrees of freedom in that direction, as shown in Fig. 1. The equivalent condensed beam elements for solid walls can be easily obtained by direct static condensation of the stiffness matrix of the walls idealized as equivalent wide columns. A more sophisticated approach is needed to express the stiffness matrix of a perforated wall as a function of its translational degrees of freedom. The most accurate procedure to obtain the condensed lateral stiffness of perforated cantilever shear walls in this case is to use a flexibility approach. Twc-dirnensional finite element models of each perforated walI are used to determine lateral flexibility. This approach has been chosen In this work and is presented below. Consider a cantilever perforated walI that has been discretized with a mesh of plane stress finite elements. It is desired to obtain the overalI stiffness matrix of the walI as a function of the average translatignaldegreespf freedom at each of the diaphragm's level. This is shown in Fig. 2, where Uj, U¡, and UA: are average displacements from al1 nodes at the floor levels due to a given applied force distribution. Apply a uniformly distributed unit lateral load at each floor level, P, where .,. 1) Define: {)l = { Vii,
l!/, Vti } the = { P, 0,
° }. ° }.
at level i, pl
displacement field associated with a load P applied
lP = { U¿, \Ir, utJ } the displacement field associated with a load P applied at level j, pJ = { 0, P, {]k = { U¡k, l{ik, Ukk } the displacement field associated with a load P applied
at level k, P = { 0, 0, P }. ~e
average lateral stiffness matrix can be written as :
[K]
February 6-8, 1992
Ku
Ki)
Kit
Kj¡
Kjj
Kjk
Kk.i
Kkj
Ku
Thursday, page 19
{KI}
-
{ KJ } {~
'" 2)
}
Student Paper
San Francisco,
EER/'s Forty-Foutth Annual Meeting
Then, for each load case, the Navier equations [ K ] { Ui
[ K ] {
lK
can be written as
pi }
... 3)
{ pi }
... 4)
} = { pk }
... 5)
} = {
n }=
] { Uk
California
By using the finite elernent rnethod, we can obtain the displacement fields u. Ui and applying their associated unit load fields pl, P' and pk. To obtain the average lateral stiffness matrix [ K ], we can rewrite the system of Eqs. 3 to 5 as :
_
u-, by
[ U ] { Ki
pi }
... 6)
[ U ] { Kj } T = { pj }
... 7)
[ U ] { Kk
where [ U ] is the displacement
[U ]
= {
}T
}T
= {
pk
... 8)
}
field matrix given by : Uii
o.
Ulei
{ Ui
n,
uj J
ukj
{ uj }
u,
-k
-k Ui
J
-le
Uk
}
... 9)
{ UI< }
which is already known. The aver~e lateral stiffness matrix is then obtained the systern of Eqs. 6 to 8 for K\ KJ and K". Cornputation
of Dynamic
by solving
Response
Once the equivalent discrete MDOF system is defined, dynamic response can be easily computed by ernploying standard solution techniques to integrate the equation of motion. The Newrnark-B method with parameters -y=1I2 and f3= 114 , corresponding to the constant average acceleration method was chosen because it is unconditionally stable numerically. The irnplernentation of Newrnark-B method is straight forward. A special purpose program is used to solve reduced MDOF dynamic problems, as is the case of the discrete MDOF dynamic rnodel proposed herein. A simple iterating algorithm could be used for this purpose. However, the Newmark-B rnethod can be more efficiently implemented by using an implicit direct integration algorithm (Ref. 6). With this approach, accelerations, velocities and displacements values are determined within a time step directly. The algorithm is computationally efficient since it does not require iterations within a time step. This algorithm is faster than the traditional iterating procedures used for the Newrnark-B method. The implicit direct integration method is a general algorithm to which several different methods are included in addition to the Newrnark-B method (Ref. 6). The equation of motion can also be solved using moda! coordinates as outlined elsewhere (Ref. 2), if the frequencies and mode shapes are determined. Response of the system can be determined from the representative modes shapes and the amount of proportional viscous damping for each representative mode. February 6-8, 1992
Thursday, page 20
Student Paper
San Francisco,
EER/'s Forty-Fourth Annual Meeting
Frequency
California
Estimates
Estimates of natural frequencies and mode shapes for the discrete MDOF dynamic model are helpful to identify dominant modes of response. The computation of frequencies and modes shapes for a reduced size system is simpler and more economical than the full system by any generalized ei -envalue solver. The Lanczos method was chosen in this work due to the availability of é special purpose program to obtain a reduced number of eigenvalues for large size systems. For a reduced size problem like this one, the Generalized Jacobi Method is al so advantageous.
Soil-Structure
Interaction
Mode1ine
Soil-structure interaction effects can be easily incorporated in the analysis by computing the average lateral diminished stiffness matrix, K, from the average lateral stiffness matrix K, and the foundation fIexibility represented by two generalized springs, as presented by Hjelmstad and Foutch (Ref. 8). A generalized spring with stiffness ke is introduced to represent the resistance of the soil-foundation system to rocking, while the other generalized spring with stiffness ks represents the resistance of the soil-foundation system to direct lateral displacement, as shown in Fig. 3. The equation
of motion for the system represented Mi.
where the average
k where :
=
1_ [ke
