Displacement-based seismic analysis for out-of-plane bending of unreinforced masonry walls.pdf

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng Struct. Dyn. 2002; 31 :833–850 (DOI: 10.1002/eqe.126)

Displacement-based seismic analysis for out-of-plane bending

of unreinforced masonry walls

K. Doherty 1 , M. C. Grith 1; ∗ ; † ,N.Lam 2 and J. Wilson 2

Department of Civil and Environmental Engineering; Adelaide University; Adelaide; SA 5005; Australia

Department of Civil and Environmental Engineering; University of Melbourne; Victoria 3010; Australia

SUMMARY

This paper addresses the problem of assessing the seismic resistance of brick masonry walls subject

to out-of-plane bending. A simplied linearized displacement-based procedure is presented along with

recommendations for the selection of an appropriate substitute structure in order to provide the most

representative analytical results. A trilinear relationship is used to characterize the real nonlinear force–

displacement relationship for unreinforced brick masonry walls. Predictions of the magnitude of support

motion required to cause exural failure of masonry walls using the linearized displacement-based

procedure and quasi-static analysis procedures are compared with the results of experiments and non-

linear time-history analyses. The displacement-based procedure is shown to give signicantly better

predictions than the force-based method. Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS : masonry; strength; displacement; bending; seismic; assessment

1. INTRODUCTION

In recent years, displacement-based (DB) design philosophies have gained popularity for the

seismic design and evaluation of ductile structures, e.g. References [1–3]. However, designers

perceive unreinforced masonry (URM) to possess very limited ductility so that its seismic

performance has been considered to be particularly sensitive to peak ground accelerations

[4]. Consequently, elastic design methods as opposed to DB design philosophies have been

thought applicable. In contrast, recent research has shown that dynamically loaded URM

walls can often sustain accelerations well in excess of their ‘quasi-static’ capabilities [5–7].

This dynamic ‘reserve capacity’ to displace out-of-plane without overturning arises because

the wall’s ‘post-cracking’ dynamic response is generally governed by stability mechanisms.

∗

Correspondence to: M. C. Grith, Department of Civil and Environmental Engineering, Adelaide University,

Adelaide, SA 5005, Australia.

†

E-mail: mcgrif@civeng.adelaide.edu.au

Contract=grant sponsor: Australian Research Council; contract=grant number: A89702060.

Received 16 November 2000

Revised 29 May 2001

Accepted 17 July 2001

834

K. DOHERTY ET AL.

That is to say, geometric instability of a URM wall will only occur when the mid-height

displacement exceeds its stability limit [8]. Indeed, research into face loaded inll masonry

panels by Abrams has shown that under dynamic loading, one of the key responses governing

wall stability is the size of the maximum displacement [9]. This suggests that DB design

philosophies could provide a more rational means of determining seismic design actions for

URM walls in preference to the traditional ‘quasi-static’ force-based approach presently in use.

Currently available static and dynamic predictive models have not been able to account for

the large displacement post-cracking behaviour and ‘reserve capacity’ of URM walls when

subjected to the transient characteristics of real earthquake excitations. Traditional ‘quasi-

static’ approaches are restricted to considerations taken at a critical ‘snapshot’ in time during

the response and hence the actual time-dependent characteristics are not modelled. As a result,

the ‘reserve capacity’ to rock is not recognized, thereby providing a conservative prediction

of dynamic lateral capacity. While such procedures may result in a reasonable design for

new structures, they may be too conservative for the seismic assessment of existing URM

structures where unacceptable economic penalty could be imposed if ‘reserve capacity’ is

ignored. In recognition of this problem, a velocity-based approach founded on the equal-

energy ‘observation’ was developed [10], which considers the energy balance of the responding

wall. The main disadvantage of this procedure is that the energy demand calculation is very

sensitive to the selection of elastic natural frequency and is only relevant for a narrow band

of frequencies. Clearly, there is a need for the development of a rational and simple analysis

procedure, encompassing the essence of the dynamic rocking behaviour and thus accounting

for the reserve capacity of the URM wall.

A major outcome of the collaborative analytical and experimental research carried out at

the Universities of Adelaide and Melbourne has been the development of a rational analysis

procedure which models the reserve capacity of the rocking wall. This procedure is based on

a linearized displacement-based (DB) approach and has been adapted for a wide variety of

URM wall boundary conditions.

The structure of this paper is as follows: A single-degree-of-freedom idealization of the

rocking behaviour of URM walls based on their force–displacement (F–) relationships is

described in detail in Section 2. This idealization applies to URM walls, such as parapet

walls and non-loadbearing (or lightly loaded) simply supported walls (i.e. possessing dierent

boundary conditions). The F– relationships have been developed in Section 3 for URM

walls behaving as rigid blocks which rock about pivot points at the fully cracked sections. In

Section 4, this idealization is relaxed by including axial and exural deformations for walls

subjected to high axial pre-compression. The sections of the wall where this deformation is

included are referred to as ‘semi-rigid’ blocks. In Section 5, the substitute structure concept

is applied to further simplify the single-degree-of-freedom (SDOF) models so the response

behaviour of URM walls can be predicted using displacement response spectra. The DB

procedure has been veried by comparing the predicted dynamic lateral capacities of simply

supported URM walls with a series of non-linear time history analyses (THA).

2. SINGLE-DEGREE-OF-FREEDOM IDEALIZATION OF URM WALLS

A cracked URM wall rocking with large horizontal displacements may be modelled as rigid

blocks separated by fully cracked cross-sections. This assumption is realistic provided that

Earthquake Engng Struct. Dyn. 2002; 31 :833–850

UNREINFORCED MASONRY WALLS

835

Figure 1. Unreinforced masonry wall support congurations.

there is little, or no, vertical pre-compression to deform the blocks. The class of URM walls

satisfying such conditions include cantilever walls (parapet walls) and simply supported walls

which span vertically between supports at ceiling and oor levels as shown in Figures 1(a)–

1(d) where the support motions can reasonably be assumed to move simultaneously. The case

of dierential support motion such as might occur in buildings with ‘exible’ oor diaphragms

[11] are also important but beyond the scope of this paper. The SDOF idealization of these

URM walls may be modelled using the displacement prole of a rocking wall (in a fashion

similar to the SDOF idealization of a multi-storey building based on the fundamental modal

deection).

From standard modal analysis principles, the equation of motion governing the rocking

behaviour of the cracked URM wall is very similar to the equation of motion governing

the response behaviour of the simple lumped mass SDOF model shown in Figure 2. Thus,

the mass of the system models the overall inertia force developed in the wall, whilst the

spring models the ability of the wall to return to its vertical position during rocking by

virtue of its self-weight. Provided that the inertia force developed in the lumped mass and

the restoring force developed in the spring are in the correct proportion, the displacement

of the lumped mass SDOF system and the wall system will always be proportional to each

other. Consequently, the response of these two systems can be related by a constant factor

at any point in time during the entire time-history of the rocking response. It can be shown

that the correct proportion is achieved if the lumped mass is equated to the eective modal

mass of the wall (calculated in accordance with the displacement prole during rocking) and

the restoring force is equated to the base shear (or total horizontal reaction) of the wall.

Earthquake Engng Struct. Dyn. 2002; 31 :833–850

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K. DOHERTY ET AL.

Non-linear spring modelling of

stabilising forces

∆

Force-Displacement relationship

Trolley modelling of

wall inertia

Dashpot modelling of

radiation damping

Base Excitations

Figure 2. Idealized non-linear single-degree-of-freedom model.

The computed displacement, velocity and acceleration of the lumped mass are dened as the

eective displacement, velocity and acceleration, respectively.

The equation of motion of the lumped mass SDOF system can, therefore, be expressed as

follows:

M e a e (t)+Cv e (t)+F( e (t))= − M e a g (t)

(1)

where a e (t) is the eective acceleration, a g (t) the acceleration at wall supports, v e (t) the eec-

tive velocity, e (t) the eective displacement, C the viscous damping coecient and F( e (t))

the non-linear spring force which can be expressed as a function of e (t)( NB : F( e (t)) is

abbreviated hereafter as F( e )).

The eective modal mass (M e ) is calculated by dividing the wall into a number of nite

elements each with mass (m i ) and displacement ( i ) and applying Equation (2) which is

dened as follows:

M e = ( i=1 m i i ) 2

i=1 m i i

(2)

For a wall with uniformly distributed mass, the eective mass for both parapet walls and

walls simply supported at their top and bottom has been calculated to be three-fourths of the

total mass, based on standard integration techniques. Thus,

M e =3=4M

(3)

where M is the total mass of the wall.

Earthquake Engng Struct. Dyn. 2002; 31 :833–850

UNREINFORCED MASONRY WALLS

837

∆ e =2 /3 t

R=F 0 /2-Mgt/2h

Inertia force distribution

pivot

F 0

F=0

h/6

F 0 /2

Mg/2

F=0

2/3 h

t/2

F 0 /2

pivots

∆ e =2 /3 t

pivot

∆ e =0

R’=F 0

Inertia force

distribution

∆ e =0

R’=F 0 /2+Mgt/2h

(a) Parapet Wall at incipient Rocking

and Point of Instability

(b) Simply-Supported Wall at Incipient Rocking

and Point of Instability

Figure 3. Inertia forces and reactions on rigid URM walls.

A similar expression, Equation (4), also derived using standard modal analysis procedures,

is used to dene the eective displacement ( e ).

e =

i=1 m i i

(4)

It can be shown from Equation (4) that

e =2=3 t (for a parapet wall) and

(5a)

e =2=3 m (for simply-supported wall)

(5b)

where t and m are the top of wall and mid-height wall displacements, respectively.

Note that both Equations (3) and (5) are based on the assumption of a triangular-shaped

relative displacement prole. This can be justied for a rocking wall where the displacements

due to rocking far exceed the imposed support displacements. The accuracy of this assumption

has been veried with shaking table tests and THA as described in Reference [12]. Thus, the

resultant inertia force is applied at two-thirds of the height of a parapet wall, and one-third of

the upper half of the simply supported wall measured from its mid-point (Figures 3(a) and

3(b)).

Earthquake Engng Struct. Dyn. 2002; 31 :833–850

i=1 m i i

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