urm

Journal of Earthquake Engineering

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AN EFFICIENT APPROACH FOR PUSHOVER ANALYSIS OF UNREINFORCED MASONRY (URM) STRUCTURES GR. G. PENELIS To cite this article: GR. G. PENELIS (2006) AN EFFICIENT APPROACH FOR PUSHOVER ANALYSIS OF UNREINFORCED MASONRY (URM) STRUCTURES, Journal of Earthquake Engineering, 10:3, 359-379, DOI: 10.1080/13632460609350601 To link to this article: http://dx.doi.org/10.1080/13632460609350601

Published online: 04 Sep 2008.

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Date: 13 June 2016, At: 02:49

Journal of Earthquake Engineering, Vol. 10, No. 3 (2006) 35S379 Q Imperial College Press

AN EFFICIENT APPROACH FOR PUSHOVER ANALYSIS OF UNREINFORCED MASONRY (URM) STRUCTURES

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GR. G. PENELIS Department of Civil Enganeering Aristotle University of Thessaloniki, Greece Received 27 October 2005 Reviewed 4 January 2005 Accepted 7 September 2005 This article presents a new method for the calculation of the moment-rotation ( M - 8 ) curves of URM elements using an analytical phenomenogical closed form solution for flexure, combined with a hybrid analytical-statistical model for shear. These are used t o define the constitutive law of nonlinear springs located at the areas of lumped inelasticity of an equivalent frame model. The method constitutes an analytical design/assessment tool, t o be used by the practicing engineer, for the pushover analysis of UFtM buildings. The proposed procedure is implemented as a pre-processor to a commercial nonlinear software codes capable of static pushover analysis using the point hinge approach.

Keywords: Moment-rotation; nonlinear; URM; pushover.

1. State of the Art

To date the general practice for the analysis of URM buildings by practising engineers has been a linear elastic analysis with either allowable stress checks or limit state analysis using a parabolic compressive stress low for masonry at critical cross sections. On the other hand the analytical procedures used for academic/research purposes have evolved following chronologically both the developments in computer industry as well the improvements in the analytical tools used mainly for reinforced concrete. So starting from the 2D equivalent frame models of the past, one can use 3D F.E. or D.E. nonlinear analysis today. The need for the application of nonlinear analysis has become obvious by experimental results showing that URM can posses considerable capacity for inelastic deformations [Abrams, 19921. 1 . l . Finite element models

There have been many different approaches for the modelling of masonry structures using the finite eIement method, which can be classified as follows: Homogeneous linear material models, Heterogeneous linear material models separating brick from mortar joints,

360

Gr. C. Penelis

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Models with nonlinear material properties available initially for mortar joints and subsequently for bricks incorporating discrete and later smeared crack patterns.

A comprehensive review of these methods is beyond the scope of this paper so the most important and most recent approaches of each category are presented. The methods based on isotropic homogeneous linear material assumption allow the modelling of large masonry walls. Analysis of this type is very useful for low levels of stress, but is very inaccurate for high levels of stress, where the stresses are influenced by nonlinearity and tensile cracking since the redistribution of stresses is not simulated; nevertheless it is still very common among practising engineers due to its simplicity. The methods based on isotropic-orthotropic homogeneous nonlanear material assumption rely mainly on homogenisation techniques through which the properties of the composite material (masonry) are derived from the two individual components (bricks and mortar). The most recent applications of this approach allow smeared cracking to simulate the nonlinearity due to cracking of the material. Characteristic applications of this approach are the ones by Page et al. [1985],Chiostrini and Vignoli [I9911 and Gambarotta and Lagomarsino [1995]. The main advantage of the method is that the nonlinear analysis of masonry structures is possible with the use of only few mechanical properties of the composite material in contradiction to the more advanced and detailed heterogeneous material approaches which use properties for each of the components as well as their interface, leading to significantly less computational time creating, thus, the ability to analyse complete structures. The drawback of tbe approach is that through homogenisation the local weaknesses of the bricks or the mortar are not simulated, leading only to ai overall simulation of the nonlinear behaviour of URM structures. Methods based on heterogeneous nonlinear materid assumption are the ones that simulate the composite material (masonry) in the most accurate way by including different elements and separate mechanical properties for the two components of masonry (bricks and mortar joints) as well as for the joint-brick interface. The most advanced of these models use a smeared cracking approach which combined with the refined discretisation required by the approach help simulate the crack pattern in an accurate way. Such approaches are the one by Ignatakis, Stavrakakis and Penelis [I9891 who developed a model for the in plane analysis of masonry and applied it on a real structure during the retrofitting of the Rotunda of Thessaloniki (19921, the approach of Shing et al. [I9921 who developed a combined approach using linear elements to model bricks and nonlinear interface elements to model mortar, the one by Ali and Page 119881 who also developed a model capable of simulating the gradual cracking of bricks and mortar separately and later by Antonucci et. al. [I9891 who developed a program for the F.E. analysis of larger scale masonry walls The main drawback of the method is that the hcretisation required makes it applicable only'for small substructures on whch only local effects can be studied.

Pushover Analysis of Unreinforced Masonry Structures

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1.2. Equivalent f r a m e models

Despite the extended research so far in modelling the inelastic behaviour of URM walls using the FEM, only in very few cases an attempt has been made for application to real structures. The main reasons are (a) the required very refined discretisation (bricks, mortar and their interface) that may exceed the capacity of commonly available hardware, and (b) the required numerous and not feasible to acquire for everyday practice input data (mechanical properties of bricks, mortar and their interface). So there has been a lot of work on modelling URR,I buildings with equivalent frames, and the reports mentioned here are only the ones of greater interest and importance. Tomazevic [I9971 developed and later improved the storey mechanism approach, which assumes a simple bilinear nonlinear relation for every pier of each storey and predicted the seismic behaviour of multi-storey masonry buildings with satisfactory accuracy in comparison to experimental results provided by literature. Penelis et al. [I9841 used a nonlinear static step by step analysis of plane frames to simulate the church of Hagios Andreas. This was probably one of the f i s t attempts to perform pushover analysis on masonry structures. Several others used available commercial software packages to analyse important monuments elastically; a detailed reference to these studies is beyond the scope of this paper. Magenes and Della Fontana [I9981 presented a n approach for the nonlinear analysis of URM buildings using an equivalent frame point hinge model which uses an elastic-plastic bilinear diagram for both flexure and shear. The behaviour is considered elastic until either moment or shear exceeds the capacity, and the elastic stiffness is controlled with the extent of the rigid offsets. The moment capacity is defined using section analysis with rectangular compression stress block, while the shear capacity using experimentally calibrated equations mentioned in Magenes and Calvi [1997].The ultimate flexural and shear deformation is defined by experimental data as 1%and 0,5% respectively. This methodoIogy presents similarities with the one proposed herein, and in the following chapters, both these and the differences will be highlighted. 1.3. Context of the present study

From the foregoing review the need for an efficient and easy to use analytical tool for the nonlinear analysis of URM is evident since the sophisticated nonlinear methods presented are not usable by the practising engineers who still rely on linear analysis for the design and assessment of U W I buildings. In the recent reports by FEMA [2000] and ATC [I9961 which give guidelines for the pushover analysis of buildings, the values provided for URM elements are indicative since they are not related to the material properties or the element orientation (pier or spandrel). Added to that the capacity curves provided by HAZUS [I9991 for URRl buildings allow

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362 Gr. G. Penelis

extremely high values (1.8%) for the ultimate drift which can only be justified for Reinforced Masonry Buildings. The objective of the present work is the development of an efficient method for the pushover analysis of URM buildings using equivalent frame models and concentrated nonlinearity at the ends of the structural elements, with a view to facilitating the use of pushover analysis of URM buildings by the practicing engineer. The nonlinearity is simulated with nonlinear rotational springs with their constitutive law defined by the moment-rotation curve of each element accounting for both flexure and shear [Penelis Gr, 2000j. 2. Analytical Calculation of Moment - R o t a t i o n Curves for U R M

The aim of the analysis is to produce a simple method of calculating the moment rotation (M-8) curve of a masonry pier under a constant axial force. This rotation is composed by a flexural and a shear component. The flexural component is calculated using plastic analysis of the cross section outlined at Sec. 2.1 while the shear component is modelled using experimental results available as described in Sec. 2.2; the two components are coupled at every step. Finally the moment rotation (M-8) diagram is created by using rotations due to flexure and shear at each increment of lateral force and by controlling the mode of failure by comparing the lateral force to the shear and flexural strength. The whole procedure is implemented in a computer code and its reliability has been checked against many experimental results in Sec. 2.4. 2.1. Inelastic M-0 diagrams due to flexure 2.1.1. Basic assumptions and input data For the creation of the inelastic Ad-8 curves the following assumptions have been made: Zero masonry tensile strength as commonly prescribed in codes of practice Parabolic distribution of compression stresses Plane sections remain plane up to failure (Fig. 1). While the following input data has been taken into account too: Minimum compressive deformation at failure E, = -2% [Tassios, 19841, (Fig. 2) Compressive strength (h) of masonry uniform, defined by the strength of bricks and mortar Modulus of elasticity Em = (550-1000) fm.Using the 1000 £invalue for the examples presented in this report as suggested in EC6. 2.1.2. Relation of curvature

(K,) to

loading (M-N)of a cross section

In the following relations between loading (M-IV) and curvature of the cross section are determined. For that, using known procedures from R/C structures, three main equations are used, two of equilibrium and one of deformation compatibility.

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363

Fig. 1. Flexural failure of masonry pier.

Consider the cross section of Fig. 1. The moment equilibrium with respect to the right edge results:

while the normal forces equilibrium on the section results:

N = D = a R . f r n - x . b = a R . frn.K,.h,b,

(2)

where x = K, . h, K, being the ratio of the compressed zone to the whole cross section: a= (YR

K,.x.

is a coefficient-to reduce the parabolic stress block into a rectangular one. In parallel the deformation compatibility condition on the cross section results:

where E, : K.

:

The strain a t the compressed edge of the cross section (Fig. 1) Curvature.

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364 Gr. G. Penelis

Fig. 2. a,-& diagram.

Taking into account that K, ranges in a narrow band with a mean value of 0.37, Eqs. (1) and (2) and the assumption of parabolic distribution of compression stresses, the following equations are derived by applying known procedures from R/C structures:

From Eqs. (3), (4), (5) and (6) it may be easily calculated that curvature n can result as a function of M and N. In fact pair values of M and N from Eq. (4) CYR is calculated, then from Eqs. (4) and ( 6 ) K, and E , are derived, and finally from Eq. (3) curvature 6 is calculated. 2.1.3. Determination of M-6 curve for flexure For a masonry pier or wall with an orthogonal cross section b . h and a length L loaded by an axial load N we calculate the M-8 curve according to the well known Mohr procedure (Fig. 3). The length L is divided in a number of equal intervals (say 10). For each incrementally increasing value of M, at each interval the curvature n is calculated

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Pwhover Analysis of Unreinforced M a s o n n ~Structures 365

M Fig. 3.

Determination of &I-0 diagram.

using the procedure developed in Sec. 2.1.2. Then by applying Betti's theorem, the rotation B results as:

In the case that K,

> 0.9 an elastic rotation 0 is calculated using:

since for K, > 0.9 the cross section has no tensile cracks and the elastic deformation calculated by Eq. (8) applies. 2.2. Shear m o d e m o m e n t

- rotation

diagram

The shear behaviour is being modelled using a hybrid method combining analytical expressions and experimental results to define the shear strength and shear deformations. 2.2.1. Shear strength The shear strength is defined in EC6 by using the following equation (Fig. 4):

where, xi = 3(h/2- M / N ) < h,

M = V . L,

(10)

and p is the coefficient of friction, fvka is the cohesion of masonry. Combining Eqs. (9) and (10) the following equation is derived (describing shear failure along bedjoints in case the section is cracked due to flexure):

where p = N / h . b and a, = M / V . h (shear ratio for cantilever).

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Gr. G.Penelas

ParaboIa fit

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M,=V,.L 0.8Vu

. . . . . . . . . . . . . . . .

b

Fig. 4. Adopted shear mode diagram.

Another approach for a cracked section proposed by Magenes and Calvi [I9971 in parallel with the above is as follows (describing diagonal cracking due to mortar joint failure) : fvko

V~=h.b.

+

'p

1+av

The notation for the above equation is given in Fig. 4. It is obvious that both Eqs. (11) and (12) should be taken into account since they are complementary to each other for the cracked section. 2.2.2. Shear defonnations The cracked and ultimate rotations due to shear; 8,, and 9,, are determined by the statistical analysis of experimental data using results from tests in Pavia and ISPRA [Magenes and Calvi, 19971. In Table 1 the results from several tests are listed in order to evaluate 19, and 8,. .

Table 1.

..

-

Results from Calvi and Magenes.

Wall

Llh

~lMPa1

MIX (+)* MI2 (-)* M13 (+)* M13 (-)* MI4 (+)* MI4 (-)* ISPl (+) I S P l (-) ISP3 (+) ISP3 (-)

1.33 1.33 2.00 2.00 2.00 2.00 1.35 1.35 1.35 1.35

1.12 0.68 1.24 1.24 0.69 0.69 0.60 0.60 1.08 1.08 Mean C.O.V.

N.B.(*):Variable axial load

(%I

......

. . . . .

Pushover Analysis of U n r e i n f o ~ e dMasonry Structures

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Regarding the ultimate rotation, the mean value calculated, 0, = 5.30%, is very reliable since it has a coefficient of variation c.0.v = m / u x = 0.109 or 10.9%. However the rotation at cracking d,, is very unreliable since it has a very high c.o.v., therefore a further analysis of the results is necessary. To tackle this problem the concept of an effective shear modulus Gef, replacing the elastic shear modulus, is introduced. The elastic shear deformation of a cantilever loaded with a , tip load Vu is described by Eq. (13):

where Gel is the elastic shear mbdulus, L is the length of the cantilever and b, h are the dimensions of the cross section. The corresponding tangent rotation is calculated as:

e

-

vu

( 14.)

"' - 5/6 . Gel . h . b' and therefore Gel = rc,/(5/6 . eel), where rc, is the shear stress

T~

(15)

= V u / ( h- b ) at cracking.

Evaluation of Gef (efective shear modulus) The high C.O.V.in the results of Pavia and Ispra experiments is mainly attributed to the flexural behaviour of some of the specimens. So in order to eliminate that parameter only the cantilever piers having a shear ratio L / H = a, < 1.5 are used. This is to ensure that the predominant deformation mode is shear, and not a combination of shear with flexure. The shear s t r e s of a cracked cross section can be calculated from Eqs. (11) and (12); in Table 2 these are shown for the previously selected specimens. Replacing Gel with G e f in Eq. (15), the following expression results:

From Eq. (15a) Gef is calculated for each of the above cases in column (5) of Table 3. In comparison to the definition of the elastic modulus as a function of fm, Gef is also defined in the same fashion as:

TabIe 2.

MI1 M12 ISPl ISP3

1.33 1.33 1.35 1.35

Shear stress calculations.

1.12 0.68 0.60 1.08

0.66 0.41 0.329 0.53

0.45 0.307 0.254 0.37

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Gr. G. Penelis Table 3. Statistically processed results. 2

I Code

@CT

( M+ V )

(%I

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M11 M12

1.2

2.35

-

Mean (%) Mean C.O.V. (%)

3 &, ( V )

5

4

6

7

10

1.044

Gef (V) fm (EC6) fm(exp) X(EC6) [MPa] [MPa] [MPa] [MPa] (MPa] 86.35 5.99 7.9 0.45 517.24

2.0445

-0.307

[%I

C.O.V.

Incl. MI2 Incl. MI2

Tcr

-

180.19

5.99-

7.9

583.89

6.69

6.54

22.9 516.61 39.4

-

12 X(exp)

[MPal 65.47

-

87.20 21.44

90.59 27,62

So, for the Pavia as well as for the Ispra tests the masonry compressive strength has been defined experimentally as 7.90 MPa and 6.20 MPa respectively. Alternatively these values can be calculated from the brick and mortar strength according to the EC6 procedure, which for the Pavia tests is:

and for the ISPRA tests is:

In TabIe 3 the rotation at cracking ,8 = &,/L due to shear (col. 3), the effective shear modulus (col. 5) and the measured masonry compressive strength. (cbl. 7) are shown for each of the selected tests. In order to reduce the c.0.v. for the data, which is 39%, the MI2 value is excluded since it is less than the lower 5% ( f C j w )which assuming a normal distribution is

where m is the mean value and mx the standard deviation. Thus the mean effective shear modulus is G e f =583.89 with a coefficient of variation c.o.v.= 0.22 or 22% which is acceptable, considering that masonry is a very unreliable material Using those data the value of the (A) parameter in Eq. (16) can be calculated and G,/ can be expressed as a function of the masonry compressive strength as follows:

P w h o v e ~Analysis of Unreinforced Masonry Structures 369

2.2.3. M-8 diagram incorporating shear deformations

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In Fig. 5 the idealised bilinear shape of the moment-rotation diagram for the shear mode behaviour is shown. The shear strength, for a cracked cross section, is determined by two different equations:

regarding bedjoint failure of the flexuraly cracked section, and

regarding diagonal cracking of an uncracked section. If x = 3(h/2 - M / N ) > h which means the section is un-cracked then Eq. (22a) is substituted:

8, is given according to Sec. 2.2.2 by the expression, 8, = 5.30%, BcT is given according to Sec. 2.2.2 also, by the expression (Eq. 15):

where Gef is obtained from Eq. 20. The two aforementioned modes, the flexural and the shear, are included in a Fortran code, which deals with them in a coupled fashion, so that at every increment of the lateral load the calculated un-cracked part of the cross section is imported as input data for the calculation of the shear strength of the element, and the total rotation (8) or corresponding top displacement (6 = 8 . L), is the sum of the flexural and shear deformation at each step. It should be noted that due to the shear span ratio of the piers, which in most URM buildings is low (squat), the shear component of the deformation is very significant, although there are exceptions as the actual building example presented in Sec. 3 for which the flexural component of the slender piers was dominant. This software code can be used as a preprocessor to any nonlinear software capable of performing pushover analysis, since the input required is each elements' geometry, the URM compressive strength, thi? coefficient of friction p, the cohesion fvko, the axial load N and the support condition (cantilever or fixed a t both ends), data that would be required even for a simple elastic analysis and design/assessment. The proposed methodology has, as mentioned in the 'introduction, similarities with the one presented by Magenes and Della Fontana [1998]. Both methods use an equivalent frame point hinge model for URM structures with openings, and the shear strength formulations and ultimate shear deformation of URM elements were developed by Magenes and Della Fontana. H6wever the proposed method introduces an analytical model to define both flexural resistance and deformations assuming

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370 Gr.

G. Penelis

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Pushover Analysis of Unreinforced Mason? Structures

371

a parabolic distribution of compressive stresses and a hybrid semi-analytical, semiempirical approach for the definition of the uncracked shear stiffness. A significant issue for modelling UFW elements is the behaviour of spandrels which is significantly different to piers [Magenes, 20001. Spandrels are usually a combination of an upper URM continuous element with a bottom simply supported wooden or R/C lintel. In the procedure adopted herein the URM part is analysed as a pier (very low axial load), ignoring the different joint orientation while the behaviour of the spandrels is a combination of the U r n 1 part regarding initial strength and stiffness and the lintel part regarding the residual strength and deformation. Another reasonable approach could be the use of an equivalent diagonal truss model similar to the one used for masonry infills in R/C structures [Kappas et al., 19981 but it is obvious that further research is needed both in the experimental and the analytical field.

2.3. Validation of the procedure The validation of the procedure has been focused on reproducing the envelopes to the cyclic and pseudo dynamic tests performed at Pavia and ISPRA. In Fig. 5 the test set up and the reproduction of the tests of four rectangular URM panels which dimensions are given in Table 1, is resented while in Fig. 6 the comparison of some key response characteristics of the pushover curves produced analytically using the proposed method with the envelopes of the cyclic tests taken as the mean of the two directions is shown. The main parameters that have to be considered in evaluating the proposed method is that these are sets of tests performed at different laboratories and under different conditions, while the proposed method combines an analytical model with a statistical evaluation of experimental data, and is not curve fitting to those tests. From Fig. 6 it is obvious that for all the specimens the ultimate strength is predicted very accurately as well as the ultimate deflection (6, = 8,. L). Regarding the initial stiffness as well as the ductility or the yield displacement there are some very large differences in the Pavia test specimens MI2 and MI4 which can be attributed to the fact that the analytical procedure uses a constant compressive strength while the experiments used a variable axial load in order to avoid premature failure. So by using a mean value of the axial force the initial stiffness is overestimated in the analytical procedure. Furthermore it should be noted that in all the Pavia tests (MI1-MI4) the vertical load has not been kept constant due to the experiment set up as it has been mentioned in the relative reports while in the ISPRA tests (ISPRA1-ISPRA2) the vertical load has been completely constant. As one can observe from the agreement between the analysis and the tests is much better in the ones from ISPRA. It should be added that a modified version 'of the method (or more accurately of the software), able to accept as input a variable axial load for each step of the lateral load, has been developed, but its application to these specific examples did not produce significant changes.

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Initial Stiffness

Ultimate Strength

Ultimate Deflection

Fig. 6. Comparative charts.

3. Application of t h e Method to 2D URM Perforated Two-Storey Walls

The developed methodology has been applied to two separate cases of URM buildings tested experimentally at the Ismes laboratory at Bergamo and at the University of Pavia, in order to attempt the reproduction of the pushover curve predicted experimentally. For that the commercial nonlinear finite element code Sap2000 has been used. In order to achieve the applicability of the methodology, and since most commercial software packages are not usually accessible for programming, an initial elastic analysis, using the vertical loads as well as an estimate of the lateral load

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Pushover Analysis of Unreinforced Masonry Stmctures

373

capacity, was performed in order to define the axial load level of each element and consequently, using the developed pre-processor independently of the main software, the h1-8curves of the plastic hinges were defined. It is obvious that should the main software (i.e. Sap2000) had been accessible this pre-processor could be built-in as a subroutine taking into account the variation of the axial load at each step. A significant modelling parameter in all point hinge models, and especially for URM buildings, is the extent of the rigid offsets depicted in Figs. 7 and 8. In a parametric elastic analysis of 2D walls and 3D buildings by Kappos and Penelis [2002] the approach with full horizontal and vertical rigid offsets was defined as the one closer to shell element analysis. This however is the outcome of elastic analysis

-

Frames Rigid zones

Pushover curves produced analytically and experimentally ]-Experiment A1 A n a l y t i c a l A 1 ]

'

Fig. 7. Ismes wall model and pushover curve.

Gr. G. Penelis

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G

-

Frames Rigid zones

Theoretical Vs experimental resulb Pavia wall D -Penelis

---

-Lagomarsino&Penna E r p e r i m e n l

.

4tmm1.

.

.

Fig. 8. Pavia wall model and pushover curve.

and is valid for buildings which are not expected to sustain serious cracking. If a highly nonlinear behaviour is expected, or a pushover analysis to failure is performed (our case) then the model with only horizontal rigid offsets is applicable because the other models with vertical offsets restrain the extent of c r a k n g unrealistically. A half scale model of a two storey URM building (prototype A l ) was tested at the shake table test facilities at Ismes under the 23/12/80 Irpinia eatthquake, a typical soft soil Mediterranean Basin excitation. The URM properties for this test have been derived experimentally as 2.20 MPa compressive strength, 0.15 MPa cohesion and an estimate (using EC6) of 0.40 for the coefficient of friction. The ~ . of that building was modelled using an equivalent hame with west w a l l ' ( ~ i 7) plastic hinges set up as described previously at both ends of each structural element (pier-spandrel), using the aforementioned method; the capacity curve for the

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Pushover Analysis of Lrnreinforced Masonry Structures

375

building calculated with the pushover analysis is overlaid upon the experimental cyclic curves. A full scale building was tested at the reaction frame facility of the University of Pavia and the wall analysed shown in Fig. 3 was modelled using the same as in the previous paragraph nonlinear equivalent frame approach. The URM properties for this test have been derived experimentally as 6.20MPa compressive strength, 0.23MPa cohesion and a coefficient of friction of 0.56. The analytically produced pushover curve is also plotted on the experimental cyclic curves. Furthermore the Pavia wall has also been modeled using the nonlinear (time history) software produced by Lagomarsino and Penna [2001] and the calculated response is also plotted in Fig. 8. The predicted behaviour of the buildings is accurate in terms of stiffness and ductility but there were some differences observed in the Pavia wall regarding the ultimate strength. This however may be considered within the commonly accepted scatter. It is also noted that the software procedure presented has also been used in the production of capacity curves and vulnerability curves for U M buildings within the RISK UE project, the so called European HAZUS. Within this project more than 40 3D pushover analysis of URM prototype buildings have been performed using the presented software, during which a serious dfference ha. been pointed out between the capacity curves proposed by HAZUS, to the ones calculated by this method and the available experimental data regarding ultimate drifts which for URM buildings are extremely optimistic in HAZUS (i.e. 1.8% for low rise, Fig. 9). Furthermore the vulnerability curves produced for single two and three storey U R M buildings resulted in predicted damage for the building stock of the Thessaloniki 1978 and Aegion 1995 quakes which was in good agreement with the observed damage (Fig. 10). Another interesting application of the proposed methodology has been in the retrofit of the National Library of Greece Building in Athens which is a neoclassic

Capacity Curves Comparison with Hams

- --

2 storey 3 storey .isme$ HANS URML W S URW

Drift Fig. 9.

Cornparison of typical capacity curves to experimental data and HAZUS proposals.

376

Gr.

C. Penelis 1st Level vulnerability curves for 2-storey mlck U r n Buildings

x

act?

x

ach

x

act3

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x act4

Fig. 10. Vulnerability curves and actual data from the Thessaloniki and Aegion quakes.

complex of buildings built in the late 19th century, which was damaged by the 1999 Athens quake. This building, which consists of three orthogonal in plan wings and has very slender URM piers, was analysed both using elastic dynamic analysis with a shell element modelling and pushover analysis with equivalent frame modelling following the proposed procedure. The results of the pushover regarding the point hinge formulation (cracking) at the performance point were in good agreement with the observed damage (Fig. 11, Penelis Gr [2001]). 4. Conclusions

The proposed methodolog has, as has been mentioned in the introduction, similarities with the method presented by Magenes and Della Fontana [1998].Both methods use the equivalent kame point hinge model for URh1 structures with openings. The proposed mode1 uses an analytical approach for flexure utilising the parabolic compression stress block assumption, a Mohr-Coulomb failure criterion for shear strength and experimentally derived shear stiffness and ultimate shear deformation. The use of the proposed methodology for practical purposes has many advantages since it allows the application of the explicit static nonlinear methods, using commercial software codes, on URM buildings, something that has been extensively used for R/C and steel buildings over the past few years, and is foreseen in modern rehabilitation guidelines. It also allows the prediction of the failure mechanism by observing the development of plastic hinges at Merent locations of the building. It should however be stressed that these plastic hinges are not the actual failure mechanism of URM (being a brittle material) but a computational simulation of the nonlinear behaviour of the whole member they correspond to. Of course the methodology has also several drawbacks since it cannot simulate the in-plane behaviour of masonry structures without openings, as is the case with walls of ancient cities, and cannot predict the crack pattern as reliably as a sophisticated finite element model with space or planar elements.

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Stresses a u (Min)

Side wing-Transverse direction

Elastic Spectra

,

Design Spectra p=3 -Pushover right wing. 85% design spectra

-

I

1 1

Fig. 11. The national library of Athens, (a) photograph, (b) elastic analysis with shell elements, (c) capacity and demand inelastic spectrum using the proposed methodology.

So taking those into account, a practical analytical tool is presented, in order t o facilitate the practising engineer to assess in a practically acceptable way the nonlinear capacity of a TJRM building. This obviously comes supplementary to the extensively used linear e~nalysis,by facilitating the more accurate definition of the behaviour factor and predicting the failure mechanism by a series of point hinge formulation.

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Acknowledgements This report has been prepared with the co-operation of the Engineering Seismology and Earthquake Engineering Section of the Civil Engineering Dept. of Imperial College and especially prof.A.Elnashai and the Reinforced Concrete Lab of the Civil Engineering Dept of Aristotle University of Thessaloniki. I would especially like to mention the assistance of professors A. Elanashai and A. Kappos on several aspects of the report. Furthermore the help of professors M. Calvi and G. Magenes of the University of Pavia who quickly supplied the available experimental data and professor S. Lagomarsino and Dr. A. Peina of the University of Genoa who supplied analytical results using their software code.

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