12TH INTERNATIONAL

BRICK/BLOCK Masonry c O N F E R E N C E

Sch

CRITICAl AXIAL lOADS FOR TRANSVERSEl Y lOADED MASONRY WAllS

A.E. Schultz', J.G. Mueffelman2 and N.J. Ojard 3

'Assoe. Prof., Dept. of Civil Engrg ., Univ. of Minnesota, Minneapolis, MN, 55455

2Struct. Engineer, Bonestroo, Rosene, Anderlik & Assoe., Roseville, MN, 55113 3Grad. Res. Assist., Dept. of Civil Engrg., Univ. of Minnesota,

Minneapolis, MN 55455

ABSTRACT

The buckling behavior of unreinforced masonry (URM) walls subjected to out-of-pla­ne lateral loads is described including numerical solutions for criticai axialload capa­city. The influence of bending on buckling capacity is illustrated, and the strongly non­linear interaction between criticai load and out-of-plane bending is discussed. A parameter study of the lateralload capacity of URM walls indicates that stability un­der transverse loads is the most likely failure mode for a wide range of common wall heights and load intensities. An experimental investigation on the stability of slender masonry walls is out/ined.

Key words: Buckling, design formulas, load tests, out-of-plane bending, post-tensio­ned masonry, stability, transverse loads, unreinforced masonry.

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INTRODUCTION

Bending arising from out-of-plane lateralloads has a dramatic impact on the sta­bility of URM walls (Fig. 1). Due to low tensile strength, flexural tension cracks masonry and reduces the effective cross-section depth. This effect renders the wall more flexible, augments lateral deflection due to out-of-plane bending, and gives rise to second-order (P-/',.) moments. These moments generate more tension which further reduces the cross-section, and this sequence is repeated until equi­librium of the member is established, or until it fails due to instability (i.e. buc­kling) .

Out-of-plane bending due to axialload eccentricity has been recognized to affect the stability of URM compression members (Angervo 1954, Sahlin 1961, Yokel 1971, Colville 1979). This research led to the development of design code provi­sions that include eccentricity as a factor in the buckling capacity of URM walls . In the U.S., a check on the buckling strength of URM walls including the effects ofaxial load eccentricity is required (MS]C 1999). However, current code provi­sions do not address the effect of bending from out-of-plane loading on the sta­bility of URM compression members.

Widespread problems in the performance of URM walls have not materialized in the U.S. because the existing URM building stock was designed according to 01-der codes that are overly conservative. Yet, severe lateral loading events such as earthquakes, hurricanes and tornadoes often generate out-of-plane collapses of URM walls. These failures are usually attributed to low tensile strength, and sta-

Figure 7. Influence af laterallaad an buckling af URM walls.

P<t.

w

P<t.

cracked zone

stress distribution

P q;

t-3

ef= MmaxlP I

bility concerns are seldom addressed. Furthermore, new masonry construction of­ten features members that are more slender and more highly stressed than those in older buildings, making stability concerns more important. Finally, these trends come at a time when design standards (Mínímum 1998) and model codes (NEHRP 1998) have adopted significant increases in wind and seismic loads.

PREVIOUS RESEARCH

Linear analysis techniques have been used to solve the governing equation for the lateral deflection of URM walls under eccentric axial load (Angervo 1954, Sahlin 1961). In these studies, computed deflections, as a function ofaxial load, were used to define criticai loads. Solutions for the eccentric buckling strength of URM compression members have been developed in the U.S. (Yokel 1971, Colville 1979). Yokel illustrated the accuracy of this approach for prediction of the buckling strength of eccentrically loaded walls tested over a wide range of variables (1971).

The impact of nonlinearity in the masonry stress-strain relation on the buckling capacity of laterally-Ioaded URM walls has been studied recently (Romano 1993, La Mendola 1995, Ganduscio 1997). They obtained analytical solutions for criti­cai axial load which require iterative solutions of multiple coupled, nonlinear equations. However, these solutions have not found their way to design practice due to their complexity. Sahlin (1961) had earlier considered the influence of out­of-plane bending on the buckling capacity of solid walls, but practical buckling solutions were not developed either.

IMPACT OF BENDING ON BUCKLlNG CAPAClTY

Yokel's formulation for the deflection of a linear, elastic URM wall, with solid cross-section and no tensile strength can be extended to include out-of-plane bending (Schultz 1999). An additional flexural eccentricity e{ was defined as the ratio of bending moment M to axial load P (Fig. 1). Differential equations were derived for moment distributions arising from four combinations of support con­ditions and lateral load, including 1) equal end-moments on a simply-supported wall, 2) uniformly distributed lateral load on a simply-supported wall, 3) concen­trated lateral load on top of a cantilever wall, and 4) uniformly-distributed lateral load on a cantilever wall.

Numerical solutions were obtained for the lateral displacement u of URM com­pression members under axial load (Fig. 1). These solutions were found to des­cribe load-displacement relationships like that shown in Fig. 2 for a simply-sup­ported wall with a uniform load at ultimate W U ' These solutions suggest that the axial force-lateral displacement response of an URM wall exhibits a strong de­pendance on out-of-plane moment, as indicated by the bending parameter b. Re­lative maxima represent points of impending instability, and the corresponding axial loads are the buckling strengths Pc-

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Figure 2. 5tabiJity curves for simp/y-supported, uniform/y /oaded wall.

0.35

0.3 13 = M/Pu

max 1

0.25

&l 0.2 o.. ã: 0.15

0.1

0.05

O O 0.2 0.4 0.6

u/u o 1

Figure 3. /nfluence of bending on buckJing capacity of URM walls.

0.3

0.25

0.2

&l o.. 0.15 -o o..

0.1

0.05

O

13 = M /P u = e/u max c 1 1

0.1 0.2 0.3 0.4

~

0.5

0.8

0.6 0.7 0.8

For each of the four load cases considered, the ratios of criticalload Pc to the equi­valent criticalload P" were correlated with the bending parameter ~ (Fig. 3). Buc­kling capacity is seen to decay quickly with out-of-plane bending. A third-order polynomial with respect to ~ was fitted through this data to define analytical ap­proximations. The numerical solutions, shown as points in Fig. 3, are approxima­ted closely by the polynomials, shown as solid lines.

Of practical interest is the use of the numerical solutions to define a buckling strength formula similar to that in the MS]C code for eccentric axial loading (1999). The following formula was obtained

[ ( ea+Àel )]3 1 - 0.577

r [1 ]

where he is the effective height kh of an equivalent simply-supported member, and the axial load eccentricity has been replaced by the sum ofaxial load eccen­tricity ea and part of the flexural eccentricity e,o With values for the constant À.

equal to 1.0, 0.905, 0.813, and 0.70, respectively, for Cases 1, 2, 3, and 4, Eq. [1] was found to provide good approximations of the computed buckling strengths.

INTERACTION OF BENOING MOMENT ANO AXIAllOAO

The solutions defined above describe a highly nonlinear interaction between buc­kling load Pc and bending moment Mmax. In Fig. 4, normalized buckling load is plotted against normalized moment for the load cases considered. Criticai axial load and moment are normalized with respect to Euler buckling load PE and ra­dius of gyration r.

As noted earlier, the magnitude of bending is seen to have a dramatic impact on criticai axialload (Fig. 4). Moreover, for a given moment, there are two values of axial load that produce instability. One of these corresponds to the "tension" re­gion (Pi PE < 0.4), for which increases in axialload reduce flexural tension without a comparable increase in p-f),. instability. But, the other load corresponds to the "compression" region (Pi PE > 0.4), for which the reduction in flexural tension do­es not offset the increased p-f),. instability associated with larger axial loads. Ho­wever, both regions represent failure due to axial compression instability, and the location of the peak moment represents the boundary.

Bending capacity, as illustrated in Fig. 4, vanishes when no axial load is present, due to the 'zero tensile strength' assumption. This assumption is necessary to ensure the existence of sim pie analytic solutions, such as Eq. [1], for the buckling strength of URM members. Yet, the accuracy of this assumption depends greatly upon the relative magnitudes of flexural tensile strength and slenderness ratio. Yokel (1971) noted that

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Figure 4. Moment-buckling load interaction for URM walls.

0.8

0.6 UI

o.. -o CL

0.4

0.2

O

~ Case 1 - o

~o - - - Case 2 ~o '- - - - - - - Case 3 .... ' '' - '' '' -

~ 0- -- ------- Case 4 ' o ,

.... .. .. 'li

" "" o-o " ""''' "" " -"

"\' " , , ,

" , , , , I

I . . ~. ~ I ,

" . -. -1 P =:rt

2E I/h 21 --o

./ ., , o

k<: ~ _o E m e -_: ..... .... .. , ~ -"'::- " ... ' .. -

~ ----"-- 0·00 ----""'F". , .

O 0.05 0.1 0.15 0.2 0.25 0.3

M /P r max E

the 'zero tensile strength' approximation can lead to gross underestimation of the buckling capacity of very slender URM walls (h/ t > 25 or h/r > 85). In Fig. 4, the 'ze­ro tensile strength' assumption is seen to distort the moment-axial load interaction, beca use URM sections have finite bending strength even if no axial load is present.

Solution of Eq. [1] in a design office environment can be cumbersome, because this equation represents a fifth-order polynomial. First, two different values for axial 10-ad are recognized to satisfy the interaction for a given bending moment (Fig. 4). Second, there is a peak moment for a wall with given material and geometric pro­perties beyond which there is no real solution to Eq. [1] . Thus, solving for maximum moment MmQx, as function of critica i load P" is computationally more expedient. For the analytical approximation given by Eq. [1], maximum moment is given by

[2]

INFlUENCE OF STRESS-STRAIN NONLlNEARITY

Ganduscio (1993) and Romano (1997) reported the solution for a cantilever wall w ith a concentrated lateral load at the top (i .e., Case 3). They used the following nonlinear stress-strain relation for masonry in compression

[3]

where E is a material constant (i.e., secant modulus at c= 1), and the exponent n controls the degree of nonlinearity. For masonry meeting current masonry speci­fications in the U.S. (MSjC 1999), the value of n is likely to exceed 0.7, and in many cases approach a linear stress-strain relation (n= 1). In the U.S., however, the elastic modulus of masonry Em is defined as

Em = -------- [4]

where f'm is compression strength, and C 0.33 and C O.05 are the strains corresponding to stresses equal to 33% and 5%, respectively, of f'm (MSjC 1999). This definition incorporates a significant portion of the nonlinearity inherent in masonry com­pression behavior. Using modulus Em defined in this manner, Eq. [1], provides re­asonable estimates of P, for nonlinear masonry materiais.

Criticai axial loads P" normalized by Euler's buckling load Pf , from Romano's so­lution (n=0.8) and the proposed elastic solution for Case 3 are compared in Fig. 5. Romano's solution is sensitive to slenderness ratio, i.e., ratio of height h to ra­dius of gyration r, whereas the elastic solution is not. Nonetheless, the elastic so­lution is seen to be a conservative approximation for ali but the stockiest walls (h/r=35). Furthermore, the elastic solution overestimates the inelastic solution for stocky walls only when there is little bending (i.e., small ~), for which case insta­bility is of little importance. Nonlinearity affects stocky walls with little bending

Figure 5. Comparison of elastic criticaI loads with nonlinear solution.

UI

o.. -o o..

1.5.....----,------.-----,---------. --- Elastic .. _~--- Inelastic, hlr=35

----0--- Inelastic, h/r=70 ---{}--- Inelastic, h/r=105

--- -fr --- I nelastic, h/r= 140

0.5~--~,-~~~r---_;---_+---~

o 0.2 0.4 0.6 0.8

f~=M /PU max 1

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beca use only under these conditions do axial compression strains reach suffi­ciently large magnitudes for nonlinearity to become important.

PARAMETER STUDV

Using the solution for Case 2, the buckling strength of simply-supported, verti­cally-spanning masonry walls subjected to uniform lateral loads was investigated. Both concrete block and hollow clay units were considered (Table 1). Axial load was assumed constant, and is reported as a vertical stress av • The lightly-Ioaded concrete masonry walls with moderate axial load eccentricity represent low-rise URM loadbearing construction. A vertical stress three times larger than that for the concrete walls was chosen for the clay walls to represent post-tensioned con­ditions. A larger compression strength was assumed for the brick masonry, and a load eccentricity equal to zero was used for concentric post-tensioning. A unit weight equal to 1,930 kg/m l (120 pcf) on the net volume was used for ali walls.

Table 7. Properties for masonry walls in paramenter study.

Para meter CMU Clay

unit thiekness, t 203 mm (8 in.) 152 mm (6 in. )

mortar bedding face shells full beds

eomp. strength, ('m 13.8 MPa (2,000 psi) 20.7 MPa (3,000 psi)

elas ti c modulus, Em 10,300 MPa (1,500 kSi) 15,500 MPa (2,250 ksi)

allowable tension, F, 1 72 kPa (25 psi) 172 kPa (25 psi)

allowable shear, F, 463 kPa (67.1 psi) 567 kPa (82.2 psi)

net area', Ao 194 em' (30 in.') 212 em' (32.8 in. ')

seetion modulus', 5 1330 em' (81.0 in. ') 788 em' (48.< in.')

moment of inertia', i 12,850 em' (308.7 in.') 5,494 em' (132 in.')

radius of gyration, r 81.5 mm (3.21 in.) 51.1 em (2.01 in.)

vertical stress, o, 0.69 MPa (100 psi) 2.07 MPa (300 psi)

load eeeentric ity, e. 25.4 mm (1.0 in.) O mm (O in.)

'Properties specified per 304.8·mm (12·in.) length of wall.

PARAMETER STUDV

Using the solution for Case 2, the buckling strength of simply-supported, verti­cally-spanning masonry walls subjected to uniform lateral loads was investiga­ted. Both concrete block and hollow clay units were considered (Table 1). Axial load was assumed constant, and is reported as a vertical stress sv. The lightly-Io­aded concrete masonry walls with moderate axial load eccentricity represent low-rise URM loadbearing construction . A vertical stress three times larger than that for the concrete walls was chosen for the clay walls to represent post-ten­sioned conditions. A larger compression strength was assumed for the brick ma­sonry, and a load eccentricity equal to zero was used for concentric post-tensio­ning. A unit weight equal to 1,930 kg/m3 (120 pcf) on the net volume was used for ali walls.

In addition to buckling, strengths for other criteria were considered, including combined axial and flexural compression, flexural tension and out-of-plane hori­zontal shear. Allowable stresses were taken from the MSjC standard (1999) and were multiplied by assumed factors of safety to convert them into strength crite­ria . For the compression criterion, a factor of safety equal to 4 was used, as noted in the MSjC standard (1999). Factors of safety equal to 4 were also used to defi­ne strengths in flexural tension and horizontal shear. In addition, a second crite­rion for each condition was defined according to the load and resistance facto r design (LRFD) approach . A load factor equal to 1.6 was used for both vertical and lateral loads, and a resistance factor equal to 0.4 was taken on the basis of the Bri­tish masonry standard (Code 1985). This simplistic LRFD formulation is included to demonstrate shortcomings of the allowable stress design method when ap­plied to buckling instability.

Instead of reporting criticai loads, the corresponding uniformly-distributed lateral load capacity at ultimate, wu, for various wall heights was computed for the assu­med vertical stresses a,. For ali walls in this study, the tension branch of the mo­ment-axial load interaction diagram (Fig . 4) dictated lateralload capacity wu, due to the relatively low leveis ofaxial stresses considered. Lateral load capacities for the clay masonry walls are shown in Fig. 6.

The clay masonry walls in the present study were subjected to larger vertical stres­ses than the concrete masonry walls, thus, the combined compression criterion becomes important for these walls. Even under these circumstances, only the

Figure 6. Lateralloads capacities for c/ay masonry walls.

wall height , h (m)

2 4 6 8

10

---o-- compression (FS=4) ____ compression (LRFD) ----fr-- tension (FS=4) ---+--- tension (LRFD) -<>-- shear (FS=4) --+- shear (LRFD) --- buckling strength

0.1

5 10 15 20 25 30 35 40 wall height, h (ft)

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LRFD version of the combined compression criterion supersedes the buckling cri­terion, and only when wall height exceeds 7.6 m (25 ft). Otherwise, for most wall heights considered (h>3 m = 10ft), the lateral load capacities for buckling are smaller than those for ali other criteria.

The LRFD criteria are seen to provide markedly lower lateral load capacities than the corresponding strength criteria derived from allowable stress concepts. This issue is important because axialload is a component of bending resistance, as no­ted in Eq. [3]. This feature may assist or detract from bending strength depending upon which branch of the buckling interaction curve (Fig. 4) controls failure.

The data for the concrete masonry walls, not shown for brevity, indicate similar observations. For a vertical stress (Jy equal to 0.69 Mpa (100 psi), which is large, but with in the range of URM loadbearing masonry walls in engineering practice, neither the flexural tension, the combined compression nor the horizontal shear criteria control for any wall height considered. Buckling controls the lateral load capacity for ali concrete walls in this study.

IMPlICATIONS FOR PRESTRESSED MASONRY

Prestress forces in post-tensioned masonry (PTM) walls have the potential to be­come driving forces in stability failures if the tendons are unrestrained (i .e., free to displace rei ative to the masonry). In the present study, the clay masonry walls were subjected to concentric vertical stresses of comparable magnitude to pres­tress in PTM construction. Thus, the computed lateral load capacities for the clay walls (Fig. 6) suggest that PTM walls with unbonded, unrestrained tendons are vulnerable to buckling instability.

There are valid behavioral reasons, however, which suggest that the URM stabi­lity mechanism may not apply to PTM walls, even if unrestrained tendons are used. First, buckling of a post-tensioned wall requ ires sufficient room within the cavity for the tendon to deflect relative to the masonry. Second, the inherent fle­xura l stiffness of post-tensioning tendons, especially if bars are used, and the ro­tational restraint provided by the anchorages may reduce the potential for buc­kling if the line of action of the prestressing force has the tendency to travei with the deflecting wall. Third, decreases in the axial stiffness of the member, upon ini­tiation of inelastic compression straining and eventual crushing of the masonry, may lead to quick decompression of this self-restrained system, and an interrup­tion of the stability failure.

EXPERIMENTAL PROGRAM

The nature and scope of the analytical results of this study underscore the need for experimental verification of the stability of slender masonry walls subjected to lateralloading. To the authors' knowledge, there are no laboratory tests available

Figure 7. Test setup for stability tests of URM walls .

...... ~_--.,.-.---- wall thickness, t

lateralload

~ wiflle Iree

on the axial load capacity of URM members that are also subjected to lateral 10-ad. Yokel (1971) verified his buckling strength formula with data from eccentric axial load tests conducted in the U.S. during the 1960's, and Romano (1993) re­fers to more recent eccentric axial load tests conducted in Germany. However, these investigations did not include lateral loads. A carefully planned experimen­tai program on laterally-Ioaded slender walls is underway at the University of Min­nesota to determine the potential for buckling instability in both URM and PTM walls .

The current research program at the University of Minnesota includes 18 wall tests organized in three series. The first series comprises URM walls subjected to axial compression and uniform lateral load (Fig. 7). Experimental variables inclu­de type of masonry (concrete block or solid clay brick), lateral load magnitude, and loading sequence. The latter issue concerns the load path in axial load-Iate­ral load space (Fig . 8) . For ali wall tests, some axial compression will be applied before any lateral load (point A in Fig. 8). Subsequently, some lateral load will be applied (point B), after which axial load can be increased to stability failure in the compression region (points C and D). Alternatively, lateral load can be increased (from point B) to stability failure in the tension region (points E and F).

The second series comprises PTM walls with unrestrained tendons subjected to la­teral loading and which will be tested in the same setup as the URM walls (Fig. 7). Experimental variable for this series include type of masonry (concrete block

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Figure 8. Calculated moment-axialload interaction for test walls.

40

35 C (f) 30 Q..

------:s:::::I 25 -o~

20 ro O

e 15

.2 10 ro

5

o

~ hollow concrete block

®r ...... / r-.~ Y"'" ..........

/ '1 V \ '" /) ~ :\ '" ,/ tê> ,,\ © ~@ r-

I/ 1 ...

\ \ =!j " r-\ .--I solid clay brick ~ 1 ~ V \

• I

1.6 ..-~

ro o... .:::Lo

1.2 '-~

3: ::::I

-o~

0.8 ro O

e 0.4 .2

ro

o o 0.2 0.4 0.6 0.8

P IP c max

and hollow clay brick), magnitude of prestress, and type of loading (monotonic and cyclic) . The third series of tests will be dedicated to the study of restraining devices for tendons in PTM walls.

CONClUSIONS

Numerical solutions to the differential equations for flexure of URM members demonstrate 1) deleterious effects on buckling strength with out-of-plane ben­ding, and 2) nonlinear interaction between moment and axial load . Due to the complexity of this interaction, moment capacity is more expediently defined as a function ofaxial load at incipient instability rather than the opposite. A parameter study of vertically-spanning, simply-supported walls indicates that a broad spectrum of URM walls have a higher potential for buckling instability failure than any other failure mode currently addressed by masonry design co­des in the U.S. Furthermore, the calculations in this study suggest a need for buckling strength calculations to be addressed using an LRFD approach. The­se observations, coupled with the lack of building code provisions for buckling of transversely loaded masonry walls, may pose a far-reaching problem in view of the exposure of URM structures to lateral loads from wind and earthquakes in the U.S.

Even though the prestress force in PTM walls with unrestrained tendons is analo­gous to vertical loading in URM walls, certain behavioral aspects of post-tensio­ned masonry may affect buckling instabil ity in such members. These aspects in­clude clear space within the cavities for relative deflection of the tendons, flexural stiffness of tendons and rotational restraint of anchorages, and decompression of

effective prestress due to inelastic compression straining of the masonry. Thus, the elastic buckling theory for URM compression members may not extend to PTM walls directly.

The potential for buckling in URM compression members, and the uncertainty su­rrounding instability behavior in PTM members with unrestrained tendons, is being addressed by an ongoing laboratory investigation. Experimental verifica­tion is essential beca use there is no data available in the technical literature to ve­rify the nature and magnitude of instability in slender masonry walls that resist out-of-plane lateral loads. The experimental investigation includes tests of URM walls, PTM walls with unrestrained tendons, and PTM walls with tendon restrai­ners.

ACKNOWLEDGEMENTS

This research was conducted with financiai support from the National Science Foundation through grant CMS-990411 O, V. J. Gopu, Program Director, and the Graduate School of the University of Minnesota, Twin Cities by means of a Grant­in-Aid of Research, Artistry and Scholarship. Additional support was received from the Minnesota Office of the International Masonry Institute (O. Bigelow), the An­chor Block Company, and the Minnesota Brick Company. This support is grate­fully acknowledged.

REFERENCES

Angervo, K., Uber die knickung und trogfohigkeit eines excentrisch gedrueckten pfeilers, Staaliche Tecnische Forschungsansalt, Helsinki, 1954.

(ode of Proctice for Use of Mosonry: Ports 7, 2 ond 3, BS 5628, British Standards Institution, Lon­don, 1985.

Colville, J. , "Stress reduction factors for masonry walls," journol of the Structurol Oivision, ASCE, 1 05(ST1 0):2035-2051, 1979.

Ganduscio, S., and F. Romano, F, "FEM and analytical solutions for buckling of nonlinear ma­sonry members," journol of Structurol Engineering, ASCE, 123(1 ):1 04-111, 1997.

Masonry Standards Joint Committee, Building (ode Requirements for Mosonry Structures, ACI 530-99/ASCE 5-99/TMS 402-99, American Concrete Institute, Farmington Hills, MI, American Society of Civil Engineers, Reston, VA, The Masonry Society, Boulder, CO, 1999.

La Mendola, L., M. Papia, and G. Zingone, "Stability of masonry walls subjected to transverse for­ces," journol of Structurol Engineering, ASCE, 121 (11 ):1581-1587, 1995.

Minimum Oesign Loods for Buildings ond Other Structures, ASCE 7-98, American Soe. of Civil Engi­neers, Reston, VA, 1998, 232 pp.

NEHRP Recommended Provisions for Seismic Regulotions for New Buildings, 1997 Ed., FEMA 302, Federal Emergency Management Agency. Washington, DC, February 1998.

Romano, F., S. Ganduscio, and G. Zingone, "Cracked nonlinear masonry stability under vertical and lateral loads," journol of Structurol Engineering, ASCE, 119(1 ):69-87, 1993.

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646

5ahlin, 5., "Transversely loaded compression members made of materiais having no tensile strength," Proceedings, International Association for Bridge and 5tructural Engineering, 21 :243-253, 1961.

5chultz, A. E. and J. G. Mueffelman, " Interaction of bending and compression in masonry mem­bers," Department of Civil Engineering, University of Minnesota, Minneapolis, 1999.

Yokel, F. Y., "5tability and capacity of members with no tensi le strength," Journal of the Struc­tural Dívísíon, A5CE, 97(7):1913-1926, 1971 .

NOTATIONS

An Net area of member cross section b Section width E Material constant for nonlinear stress-strain relation Em Modulus of elasticity of masonry e Eccentricity ofaxial load ea Actual eccentricity ofaxial load at end of member e, Effective flexural eccentricity FS Factor of safety for allowable stress desígn F, Allowable flexural tensile stress for masonry F, Allowable shear stress for masonry f'm Masonry compression strength h Member height h. Effective height of equivalent pin-supported member I Moment of inertia of member cross-section about weak axis M Bending moment due to out-of-plane lateralload Mmax Maximum bending moment due to out-of-plane lateral load n Exponent for nonlinear stress-strain relation P Externai axial load Pf Euler buckling load p.c Equivalent criticai load based on cracked section depth Pc Criticai (buckling) load r Radius of gyration for member cross-section about weak axis S Section modulus for member

Member thickness U Lateral deflection of compression face Ua Maximum lateral deflection of compression face U, Lateral deflection of compression face at member end w Lateral load intensity W u Factored lateral load intensity (at ultimate) ~ Bending parameter (Mm.,/Pu,) t Vertical compression strain in masonry 'A Constant for approximate criticai load formula (cases 1 -3) (J, Vertical compression stress in masonry