NBSIR 75-703

A Failure Hypothesis for

Masonry Stiearwails

Felix Y. Yokel and S. George Fattal

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

May 1975

Final Report

Prepared for Cooperative Masonry Project:

Center for Building Technology

Brick Institute of America

Masonry Institute of America

National Concrete Masonry Association

Tri-Service Building Materials Investigational Program
, ;

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'

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J, .... •
NBSIR 75-703

A FAILURE HYPOTHESIS FOR

MASONRY SHEARWALLS

Felix Y. Yokel and S. George Fattal

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

May 1975

Final Report

Prepared for

Cooperative Masonry Project: Center for Building Teclinology

Brick Institute of America

Masonry Institute of America

National Concrete Masonry Association

Tri-Service Building Materials Investigational Program

" %

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistartt Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Table of Contents

Page

List of Figures , . ii

List of Symbols iii

Abstract 1

1. Introduction , . . . 1

2. Scope 1

3. Loading Condition 1

4. Stress Distribution 3

5. Failure Hypotheses 5

5.1 General 5

5.2 Failure by Critical Normal Stress , . . . 11

5.3 Failure by Critical Combination of Normal

Principal Stresses 13

5.4 Failure by Critical Tensile (Extensional) Strain 16

5.5 Failure by Joint Separation 16

6. Analysis of the Test Results 17

6.1 Test Data • • 17

6.2 Comparison of the Test Results with Load Capacities

Predicted by the Failure Hypotheses • • • 17

6.2.1 Masonry Type AC 17

6.2.2 Masonry Type AH 21

6.2.3 Masonry Type BH . . . 21

6.2.4 Masonry Type SH 21

6.3 Interpretation of Test Results 25

6.4 Prediction of Load Capacity 27

7. Summary 27

8 . Acknowledgement 28

9. Bibliography 29

Appendix 30

A.l General 30

A. 2 Failure by Critical Normal Stress 31

A. 3 Failure by Critical Combination of Normal Principal Stresses. . . 32

A. 4 Failure by Critical Extensional Strain 33

•I
List of Figures:

Page

Fig. 1 Loading Condition * 2 Fig. 2 Stress Distribution 4 Fig. 3 Calculated State of Stress in the Center of a Panel 6

Fig. 4 Failure Modes of Specimens Type AC 8

Fig. 5 Failure Modes of Specimens Type AH. ....... 9

Fig. 6 Failure Modes of Specimens Type BH and SH . 10

Fig. 7 Failure Hypotheses for Splitting 12

Fig. 8 Strength Envelopes 14

Fig. 9 Test Series AC 20

Fig. 10 Test Series AH 22

Fig. 11 Test Series BH 23

Fig. 12 Test Series SH 24

\

•

, ,

^

. ,

1

'

11
List of Symbols

The following symbols are used in this paper:

b width of panel

unconfined compressive strength of masonry

f" calculated strength parameter defined as the apparent splitting

s strength if = 0.

flexural tensile strength of horizontal masonry cross section

diagonal compressive load

diagonal load capacity when P^ = 0 resultant vertical edge load

R' ratio f'/x' m d

R ratio f'/f" m s t thickness of panel

a angle of with horizontal

Bp unconfined compressive strength of concrete

a„ P /bt V V

y coefficient in the joint separation equation

V Poisson's Ratio

a-^ principal tensile (or smaller compressive) stress in-plane of panel

principal compressive stress in plane of panel

^2 principal stress normal to plane of panel

Oy nominal vertical stress (perpendicular to bed joints)

T nominal shear stress (parallel to bed joints)

0.707 P' tI = nominal shear strength when P =0

^ bt ^ V

calculated strength parameter

T shear strength at a =0 o y

calculated strength parameter in an approximate equation for in-plane

splitting failure

Subscript "c" denoting "critical," identifies stress at failure.

"Prim.e" as in or denotes a measured strength term.

"Double prime" such as in f" denotes a calculated strength parameter

s ill

A FAILURE HYPOTHESIS FOR MASONRY SHEAR WALLS*

by

Felix Y. Yokel and S. George Fattal

Various failure hypotheses for wall panels subjected simultaneously to

diagonal compressive load and to vertical compressive edgeload are compared

with the results of thirty two tests on four types of brick masonry walls

which were published elsewhere. It is concluded that failure can occur by

joint separation or by splitting. A failure hypothesis is advanced which

is shown to be in good agreement with the test results examined.

Key Words ; Brick; failure; failure theories; masonry; shear walls; shear

strength; shear test; stresses; stress distribution; structural

engineering.

1. Introduction

This study deals with the load capacity of clay masonry walls subjected

to a diagonal compressive load combined with a compressive edge load acting

in the plane of the wall and normal to the direction of the mortar bed

joints. The loading and boundary conditions of these walls are similar to

those encountered in certain shear wall elements in buildings.

2. Scope

Various failure hypotheses are compared with the results of thirty-two

tests of 48 in. x 48 in. (122 cm. x 122 cm.) single-wythe clay-brick walls

which were published elsewhere [4]—'^. A failure hypothesis is advanced which

accounts for the observed failure modes.

3. Loading Condition

The loading condition studied is shown in figure 1. the diagonal

compressive load and is the resultant of the distributed edge load. For the

purpose of this study, the following nominal stresses are introduced: x, the

—Numbers in brackets refer to literature references in Appendix I. Research performed by the Cooperative Masonry Project: Center for Building

Technology, Brick Institute of America, Masonry Institute of America, National

Concrete Masonry Association, Tri-Service Building Materials Investigational

Program Committee.

1
2
nominal shear stress, is the horizontal component of divided by the horizontal cross sectional area, bt; o^, the nominal vertical stress, is the resultant

vertical component of the applied load which consists of and the vertical

component of P^, divided by the horizontal cross sectional area. In the

actual tests examined in this report wall width, b, was approximately 48

in. (122 cm) and wall thickness, t, was approximately 3.7 in. (9.4 cm.). Load

P^ was transmitted by a triangular loading shoe with a 6 in. (15 cm) side

length and P^ was transmitted by three 11 in. (28 cm) long loading blocks.

4. Stress Distribution

In this analysis it is assumed that strength can be correlated with

hypothetical stress levels at failure, calculated using an isotropic linearlyelastic model. In view of the lack of specific information on the directional

variation of the elastic properties of the test specimens an anisotropic model

was not feasible. However, a numerical analysis using a 2 to 1 ratio between

the orthogonal elastic moduli indicated that the results of the analysis presented in this study will not be significantly altered by considerations of

orthotropy. Actual local stresses resulting from discontinuities and

material flaws are likely to differ from the calculated stresses.

The elastic stress distribution on a square plate loaded by P^ alone

was approximately calculated by Frocht [3] who also documented reasonable

agreement between calculated stresses and those observed in a photoelastic

model. Frocht simplified his equations by assiaming Poisson's Ratio to be zero.

In figure 2, Frocht 's solution is compared with the results of a finite

element analysis using 78 elements in the first quadrant (y = 0 to h, x = 0

to h) , with element sizes of h/16 near the center and the corners and h/8

elsewhere. The calculated stress distribution is shown for v = 0 and v = 0.3, in which v is Poisson's Ratio. For the case v = 0 the principal

tensile stress in the center of the panel calculated by the finite element

method was 4 percent less, and the principal compressive stress 3 percent

more than the corresponding stresses calculated by Frocht. For the case

V = 0.3, the tensile stress calculated by the finite element method approximately equaled Frocht 's solution and the compressive stress was 4 percent

less. On the basis of the finite element investigation it was concluded

that the effect of Poisson's ratio on the magnitude and distribution of

calculated stresses is not significant enough to warrant consideration in

this study.

In figure 2, principal stresses are shown non-dimensionally as multiples

of T. In the following discussion always denotes the principal tensile

stress or the smaller principal compressive stress in the plane of the

3

wall, denotes the (larger) principal compressive stress in the plane of

the wall and the principal stress normal to the plane of the wall, is

assumed 0.

Frocht calculated the following principal stresses in the center of

the wall in figure 2:

= 0.7336T .... (1) = -2.380T .... (2)

in which compressive stresses are negative, tensile stresses are positive,

and T is assxmed positive. A Mohr Circle representation of the corresponding state of stress is shown in figure 3 (a) , which also shows the

stesses on an infinitesimal cube whose sides are parallel to the sides of

the wall. The orientation of the plane normal to with respect to the

horizontal (the direction of bed joints is assumed "horizontal") is at an angle a = 45".

Figure 3 (b) shows the state of stress calculated in the center of a panel subjected simultaneously to and P^. Superposition is used by

P

adding stress a = — to the stresses acting on the infinitesimal cube ^ bt

shown in figure 3. The following expressions can be derived from Mohr's

Circle shown in figure 3 (b)

a / , a2 .

(J = -0.823T + — + 1/(1.556^)^ + — (3) ^ 2 r—'

^

a = -0.823T + — - \(1.556t)^ + —

2 ' 4 1 ^-1 , °v X (5) o = — cot ( ) ^

' 3 .112T ons aV

is tensile. t is always assumed positive.

in these expressions is negative if P^ is compressive and positive if P^

5. Failure Hypotheses

5.1 General

Not all the failure hypotheses that have been advanced for materials similar

to masonry are investigated in detail, since the test information considered

does not provide an adequate basis for such an approach. Three failure hypotheses for splitting and one hypothesis for joint separation are considered.

5
T

(a) PANEL LOADED BY Pd ALONE

(b) PANEL LOADED BY P^ and Py

FIGURE 3 CALCULATED STATE OF STRESS IN THE CENTER OF A PANEL

6
The hypotheses for splitting are: failure by critical normal stress; failure

by a critical biaxial combination of normal principal stresses, a concept,

recently corroborated for concrete by test results [8] , which covers a broad

spectrum of different failure hypotheses advanced in the past; failure at a critical in-plane tensile (extensional) strain, a hypothesis which could be

utilized in a single failure criterion which would account for "tensile", as well as "compressive" failures.

Failures that are actually documented for the tests discussed in this

paper are shown in figures 4 through 6 and can be placed in three general

categories:

1. Separation along mortar joints. (See, for instance, figure 4.,

Specimen AC 2)

2. Splitting, generally in the direction of (the "direction

of splitting" is taken as the direction in which material

particles separate, which is normal to the direction of crack

propagation) , in a region along the loaded diagonal which

includes the center of the panel. (See, for instance, figure 5,

AH2-4)

3. Splitting, approximately in the direction of a^, most severe in

the vicinity of the loading fixtures and not necessarily

including the center of the panel. (See, for instance, figure 5,

AH8) , No evidence of failures by crushing of masonry units or mortar joints or by

splitting in the direction normal to the plane of the wall is observed in

the records available.

The following investigation of failure hypotheses related to splitting

is entirely based on the elastic state of stress in the center of the

panel. This approach provides a valid predictive model if two conditions

are satisfied:

1. The failure is brittle (there is no substantial re-distribution

of stresses at load levels lower than those defined as failure

loads)

2. The most critical condition occurs at or near the center of

the panel.

7
/ \ \ \ / III/

TT , , , ,

1 r

I I I T

III 1^

AC2

1 1 1 /

2

TTT

TTTIII

1 1 i

ACS

in /

It

^

1 1

ACS

FIGURE i| FAILURE MODES OF SPECIMENS TYPE AC

8
/ \ \ \ / lit/

iz

z

AH2

I I I T

TTT

TTT

1=1

I I t

AH3

2

eg

1 1

AH4

I I I

5

^

t t t

AH5

Illy

I t I

AH7

III/

I . I TTT

t t I

AH8

FIGURE 5 FAILURE MODES OF SPECIMENS TYPE AH

9
BH 3 BH 7
For splitting failures (categories 2 and 3) the first condition is reasonably

satisfied. The second condition is satisfied for failure category 2 previously

discussed in this section, but not necessarily for failure category 3 where

the most critical conditions may occur in locations other than the center

of the panel.

The interpretation of compression failures near the corner fixtures

and the loading blocks would have to account for the complex confining

effect of these devices. This was not attempted since the load transfer

conditions causing category 3 failures in test specimens are not normally

anticipated in shearwall elements of buildings.

5.2 Failure by Critical Normal Stress

According to this hypothesis the wall would fail either if exceeds

the tensile strength or if a, exceeds the compressive strength. Curve 1 in

2/ - - figure 1—' shows critical combinations of t and that would cause failure

in accordance with this hypothesis, t and are given as multiples of

0.707 Pi

f' = .... (6) ^ bt

in which is the diagonal failure load when P^ = 0 (t/t^ = 1, ^y/'^^j = -1) . Curve 1 was calculated on the basis of the premise that at t/t^ = 1, and

Oy/x^ = -1, equaled the tensile strength of the material. Curves 1(a) and

1(b) represent compressive failures in the center of the panel corresponding

to two ratios: R' = "8.3, and R' = -9.7 typical for the test specimens in

which

J

,

R' = -21 (7)

and f^ is the (unconfined) compressive strength. As previously noted, in the

specific tests examined, compressive failure apparently did not originate in

the center of the panel. The information conveyed by curves 1(a) and 1(b) is

therefore not applicable to the test results.

Curve 1 was calculated without specific consideration of directional

variations in splitting strength. Such variations are known to exist.

Johnson and Thompson [6] reported an increase in splitting strength as the

^

The equations for all the failure envelopes shown in figure 7 are derived

in the appendix.

11
12
orientation of the splitting crack with respect to the horizontal (direction of bed joints), (a), increases from 0° to 90°. For a change from 45° to

90°, the range occuring between a /xl = -1 and a /xl = f'/^i, the reported

y a y a m a strength increase ranged from 6 percent for masonry with a 90°-splitting

strength of 490 psi (3.38 MPa) to 25 percent for masonry with a 90°-

splitting strength of 275 psi (1.9 MPa). Consideration of such a change

in strength would increase the calculated strength when o^/x^ < -1. The

calculated strength when ^y/^^ > -1 would decrease considerably if the

substantial decrease in splitting strength from a = 45° to a = 0° were

considered. This range is not examined in this paper, since no pertinent

test results are available.

5.3 Failure by a Critical Combination of Normal Principal Stresses

This hypothesis assumes that both principal stresses in the plane of the

wall contribute to the failure as suggested by Griffith for brittle fracture [5] . Recent experimental work by Kupfer et al. [8] corroborates this

concept for concrete. One of Kupfer 's strength envelopes for concrete is

shown in figure 8 (a) . A strength envelope determined by Khoo and Hendry

[7] from biaxial tests on 1 in. -square by 3 in, (2.5 cm-square x 7.6 cm)

specimens cut from 1/3-scale model brick is shown in figure 8(b). Khoo's

interaction curve does not reflect the effect of the mortar in the brick

masonry since the specimens were cut from a brick unit. The assumed strength

envelope for masonry used in this investigation is shown in figure 8 (c) . In

the range ^ ^ ^ ^ -^i the simplified linear strength envelope in the

second quadrant of figure 8(c) satisfies "Mohr's Failure Theory" [12], and

is uniquely defined by two points: one corresponding to the state of

stress at failure under vertical compressive load, = 0, = f^^; and the

other corresponding to the previously-defined state of stress under failure

load P^, = 0.7336 x^, = -2 .380 x^ . Parameter f", the apparent major

principal stress at failure when = 0, is derived as:

f

• f" = 0.7336 t' 1— (8)

^ ° (f + 2.38 t') m d

The nondimensional parameter R defined by

f

^

f.. (9) s and introduced in figure 8(c) for convenience, is related to the measured

f

• quantity R' = — by the following equation:

^d

13
1.2 IjO .8 .6 .4 .2 —.2

--.2

--.4

--.6

H.8

1.0

1—1.2

= UNCONFINED COMPRESSIVE

STRENGTH OF CONCRETE

(a) CONCRETE (KUPFER et AL) (b) MODEL BRICK (KHOO & HENORY)

APPROXIMATE RANGE

OF TEST DATA

CURVES 2,3,and 4 (fig?)

(c) ASSUMED STRENGTH ENVELOPE

FIGURE 8 STRENGTH ENVELOPES

14
R = 1.36 R' + 3.24 (10)

In accordance with the strength envelope in figure 8(c) failure would occur

at the following critical stresses:

if < 0 <

a. = f" -

1 s

if = 0

3 _ m 3 . , . , , (11)

R R

a. = f • (12)

J in

if < 0

a- = f ' + a. > 1.2 f- (13)

3 m 1 - iti

In eq. (11) is positive, in eq. (13) is negative, and o^, f^

and ratio R are negative

The data in figure 8 indicate that the interaction curve between

point = f^, = 0 and point = 0, ^-^ = ^m^^ necessarily be

linear. It is also know that the splitting strength changes with the

orientation of a^, and tends to decrease substantially as becomes normal

to the mortar bed joints [6] . It is assumed that for the loading and

boundary conditions considered in this study, figure 8(c) which is used to

derive equations (11) through (13) should lead to an approximate estimate

of strength in the range o^/t^ < -1 . When extrapolated to the range

a^/x^ > -1 which corresponds to a tensile edge load P^, for which no test

data are available, the estimate is probably too high, with the discrepancy

increasing as a ^ 0. For more general application of this hypothesis the

directional variation in the splitting strength would have to be considered,

so that:

at failure = /(a^,a) (14)

In figure 7 the interaction curves, calculated by this failure hypothesis,

are shown as curves 2, 3, and 4 for values of R = -8, R = -10, and R = -16,

respectively. The previously discussed qualifications for compressive

failure also apply to the descending part of these curves, which signify

"compressive failures." As in previous cases, calculated nominal stresses

at failure are based on the premise, f f that at a /x' = -1, and t/tI = 1, failure

y' d ' ^ d

' occurs by the failure criteria of this hypothesis. As previously noted, the

extension of the curves to o^/t^ > -1 probably overestimates strength.

15
5.4 Failure by Critical Tensile (Extensional) Strain

In accordance with this hypothesis failure occurs when a critical tensile

strain is exceeded:

-va^ = constant (15)

In this case Poisson's Ratio (v) has a considerable effect on calculated

load capacity. Not much information is available on Poisson's Ratio for

clay masonry. For concrete a range from 0.16 to 0,25 was reported [9]

Blume et al, [2] estimated Poisson's Ratio of reinforced brick masonry used

in their test program to range from 0,10 to as much as 0.25 at high stresses,

and Sahlin [10] reports measurements ranging from 0.2 at initial loading to

0.35 at ultimate load on 3 x 3 x 6 cm prisms cut from brick units. For

this study a ratio of 0.2 was assumed. Curve 5 in figure 7 was calculated,

assuming that, at failure;

- 0,2 0^ = constant, . , , ^ (16)

As for the previous failure hypotheses, curve 5 is normalized so that

failure would occur at t/x^ = 1, '^y/^^ = -1. It can be seen from curve 5 that, by this hypothesis, failure under edge load P alone would occur at

a^/T^ = -6,05. However P^ could be increased if a diagonal load is simultaneously applied. This apparent strength increase can be explained by the

fact that, within certain limits, principal extensional strains decrease

when diagonal load P^ is superposed on edge load P^.

It is noted that in the range a, < 0 < a. the critical tensile strain ~ ~ 1 hypothesis is identical to the strength envelope in figure 8(c) if v = --, For the three typical values observed for the specimens examined herein

of R = -8, R = -10 and R = -16, the corresponding v values would be 0.125, 0.1

and 0,063, These ratios appear to be low when compared with available

information, even though that information is very limited,

5.5 Failure by Joint Separation

A typical joint separation failure is shown in figure 4 for specimen

AC2. A criterion for joint separation, reported by many investigators [11]

can be formulated as follows:

= - ... .(17)

in which t is the nominal shear stress at failure,

c

'* is the nominal shear stress at failure when a =0, and y is a o y coefficient, viewed by some authors to be attributable to friction

[111 .
Since this criterion is stated in terms of nominal stresses it can be

postulated that the failure load is not sensitive to the stress distribution producing the nominal stresses as long as a change in stress distribution does not precipitate a change in the failure mode.

6. Analysis of the Test Results

6.1 Test Data

The test data taken from reference [4] are summarized in Tables 1, 2,

and 3. Three brick types and two mortar types were used. Table 1 shows the

properties of the brick units which were measured in accordance with ASTM

C67 [1] . The brick were used to produce four types of masonry designated

AC, AH, BH, and SH, in which the first letter designates the type of brick

used and the second letter the type of mortar. "C" stands for conventional

mortar, "H" stands for high-bond mortar which had a polyvinylidene chloride

admixture to increase the flexural strength of the masonry.

Table 2 gives the measured unconfined compressive, and flexural tensile

strengths of a horizontal cross section for each type of masonry.

The test results are summarized in Table 3. In columns 7 and 8 of

the table nominal failure stresses are given as multiples of the average

value of T^. Specimens ACl, AC2, and AC3 failed by joint separation (see

figure 4) and it is reasoned that would therefore not be a meaningful

measure of splitting strength. A parameter was therefore calculated,

using specimen AC4 which was arbitrarily selected, being the specimen with

the lowest vertical edge load which gives no visual indication of joint

separation failure, and the strength envelope in figure 3(c) with an R

value of -16. was calculated to be approximately 1.3 x^.

6.2 Comparison of the Test Results with Load Capacities Predicted by the

Failure Hypotheses.

(

6.2.1 Masonry Type AC

Figure 9 shows a plot of Test Series AC together with curves 1, 4, and

5 for the three failure hypotheses for splitting (refer also to figure 7)

The failure modes of the Type AC specimens are shown in figure 4. As previously noted, specimens ACl, AC2, and AC3 definitely failed by joint separation. A straight line connecting ac3 with the average of ACl and AC2 is

approximated by the equation:

17
Table 1. Brick Units

Designation

A

B

C

Unit Compressive

Strength in psi

14.480

20,660

17,560

Modulus of

Rupture in psi

848

761

741

Initial Rate

of Absorption

Type of

Unit

g/ (min. x 3 0 in. )

6 . 2 cored

2.6 cored

19.8 so] id

Note! I psi = 6-9 kPa 1 in.^ = 6.45 cm.^

Table 2. Masonry Properties (based on gross area)

Type Compressive Strength f in psi m

AC . ' -3,190

AlH ; -4,830

-mt-^y^^-' ''.'^-^'-r^r- -5,170

SH -6,100

Note: 1 psi = 6.9 Pa

Flexural Strength

f^ in psi

80

360

452

220

18
Table 3. Test Results

Type

AC

AH

BH

SH

scimen PV

P

,

d

-a

y

X a

- _y

X In in in in

kip kip psi psi d

ACl 0 44 175 175 .753- .753-

AC2 0 45.5 181 181 . 780 .780

AC3 25 68 410 270 1.766 1. 162

AC4 50 89.2 636 355 2.739 1.529

ACS 75 115 878 457 3.782 1.968

AC6 100 127 1068 506 4. 600 2.179

AC7 125 133 1232 530 5.306 2.283

ACS 150 141 1409 560 6.073 2.412

AHl 0 141 560 560 1.101 1.101

AH2 0 115 457 457 0. 898 0. 898

AH3 50 168. 5 951 670 1. 870 1. 318

AH4 100 178 1270 708 2.498 1.392

AH5 150 238 1778 946 3. 497 1. 860

AH6 200 209 .1953 833 3. 841 1.638

AH7 250 258 2420 1020 4.759 2.006

AH8 450 233 3456 926 6. 796 1. 821

BHl 0 161 608 608 0.9''3 .973

BK2 0 167.5 642 642 1. 027 1. 027

BH3 50 245 1191 924 1. 906 1.478

BH4 50 231 1139 872 1.822 1.395

BH5 100 205 1308 774 2. 093 1. 238

BH6 100 246 1728 928 2. 765 1.485

BH7 200 261 2050 984 3. 280 1. 574

BH8 300 280 2651 1056 4 . 242 1.690

SHI 0 167 690 690 . 932 .932

SH2 0 175 724 724 . 978 . 978

SH3 0 195 806 806 1. 089 1. 089

SH4 50 217 1191 898 1.609 1.214

SH5 100 257 1646 1060 2.224 1.432

SH6 125 274 1862 1130 2.516 1.527

SF7 150 297 2109 1230 2.850 1.662

SH8 200 306 2452 1280 3.313 1.730

a/

Note: 1 kip = 4.45 kN 1 psi = 6.9 kPa —^In AC specimens is substituted for

19
20
. (18)

It is observed that an extension of this line also closely predicts the

failure loads of specimens AC4 and ACS. This observation suggests that in

these two specimens, even though the failure had the visual appearance of

splitting, this splitting failure was triggered by joint separation. This

proposition is reinforced by an examination of the failure mode of specimen

AC3 which exhibits characteristics of both joint separation and splitting.

Specimens AC6, AC7, and ACS plot in the vicinity of curve 4 which represents the strength envelope hypothesis for type AC masonry. However, the

uncertainty in the determination of raises doubt about the correct

position of the plotted test points for specimens AC relative to curves 1,

4 , and 5

6.2.2 Masonry Type AH

The AH test series is plotted in figure 10 together with curves 1, 3,

and 5 from figure 7. Figure 5 shows the observed failure modes. For all

specimens the failure mode appears to be splitting, and for the specimens

subjected to a high the splitting is pronounced near the loading fixtures and does not always extend through the center of the panel.

The plot of the test results shows reasonable agreement with curve 3 for the strength-envelope hypothesis. The relatively low strength of specimen

AH8 appears to be attributable to failure in locations other than the

center of the panel and therefore is not adequately predicted on the basis

the state of stress in the center of the panel.

6.2.3 Masonry Type BH

Figure 11 shows a plot of test series BH. Figure u shows the failure

modes of specimens BH3 and BH7 which appear to be splitting that includes

the region near the center of the panel. The plot in figure 11 shows

reasonable agreement with curve 2 for the strength-envelope hypothesis.

6.2.4 Masonry Type SH

Figure 12 shows a plot of SH test series. The failure mode of specimen

SH5 is shown in figure 6.

21
FIGURE 10 TEST SERIES AH

22
23
3

FIGURE 12 TEST SERIES SH

24
Tests SHI through SH7 plot approximately on a straight line defined by

the equation:

T CO= T - 0. 37 Ov .... (19)

J

Specimen SH8 falls below this line. The correlation between the tests

suggest that the failure may have been caused, or triggered, by joint

separation. Close examination of specimen SH5 in figure 6 reveals partial

joint separation.

A contributing factor to this failure mode may have been the surface

condition of the units: S units were solid, while A and B units were

cored. The break in the trend of the test data evident from the relationship between specimens SH7 and SH8 suggests that at the load level of

specimen SH7 there may have been a transition from joint-separation to

splitting. Unfortunately SH5 (figure 6) is the only available intelligible

record of a failure mode for these specimens.

Curve 1 in figure 12 provides another possible explanation for the

load capacity of the SH specimens. This curve, which was calculated for

failure by critical normal stress, is closely approximated by equation (18)

within the range of the SH tests. It is noted, however, that the test

points in this figure were arbitrarily plotted on the basis of rather

than T^. Their true position relative to curves 1, 2, and 5 is not known,

since the available visual failure observation shown in figure 3 does not

provide evidence of a splitting failure.

6.3 Interpretation of Test Results

The three splitting-failure hypotheses are mutually exclusive, however both

splitting and joint separation could occur simultaneously in any one specimen

or, for the same type of masonry, either splitting or joint separation could

occur in different ranges of o^/t^.

Whether joint separation or splitting governs depends on the relative

magnitude of coefficient y in eq. (18) as well as on the resistance of the

mortar joints to separation when = 0. Coefficient y was found to be

approximately 0.4 for types AC and SH masonry. The resistance of the

mortar joints to separation depends on the tensile and shear strength of

the mortar joints and on the bond between mortar and masonry units. It is

reasoned that f^, the flexural strength of a horizontal section, which

depends on the tensile strength of the mortar and on the bond between the

25
mortar joint and the masonry unit should be related to the resistance to

separation of mortar joints. Accordingly, the ratio T^/f^ should give an indication whether the failure under load is attributable to joint

separation or to splitting. These ratios are calculated below for the four

types of masonry:

Type Average f^, psi

- psi

AC 178 80 2.23

AH 508 360 1.41

625 452 1.38

SH 740 220 3.36

NOTE: 1 psi = 6.9 kPa

The calculated ratios are substantially higher in type AC and SH masonry,

which adds substance to the proposition that in these types of masonry

failure under occured by joint separation. On the other hand, the ratios

for AC and SH differ among themselves.

The correlation between the splitting hypotheses and the test results

is examined on the basis of the AH and BH tests which failed by splitting.

An examination of figures 10 and 11 shows that:

1. most test results fall below curve 1 and above curve 5;

2. the curves based on the strength envelope in figure 7 reasonably

approximate the test results; and

3. for large compressive edge loads test results tend to fall

below these curves.

On this basis it is concluded that the failure hypothesis based on the

strength envelope in figure 8(c) provides the best predictive model for the

test results. For the type of load-transfer mechanism used in these tests

curves 2, 3, and 4 apply in the approximate region of a /xi > -5. For

higher compressive edge loads failure can not be predicted on the basis of

the calculated state of stress in the center of the panel.

It can be shown that, for the same type of masonry, joint separation

could occur below a certain compressive edge load (P^) , and splitting for

higher values of P^.

The approximate equation derived from the experimental results for

joint separation is:

26
= " '^y .... (18)

Curves 2-4 for the range -1 > ^y/^^ > ~5 fall approximately between the

following boundaries:

T coy = t" - 0.175 a for R = -8 .... (20)

T = T" - 0.225 a for R = -16 .... (21) c o _ _ y

in which x" > t' if failure at t' was by joint separation,

o - o d

^

If failure at occurred by joint sepration the line defined by eq. (18)

could intersect the linesdefined by eqs . (20 or (21) at a value of (^y/t^ > -5,

causing the failure mode to change from joint separation to splitting.

This leads to the important conclusion that the equation for joint separation should not be extrapolated beyond the range within which it is known

to be valid.

6.4 Prediction of Load Capacity

If t' and f are determined by test, and if the failure at t1 is by d m _ ^ ' d

diagonal splitting, can be determined by extrapolation within the range

-1 > cTy/t^j^ > ~5 using curves 2, 3, or 4 or approximate eqs. (20) or (21). If

is determined by test and failure is by joint separation, the result cannot

be extrapolated without additional information on the value of y , and on the

range of cf^/x^ within which joint separation failure occurs. Such information could be obtained by tests similar to those examined herein.

Whether the failure at x^ was by splitting or by joint separation may

be difficult to determine. Visual inspection alone may be misleading in cases similar to those discussed for specimens AC3, 4 and 5. Ratio x^/f^

could serve as an indicator of the failure type, however the information in

this paper would have to be supplemented by more test data to provide a sufficient basis for such an approach. A more reliable approach would be

to provide independent information on the splitting strength envelope,

coefficient y, and the resistance to joint separation, using other test

methods in addition to diagonal compressive tests. Presently used standard

tests do not provide this information.

7 . Summary

1. Failure under combinations of compressive diagonal and edge loads

(figure 1) can occur by joint separation or splitting.

27
For the same type of masonry, failure may occur by joint separation

under small compressive edge loads (P^) and change to splitting above a certain level of P^. It is therefore not advisable to extrapolate data

obtained for joint separation at low levels of P^ to estimate load capacity

at higher levels of P^.

2. Those test data which provide indication of joint separation

failure are in good agreement with equation (17) : ~ ^o

~ ^^y ^-"-^^ approximate y value of 0.4.

3. Where failure is caused by splitting, load capacity is reasonably

predicted on the basis of the assumption that, within the range -1 > o^/t^

failure originates in the center of the panel and is caused by a critical

combination of principal normal stresses.

Within the same range, load capacity is overestimated by the hypothesi

that failure occurs when a critical principal tensile stress is exceeded,

and underestimated by the hypothesis that failure occurs at a critical

extensional strain assuming a Poisson Ratio of 0.2.

'

. \

'

. 8. Acknowledgement

The authors are indebted to Louis E. Cattaneo from the Center for

Building Technology, National Bureau of Standards, who made available

unpublished information related to Reference [4] and critically reviewed

this report, and to the Brick Institute of America, the Masonry Institute

of America, the National Concrete Masonry Association and the Tri-Service

Building Materials Investigational Program Committee who within the framework of the "Cooperative Masonry Program" participated in the funding of an investigation of masonry shear walls which included this study.

Departments of the Army, Navy and Air force.

28
9. Bibliography

[1] ASTM Designation C67-66, "Methods of Sampling and Testing Brick",

American Society for Testing and Material, Philadelphia, Pa., 1966.

[2] Blume, T. A., and Prolux, J., "Shear in Grouted Brick Masonry

Wall Elements," Western States Clay Products Association, San

Francisco, Cal., 1968.

[3] Frocht, M.M. , "Recent Advances in Photoelasticity , " Transactions

ASME , Vol. 55, pp. 135-153, Sept. - Dec. 1931.

[4] Grenley, D. G. and Cattaneo, L. E., "The Effect of Edgeload on Racking Strength of Clay Masonry," Proceedings, Second International

Brick Conference, -Stoke on Trent, G. B. , The British Ceramic

Research Association, Stoke-on-Trent, ST4 7LQ, April 1970.

[5] Griffith, A. A., "The Phenomenon of Rupture and Flow in Solids,"

Phil. Trans. Roy. Soc

, A221, 163, London 1921.

[6] Johnson, F. B. and Thompson, T. M., "Development of Diametral

Testing Procedures to Provide a Measure of Strength Characteristics

of Masonry Assemblages," Design Engineering and Construction of

Masonry , F. B. Johnson, ed. , Gulf Publishing Co., Houston, Texas

1969.

[7] Khoo, C. L. and Hendry, A. M., "A Failure Criterion for Brickwork

in Axial Compression," Preprint 2b2 , 3rd International Masonry

Conference, Essen, Germany, 1973.

[8] Kupfer, H., Hilsdorf, H. K., and Rusch, H., "Behavior of Concrete

under Biaxial Stresses," J., American Concrete Institute , Vol. 66,

No. 8 August 1969.

[9] Reichard, T. W., "Creep and Drying Shrinkeage of Lightweight and

Normal Weight Concrete," NBS Monograph 74 , National Bureau of

Standards, Washington, D. C, (1964).

[10] Sahlin, S., "Chapter D", Structural Masonry , Prentice-Hall, Inc.,

Clifton, N.J. 1971.

[11] ibid. , "Chapter G"

[12] Timoshenko, S., "Chapter 10, pp. 459-462," Strength of Materials , Part II , 3rd Ed., Van Norstrand, N.Y., 1956.

29
APPENDIX

- Derivation of Equations for Splitting Failure Hypotheses

A. 1 General

For brevity, the symbol is introduced in the following derivations to

denote stresses normalized with respect to which is defined by eq. (6)

Accordingly, the general (biaxial, in this c state of stress at the center

of a square panel under diagonal and edge loading (figure 1) is described by

reference to figure 3(b) as follows,

= -0. 823 T r . (A.l)

'

: : = - 0.823 T (A. 2)

^ , = 1.557 T (A. 3) xy

where a = a a = a /t^ and so on. Likewise, division of eqs. (3) and (4) _ X x'^ d y y d ' ^ v ' by yields the corresponding equations for the non-dimensionalized principal

stresses

a I ~F

O " A 0^ = -0.823 T + — + \ (1.556 t) + — . . (A. 4)

. ^ 2 ' 4_

~ « ^v ^/ - 2 ^v . a-. = -0.823 T + — - 1/ (1.556 t) + — . . (A. 5)

2 ' 4 The equation

5 = - ? ....... (A.6)

y V

which follows from the definitions of nominal stresses given in Section 3, may

be used to eliminate from eqs. (A. 4) and (A. 5). The resulting equations

a, = -0.323 T + JL +J2.6I + 0.5 to + 0.25 ... (A. 7) 1 .. 2 \ y y

a, = -0.323 T + i -^2.61 + 0.5 Ta„ + 0.25 a?. . . . (A. 8)

2 simplify the subsequent formulations of t as an explicit function of a^.

30
A. 2 Failure by Critical Normal Stress

According to this hypothesis, failure occurs when either or attains

its critical stress level for a given configuration of diagonal and edge load

intensities. Under the assumption that critical tensile stress is attained

first, eq. (1) becomes

a. = 0.734 T' or (A. 9) = 0.734

The equation for the tensile failure envelope (curve 1 in figure 8) is derived

by solving eq. (A. 7) for t after substituting the value of from eq. {A.9)

a +-Jo.i

y 1

T = 0.0922 - 0.1601 a.. +-f/0.0256 - 0.3152 + 0.2181 . . (A. 10)

The case of zero edge load (a^ = 0) which should correspond to failure under

diagonal load alone, is represented by the point (a^ = -1, t = 1) , as may be

readily verified by substituting these values in eqs. (A. 6) and (A. 10) . The

T-axis intercept of curve 1 in figure ^ (ay ss 0, x = 0.559) calculated from

equation (A. 10) corresponds to the case of a positive (tensile) edge load which

is equal (and opposite) to the vertical component of the diagonal load at

failure, as may be ascertained from eq. (A.6) . The maximum positive edge load

for which eq. (A. 10) is valid may be calculated by equating the expression under

the radical sign to zero, solving for a^, substituting it back into eq. (A. 10)

and solving for t. The resulting values are (a^ = 0.635, t = 0.159) which,

upon substitution into eq. (A.6) yield a =0. This places a lower bound on curve 1 in figure 7 so that the following should hold if eq. (A. 10) were to be

valid,

a > 0.635, T = 0.159 and a > 0.794

y — v —

As the edge load increases in the negative sense (compression) , curve 1

approaches the asymptote 5^/1 = -3.123 which may be verified from eq. (A. 7)

by setting = 0.

Noting that is equal to R' defined by eq. (7) , the equation for the

compressive failure envelope is derived from eq. (8) after substituting R' for

^^3'

T = 0.1257R' - 0.1601 ay + ^.4050 (R')^ - 0.4294R' + 0.0256 . . . (A. 11)

31
Curves 1(a) and 1(b) in figure 7 were constructed using R' = -8.3 and

R' = -9.7 in eq. (A. 11) respectively, which were typical for the test specimens at hand (Section 5.2). Depending on the value of R', one or the other

of these curves together with curve 1 define the failure envelope for the

critical normal stress hypothesis. Note that curve 1 and the a^-axis define

the physical bounds within which eq. (A. 11) is valid while curve 1(a) or 1(b)

imposes an upper bound on eq. (A. 10) . These junction points (of discontinuity)

delineate a state of stress at which tension and compression attain their

critical values simultaneously.

A. 3 Failure by Critical Combination of Normal Principal Stresses

On the basis of the idealized strength envelope depicted in figure 8(c),

the bilinear relationship between and as prescribed by eqs. (11) and (13)

describes the critical stress combination for which failure is imminent

according to this hypothesis. The failure envelope corresponding to the

region in the second quadrant of figure 8(c), is derived by normalizing

eq. (11) as follows,

. , . . ^1 = I

" °3^ (A. 12)

and combining it with eqs. (A. 7) and (A. 8) . The resulting equation

[2.6742 (R - 1)^ - 0.1043 (R + 1)^]t^

+ [0.5 (R - l)^ +0.323 (R + I) ^ - 0 . 646 R ' (R + 1)

]

t

+ [R'(R + 1) 0^ - Ra^ - (R')2] =0 (A. 13)

was used to construct the ascending portions of curves 2, 3 and 4 of figure 7

,

for the respective R' values of -8.3, -9.7 and -14.1, which are representative of three types of masonry used in the tests. The corresponding values of

R, namely -8, -10 and -16, used in the same equation, were calculated in

accordance with equation (10)

The equation for that portion of the failure envelope falling in the third

quadrant of figure 8(c) is obtained by normalizing eq. (13) as follows

- - v - = - R' (A. 14)

and combining it with eqs. (A. 7) and (A. 8) . The resulting equation

+ (0.187 Oy) T + 0.0935 [aj - (R')^] =0 ... (A. 15)

32
was used to construct the descending portions of curves 2, 3 and 4 (the latter

falls outside the range shown in figure 8) for the same values of R' and R as used in eq. (A.13)

A. 4 Failure by Critical Extensional Strain

According to this hypothesis, failure will occur when the principal extensional strain attains a critical magnitude K. Stated mathematically,

- va^ = K (A. 16)

where v is the Poisson's ratio. The constant K = 1.21 is evaluated by substituting = 0.734 and = -2.380 obtained from eqs. (1) and (2) for the

critical state at zero edge load together with v = 0.2 used in this study into

eq. (A. 16) . This gives

- 0.2 = 1.21 (A. 17)

which, when used in combination with eqs. (A. 7) and (A. 8), yields the equation

for the failure envelope of this hypothesis. Noting, however, the similarity

of eqs. (A. 17) and (A. 12) , the governing equation in this case may be readily

obtained from eq. (A.13) . From eqs. (A. 12) and (A. 17)

R' ^3

I

i = 1.21 + 0.2 a3 (A. 18)

which, together with eq. (10), yields R = -5 and R' = -6.05. Substitution of

these values in eq. {A.13) gives

T = 0.0826 - 0.1224 a + /o.3937 - 0.276 a - 0.0378 . (A. 19)

y y y

or

= -2.42 - 2.3168 x - /l3.177 + 14.34 t - 13.553 . . (A. 20)

Curve 5 in figure 7 was constructed using either equation. Since the bounds

of this curve fall outside the quadrant of interest shown in figure 8 , they

need not be established.

33

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2, Ciov't Acces.sion No.

A Failure Hypothesis for Masonry Shearwalls

3. Recipient'.s Acces.sion No.

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7. AUTHOR{S)

Felix Y. Yokel and S. George Fattal

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16. ABSTRACT (A 200-word or /ess factual summary of most sigriificant information. If document includes a significant

bibliography or literature survey, mention it here.) Various failure hypotheses for wall panels subjected simultaneously to diagonal

compressive load and to vertical compressive edgeload are compared with the

results of thirty-two tests on four types of brick masonry walls which were published elsewhere. It is concluded that failure can occur by joint separation

or by splitting. A failure hypothesis is advanced which is shown to be in good

agreement with the test results examined.

7. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Brick; failure; failure theories; masonry; shear walls;

shear strength; shear test; stresses; stress distribution; structural engineering.

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