SLENDERNESS OF MASONRY BLOCK WALLS
By
ERIC MAN
B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1981
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE
OF MASTER OF APPLIED SCIENCE
i n the Department
of
Civil
We accept
this
Engineering
t h e s i s as conforming
to the r e q u i r e d
standard
THE UNIVERSITY OF BRITISH COLUMBIA
©
E r i c Man, 1981
In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l
fulfilment
of the
requirements
f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I
the L i b r a r y s h a l l make i t
further
freely available
agree
f o r r e f e r e n c e and s t u d y .
agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s
that
I
for
s c h o l a r l y purposes may be granted by the Head of my department or by h i s
o r her r e p r e s e n t a t i v e s .
It
i s understood t h a t c o p y i n g or p u b l i c a t i o n of
t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my w r i t t e n
permission.
E r i c H.Y. Man
Department of C i v i l E n g i n e e r i n g
The U n i v e r s i t y of B r i t i s h Columbia
2324 Main M a l l
Vancouver, B . C . , Canada
V6T 1W5
-
ii
-
ABSTRACT
The s l e n d e r n e s s e f f e c t
of masonry w a l l s
i s examined through
combination o f e x i s t i n g a n a l y t i c a l methods and e x p e r i m e n t a l
masonry w a l l s .
the
results
for
The c u r r e n t code d e s i g n c a p a c i t i e s , when compared w i t h
theoretical
v a l u e s , are found t o be i n c o n s i s t e n t and o v e r - c o n s e r v a t i v e .
Limit
Design approach f o r p l a i n and r e i n f o r c e d masonry w a l l s
State
proposed based on the Moment M a g n i f i e r
d e s i g n format f o r
Method, which i s the c u r r e n t
s l e n d e r c o n c r e t e columns.
is
ACI
the
A
- i i iTABLE OF CONTENTS
Page
ABSTRACT
- i -
TABLE OF CONTENTS
- i i i -
LIST OF FIGURE
-v-
LIST OF TABLES
- v i i -
ACKNOWLEDGEMENTS
-ix-
I.
INTRODUCTION
1
1.1
Background
1
1.2
L i t e r a t u r e Review
1
1.3
Scop and
4
9
II.
EQUIVALENT STRESS-STRAIN CURVES FOR
MASONRY WALLS
2.1
General
11
2.2
The Computer Program f o r S i m u l a t i o n o f the Wall's Behaviour
11
2.3
E q u i v a l e n t S t r e s s - S t r a i n Curve
14
2.4
E x p e r i m e n t a l Data
15
2.5
Development of the S t r e s s - S t r a i n Curves
20
2.6
F i n a l S t r e s s - S t r a i n Curves f o r P l a i n and P a r t i a l l y
Masonry Walls
2.7
E f f e c t of T e n s i l e Strength
Purpose
PLAIN AND
REINFORCED
Grouted
11
25
26
I I I . VERIFICATION OF THE ANALYSIS
28
3.1
Theoretical
28
3.2
Comparison o f the E x p e r i m e n t a l and T h e o r e t i c a l R e s u l t s f o r
F a c e - S h e l l C o n s t r u c t e d P l a i n Masonry W a l l s
28
3.2.1
3.2.2
33
34
3.3
I n t e r a c t i o n Diagrams
Short Wall C a p a c i t y
F u l l Size Walls
Comparison of the E x p e r i m e n t a l and the T h e o r e t i c a l
f o r P a r t i a l l y Grouted R e i n f o r c e d Walls
3.3.1
3.3.2
Short Wall C a p a c i t y
F u l l Size Walls
Results
37
37
39
- iv -
3.4
Joint
IV.
EVALUATION OF THE CODE DESIGN METHOD
4.1
Introduction
4.2
Code Design E q u a t i o n s
4.3
Comparison of the T h e o r e t i c a l C a p a c i t i e s and the Code
Design Values f o r Masonry Walls
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
Reinforcement
General
The T h e o r e t i c a l Walls
C a l c u l a t i o n of the Code Design Values
Comparison of the R e s u l t s
General Remarks
V.
MODIFICATION OF THE MOMENT MAGNIFIER METHOD FOR MASONRY
DESIGN
5.1
5.2
5.3
The Moment M a g n i f i e r Method
The Modulus o f E l a s t i c i t y of Masonry Assemblages
General F u n c t i o n f o r the R i g i d i t y Reduction F a c t o r (X)
5.4
The F u n c t i o n ( K )
5.5
The E x p r e s s i o n f o r the Load I n f l u e n c e F a c t o r K (P/P )
p
o
5.6
Improvement
5.7
D i s c u s s i o n s on the Implied I n t e r a c t i o n Diagrams
5.8
A p p l i c a t i o n f o r U n r e i n f o r c e d Masonry Walls
5.9
Design o f Slender Masonry Walls
VI.
CONCLUSIONS AND
g
REFERENCES
of the E s t a b l i s h e d F u n c t i o n s
RECOMMENDATIONS
- v Page
LIST OF FIGURES
1.1
I n t e r a c t i o n Curve f o r a Pinned Column With Equal End
Eccentricities
3
2.1
A f a m i l y of Load-Moment I n t e r a c t i o n Curves f o r V a r i o u s Heights
of the Member
13
2.2
T y p i c a l S e c t i o n of Masonry Walls ( P l a i n and P a r t i a l l y
17
2.3a E f f e c t i v e
Grouted)
S e c t i o n f o r 8" P l a i n Wall Masonry and i t s C o o r d i n a t e s
19
2.3b E f f e c t i v e S e c t i o n f o r 8" ( P a r t i a l l y Grouted) R e i n f o r c e d Masonry
W a l l and i t s C o o r d i n a t e s
19
2.4
E x p e r i m e n t a l and Computed D e f l e c t e d Shapes of a F a c e - S h e l l
C o n s t r u c t e d 8" P l a i n Masonry W a l l
21
2.5
E x p e r i m e n t a l and Computed D e f l e c t e d Shapes o f a 8" R e i n f o r c e d
Masonry Wall
23
2.6
S t r e s s - S t r a i n Curves f o r Masonry
24
3.1
I n t e r a c t i o n Diagram of 8" F a c e - S h e l l C o n s t r u c t e d P l a i n Masonry
W a l l and the E x p e r i m e n t a l R e s u l t s
29
3.2
T h e o r e t i c a l I n t e r a c t i o n Diagram f o r 8" R e i n f o r c e d Masonry Walls
(3-#3 @ <£) With the A s s o c i a t e d E x p e r i m e n t a l R e s u l t s
30
3.3
T h e o r e t i c a l I n t e r a c t i o n Diagram f o r 8" R e i n f o r c e d Masonry Walls
(3-#6 @ C_) w i t h A s s o c i a t e d E x p e r i m e n t a l R e s u l t s
31
3.4
T h e o r e t i c a l I n t e r a c t i o n Diagram f o r , 8 " R e i n f o r c e d Masonry Walls
(3-#9 @ <£) With the A s s o c i a t e d E x p e r i m e n t a l R e s u l t s
32
4.1
T h e o r e t i c a l I n t e r a c t i o n Diagram f o r 12" P l a i n Masonry Walls
48
5.1
R e l a t i o n s h i p Between Normalized
the Slenderness R a t i o s ( L / r )
74
5.2
Function K
5.3
R e l a t i o n s h i p Between K and the Load R a t i o s ( P / P ) f o r
R e i n f o r c e d Masonry Walls
80
5.4
Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagram f o r
8" R e i n f o r c e d Masonry Wall (3-#3 @ <£)
85
5.5
R e l a t i o n s h i p of the C u t - o f f Value f o r K
Reinforcement R a t i o (p)
88
s
R i g i d i t y Reduction F a c t o r and
and the Slenderness R a t i o s ( L / r )
p
75
Q
p
and the V e r t i c a l
-
v i
-
Page
5.6
Comparison of T h e r o e t i c a l and Implied I n t e r a c t i o n Diagram f o r
8" R e i n f o r c e d Masonry Walls (3-#3 @ C_) @ Low ( P / P ) R a t i o s
90
5.7
Comparison of T h e o r e t i c a l and Implied I n t e r a t i o n Diagrams
f o r 8" R e i n f o r c e d Masonry Walls (3-#3 @ CJ
91
5.8
Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams
f o r R e i n f o r c e d Masonry Walls (3-#3 @ C_) @ Low ( P / P )
92
Q
Q
Ratios
5.9
Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams
f o r 8" R e i n f o r c e d Masonry Walls (3-#6 @ G_)
93
5.10 Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams
f o r 8" R e i n f o r c e d Masonry Walls (3-#9 @ <£) @ Low ( P / P )
94
Q
Ratios
5.11 Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams
f o r 8" R e i n f o r c e d Masonry Walls (3-#9 @ <£)
95
5.12 A P l o t of Normalized R i g i d i t y Reduction F a c t o r s and
Slenderness R a t i o s ( L / r ) f o r P l a i n Masonry W a l l s
97
5.13 R e l a t i o n s h i p . o f
98
K
p
and ( P / P ) f o r 8" P l a i n Masonry Walls
5.14 T h e o r e t i c a l and Implied
Masonry Walls
Q
I n t e r a c t i o n Diagrams f o r 8" P l a i n
5.15 T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams f o r 12" P l a i n
Masonry W a l l s
102
103
- viiPage
LIST OF TABLES
3.1
Comparison of E x p e r i m e n t a l R e s u l t s and T h e o r e t i c a l Values o f
Short Wall C a p a c i t i e s o f F a c e - S h e l l C o n s t r u c t e d P l a i n Masonry
Walls
35
3.2
Comparison of E x p e r i m e n t a l R e s u l t s and T h e o r e t i c a l Values f o r ,
F u l l S i z e , F a c e - S h e l l C o n s t r u c t e d P l a i n Masonry W a l l s
36
3.3
Comparison of Test R e s u l t s and T h e o r e t i c a l C a p a c i t i e s f o r
A x i a l l y Loaded 8" R e i n f o r c e d Masonry W a l l s
38
3.4a Comparison of T e s t and T h e o r e t i c a l C a p a c i t i e s f o r F u l l
R e i n f o r c e d Masonry W a l l s (3-#9 @ C_)
Size
40
3.4b Comparison of T e s t and T h e o r e t i c a l C a p a c i t i e s f o r 137.0 i n .
H e i g h t s , 8 i n . R e i n f o r c e d 8" Masonry W a l l s
41
4.1
Theoretical
Capacities for
8" P l a i n Masonry Walls
49
4.2
Theoretical
Capacities for
12" P l a i n Masonry Walls
49
4.3
Theoretical
@ £)
Capacities for
8" R e i n f o r c e d Masonry Walls (3-#3
50
4.4
Theoretical
@ £)
Capacities for
8" R e i n f o r c e d Masonry Walls (3-#6
51
4.5
Theoretical
Capacities for
8" R e i n f o r c e d Masonry Walls (3-#9
52
4.6
A l l o w a b l e Loads f o r 8" P l a i n Masonry Walls
54
4.7
A l l o w a b l e Loads f o r 12'' P l a i n Masonry Walls
54
4.8
A l l o w a b l e Loads f o r 8" R e i n f o r c e d Masonry Walls (3-#3 @
55
4.9
A l l o w a b l e Loads f o r 8" R e i n f o r c e d Masonry Walls (3-#6 @
56
@9
4.10 A l l o w a b l e C a p a c i t i e s f o r R e i n f o r c e d Masonry W a l l s
(3-#9 @ <£)
57
4.11 F a c t o r s o f S a f e t y f o r 8" P l a i n Masonry Walls i n Current Design <£
59
F a c t o r s of S a f e t y f o r 12" P l a i n Masonry Walls i n Current Masonry
D e s i g n Code
59
4.12
4.13 F a c t o r of S a f e t y f o r 8'' R e i n f o r c e d Masonry Walls
C u r r e n t Masonry Design Code
(3-#3 @ <£) i n
4.14 F a c t o r of S a f e t y f o r 8'' R e i n f o r c e d Masonry Walls
C u r r e n t Masonry Design Code
(3-#6
@ «2) i n
60
61
- viii
Page
4.15 F a c t o r of S a f e t y f o r 8" R e i n f o r c e d Masonry Walls (3-#9 @ C_) i n
C u r r e n t Masonry Design Code
62
5.1
R i g i d i t y Reduction F a c t o r s f o r 8" P l a i n Masonry Walls
68
5.2
R i g i d i t y Reduction F a c t o r s f o r 12" P l a i n Masonry Walls
69
5.3
R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry Walls
(3-#3 @ <E)
70
5.4
R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry Walls
(3-#6 @ <E)
71
5.5
R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry Walls
(3-#9 @ G_)
72
5.6
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d
Masonry Walls (3-#3 @ (£)
76
5.7
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d
Masonry W a l l s (3-#6 @ <£)
77
5.8
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d
Masonry W a l l s (3-#9 @»<Q
78
5.9
O v e r a l l Average Values of Normalized R i g i d i t y Reduction
F a c t o r s f o r 8" R e i n f o r c e d Masonry Walls
82
5.10 Average Values of K
p
f o r R e i n f o r c e d Masonry W a l l s (3-#3 @ <£)
82
5.11 Average Values of K
p
f o r R e i n f o r c e d Masonry W a l l s (3-#6 @ <£)
83
5.12 Average Values of K
p
f o r R e i n f o r c e d Masonry Walls (3-#9 @ C_)
83
5.13 Values of K
g
f o r 8" P l a i n Masonry Walls
100
5.14 Values of K
g
f o r 12" P l a i n Masonry Walls
100
5.15 Average Values of Kp f o r 8" P l a i n Masonry Walls
101
5.16 Average Values of K
101
f o r 12" P l a i n Masonry Walls
- ix-
ACKNOWLEDGEMENT
The
author
wishes to thank Drs-
N.D. Nathan and D.L. Anderson f o r
t h e i r a d v i s e and a s s i s t a n c e i n the p r e p a r a t i o n o f t h i s t h e s i s .
support
was provided by the N a t i o n a l Science and E n g i n e e r i n g
C o u n c i l through
grant number 67-7603.
Financial
Research
1
I.
1.1
INTRODUCTION
Background
The i d e a of u s i n g masonry as a b u i l d i n g m a t e r i a l has been around
centuries.
In t h e ' p a s t
two decades, with economy on i t s
m a t e r i a l has been w i d e l y
warehouses,
employed i n many i n d u s t r i a l
and shopping m a l l s .
side,
and o f f i c e
construction material,
masonry s t r u c t u r e s
i n c r e a s e s , the demand f o r a s o p h i s t i c a t e d
basic
of
design
inexorable.
the p r e s e n t Canadian masonry d e s i g n c o d e
The d e s i g n of l o a d b e a r i n g members i s s t i l l
allowable
of
As the degree of c o m p l e x i t y
When compared w i t h d e s i g n codes f o r other m a t e r i a l s
concrete),
the number
b u i l d i n g s , with masonry as the
i s ever i n c r e a s i n g .
procedure f o r masonry i s
structures,
construction
and the p l e a s i n g appearance p r e s e n t e d by masonry s t r u c t u r e s ,
h i g h - r i s e apartments
masonry
low-rise
Due t o both the economy of
for
1
(eg:
i s not
based on an
steel
and
up-to-date.
empirical
s t r e s s d e s i g n a p p r o a c h , and some important a s p e c t s i n d e s i g n ,
such as s e c t i o n geometry
and support c o n d i t i o n s , have been n e g l e c t e d .
order t o keep pace with the other
In
d e s i g n c o d e s , the next i m p o r t a n t step
the e v o l u t i o n of"masonry d e s i g n i s the i n t r o d u c t i o n
of l i m i t
in
states
design.
The l i m i t s t a t e s
d e s i g n approach has a l r e a d y
and some European codes f o r masonry.
designer to e x p l o i t
been adopted i n
The d e s i g n method a l l o w s
the p o t e n t i a l of the m a t e r i a l ,
and d e a l
w i t h each aspect of d e s i g n , such as m a t e r i a l s t r e n g t h ,
end c o n d i t i o n s , and s l e n d e r n e s s e f f e c t s .
design approach o v e r a l l
design.
the
individually
loading
T h i s g i v e s the
Mexican
limit
conditions,
states
supremacy over the c o n v e n t i o n a l a l l o w a b l e
stress
When a c c o u n t i n g f o r
limitation
slenderness e f f e c t s
i n masonry w a l l d e s i g n , the
of the s l e n d e r n e s s r a t i o i n the c u r r e n t code may be over
conservative.
R e c e n t l y , the development of c o n c r e t e c o n s t r u c t i o n
t e c h n o l o g y has been v e r y r a p i d , and an improvement i n the e v a l u a t i o n
slenderness e f f e c t s
c o u l d govern the c h o i c e of masonry or c o n c r e t e
wall construction.
With t h i s
important
economic a s p e c t i n mind,
study w i l l d e a l with the e v a l u a t i o n of the s l e n d e r n e s s e f f e c t
of
tilt-up
this
i n masonry
walls.
When d e s i g n i n g l o a d b e a r i n g members, i t
i s important
t h a t the c a p a c i t y of the members depends on the c r i t e r i a
ure and i n s t a b i l i t y
by N a t h a n .
2
failure.
F i g . 1.1
This particular
shows the i n t e r a c t i o n
The l i n e 0-A d e f i n e s
However the midspan moment
Material failure
o c c u r s at P o i n t B, when
At g r e a t e r .
the midspan d e f l e c t i o n i n c r e a s e s t o a p o i n t such as D and
the member becomes u n s t a b l e .
failure,
discussed
and the c o r r e s p o n d i n g load-moment
the end c o n d i t i o n s are as i n d i c a t e d a t P o i n t C .
eccentricities,
fail-
curve f o r a p i n n e d column w i t h
the r e l a t i o n s h i p between l o a d and end moment.
r e l a t i o n s h i p i s d e f i n e d by 0 - B .
of m a t e r i a l
aspect- i s c l e a r l y
equal end moments t e n d i n g t o cause s i n g l e c u r v a t u r e .
i s m a g n i f i e d by the member d e f l e c t i o n ,
t o bear i n mind
D e f l e c t i o n then i n c r e a s e s suddenly t o
and the end c o n d i t i o n s at maximum l o a d are g i v e n by E .
Thus
MOMENT
F i g . 1.1
Interaction Curve f o r a Pinned Column with
Equal End E c c e n t r i c i t i e s
in this instability
c a s e , maximum l o a d i s a c t u a l l y
short-column i n t e r a c t i o n
clear,
therefore,
independent of
curve — independent of m a t e r i a l
the
failure.
It
t h a t the moment m a g n i f i e r p r o c e d u r e , i n which the d e s i g n
moment i s m a g n i f i e d and compared w i t h the s h o r t column moment, i s
d i v o r c e d from the r e a l i t y
w a l l s with low a x i a l
fail
familiar
i n cases of i n s t a b i l i t y
failure.
quite
In most c a s e s ,
l o a d s and h i g h moments a r i s i n g from wind p r e s s u r e , do
i n the i n s t a b i l i t y
an a r t i f i c i a l
is
mode.
Use of the moment m a g n i f i e r method i s
e m p i r i c a l d e v i c e with but two j u s t i f i c a t i o n s ;
1)
it
t o E n g i n e e r s , 2) we do not as y e t have any o t h e r way of
then
is
handling
the problem by a simple d e s i g n o f f i c e p r o c e d u r e .
1.2
Literature
Reivew
In the p a s t decade, a s u b s t a n t i a l amount of masonry r e s e a r c h has been
performed i n Canada.
The u n d e r s t a n d i n g o f the m a t e r i a l
behaviour
in
masonry has improved and much of the e n g i n e e r i n g knowledge has been p u t
i n t o upgrading masonry d e s i g n methods.
rational
A few d e s i g n methods based on
a n a l y s i s have been put forward and a b r i e f
review of work done i n
t h i s area i s p r e s e n t e d i n t h i s s e c t i o n . O j i n a g a and T u r k s t r a
recommended a d e s i g n method which was based on
the c o n v e n t i o n a l d i r e c t P - A a p p r o a c h .
effective
moment of i n e r t i a
condition.
It
involved estimating
of the s e c t i o n a c c o r d i n g t o the
(the
1 6
bending
f o r c e s a c t i n g through
the
s o - c a l l e d P - A moment).
The s h o r t column i n t e r a c t i o n
Dikker
loading
The maximum moment was found by combining the primary
moments w i t h the bending moments due t o a x i a l
deflection
the
e q u a t i o n s d e r i v e d by Y o k e l , Mathey and
f o r p l a i n masonry w a l l s with no t e n s i l e
calculating section capacities.
s t r e n g t h were used f o r
When no t r a n s v e r s e l o a d a c t s ,
the
5.
upper l i m i t f o r the use o f the s h o r t column c a p a c i t y was a s s e s s e d a s :
L / r = 35 - 17.5 - ^ / 2
f °
* l 2 * ^
( f ° s i n g l e curvature)
e
and
L / r = 35 -
35 e - j . /
e
r
f °
e
2
0
r
0
e
/ / e
r
* l^ 2
e
e
(1.1.1)
^
double
*
o
curvature)
r
(1.1.2)
The recommended minimum e c c e n t r i c i t y was 1/12 of the w a l l t h i c k n e s s
and the maximum s l e n d e r n e s s r a t i o
If
the s l e n d e r n e s s e f f e c t
of i n e r t i a
I
and
eff
I ^
eff
had t o be c a l c u l a t e d , the e f f e c t i v e
moment
was determined a s :
= (I
end 1
+ i
end 2
0/4
for
0 < e /e
1 2
o
< 0 (1.2.1)
= min f - ( I
,
+ I)/4
- end 1
( I
- (-(I
I
I
( L / r ) allowed was 8 0 . 0 .
(1.2.2)
+
n
+ I)/4
.
end 2
f o r -1
< e , / e „ * 0 (1.2.3)
1 2
, , and I
, „ = uncracked moment of i n e r t i a ,
end 1
end 2
I = uncracked moment of
cracked i f
appropriate,
inertia.
The d e f l e c t i o n s were c a l c u l a t e d u s i n g an e l a s t i c d e f l e c t i o n
and the primary bending moment.
due t o end moments, t r a n s v e r s e
equation
S u p e r - i m p o s i t i o n of the moment
l o a d s and the v e r t i c a l
diagram
load acting
through
the d e f l e c t e d shape gave the maximum moment f o r d e s i g n .
T u r n i n g from p l a i n t o r e i n f o r c e d masonry w a l l s ,
recommended the same f o r m u l a t i o n
O j i n a g a and T u r k s t r a
and procedure i n "The Design o f
Reinfor-
D
ced Masonry W a l l s and Columns."
Since the a n a l y s i s on which the method depends c o n s i d e r s o n l y material
failure,
a l i m i t on s l e n d e r n e s s had t o be imposed t o a v o i d
cases o f i n s t a b i l i t y .
potential
F o r a s o l i d s e c t i o n , the imposed maximum s l e n d e r -
ness r a t i o o f 80 i s more c o n s e r v a t i v e than the c u r r e n t
appears t o be unecessary a t s m a l l e c c e n t r i c i t i e s
code p r o v i s i o n , and
and h i g h l o a d s .
However,
when a member i s loaded w i t h a combination of h i g h end-moment and low
vertical
load, instability
l e s s than 80.
failure
can occur at a s l e n d e r n e s s r a t i o much
6.
Yokel and Dikker studied the p o s s i b i l i t y of adopting the
Magnifier Method presently used i n concrete design for masonry design i n
Strength of Load Bearing Masonry Walls, .
5
In the paper, a l i n e a r stress-
s t r a i n r e l a t i o n s h i p was assumed, f l e x u r a l compressive strength (as defined
by the Uniform Building Code) was replaced by the a x i a l compressive
strength (again as defined by the Uniform Building Code), and a s t r a i n
gradient e f f e c t was considered at high e c c e n t r i c i t y .
Based on the experimental
results of 76 concrete masonry walls, some
hollow and some reinforced, Yokel and Dikker believed that at high
e c c e n t r i c i t y , the f l e x u r a l compressive strength was 1.6 times the a x i a l
prism compression strength.
When computing the short wall interaction
diagram 1.6 fm' was used as the compressive strength of the material when
the e c c e n t r i c i t y exceeded 1/6 the wall thickness.
For pure a x i a l load,
the prism strength (fm') was used; and for e c c e n t r i c i t i e s between 0 and
1/6 the wall thickness a l i n e a r i n t e r p o l a t i o n between these values was
recommended.
When c a l c u l a t i n g the slenderness e f f e c t , the buckling load had to be
computed.
The authors recommended
fixed values for the r i g i d i t y reduction
factors, such that,
FT
(
E
I
)
e f f = -
( 1
'
3 )
where EI = f u l l section r i g i d i t y
A
=3.5
for p l a i n masonry walls
A
= 2.5
f o r reinforced masonry walls
These values were used i n a formula similar to that used by the ACI code
for concrete.
7.
Because o f the use o f constant
slenderness
rigidity
e f f e c t cannot be m o d i f i e d
Yokel
f a c t o r s , the
to f i t a l l c i r c u m s t a n c e s .
found t h a t the r e s u l t i s v e r y c o n s e r v a t i v e
becomes u n c o n s e r v a t i v e a t l a r g e
reduction
It i s
a t s m a l l e c c e n t r i c i t i e s , but
eccentricities.
made a s t a b i l i t y and c a p a c i t y a n a l y s i s o f compression members
6
with no t e n s i l e
strength.
He assumed that the m a t e r i a l s t r e s s - s t r a i n
r e l a t i o n s h i p was l i n e a r i n the compression range, and the magnitude o f
d e f l e c t i o n was s m a l l , so t h a t a s m a l l d e f l e c t i o n a p p r o x i m a t i o n c o u l d be
applied.
The s e c t i o n o f the member was assumed t o be s o l i d and
rectangular,
P
cr
where b
h
and the r e s u l t a n t c r i t i c a l
=
0.64 n % bu / h
1
3
l o a d was computed a s :
(1.4)
2
= width o f w a l l
= height
of w a l l
u^ = d i s t a n c e between t h r u s t l i n e and the
compression edge o f s e c t i o n , o r ,
= ( t / 2 - e)
E
Applying
= i n i t i a l modulus of m a t e r i a l
the above r e s u l t to a s o l i d masonry w a l l , the g e n e r a l
e q u a t i o n s f o r w a l l c a p a c i t y as a f r a c t i o n o f P
give:
0.8 u /u ( r e l a t e d to m a t e r i a l compr
o 1
sion f a i l u r e :
1.4.1)
At e = t/6:
= min
192/(h/t) ( r e l a t e d to i n s t a b i l i t y
failure:
u
Q
= distance
1.4.2)
between l i n e o f a c t i o n o f compressive l o a d and
compressive f a c e o f member a t mid-height
8.
0.4 u / u ^
Q
(compression
f a i l u r e : 1.4.3)
At e = t / 3 : p— = min
L_24/(h/t)
2
( i n s t a b i l i t y f a i l u r e : 1.4.4)
H a t z i n i k o l a s , Longworth, and Warwaruk ' , adopted the moment
7
m a g n i f i e r method
i n masonry d e s i g n .
8
When e v a l u a t i n g the s l e n d e r n e s s
e f f e c t , they performed a s t a b i l i t y a n a l y s i s s i m i l a r to the work by
and
a r r i v e d a t the same r e s u l t .
obtained
P
They f u r t h e r s i m p l i f i e d
Yokel
6
e q u a t i o n 1.4 and
the f o l l o w i n g r e s u l t .
n
= 8(1/2 - e / t ) E
3
cr
2
I
o
/h
(1.5)
2
bt
= -JTJ", moment i n e r t i a o f the ( r e c t a n g u l a r ) s e c t i o n .
3
where I
Q
They b e l i e v e d t h a t t e n s i l e s t r e n g t h was important
f o r a n a l y s i s of
l o a d b e a r i n g masonry members, and i n c l u d e d t h i s i n the d e s i g n f o r m u l a t i o n ,
which became:
P
= n
cr
I
where
±
2
E
I
1
/h
= 8(0.5 - e / t + - ^ )
«
"
5
= 2tP/A f
5 f
t
, / f
1
3
I
(1.6.2)
Q
(1.6.3)
max
(1.6.4)
max
2e.
max
f
(1.6.1)
2
(1.6.5)
t
= t e n s i l e s t r e n g t h of masonry
I f the member was loaded
was more c o m p l i c a t e d ;
i n double
c u r v a t u r e bending,
an " e q u i v a l e n t stepped
column", with a jump i n the
v a l u e o f the moment o f i n e r t i a was i n t r o d u c e d .
then:
the d e s i g n method
The b u c k l i n g l o a d
was
9.
P
aE
cr
I /h
o
(1.7)
2
where ot = b u c k l i n g c o e f f i c i e n t f o r the stepped column or w a l l ( i t i s a
f u n c t i o n o f l o c a t i o n o f i n f l e c t i o n p o i n t and r a t i o o f
e f f e c t i v e moment o f i n e r t i a and g r o s s - s e c t i o n i n e r t i a )
They a l s o proposed a lower
l i m i t on the f l e x u r a l s t i f f n e s s f o r a
r e i n f o r c e d masonry member, namely,
E I (0.5 - e/t)
o
'
v
The
(1.8)
> 0.10 E I
o
proposed method was compared to a s e r i e s o f experiments they had
performed, and the r e s u l t was s a i d to be s a t i s f a c t o r y .
draw-backs t o t h i s d e s i g n method.
There are some
The a n a l y s i s was o r i g i n a l l y based on
the assumption t h a t the s e c t i o n i s s o l i d and r e c t a n g u l a r and i t does not
seem j u s t i f i a b l e to a p p l y the same a n a l y s i s t o p l a i n masonry w a l l s w i t h a
hollow
s e c t i o n , which have a completely
combination
account
1.3
different
s e c t i o n geometry.
At a
o f h i g h moment and low a x i a l l o a d , the d e s i g n method does not
f o r the c o n t r i b u t i o n o f v e r t i c a l
reinforcement.
Scope and Purposes
The
c u r r e n t ACI format
f o r c o n c r e t e d e s i g n works w e l l f o r l o a d
b e a r i n g members with h i g h a x i a l l o a d and low s l e n d e r n e s s r a t i o .
Since
masonry w a l l s u s u a l l y serve as panel elements i n s t r u c t u r e s , the a x i a l
a p p l i e d l o a d can be low with h i g h a p p l i e d bending
moment, and the
s l e n d e r n e s s r a t i o i s u s u a l l y h i g h e r than i n c o n c r e t e l o a d b e a r i n g members,
so t h a t i t i s not r e a s o n a b l e
to a p p l y the ACI d e s i g n format
f o r concrete
d i r e c t l y t o masonry d e s i g n .
R e c e n t l y , a g r e a t number o f t e s t s have been performed on masonry
w a l l s w i t h v a r i o u s combinations
o f a x i a l l o a d and moment.
With the
ID.
availability
of these e x p e r i m e n t a l r e s u l t s and some e s t a b l i s h e d t h e o r y ,
r a t i o n a l a n a l y s i s can be made to e v a l u a t e the s l e n d e r n e s s e f f e c t
masonry w a l l s .
S i n c e the ACI d e s i g n method i s f a m i l i a r
e n g i n e e r s and t h e r e i s no new dependable a l t e r n a t i v e ,
to a l l design
t h i s study w i l l
c o n c e n t r a t e on m o d i f y i n g the c u r r e n t ACI d e s i g n f o r m a t ,
a p p l i c a b l e f o r masonry d e s i g n .
in
such t h a t i t
is
11.
II.
DEVELOPMENT OF STRESS-STRAIN CURVES FOR PLAIN AND
REINFORCED MASONRY WALLS
2.1
General
The c o n s t i t u t i v e p r o p e r t i e s of a m a t e r i a l
external
influences.
Knowing the c o n s t i t u t i v e
govern i t s behaviour under
p r o p e r t i e s of the
and the s t a t e of s t r e s s , d e f o r m a t i o n and f a i l u r e
criteria
can be p r e d i c t e d with the c o r r e c t mechanism or t h e o r y .
procedure can be a p p l i e d :
properties i f
if
f o r the
element
The r e c i p r o c a l
we can work backwards and f i n d the
we know the c o r r e c t mechanism and the s t a t e
For masonry w a l l s ,
material
material
of the member.
we know the deformations of the w a l l s under l o a d s
and the f a i l u r e
c o n d i t i o n s , we can deduce the p r o p e r t i e s of the masonry
wall unit quite
accurately.
2.2
The Computer Program f o r E v a l u a t i n g the C a p a c i t y o f Masonry W a l l s
Under C o n c e n t r i c o r E c c e n t r i c Loads
A computer program i s used t o e v a l u a t e
axial
loads and b e n d i n g .
the behaviour of w a l l s
The program was o r i g i n a l l y
developed f o r
p r e s t r e s s e d , or r e i n f o r c e d c o n c r e t e w a l l s or c o l u m n s ' '
2
9
advantage of the c a p a c i t y of p r e s e n t computer t e c h n o l o g y ,
integration
and r e g r e s s i o n r o u t i n e s are used f r e q u e n t l y
compute s t r e s s d i s t r i b u t i o n ,
p r e d i c t e d by n u m e r i c a l l y i n t e g r a t i n g
failure
1 0
'
1 1
.
By t a k i n g
numerical
i n t h i s program t o
f o r c e s , moments, and d e f l e c t e d shapes o f
members with v a r i o u s c r o s s - s e c t i o n s and h e i g h t s .
and i n s t a b i l i t y
under
Material
failure
is
the s t r e s s e s over the c r o s s - s e c t i o n
i s monitored by c o n s t r u c t i n g the column d e f l e c t i o n
curves to l o c a t e the maximum end moment c a p a c i t y .
the method i s as f o l l o w s :
A brief
d e s c r i p t i o n of
With the assumption that plane sections remain plane under bending
action, the s t r a i n d i s t r i b u t i o n , and hence the stresses (by r e f e r r i n g to
the s t r e s s - s t r a i n curve), are found at various combinations of the curvature and the neutral axis depth.
The a x i a l load and moment are evaluated
for each curvature and neutral axis depth, by numerical integration of the
stresses, and a contour of constant curvature may be plotted on the loadmoment interaction diagram.
By means of such contour l i n e s , the moment-
curvature relationship for any value of load may be constructed.
Then the
column d e f l e c t i o n curves can be computed for each a x i a l load, and they are
used to f i n d the maximum end moment for each of the number of chosen
member lengths.
represents
F i g . 2.1 shows an example of a family of curves, which
the maximum moment and load that the member can carry at
various given
lengths.
Application of the above computer program to masonry wall analysis
requires no s i g n i f i c a n t a l t e r a t i o n . The basic theory and method of computation are the same. Due to the differences i n material properties, the
s t r e s s - s t r a i n curve of concrete w i l l be replaced by a s t r e s s - s t r a i n curve
which represents the properties of the masonry wall unit.
13.'
F i g . 2.1 A Family o f Load-Moment I n t e r a t i o n
V a r i o u s Heights o f the Member
Curves f o r
14.
2.3
S t r e s s - S t r a i n Curve
The s t r e s s - s t r a i n curve of a m a t e r i a l c o n t a i n s the f o l l o w i n g
informa-
tion:
the shape of the curve ( l i n e a r / n o n - l i n e a r )
which governs
p r o p o r t i o n a l i t y of s t r e s s and s t r a i n of the m a t e r i a l ;
the peak s t r e s s and s t r a i n which l i m i t s
corresponding deformation;
the
the u l t i m a t e c a p a c i t y and
and the g r a d i e n t of the curve which d e s c r i b e s the e l a s t i c i t y
m a t e r i a l a t any s t a t e of s t r e s s .
A masonry w a l l c o n s i s t s of b l o c k s and m o r t a r ,
r e i n f o r c e d masonry w a l l ,
t o combine a l l
an e f f e c t i v e
It
i s very
c u r v e , deduce the response of the w a l l ,
r e s u l t s under s i m i l a r
approach; i t
stress-strain
experimental
The method i s a " t r i a l
and
i n v o l v e s u s i n g a sample s t r e s s - s t r a i n curve and
shape or u l t i m a t e c a p a c i t y of the w a l l w i t h an
e s t a b l i s h e d computer program.
The computed behaviour of the w a l l u n i t
compared w i t h the e x p e r i m e n t a l l y
The f i n a l
is
interaction.
and compare t h i s with
loading conditions.
computing the d e f l e c t e d
t o the h y p o t h e t i c a l
equivalent
obtain
because i t
the c o n t r i b u t i o n of each m a t e r i a l and t h e i r
The a l t e r n a t i v e i s t o c o n s t r u c t a h y p o t h e t i c a l
difficult
each m a t e r i a l t o
s t r e s s - s t r a i n curve f o r the masonry w a l l u n i t ,
hard t o e v a l u a t e
error"
for
the
and i n the case of a
of grout and s t e e l as w e l l .
the s t r e s s - s t r a i n r e l a t i o n s h i p s
of
observed b e h a v i o u r ,
is
and changes are made
s t r e s s - s t r a i n curve to improve the c o r r e s p o n d e n c e .
s t r e s s - s t r a i n curve w i l l r e f l e c t
masonry w a l l as an e q u i v a l e n t
"monolithic"
the p r o p e r t i e s
the
material.
Since the b a s i c m a t e r i a l f o r masonry w a l l s
i s concrete, a non-linear
s t r e s s - s t r a i n curve as i n c o n c r e t e was expected f o r
s t r e s s - s t r a i n curve of the masonry w a l l u n i t .
s h i p f o r prisms under a x i a l
of
the
"equivalent"
The s t r e s s - s t r a i n
relation-
compression r e c o r d e d by D r y s d a l e and H a m i d
1 2
• 15.
was used as the s t a r t i n g point.
Improvements were made as described
above.
At t h i s point, the author would l i k e to emphasize that the f i n a l
s t r e s s - s t r a i n curve for the masonry wall unit may be d i f f e r e n t from the
curve recorded from an a x i a l l y compressed prism t e s t .
Governed by the
loading condition, the s t r e s s - s t r a i n relationship i s affected by the
s t r a i n gradient e f f e c t .
1 0
The s t r e s s - s t r a i n relationship may
also be
governed by the number of grouted cores, as section geometry may be an
important factor.
2.4
Experimental Data
When searching for the s t r e s s - s t r a i n relationship of masonry mater-
i a l , experimental r e s u l t s were required for comparison.
For s i m p l i c i t y ,
the experimental results for concrete masonry walls recorded i n Ref. (7)
were used throughout this study.
A l l walls were loaded v e r t i c a l l y with
top and bottom pinned tb a r i g i d support to ensure no l a t e r a l movement at
either end.
The basic units used i n constructing a l l test specimens were 8" x 8"
x 16" stretcher blocks, 8" x 8" x 16" end blocks, and 8" x 8" x 8" half
blocks with an average strength of 2350 p s i . Type S mortar was
used
throughout, and a l l walls were b u i l t to 2 1/2 blocks wide (39.625") i n
running bond.
Face-shell construction was used on a l l p l a i n walls.
For
the reinforced walls, three bars with ultimate strength of 60 k s i were
used as v e r t i c a l reinforcement at the center l i n e of the wall, one i n each
alternate hole and these holes with v e r t i c a l reinforcement were f i l l e d
with grout of mean compressive
strength 2380 p s i . The bar sizes used were
16.
#3,
#6,
on F i g .
and #9.
T y p i c a l s e c t i o n s of p l a i n and r e i n f o r c e d w a l l s are
shown
2.2.
Data e x t r a c t e d were:
Deflected
ings
kips)
(20
shapes of a p l a i n w a l l ,
105"
k i p s , 40 k i p s , 60 k i p s , 80 k i p s ,
at a c o n s t a n t
Deflected
end e c c e n t r i c i t y
shapes of a r e i n f o r c e d
l o a d i n g s of 20 k i p s , 40 k i p s ,
eccentricity
of
2.54";
of
in height,
100 k i p s ,
with v a r i o u s
120 k i p s ,
load-
and 140
1.27";
w a l l 185"
in height,
with various
60 k i p s and 200 k i p s at a c o n s t a n t
end
17.
8"X8"X 16" SINGLE
CORNER
, / , MORTARED
///,
AREA
MORTAR JOINT
^ 8 " X 8 " X 16" STRETCHER
.T
m|oo
IS
V?777r//7777?/A\?77777//77777/,
5|"
Jn
8 X 8 X 8
HALF BLOCK
39 | -
T Y P I C A L PLAIN WALL S E C T I O N ( F A C E - S H E L L CONSTRUCTED)
VERTICAL
REINFORCEMENT
GROUT
A•A
• •A
MA. A
AV
77777y^7777rXS77777'A
}
39 |
T Y P I C A L REINFORCED WALL SECTION ( PARTI ALLY GROUTED)
F i g . 2.2
Typical Section of Masonry Walls ( P l a i n and
P a r t i a l l y Grouted)
18.
Failure
l o a d s f o r w a l l s of v a r i o u s h e i g h t s loaded at v a r i o u s end
eccentricities.
S i n c e the f a c e - s h e l l area i s the e f f e c t i v e
load for a p l a i n w a l l ,
area i n r e s i s t i n g
vertical
when e n t e r i n g the c r o s s - s e c t i o n of the p l a i n
wall
i n t o the computer program, the c r o s s - s e c t i o n can be modelled as a s e r i e s
of box s e c t i o n s .
Since the assumption i s made t h a t p l a n e s e c t i o n s remain
p l a n e , the web elements are a l l
similarly
s t r e s s e d , and the
i s i n c a p a b l e of d i s t i n g u i s h i n g between box and I - s e c t i o n s .
computation
Thus,
for
c o n v e n i e n c e , an I - s e c t i o n as shown i n F i g . 2.3a was a c t u a l l y used i n
the
present c a l c u l a t i o n s .
The net c r o s s - s e c t i o n s f o r r e i n f o r c e d w a l l s were not d e f i n e d
i n Ref.
(7).
When m o d e l l i n g the r e i n f o r c e d w a l l s ,
d e f i n e d as f o l l o w s :
plain wall,
clearly
the c r o s s - s e c t i o n was
The f l a n g e i s the same as the f a c e - s h e l l area i n a
and the web area i s c a l c u l a t e d by s u b t r a c t i n g the f l a n g e
from the t o t a l net area d e r i v e d by d i v i d i n g the f a i l u r e
average s t r e s s e s noted on Page 148 o f R e f .
(7).
by d i v i d i n g the web area by the known web d e p t h .
area
l o a d by the
Width of the web i s
The modelled net
s e c t i o n and the c o r r e s p o n d i n g c o o r d i n a t e s are shown i n F i g .
2.3b.
found
cross-
19.
Y
A
(0. , 7.63)
(39.63 ,7.63)
t
(0., 6.13)
( 19.73,6.13)
(19.91,6.13)
(39.63,6.13)
(0.,
(19.73, 1.5)
(19.91, 1.5)
( 3 9 . 6 3 , 1.5)
(0.,
1.5)
t
0.)
(39.63 , 0. )
Fig. 2.3a
X
Effective Section fer 8" Plain Masonry Wall
and its Coordinates
Y
A
(0.,
7.63)
( 3 9 . 6 3 , 7 . 63 )
3
( 0. , 6.13)] ( 3 . 6 , 6.13 )
(36.02,6.13)
( 3 9 . 6 3 , 6.13)
( 0 . , 1.5 )
( 3 6 . 0 2 , 1.5 )1
(3|.63,
(3.6,
I
( 0 . , 0. )
Fig. 2.3b
1.5)
1.5)
(39.63, 0. ^
Effective Section for 8" (Partially Grouted)
Reinforced Masonry Wall and its Coordinates
X
20.
2.5
Development of the S t r e s s - S t r a i n Curves
The
procedure f o r f i n d i n g the e q u i v a l e n t
p l a i n masonry w a l l s was
as
follows:
A sample s t r e s s - s t r a i n curve,
in
F i g . 2.3a
or 2.3b), w a l l h e i g h t
i n t o the computer program.
defined
reported
the
in
and
The
loads were computed and
i n Ref.
shape of the
(7).
The
along with
and
can
tell
section coordinates
plotted against
s t i f f n e s s of the modelled w a l l i s governed
i f the modelled w a l l i s too s o f t o r too
d e f l e c t i o n s i s shown i n F i g . 2.4.
cores —
the
i n the p r e v i o u s
a x i a l l o a d and
was
s t r a i n of the curve were v a r i e d
failure
load.
of c r o s s - s e c t i o n between a
paragraph was
s t r e s s - s t r a i n curves f o r p l a i n and
bending, are shown on F i g . 2.6.
plain
i n alternate
another s t r e s s - s t r a i n curve i s needed f o r the l a t t e r .
final
final
the same s t i f f n e s s .
used, and
p l o t of the superimposed d e f l e c t e d shapes are shown i n F i g .
The
until
experimental
a r e i n f o r c e d w a l l - the presence of grout
procedure as d e s c r i b e d
The
s t r e s s - s t r a i n curve
the e x p e r i m e n t a l
to the d i f f e r e n c e i n composition
masonry w a l l and
accordingly
deflection.
the r e a l w a l l had
the u l t i m a t e
u n t i l the f a i l u r e l o a d e q u a l l e d
stiff,
Once the d e f l e c t e d shapes matched, i t
assumed t h a t the shape of the e q u i v a l e n t
c o r r e c t , as the modelled w a l l and
by
By l o o k i n g at the d i f f e r e n c e s
comparison p l o t between the computed d e f l e c t i o n s and
Due
the
the d e f l e c t e d shapes
the slope of the s t r e s s - s t r a i n curve can be a d j u s t e d
F i n a l l y the peak s t r a i n and
entered
d e f l e c t e d shapes of the w a l l under
the computed d e f l e c t i o n matches the experimental
was
(shown
the l o a d s of i n t e r e s t were
input s t r e s s - s t r a i n c u r v e .
d e f l e c t i o n , one
s t r e s s - s t r a i n curve f o r
The
the
same
final
2.5.
r e i n f o r c e d w a l l s , under
The
developed
equivalent
21.
-0.15
F i g . 2.4
0
0.15
0.30
D E F L E C T I O N , in.
0.45
Experimental and Computed Deflected Shapes of a
Face-Shell Constructed 8" P l a i n Masonry Wal1
0.60
22.
s t r e s s - s t r a i n curves may be a p p l i c a b l e
cross-section.
to w a l l s w i t h the
same n a t u r e
23.
-0.5
0
£ig.. 2.5
0.5
1.5
2.5
D E F L E C T I O N , in.
3.5
Experimental and Computed Deflected Shapes of a
8" Reinforced Masonry Wall
4.5
24.
3.0
STRAIN
F i g . 2.6 Stress-Strain Curves f o r Masonry
25.
2.6
S t r e s s - S t r a i n Curves f o r P l a i n and P a r t i a l l y Grouted Masonry W a l l s
Fig.
2.6
shows the f i n a l p l o t f o r the s t r e s s - s t r a i n curves f o r
p l a i n and p a r t i a l l y
grouted masonry w a l l s .
The s t r e s s - s t r a i n c u r v e s
r e c o r d e d by D r y s d a l e and H a m i d , and H a t z i n i k o l a s
1 2
under a x i a l compression t e s t
both
7
f o r ungrouted p r i s m s
are a l s o p l o t t e d .
Both d e r i v e d s t r e s s - s t r a i n curves are n o n - l i n e a r as e x p e c t e d .
p l a i n w a l l s , the peak s t r e s s i s 2.1
of 0.0022, and the i n i t i a l
k s i with the c o r r e s p o n d i n g peak
strain
modulus of e l a s t i c i t y and u l t i m a t e s t r a i n
1389 k s i and 0.0025 r e s p e c t i v e l y .
s t r e s s and s t r a i n are 2.35
For
For p a r t i a l l y
grouted w a l l s ,
are
the peak
k s i and 0.0022 r e s p e c t i v e l y .
When comparing the d e r i v e d s t r e s s - s t r a i n c u r v e s , the one f o r
partial-
l y grouted w a l l s has h i g h e r modulus of e l a s t i c i t y and peak s t r e s s , w i t h
the same peak s t r a i n and u l t i m a t e s t r a i n .
wall a greater
tions.
lished.
It
g i v e s the p a r t i a l l y
grouted
c a p a c i t y than the p l a i n w a l l under e c c e n t r i c l o a d i n g c o n d i -
C u r r e n t l y , the e f f e c t
D r y s d a l e and Hamid
grouted p r i s m s e x h i b i t
On the c o n t r a r y ,
of grout on masonry w a l l s i s not w e l l e s t a b report
t h a t i n a x i a l compression t e s t s ,
lower compressive s t r e n g t h than ungrouted p r i s m s .
B o u l t ^ reports that i f
o f both b l o c k and grout are s i m i l a r ,
the modulus and l i m i t i n g
the r e s u l t i n g p r i s m has u l t i m a t e
p r o p e r t i e s i n excess of the i n d i v i d u a l e l e m e n t s .
t h a t masonry assemblages have d i f f e r e n t
It
was a l s o r e p o r t e d
modes o f f a i l u r e
under
loaded and e c c e n t r i c a l l y l o a d e d c o n d i t i o n s ; the former f a i l
s p l i t t i n g of the u n i t
and the l a t t e r
fail
axially
by t e n s i l e
when the s t r e s s of the compres-
s i o n s i d e reaches the s t r e n g t h of the m a t e r i a l s .
w i t h the remarks made by B o u l t
T h i s comment t o g e t h e r
indicate that p a r t i a l l y
grouted w a l l s may
be expected to behave as though s t r o n g e r than ungrouted w a l l s i n
but more e x p e r i m e n t a l
strain
s t u d i e s are needed t o c o n f i r m t h i s .
flexure,
26.
The s t r e s s - s t r a i n r e l a t i o n s h i p
from t e s t s
in axial
compression one and o n e - h a l f
h i g h , of t e n ungrouted specimens.
strain
r e c o r d e d by H a t z i n i k o l a s
relationship,
b l o c k s wide,
Fig.
2.6.
s p i t e of the d i f f e r e n c e
s t r e s s - s t r a i n curve f o r p l a i n w a l l s
The moduli of e l a s t i c i t y
curves are the same a t
five
blocks
stress-
T h i s s t r e s s - s t r a i n curve i s compared w i t h
shapes and u l t i m a t e loads)
In
derived
i s 1120 k s i , and the
d e r i v e d s t r e s s - s t r a i n c u r v e s , which are based on experimental
(deflection
was
With the assumption of a l i n e a r
the modulus of e l a s t i c i t y
ure s t r e s s i s 2.056 k s i .
7
of e c c e n t r i c a l l y
failthe
data
loaded w a l l s ,
i n l o a d i n g c o n d i t i o n s , the
derived
does resemble the p r i s m t e s t
result.
are almost the same, and the peak s t r e s s f o r
2.1
ksi.
c o n d i t i o n s have no s i g n i f i c a n t
in
The s i m i l a r i t i e s
effect
suggest t h a t
both
loading
on the s t r e n g t h of a f a c e - s h e l l
p l a i n masonry assemblage.
2.7
E f f e c t of T e n s i l e
The a l l o w a b l e
Strength
tensile
d e s i g n of masonry w a l l s
and f u l l y
strength permitted
i s 23 p s i and 36 p s i f o r
bedded w a l l s r e s p e c t i v e l y ,
s t r e n g t h of 350 p s i and i t
of the w a l l c a p a c i t y .
i n the c u r r e n t
Ref.
(7)
face-shell
code
tensile
for
computation
But i n most s t u d i e s , s i n c e the magnitude of
s t r e n g t h i s s m a l l compared t o compressive s t r e n g t h ,
for
constructed
r e c o r d e d a mean
was i n c l u d e d i n the f o r m u l a t i o n
1
the t e n s i l e
tensile
strength
f o r masonry m a t e r i a l i s assumed t o be z e r o .
In order to study the c o n t r i b u t i o n s of t e n s i l e
capacity,
two s e t s of computations were made on a f a c e - s h e l l
p l a i n masonry w a l l ,
140 k i p s at
sile
s t r e n g t h to the
105"
i n height
an e c c e n t r i c i t y
s t r e n g t h and the other
and 2 1/2
of 1 . 2 7 " .
b l o c k s wide,
wall
constructed
with a l o a d of
One s e t was computed w i t h no t e n -
had an assumed t e n s i l e
s t r e n g t h of 360
psi
27.
w i t h a c o r r e s p o n d i n g t e n s i l e s t r a i n of 0.00017.
The v a l u e of 360 p s i was
based on the recomendation of the c o n c r e t e d e s i g n c o d e
1 5
.
(ie:
f
=
t
7.5/f
c
')
and t h i s v a l u e agreed w i t h the one r e p o r t e d i n R e f .
The computed d e f l e c t e d shapes and s h o r t w a l l load-moment
c a p a c i t i e s were compared.
It
interaction
was found t h a t t h e r e was no n o t i c e a b l e
d i f f e r e n c e between the two s e t s of r e s u l t s .
T h i s i m p l i e d t h a t the
of t e n s i l e s t r e n g t h i s i n s i g n i f i c a n t i n the computation of the
capacity.
(7).
wall
effect
28.
I I I . VERIFICATION OF THE
3.1
ANALYSIS
T h e o r e t i c a l I n t e r a c t i o n Diagrams
As the f i n a l
s t r e s s - s t r a i n r e l a t i o n s h i p s were developed
v i o u s chapter f o r p l a i n and p a r t i a l l y grouted masonry w a l l s ,
i n t e r a c t i o n diagrams can now
I t was
found
be produced
t h a t (as suggested
i n the p r e different
f o r the w a l l s of those t y p e s .
i n S e c t i o n 2.5)
the s t r e s s - s t r a i n
rela-
t i o n s h i p depends upon the c r o s s - s e c t i o n geometry o f the w a l l , s i n c e two
different
s t r e s s - s t r a i n curves were o b t a i n e d s e p a r a t e l y f o r f a c e - s h e l l
c o n s t r u c t e d and p a r t i a l l y grouted w a l l s .
a c t i o n diagram
i s governed
s t r e n g t h of the components.
With t h i s l i m i t a t i o n , the
inter-
by the c o m p o s i t i o n of the c r o s s - s e c t i o n and
S i n c e the purpose
the
of t h i s study i s to model
the g e n e r a l behaviour of masonry w a l l s and the s l e n d e r n e s s e f f e c t ,
the
a v a i l a b l e i n f o r m a t i o n i s adequate to meet the t a s k .
The
t h e o r e t i c a l i n t e r a c t i o n diagrams f o r p l a i n and r e i n f o r c e d w a l l s
were produced
w i t h the computer program d e s c r i b e d i n S e c t i o n 2.2
s l e n d e r n e s s e f f e c t was
i n c l u d e d i n these i n t e r a c t i o n diagrams w i t h the
w a l l h e i g h t s v a r y i n g from 95.6"
f o r c o n s i d e r a t i o n . The
those s p e c i f i e d
F i g . 3.1,
3.2,
to 375.6", which was
a reasonable
c o m p o s i t i o n of the w a l l s modelled
i n S e c t i o n 2.4.
3.3
and 3.4,
The
was
similar
and the c o r r e s p o n d i n g e x p e r i m e n t a l
to
results
Each datum i s
r e p r e s e n t e d by a s o l i d dot w i t h an arrow l e a d i n g to the r e l a t e d
3.2
range
i n t e r a c t i o n diagrams a r e shown on
r e c o r d e d i n Ref. (7) are a l s o p l o t t e d f o r comparison.
tical
The
theore-
location.
Comparison o f the E x p e r i m e n t a l and the T h e o r e t i c a l V a l u e s f o r FaceS h e l l C o n s t r u c t e d P l a i n Masonry W a l l
F i g . 3.1
r e p r e s e n t s the t h e o r e t i c a l load-moment i n t e r a c t i o n c a p a c i -
29.
0
F i g . 3.1
5
10
15
20
25
MOMENT, kip-ft.
30
35
Theoretical Interaction Diagram f o r 8" Face-Shell Constructed
P l a i n Masonry Walls and the Associated Experimental Results
30.
F i g . 3.2
Theoretical Interaction Diagram for 8" Reinforced
Masonry Walls (3-#3 @ ([} with the Associated Experimental
Results
31.
900i
800
•
10
Fig.
3.3
20
Experimental
result
30
40
MOMENT, kip-ft.
Theoretical Interaction
Masonry Walls (3-#6 @
Experimental R e s u l t s
1
50
60
Diagram f o r 8" R e i n f o r c e d
with the A s s o c i a t e d
32.
900
Fig.
3.4
T h e o r e t i c a l I n t e r a c t i o n Diagram f o r 8" R e i n f o r c e d
Masonry Walls (3-#9 @ C) with the A s s o c i a t e d
Experimental R e s u l t s
33.
t i e s of the 2 1/2
b l o c k s ' wide f a c e - s h e l l c o n s t r u c t e d masonry w a l l s w i t h
various lengths.
A l l p l a i n w a l l s used
f a c e - s h e l l c o n s t r u c t e d w i t h no j o i n t
3.2.1
f o r t h i s comparison purpose are
reinforcement.
Short W a l l C a p a c i t y
U n r e i n f o r c e d s h o r t w a l l s , 39.625" i n h e i g h t , were loaded a x i a l l y
e c c e n t r i c a l l y to r e s u l t i n f a i l u r e .
and P. 118
comparison.
ponding
The
r e s u l t s were r e c o r d e d on P.
of Ref. (7) r e s p e c t i v e l y and were a l s o p l o t t e d i n F i g . 3.1
For c l a r i t y , Table 3.1
and
117
for
shows the comparison of the c o r r e s -
e x p e r i m e n t a l f a i l u r e l o a d s and
the t h e o r e t i c a l l y p r e d i c t e d v a l u e s .
The
spread between repeated e x p e r i m e n t a l r e s u l t s was
as h i g h as 33% which
was
r e g i s t e r e d w i t h the specimens loaded a t an e c c e n t r i c i t y of t/3
(e =
2.54").
The
comparison i n d i c a t e s t h a t the t h e o r e t i c a l s h o r t w a l l i n t e r a c t i o n
curve does p r e d i c t
percentage
tions.
the c a p a c i t y of the w a l l a c c u r a t e l y with the h i g h e s t
e r r o r of 8.9%
It i s interesting
which was
recorded under a x i a l l o a d i n g c o n d i -
to note t h a t the t h e o r e t i c a l v a l u e
over-
e s t i m a t e s the e x p e r i m e n t a l f a i l u r e l o a d s under c o n c e n t r i c l o a d i n g c o n d i t i o n s , but the d i f f e r e n c e s are q u i t e s m a l l (8.9% d i f f e r e n c e between the
t h e o r e t i c a l v a l u e and
the average
spread i n the e x p e r i m e n t a l d a t a .
d i f f e r e n c e s mentioned may
mental specimens —
eg:
of the t e s t r e s u l t s ) compared to the
At t h i s p o i n t , i t i s noted
be caused
by the i m p e r f e c t i o n s of the e x p e r i -
i n i t i a l deflections.
The
e f f e c t of
g r a d i e n t r e p o r t e d i n most s t u d i e s (eg: R e f s . 5, 7, and
apparent.
I t suggests
t h a t the
13)
strain
i s not
t h a t the e f f e c t s of s t r a i n g r a d i e n t are
cant f o r f a c e - s h e l l c o n s t r u c t e d u n r e i n f o r c e d masonry w a l l s .
insignifi-
34.
3.2.2
F u l l S i z e Walls
W a l l s , w i t h h e i g h t s of 105", 121", 137", and 187" were loaded
and e c c e n t r i c a l l y u n t i l
f a i l u r e and l o a d i n g c o n d i t i o n s v a r i e d
c o n c e n t r i c to an e c c e n t r i c i t y of 3.0".
axially
from
The r e s u l t s were r e c o r d e d on P.
148 and P. 149 of Ref. (7) and are p l o t t e d i n F i g . 3.1 a c c o r d i n g l y .
Table
3.2 shows the comparison between the t h e o r e t i c a l l y p r e d i c t e d v a l u e s and
the e x p e r i m e n t a l r e s u l t s .
There i s an e r r o r of 39.5% f o r the 105" h i g h
w a l l w i t h 2.54" e c c e n t r i c i t y , but otherwise e r r o r s a r e 17% o r l e s s g e n e r a l l y l e s s than the spread i n the d u p l i c a t e d e x p e r i m e n t a l r e s u l t s f o r
the
short-wall.
I t i n d i c a t e s t h a t the present a n a l y t i c a l method i s
capable of m o d e l l i n g the s l e n d e r n e s s e f f e c t and p r e d i c t i n g
combinations of l o a d s and moments f o r p l a i n masonry w a l l s .
the f a i l u r e
35.
Load
Eccentricity
(in.)
Experimental
Capacity
. . . . (kip)
0.00
215.5
0.00
249.1
232.3 (Ave.)
1.27
196.9
1.27
150.1
173.5 (Ave.)
2.54
Experimental
Discrepancy
(%)
Theoretical
Capacity
(kip)
Error
.. 15.6
253.
8.9
31.2
178.
2.6
33.0
137.
-1.4
119.3
158.7
139.0 (Ave.)
Table 3.1
Comparison of Experimental Results and Theoretical
Values for Short Wall Capacities of Face-Shell
Constructed Plain Masonry Walls.
36.
Height of
Wall (in.)
(in.)
Load
Eccentricity
(in.)
Experimental
Capacity
(kips)
Theoretical
Capacity
(kips)
Error
105.
0
262.5
238.6
227.
- 9.4
105.
1.27
159.2
154.
- 3.3
105.
2.54
80.3
112.
39.5
105.
3.00
26.1
30.
15.
121.
0.00
190.0
223.
17.4
137.
0.00
218.3
218.
- 0.1
185.
0.00
207.8
200.
- 3.8
185.
1.27
120.
135.
12.5
Table 3.2
Comparison of Experimental Results and Theoretical
Values for F u l l Size, Face-Shell Constructed Plain
Masonry Walls.
37.
3•3
Comparison of the Experimental and the Theoretical Results f o r
P a r i t i a l l y Grouted Reinforced Walls
As described e a r l i e r , a l l reinforced walls tested were 2 1/2 blocks
wide with 3 alternate cores f i l l e d with grout.
placed at the centre of each grouted core.
Reinforcing bars were
There were three types of
reinforced walls examined, and their differences were i n the size of the
r e i n f o r c i n g bars - 3-#3. 3-#6, 3-#9.
The corresponding t h e o r e t i c a l
interaction diagrams are shown on Figs. 3.2, 3.3, and 3.4.
3.3.1
Short Wall Capacity
Unfortunately, no reinforced short wall was loaded e c c e n t r i c a l l y to
f a i l u r e i n Ref. (7). There are thus no experimental results to study the
accuracy of the t h e o r e t i c a l short wall interaction curves.
Data for a x i a l l y loaded reinforced short walls (39.625") are
available and the f a i l u r e s were recorded on P. 120, Ref. (7). Table 3.3
shows the comparison between the predicted a x i a l capacities and the
experimentally recorded r e s u l t s , for the three types of reinforced walls.
It was found that the experimental results were a l o t less than the
predicted capacities.
They are 48%, 42.2%, and 40% of the capacities
calculated from the f l e x u r a l compressive strength of the masonry u n i t .
It i s very possible that the loss i n capacity i s related to the
t e n s i l e stress induced on v e r t i c a l planes by l a t e r a l deformation of the
mortar.
Vertical
Reinforcement
3 - #3 @ <£
(1)
Expe r imenta1
Capacity
(2)
Theoretical
Capacity (kips)
d)/(2)
348.3
280.3
314.3
3 - #6 @ <£
(Ave.)
651.6
0.48
299.7 (Ave.)
711.0
0.42
811.8
0.40
334.1
265.3
3 - #9 @ <£
386.6
275.3
330.9.(Ave.).
T a b l e 3.3
Comparison o f T e s t and T h e o r e t i c a l C a p a c i t i e s f o r
Loaded 8" R e i n f o r c e d Masonry W a l l s
Axially
39.
The e f f e c t does not appear to be present i n p l a i n walls of face-shell
construction, so i t could be due to the interaction of blocks, mortar, and
grout as suggested by Turkstra and Thomas
1
. The l a t e r a l displacement of
the grout and the mortar may increase the l a t e r a l t e n s i l e stress i n the
blocks near the j o i n t and i t may cause the change of f a i l u r e mode and the
s i g n i f i c a n t loss i n capacity.
A detailed study on t h i s matter i s
recommended i n order to have a better understanding of the f a i l u r e
mechanism.
3.3.2
F u l l Size Walls
F u l l size walls reinforced with 3-#9 bars, of 105",
127", 137", and
185" heights were loaded at end e c c e n t r i c i t i e s of 1.27", 2.54", 3.00", and
3.50"; while the walls reinforced with 3-#3, and 3-#6 bars were tested
under eccentric loads at one wall height of 137".
The values of f a i l u r e
loads were recorded on P. 148 and P. 172 of Ref. (7).
Table 3.4(a and b) shows the comparison of the experimental
and the predicted values.
results
As i n the case of short walls, the t h e o r e t i c a l
values tend to over estimate the capacities by a big margin under pure
a x i a l compression.
When ignoring the results for the a x i a l loading
condition, the correlation between the test results and the predicted
values i s good, and most of the errors are within 10%.
The results shown
on Table 3.4b f o r walls reinforced with 3-#6 and 3-#3 bars
40.
Height o f
Wall (in.)
105.0
121.0
137.0
185.0
T a b l e 3.4a
Load
Ecc.
Experimental
Capacity (kips)
Theoretical
Capacity (kips)
1.27
320.0
314.0
2.54
140.0
146.0
4.3
3.00
155.0
124.0
-20.0
3.50
114.9
106.5
-
0.00
315.0
620.0
1.27
249.6
278.0
2.54
125.0
131.5
3.00
122.5
112.0
3.50
90.0
97.5
8.3
0.00
400.0
580.0
45.0
1 .27
200.0
246.5
23.0
2.54
108.8
117.5
8.0
3.00
94.5
101.5
7.4
3.50
83.0
89.0
7.2
0.00
383.5
445.0
16.0
1.27
150.0
176.0
17.0
2.54
90.0
84.0
-
6.7
3.00
80.0
74.0
-
7.5
3.50
73.3
66.7
-
9.0
(in.)
Error
-
1.9
7.3
97.0
.
11.4
5.2
-
Comparison o f T e s t and T h e o r e t i c a l C a p a c i t i e s f o r F u l l
R e i n f o r c e d Masonry W a l l s
(3 - #9 g )
8.6
Size
41.
Vertical
Reinf.
3-#6
3-#3
@ <£
@ <£
T a b l e 3.4b
Load
Ecc.
(in.)
Experimental
Capacity (kips)
Theoretical
Capacity (kips)
Error
(%)
0.00
375.2
540.0
1.27
259.5
254.0
2.54
86.3
98.0
13.6
3.00
65.1
82.0
26.0
3.50
56.0
69.5
24.0
0.00
305.0
515.0
68.9
1.27
217.0
259.0
19.4
2.54
54.0
68.5
26.9
3.00
32.9
44.0
34.0
3.50
24.3
32.5
34.0
44.0
-
2.1
Comparison o f T e s t and T h e o r e t i c a l C a p a c i t i e s f o r
137.0 i n H e i g h t , 8 i n . R e i n f o r c e d Masonry W a l l s
are misleading.
Due to the small absolute magnitude, with the f a i l u r e
load small compared to the accuracy of the experimental set-up, a d i f f e r ence of a few kips caused by inaccurate reading w i l l be shown as a b i g
percentage error, when comparing i t with the actual f a i l u r e load i t s e l f .
This explains the high percentage error (as much as 34%) recorded for
walls reinforced with 3-#6
and 3-#3
bars.
Figs. 3.3 and 3.4 give a better
picture of the comparison and the results appear to give good correlations.
3.4
Joint
Reinforcement
The normal function of the j o i n t (horizontal) reinforcement i s to
provide a two-way action for r e s i s t i n g l a t e r a l forces.
The presence of
j o i n t reinforcement can also r e s t r i c t l a t e r a l expansion of mortar joints
under in-plane compressive
c i t y of the wall.
forces and hence increase the compression
capa-
The e f f e c t of the j o i n t reinforcement has been i n v e s t i -
gated i n both Refs. (7) and (8).
1 9
Supported by the results of the prism tests, Dyrsdale and Hamid ^
1
reported that j o i n t reinforcement of normal gauge wires did not provide
much b e n e f i c i a l confining e f f e c t or any- deterimental e f f e c t due to stress
concentration.
In Ref. (7) Hatzinikolas, Longworth, and Warwaruk,
reported that there were reductions i n strength for walls with normal
joint-reinforcement compared to the walls without joint reinforcement.
The reduction of capacities, for walls without v e r t i c a l reinforcement,
varied from 18% to 22% and the reduction of capacities for reinforced
short walls was 6%.
Both papers asserted that flattened j o i n t reinforce-
ment was more b e n e f i c i a l than normal gauge wire joint reinforcement.
I t i s d i f f i c u l t to introduce the e f f e c t of j o i n t reinforcement i n the
p r e s e n t a n a l y s i s , a n d , s i n c e both papers are i n favour of the w a l l s w i t h
no j o i n t
reinforcement,
all
e x p e r i m e n t a l r e s u l t s chosen f o r comparison
p u r p o s e s , and the a n a l y t i c a l r e s u l t s , are based on w a l l s without
reinforcement.
joint
44.
IV.
EVALUATION OF THE CODE DESIGN METHOD
4.1
Introduction
The c u r r e n t
Canadian Design Code f o r Masonry W a l l s and Columns has
adopted a l l o w a b l e
stress design.
The p r o v i s i o n s are c l e a r l y
"Masonry Design and C o n s t r u c t i o n f o r B u i l d i n g s . "
stated
(CAN3-S304-M78).
in
The
code i s mainly based on the work done by the B r i c k I n s t i t u t e of A m e r i c a ,
and g i v e s p r o v i s i o n s f o r many a s p e c t s o f d e s i g n and c o n s t r u c t i o n
Material properties,
covered.
construction requirements,
and d e s i g n procedures
are
The t o p i c t o be f o c u s e d on here i s the d e s i g n equations used f o r
c a l c u l a t i n g the c a p a c i t y of l o a d b e a r i n g members or
4.2
detail.
walls.
Code Design P r o v i s i o n s
In the c o d e , the d e s i g n s e c t i o n c a p a c i t y of a masonry l o a d
1
member i s dependent on the a l l o w a b l e
net s e c t i o n a r e a , where the a l l o w a b l e
s t r e s s of the masonry u n i t and the
stress for
type of s t r e s s and the p r i s m a x i a l s t r e n g t h
allowable
magnitude
d e s i g n i s based on the
of the masonry u n i t .
d e s i g n l o a d i s o b t a i n e d by m u l t i p l y i n g
reduction faction
and p a t t e r n of the a p p l i e d end moment,
A brief
the e f f e c t of
and the
d e s c r i p t i o n of the d e s i g n
P = C C f
A
e s m n
P = C C (f +0.8
e s m
exceeds
equations
t/3,
f o r p l a i n masonry w a l l s
p f
s
)A
n
4.1
f o r r e i n f o r c e d masonry
walls.
the
slenderness
follows:
When the maximum v i r t u a l e c c e n t r i c i t y
The
the c a p a c i t y by two
f a c t o r s C and C , a c c o u n t i n g f o r
e
s
effect respectively.
bearing
4.2
45.
where
e = maximum v i r t u a l e c c e n t r i c i t y which i s d e f i n e d as the maximum
v a l u e of the primary moment a l o n g the member d i v i d e d by the
applied v e r t i c a l load.
t = effective
t h i c k n e s s of the member
P = a l l o w a b l e working
f
m
= allowable
load
compressive s t r e s s f o r masonry
A = net c r o s s - s e c t i o n area
n
p = reinforcement
f
C
s
e
= allowable
ratio
steel
= eccentricity
C = slenderness
s
stress
coefficient
coefficient
The s l e n d e r n e s s c o e f f i c i e n t ,
C , i s a f u n c t i o n of the h / t
s
t h e end e c c e n t r i c i t y r a t i o , -^/ 2'
e
at both ends w i t h
> e^.
w
e
n
e
r
e
e
^
a n (
i 2
e
a
r
e
the
r a t i o and
eccentricities
The e m p i r i c a l e q u a t i o n f o r C
g
is:
e
C
= 1.20 - ^ | (5.75 + (1.5+-i) ) < 1.0
s
300
e
4.3
2
2
The e c c e n t r i c i t y
virtual
eccentricity,
eccentricity
ratio,
e
coefficient,
C e , i s a f u n c t i o n of the maximum
e , the e f f e c t i v e
^/ 2•
e
T l i e
thickness, t,
and the end
f u n c t i o n i s d e s c r i b e d by two
governed by the magnitude of the maximum e c c e n t r i c i t y .
C
e =
ITTTiA)
+
H ~
z0
) ( 1
" IT- * «
e
C = 1.95(i - §) + i ( f - ^ ) ( 1 — - )
e
2.
t
z t
20
e^
t/20 < e < t/6
f o r t/6
< e < t/3
equations
The e q u a t i o n s
4.4
4.5
are:
46.
When the maximum v i r t i a l
s e c t i o n c a p a c i t y of the
e c c e n t r i c i t y exceeds t / 3 ,
(plain
or r e i n f o r c e d )
formed s e c t i o n and the assumption of l i n e a r
the
allowable
w a l l i s based on the
stress d i s t r i b u t i o n .
s e c t i o n c a p a c i t y i s m o d i f i e d by the s l e n d e r n e s s c o e f f i c i e n t ,
transThen the
C , to
s
obtain
the a l l o w a b l e working l o a d .
When d e s i g n i n g the l o a d b e a r i n g member, a l i m i t a t i o n
ness of the member i s imposed such t h a t the r a t i o
the v a l u e o f 10(3-e^/e^).
of h / t
on the
slender-
does not exceed
And a minimum e c c e n t r i c i t y of t / 2 0
i s recom-
mended i n the code t o account f o r the i m p e r f e c t i o n of the member.
4.3
Comparison o f the T h e o r e t i c a l C a p a c i t i e s and the Code Design V a l u e s
f o r Masonry Walls
4.3.1
General
In order t o examine the e f f e c t i v e n e s s of the c u r r e n t code d e s i g n
method, a comparison t e s t was made between the code d e s i g n v a l u e s and the
theoretically
"exact" v a l u e s e v a l u a t e d with the computer programme
described i n Section
4.3.2
2.2.
The T h e o r e t i c a l V a l u e s
The t h e o r e t i c a l
masonry w a l l s
interaction
shown on F i g s .
m i n i n g the t h e o r e t i c a l
2.6,
diagrams f o r 8" p l a i n and r e i n f o r c e d
3.1, 3 . 2 ,
3.3,
and 3.4
c a p a c i t y of the w a l l s .
were used f o r
deter-
As suggested i n s e c t i o n
the w a l l s w i t h s i m i l a r c o m p o s i t i o n of the c r o s s - s e c t i o n can have
similar
c r o s s - s e c t i o n p r o p e r t i e s and t h e r e f o r e
stress-strain relationships.
can be governed by the same
For f a c e - s h e l l c o n s t r u c t e d p l a i n w a l l s ,
the
c r o s s - s e c t i o n geometries f o r w a l l s c o n s t r u c t e d w i t h v a r i o u s b l o c k s i z e s
are s i m i l a r and t h e i r
effective
concrete with d i f f e r e n t
s e c t i o n s are r e p r e s e n t e d by two s t r i p s o f
s p a c i n g s between them.
If
the m a t e r i a l s
of
c o n s t r u c t i o n a r e the same, the i n t e r a c t i o n diagram f o r the 12" f a c e - s h e l l
c o n s t r u c t e d masonry w a l l can be c o n s t r u c t e d by u s i n g
curve
developed f o r the 8" f a c e - s h e l l c o n s t r u c t e d
the s t r e s s - s t r a i n
plain wall.
Based on
t h i s argument, the t h e o r e t i c a l load-moment i n t e r a c t i o n diagram f o r the 12"
p l a i n masonry w a l l i s produced and shown on F i g . 4.1.
T h i s diagram w i l l
be used along with F i g s . 3.1, 3.2, 3.3, and 3.4 i n the l a t t e r course of
t h i s study,
and i t w i l l p r o v i d e more i n f o r m a t i o n when s t u d y i n g
the s l e n -
derness e f f e c t on masonry w a l l s .
Straight
l i n e s r e p r e s e n t i n g constant
end e c c e n t r i c i t i e s were drawn on
the t h e o r e t i c a l load-moment i n t e r a c t i o n diagrams.
the s t r a i g h t l i n e and the i n t e r a c t i o n curve,
Each i n t e r s e c t i o n of
representing a particular
w a l l h e i g h t , gave the f a i l u r e c o n d i t i o n as a combination of l o a d and
moment.
A l l of the t h e o r e t i c a l c a p a c i t i e s
f o r walls with various wall
h e i g h t s were e x t r a c t e d i n t h i s manner and were recorded
4.2, 4.3, 4.4, and 4.5.
i n Tables
4.1,
48.
0
10
20
30
40
50
60
MOMENT, kip-ft.
Fig.
4.1
Theoretical
Walls
I n t e r a c t i o n Diagram f o r
12" P l a i n Masonry
49.
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
0
232.0
218.0
205.0
185.0
160.0
133.0
110.0
92.0
t/20
207.0
194.0
181.0
162.0
139.0
117.0
97.0
82.0
t/15
199.0
187.0
175.0
154.0
133.0
111.0
93.0
77.0
t/10
187.0
175.0
162.0
142.0
122.0
100.0
83.0
69.0
t/6
165.0
154.0
141.0
121.0
98.0
80.0
66.0
54.0
t/5
156.0
145.0
132.0
110.0
87.0
70.0
57.0
48.0
t/4
143.0
134.0
113.0
88.0
68.0
53.0
43.0
32.0
t/3
121.0
88.0
59.0
45.0
25.0
20.0
.15.0
7.0
Height
(in.)
Ecc.
Table 4.1
Theoretical Capacities ( i n kips) for 8" P l a i n
Masonry Walls
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
0
251.0
250.0
239.0
232.0
222.0
212.0
200.0
175.0
t/20
224.0
. 217.0
208.0
202.0
192.0
182.0
166.0
155.0
t/15
214.0
207.0
200.0
192.0
184.0
173.0
160.0
150.0
t/10
197.0
192.0
185.0
176.0
168.0
161.0
150.0
140.0
t/6
175.0
170.0
162.0
156.0
151.0
143.0
134.0
123.0
t/5
167.0
161 .0
155.0
150.0
144.0
136.0
125.0
111.0
t/4
155.0
150.0
145.0
140.0
133.0
122.0
104.0
87.0
t/3
138.0
134.0
127.0
112.0
80.0
68.0
58.0
41.0
Height
(in.)
Ecc.
Table 4.2
Theoretical Capacities ( i n kips) f o r 12" Plain
Masonry Walls
50....
Theoretical
Capacities
(kips)
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
0
570.0
520.0
439.0
362.0
302.0
252.0
210.0
163.0
t/20
484.0
434.0
361.0
296.0
240.0
194.0
160.0
137.0
t/15
462.0
405.0
339.0
271.0
220.0
178.0
147.0
127.0
t/10
419.0
359.0
291.0
229.0
184.0
150.0
126.0
103.0
t/6
332.0
265.0
203.0
159.0
124.0
102.0
80.0
57.0
t/5
286.0
220.0
163.0
128.0
100.0
81.0
52.0
40.0
t/4
215.0
152.0
110.0
86.0
64.0
36.0
29.0
20.0
t/3
110.0
70.0
44.0 .
30.0
21.0
16.0
13.0
11.0
51/12
60.0
. 38.0
26.0
20.0
16.0
13.0
11.0
8.0
Height
(in.)
Eccentricity
Table 4.3
Theoretical
C a p a c i t i e s f o r 8" R e i n f o r c e d Masonry W a l l s
( 3 - #3 @ £ )
51...
Theoretical
Capacities
(kips)
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
0
610.0
560.0
460.0
377.0
308.0
250.0
204.0
173.0
t/20
512.0
442.0
360.0
297.0
239.0
191.0
159.0
137.0
t/15
484.0
411.0
332.0
274.0
218.0
172.0
146.0
126.0
t/10
433.0
356.0
287.0
228.0
181.0
145.0
123.0
103.0
t/6
334.0
256.0
197.0
150.0
120.0
100.0
80.0
50.0
t/5
282.0
210.0
156.0
121.0
97.0
73.0
60.0
32.0
t/4
212.0
150.0
108.0
79.0
61.0
50.0
30.0
24.0
t/3
143.0
98.0
70.0
52.0
42.0
31.0
24.0
19.0
5t/12
107.0
75.0
54.0
42.0
33.0
25.0
20.0
17.0
• t/2
87.0
62.0
45.0
36.0
27.0
23.0
18.0
14.0
Height
(in.)
Eccentricity
Table 4.4
Theoretical
C a p a c i t i e s f o r 8" R e i n f o r c e d Masonry Walls
( 3 - #6 @ £ )
52.
Theoretical
Capacities
(kips)
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
0
690.0
600.0
463.0
384.0
311.0
255.0
207.0
182.0
t/20
557.0
465.0
372.0
300.0
243.0
196.0
160.0
135.0
t/15
520.0
426.0
338.0
273.0
220.0
175.0
144.0
123.0
t/10
454.0
354.0
288.0
227.0
175.0
145.0
119.0
103.0
t/6
343.0
261.0
200.0
153.0
123.0
101.0
80.0
68.0
t/5
276.0
204.0
151.0
117.0
93.0
74.0
60.0
50.0
t/4
211 .0
156.0
115.0
87.0
68.0
58.0
43.0
38.0
t/3
154.0
118.0
90.0
67.0
55.0
46.0
33.0
31.0
5t/12
124.0
99.0
77.0
58.0
50.0
39.0
31.0
28.0
Height
(in.)
Eccentricity
Table 4.5
Theoretical
C a p a c i t i e s f o r 8" R e i n f o r c e d Masonry Walls
( 3 - #9 @
)
53.
4.3.3
C a l c u l a t i o n of the Code Design V a l u e s
In c o o r d i n a t i o n w i t h the t h e o r e t i c a l
c a p a c i t i e s , the a l l o w a b l e
f o r p l a i n and r e i n f o r c e d w a l l s w i t h h e i g h t s v a r y i n g from 95.625"
375.625" — with i n t e r v a l s
of 40"
t o the p r e s e n t code p r o v i s i o n s .
from 0 t o t / 3
and
4.10.
to
(5 courses) — were c a l c u l a t e d a c c o r d i n g
The equal end e c c e n t r i c i t i e s were v a r i e d
f o r p l a i n w a l l s and from 0 t o 5t/12
corresponding allowable
loads
reinforced walls.
c a p a c i t i e s are shown i n t a b l e s 4 . 6 ,
4.7,
4.8,
The
4.9
54.
Allowable
Loads ( k i p s )
95.63
135.63.
175.63*
t/20
26.18
18.3
10.45
2.58
t/15
24.31
17.0
9.70
2.40
t/10
21.27
14.88
8.49
2.10
t/6
17.02
11.90
6.79
1.68
t/5
15.32
10.71
6.11
1.51
t/4
12.76
8.93
5.09
1.26
t/3
8.51
5.95
3.40
0.84
Height
(in.)
215.63*
Eccentricity
0 -
T a b l e 4.6
Allowable
Loads f o r 8" P l a i n Masonry
Walls
95.63
135.63
t/20
32.64
27.49
22.44
17.17
12.01
6.85
1.69
t/15
30.31
25.52
20.73
15.94
11.15
6.36
1.57
t/10
26.52
22.33
18.14
13.95
9.76
5.57
1.38
t/6
21.22
17.87
14.51
11.16
7.81
4.45
1.10
t/5
19.10
16.08
13.06
10.04
7.03
4.01
0.99
t/4
15.91
13.40
10.88
8.37
5.85
3.34
0.83
t/3
10.61
8.93
7.26
5.58
3.90
2.23
0.55
Height
(in.)
175.63 215.63 255.63* 295.63* 335.63*
Eccentricity
0 -
Table 4.7
A l l o w a b l e Loads f o r 12" P l a i n Masonry
* Values not p e r m i t t e d i n code
Walls
55.
A l l o w a b l e Loads ( k i p s )
95.63
135.63
175.63*
t/20
70.28
49.16
28.05
6.93
t/15
65.26
45.65
26.04
7.47
t/10
57.10
39.94
22.79
5.63
t/6
45.68
31.95
18.23
4.51
t/5
41.12
28.76
16.41
4.06
t/4
34.26
23.97
13.67
3.38
t/3
22.84
15.98
9.11
2.25
51/12
11.77
8.23
4.70
1.16
Height
(in.)
215.63*
Eccentricity
0 -
Table 4.8
A l l o w a b l e Loads f o r 8" R e i n f o r c e d Masonry
W a l l s ( 3 - #3 <§ g )
Allowable Loads (kips)
95.63
135.63
175.63*
215.63*
0 - t/20
83.52
58.42
33.33
8.24
t/15
77.55
54.25
30.95
7.65
t/10
67.86
47.47
27.08
6.69
t/6
54.29
37.97
21.66
5.36
t/5
48.86
34.17
19.56
4.82
t/4
40.72
28.48
16.25
4.02
t/3
27.14
18.99
10.83
2.68
5t/12
14.63
14.63
5.84
1.44
Height (in.)
Eccentricity
Table 4.9
Allowable Loads for 8" Reinforced Masonry
Walls ( 3 - #6 @ g )
Allowable
Loads ( k i p s )
95.63
135.63
175.63*
215.63*
t/20
106.23
74.32
42.39
10.48
t/15
98.65
69.00
39.36
9.7
t/10
86.32
60.37
34.44
8.52
t/6
69.05
48.30
27.55
6.81
t/5
62.15
43.47
24.80
6.13
t/4
51.79
36.22
20.67
5.11
t/3
34.53
24.15
13.78
3.41
5t/12
15.64
10.94
6.24
1.54
Height
(in.)
Eccentricity
0 -
T a b l e 4.10
.
A l l o w a b l e Loads f o r 8" R e i n f o r c e d Masonry
W a l l s ( 3 - #9 @ £ )
58.
4.3.4
Comparison of the R e s u l t s
Tables 4.11,
4.12,
i n v o l v e d i n the c u r r e n t
as the r a t i o
4.13,
4.14,
and 4.15
code d e s i g n method.
of the t h e o r e t i c a l
r e c o r d the s a f e t y
factors
The s a f e t y f a c t o r
is
defined
c a p a c i t y t o the c o r r e s p o n d i n g a l l o w a b l e
design l o a d .
S i n c e the s l e n d e r n e s s r a t i o
10(3
of each w a l l i s l i m i t e d t o not e x c e e d i n g
- e ^ / e ^ ) , with equal end e c c e n t r i c i t i e s
limitation
f o r 8" w a l l s
is
160",
the number of c o r e s g r o u t e d .
and f o r
(e^=e^), the
height
12" w a l l s i s 240",
However, the e q u a t i o n f o r C
regardless of
quoted from the
s
code does a l l o w one t o c a l c u l a t e l o a d s f o r h e i g h t s up t o 229"
walls,
and 349"
for
T a b l e s 4.11
walls.
12"
for
walls.
and 4.12
show the s a f e t y f a c t o r s f o r the 8" and 12"
W i t h i n the range p e r m i t t e d by the code, the s a f e t y f a c t o r
from 7.91
to
14.79.
Beyond the s l e n d e r n e s s r a t i o
c o d e , the s a f e t y f a c t o r v a r i e s from 17.92
i s found on 12" p l a i n w a l l s .
plain
varies
l i m i t recommended i n
to 72.85.
The v a l u e s
i n c o n s i s t a n t and they i n c r e a s e w i t h h e i g h t and e c c e n t r i c i t y .
pattern
8"
are
The same
The s a f e t y f a c t o r s vary from 6.83
20.07 w i t h i n the range p e r m i t t e d by the c o d e , and when the s l e n d e r n e s s
ratio
exceeds the l i m i t a t i o n ,
Tables 4.13,
partially
4.14,
the v a l u e s range from 15.99
and 4.15
126.26.
show the comparison of r e s u l t s f o r
grouted w a l l s r e i n f o r c e d w i t h 3-#3,
respectively.
to
3-#6,
and 3-#9
to
10.58
8"
bars
W i t h i n the s l e n d e r n e s s r a t i o p e r m i t t e d by the c o d e ,
v a l u e s vary from 4.38
the
the
to
Factors of Safety
95.63
135.63
175.63*
215.63*
0
8.86
11.91
19.62
71.71
t/20
7.91
10.60
17.32
62.79
t/15
8.19
11.00
i
18.04
64.17
t/10
8.79
11.76
19.09
67.68
t/6
9.70
12.94
20.77
72.02
t/5
10.19
13.54
21.60
72.85
t/4
11.13
15.01
22.19
69.70
t/3
14.22
14.79
17.38
53.62
Height ( i n . )
Eccentricity
Table 4.11
F a c t o r s of S a f e t y f o r 8" P l a i n Masonry Walls
i n Current Design Code
F a c t o r s of
Safety
96.63
135.63
175.63
215.63
255.63*
295.63* 335.63*
0
7.69
9.09
10.70
13.51
18.48
30.95
118.34
t/20
6.86
7.89
9.31
11.76
15.99
26.57
98.22
t/15
6.83
8.11
9.65
12.05
16.50
27.20
101.91
t/10
7.43
8.60
10.20
12.62
17.21
28.90
108.70
t/6
8.25
9.51
11.16
13.98
19.33
32.13
121.82
t/5
8.74
10.01
11.87
14.94
20.48
33.92'
126.26
t/4
9.74
11.19
13.33
16.73
22.74
36.53
125.30
t/3
13.01
15.01
17.49
20.07
20.51
30.49
105.45
Height ( i n . )
Eccentricity
Table 4.12 F a c t o r s o f S a f e t y f o r 12" P l a i n Masonry Walls
i n Current Masonry Design Code
* Values not p e r m i t t e d i n code
60..
Factors of
Height
(in.)
95.63
135.63
0
8.11
t/20
Safety
175.63*
215.63*
10.58
15.65
52.24
6.89
8.83
12.87
42.71
t/15
7.08
8.87
13.02
36.28
t/10
7.34
8.99
12.77
40.67
t/6
7.27
8.29
11.14
35.28
t/5
6.96
7.65
9.93
31.53
t/4
6.28
6.34
8.05
25.44
t/3
4.82
4.38
4.83
13.33
5.10
4.62
5.53
17.24
Eccentricity
5t/12.
Table 4.13
.
F a c t o r s o f S a f e t y f o r 8" R e i n f o r c e d Masonry
W a l l s ( 3 - #3 § ^ ) i n C u r r e n t Masonry
Design Code
Factors of Safety
95.63
135.63
0
7.30
t/20
Height ( i n . )
175.63*
215.63*
9.59
13.80
45.75
6.13
7.57
10.80
36.04
t/15
6.54
7.58
10.73
35.82
t/10
6.38
7.50
10.60
34.08
t/6
6.15
6.74
9.10
27.99
t/5
5.77
6.15
8.00
25.10
t/4
5.21
5.27
6.65
19.65
t/3
5.27
5.16
6.46
19.40
5t/12
7.31
5.13
9.23
29.17
Eccentricity
T a b l e 4.14
F a c t o r s o f S a f e t y f o r 8" R e i n f o r c e d Masonry
Walls ( 3 - #6 @ £ ) i n C u r r e n t Masonry
Design Code
Factors of
95.63
135.63
0
6.6,0
t/20
Safety
175.63*
215.63*
8.08
10.92
36.64
5.24
6.26
18.78
28.63
t/15
5.27
6.17
18.59
28.05
t/10
5.26
5.86
18.36
26.66
t/6
4.97
5.40
7.26
22.46
t/5
4.44
4.69
6.09
19.08
t/4-
4.07
4.31
5.56
17.03
t/3
4.46
4.89
6.53
19.67
5t/12
9.55
10.90
14.85
45.26
Height
(in.)
Eccentricity
T a b l e 4.15
.
F a c t o r s o f S a f e t y f o r 8" R e i n f o r c e d Masonry
W a l l s ( 3 - #9 @ £ ) i n C u r r e n t Masonry
Design Code
63.
f o r w a l l s r e i n f o r c e d with 3 -
#3 b a r s , from 5.13
r e i n f o r c e d w i t h 3 - #6 bars and from 4.31
with 3 -
#9 b a r s .
limitation,
6.46
#6,
to
t o 9.59
10.90
for
for walls
walls
reinforced
At w a l l h e i g h t s e x c e e d i n g the code s l e n d e r n e s s
the v a l u e s of the s a f e t y f a c t o r s vary from 4.83
t o 45.75 and from 5.56
to 52.24,
from
t o 45.26 f o r w a l l s r e i n f o r c e d w i t h 3 - #3,
and 3 - #9 bars r e s p e c t i v e l y .
The ranges of d i s c r e p a n c y are
3 -
smaller
than those e x h i b i t e d i n p l a i n masonry w a l l s .
The s a f e t y f a c t o r s
increase
with h e i g h t of the w a l l or s l e n d e r n e s s r a t i o ,
but decrease with the
magnitude of the end e c c e n t r i c i t y .
4.3.5
In
Remarks
g e n e r a l , the r e s u l t s above show t h a t the c u r r e n t code d e s i g n
method g i v e s i n c o n s i s t a n t r e s u l t s when compared with t h e o r e t i c a l
and the r e s u l t s f o r the p l a i n w a l l s are more c o n s e r v a t i v e than
reinforced walls.
the
They a l s o show t h a t the code d e s i g n c a p a c i t y ,
by the recommended s l e n d e r n e s s c o e f f i c i e n t ,
r e s u l t s f o r t a l l masonry w a l l s ,
slenderness r a t i o .
values,
gives over-conservative
as the s a f e t y f a c t o r
The l i m i t a t i o n
modified
i n c r e a s e s with the
on s l e n d e r n e s s r a t i o
recommended by the
code i s found t o be a v e r y c o n s e r v a t i v e measure, s i n c e t h e r e appears t o be
a u s e f u l l o a d c a p a c i t y at h e i g h t s g r e a t e r
Furthermore,
the use of the h / t
a bad c h o i c e , s i n c e i t
does not r e f l e c t
members, which i s an important
effect.
ratio
factor
Compared with a p a r t i a l l y
masonry w a l l may have d i f f e r e n t
s l e n d e r n e s s of the member.
than i m p l i e d
as the s l e n d e r n e s s parameter
the s e c t i o n geometry of
is
the
i n e v a l u a t i n g the s l e n d e r n e s s
grouted w a l l ,
a f a c e - s h e l l constructed
c h a r a c t e r i s t i c s when c o n s i d e r i n g the
The use of the r a t i o
t o the r a d i u s of g y r a t i o n of the member, h / r ,
parameter
thereby.
s h o u l d be an improvement.
of the e f f e c t i v e
(or L / r )
height
as the s l e n d e r n e s s
64.
V.
MODIFICATION OF THE MOMENT MAGNIFIER METHOD
FOR MASONRY WALL
5.1
DESIGN
The Moment M a g n i f i e r Method
The Moment M a g n i f i e r Method was o r i g i n a l l y developed
steel.
for structural
I t has been w e l l adapted by the ACI code f o r the d e s i g n o f
concrete s t r u c t u r e s , accounting
the presence
f o r the e f f e c t s o f m a t e r i a l p r o p e r t i e s ,
o f c r a c k s and r e i n f o r c e m e n t ,
b e a r i n g member.
and the s l e n d e r n e s s o f the l o a d
T h i s method c o n s i d e r s e c c e n t r i c i t y and the s l e n d e r n e s s
e f f e c t s by magnifying
end moments.
The i n c r e a s e d moment and c o r r e s p o n d i n g
l o a d a r e then checked a g a i n s t the l i m i t imposed by the s h o r t column ( o r
wall) capacity.
The m a g n i f i e r has the form:
C
" 1.0
0
where
- P/cJP
3 , J
cr
C = 0.6 +'0.4 e , / e
m
1 2
TT ( E I )
-
5.1.1
0
2
P
e
= ,...2 y
5.1.2
&
cr
l> 2
( k L ) A
= end e c c e n t r i c i t i e s
e
k = e f f e c t i v e length factor
(EI)
The
e
= e f f e c t i v e r i g i d i t y of s e c t i o n ,
capacity reduction factor,
<|>, w i l l be omitted a t t h i s stage o f the
discussion.
(EI)
i n l i g h t l y r e i n f o r c e d s e c t i o n s such a s a r e t y p i c a l o f masonry
e
may be expressed a s :
(EI)
where
e
= EI/X
5.1.3
E
= i n i t i a l modulus o f masonry assemblage.
I
= net s e c t i o n uncracked
X
= a rigidity
moment o f i n e r t i a .
reduction factor.
65.
5.2
The Modulus o f E l a s t i c i t y of Masonry
Assemblages
In u l t i m a t e s t a t e masonry d e s i g n , the s t r e n g t h and d e f l e c t i o n a r e
governed by the modulus o f e l a s t i c i t y o f the m a t e r i a l .
At t h i s s t a g e o f
time, l i t t l e a t t e n t i o n has been p a i d to the e l a s t i c modulus o f masonry and
experimenal i n f o r m a t i o n i s l a c k i n g .
The e q u i v a l e n t modulus o f e l a s t i c i t y
of masonry assemblages, b e s i d e s being i n f l u e n c e d by the moduli o f masonry
c o n s t i t u e n t s (namely, b l o c k , mortar, and g r o u t —
depends on the geometry
i f applicable),
also
of the s e c t i o n , the t h i c k n e s s o f the mortar
and the l o a d i n g c o n d i t i o n .
joints
In-depth i n v e s t i g a t i o n o f t h i s parameter i s
out o f the scope of t h i s paper, but i t w i l l be an i n t e r e s t i n g
topic for
f u t u r e study.
In the p r e c e d i n g c a l c u l a t i o n s , the i n i t i a l g r a d i e n t o f the s t r e s s s t r a i n curve shown i n F i g . 2.6 was
masonry.
used as the modulus of e l a s t i c i t y o f
The v a l u e s used were 1389 k s i and 2083 k s i f o r p l a i n and
p a r t i a l l y grouted w a l l s r e s p e c t i v e l y .
p a r t i a l l y grouted w a l l i s s t i f f e r
Thus under f l e x u r a l type l o a d i n g , a
than a p l a i n masonry w a l l .
S i n c e the
a v a i l a b l e e x p e r i m e n t a l data are l i m i t e d , i t i s i m p o s s i b l e to r e l a t e the
e f f e c t i v e e l a s t i c modulus o f the masonry assemblage
to i t s s e c t i o n
geometry.
Under the p r o v i s i o n s o f the c u r r e n t Canadian Masonry Code S-304-M78,
the modulus o f masonry depends s o l e l y on the compressive s t r e n g t h o f
masonry, which i s governed by the s t r e n g t h o f the u n i t and mortar type
used (E = lOOOf'm).
m
With the average u n i t
s t r e n g t h of 2350 p s i and S
type mortar being used, the nominal compressive s t r e n g t h of the masonry
(fm') i s 1520 p s i and the recommended modulus i s 1520 k s i . '
E s k e n a z i , O j i n a g a , and T u r k s t r a
1 7
suggest t h a t E
731,000 p s i , and the modulus of e l a s t i c i t y would
m
= 440 x fm' +
then be 1400 k s i .
66,.
Hatzinikolas
7
recommends a lower m u l t i p l i e r
than the c o d e ,
(ie:
E
=
m
750 f m )
and h i s recommended v a l u e of e l a s t i c modulus would be 1140
ksi.
The e m p i r i c a l e x p r e s s i o n s above f o r e v a l u a t i n g the modulus of
elasticity
of masonry are a l l
ungrouted p r i s m s .
the same u n i t
based on compression t e s t s of a x i a l l y
These v a l u e s are used f o r a l l masonry assemblages w i t h
s t r e n g t h and mortar type r e g a r d l e s s of the d i f f e r e n c e s
c r o s s - s e c t i o n composition ( p l a i n ,
It
loaded
partially
g r o u t e d , or f u l l y
in
grouted).
i s i n t e r e s t i n g t o note t h a t the modulus f o r f a c e - s h e l l c o n s t r u c t e d
p l a i n w a l l s o b t a i n e d from the s t r e s s - s t r a i n diagram d e r i v e d i n Chapter
II,
does agree with the v a l u e s p r e d i c t e d w i t h the e x p r e s s i o n suggested by
E s k e n a z i , O j i n a g a , and T u r k s t r a , w h i l e the e l a s t i c modulus of
9
grouted w a l l s i s about 1.5
times the modulus f o r p l a i n w a l l s .
absence of s u f f i c i e n t knowledge on the i n t e r a c t i o n
it
i s b e t t e r t o use d i f f e r e n t
partially
In
the
of masonry components,
moduli f o r p l a i n and p a r t i a l l y
grouted
masonry w a l l s .
5.3
G e n e r a l F u n c t i o n f o r the R i g i d i t y Reduction F a c t o r
(X)
The f o l l o w i n g a n a l y s i s c o n c e n t r a t e s on d e v e l o p i n g a l i m i t
d e s i g n method based on the moment-magnifier
state
procedure.
For s i m p l i c i t y , a l l w a l l s were assumed t o be loaded w i t h equal end
eccentricities:
e, = e„
1
2
Then e q u a t i o n 5.1
and
C = 1.0
m
becomes:
By s u b s t i t u t i n g 5.1.2
o=
and 5.1.3
1
_
into
—
cr
5.2, we have:
5.2
67.
6=
i
If
the top and
5.3
[Pk L A/(Tf EI)]
1 -
2
2
2
bottom are assumed to be pinned,
k i s e q u a l to 1,
and
the above e q u a t i o n becomes
6
1
1 l e a d i n g to 1/6
or
(PL X)/(n EI)
2
2
- ( P L A ) / ( TT EI)
= 1.0
2
2
X = ( TT EI)(1-1/6)/(PL )
2
By f i n d i n g
5.4
2
the r i g h t r i g i d i t y
r e d u c t i o n f a c t o r the m a g n i f i e r can
e v a l u a t e d and
the reduced
be o b t a i n e d .
S i n c e the r i g i d i t y E I and
particular wall,
c a p a c i t y of w a l l due
to s l e n d e r n e s s e f f e c t
l e n g t h L, a r e c o n s t a n t
1/6
c a l c u l a t e d by deducing
a t any a s s i g n e d l o a d v a l u e , P.
the m a g n i f i c a t i o n f a c t o r
Values
<$ d i r e c t l y
retical
i n t e r a c t i o n curves o b t a i n e d
lations
f o r X f o r each s e c t i o n are shown i n Table 5.1,
can
for a
the v a l u e s of f a i l u r e l o a d and moment g i v e the
s h i p between X and
relation-
f o r X were
from the
from the p r e v i o u s c h a p t e r .
5.2,
be
theo-
The
5.3,
calcu-
5.4
and
5.5.
In the c o n c r e t e codes, the r i g i d i t y has been reduced
depending on the s t e e l r a t i o , but
applied load l e v e l
ratio,
be r e l a t e d
factors
to the r a t i o s
of
to the a x i a l c a p a c i t y of s e c t i o n , ( P / P ) which i s a
means of m o n i t o r i n g
derness
X may
by
Q
the degree of c r a c k i n g of the s e c t i o n , and
( L / r ) , where r i s the r a d i u s of g y r a t i o n o f the
the
slen-
section.
I f the r i g i d i t y r e d u c t i o n f a c t o r can be assumed to be a s e p a r a b l e f u n c t i o n
of s l e n d e r n e s s r a t i o
( L / r ) and
can be w r i t t e n as a product
the l o a d r a t i o
of two
(P/Po),
the e x p r e s s i o n f o r X
f u n c t i o n s , such t h a t :
X = K ( L / r ) * K (P/P )
s
p
o
W i t h the above assumptions, the s l e n d e r n e s s e f f e c t and
effect
can be d e a l t with
individually.
the l o a d
R i g i d i t y Reduction F a c t o r s ( A)
95.63
135.75
175.63
215.63
255.63
295.63
335.63
375.63
30.92
43.85
56.78
69.71
82.65
95.58
108.51
121.44
0.080
9.78
6.25
4.56
4.12
3.32
2.92
2.50
2.35
0.099
8.24
5.23
3.80
3.42
2.82
2.35
2.12
1.90
0.119
5.52
4.04
2.98
2.62
2.32
2.01
1.77
1.66
0.159
5.23
3.35
2.44
2.06
1 .78
1.64
1.49
1.38
0.199
4.31
3.03
2.34
1.91
1.67
1.53
1.41
1.29
0.239
3.57
2.54
2.07
1.74
1.55
1.42
1.33
1.24
0.318
2.48
2.02
1.73
1.58
1.44
1.36
1.27
1.20
0.398
2.16
1.89
1.61
1.55
1.44
1.36
1.25
0.477
2.36
2.00
1.77
1.62
1.42
1.33
0.557
1.89
1.79
1.70
1.58
1.45
0.636
2.15
1.98
1.72
1.60
Height
(in.)
(L/r)
(P/P )
Q
Table 5.1
R i g i d i t y Reduction F a c t o r s f o r 8" P l a i n Masonry W a l l s
R i g i d i t y Reduction F a c t o r s (X)
Height
(in.)
(L/r)
95.63
135.75
175.63
215.63
255.63
295.63
335.63
375.63
18.82
26.69
34.57
42.44
50.31
58.18
66.06
73.93
(P/P )
Q
0.080
14.68
8.51
6.53
5.29
4.45
3.84
3.38
3.01
0.119
14.84
8.47
6.03
4.65
3.77
3.16
2.72
2.28
0.159
8.21
5.34
4.12
3.23
2.74
2.38
2.10
1.88
0.199
6.10
4.25
3.26
2.72
2.34 •
2.05
1.82
1.66
0.239
5.66
3.80
3.10
2.45
2.06
1.87
1.70
1.60
0.318
3.44
2.69
2.53
2.35
2.06
1.85
1.72
1.56
0.398
2.79
2.40
2.13
1.88
1.76
1.67
1.60
1.52
0.477
3.16
2.69
2.21
1.98
1.80
1.69
1.59
1.50
0.596
2.35
2.34
2.24
1.95
1.81
1.69
1.64
1.54
Table 5.2
R i g i d i t y Reduction F a c t o r s f o r 12" P l a i n Masonry W a l l s
Rigidity Reduction Factors (A)
Height (in.)
95.63
135.75
175.63
215.63
255.63
295.63
335.63
375.63
(L/r)
41.83
59.34
76.84
94.34
111.84
129.34
146.84
164.34
(P/PQ)
0.031
43.35
28.31
22.06
17.31
13.86
10.90
9.04
7.63
0.046
28.64
19.76
15.18
11.27
8.97
6.93
5.67
4.80
0.061
21.74
15.36
11.26
8.33
6.56
5.17
4.24
3.62
0.092
14.34
10.11
7.17
5.33
4.10
3.36
2.84
2.47
0.123
11.36
7.40
5.34
3.93
3.10
2.53
2.19
1.90
0.154
10.89
6.84
4.79
3.50
2.74
2.24
1.89
1.64
0.200
6.51
4.55
3.35
2.61
2.15
1.81
1.55
1 .36
0.246
4.96
3.69
2.79
2.24
1.87
1.60
1.40
1.25
0.307
3.86
2.99
2.39
1.97
1.67
1.43
1.25
0.384
3.15
2.52
2.10
1.80
1.53
1.31
0.460
2.65
2.33
1.96
1.68
1.45
Table 5 . 3
Rigidity Reduction Factors for 8 " Reinforced Masonry Walls ( 3 - #3 @ g )
Rigidity
Height
(in.)
(L/r)
Reduction F a c t o r s
( X)
95.63
135.75
175.63
215.63
255.63
295.63
335.63
375.63
41.83
59.34
76.84
94.34
111.84
129.34
146.84
164.34
(P/P )
Q
0.028
15.67
17.26
16.85
16.19
13.35
11.38
10.02
8.70
0.042
13.54
14.62
14.11
11.99
9.96
8.42
6.99
5.83
0.056
12.54
13.01
11.80
9.60
7.65
6.25
5.20
4.34
0.084
10.72
10.33
8.61
6.84
5.10
4.04
3.26
2.75
0.113
8.79
8.34
6.58
4.92
3.72
2.98
2.44
2.04
0.141
7.53
6.91
5.13
3.81
2.94
2.37
1.98
1.67
0.183
5.87
5.03
3.78
2.88
2.30
1.89
1.60
1.38
0.225
4.84
4.46
3.03
2.38
1 .96
1.65
1.42
1.22
0.281
3.96
3.19
2.52
2.02
1.69
1.46
1.26
0.422
2.71
2.40
2.02
1.69
1.44
0.492
..2.52.
1.98
1.62
Table 5.4
Rigidity
..
2.32
.
Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry W a l l s
( 3 - #6 @
)
Rigidity Reduction Factors (X)
Height (in.)
95.63
135.75
175.63
215.63
255.63
295.63
335.63
375.63
(L/r)
41.83
59.34
76.84
94.34
111.84
129.34
146.84
164.34
0.025
9.21
9.38
8.84
9.06
9.02
8.63
8.37
7.65
0.049
8.49
7.81
7.58
7.49
6.76
6.04
5.28
4.30
0.099
6.95
6.48
5.97
5.08
3.99
3.15
2.55
2.11
0.160
5.59
5.03
4.08
3.08
2.43
1.96
1.64
1.40
0.197
4.97
4.21
3.28
2.53
2.04
1.67
1.42
1.22
0.246
4.16
3.38
2.61
2.07
1.71
1.44
1.25
0.308
3.41
2.79
2.20
1.78
1.50
1.29
0.369
2.91
2.53
2.00
1.67
1.43
0.431
2.73
2.41
1.94
1.61
0.493
2.51
2.28
1.91
(P/P )
Q
Table 5.5
Rigidity Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #9 @ £ )
5.4
The F u n c t i o n (K )
s
In
order to f i n d a g e n e r a l f u n c t i o n f o r K ,
X had f i r s t
f o r each v a l u e o f ^l^ -
to be normalized
Q
5.8 c o n t a i n a l l v a l u e s o f X normalized
slenderness r a t i o
for
the t h e o r e t i c a l v a l u e s o f
g
( L / r ) i s 76.86.
Tables 5.6, 5.7, and
with r e s p e c t to the v a l u e where the
The average v a l u e s o f the n o r m a l i z e d
X
a l l (P/P ) r a t i o s are shown on Table 5.4 along with the c o r r e s p o n d i n g
o
(L/r)
values.
The above r e s u l t s were p l o t t e d on F i g . 5.1, and, by u s i n g a
U.B.C. n o n - l i n e a r f i t t i n g r o u t i n e "LQF" with 3 parameters requested, a
q u a d r a t i c e q u a t i o n was f i t t e d
through the average v a l u e s o f T a b l e s 5.4.
The approximate f u n c t i o n was found
K ( L / r ) = -0.294 + 1 3 0 ( L / r ) ~
s
The
complete p l o t o f the K
Fig.
fitted
g
to be:
1
- 2325(L/r)"
5.5
f u n c t i o n i s shown on F i g . 5.2.
5.1 shows the p l o t o f n o r m a l i z e d
function of K .
s
2
X vs ( L / r ) r a t i o along with the
There i s a l o t more s c a t t e r a t low ( L / r ) r a t i o s .
The data f o r w a l l s w i t h r e i n f o r c e m e n t r a t i o o f 0.00123 and 0.00491 bunch
t o g e t h e r w e l l a t ( L / r ) r a t i o more than 76.92, while w a l l s with
r e i n f o r c e m e n t r a t i o o f 0.01116 tend to s c a t t e r more a t t h a t range.
At the
low range o f ( L / r ) v a l u e s , the s c a t t e r i n g c h a r a c t e r i s t i c i s i n d i f f e r e n t
for
a l l walls.
By o b s e r v a t i o n , the d e r i v e d f u n c t i o n does d e s c r i b e the
i n f l u e n c e o f s l e n d e r n e s s r a t i o on the r e d u c t i o n f a c t o r r e a s o n a b l y
The
well.
complete p l o t o f the f u n c t i o n K on F i g . 5.2, becomes n e g a t i v e a t
s
s l e n d e r n e s s r a t i o s above 500 and below 20.
75.
Normalized R i g i d i t y
Reduction F a c t o r
41.84
59.34
76.84
94.34
111 .84
129.34
146.84
164.34
0.031
1.97
1.28
1.00
0.78
0.63
0.49
0.41
0.35
0.046
1.89
1.30
1.00
0.74
0.59
0.46
0.37
0.32
0.061
1.93
1.36
1.00
0.74
0.58
0.46
0.38
0.32
0.092
2.00
1.42
1.00
0.74
0.57
0.47
0.40
0.34
0.123
2.13
1.38
1.00
0.74
0.58
0.47
0.41
0.36
0.154
2.27
1.43
1.00
0.73
0.57
0.47
0.40
0.34
0.200
1.94
1.36
1.00
0.78
0.64
0.54
0.46
0.41
0.246
1.78
1.32
1.00
0.80
0.67
0.57
0.50
0.45
0.307
1.61
1.25
1.00
0.82
0.79
0.60
0.49
0.384
1.50
1.20
1.00
0.85
0.73
0.62
0.460
1.35
1.19
1.00
0.86
0.74
(L/r)
(P/P )
Q
Table 5.6
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry
Walls ( 3 - #3 @ £ )
Normalized R i g i d i t y
Reduction F a c t o r
146.84
164.34
41.84
59.34
76.84
94.34
111.84
129.34
0.028
0.93
1.02
1.00
0.96
0.80
0.68
0.60
0;52
0.042
0.96
1.04
1.00
0.85
0.71
0.60
0.50
0.41
0.056
1.06
1.10
1.00
0.81
0.65
0.53
0.44
0.37
0.084
1.25
1.20
1.00
0.80
0.60
0.47
0.38
0.32
0.113
1.34
1.27
1.00
0.75
0.57
0.45
0.37
0.31
0.141
1.47
1.35
1.00
0.74
0.57
0.46
0.39
0.33
0.183
1.55
1.33
1.00
0.76
0.61
0.50
0.42
0.37
0.225
1.60
1.47
1.00
0.79
0.65
0.54
0.47
0.41
0.281
1.57
1.27
1.00
0.80
0.67
Q.58
0.50
0.422
1.34
1.19
1.00
0.84
0.71
0.492
1.27
1.19
1.00
0.82
(L/r)
(P/P )
Q
Table 5.7
——
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry
Walls ( 3 - #6 @ %)
Normalized R i g i d i t y
Reduction F a c t o r
146.84
164.34
41.84
59.34
76.84
94.34
111 .84
129.34
0.025
1.04
1.06
1.00
1.02
1.02
0.98
0.95
0.87
0.049
1.12
1.03
1.00
0.99
0.89
0.80
0.70
0.57
0.099
1.16
1.09
1.00
0.85
0.67
0.53
0.43
0.35
0.160
1.37
1.23
1.00
0.76
0.60
0.48
0.40
0.34
0.197
1.51
1.28
1.00
0.77
0.62
0.51
0.43
0.37
0.246
1.60
1.30
1.00
0.80
0.66
0.55
0.48
0.308
1.55
1.27
1.00
0.81
0.68
0.58
0.370
1.45
1.27
1.00
0.84
0.72
0.431
1.41
1.24
1.00
0.83
0.493
1.31
1.20
1.00
(L/r)
(P/P )
Q
Table 5.8
. — ~ . . -.-—- .
Normalized R i g i d i t y Reduction F a c t o r s f o r 8" R e i n f o r c e d Masonry
Walls ( 3 - #9 @ £ ).
T h i s e f f e c t can be
i s way
over
the p r a c t i c a l
r a t i o of less
order
ignored as a t s l e n d e r n e s s
r a t i o more than 500,
l i m i t a t i o n on s l e n d e r n e s s , and
at a
the w a l l
slenderness
than 20 the w a l l i s g e n e r a l l y t r e a t e d as a s h o r t w a l l .
to a v o i d having n e g a t i v e numbers f o r K
, i t i s helpful
In
to s e t a
s
sensible limit
The
for K
g
which i s p i c k e d
to be 40 < L / r <
400.
f u n c t i o n of Ks does c o r r e l a t e r e a s o n a b l y w e l l w i t h the somewhat
s c a t t e r e d data and
i t does r e f l e c t
the i n f l u e n c e o f s l e n d e r n e s s on
the
r i g i d i t y r e d u c t i o n f a c t o r i n the s i m p l e s t form.
5.5
The
E x p r e s s i o n f o r Load I n f l u e n c e F a c t o r , K
Accepting
K
P
(P/P ) was
o
the above f u n c t i o n f o r Kg,
(P/P )
o
P
the c o r r e s p o n d i n g
s o l v e d e a s i l y with s i m i l a r a n a l y t i c a l
d e f i n e d p r e v i o u s l y , the r i g i d i t y r e d u c t i o n f a c t o r ,
two
functions, K
dividing
s
( L / r ) , and
each t h e o r e t i c a l
K
X by
( P / P ).
o
p
The
for ^
p
computed average v a l u e s
5.11, and
the best
q
the e x p r e s s i o n had
K
p
the
(P/P ) = a + b
q
= a + b
The
For s i m p l i c i t y , an i n v e r s e f u n c t i o n was
5.12.
expression
exponential
chosen
and
form:
(p/p )
- 1
o
(p /p)
Q
e x p r e s s i o n f o r K^ was
r o u t i n e "LQF"
by
calculated
g
( P / P ) i s seen to be not a q u a d r a t i c f u n c t i o n , but an
or inverse f u n c t i o n .
of
were o b t a i n e d
v a l u e of K
vs ( P / P ) i n F i g . 5.3,
q
As
X, i s the product
f o r each ( P / P ) r a t i o a r e shown i n T a b l e s 5.10,
o
p
When s t u d y i n g the p l o t of
ting
section.
techniques.
v a l u e s of K
the c o r r e s p o n d i n g
from the e x p r e s s i o n i n the p r e v i o u s
of K
The
function for
found by u s i n g the same l e a s t
as i n the p r e v i o u s s e c c t i o n .
square
Instead of u s i n g
v a l u e o f ( P / P ), the i n v e r s e o f the v a l u e s were e n t e r e d as
o
fit-
the
the v a r i a b l e s
30
0
0.1
0.2
0.3
P/P
Fig.
5.3
0.4
0
R e l a t i o n s h i p Between Kp and the Load Ratios
(P/P )
Q
for
R e i n f o r c e d Masonry Walls
0.5
81..
a l o n g w i t h the average v a l u e s of
requesting 2 parameters,
K
p
as the dependent v a r i a b l e s .
the r e s u l t i n g e x p r e s s i o n f o r
= 1.207 + 0.457 (p / p )
r
o
r
By
i s shown below:
5.7
(L/r)
Average Normalized
R i g i d i t y Reduction
Factors
Table 5.9
41.84
59.34
76.84
94.34
111.84
129.34
146.84
164.34
1.48
1.23
1.00
0.82
0.68
0.57
0.48
0.41
O v e r - a l l Average Values o f R i g i d i t y Reduction F a c t o r s f o r
8" R e i n f o r c e d Masonry Walls
(P/P )
0.031 . 0.046
0.061
0.092
0.123
0.154
0.200
0.246
0.307
0.384
0.461
Average Value
21.56
14.15
10.62
7.10
5.25
4.74
3.44
2.93
2.48
2.17
1.97
0
T a b l e 5.10
Average Values o f
f o r R e i n f o r c e d Masonry Walls ( 3 - #3 @ £ )
(P/PcJ
0.028
0.042
0.056
0.084
0.113
0.141
0.183
0.225
0.281
0.422
0.492
Average Value
of K
17.88
13.50
10.78
7.59
5.76
4.65
3.63
3.07
2.56
2.00
1.89
p
Table 5.11
Average Values of K
p
f o r R e i n f o r c e d Masonry Walls
( 3 -
#6 @
)
(P/P >
0.025
0.049
0.099
0.160
0.197
0.246
0.308
0.370
0.431
0.493
0.554
Average Value
of K
14.17
8.85
5.50
3.69
3.12
2.62
2.24
2.03
1.93
1.82
1.83
0
p
T a b l e 5.12
Average Values of K,* f o r R e i n f o r c e d Masonry Walls
( 3 - #9 @
)
84.
F i g . 5.3 shows t h a t a t ( P / P ) > 0.2, v a l u e s o f K
Q
and
p
can be f i t t e d w e l l by t h e e x p r e s s i o n found above.
a l l bunch t o g e t h e r
At ( P / P ) < 0 . 2 ,
s c a t t e r i n g o f d a t a becomes severe as ( P / P ) d e c r e a s e s .
q
developed
above by u s i n g the average v a l u e s o f
Q
The e x p r e s s i o n
f o r each type o f r e i n -
f o r c e d w a l l c a n o n l y r e p r e s e n t a f u n c t i o n d e s c r i b i n g the g e n e r a l
of the data.
behavior
The use o f t h i s e x p r e s s i o n f o r d e s i g n purposes w i l l
unconservative r e s u l t s
l e a d to
f o r most w a l l s , e s p e c i a l l y a t low (P/P ) r a t i o s .
o
The
tical
above problem can be i l l u s t r a t e d
c l e a r l y by comparing the t h e o r e -
i n t e r a c t i o n c u r v e s w i t h the i n t e r a c t i o n c u r v e s reproduced
by a p p l y -
i n g the d e r i v e d f u n c t i o n s of K
( L / r ) and K
( P / P ) to the moment magnis
p
o
f i e r method. F i g . 5.4 shows the comparison f o r w a l l s w i t h 3 - #3
reinforcing bars.
I t shows good c o r r e l a t i o n but i s u n c o n s e r v a t i v e i n the
r e g i o n where the ( P / P ) r a t i o i s low. As mentioned i n the i n t r o d u c t i o n o f
o
t h i s Chapter,
masonry w a l l s o f t e n serve as p a n e l elements i n s t r u c t u r e s ,
and a r e r e q u i r e d to c a r r y h i g h e r r a t i o s o f moment t o v e r t i c a l l o a d
g e n e r a l l y occur i n column d e s i g n .
F o r t h a t p a r t i c u l a r reason,
of i n t e r a c t i o n c u r v e s w i t h low ( P / P ) r a t i o i s important
o
than
the p o r t i o n
f o r masonry w a l l
d e s i g n and the p r e s e n t l y d e r i v e d f u n c t i o n s a r e not s a t i s f a c t o r y .
Another problem i n u s i n g the p r e s e n t f u n c t i o n i s t h a t when the r a t i o
of ( P / P ) approaches z e r o , the v a l u e of K
o
p
85.
800
THEORETICAL
700
CURVE
IMPLIED CURVE
600
500
OL
<
O
400
300
200
100
20
30
40
MOMENT , kip-ft.
Fig.
5.4
50
60
Comparison o f T h e o r e t i c a l and Implied I n t e r a c t i o n ' D i a g r a m
f o r 8" R e i n f o r c e d Masonry Walls ( 3-#3 @ Q_)
86.
becomes i n f i n i t e .
In a p p l y i n g t h i s to the moment m a g n i f i e r method, the
becomes zero and the r e s u l t a n t m a g n i f i e r 6
r e s u l t i s v e r y u n s t a b l e as P
approaches i n f i n i t y , w h i l e the t h e o r e t i c a l i n t e r a c t i o n curves show t h a t
the m a g n i f i e r becomes 1. as (P/P ) r a t i o approaches z e r o .
o
This w i l l pro-
v i d e v e r y i n a c c u r a t e r e s u l t s a t v e r y low ( P / P ) r a t i o s .
0
5.6
Improvement o f the E s t a b l i s h e d F u n c t i o n s
F o r the two f u n c t i o n s d e r i v e d c u r r e n t l y , the K ( L / r ) f u n c t i o n i s a
s
g e n e r a l f u n c t i o n which o n l y r e f e r s to the e f f e c t o f s l e n d e r n e s s on r i g i d ity.
Since the two problems encountered
concern
i n the p r e v i o u s s e c t i o n m a i n l y
the l o a d r a t i o e f f e c t , there i s no obvious
change i n the K f u n c t i o n .
s
The K
p
reason
to make any
(P/P ) e x p r e s s i o n i s the one which has
o
to be improved.
In a t t a c k i n g the f i r s t
problem, F i g . 5.3 shows the complete p i c t u r e
which has t o be contended w i t h , as i t i s o l a t e s the e f f e c t o f l o a d r a t i o on
the r e d u c t i o n f a c t o r .
it
By m o d i f y i n g
the present e x p r e s s i o n f o r K
to make
an upper bound f o r a l l data i n F i g . 5.3, the p r e d i c t e d v a l u e s of
are
always h i g h e r than the t h e o r e t i c a l v a l u e s , and the r e s u l t a n t v a l u e s o f the
magnification factors are conservative.
by s h i f t i n g the present curves
The above o b j e c t i v e i s f u l f i l l e d
to the r i g h t i n order to cover most o f the
data p o i n t s a t low P/P r a t i o , and the improved e x p r e s s i o n i s :
o
K
(P/P ) = 0.7 + 0.75 (P/P )
5.8
p
o
o
-
The
p l o t i s superimposed on F i g . 5.3.
The
second problem which i s about a mismatch o f the o r i g i n was s o l v e d
by a p p l y i n g a c u t - o f f p o i n t to the
(P/P
1
f u n c t i o n , so t h a t a t v a l u e s o f
) from the c u t - o f f p o i n t t o z e r o , K i s a c o n s t a n t i n s t e a d o f a v a l u e
o'
p
approaching
infinity.
87.
i
On the load-moment i n t e r a c t i o n
value i n
at
diagram, the i n t r o d u c t i o n
produces an a p p r o x i m a t e l y s t r a i g h t
the c u t - o f f
of a
line joining
cut-off
the p o i n t
v a l u e o f ( P / P ) t o the pure moment c a p a c i t y o f the s h o r t
q
wall.
To
select
a proper c u t - o f f
p o i n t , i t i s important t o r e f e r
reproduced i n t e r a c t i o n diagram.
I f the c u t - o f f
value of K
to the
i s too low,
P
the
resultant
interaction
o c c u r s i f the c u t - o f f
curves w i l l be u n c o n s e r v a t i v e ; the o p p o s i t e
value of K
i s too h i g h .
I t i s found t h a t
different
(P/P ) r a t i o s .
For w a l l s
P
s t e e l ratios require cut-off
points at d i f f e r e n t
w i t h a h i g h s t e e l r a t i o , the c u t - o f f
q
p o i n t should be a t a h i g h ( P / P )
Q
ratio.
Through o b s e r v a t i o n , an e m p i r i c a l
cut-off
v a l u e o f ( P / P ) t o the s t e e l r a t i o i s found as
(P/P
where
)
O CO
(P/P )
O CO
p
The
linear function
r e l a t i n g the
follows:
= 0.0045 + 3.57 (p)
= v a l u e o f ( P / P ) r a t i o a t the c u t - o f f
o
Kp f u n c t i o n
= v e r t i c a l reinforcement
corresponding plot
i s shown on F i g .
5.5.
ratio
i n the
A
0
0.005
0.010
STEEL
Fig.
5.5
0.015
0.020
RATIO , • p
R e l a t i o n s h i p o f the C u t - o f f Value f o r
and V e r t i c a l
Reinforcement R a t i o
(p)
89.
5.7
D i s c u s s i o n o f the Implied I n t e r a c t i o n Diagrams
Figs
5.6
f o r s m a l l (P/P
and
5.7
show the blown up s e c t i o n of i n t e r a c t i o n diagrams
) r a t i o s and
o
the complete i n t e r a c t i o n curves f o r a 40"
w a l l w i t h a l t e r n a t e c o r e s grouted and w i t h 3-#3
The
vertical
wide
reinforcement.
s o l i d dark l i n e s are the t h e o r e t i c a l i n t e r a c t i o n curves f o r v a r i o u s
l e n g t h s , w h i l e the d o t t e d l i n e s a r e the c o r r e s p o n d i n g ones developed
the c u r r e n t e q u a t i o n s .
f o r the K
p
(P/P
o
using
With a s t e e l r a t i o of 0.00123, the c u t - o f f p o i n t
) f u n c t i o n i s 0.009, w i t h the a s s o c i a t e d K
p
v a l u e of
87.9.
I t i s seen t h a t the i m p l i e d i n t e r a c t i o n curves a r e c o n s e r v a t i v e f o r w a l l
h e i g h t s above 135.6".
For w a l l s which are 95.63" or s h o r t e r the
are s l i g h t l y u n c o n s e r v a t i v e , but the d e v i a t i o n i s s m a l l and
ted.
At h i g h v e r t i c a l l o a d s , the reproduced
curves become
results
can be n e g l e c unconserva-
t i v e , but the problem v a n i s h e s when the minimum e c c e n t r i c i t y of
t/20,
recommended i n the c u r r e n t code, i s i n t r o d u c e d .
F i g s . 5.8,
5.9,
5.10,
and 5.11
f o r c e d with 3 - #6 or 3 - #9.
show s i m i l a r p l o t s f o r w a l l s r e i n -
The e f f e c t of the c u t - o f f v a l u e f o r K
is
P
more pronounced a t these r e i n f o r c e m e n t r a t i o s .
v a l u e s of the K
P
Due
to the imposed
cut-off
f u n c t i o n , the i m p l i e d curves a r e a n g u l a r a t the c u t - o f f
( P / P ) r a t i o , but they do match the pronounced double c u r v a t u r e c h a r a c t e r Q
istic
of the t h e o r e t i c a l i n t e r a c t i o n curves due
a t the h i g h r e i n f o r c e m e n t
90.
0
Fig.
5.6
10
20
MOMENT, kip-ft.
Comparison o f T h e o r e t i c a l and Implied
f o r 8" R e i n f o r c e d Masonry Walls (3-#3
30
I n t e r a c t i o n Diagram
@ Cj @ Low ( P / P ) R a t i o
Q
Fig.
5.7
Comparison o f T h e o r e t i c a l and Implied I n t e r a c t i o n
f o r 8" R e i n f o r c e d Masonry Walls (3-#3 @ (Q
Diagrams
MOMENT, kip-ft.
Fig.
5.8
Comparison- of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams f o r 8"
Masonry Walls (3-#6 @ CJ at Low ( P / P ) R a t i o s
Reinforced
93,
900
THEORETICAL
800
IMPLIED
20
Fig.
5.9
CURVE
CURVE
30
40
MOMENT, kip - ft.
50
60
Comparison o f T h e o r e t i c a l and Implied I n t e r a c t i o n
Diagrams f o r 8" R e i n f o r c e d Masonry Walls (3-#6 @ Gj
0
10
20
30
40
MOMENT, kip-ft.
Fig.
5.10
Comparison of T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams f o r 8" R e i n f o r c e d Masonry
Walls (3-#9 @ Gj at Low ( P / P ) Ratios
Q
50
95.
900
0
Fig.
5.11
10.0
20.0
30.0
40.0
50.0
MOMENT , kip-ft.
60.0
Comparison o f T h e o r e t i c a l and Implied I n t e r a c t i o n
f o r 8" R e i n f o r c e d Masonry Walls (3-#9 @ Gj
70.0
Diagrams
96.
ratio.
The
One
i m p l i e d v a l u e s are a l l c o n s e r v a t i v e f o r these two
types of w a l l s .
drawback to the above e x p r e s s i o n i s t h a t the degree of
t i s m v a r i e s w i t h the amount of r e i n f o r c e m e n t .
The
implied
conserva-
interaction
diagrams f o r w a l l s r e i n f o r c e d with 3 - #9 bars are more c o n s e r v a t i v e
those f o r o t h e r q u a n t i t i e s o f r e i n f o r c e m e n t .
o n l y one
e x p r e s s i o n of K
than
T h i s must be accepted i f
i s to be used f o r a l l w a l l s i n a trade o f f of
P
accuracy
5.8
for simplicity.
A p p l i c a t i o n to P l a i n Masonry W a l l s
In the f o l l o w i n g s e c t i o n , the a p p l i c a t i o n of these formulas
unreinforced walls i s i n v e s t i g a t e d .
to
I f p o s s i b l e , the same f o r m u l a t i o n
w i l l be used to compute the r i g i d i t y r e d u c t i o n f a c t o r , so t h a t the
pro-
posed moment m a g n i f i e r method f o r d e s i g n of c o n c r e t e masonry w a l l s can
s i m p l e , s t r a i g h t - f o r w a r d and
The
The
be
complete.
same a n a l y t i c a l approach as i n the p r e v i o u s s e c t i o n s i s employed.
r i g i d i t y reduction factor
tions, K
(P/P
p
) and K
o
X = K
X i s assumed to be the product of two
func-
( L / r ) , such t h a t :
s
p
v
(P/P ) * K
o'
s
v
(L/r)
'
S i n c e the p l a i n w a l l s have d i f f e r e n t s e c t i o n geometry from r e i n f o r c e d
w a l l s , the r a d i u s of g y r a t i o n i s changed.
t e r i s t i c s of the s e c t i o n , F i g . 5.12
to a ( L / r ) r a t i o o f 56.8.
the r e i n f o r c e d w a l l s .
same f u n c t i o n f o r K
The
In order to study the
shows a p l o t of
t h a t was
X vs ( L / r ) n o r m a l i z e d
r e s u l t s shows t h a t i t has
For the sake of convenience
charac-
and
the same trend as
simplicity,
the
d e r i v e d f o r r e i n f o r c e d w a l l s i s used.
s
With e q u a t i o n 5.5,
( L / r ) r a t i o s are g i v e n
the v a l u e s of K
g
were c a l c u l a t e d f o r p a r t i c u l a r
97.
2.0
I .5
,< I . 0 —
I
0.51—
0.04
Fig.
5.12
|;A P l o t o f Normalized R i g i d i t y Reduction F a c t o r s and
S l e n d e r n e s s R a t i o s ( L / r ) f o r P l a i n Masonry Walls
99.
i n T a b l e s 5.13
The
average of
5.15
and 5.16
and
5.14.
f o r 8" and
12"
respectively.
average v a l u e s of K
reinforced
The v a l u e s of K
p
w a l l s was
vs (P/P
also
o
P
were e v a l u a t e d from K
P
p l a i n masonry w a l l s are shown on
F i g . 5.13
shows the c o r r e s p o n d i n g
) ratio.
The
plotted
and
expression for K
p
= X/K
s
.
Tables
plot
of
used f o r
superimposed on the same diagram f o r
comparison.
I t i s found
obtained
t h a t the t h e o r e t i c a l v a l u e s a r e s m a l l e r than the
from the p r e v i o u s e x p r e s s i o n f o r K , and
P
upper bound of a l l the d a t a .
Thus i t i s tempting
the e q u a t i o n g i v e s an
Thus the same e x p r e s s i o n used f o r
w a l l s w i l l produce c o n s e r v a t i v e r e s u l t s
to use
the same e x p r e s s i o n f o r p l a i n w a l l s , but
the c h o i c e must depend upon the a c c u r a c y of
implied
curves.
The
reinforced
for plain walls.
i n the l a s t r e s o r t ,
interaction
values
i n t e r a c t i o n diagrams f o r 8" and
12"
the
p l a i n masonry w a l l s w i t h
v a r i o u s s l e n d e r n e s s r a t i o s as i m p l i e d by the present moment m a g n i f i e r
method were superimposed on the t h e o r e t i c a l
on F i g s .
5.14
and
i n t e r a c t i o n diagrams as shown
5.15.
F a i r l y good agreement i s found between the i m p l i e d diagrams and
t h e o r e t i c a l diagrams.
Most v a l u e s computed by the moment m a g n i f i e r method
a r e c o n s e r v a t i v e , except
100"
f o r the 12" w a l l s w i t h w a l l h e i g h t s of l e s s
than
i n which the computed v a l u e exceed the t h e o r e t i c a l l y p r e d i c t e d v a l u e .
In the c u r r e n t code, w a l l s with ( h / t ) r a t i o of 8 or l e s s , such
of
the
100"
h e i g h t and
unconservative
12"
t h i c k n e s s are designed
as showt w a l l s , and
r e s u l t i n t h i s range i s of no concern.
i s r e a s o n a b l y c o n s e r v a t i v e and
satisfactory.
as a w a l l
The
overall
the
result
Height
(in.)
(L/r)
K
s
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
30.92
43.85
56.78
69.71
82.65
95.58
108.51
121.44
1.47
1.46
1.27
1.09
0.94
0.81
0.70
0.61
T a b l e 5.13
Height.(in.)
(L/r)
K
s
.
.
Values of K
s
f o r 8" P l a i n Masonry W a l l s
95.63
135.63
175.63
215.63
255.63
295.63
335.63
375.63
18.82
26.69
34.57
42.44
50.31
58.18
66.06
73.93
0.04
1.31
1.52
1.48
1 .37
1.25
1.14
1.04
T a b l e 5.14
Values o f K
s
f o r 12" P l a i n Masonry W a l l s
(P/P )
Q
Average Value
0.080
0.100
0.119
0.159
0.199
0.239
0.318
0.398
0.477
0.557
0.636
4.11
3.42
2.68
2.25
2.05
1.84
1.61
1.50
1.51
1.37
1.45
Table 5.15
(P/P )
0
Average Value
Of Kp
Average Values of K
ir
f o r 8" P l a i n Masonry W a l l s
0.080
0.119
0.159
0.199
0.239
0.318
0.398
0.477
0.596
4.08
3.35
2.36
1.97
1.80
1.61
1.43
1.48
1.46
Table 5.16
Average Values of K
f o r 12" P l a i n Masonry W a l l s
102.
MOMENT, kip-ft.
Fig.
5.14
T h e o r e t i c a l and Implied I n t e r a c t i o n
P l a i n Masonry Wal 1 s
Diagrams f o r 8"
103. '
0
10
20
30
40
50
60
MOMENT, kip-ft.
Fig.
5.15
T h e o r e t i c a l and Implied I n t e r a c t i o n Diagrams
12" P l a i n Masonry Walls
for
104..
5.9
Design of Slender Masonry Walls
As discussed i n the previous sections, the capacity of a masonry wall
does not only depend on the strength of the material i t s e l f , but also on
the effects of slenderness.
Although i t i s of questionable a p p l i c a b i l i t y
in the case of l i g h t l y loaded wall panels, the ACI moment magnifier
approach has been used here, since i t i s f a m i l i a r to most p r a c t i c i n g
engineers.
For walls i n single curvature bending, the maximum mid-height
moment capacity i s obtained by multiplying the maximum applied moment by
the moment magnifier 6, as shown i n Eq. 5 . 1 .
The magnified moment i s then
compared with the short column interaction curve.
bending,
6 i s modified by a factor C
In double curvature
depending on the r a t i o of end
m
moments.
The t e n s i l e strength of the masonry material was found to have no
s i g n i f i c a n t effect on wall behaviour.
I t i s important to point out that
the production of the correct short wall interaction diagram i s very
c r i t i c a l for the design of masonry walls.
Unfortunately, due to the
" s t r a i n gradient e f f e c t " , the construction of a correct interaction
diagram i s a very d i f f i c u l t task, as the compressive
strength of the
material appears to increase as the loading condition varies from a x i a l to
eccentric.
The effect i s very severe f o r p a r t i a l l y or f u l l y grouted
walls.
The suggested procedure for the design of slender masonry walls i s
given i n the step-by-step outline, where most of the equations are
repeated for c l a r i t y .
1.
Compute the required design load and moment.
2.
Compute the effective r i g i d i t y of the section:
105.
(EI)
=
EI/X
=
1389 ( f o r p l a i n masonry w a l l )
=
2083 ( f o r p a r t i a l l y grouted
I
=
moment o f i n e r t i a o f net s e c t i o n , i n c l u d i n g
X
=
(K )(K )
P
s
K
=
-0.294 + 130/(L/r) - 2 3 2 5 / ( L / r )
where
E
where
g
e
2
=
0.7 + 0.75/p
P
=
P/Po, but not l e s s than 0.0045 + 3.57p
p
=
s t e e l r a t i o , based
P
n
4.
grout
40 < L / r < 400
K
3.
wall)
Compute
Compute
P
cr
6
on net s e c t i o n
(*D
2
e
r—
=
(kL)
Cm
= i _ p/ $p
2
cr
see ACI code f o r v a l u e s of k, Cm and cj.
5.
Check whether P and SM f a l l w i t h i n the i n t e r a c t i o n curve f o r s h o r t
walls.
An
i s based
i l l u s t r a t i o n o f the d e s i g n procedure
i s shown below.
The example
on a 40" wide and 7.265" t h i c k f a c e - s h e l l c o n s t r u c t e d p l a i n
The w a l l i s 150" h i g h and the b l o c k s t r e n g t h i s 2350 w i t h S type
The w a l l i s pinned
t i e s o f 1.5".
wall.
mortar.
and c a r r i e s a l o a d o f 100 k i p s a t equal end e c c e n t r i c i -
The e l a s t i c modulus o f the w a l l u n i t i s 1389 k s i , and the
s h o r t w a l l i n t e r a c t i o n diagram shown on F i g . 5.13 i s used
f o r the f o l l o w -
i n g c a l c u l a t i o n w i t h the a x i a l c a p a c i t y (P ) o f 251.5 k i p s .
106.
Step 1:
P = 100
3.09
r
Step 2:
K
in
Mn
2
= 12.5
I = 1137.
kips-ft.
±n
P
k
o
= 251.5
= -.294 + 130(3.09/150) - 2325(3.09/150)
g
=
P/P
kips
kips
2
1.397
= 100/251.5 = 0.389 > .0045 + 3.57 p
o
K
= .7 + .75/.398
P
= 2.587
X = (1.39)(2.587) = 3.61
TT ( 1388) (1137)
2
Step 3:
P
c r
(150) (3.61)
2
= 192
Step 4:
6=
kips
!_ o/(192)(0.7)
10
= 3.91
Step 5:
<SM = (3.91)(12.5) = 48.8
Since Mu
kip-ft.
from s h o r t w a l l i n t e r a c t i o n curve i s 27.4
l e s s than the a p p l i e d moment (48.8
kip-ft),
the
k i p - f t , and
s e c t i o n has
to
is
be
redesigned.
In the example above, s i n c e the c a p a c i t y r e d u c t i o n
the ACI
design
format f o r concrete
m a t e r i a l , the v a l u e
illustration.
design
to be determined f o r masonry
was
used to complete
For w a l l s with v e r t i c a l r e i n f o r c e m e n t ,
at low
P/P
0
ratio,
the c u t - o f f p o i n t i s imposed on the
section.
according
the
the procedure f o r
A d i f f e r e n t s h o r t w a l l i n t e r a c t i o n c u r v e has
f o r the l o a d i n f l u e n c e f a c t o r , K^,
ment i n the
yet
recommended f o r concrete
is similar.
used, and
has
f a c t o r (<(>) used i n
to
be
function
to the amount of r e i n f o r c e -
107.
VI
Using n u m e r i c a l
CONCLUSIONS AND RECOMMENDATIONS
i n t e g r a t i o n techniques
to determine the s e c t i o n capa-
c i t y and the column d e f l e c t i o n c u r v e s , and t h e r e f o r e the i n s t a b i l i t y
f a i l u r e c o n d i t i o n s , a t h e o r e t i c a l a n a l y s i s was performed on masonry w a l l s .
A moment m a g n i f i e r method was developed
the above a n a l y s i s .
f o r masonry w a l l d e s i g n based on
I n the moment m a g n i f i e r method, the s l e n d e r n e s s
e f f e c t i s s i m u l a t e d by i n t r o d u c i n g the r i g i d i t y r e d u c t i o n f a c t o r which i s
i n t u r n a f u n c t i o n of the s l e n d e r n e s s r a t i o
r a t i o (P/P ) .
o
vative.
The d e s i g n method was found
( L / r ) and the a p p l i e d l o a d
to be s a t i s f a c t o r y and conser-
For s i m p l i c i t y , t h e r e i s o n l y one g e n e r a l e x p r e s s i o n used
accounting
f o r the s l e n d e r n e s s e f f e c t
( p l a i n or r e i n f o r c e d ) .
forward
f o r a l l types o f masonry w a l l d e s i g n
In g e n e r a l , the d e s i g n method i s s i m p l e ,
straight
and adequate f o r d e s i g n purposes, although i t may be c o n s i d e r e d
o v e r - c o n s e r v a t i v e i n some c a s e s .
The
c u r r e n t code, based on a l l o w a b l e s t r e s s e s , was found
s i s t e n t and o v e r - c o n s e r v a t i v e i n most c a s e s .
t o be i n c o n -
The d e s i g n method i s not
capable o f d e a l i n g i n d i v i d u a l l y w i t h d i f f e r e n t d e s i g n a s p e c t s .
The use o f
the h / t r a t i o as the s l e n d e r n e s s parameter does not d i s t i n g u i s h the
d i f f e r e n c e between a f a c e - s h e l l c o n s t r u c t e d p l a i n w a l l and a f u l l y
wall.
grouted
The l i m i t a t i o n on the s l e n d e r n e s s r a t i o s t a t e d i n the present
code
apears t o be an o v e r c o n s e r v a t i v e measure.
Since the magnitude o f the t e n s i l e s t r e n g t h o f masonry m a t e r i a l i s
s m a l l , i t has no s i g n i f i c a n t e f f e c t on masonry w a l l d e s i g n .
deflected
shapes and the f a i l u r e loads of some e x i s t i n g
Based on the
experimental
r e s u l t s , the w a l l s f i l l e d w i t h grout o f s t r e n g t h s i m i l a r t o t h a t o f the
b l o c k tend to have h i g h e r f l e x u r a l compressive s t r e n g t h and o v e r a l l
: .103.:
e l a s t i c modulus than the f a c e - s h e l l c o n s t r u c t e d p l a i n w a l l s w i t h the same
block
strength.
Comparing a n a l y t i c a l t o e x p e r i m e n t a l r e s u l t s , the
effect'
appears t o be more pronounced i n p a r t i a l l y
f a c e - s h e l l constructed p l a i n walls.
In
fact,
'strain
gradient
grouted w a l l s than
in
t h e r e was no s i g n i f i c a n t -
d i f f e r e n c e when comparing the e x p e r i m e n t a l pure a x i a l
l o a d c a p a c i t y of
the
f a c e - s h e l l c o n s t r u c t e d w a l l s w i t h the t h e o r e t i c a l a x i a l c a p a c i t y based on
the f l e x u r a l
compressive s t r e n g t h .
But f o r p a r t i a l l y
grouted w a l l s ,
the
e x p e r i m e n t a l pure a x i a l c a p a c i t y was a p p r o x i m a t e l y o n e - h a l f of the
theore-
tical
wall
c a p a c i t y e v a l u a t e d from f l e x u r a l
compressive s t r e n g t h of the
assemblages.
Due t o the tremendous d i f f e r e n c e i n c a p a c i t y caused by the
gradient e f f e c t " ,
the c o n s t r u c t i o n of an a c c u r a t e s h o r t w a l l
diagram i s d i f f i c u l t
e s p e c i a l l y f o r grouted w a l l s .
interaction
F u r t h e r study s h o u l d
be d i r e c t e d towards r e l a t i n g the " s t r a i n g r a d i e n t e f f e c t "
geometry of masonry w a l l assemblages.
"strain
with the s e c t i o n
109'.
REFERENCES
1.
Canadian Standard . A s s o c i a t i o n , 1978, "Masonry Design and C o n s t r u c t i o n
f o r B u i l d i n g s . " N a t i o n a l Standard o f Canada. Can 3 - 5304 - M 18,
Rexdale, Ontario.
2.
Nathan, N.D. " S l e n d e r n e s s o f P r e s s t r e s s e d Concrete Beam-Columns."
PCI J o u r n a l , V o l . 17 - #6.
N o v . - D e c . 1972. p p . 45-57.
3.
O j i n a g a , J . and T u r k s t r a , C . "The Design o f P l a i n Masonry
Dept. o f C i v i l E n g i n e e r i n g . M c G i l l U n i v e r s i t y , M o n t r e a l .
4.
O j i n a g a , J . and T u r k s t r a C . "The Design o f R e i n f o r c e d Masonry W a l l s
and Columns. I - c o n c e n t r i c L o a d i n g and Minor A x i s B e n d i n g . " Dept. of
C i v i l E n g i n e e r i n g , M c G i l l U n i v e r s i t y , S e p t . 1979.
5.
Yokel, Dikker.
" S t r e n g t h o f Load B e a r i n g Masonry W a l l s . "
D i v i s i o n , ASCE J o u r n a l , May 1971, p p . 1593-1609.
6.
Yokel, Dikker.
s i l e Strength."
1913-1925.
7.
H a t z i n i k o l a s , Longworth, and Warwaruk, "Concrete Masonry W a l l s . "
P h . D . T h e s i s , S t r u c t u r a l E n g i n e e r i n g r e p o r t No. 70, D e p t . o f C i v i l
E n g i n e e r i n g , U n i v e r s i t y o f A l b e r t a , Edmonton, S e p t . 1978.
8.
H a t z i n i k o l a s , M . , Longworth, J . and Warwaruk, J . "The A n a l y s i s o f
E c c e n t r i c a l l y Loaded Masonry W a l l s by t h e Moment M a g n i f i e r Method."
P r o . 2nd Canadian Masonry C o n f e r e n c e , 1980. p p . 245-252.
9.
Nathan, N.D. and Chaudwani, R . , " P r e c a s t P r e s t r e s s e d S e c t i o n s Under
A x i a l Load and B e n d i n g . " PCI J o u r n a l V o l . 16-#3. May-June 1971, p p .
10-19.
10.
Nathan, N.D. " A p p l i c a t i o n o f ACI S l e n d e r n e s s Computatins t o P r e s t r e s s e d Concrete S e c t i o n s . "
PCI J o u r n a l , V o l . 20-#3, May-June,
1975, p p . 68-75.
11.
A l c o c k , W . J . , and Nathan, N.D. "Moment M a g n i f i c a t i o n T e s t s o f P r e s t r e s s e d Concrete Columns." PCI J o u r n a l , V o l . 22-#4, J u l y - A u g u s t ,
1977.
12.
D r y s d a l e , R . G . and Hamid, A . A . "Behaviour o f Concrete Block Masonry
Under A x i a l C o m p r e s s i o n . " T e c h n i c a l P a p e r .
ACI J o u r n a l , June 1979.
p p . 707-721.
13.
T u r k s t r a , C , and Thomas, G . S t r a i n G r a d i e n t E f f e c t s i n Masonry.
S t r u c t u r a l Masonry s e r i e s No. 7 8 - 1 , D e p t . o f C i v i l E n g i n e e r i n g and
A p p l i e d M e c h . , M c G i l l U n i v e r s t i y , M o n t r e a l , A p r i l 1978.
Walls."
Structural
" S t a b i l i t y and Load C a p a c i t y o f Members w i t h No T e n S t r u c t u r a l D i v i s i o n , ASCE J o u r n a l , J u l y , 1971, p p .
110.
References Cont'd...
14.
B o u l t , B.F.
Concrete Masonry P r i s m T e s t i n g , ACI
pp. 513-535.
Journal, A p r i l
1979,
15.
Canadian Standard A s s o c i a t i o n .
Code f o r the Design of Concrete
Structures for Buildings.
N a t i o n a l Standard of Canada, CAN 3 - A
23.3 - M 77.
16.
Yokel, F.Y., Mathey, R.G. and D i k k e r s , R.D.,
" S t r e n g t h of Masonry
Walls Under Compressive and T r a n s v e r s e Loads", N a t i o n a l Bureau of
Standards, B u i l d i n g Science S e r i e s 34, March 1981.
17.
E s k e n a z i , A., Ojinaga, J . and T u r k s t r a , C.J., "Some M e c h a n i c a l
P r o p e r t i e s of B r i c k and B l o c k Masonry". I n t e r i m Report, Dept. of
C i v i l E n g i n e e r i n g and A p p l i e d Mechanics, M c G i l l U n i v e r s i t y , M o n t r e a l ,
S t r u c t u r a l Masonry S e r i e s 75-2, pp. 75,
1975.