GENERALIZED DESIGN METHOD OF LATERALL Y

LOADED MASONRY W ALLS

Olof Sj6strand I)

1. ABSTRACT

This design method refer to the ultimate state and is based on the yield line theory modified with regard to varying moment capacities in horizontal and vertical direc­tions for unreinforced as well as for reinforced or prestressed masonry.

2. NOTATIONS

I length of masonry panel

h height of masonry panel

a,b coordinates for yield lines

p design lateral load in ultimate state

1113

1214 Y 13 Y 1 Y24Y2 mh mv mha mva

vertical edge fixities

horizontal edge fixities

fixity pararneters of vertical edges

fixity pararneters of horizontal edges

horizontal moment capacity in ultimate state }

vertical moment capacity in ultimate state

horizontal moment capatity in ultimate state j vertical moment capacity in ultimate state

for unreinforced masonry

for reinforced masonry

Keywords: Masonry; Walls ; Design Method; Horizontal Load; Yield Line Theory.

1) Civil Engineer, M.Sc., J&W Consulting Engineers , Hebsackersgatan 24, S-254 37 Helsingborg, Sweden

151

3. YIELD UNE THEORY FOR RECTANGULAR PANELS

The design method of laterally loaded unreinforced masonry walls used in Code of Practice BS 5628, Part 1 is derived from yield line theory with the equations based on the ortogonal ratio)1 = mh/mv and the aspect ratio A = (hll)/ (li'

Design following the method in the unreinforced masonry Code BS 5628 leads to a complicated and non-flexible processo The disadvantages are discussed in (1) and surnrnarized here:

1) The panel strength (moment capacities) must be guessed . If the chosen capacities doesn't satisfy the load conditions, the designing means a timeconsuming iterative processo

2) The Code method may result in wrong conclusions, as it can give the im­pression of different load capacities in horizontal and vertical direction.

3) All combinations of common edge support conditions are not covered by the Code method.

4) The Code method deals only with unreinforced panels and it is compli­cated to extend the design to reinforced and prestressed masonry panels.

This paper is an adaptation of (1) and presents a simplified and generalized design method of rectangular masonry panels laterally loaded by distributed loads. To create conditions for correct design, direct solutiollS of the yield line equa-tions have been developed, based on following parameters: design load p, panel dimensions lxh, support conditions and ultimate moments in horizontal and veni­cal direction, fi&...L The design wind load is denoted p = ílf wk·

p design load 11 ! ! ! ! !

p

Edge r ixi I ies Suppor I cond i I i ons

General sign ror supporled edge

i = -1 Unsuppar I ed « ree I edge

i = O Sil1lJly supported (pinnedl

i = 1 Cont i naus « ixedl LLC:./ LÜ.

O < i < 1 ParI i al r i xed edge /// // /, <" « ,

i > 1 Superf i xed edge xxx.x.x.>v

Fixity para-neters

Yll = .[1:i, + ~

Y2, = ~ + {1:i;

edge

edge

Fig 1. Design parameters. Support conditions and fixities. Fixity is defined by the ratio between the capacities of edge moment and field moment in the ultimate state.

152

Ou! of the yield line equations generalized design charts have been drawn to cover ali possible variations in above mentioned parameters , Desilm chart 1-4. Ali kinds of panel strength can be treated - unreinforced as well as reinforced or prestress­ed in each of the horizontal and vertical directions. Panels supported on four , three or two edges can be treated with ali possible combinations of support con­ditions , fuL.1.

Free edge V

13

12 12

Y13 = Y, = [1":1;

Fig 2. Fixities and flxity parameters by three edge supported panels.

Horizontal yield Line systems d1 a2 B} l l ., '1

OH

À ~ Free edge

[(?S]

Vertical yield line systems

~lb7fh ~IbJ W-J

OV Free ed e

UH UV

~> Free edge

CH

]z' 1 Free edges

LH LV

Fig 3. Yield line patterns in rectangular panels supported in 4 ,3 Of 2 edges and under uniform distributed load, UDL.

153

4. YIELD UNE EQUATIONS FOR ULTIMATE STATE

In each of the four panel types , 0- , U-, C- or L-shaped, the yield line equa­tions are derived for two principie yield line patterns each, fig 3. The appearance of the yield lines are described by the coordinates a and b and the limits of the two principie patterns. The yield line equations are solved in generalized form of the specific rnornents

2 2 rnh/pl and rn)ph

Observe the resernblance between the equations for OV, UV and CH resp. For CH the coordinates however are reversed. Also for the equations for UH, CV, LH and LV there are resernblances with reversed coordinates.

4 .1 Explicit solutions of yield line equations

YleLd Llne pottern OH Yl eLd Ll ne pottern UH

mh __ 3_ ( 2. m, f (1 ) mh 3 (1 - 21 2 ·~r -pL2 - SY,l 1 - 2Y24 lJii2

-;;õT =. p (1) pL SY,l 16m,

+ '1Jii2 a, +02 _ J 6Y 2. mh

L - 1l-pL2 (2)

o, +a2 _ ~-m-h-- (S) L Iml t L - 6Yll ' ( 6m , )

pl2 1+-;;d-ph

m, 1 (3) -pL2 .;: 6Y-;[ Llml t

m, ~ 1 (9)

Yleld llne pattern OV lJii2 6(2+12)

mh 1 ( J 2 m, \ -pL2 = 6y

1l2 3 - 24 Y24 '1Jii2 ) (4) Yl eld tine pottern lN

mh _ 1 ( 3 J 22m, ) ( 10)

b,+b2 _ ~m, -pL2 - 6Y,/ - 4Y2 '1Jii2

h - 24 "jii12 (5)

L Iml t b ~--;n;-- = 6Y2'~ h ph (11)

m, " 1 "jii12 6Y2,2 (6)

Llml t

m, 1 "jii12';: 6Y/

(12)

154

)'Ield llne parlern CH Observe Ihat the Independent varlable IS m"/pl2 .

~ = 6)':,2 (3 -J 24'1', L~fz- )

L Iml r

)'Ield llne pallern (V

Observe Ihal lhe Independent varlable 15 mh/pl2 .

6)' 2 mv 2" (&n) ph2 1+~ . pl

L Iml I

)'Ield llne pattern LH

(1 _ 212 .~)2 mh _ 3 ph

Pf2' - W ' 1 + 6'~ ph2

- 21 2'~ a = l...2l . __ -::Po.ch_ .;; l

• 6 '~ ph2

)'Ield llne pattern LV Observe Ihal lhe I ndependenl var I able I S mh/pl2.

mh

b _ 2!!.. 1 - 21,·pr - 2 " h

+ 6'~ pl

155

( 13)

( 14)

( 15)

( 16)

(17)

( 18)

(19)

(20)

(21)

(22)

5. DESIGN CHARTS

To simplify the design work, design charts 1-4 have been drawn in further genera­lized form involving also edge fixities i C i4 and fixity parameters Y 13' Y l' Y 24 and Y 2' as defined in fig 1 and 2.

Along the axis in the design charts the generalized moments are denoted

as mv Y 2/ Iph2

or m/Ph2

2 2 2 as mh Y 13 IpI or mh/pI

01 +D2

0.5 . Y24

2

I, =0 1.000

h

i 1

O 1 116 i1=0~i3=0 [ 4--J

0.5

0.4

0.3

0.2 116

0.1

o

I\--

~ \

x

\ OV

~ /

~

0 . 1 116 0.2

l 1

L

OH

OH

~

0.3

l , L

OV

"r--l J1

_ mh 2 Y-PL"Z' Y13

0.2 116

0.1

0.4 _ mv 2

O . 5 x- ph2 . Y24

- / 1/

./ d1 +d 2 0.5 1. O L

Design Chart 1. Panel supported in alI four edges . YieId line pattem OH and OV . N .B. Field reinforcement crossing inclined yield lines does not considerably contribute to the panel bearing capacity in four edge supported brick panels without speciaI grooves in the bedjoints . With mh 'nf d Imh 'nf fi Id = i1 > 1 and i3 > 1 resp reI e ge umel le

(i.e. superfixity), the lacking bearing effect of fieId reinforcement in ca.se O~ resp case OV for (b1 + b2)!h > 0.5 wilI be taken into conslderatIon.

See paragraph 6 and example 7.2.

156

_ mh 2 Y-p-LZ' Y13

A À [JJIb} Y,l

E:]} il=O[G] i) =O

l l l t-W .., 'I

0.5 UH UV il=1~i3=0

0.4

il=1E]i)=1

mv o 0.1 0.2 0.3 0.4 0.5 1 O X=-2 . ph

Design Chart 2 . Panel with one horizontal edge unsupported. Yield line pattem UH and UV .

157

Y/

[ 1 J i2 =0

QJ 12 = 1

mh y=--pL2

1. O

0.5

0.4

0.3

0.2

0.1

O 0.1

Design Chart 3.

Cl Y1 2

{----i [dlbJ ~} í1=O[2J Ib 1

J l l W 1

CH CV

í1=1U

mv 2

0.2 0.3 0.4 0.5 x= ph 2 . Y24

Panel with one vertical edge unsupported. Yield tine pattem CH and CV.

158

YC4C

í 4 =0

[IJ í2=0

í 4 =0

[~.8~J 12 =1

W 12 = 1

i l =1[ ® ]

o . 4 -t---fl---+\--J i2 =0

o . 3 -t-t\-?~-+---\-J

Hor izon tal edge p i nned i 2 =0

O. 111x~~:t:=-t-hRlrP=-t-=j----:í",======~

O 0.1 0.2 0.3 0.4 0 .5

Horizonta l edge f i xed iz=1

Design Chart 4 . Panel supported on two perpendicular edges. Yield line pattem LH and LV.

6. ULTIMATE STATE DESIGN

If the coordinate lines of actual generalized moment capaclties IDlersect below the curve, e .g. point 1 fig 4 , it can be concluded that the panel strength is inadequate . One or both capacities have to be increased up to some point on the curved line between point 2 and 3.

According to investigations by Cajdert (2) of four edge supported brick panels, without special grooves in the bedjoints , the horizontal field reinforcement crossing inclined yield !ines does not considerably contribute to the panel bearing capacity due to the lacking torsional strength of brick masonry panels . Field reinforcement across vertical yield line in four edge supported brick panel - as in the pattem OV - does however contribute to the bearing capacity . In practical calculations this field reinforcement contribution will be taken into cosideration in case OV, if (b 1 +b2)/b is smaller than say 0.5.

For blockwork does not this restriction apply . In blockwork, with its rough con­tact surfaces between block and mortar, and with the reinforcement bars mostly placed in special grooves in the bed joints , the joint reinforcement is quite effective in increasing the ultimate load . Cajdert (2).

us/1247 159

b,+b 2

h 0.95 (í) 1 . O +--.---,--~

O . 5 ~+--+---1H

_ mh 2 0.1 1/6 i1=0~iJ=0

Y-p-LT' Y13

0.477 _0_. _5 -+-__ ---.~

0.4

0.3 OV r'\

ExampLe

D~ &TI

_ mh y 2 -Y-p-LT' 13

0.2 OH 0.2 O. 178 -.1/"6c+----~/ @1/6

I~~~--~~--------

0.12!L0

. 1

o 0.151; 0.220

0.191

0.4

Fig 4. Example using Design Chart 1.

7. EXAMPLE

0.1

mv Y 2 O . 5 x= ph2 . 24

-

/

/ / a, +a2

l

Panel in fig 4 supported on alI four edges to be designed for design wind load p = 1.3 kN/m2. Design data: 200 mm Concrete hollow block Unreinforced moment capacities: mh = 1.24 kNrnlm m = 0.80 kNrnlm (incl self-weight at midheight) MYnimum reinforced moment capacities: mh = 2.45 kNrnlm (2 double bars in every 6th bed joint) mv~ = 2.70 kNrnlm (1 double bar in every block ele 600 mm)

Fixity parameters

2 2 Y13 =8 Y24 =4

7.1 Control (1)

Coordinates of actual generalized moment capacities in design chart 1

y = mh'Y 13 2

/pl2 = 1.24'8/1.3'8.02

= 0.120) 2 2 2 x = mv'Y24 Iph = 0.8-4/1.3-4.0 = 0.154

160

The coordinates are crossing in point (1), below the curved line. Thus the strength of the wall is inadequate .

If the generalized moment capacities x and y are increased proportionally up to the curved !ine, point IA, one can see that the yield pattem is OH and aI + a2 = 0.90 l.

One of following measures may be tried to increase the wall strength.

7.2 Measure (2)

Add horizontal reinforcement in the bedjoints, so point (2) wiU be reached , i.e . Yreq 2.. 0.178.

Calculate required horizontal reinforcement capacity 2 2 2

mhareq = y.pl /Y13 = 0.178·1.3-8 .0/8 = 1.85

With minirnum reinforcement two double bars in every 6th joint the actual moment

capacity is

mha = 2.45 > mha req = 1.85 kNmlm.

Thus the strength of the reinforced wall is well adequate.

7.3 Measure (3)

Add vertical reinforcement in vertical centroid holes instead of horizontal rein­forcement so that point (3) will be reached , i.e . x 2.. 0.220.

Calculate required moment capacity

fi = xo ph2/Y242 = 0 .220.1.3-4.02/4 = 1.14 kNmlm v req

With 1 double bar in every block ele 600 the actual moment capacity is

fiva = 2.70 > mv req = 1.14 kNmlm

Thus the strength of the wall is well adequate.

7.4 Measure (4)

Provide extra vertical support instead of horizontal reinforcement. With x = 0.154 as above and

2 2 2 Y = mh-Y 13 IpI = 1.24-8/1.3-4.0 = 0.477

the coordinates are crossing in point (4) well above the curved !ine and thus the strength of the wall is well adequate.

161

7.5 Measure (5)

Calculate the effect if the moment capacity due to cracked section at the bottom edge with DPC is taken into consideration

mv cracked = 0.48 kNrnlm

i2 = mv cracked/mv = 0.48/0.80 = 0.6 (partial fixity)

y 242=( Yl+0.6' + I{l+ol = 5.13

With Y = 0.120 as in 7.1 and

222 x = m';Y24 /ph = 0.8-5 .13/ 1.3<4.0 = 0.197

the coordinates still are crossing below the curved tine and the strength of waIl is not adequate. With extra permanent load from roof, floors, waIls the vertical crack­ed moment capacity in the bottom edge may satisfy the waIl strength.

8. RETAINING WALLS

For earth pressures the load generally is a partial uniforrnly distributed load PUDL, consisting of a uniformly distributed load UDL and a triangularly distributed load TDL. It has been shown (3), (4) that a TDL or PUDL replaced by a fuIl UDL of the sarne total force will give the upper bound to the exact yield line solution. Thus the design charts 1-4 can be used for a conservative design of retaining waIls by replacing a TDL or PUDL with the fuIl UDL of the same total force.

9. REFERENCES

(1)

(2)

(3)

(4)

usl1246

Golding, J .M. , "Practical design of laterally loaded masonry panels" . Construction & Building MateriaIs, Vol 5 No 3, Sept 1991.

Cajdert, A. , "Laterally loaded masonry waIls" , (dissertation) . Chalmers University of Technology, Division of Concrete Structures, 1980. Pub­lication No 80:5. G6teborg, Sweden.

Johansen, K W, "Yieldline formulae for slabs". Eyre and Spottiswoode , 1972.

Pamell , F N., "The general principie of superposition in the design of rigid-plastic plates" . Concrete and Constructional Engineering , Septem­ber 1966, p323 .

162