D.1 INTRODUCTION
The object of a yield-line analysis is to postulate a yield-line pattern from which the ultim ate m om ent can be determ ined by:
• considering the equilibrium of the slab elements, or
• using the work equation (W ood, 1967).
The inherent difficulty of a yield-line analysis is to determ ine the yield-line pattern that will give a minimum value of the ultim ate load for a given arrangem ent of reinforcement. In general, an upper bound solution is obtained (correct or unsafe) (W ood, 1967). However, owing to m em brane effects, it can be shown that much higher ultim ate loads than those predicted by a yield-line analysis can be obtained. Thus the yield-line theory may be used in the design office if a reasonable attem pt is m ade to find the minimum ultim ate load. The general proce
dure for the analysis of a square slab with simply supported edges is outlined below (case 1, Figure D .l), and the formulae only will be given for all other cases (cases 2 -6 , Figure D.2). A com prehen
sive tabulation of design form ulae for simple slabs of various shapes, including flat slab floors, is given in the Unesco (1971) manual.
D.2 SQUARE SLAB, SIMPLY SUPPORTED, UNIFORMLY DISTRIBUTED LOADING
The assumed yield-line pattern is shown in Figure D .l and further assumptions are as follows:
1. The external work done by the slab elem ents under unit displacem ent (8 = 1 at point O) is
equated to the com ponents of internal work expressed as (m om ent per unit length along axis of rotation) x (length of projection) x (rotation about the axis under consideration).
2. Each yield line is considered as an infinite succession of straight lines disposed stepwise (Unesco, 1971).
3. The ratio of the ultim ate m om ents per unit length in the x and y directions (orthogonal axes) is given by p = m ylm x.
4. The area of tensile reinforcem ent at any point or in any direction is such that the value of xld does not exceed 0.25; see C hapter 3.
For unit displacement of the slab at O, it can then be considered as four plate elem ents, the centre of gravity of each plate elem ent undergoing a displace
m ent of one-third. Then
external work for plate O A B - w x (L 2/4) x (1/3)
= w L m i and thus
total external work = 4 x w L2/12 = w L2/3 For plate O A B, there is rotation about the y axis only. The rotation 0 of the slab is 2/L, the m om ent per unit length is m x and the length of the projec
tion about the y axis (A B ) is L. Thus internal work for plate OAB
= m x x L x (2/L)
= 2 mx
bottom (+ve) reinforcement
J J _ l _ - ( f l -
i rr
top (-ve) reinforcement
free edge
/ / / / / / / / / / / simply supported edge
continuous (built in) edge
Case 1 (U.D.L = w) mx = mv = m m =
7 wL2
2 4
Figure D .l Yield-line analysis - general notation and case 1.
SQUARE SLAB, SIMPLY SUPPORTED, UNIFORMLY DISTRIBUTED LOADING 177
Case 2 Square Slab, continuous edges, Isotropic reinforcement - Top, Bottom, U.D.L. = W
m = wL2 48
--- +ve Yield Line
— — — -ve Yield Line
Case 3 Rectangular Slab, continuous edges, Isotropic reinforcement pm (top) and m (bottom), U .D .L -W
m = W • Li • L2 8(1 +p)
Case 4 1-way continuous slab m (+ve) and pm (-ve),
m = wL2 (1 + p f t -1
m = wL2 8(1 + p)
P m pm
0.5 wL2 wL2
10 20
1.0 wL2 wL2
11.6 11.6
1.5 wL2 wL2
13.4 9
2.0 wL2 wL2
15 7.5
m (+ve) and pm (-v
>pan BC
P m pm
1 wL2 wL2
16 16
1.5 wL2 wL2
20 13.3
2.0 wL2 wL2
24 12
m
- N - - M -
i i pm
Case 6 Fan pattern, Isotropic reinforcement pm (top) and m (bottom), concentrated load P
P = 2 n m (1 + j l i)
= 47tm for p = 1
For application of fan pattern to flat slabs, see references
Figure D.2 Yield-line analysis - design formulae for cases 2-6.
The internal work for plate O C D is 2m x and that for plates O BC and O A D is 2m y. Thus
total internal work = 2 (2 m x + 2 m y) Thus
m x + m y = w L2/12 If m x - m y - m, then
m = wL2/24 or
w = 24 m l I 2
It m ust be em phasized that the yield-line pattern shown in Figure D .l is merely a postulation and it is necessary to examine alternative patterns which may result in a lower value of w than 24ra/L2. It is possible for a yield line to fork before it reaches the slab edge (W ood, 1967; Jones and W ood, 1967), forming what is known as a corner lever (see Figure D .l). Yield-line patterns that have corner levers are m ore critical, but the error involved is small for square slabs and is related to the ratio of the top to bottom reinforcem ent \xmlm = jul. The following results apply to a square slab with uniform ly distrib
uted loading (w):
In many cases, the general form of the work eq u a
tion can be expressed as:
m = ulv
where u and v are functions of x. Thus dm/dx = (v du/dx - u dv/dx)/v2 Putting
dm/dx = 0 then
ulv = (duldx)/(dv/dx)
This expression has a num ber of applications, e.g.
to rectangular slabs (see W ood, 1967; Jones and W ood, 1967).
REFERENCES
Jones L.L. and W ood R.H. (1967) Yield L ine Analysis o f Slabs, C hatto and W indus/Tham es and H udson, London.
U nesco (1971) Reinforced Concrete, A n International M anual, English translation, B utterw orths, London.
W ood R.H. (1967) Plastic and Elastic Design o f Slabs and Plates, Tham es and H udson, London.
fji m
0 w LV 22
0.25 w L 2/23 0.50 w L 2/23.6 1.0 w L 2/ 24
Appendix E
THE STRIP METHOD