E.1 INTRODUCTION
In section 3.4.1(a), it was indicated that the strip m ethod (after Hillerborg, 1975) is a useful approach to the analysis of slabs of certain configurations at the ultim ate limit state. The strip m ethod is a design approach in which the load is carried by a series of strips assumed to act like beams. The strips are rein
forced along their full length like beams carrying the same load. Thus it is possible to design a slab with variable reinforcem ent, in contrast to the yield-line theory, which, in general, is used to analyse a slab with an assum ed arrangem ent of uniform rein
forcem ent. The slab may be divided into a series of x and y strips, thus reducing the equilibrium equa
tion of a slab elem ent to the form
d2M J d x 2 + d M J d 2 = - ( Wx + Wy) = - W
It is assumed that plasticity allows the deflection of perpendicular strips to become com patible. Division of the load into x and y strips is achieved by intro
ducing load dispersion lines (Figure E .l(a )). This implies that trapezoidally loaded areas are formed, which m eans that continuously varying reinforce
m ent is required, and this is not practicable. W ood and A rm er (1970) suggest the adoption of discontin
uity lines as shown in Figure E .l(b ) and thus the reinforcem ent may be placed in bands of convenient width. Continuity over supports is dealt with by introducing zero m om ent lines; see Figure E .l(c).
The position of the zero m om ent lines should be chosen such that the ratio of support to span m om ent does not depart too far from that obtained by an elastic analysis. A large departure may result in serviceability problem s; see C hapter 4. The
general approach is considered in the following outline design for the ultim ate limit state.
E.2 EXAMPLE OF USE OF STRIP METHOD
A reinforced concrete tank wall (see Figure E.2(a)) built in on three sides and free on the fourth is subjected to an equivalent fluid pressure of 5 kN/m 2 per m etre height of wall. yf will be taken as 1.5 for ultim ate limit state.
E.2.1 Loading (ULS)
The first step is to choose the position of the load dispersion and zero m om ent lines. This choice should be m ade bearing in mind the elastic distrib
ution of bending m om ents at service loads, o th er
wise crack widths at service loads may be excessive.
Tim oshenko and W oinowsky-Krieger (1959) give tabulated data for elastic analysis. Two typical strips will be considered; see Figure E.2(a). Strip 1-1 may be taken as a beam built in at both ends, and an elastic analysis would give the point of contraflexure at a distance of 0.21 x 6 = 1.26 m from the supports.
Thus for the ultim ate limit state, the distance of the zero m om ent line from the support should not be too far from 1.26 m and a value of 1.0 m has been assumed. The centre line of strip 1-1 (1.0 m wide) has a loading intensity of 5 x 5 = 25 kN/m 2, and this will be taken as constant over the full width of the strip; see Figure E.2(b). The loading on strip 2 -2 is shown in Figure E.2(c). In order to m aintain equi
librium, a reaction R 2 is required at the free edge (for which additional reinforcem ent is required).
< - X
BMD Strip 3 - 3 82 Mx/8x2 = -w
3 \ JV\ 3
| 2
1
1 2
| W
J
1 1 1 |
Aw w x K
(a)
BMD Strip 1 -1 S2 My/8y2 = -w
P 7
BMD Strip 2 - 2 82 MX/5X2 = -w
/ // // // // // /
/ / / / / / / / / / / / / / / / / / / / / / / / / / / ,
iii ii
j
___ i
L11
I I
/7 V 7 7 7 7 7 7 7 7 7 T 7 7 7 7 7 7 7 7 7 7 7 7 Load dispersion line
chosen to give banded reinforcement layout
(b)
Free Edge
BMD Strip 2 - 2
. / . J / / . Simply supported edge w s a v Built in
edge
________ Free edge
(c)
BMD Strip 1 -1
Figure E .l The basis of the strip method.
EXAMPLE OF USE OF STRIP METHOD 181
(a)
6.0m
I 1.0m r<— >l Free Edge
(b) 1.0m I.. 4.0m 1.0m,I
Zero moment line
— Load dispersion line
25kN/m T~
\ /
25kN/m
\
(8 , = 1.5 omitted)
\ /
(C ) 15 25
/ V R, 1.5m I 1.5m
- > K --- > N -
5.0m
/N Rp
Figure E.2 Application of the strip method.
Strip 1-1
Between the zero m om ent lines, the strip is designed as a simple beam. Thus the maximum bending m om ent M and reaction at the zero m om ent line are
M = 0.125 x 25 x 1.5 x 42 = 75 kN m/m V = 0.5 x 25 x 1.5 x 4 = 75 kN
The m om ent M and shear V at the slab edge are M = 75 x 1.0 + 25 x 1.0 x 0.5 = 87.5 kN m/m V = 75 + 1.0 x 25 x 1.5 = 112.5 kN
Strip 2 - 2
R 2= 25 x 1.5 x 1.5 x 0.75/6.5 + 3.75 x 1.5 x 1.5 x 0.5/6.5
= 7.14 kN
/?,= 25 x 1.5 x 1.5 x 5.75/6.5 + 3.75 x 1.5 x 1.5 x 6.0/6.5
= 57.55 kN
The bending m om ent at various positions along strip 2 - 2 can now be evaluated. As with strip 1 -1 , the strip is assum ed to be uniformly loaded across its m etre width. Thus
M 5 = R 2 x 5 = 7.14 x 5 = 35.7 kN m/m M s = 57.55 x 1.5 + 32.5 x 1.5 x 1.5 x 0.75
+ 3.75 x 1.5 x 1.5 x 0.5
= 86.33 + 54.84 + 4.22
= 145.39 kN m/m
The shear force at the base of the wall for strip 2 -2 is
V8 = 57.55 + 32.5 x 1.5 x 1.5 + 3.75 x 1.5 x 1.5
= 139.11 kN
E.2.3 Section analysis (ULS)
For brevity, M 8 and V's only will be considered with the following data: h = 250 mm, d = 210 mm, / ck = 35 N/m m2 and / yk = 400 N/mm2. From Figure 4.3 (or B.2)
E.2.2 Member analysis (ULS)
M 8/bwd2= 145.39 x KT/IO3 x 2102 = 3.3
Thus for / ck = 35 N/m m 2, xld = 0.23. (The strip m ethod is a plastic analysis and thus the upper limit of xld should be taken as 0.25.) Thus
x = 0.23 x 210 = 48.3 d - 0.4 x = 190.68 mm Thus
A W y k W - 0 -4 X )
= 145.39 x 106 x 1.15/400 x 190.68
- 2192 m m 2/m (20 <p - 140 = 2243 m m 2/m) and
p = 2243/210 x 103 = 0.0107
From Table 1.6 (or B.2), the basic shear stress TRd is 0.37 N/mm2 for / ck = 35 N/m m 2. VRdl is given by equation (4.2) as
^ R d l “ T Rd ^ ( 1 * 2 + 40 p x) b^d and
fc = 1.6 - 0.21 = 1.39 p x = 0.0107
Thus
y Rdl = 0.37 x 1.39 (1.2 + 40 x 0.0107) x 210
= 175.83 kN
Thus the section is adequate in shear.
A serviceability check should be carried out in accordance with C hapter 4 (section 4.8). F or equiv
alent fluid pressure, it is suggested th at *¥ should be taken as unity. A rm er and M oore (1989) should be consulted for further inform ation on the strip approach and, in particular, the advanced strip m ethod, which is intended for the design of slabs supported as a whole, or in part, by columns.
REFERENCES
H illerborg A. (1975) Strip M ethod o f Design, Viewpoint.
W ood R.H. and A rm er G.S.T. (1970) The Strip M ethod o f Designing Slabs, Building R esearch Establishm ent C urrent Paper 39/70.
Tim oshenko S. and W oinowsky-Krieger S. (1959) Theory o f Plates and Shells, M cGraw-Hill, New York.
A rm er G.S.T. and M oore D.B. (eds) (1989) Frame and Slab Structures (chapter 7, L.L. Jones, H illerborg’s advanced strip m ethod - a review and extensions), B utterw orths, London.