Civil Engineering Reference
In-Depth Information
Adding the results of Eqs (i)-(v) we obtain the total internal virtual work, i.e.
14
.
07
+
5
x
VW
(
m
)
=
(vi)
1
+
x
The external virtual work may be found by dividing the slab into rectangles
enpf and ekqf and triangles ane, ekd and ade. Since the displacement of e is
unity the displacement of each of the centroids of the rectangles will be 1/2 and
the displacement of each of the centroids of the triangles will be 1/3. The total
virtual work due to the applied load is then given by
10
2(4
x
)
2
2
2
2
x
)
1
2
1
3
x
VW
(
w
)
=
−
+
+
2(2
+
3
x
)
1
2
1
3
+
4
.
5
×
0
.
89(1
+
which simplifies to
VW
(
w
)
=
10(9
.
33
−
0
.
67
x
)
(vii)
Equating internal and external virtual works, Eqs (vi) and (vii), we have
10(1
+
x
)(9
.
33
−
0
.
67
x
)
m
=
(viii)
14
.
07
+
5
x
The value of
x
corresponding to the maximum value of
m
may be found by
differentiating Eq. (viii) with respect to
x
and equating to zero. Alternatively, a
series of trial values of
x
may be substituted in Eq. (viii) and the maximum value
of
m
obtained. Using the former approach gives
x
=
2
.
71m from which
m
=
10
.
09 kNm
/
m
19.2 D
ISCUSSION
The method presented here for the analysis of reinforced concrete slabs gives, as
we have seen, upper bound values for the collapse loads of slabs. However, in rela-
tively simple cases of slab geometry and loading, the yield line method can be used
as a design method since the fracture pattern can be obtained with reasonable accu-
racy. Also, in practice the actual collapse load of a slab may be above the calculated
value because of secondary effects such as the kinking of the reinforcing steel in the
vicinity of the fracture line and the effect of horizontal edge restraints which induce
high compressive forces in the plane of the slab with a consequent increase in load
capacity.
An alternative to yield line theory is the
strip method
proposed by A. Hillerborg at
Stockholm in 1960. This method is a direct design procedure as opposed to yield line
theory which is analytical and therefore will not be investigated here.
