The bending moments of the following floor slab are computed using direct calculations of bending moments coefficients using equations 14 to 18 of BS 8110-1:1997
It should be noted that the bending moments coefficients of equations 16 to 18 of BS 8110-1:1997 were developed using the modified yield line methods to enable rapid analysis and design of reinforced concrete slabs
Loading
Using slab
of thickness = 150mm
Deadload load
Self
weight = 0.15 × 24 =
3.6 kN/m2
Finishes
& partitions = =
1.5 kN/m2
Characteristic
dead load; gk = 5.1
kN/m2
Imposed load
Imposed
load =
1.5 kN/m2
Characteristic
imposed load; qk = 1.5 kN/m2
Design
load, n = 1.4 gk + 1.6 qk = 1.4 × 5.1 + 1.6 × 1.5 = 9.54
kN/m2
Considering
panel A, it can be seen that two of its edges are discontinuous while the other
two edges are continuous.
βy
= (24 + 2Nd + 1.5 Nd2)/1000……………………………………..equation
16
where Nd
is the number of discontinous edges
Here the
number of discontinouos edges Nd is equal to two
Therefore
βy = (24 + (2 × 2) + (1.5 ×22)/1000 = 0.034
Therefore
the long span bending moment coefficient = 0.034
And the short
edge bending moment coefficient β2 = (4/3) × 0.034 = 0.045
For the
remaining bending moment coefficients use is to be made of equations 17 and 18
g = (2/9) [3-Ö18(lx/ly)
{Ö(βy+ β1) + Ö(βy
+ β2)}]…………………… equation 17
From our
previous calculation, βy = 0.034 and from inspection of the support
conditions of the slab
we find that β1
= 0 and β2 = 4βy/3 , substituting for these values
g = (2/9) [3-Ö18(4.5/5) {Ö(0.034+
0) + Ö(0.034 + 0.045)}]
g = 0.27171
In order
to calculate the value of βx, the value of the short
span bending moment coefficient,
we make use of equation 18
Ög = Ö(βx+ β3) + Ö(βx
+ β4)………………………………………….. equation 18
from
inspection of the support conditions of the slab
we find that β3
= 0 and β4 = 4βx/3 , and we now know that g =
0.27171
substituting
for these values
Ö0.27171 = Ö(βx+ 0) + Ö(βx
+ 4βx/3)
0.52125 = Ö βx
(Ö1+Ö(1+(4/3))
0.52125 = Ö (βx)
(2.527525)
Ö βx = (0.52125/2.527525)
Squaring
both sides
βx = 0.042 and the long edge moment
coefficient = β4 = (4/3) × 0.042 = 0.056
Span moments
Short span
= βxnlx2 = 0.042 × 9.54 × 4.52 =
8.11 kNm
Long span
= βynlx2 = 0.034 × 9.54 × 4.52 = 6.57
kNm
Support edge moments
Short edge
= β2nlx2 = 0.045 × 9.54 × 4.52 = 8.69
kNm
Long span
= β4nlx2 = 0.056 × 9.54 × 4.52 = 10.81
kNm
The above
calculations will seem to be tedious to the first time observer, but it is not
so. These calculations are the bases of the values presented in table 3.14 of
BS 8110-1:1997. For the university student and the practicing engineer, they are
very valuable tools for carrying out day-to-day work involving the design of
slabs.
In the design office, these calculations could easily be resolved each time with the use of spreadsheet software that are installed in almost every computer. This can be achieved with very minimal computer programming knowledge. Also, there will be no need for carrying out interpolation for intermediate values of aspect ratios ly/lx .
All values of slab bending coefficients are computed directly and used immediately. For those engaged in civil structural design offices, automation of the above process will make life easier.
In the
coming posts, we will learn how to implement this very easily with a
spreadsheet.
