Design of a Residential Building Floor by Direct Calculations of Bending Moments Coefficients using Equations 14 to 18 of BS 8110-1:1997

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

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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; g_k         = 5.1 kN/m²

Imposed load

Imposed load                          = 1.5 kN/m²

Characteristic imposed load; q_k   = 1.5 kN/m²

Design load, n = 1.4 g_k + 1.6 q_k = 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.

β_y = (24 + 2N_d + 1.5 N_d²)/1000……………………………………..equation 16 

where N_d is the number of discontinous edges

Here the number of discontinouos edges N_d is equal to two

Therefore β_y = (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(l_x/l_y) {Ö(β_y+ β₁) + Ö(β_y + β₂)}]…………………… equation 17 

From our previous calculation, β_y = 0.034 and from inspection of the support conditions of the slab

we find that β₁ = 0 and β₂ = 4β_y/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 β_x, the value of the short span bending moment coefficient,

 we make use of equation 18

Ög = Ö(β_x+ β₃) + Ö(β_x + β₄)………………………………………….. equation 18 

from inspection of the support conditions of the slab

we find that β₃ = 0 and β₄ = 4β_x/3 , and we now know that g = 0.27171

substituting for these values

Ö0.27171 = Ö(β_x+ 0) + Ö(β_x + 4β_x/3)

0.52125 = Ö β_x (Ö1+Ö(1+(4/3))

0.52125 = Ö (β_x) (2.527525)

 Ö β_x = (0.52125/2.527525)

Squaring both sides

 β_x = 0.042 and the long edge moment coefficient = β₄ = (4/3) × 0.042 = 0.056

Span moments

Short span = β_xnl_x² = 0.042 × 9.54 × 4.5² = 8.11 kNm

Long span = β_ynl_x² = 0.034 × 9.54 × 4.5² = 6.57 kNm

Support edge moments

Short edge = β₂nl_x² = 0.045 × 9.54 × 4.5² = 8.69 kNm

Long span = β₄nl_x² = 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 l_y/l_{x .}
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.