Question

For economical spacing of roof truss if t, p and r are the cost of truss, purlin and roof coverings respectively , then

• A. t = p + r
• B. t = p + 2r
• C. t = p + 3r
• D. t =2p + r

Answer 1

The correct option is D t =2p + r

Let,
s = spacing of the trusses
t = cost of the trusses per unit area
p = cost of the purlin per unit area
r = cost of the roof coverings per unit area
x = overall cost of the roof system per unit area

The cost of the truss is inversely proportional to the spacing of the trusses,

$t\propto \frac{1}{s}$

$\Rightarrow t=\frac{C_1}{s}$

The cost of the purlin is directly proportional to the square of spacing of trusses,

$p\propto S^2$

$p=C_2{S}^2$

The cost of roof coverings us directly proportional to the spacing of trusses,

$r\propto S$

$r=C_{3}S$

The overall cost,

x = t + p + r

$=\frac{C_1}{S}+C_2{S}^2+C_{3}S$

For the overall cost to be minimum

$\frac{dx}{ds}=0$

$\Rightarrow \frac{dx}{ds}=0=-\frac{C_1}{S^2}+2C_{2}S+C_3$

$\Rightarrow -\frac{C_1}{S^2}+2C_{2}S+C_3=0$

$\Rightarrow -\frac{C_1}{S}+2C_2{S}^2+C_{3}S=0$

$\Rightarrow -t+2p+r=0$

$\Rightarrow t=2p+r$

Therefore, for economic spacing of roof trusses, the cost of trusses should be equal to twice the cost of purlins plus the cost of roof coverings.