Question

The forces acting on member 80 are shown in the free-body diagram. a = 2 m, b-2m, c-sm, d-2.5m, ITall-llTnell-liit, Foyl 20.6 kN, Foll -0.48 kN. Express each force acting on 8D in terms of rectangular component vectors. Exercise 2.5.65 BC TBA d-써 Weighs 5.0 kN TBC Fpy D TBA D: FE

Answer 1

$A(a,0,b),B(0,c+d,0),C(-a,0,b),D(0,0,0),E(0,c,0)$

$\bold{BC}=-a\hat{x}-(c+d)\hat{y}+b\hat{z}$

$\hat{u}_{BC}=\frac{\bold{BC}}{BC}=\frac{-a\hat{x}-(c+d)\hat{y}+b\hat{z}}{\sqrt{a^2+(c+d)^2+b^2}}=\frac{-2\hat{x}-7.5\hat{y}+2\hat{z}}{\sqrt{2^2+(7.5)^2+2^2}}=\frac{-2\hat{x}-7.5\hat{y}+2\hat{z}}{8.0156}$

$\bold{BA}=a\hat{x}-(c+d)\hat{y}+b\hat{z}$

$\hat{u}_{BA}=\frac{\bold{BA}}{BA}=\frac{a\hat{x}-(c+d)\hat{y}+b\hat{z}}{\sqrt{a^2+(c+d)^2+b^2}}=\frac{2\hat{x}-7.5\hat{y}+2\hat{z}}{\sqrt{2^2+(7.5)^2+2^2}}=\frac{2\hat{x}-7.5\hat{y}+2\hat{z}}{8.0156}$

Forces acting on $BD$ in vector notation are:

$\bold{F}_{Dy}=(0\hat{x}+20.6\,\hat{y}+0\hat{z})\,k N$

$\bold{F}_{Dz}=(0\hat{x}+0\,\hat{y}-0.48\hat{z})\,k N$

$\bold{F}_{E}=(0\hat{x}+0\,\hat{y}-5.0\hat{z})\,k N$

$\bold{T}_{BA}=T_{BA}\hat{u}_{BA}=11k\left ( \frac{2\hat{x}-7.5\hat{y}+2\hat{z}}{8.0156} \right )=(2.74\hat{x}-10.29\,\hat{y}+2.74\hat{z})\,k N$

$\bold{T}_{BC}=T_{BC}\hat{u}_{BC}=11k\left ( \frac{-2\hat{x}-7.5\hat{y}+2\hat{z}}{8.0156} \right )=(-2.74\hat{x}-10.29\,\hat{y}+2.74\hat{z})\,k N$