Find the reactions:
- Draw a free body diagram of the entire truss.
- Get scalar components of loads B and C for later use.
Bx=-273.62 kNBy=-751.75 kNCx=0 kNCy=-100 kN
- Take moments at A to find reactions at D. Note 1-2-√3 triangle.
ΣMA=02√3D=– 1Bx+√3By+√3CD=– Bx2√3+By2+C2D=12(−273.6√3+751.8+100)=346.89 kN up.
- Apply ΣFx and ΣFy equations to get the components of the reaction force at A.
ΣFx=0Ax=Bx=273.62 kNto the right.
ΣFy=0Ay=Cy−Dy+ByAy=100.00−346.89+751.75=504.86 kNup.
Analyze the joints:
- Draw a FBDs of the joints showing your assumed directions for the forces in the members. Members in tension pull away from the joint, compression push towards the joint.
GeoGebra applet loading...
- Pick a joint with two unknowns, joint D for example. Joint A would also work.
- Apply equations of equilibrium to joint D to find forces in members BD and CD.
\begin{align}\\D: \Sigma F_y &= 0\\ BD_y &= D\\ BD &= \dfrac{D}{\sin 30°}\\ &= \dfrac{\var{display(magD)}}{0.5}\\ &= \var{display(BD)} \quad \text{Compression.}\end{align}
\begin{align}\\D: \Sigma F_x &= 0\\ CD &=BD_x\\ &= BD \cos 30° \\&= \var{display(BD)} \cos 30° \\&= \var{display(CD)} \quad \text{Tension.}\end{align}
- Move to an adjacent joint with two unknowns, joint C in this case. Joint C is a special case.
\begin{align}\\ C: \Sigma F_y &= 0\\BC &= C\\ &= \var{display(BC)} \quad \text{Tension.}\end{align}
\begin{align}\\ C: \Sigma F_x &= 0\\AC &= CD\\ &= \var{display(AC)} \quad \text{Tension.}\end{align}
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Move to joint B and solve for AB.
B: \Sigma F_y=0
-AB_y - BC + BD_y \, \mathbf{\var{if(sign(vecB[1])=1,'+','–')}}\, B_y = 0
AB_y= \var{vecC[1]} + \var{precround(vecD[1],1)} \, \var{if(sign(vecB[1])=-1,'','+')} \, \var{precround(vecB[1],1)}= \var{display(-ABy)}
\begin{align}AB &= \dfrac{AB_y}{\sin 30°}\\ &= \var{display(-2ABy)}\\ &= \var{display(AB)} \var{if(sign(ABy)=-1,' Tension.', ' Compression.')}\end{align}
Check your work:
At this point you should check your work by verifying that joint A is in equilibrium. If it is not you have made a mistake somewhere above.
A: \Sigma F_x \stackrel{?}{=} 0
-A_x + AB_x + AC = \var{precround(vecA[0],1)}+ \var{precround(-vecAB[0],1) } + \var{precround(-vecCD[0],1) } = \var{vecA[0] -vecAB[0] - vecCD[0]}
Close Enough
A: \Sigma F_y \stackrel{?}{=} 0
A_y + AB_y = \var{precround(vecA[1],1)}+ \var{precround(-vecAB[1],1) } = \var{vecA[1] -vecAB[1]}
Close Enough