List the given data and the goal
We are given the forces $P_1 = 9,mathrm{kN}$, $P_2 = 12,mathrm{kN}$, and $P_3 = 6,mathrm{kN}$, with $FE = AB = BC = CD = FB = EC = 3,mathrm{m}$. We must find the force in members $EF$, $BE$, $BC$ and $BF$ and state tension or compression.
Draw the free-body diagram of the truss
A negative value of force represents compression and a positive value represents tension.
Find the support reaction at D
Taking the moment about $A$ equal to zero: [ -P_1(AB) - P_3(FB) - P_2(AB+BC) + N_D(AB+BC+CD) = 0 ] [ -(9)(3) - (6)(3) - (12)(6) + N_D(9) = 0 ] [ N_D = 13,mathrm{kN} ]
Cut the truss at section a-a
Expose the unknown member forces $F_{EF}$, $F_{BE}$ and $F_{BC}$.
Length of BE
[ BE = sqrt{(CE)^2 + (BC)^2} = sqrt{3^2 + 3^2} = 3sqrt{2},mathrm{m} ]
Force in member EF
Moment about $B$: [ -P_2(BC) + N_D(BC+CD) + F_{EF}(EC) = 0 ] [ -(12)(3) + (13)(6) + F_{EF}(3) = 0 ] [ F_{EF} = -14,mathrm{kN} Rightarrow 14,mathrm{kN};(mathrm{C}) ]
Force in member BC
Moment about $E$: [ N_D(CD) - F_{BC}(EC) = 0 ] [ (13)(3) - F_{BC}(3) = 0 Rightarrow F_{BC} = 13,mathrm{kN};(mathrm{T}) ]
Force in member BE
Sum forces along $y$: [ N_D - F_{BE}cosangle BEC - P_2 = 0 ] with $cosangle BEC = tfrac{3}{3sqrt2}$: [ 13 - F_{BE}!left(tfrac{3}{3sqrt2}right) - 12 = 0 Rightarrow F_{BE} = sqrt2 = 1.41,mathrm{kN};(mathrm{T}) ]
Force in member BF (joint B)
At joint $B$, sum forces along $y$: [ F_{BF} + F_{BE}cosangle BEC - P_1 = 0 ] [ F_{BF} + (1.41)!left(tfrac{3}{3sqrt2}right) - 9 = 0 Rightarrow F_{BF} = 8,mathrm{kN};(mathrm{T}) ]
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