Question 6.SP.5: Determine the components of the forces acting on each member......
Determine the components of the forces acting on each member of the frame shown.
STRATEGY: The approach to this analysis is to consider the entire frame as a free body to determine the reactions, and then consider separate members. However, in this case, you will not be able to determine forces on one member without analyzing a second member at the same time.
Final Answer
MODELING and ANALYSIS: The external reactions involve only three unknowns, so compute the reactions by considering the free-body diagram of the entire frame (Fig. 1).
+↺ \Sigma M_E=0: \quad-(2400 \mathrm{~N})(3.6 \mathrm{~m})+F(4.8 \mathrm{~m})=0
F=+1800 \mathrm{~N} \quad \mathrm{~F}=1800 \mathrm{~N} \uparrow
+\uparrow \Sigma F_y=0: \quad-2400 \mathrm{~N}+1800 \mathrm{~N}+E_y=0
E_y=+600 \mathrm{~N} \quad \mathbf{E}_y=600 \mathrm{~N} \uparrow
\stackrel{+}{\rightarrow} \Sigma F_x=0: \quad \mathbf{E}_x=0
Now dismember the frame. Since only two members are connected at each joint, force components are equal and opposite on each member at each joint (Fig. 2).
Free Body: Member BCD.
+↺ \Sigma M_B=0: \quad-(2400 \mathrm{~N})(3.6 \mathrm{~m})+C_y(2.4 \mathrm{~m})=0 \quad C_y=+3600 \mathrm{~N}
+↺ \Sigma M_C=0: \quad-(2400 \mathrm{~N})(1.2 \mathrm{~m})+B_y(2.4 \mathrm{~m})=0 \quad B_y=+1200 \mathrm{~N}
\stackrel{+}{\rightarrow} \Sigma F_x=0: \quad-B_x+C_x=0
Neither B_x \text { nor } C_x can be obtained by considering only member BCD; you need to look at member ABE. The positive values obtained for B_y \text { and } C_y indicate that the force components B_y \text { and } C_y are directed as assumed.
Free Body: Member ABE.
+↺ \Sigma M_A=0: \quad B_x(2.7 \mathrm{~m})=0 \quad B_x=0
\stackrel{+}{\rightarrow} \Sigma F_x=0: \quad+B_x-A_x=0 \quad A_x=0
+\uparrow \Sigma F_y=0: \quad-A_y+B_y+600 \mathrm{~N}=0
-A_y+1200 \mathrm{~N}+600 \mathrm{~N}=0 \quad A_y=+1800 \mathrm{~N}
Free Body: Member BCD. Returning now to member BCD, you have
\stackrel{+}{\rightarrow} \Sigma F_x=0: \quad-B_x+C_x=0 \quad 0+C_x=0 \quad C_x=0
REFLECT and THINK: All unknown components have now been found. To check the results, you can verify that member ACF is in equilibrium.
+↺ \Sigma M_C=(1800 \mathrm{~N})(2.4 \mathrm{~m})-A_y(2.4 \mathrm{~m})-A_x(2.7 \mathrm{~m})
=(1800 \mathrm{~N})(2.4 \mathrm{~m})-(1800 \mathrm{~N})(2.4 \mathrm{~m})-0=0 (checks)
