Analyzing a Frame The frame in Fig. 6.36 is subjected to a 200-N-m couple. Determine the forces and couples on its members.

Question

Analyzing a Frame

The frame in Fig. 6.36 is subjected to a 200-N-m couple. Determine the forces and couples on its members.

Accepted Answer

Determine the Reactions at tiie Supports We draw the free-body diagram of the entire frame in Fig. a. The term [latex]M_{A}[/latex] is the couple exerted by the built-in support. From the equilibrium equations

[latex]\sum{F_{x} } = A_{x} = 0,[/latex]

[latex]\sum{F_{y} } = A_{y}+ C = 0,[/latex]

[latex]\sum{M_{point A} } =M_{A} - 200 + (1)C = 0,[/latex]

we obtain the reaction [latex] A_{x} = 0.[/latex] We can't determine [latex] A_{y} , M_{A}[/latex] .or C from this free-body diagram.

Analyze the IVIembers We "disassemble" the frame to obtain the freebody diagrams of the members in Fig. b. The equilibrium equations for member BC are

[latex]\sum{F_{x} } = -B_{x} = 0,[/latex]

[latex]\sum{F_{y} } = -B_{y}+ C = 0,[/latex]

[latex]\sum{M_{point B} } = - 200 + (0.4)C = 0,[/latex]

Solving these equations, we obtain [latex]B_{x} = 0 , B_{y} = 500 N , and C = 500 N.[/latex] The equilibrium equations for member AB are

[latex]\sum{F_{x} } = A_{x} + B_{x} = 0 ,[/latex]

[latex]\sum{F_{y} } = A_{y} + B_{y} = 0 ,[/latex]

[latex]\sum{M_{point A} } = M_{A} + (0,6)B_{y} = 0.[/latex]

Because we already know [latex]A_{x} , B_{x} , and B_{Y}[/latex] we can solve these equations for [latex]A_{y} = -500 N and M_{A} = -300 N-m [/latex]. This completes the solution (Fig. c).

Discussion

We were able to solve the equilibrium equations for member BC without having to consider the free-body diagram of member AB. We were then able to solve the equilibrium equations for member AB. By choosing the members with the fewest unknowns to analyze first, you will often be able to solve them sequentially, but in some cases you will have to solve the equilibrium equations for the members simultaneously.

Even though we were unable to determine the four reactions [latex]A_{x}, A_{y} , M_{A}[/latex] and C with the three equilibrium equations obtained from the free-body diagram of the entire frame, we were able to determine them from the free-body diagrams of the individual members. By drawing free-body diagrams of the members, we gained three equations because we obtained three equilibrium equations from each member but only two new unknowns, [latex]B_{x} and B_{y}[/latex].

Question 6.6: Analyzing a Frame The frame in Fig. 6.36 is subjected to a 2...

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