Applying the Method of Joints Determine the axial forces in the members of the truss in Fig. 6. 12.

Question

Applying the Method of Joints

Determine the axial forces in the members of the truss in Fig. 6. 12.

Accepted Answer

Determine the Reactions at the Supports We first draw the free-body diagram of the entire truss (Fig. a). From the equilibrium equations,

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

[latex]\sum{F_{y} } = A_{y} - 2 = 0[/latex],

[latex]\sum{M_{(point B )} } = -6A_{x} - (10)(2) = 0[/latex].

we obtain the reactions [latex]A_{x} = -3.33 kN , A_{y} = 2 kN [/latex] , and B = 3.33 kN.

Identify Special Joints Because joint C has three members, two of which are coUinear, and no load, the axial force in member BC is zero, [latex]T_{BC} = 0[/latex], and the axial forces in the collinear members AC and CD are equal, [latex]T_{AC} = T_{CD}[/latex].

Draw Free-Body Diagrams of the Joints We know the reaction exerted on joint A by the support, and joint A is subjected to only two unknown forces, the axial forces in members AB and AC. We draw its free-body diagram in Fig. b. The angle α = arctan(5/3) = 59.0°. The equilibrium equations for joint A are

[latex]\sum{F_{x} } = T_{AC}\sin α - 3.33 = 0[/latex]

[latex]\sum{F_{y} } = 2 - T_{AB} - T_{AC} \cos α = 0[/latex]

Solving these equations, we obtain [latex]T_{AB} = 0 and T_{AC} = 389 kN.[/latex] Because the axial forces in members AC and CD are equal, [latex]T_{CD} = 3.89 kN[/latex].

Now we draw the free-body diagram of joint B in Fig. c. (We already know that the axial forces in members AB and BC are zero.) From the equilibrium equation

[latex]\sum{F_{x} } = T_{BD} + 3.33 = 0[/latex],

we obtain [latex]T_{BD} = -3.33 kN[/latex]. The negative sign indicates that member BD is in compression.
The axial forces in the members are

AB: 0,
AC: 3.89 kN in tension (T),
BC: 0,
BD: 3.33 kN in compression (C),
CD: 3.89 kN in tension (T).

Question 6.1: Applying the Method of Joints Determine the axial forces in ...

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