Reactions at Properly Aligned Hinges The plate in Fig. 5.34 is supported by hinges at A and B and the cable CE. The properly aligned hinges do not exert couples on the plate, and the hinge at A does not exert a force on the plate in the direction of the hinge axis. Determine the reactions at the

Question

Reactions at Properly Aligned Hinges

The plate in Fig. 5.34 is supported by hinges at A and B and the cable CE. The properly aligned hinges do not exert couples on the plate, and the hinge at A does not exert a force on the plate in the direction of the hinge axis. Determine the reactions at the hinges and the tension in the cable.

Accepted Answer

Draw the Free-Body Diagram We isolate the plate and show the reactions at the hinges and the force exerted by the cable (Fig. a). The term T is the force exerted on the plate by cable CE.

Apply the Equilibrium Equations Since we know the coordinates of points C and E, we can express the cable force as the product of its magnitude T and a unit vector directed from C toward E. The result is

r( -0.842 i + 0.337j + 0.421k).

The sums of the forces in each coordinate direction equal zero:

[latex]\sum{F{x} } =A_{x} + B_{x} - 0.8427T = 0,[/latex]

[latex]\sum{F{y} } =A_{y} + B_{y} + 0.337T - 400 = 0,[/latex]

[latex]\sum{F{z} } =B_{z} - 0.4217T = 0,[/latex]

If we sum the moments about B. the resulting equations will not contain the three unknown reactions at B. The sum of the moments about B is

[latex]\sum{M{(point B)} } =\left|\begin{matrix} i & j & k \ 0.2 & 0 & 0 \ -0.842T & 0.337T & 0.421T \end{matrix} ight|+ \left|\begin{matrix} i & j & k \ 0 & 0 & 0.2 \ A_{x} & A_{y} & 0 \end{matrix} ight| + \left|\begin{matrix} i & j & k \ 0.2 & 0 & 0.2 \ 0 & -400 & 0 \end{matrix} ight| = (-0.2A_{y} + 80 )i + (-0.0842T + 0.2A_{x} )j + (0.0674T - 80)k =0. [/latex]

The scalar equations are

[latex]\sum{M_{x} } = -0.2A_{y} + 80 =0 ,[/latex]

[latex]\sum{M_{y} } = -0.0842T + 0.2A_{x} =0 ,[/latex]

[latex]\sum{M_{z} } = 0.0674T - 80 =0 .[/latex]

Solving these equations, we obtain the reactions

[latex]T= 1187 N, A_{x} = 500 N, A_{y} = 400 N.[/latex]

Then from Eqs. (5.24), the reactions at B are

[latex]B_{x}= 500 N, B_{y} = -400 N, B_{z} = -500 N.[/latex]

Discussion
If our only objective had been to determine the tension T, we could have done so easily by setting the sum of the moments about the line AB (the z axis) equal to zero. Since the reactions at the hinges exert no moment about the z axis, we obtain the equation

(0.2)(0.337r) - (0.2)(400) = 0,

which yields the result T = 1 187 N.

Question 5.10: Reactions at Properly Aligned Hinges The plate in Fig. 5.34 ...

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