Calculate all the reactions and draw the moment diagrams.
Answer to Problem 6P
The vertical reaction at A is
The vertical reaction at B is
The vertical reaction at C is
The bending moment at A is
The bending moment at B is
The bending moment at C is
Explanation of Solution
Restrained structure: Clamp joint 2:
Show the free body diagram of the restrained structure as in Figure (1).
Refer Figure (1),
Find the fixed end moment of the member 1-2
Find the fixed end moment of the member 2-3
Find the moment in clamp using the relation;
Show the free body diagram of the restrained structure with FEM and moment as in Figure (2).
Show the free body diagram of the structure with unit rotation at 2 Figure (3).
Moments due to unit rotation at 2:
Find the moment at the member 1-2 using the relation;
Find the moment at the member 2-1 using the relation;
Find the moment at the member 2-3 using the relation;
Find the moment at the member 3-2 using the relation;
Find the stiffness coefficient
Find the rotation at joint 2;
Take summation moment about joint 2.
Find the moment at the member 1-2 using the relation;
Find the moment at the member 2-1 using the relation;
Find the moment at the member 2-3 using the relation;
Find the moment at the member 3-2 using the relation;
Show the free body diagram of the beam as in Figure (4).
Refer Figure (4),
Consider AB:
Find the vertical reaction at A;
Take summation moment about B.
Thus, the vertical reaction at A is
Find the vertical reaction at B;
Summation of forces about y axis is equal to zero.
Consider BC:
Find the vertical reaction at B;
Take summation moment about C.
Find the vertical reaction at C;
Summation of forces about y axis is equal to zero.
Thus, the vertical reaction at C is
Find the total vertical reaction at B;
Thus, the vertical reaction at B is
Show the free body diagram of the beam as in Figure (5).
Refer Figure (5),
Shear force calculation:
Find the distance when shear force is zero (x):
Equate the equation of shear force to zero.
Bending moment calculation:
Thus, the bending moment at A is
Thus, the bending moment at B is
Thus, the bending moment at C is
Show the shear force and bending moment diagram as in Figure (6).
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Chapter 14 Solutions
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