Determine the reactions at the suooorts, then draw the shear and moment diagrams. Assume the support at A is fixed and B is a roller, EI is constant.
The reactions at the supports and to draw the shear and moment diagrams.
Answer to Problem 9.1P
The vertical reaction at support A is
The horizontal reaction at support A is
The reaction moment at support A is
The vertical reaction at support B is
The shear diagram is shown below.
The moment diagram is shown below.
Explanation of Solution
Concept Used:
Write the expression for the net force balance in the vertical direction of the beam.
Here,
Write the expression for the net force balance in the horizontal direction in the beam.
Here,
Write the expression for the net moment about end
Here,
Calculations:
The free body diagram for the beam is shown below.
Figure (1)
Here, the vertical reaction at point
Calculate the support reactions using Equation (II).
The uniformly varying load is replaced by a concentrated force of magnitude
Figure (2)
Consider the moment at point A using Equation (III).
The displacement of the beam for the given load is shown below.
Figure (3)
Calculate the displacement of the beam for the given load.
Here, the displacement of the beam for the given load is
The displacement of the beam for the reaction at point B. is shown below.
Calculate the displacement of the beam for the given load.
Here, the displacement of the beam for the reaction at point B is
Add Equation (VI) and Equation (VII) for the displacement values according to the compatibility condition.
Substitute
Calculate the vertical reaction at A using Equation (IV).
Substitute
Calculate the bending moment at A using Equation (V).
Substitute
Consider the beam as shown below.
Figure (5)
Write the equation to determine the shear in the beam.
Calculate shear force at a distance
Substitute
Calculate shear force at a distance
Substitute
Calculate shear force at a distance
Substitute
Calculate shear force at a distance
Substitute
Calculate shear force at a distance
Substitute
Substitute
Write the expression for the bending moment.
Calculate moment at a distance
Substitute
Calculate moment at a distance
Substitute
Calculate moment at a distance
Substitute
Calculate moment at a distance
Substitute
Calculate moment at a distance
Substitute
Calculate moment at a distance
Substitute
Conclusion:
The vertical reaction at support A is
The horizontal reaction at support A is
The reaction moment at support A is
The vertical reaction at support B is
The shear diagram is shown below
Figure (6)
The moment diagram is shown below
Figure (7)
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