Concept explainers
For the beam and loading shown, determine the equations of the shear and bending-moment curves and the maximum absolute value of the bending moment in the beam, knowing that (a) k = 1, (b) k = 0.5.
Fig. P5.155
(a)
The equations of the shear and bending moment curves and the maximum absolute value of the bending moment when
Answer to Problem 155RP
The equation of the shear force curve is
The equation of the bending moment curve is
The maximum absolute bending moment in the beam is
Explanation of Solution
Find the load function
By definition;
Substitute
Integrate the equation to find V.
By definition;
Substitute
Integrate the equation to find M.
Apply the boundary condition;
Substitute 0 for M and 0 for x in Equation (2).
Substitute 0 for
Apply the boundary condition;
Substitute 0 for V and 0 for x in Equation (1).
Substitute 0 for
Substitute 0 for
When
Substitute 1 for k in Equation (4).
Substitute 1 for k in Equation (5).
Maximum bending moment;
The maximum bending moment occurs at the fixed end.
Substitute L for x in Equation (6).
Therefore,
The equation of the shear force curve is
The equation of the bending moment curve is
The maximum absolute bending moment in the beam is
(b)
The maximum absolute value of the bending moment when
Answer to Problem 155RP
The equation of the shear force curve is
The equation of the bending moment curve is
The maximum absolute bending moment in the beam is
Explanation of Solution
When
Substitute 0.5 for k in Equation (4).
Substitute 0.5 for k in Equation (5).
The maximum bending moment occurs where the shear force changes sign. i.e.,
Substitute 0 for V in Equation (8).
Substitute
Therefore,
The equation of the shear force curve is
The equation of the bending moment curve is
The maximum absolute bending moment in the beam is
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Chapter 5 Solutions
Mechanics of Materials, 7th Edition
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