Concept explainers
3–48 and 3–49 The beam shown is loaded in the xy and xz planes.
- (a) Find the y- and z-components of the reactions at the supports.
- (b) Plot the shear-force and bending-moment diagrams for the xy and xz planes. Label the diagrams properly and provide the values at key points.
- (c) Determine the net shear-force and bending-moment at the key points of part (b).
- (d) Determine the maximum tensile bending stress. For Prob. 3–48, use the cross section given in Prob. 3–34, part (a). For Prob. 3–49, use the cross section given in Prob. 3–39, part (b).
(a)
The
The
Answer to Problem 48P
The
The
Explanation of Solution
The following figure shows the forces acting on the beam along x-z plane.
Figure-(1)
Write the equilibrium equation of force applied on the beam in
Write the bending moment at the support
The following figure shows the forces acting on the beam along x-y plane.
Figure-(2)
Write the equilibrium equation of force applied on the beam in
Write the bending moment at the support
Conclusion:
From Equation (II),
Substitute
Thus, the
Simplify Equation (IV).
Substitute
Thus, the
(b)
The shear force and bending moment diagram of the beam.
Explanation of Solution
Write the expression for the moment at point A.
Write the expression for the moment at point B.
Write the equation for the shear force.
Write the expression for the maximum bending moment.
Write the expression for the bending moment at B.
Write the equation for the shear force.
Write the expression for the maximum bending moment.
Conclusion:
Substitute
Substitute
The maximum bending moment occur at
Substitute
Substitute
The following figure shows the shear force and bending moment diagram of the beam along x-z plane.
Figure-(3)
Substitute
The maximum bending moment occur at
Substitute
Substitute
The following figure shows the shear force and bending moment diagram of the beam along x-y plane.
Figure-(4)
(c)
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
Answer to Problem 48P
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
The net bending moment at
The net shear force at
Explanation of Solution
Write the expression for the net shear force on the beam.
Write the expression for the net bending moment on the beam.
Conclusion:
The following table shows the magnitudes of bending moment and shear force along x-y and x-z plane from the shear force and bending moment diagram.
At
Substitute
Thus, the net shear force at
Substitute
Thus, the net bending moment at
At
Substitute
Thus, the net shear force at
Substitute
Thus, the net bending moment at
At
Substitute
Thus, the net shear force at
Substitute
Thus, the net bending moment at
At
Substitute
Thus, the net shear force at
Substitute
Thus, the net bending moment at
(d)
The maximum tensile bending stress on the beam.
Answer to Problem 48P
The maximum tensile bending stress on the beam is
Explanation of Solution
The following diagram shows the cross-section of the beam.
Figure-(5)
Write the expression for the second moment of area of the section about z-axis.
Here, the second moment area of the area
Write the moment of inertia of the rectangular section.
Here total height of the section is
Write the expression for the moment of inertia about y-axis.
Here, the height of the flange is
Write the expression for maximum tensile bending stress.
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the maximum tensile bending stress on the beam is
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