Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 7.3, Problem 7.88P

(a)

To determine

To write the equations of the shear and bending moment curves.

(a)

Expert Solution
Check Mark

Answer to Problem 7.88P

The equation for shear curve is V(x)=w0Lπcos(πxL) and the equation for bending moment curve is M=w0L2π2sin(πxL).

Explanation of Solution

Refer Fig P7.88.

Write the equation for change in shear moment with distance.

dVdx=w (I)

Here, V is the shear moment.

Write the equation for change in bending moment with distance.

dMdx=V (II)

Here, M is the bending moment.

Conclusion:

Calculate the change in shear moment with distance using equation (I) and thus, find the equation for the shear moment curve.

dVdx=w=w0sinπxLV=w0sinπxLdx=w0LπcosπxL+C1 (III)

Calculate the change in bending moment with distance using equation (II) and thus find the equation for the bending moment curve.

dMdx=V=w0LπcosπxL+C1M=Vdx=(w0LπcosπxL+C1)dx=w0L2π2sinπxL+C1x+C2 (IV)

Apply boundary condition to find C1.

At x=L

M=0C1=0 (V)

Apply boundary condition to find C2.

At x=0

M=0C2=0 (VI)

Use equation (V) to find the equation for shear curve.

V(x)=w0Lπcos(πxL)

Use equation (VI) to find the equation for bending moment curve.

M=14ω0L2[(xL)32(xL)2+(xL)]

Therefore, the equation for shear curve is V=14ω0L[3(xL)24(xL)+1] and the equation for bending moment curve is M=w0L2π2sin(πxL).

(b)

To determine

The magnitude and location of the maximum bending moment.

(b)

Expert Solution
Check Mark

Answer to Problem 7.88P

The magnitude of the maximum bending moment is w0L2π2 and the location is at x=L3.

Explanation of Solution

The maximum absolute value of the bending moment can be found using the equation for the bending moment curve.

Write the equation to find the maximum bending moment.

dMdx=V=0

Conclusion:

Calculate the location of the maximum bending moment.

x=L2 (VII)

Calculate the maximum bending moment.

M=w0L2π2sin(πxL)=w0L2π2sin(πLL2)=w0L2π2sin(π2)=w0L2π2 (VIII)

Therefore, the magnitude of the maximum bending moment is w0L2π2 and the location is at x=L3.

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Chapter 7 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 7.1 - A semicircular rod is loaded as shown. 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Understanding Shear Force and Bending Moment Diagrams; Author: The Efficient Engineer;https://www.youtube.com/watch?v=C-FEVzI8oe8;License: Standard YouTube License, CC-BY
Bending Stress; Author: moodlemech;https://www.youtube.com/watch?v=9QIqewkE6xM;License: Standard Youtube License