VECTOR MECH. FOR EGR: STATS & DYNAM (LL
VECTOR MECH. FOR EGR: STATS & DYNAM (LL
12th Edition
ISBN: 9781260663778
Author: BEER
Publisher: MCG
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Chapter 7.3, Problem 7.86P

For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b) determine the magnitude and location of the maximum bending moment.

Chapter 7.3, Problem 7.86P, For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b)

(a)

Expert Solution
Check Mark
To determine

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

Answer to Problem 7.86P

The equation for shear curve is V=w0L[13xL+12(xL)2] and the equation for bending moment curve is M=w0L2[13(xL)12(xL)2+16(xL)3].

Explanation of Solution

Refer Fig P7.86 and figure 1.

VECTOR MECH. FOR EGR: STATS & DYNAM (LL, Chapter 7.3, Problem 7.86P , additional homework tip  1

Write the equation for the distributed load.

w=w0(1xL) (I)

Here, w0 is the initial load, w is the load, and L is the length of the load.

Write the condition used to find the reaction force B.

ΣMA=0 (II)

Here, ΣMA is the sum of the moments acting on A acting counter clockwise.

Write the condition used to find the reaction force A along y direction.

ΣFy=0 (III)

Here, ΣFy is the sum of the forces acting along y direction.

Write the equation for change in shear moment with distance.

dVdx=w (IV)

Here, V is the shear moment.

Write the equation for change in bending moment with distance.

dMdx=V (V)

Here, M is the bending moment.

Conclusion:

Use equation (I) to calculate the reaction force at B.

ΣMA=0L3(12w0L)LB=0B=w0L6upwards

Use equation (II) to calculate the reaction force at B.

ΣFy=0Ay12w0L+w0L6=0Ay=w0L3upwards

Sketch the shear diagram.

VECTOR MECH. FOR EGR: STATS & DYNAM (LL, Chapter 7.3, Problem 7.86P , additional homework tip  2

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

dVdx=w=VA0xw0(1xL)dxV=(w0L3)w0x+12w0Lx2=w0L[13xL+12(xL)2]

Sketch the bending moment diagram.

VECTOR MECH. FOR EGR: STATS & DYNAM (LL, Chapter 7.3, Problem 7.86P , additional homework tip  3

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

dMdx=VM=0xVdx=L0x/LV(xL)d(xL)=w0L20x/L[13xL+12(xL)2]d(xL)=w0L2[13(xL)12(xL)2+16(xL)3]

Therefore, the equation for shear curve is V=w0L[13xL+12(xL)2] and the equation for bending moment curve is M=w0L2[13(xL)12(xL)2+16(xL)3].

(b)

Expert Solution
Check Mark
To determine

The magnitude and location of the maximum bending moment.

Answer to Problem 7.86P

The magnitude of the maximum bending moment is 0.06415w0L2 and the location is at x=0.423L.

Explanation of Solution

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

The distance of the maximum bending moment is measured from the left hand side.

Conclusion:

At x=L,

V=w0L6

If V=0 at (xL)22(xL)+23, then,

xL=113

The magnitude of the maximum bending moment.

Mmax|xL=113=0.06415w0L2

According to figure 2, find the location of the maximum bending moment.

x=0.423L

Therefore, the magnitude of the maximum bending moment is 0.06415w0L2 and the location is at x=0.423L.

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Chapter 7 Solutions

VECTOR MECH. FOR EGR: STATS & DYNAM (LL

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