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Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

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Chapter
Section
BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 9, Problem 1P
Textbook Problem
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For the beam of Problem 8.6, determine the maximum negative bending moment at point B due to a 75 kN concentrated live load.

Chapter 9, Problem 1P, For the beam of Problem 8.6, determine the maximum negative bending moment at point B due to a 75 kN

FIG. P8.6, P8.61

To determine

Find the maximum negative bending moment at point B.

Answer to Problem 1P

The maximum negative bending moment at point B is 150kN-m_.

Explanation of Solution

Given Information:

The concentrated live load (P) is 75 kN.

Calculation:

Apply a 1 kN unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis, Chapter 9, Problem 1P , additional homework tip  1

Refer Figure 1.

Find the equation of support reaction (Cy) at C using equilibrium equation:

Take moment about point A.

Consider moment equilibrium at point A.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point A is zero.

ΣMA=0Cy(14)1(x)=014Cy=xCy=x14        (1)

Find the equation of support reaction (Ay) at A using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive (+) and downward force as negative ().

Ay+Cy=1

Substitute x14 for Cy.

Ay+x14=1Ay=1x14        (2)

Find the equation of moment at B.

Apply 1 kN at just left of B (0x7m).

Sketch the free body diagram of the section AB as shown in Figure 2.

Structural Analysis, Chapter 9, Problem 1P , additional homework tip  2

Refer Figure 2.

Consider moment at B.

Consider clockwise moment as positive and anticlockwise moment as negative.

MB=Ay(7)(1)(7x)

Substitute 1x14 for Ay.

MB=(1x14)(7)(1)(7x)=7x27+x=x2

Apply 1 kN at just right of B (7m<x18m).

Sketch the free body diagram of the section BD as shown in Figure 3.

Structural Analysis, Chapter 9, Problem 1P , additional homework tip  3

Refer Figure 3.

Consider moment at B.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion BC (7m<x18m).

MB=Cy(7)(1)[7(14x)]=7Cy7+(14x)=7Cy+7x

Substitute x14 for Cy.

MB=7(x14)+7x=x2+7x=7x2

Thus, the equations of the influence line for MB are,

MB=x2, 0x7m        (3)

MB=7x2, 7mx18m        (4)

Find the value of influence line ordinate of moment at various points of x using the Equations (3) and (4) and summarize the value as in Table 1.

x (m)MB (kN-m/kN)
00
772
140
28–2

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 4.

Structural Analysis, Chapter 9, Problem 1P , additional homework tip  4

Refer Figure 4,

The maximum negative influence line ordinate of bending moment at B is 2kN-m/kN.

Find the maximum negative bending moment at point B using the equation.

Maximum Negative MB=P×(maximum negative ILD of MB)

Substitute 75 kN for P and 2kN-m/kN for maximum negative ILD of MB.

Maximum Negative MB=75×(2)=150kN-m

Therefore, the maximum negative bending moment at point B is 150kN-m_.

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