Fundamentals Of Structural Analysis:
Fundamentals Of Structural Analysis:
5th Edition
ISBN: 9781260083330
Author: Leet, Kenneth
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 5, Problem 20P
To determine

Sketch the shear and moment curves and label the maximum values of shear and moment, locate points of inflection, and draw the deflected shape.

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Explanation of Solution

Apply the sign conventions for calculating reactions using the three equations of equilibrium as shown below.

  • For summation of forces along x-direction is equal to zero (Fx=0), consider the forces acting towards right side as positive (+) and the forces acting towards left side as negative ().
  • For summation of forces along y-direction is equal to zero (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
  • For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as positive and the counter clockwise moment as negative.

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

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  1

Refer to Figure 1.

Calculate the length of the beam AB as shown in Figure 1.

LAB=32+42=5ft

Use equilibrium equations:

Summation of forces along x-direction is equal to 0.

+Fx=0Dx=0

Summation of forces along y-direction is equal to 0.

+Fy=0Dy+By122×402×5=0Dy+By=102        (1)

Summation of moments about B is equal to 0.

MB=0Dy×40MD+2×40×4022×5×4212×4=040Dy+MD=1532        (2)

Sketch the free body diagram of the segment CD as shown in Figure 2.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  2

Refer to Figure 2.

Summation of moments about C is equal to 0.

MC=0Dy×20MD+2×20×202=020Dy+MD=400        (3)

Solving Equations (2) and (3) to get the value of Dy as shown below.

Dy=56.6kips

Substitute 56.6 kips for Dy in Equations (3).

20×56.6+MD=400MD=732kft

Substitute 56.6 kips for Dy in Equations (1).

56.6+By=102By=45.4k

Sketch the free body diagram of the segment AB as shown in Figure 3.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  3

Refer to Figure 3.

Summation of forces along x-direction is equal to 0.

+Fx=0Bx=0

Summation of forces along y-direction is equal to 0.

+Fy=0By122×5=0By=22k

Summation of moments about B is equal to 0.

MB=0MB2×5×4212×4=0MB=68kft

Calculate the axial forces as shown below.

125=AA3AA=7.2 k2×55=AB3A@BLeft=6 kAB=6+7.2    =13.2 k

Calculate the shearing forces as shown below.

125=VA4VA=9.6 k2×55=VB4VB=8 k

Calculate the shear at each point as shown below.

For segment AB.

VA=9.6kV@BLeft=9.68          =17.6k

For segment BCD.

VB=45.422=23.4 kVD=45.4222×40=56.6k

Calculate the moment at each point as shown below.

For segment AB.

MA=0MB=12×42×5×42=68kft

For segment BCD.

MB=68kftM@ Left of D=6822×40+45.4×402×40×402               =732kft

Calculate the point of zero shear force as shown below.

Consider a section at a distance x from B.

Sketch the Free Body Diagram of BCD as shown in Figure 4.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  4

Refer to Figure 4.

Calculate the location of zero shear force as shown below.

Summation of shear at section x is equal to 0.

45.4222×x=02x=23.4x=11.7ft from B

Maximum bending moment occurs at the point of zero shear.

Mmax=6822×11.7+45.4×11.72×11.7×11.72=68.9kft

Point of inflection is the location at which the bending moment changes its sign.

Here the BM is zero sign between BC and CD.

Refer to Figure 4.

Calculate the location of point of inflection where BM is zero as shown below.

Summation of moment at section x-x is equal to 0.

6822×x+45.4×x2×x×x2=0x223.4x+68=0x=3.4ft and 20ft from B

Hence, the point of inflection occurs at C and a distance of 3.4 ft from B.

Sketch the shear and moment curves for segment AB as shown in Figure 5.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  5

Sketch the shear and moment curves for segment BCD as shown in Figure 6.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  6

Sketch the deflected shape and the location of points of inflection of the beam as shown in Figure 7.

Fundamentals Of Structural Analysis:, Chapter 5, Problem 20P , additional homework tip  7

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