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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 13, Problem 8P
Textbook Problem
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Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P13.1-P13.4 by using the method of consistent deformations. Select the reaction moment at the fixed support to be the redundant.

Chapter 13, Problem 8P, Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs.

FIG. P13.4, P13.8

To determine

Find the reactions and sketch the shear and bending moment diagrams for the given beam using method of consistent deformation.

Explanation of Solution

Given information:

Apply the sign conventions for calculating reactions, forces and moments 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 negative and the counter clockwise moment as positive.

Calculation:

Show the beam as shown in the Figure 1.

Refer Figure 1.

Consider the horizontal and vertical reaction at C are denoted by Cx and Cy.

Consider the moment at C is denoted by MC.

Consider the vertical reaction at C is denoted by Cy.

The reactions acting in the beam is 4.

The number of Equilibrium reaction is 3. Then,

The degree of indeterminacy of the beam is 1.

Take the moment at C as the redundant.

Replace the fixed support by hinged support and apply a 1 k-ft counterclockwise moment.

Modify the Figure 1 as shown in Figure 2.

Refer Figure 2.

The slope at C due to the external loading is denoted by θCO.

The flexibility coefficient representing the slope at C due to unit value of redundant MC is fCC.

Slope of the Primary Beam:

Show the expression for the slope at the hinge support of a simply supported beam due to the point load P as follows:

θ=Pab6lEI(2lb)        (1)

Here, l is the length of the beam, b is the distance of the load from the roller support, E is the modulus of Elasticity, a is the distance from the hinge support, and I is the moment of inertia.

Calculate the slope of the primary beam due to the action of 60kN load alone as follows:

Substitute 10ft for b, 20ft for a, 30ft for l, and 60k for P in Equation (1).

θ1=60×20×106×30EI(2×3010)=3,333kft2EI

Show the expression for the slope at the hinge support of a simply supported beam due to the uniformly distributed load of intensity w as follows:

θ=wl324EI        (2)

Here, l is the length of the beam, E is the modulus of Elasticity, and I is the moment of inertia.

Substitute 3k/ft for w, 30ft for l in Equation (2).

θ2=3×30324EIθ2=3,375kft2EI

Calculate the slope of the primary beam as follows:

θCO=θ1+θ2θCO=3333kft2EI3375kft2EIθCO=6,708kft2EI        (3)

Calculate the value of fCC as follows:

Show the expression for the slope at the hinge support of a simply supported beam due to the unit moment M as follows:

fCC=Ml6EI        (4)

Substitute 1kft for M, 30ft for l in Equation (4).

fCC=1×303EIfCC=10kft2/kftEI        (5)

Show the compatibility Equation as follows:

θCO+fCCMC=0        (6)

Modify Equation (6) using Equation (3) and (5).

6708kft2EI+10kft2/kftEI×MC=0MC=6708kft2EI×EI10kft2/kftMC=670

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Chapter 13 Solutions

Structural Analysis
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Ch. 13 - 13.9 through 13.12 Determine the reactions and...Ch. 13 - Determine the reactions and draw the shear and...Ch. 13 - Determine the reactions and draw the shear and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - Determine the reactions and draw the shear and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.13 through 13.25 Determine the reactions and...Ch. 13 - 13.26 through 13.29 Determine the reactions and...Ch. 13 - 13.26 through 13.29 Determine the reactions and...Ch. 13 - 13.26 through 13.29 Determine the reactions and...Ch. 13 - 13.26 through 13.29 Determine the reactions and...Ch. 13 - 13.30 through 13.33 Solve Problems 13.9 through...Ch. 13 - 13.30 through 13.33 Solve Problems 13.9 through...Ch. 13 - 13.30 through 13.33 Solve Problems 13.9 through...Ch. 13 - 13.30 through 13.33 Solve Problems 13.9 through...Ch. 13 - 13.34 through 13.36 Determine the reactions and...Ch. 13 - 13.34 through 13.36 Determine the reactions and...Ch. 13 - 13.34 through 13.36 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.37 through 13.45 Determine the reactions and...Ch. 13 - 13.46 and 13.47 Determine the reactions and the...Ch. 13 - 13.46 and 13.47 Determine the reactions and the...Ch. 13 - Determine the reactions for the beam shown in Fig....Ch. 13 - Solve Problem 13.1 for the loading shown and a...Ch. 13 - Solve Problem 13.9 for the loading shown in Fig....Ch. 13 - Solve Problem 13.12 for the loading shown in Fig....Ch. 13 - Solve Problem 13.27 for the loading shown in Fig....Ch. 13 - Solve Problem 13.37 for the loading shown in Fig....Ch. 13 - Solve Problem 13.39 for the loading shown in Fig....Ch. 13 - Determine the reactions and the force in each...Ch. 13 - Determine the reactions and the force in each...Ch. 13 - Determine the reactions and the force in each...Ch. 13 - Solve Problem 13.15 by the method of least work....Ch. 13 - Solve Problem 13.16 by the method of least work....Ch. 13 - Solve Problem 13.35 by the method of least work....Ch. 13 - A beam is supported by a fixed support B and a...

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