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Solve Problem 13.12 for the loading shown in Fig. P13.12 and the support settlements of 10 mm at A , 65 mm at B , and 40 mm at C . 13.9 through 13.12 Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P13.9–P13.12 using the method of consistent deformations. Select the reaction at the interior support to be the redundant.

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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 51P
Textbook Problem
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Solve Problem 13.12 for the loading shown in Fig. P13.12 and the support settlements of 10 mm at A, 65 mm at B, and 40 mm at C.

13.9 through 13.12 Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P13.9–P13.12 using the method of consistent deformations. Select the reaction at the interior support to be the redundant.

Chapter 13, Problem 51P, Solve Problem 13.12 for the loading shown in Fig. P13.12 and the support settlements of 10 mm at A,

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:

The settlement at support A (ΔA) is 10 mm.

The settlement at support B (ΔB) is 65 mm.

The settlement at support C (ΔC) is 40 mm.

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 A are denoted by Ax and Ay.

Consider the vertical reaction at B and C is denoted by By and Cy.

The reactions acting in the beam is 4.

The number of Equilibrium reaction is 3.

The degree of indeterminacy of the beam is 1.

Take the vertical reaction at B as the redundant.

Modify the Figure 1 as shown in Figure 2.

Refer Figure 2.

Consider a section at a distance x from both the supports A and C.

The deflection at B due to the external loading is denoted by ΔBO.

The flexibility coefficient representing the deflection at B due to unit value of redundant By is fBB.

Sketch the settlement at supports as shown in Figure 3.

Refer to Figure 3.

The settlement at B ΔB=6520=45mm

Primary beam subjected to unit value of redundantBy.

Calculate the value of reaction at C as follows:

MA=0Cy×30+1×10=0Cy=13

Calculate the value of reaction at A as follows:

V=0Ay+Cy=1Ay+13=1Ay=23

Calculate the value of deflection fBB as follows:

fBB=1EI010(23x)2dx+12020(13x)2dx=1EI01049x2dx+1202019x2dx=1EI[(49×x33)010+(12×19×x33)020]=1EI[(49×1033)+(12×19×2033)]

  =1EI[148.1481+148.1481]=296.3kNm3/kNEI        (1)

Substitute 200GPa for E and 500×106mm4 for I in Equation (1).

fBB=296.3kNm3/kN200GPa×106kN/m21GPa×500×106mm4×(1m1,000mm)4=296.3100,000=0.00296m/kN×1,000mm1m=2.96mm/kN

Primary beam subjected to external loading.

Calculate the value of reaction at A and C as follows:

Ay=Cy=25×302Ay=Cy=375kN

Calculate the value of deflection ΔBO as follows:

ΔBO=1EI[010(375x25x22)(23x)dx+12020(375x25x22)(13x)dx]=1EI[010(250x2+50x36)dx+12020(125x2+25x36)dx]=1EI[(250×x33+50x424)010+12(125x33+25x424)020]=1EI[(250×1033+50×10424)+12(125×2033+25×20424)]

  =1EI[(250×1033+50×10424)+12(125×2033+25×20424)]=1EI[62,50083,333

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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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