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Structural Analysis

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

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Section
BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 13, Problem 11P
Textbook Problem
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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 11P, 13.9 through 13.12 Determine the reactions and draw the shear and bending moment diagrams for the

FIG. P13.11, P13.32

To determine

Calculate the support reactions for the given structure using method of consistent deformation.

Sketch the shear and bending moment diagrams for the given structure.

Explanation of Solution

Given information:

The structure is given in the Figure.

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 positive when moment taken from left and the counter clockwise moment as positive when moment taken from right.
  • Consider the positive sign indicates the clockwise moment the negative sign indicates the counterclockwise moment.

Calculation:

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

Refer Figure 1.

Find the degree of indeterminacy of the structure:

Degree of determinacy of the beam is equal to the number of unknown reactions minus the number of equilibrium equations.

The beam is supported by 4 support reactions (r) and the number of equilibrium equations (e) is 3.

i=re=43=1

Therefore, the degree of indeterminacy of the beam is i=2.

Select the vertical reaction Cy at the supports C as redundant.

Release the support C and consider the downward deflection due to external load at C as ΔCO

Sketch the loading diagram with deflected shape as shown in Figure 2.

Refer Figure 2.

Find the vertical reaction at A and C.

The loading pattern of the beam is symmetric.

AyO=EyO=50k

Find the moment at A. (hinged support):

MAO=0

The moment of inertia of portion AB is doubled. The moment of the portion AC is the half of the moment of the portion CE. Divide the moment by 2 in the portion AC.

Find the moment at B from A. (I=2I).

MBO=(AyO×8)2

Substitute 50 k for AyO.

MBO=(50×8)2=200k-ft

Find the moment at C from A. (I=2I).

MCO=(AyO×16)50(8)2

Substitute 50 k for AyO.

MCO=(50×16)50(8)2=200k-ft

Find the moment at E. (roller support):

MEO=0

Find the moment at D from E.

MDO=EyO(8)

Substitute 50 k for EyO.

MDO=50(8)=400k-ft

Find the moment at C from E.

MCO=EyO(16)50(8)

Substitute 50 k for EyO.

MCO=50(16)50(8)=400k-ft

Sketch the MEI diagram using moment values as shown in Figure 3.

Refer Figure 3.

Find the vertical reaction at A in the MEI diagram.

Summation of moment at E is zero.

[AyEI(16)+12(8)(200EI)(83+24)+(8)(200EI)(82+16)+(8)(400EI)(82+8)+12(8)(400EI)(23×8)]=032AyEI+1EI(21,333.33+32,000+38,400+8,533.33)=032AyEI=1,00,266.66EIAyEI=3,133.33EI

Find the downward deflection at C.

The moment at C in the MEI diagram is the equal to the downward deflection at point C.

ΔCO=AyEI(16)+12(8)(200EI)(83+8)+(8)(200EI)(82)=(3,133.33EI)(16)+8,533.33EI+6,400EI=35,200EI

Find the upward deflection at B.

Remove the external load and applied unit upward load at C

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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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