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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 41P
Textbook Problem
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13.37 through 13.45 Determine the reactions and draw the shear and bending moment diagrams for the structures shown in Figs. P13.37–P13.45 using the method of consistent deformations.

Chapter 13, Problem 41P, 13.37 through 13.45 Determine the reactions and draw the shear and bending moment diagrams for the

FIG. P13.41

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:

The beam and the loading are symmetric since only analyze only the right half DG of the beam.

Sketch the free body diagram of the right half DG 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 5 support reactions and the number of equilibrium equations is 3.

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

Select the vertical reaction Fy and Gy at the supports F and G as redundant.

Release the support F and G and consider the notation of moments due to external load as MO.

Sketch the free body diagram of primary beam subjected to external loading without support F and G as shown in Figure 2.

Refer Figure 2.

Find the reactions at the supports without considering support F and G using equilibrium equations:

Summation of moments of all forces about D is equal to 0.

MD=0MDO=50(15)2(30)(302+30)=3,450k-ftMDO=3,450k-ft()

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

+Fx=0DyO502(30)=0DyO=110k()

For unit value of the unknown redundant Fy:

Consider the notation of moments due to external load as mF.

Sketch the free body diagram of primary beam Subjected to unit value of redundant Fy as shown in Figure 3.

Refer Figure 3.

Find the support reaction and moment at D when 1 k vertical load applied at C.

Summation of moments of all forces about A is equal to 0.

MD=0MDF=1(30)MDF=30k-ft()

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

+Fy=0DyF+1=0DyF=1k()

For unit value of the unknown redundant Gy:

Consider the notation of moments due to external load as mG.

Sketch the free body diagram of primary beam Subjected to unit value of redundant Gy as shown in Figure 4.

Refer Figure 4.

Find the support reaction and moment at D when 1 k vertical load applied at C.

Summation of moments of all forces about A is equal to 0.

MD=0MDG=1(60)MDF=60k-ft()

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

+Fy=0DyG+1=0DyG=1k()

Find the moment equation of the beam for different sections on the beam.

Consider a section XX in the portion DE of the primary structure at a distance of x from D.

Refer Figure 1.

Draw the primary structure with section XX as shown in Figure 5.

Refer Figure 5.

Portion AB.

Consider origin as A. (0x15ft).

Find the moment at section XX in the portion DE as shown in Figure 5.

MO=3,450+110(x)=3,450+110x

Similarly calculate the moment of the remaining section in the external loading and redundant loading structures.

Tabulate the moment equation of different segment of beam as in Table 1.

Segmentx-coordinateMO (k-ft)mFY (k-ft/k)mGY (k-ft/k)
OriginLimits (ft)
DEA0153,450+110x301(x)601(x)
EFE15303,450+110(x)50(x15)301(x)601(x)
FGG0302x22=x20x

The vertical deflection at point F due to external loading is ΔFO, vertical deflection at point G due to external loading is ΔGO,  and the flexibility coefficient representing the deflection at F due to unit value of redundant Fy is fFF, the redundant Gy is fGG, and the both redundant Fy and Gy is fFG=fGF.

Calculate the value of ΔFO using the equation as follows:

ΔFO=ΣMOmFYEIdx=1EI{015(3,450+110x)(30x)dx+1530[3,450+110(x)50(x15)][30x]dx+030(x2)(0)dx}=1EI[916,875168,750+0]=1,085,625EIk-ft3

Calculate the value of ΔGO using the equation as follows:

ΔGO=ΣMOmGYEIdx=1EI{015(3,450+110x)(60x)dx+1530[3,450+110(x)50(x15)][60x]dx+030(x2)(x)dx}=1EI[2,098,125776,2502,02,500]=3,076,875EIk-ft3

Calculate the value of fFF using the equation as follows:

fFF=ΣmFY2EIdx=1EI{015(30x)2dx+1530(30x)2dx+030(0)2dx}=1EI[7,875+1,125+0]=9,000EIk-ft3/k

Calculate the value of fGG using the equation as follows:

fGG=Σ

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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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