Fundamentals of Structural Analysis
Fundamentals of Structural Analysis
5th Edition
ISBN: 9780073398006
Author: Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher: McGraw-Hill Education
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Chapter 9, Problem 1P
To determine

Find the reactions and sketch the shear and moment diagram.

Locate the point of maximum deflection for the beam.

Expert Solution & Answer
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Answer to Problem 1P

The reaction at A and C are RA=20.45k_ and RC=15.55k_.

The bending moment at A is MA=90.75kipsft(Anticlockwise)_.

Explanation of Solution

Given information:

EI is constant.

Calculation:

Consider the vertical reaction at C is denoted by RC.

Sketch the beam as shown in Figure 1.

Fundamentals of Structural Analysis, Chapter 9, Problem 1P , additional homework tip  1

Refer Figure 1.

The horizontal reaction at A is Ax=0.

The vertical reaction at C is the constraint in the structure.

Select the vertical reaction at C as the redundant.

Remove the redundant from the beam to get the released structure.

Calculate the gap (ΔCO) produced by the applied load on the released structure using the relation;

ΔCO=wL133EI+(wL122EI×L2)=36×933EI+(36×922EI×6)=8,748EI+8,748EI=17,496EI

Calculate the gap δCC produced by the unit load w=1kip applied on the released structure using the relation:

δCC=wL33EIδCC=(1)×1533EI=1,125EI

The deflection ΔC of the beam at the hinge support is zero.

Refer Figure 1.

Take the upward deflection as positive and downward deflection as negative.

Apply the compatibility Equation as follows:

ΔC=ΔCO+ΔCCΔC=ΔCO+δCC×RC0=17,496EI+1,125EIRCRC=17,496EI×EI1,125RC=15.55k

Thus, the reaction at C is RC=15.55k_.

Find the reaction RA at A as follows:

RA+RC=36RA+15.55=36RA=20.45k

Thus, the reaction at A is RA=20.45k_.

Find the moment at A as follows:

MA=0MA=RC×1536×9=15.55×1536×9=90.75kipft

Thus, the bending moment at A is MA=90.75kipft(Anticlockwise)_.

Find the moment at B as follows:

MB=0MB=RC×6=15.55×6=93.3kipsft

The bending moment Mx at the point of contra flexure is zero.

Consider the point of contra flexure is at a distance x from the fixed support A as follows:

MxMA=RA(x)0(90.72kipft)=(20.45kips)×xx=90.7220.45x=4.436ft

The bending moment MC at the free end C is zero.

Find the shear-force at A, B, and C as follows:

VA=RA=20.45k

VB,left=RA=20.45k

VB,right=RA36=20.4536=15.55k

VC=RA36+15.55=20.4536+15.55=15.55+15.55=0

Show the shear force, bending moment diagram as shown in Figure 2.

Fundamentals of Structural Analysis, Chapter 9, Problem 1P , additional homework tip  2

Refer Figure 2.

The point of maximum deflection occurs at 9 ft from the left support.

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