Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
11th Edition
ISBN: 9780073398242
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 10.1, Problem 10.53P
To determine

Find the force representing the reaction at A using the virtual work method.

Find the couple representing the reaction at A using the virtual work method.

Expert Solution & Answer
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Answer to Problem 10.53P

The force representing the reaction at A is 250N()_.

The couple representing the reaction at A is 450Nm(Counterclockwise)_.

Explanation of Solution

Find the force at A;

Show the free-body diagram of the continuous beam as in Figure 1.

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 10.1, Problem 10.53P , additional homework tip  1

Consider the member AB is horizontal.

The vertical displacement at point A is δyA.

Find the vertical displacement (δyA) at A with respect to vertical displacement (δyB) at B as follows;

δyA=δyB

Find the vertical displacement (δyC) at C with respect to vertical displacement (δyB) at B as follows;

δyB2.4=δyC1.5δyC=1.52.4δyB=58δyB

Find the vertical displacement (δyE) at E with respect to vertical displacement (δyB) at B as follows;

δyB2.4=δyE1.2δyE=1.22.4δyB=δyB2

Find the vertical displacement (δyF) at F with respect to vertical displacement (δyB) at B as follows;

δyE1.8=δyF1.5δyF=1.51.8δyE=56(δyB2)=512δyB

Use the concept of virtual work;

δU=0;AδyA+800δyC600δyF=0

Substitute δyB for δyA, 58δyB for δyC, and 512δyB for δyF.

AδyB+800(58δyB)600(512δyB)=0A+500250=0A=250N()

Therefore, the force representing the reaction at A is 250N()_.

Find the couple at A;

Show the free-body diagram of the continuous beam as in Figure 2.

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 10.1, Problem 10.53P , additional homework tip  2

Rotate the member AB in the beam through δθ at the point A.

Find the vertical reaction (δyB) at point B as follows;

δθ=δyB1.8δyB=1.8δθ

Find the vertical displacement (δyC) at C with respect to vertical displacement (δyB) at B as follows;

δyB2.4=δyC1.5δyC=1.52.4δyB=58(1.8δθ)=1.125δθ

Find the vertical displacement (δyE) at E with respect to vertical displacement (δyB) at B as follows;

δyB2.4=δyE1.2δyE=1.22.4δyB=12(1.8δθ)=0.9δθ

Find the vertical displacement (δyF) at F with respect to vertical displacement (δyB) at B as follows;

δyE1.8=δyF1.5δyF=1.51.8δyE=56(0.9δθ)=0.75δθ

Use the concept of virtual work;

δU=0;MAδθ+800δyC600δyF=0

Substitute 1.125δθ for δyC and 0.75δθ for δyF.

MAδθ+800(1.125δθ)600(0.75δθ)=0MA+900450=0MA=450Nm(Anticlockwise)

Therefore, the couple representing the reaction at A is 450Nm(Counterclockwise)_.

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Chapter 10 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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