   # For the cantilever beam of Problem 8.5, determine the maximum upward vertical reaction and the maximum counterclockwise reaction moment at support A due to a concentrated live load of 25 k, a uniformly distributed live load of 2 k/ft, and a uniformly distributed dead load of 0.5 k/ft. FIG. P8.5, P8.58

#### Solutions

Chapter
Section
Chapter 9, Problem 5P
Textbook Problem
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## For the cantilever beam of Problem 8.5, determine the maximum upward vertical reaction and the maximum counterclockwise reaction moment at support A due to a concentrated live load of 25 k, a uniformly distributed live load of 2 k/ft, and a uniformly distributed dead load of 0.5 k/ft. FIG. P8.5, P8.58

To determine

Find the maximum upward vertical reaction and the maximum counterclockwise reaction moment at support A.

### Explanation of Solution

Given Information:

The concentrated live load (P) is 25 k.

The uniformly distributed live load (wL) is 2 k/ft.

Calculation:

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

Apply a 1 k unit moving load at a distance of x from left end A.

Refer Figure 1.

Find the equation of support reaction (Ay) at A using equilibrium equation:

Consider upward force as positive (+) and downward force as negative ().

ΣFy=0Ay1=0Ay=1k

The value of influence line ordinate of reaction is 1 k/k applied throughout the length of the beam.

Draw the influence line diagram for the vertical reaction at support A as shown in Figure 2.

Refer Figure 1.

The maximum influence line ordinate due to upward reaction at A is 1 k/k.

Find the area of the influence line diagram (A) for the vertical reaction at support A.

A=L(1)

Here, L is the length of the beam.

Substitute 36 ft for L.

A=(36)(1)A=36ft

Find the maximum upward vertical reaction at support A using the equation.

MaximumupwardAy=[P(maximuminfluencelineordinate)+wL(A)+wD(A)]

Substitute 25 k for P, 1 k/k for maximuminfluencelineordinate, 2 k/ft for wL, 36 ft for A, and 0.5 k/ft for wD.

MaximumupwardAy=25(1)+2(36)+0.5(36)=115k()

Therefore, the maximum upward vertical reaction at support A is 115k_.

Find the equation of reaction moment (MA) at support A.

Consider moment equilibrium at point A.

Consider clockwise moment as negative and anticlockwise moment as positive.

Sum of moment at point A is zero.

ΣMA=0MA1(x)=0MA=x        (1)

Consider Equation (1)

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