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Structural Analysis

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

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Chapter
Section
BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 8, Problem 19P
Textbook Problem
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Draw the influence lines for the vertical reactions at supports A, B, and E and the shear at internal hinge D of the frame shown in Fig. P8.19.

Chapter 8, Problem 19P, Draw the influence lines for the vertical reactions at supports A, B, and E and the shear at

FIG. P8.19

To determine

Draw the influence lines for the vertical reactions at support A, B, and E.

Draw the influence line for the shear at internal hinge D.

Explanation of Solution

Calculation:

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

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

Refer Figure 1.

Consider the unit load at a variable position x to the left hinge D. (placed portion BD of the beam 0x<30ft).

Find the vertical support reaction (Ey) at E for portion BD.

Take Moment at hinge D from right end E.

Consider clockwise moment as negative and anticlockwise moment as positive.

ΣMDDE=0Ey(15)=0Ey=0

Consider the unit load at a variable position x to the right hinge D. (Place portion DE of the beam 30ft<x45ft).

Find the vertical support reaction (Ey) at E for portion DE.

Take Moment at hinge D from right end E.

Consider clockwise moment as negative and anticlockwise moment as positive.

ΣMDDE=0Ey(15)1(x30)=015Ey=x30Ey=x152

Thus, the equations of the influence line ordinate for Ey are,

Ey=0, 0x<30ft (1)

Ey=x152, 30ft<x45ft (2)

Find the influence line ordinate of Ey using Equation (1) and (2) and summarize the values in Table 1.

x (ft)PointsInfluence line ordinate of Ey (k/k)
0B0
15C0
30D0
45E1

Sketch the influence line diagram for vertical reaction at support E as shown in Figure 2.

Refer Figure 1.

Find the equation of influence line ordinate for the vertical reaction (Ay) using equilibrium equation.

Apply moment equilibrium at B.

Consider clockwise moment as negative and anticlockwise moment as positive.

ΣMB=0Ey(45)+Ay(15)1(x)=015Ay=x45EyAy=x153Ey (3)

Find the influence line ordinate of vertical reaction (Ay) for portion BD (0x30ft).

Substitute 0 for Ey in Equation (3).

Ay=x153(0)=x15 (4)

Find the influence line ordinate of vertical reaction (Ay) for portion DE (30ft<x45ft).

Substitute x152 for Ey in Equation (3).

Ay=x153(x152)=x15x5+6=x3x15+6=62x15 (5)

Find the influence line ordinate of Ay using Equation (4) and (5) and summarize the values in Table 2.

x (ft)PointsInfluence line ordinate of Ay (k/k)
0B0
15C1
30D2
45E0

Sketch the influence line diagram for vertical reaction at support A as shown in Figure 3.

Refer Figure 1.

Find the equation of influence line ordinate for the vertical reaction (By) using force equilibrium equation.

Consider the vertical forces equilibrium condition, take the upward force as positive (+) and downward force as negative ().

ΣFy=0By+Ay+Ey=1By=1AyEy (6)

Find the influence line ordinate of vertical reaction (By) for portion BD (0x<30ft)

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Chapter 8 Solutions

Structural Analysis
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