# Draw the influence lines for the reaction moment at support A , the vertical reactions at supports A and F , and the shear and bending moment at point E of the frame shown in Fig. P8.37. FIG. P8.37

### Structural Analysis

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

Chapter
Section

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 8, Problem 37P
Textbook Problem
133 views

## Draw the influence lines for the reaction moment at support A, the vertical reactions at supports A and F, and the shear and bending moment at point E of the frame shown in Fig. P8.37.FIG. P8.37

To determine

Draw the influence lines for the reaction moment at support A, the vertical reactions at supports A and F and the shear and bending moment at point E.

### Explanation of Solution

Calculation:

Influence line for moment at support A.

Apply a 1 kN unit moving load at a distance of x from left end C.

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

Refer Figure 1.

Apply 1 kN load just left of C (0x5m).

Take moment at A from B.

Consider clockwise moment as positive and anticlockwise moment as negative.

MA=1(5x)=x5

Apply 1 kN load just right of C and just left of D (5mx10m).

Take moment at A from D.

MA=1(x5)=x5

Apply 1 kN load just right of D and just right of F (10mx20m).

Take moment at A from F.

MA=1(x5)Fy(15)=x515Fy

Thus, the equation of moment at A as follows,

MA=x5, (0x10m) (1)

MA=x515Fy, (10mx20m) (2)

Find the influence line ordinate of MA at B using Equation (1).

Substitute 0 for x in Equation (1).

MA=05=5kN-m

Find the influence line ordinate of MA at F using Equation (2).

The vertical reaction at F is 1 kN when 1 kN applied at F.

Substitute 20 m for x and 1 kN for Fy in Equation (2).

MA=20515(1)=1515=0

Thus, the influence line ordinate of MA at F is 0kN-m/kN.

Similarly calculate the influence line ordinate of MA at various points on the frame and summarize the values in Table 1.

 x (m) Points Influence line ordinate of MA(kN-m/kN) 0 B ‑5 5 C 0 10 D 5 20 F 0

Sketch the influence line diagram for the moment at support A using Table 1 as shown in Figure 2.

Influence line for vertical reaction at support F.

Apply a 1 kN unit moving load at a distance of x from left end C.

Refer Figure 1.

Find the vertical support reaction (Fy) at F using equilibrium equation:

Apply 1 kN load just left of D (0x10m).

Consider section DF.

Consider moment equilibrium at point D.

Consider clockwise moment as positive and anticlockwise moment as negative

ΣMD=0Fy(10)=0Fy=0

Apply 1 kN load just right of D (10mx20m).

Consider section DF.

Consider moment equilibrium at point D.

Consider clockwise moment as positive and anticlockwise moment as negative

ΣMD=0Fy(10)+1(x10)=010Fy=x10Fy=x101

Thus, the equation of vertical support reaction at F as follows,

Fy=0, (0x10m) (3)

Fy=x101, (10mx20m) (4)

Find the influence line ordinate of Fy at F using Equation (2).

Substitute 20 for x in Equation (1).

Fy=20101=21=1kN

Thus, the influence line ordinate of Fy at F is 1kN/kN.

Similarly calculate the influence line ordinate of Fy at various points on the frame and summarize the values in Table 2.

 x (m) Points Influence line ordinate of Fy(kN/kN) 0 B 0 5 C 0 10 D 0 15 E 0.5 20 F 1

Sketch the influence line diagram for the vertical reaction at support F using Table 2 as shown in Figure 3.

Influence line for vertical reaction at support A.

Apply a 1 kN unit moving load at a distance of x from left end C.

Refer Figure 1.

Apply vertical equilibrium in the system.

Consider upward force as positive and downward force as negative.

Ay+Fy1=0Ay=1Fy (5)

Find the equation of vertical support reaction (Ay) from B to C (0x10m) using Equation (5).

Substitute 0 for Fy in Equation (5).

Ay=1(0)=1kN

Find the equation of vertical support reaction (Ay) from C to F (10mx20m) using Equation (5).

Substitute x101 for Fy in Equation (5).

Ay=1(x101)=1x10+1=2x10

Thus, the equation of vertical support reaction at A as follows,

Ay=1kN, (0x10m) (6)

Ay=2x10, (10mx20m) (7)

Find the influence line ordinate of Ay at F using Equation (7).

Substitute 20 m for x in Equation (7).

Ay=22010=22=0

Thus, the influence line ordinate of Fy at F is 1kN/kN.

Similarly calculate the influence line ordinate of Ay at various points on the beam and summarize the values in Table 3.

 x (m) Points Influence line ordinate of Ay(kN/kN) 0 B 1 5 C 1 10 D 1 15 E 0.5 20 F 0

Sketch the influence line diagram for the vertical reaction at support A using Table 3 as shown in Figure 4.

Influence line for shear at point E.

Sketch the free body diagram of the section BD as shown in Figure 5.

Refer Figure 5.

Apply equilibrium equation of forces.

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

ΣFy=0

AySE1=0SE=Ay1 (8)

Find the equation of shear force at E of portion BD (0x10m)

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