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Draw the influence lines for the reaction moment at support A, the vertical reactions at supports A and B and the shear at the internal hinge C.
Explanation of Solution
Calculation:
Influence line for vertical reaction at support B.
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.
Apply 1 k load just left of C
Consider section BC.
Consider moment equilibrium at C.
Take moment at C from B.
Consider clockwise moment as positive and anticlockwise moment as negative.
Apply 1 k load just right of C
Consider section BC.
Consider moment equilibrium at C.
Take moment at C from B.
Consider clockwise moment as positive and anticlockwise moment as negative.
Thus, the equation of vertical support reaction at B as follows,
Find the influence line ordinate of
Substitute 0 for
Thus, the influence line ordinate of
Similarly calculate the influence line ordinate of
x (ft) | Points | Influence line ordinate of |
0 | B | 1 |
10 | C | 0 |
20 | D | 0 |
30 | E | 0 |
40 | F | 0 |
Sketch the influence line diagram for vertical support reaction at B using Table 1 as shown in Figure 2.
Influence line for vertical reaction at support A.
Apply a 1 k unit moving load at a distance of x from left end C.
Refer Figure 1.
Find the vertical support reaction
Apply 1 k load just left of E
Consider section EF.
Consider moment equilibrium at point E.
Consider clockwise moment as positive and anticlockwise moment as negative
Apply 1 k load just right of E
Consider section EF.
Consider moment equilibrium at point E.
Consider clockwise moment as positive and anticlockwise moment as negative
Thus, the equation of vertical support reaction at F as follows,
Apply a 1 k unit moving load at a distance of x from left end B.
Refer Figure 1.
Apply vertical equilibrium in the system.
Consider upward force as positive and downward force as negative.
Find the equation of vertical support reaction
Substitute
Find the equation of vertical support reaction
Substitute
Find the equation of vertical support reaction
Substitute
Thus, the equation of vertical support reaction at A as follows,
Find the influence line ordinate of
Substitute 40 ft for
Thus, the influence line ordinate of
Similarly calculate the influence line ordinate of
x (ft) | Points | Influence line ordinate of |
0 | B | 0 |
10 | C | 1 |
20 | D | 1 |
30 | E | 1 |
40 | F | 0 |
Sketch the influence line diagram for the vertical reaction at support A using Table 3 as shown in Figure 3.
Influence line for moment at support A.
Apply a 1 k unit moving load at a distance of x from left end B.
Refer Figure 1.
Apply 1 k load just left of C
Take moment at A from B.
Consider clockwise moment as positive and anticlockwise moment as negative.
Substitute
Apply 1 k load just right of C to just left of D
Take moment at A from B.
Consider clockwise moment as positive and anticlockwise moment as negative.
Substitute
Apply 1 k load just right of D to just left of E
Take moment at A from F.
Consider clockwise moment as positive and anticlockwise moment as negative.
Substitute
Apply 1 k load just right of E
Take moment at A from F.
Consider clockwise moment as positive and anticlockwise moment as negative.
Substitute
Thus, the equation of moment at A as follows,
Find the influence line ordinate of
Substitute 0 for
Thus, the influence line ordinate of
Similarly calculate the influence line ordinate of
x (ft) | Points | Influence line ordinate of |
0 | B | 0 |
10 | C | ‑10 |
20 | D | 0 |
30 | E | 10 |
40 | F | 0 |
Sketch the influence line diagram for the moment at support A using Table 3 as shown in Figure 4.
Influence line for shear at point C.
Find the equation of shear force at C of portion BC
Sketch the free body diagram of the section BC when 1 k load placed between BC as shown in Figure 5.
Refer Figure 5.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Find the equation of shear force at C of portion CF
Sketch the free body diagram of the section BC when 1 k load placed between CF as shown in Figure 6.
Refer Figure 5.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Thus, the equations of the influence line for
Find the influence line ordinate of
Substitute 10 m for
Thus, the influence line ordinate of
Find the shear force of
x (ft) | Points | Influence line ordinate of |
0 | B | 0 |
10 | ‑1 | |
20 | 0 | |
30 | E | 0 |
40 | F | 0 |
Draw the influence lines for the shear force at point C using Table 4 as shown in Figure 7.
Therefore, the influence lines for the moment at support A and the vertical reactions at supports A and B and the influence lines for the shear hinge C are drawn.
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