Concept explainers
Draw the influence lines for the shear and bending moment at point F.
Explanation of Solution
Equation of influence line ordinate of support B:
Apply a 1 k unit moving load at a distance of x from left end D.
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 AC of the beam
Find the vertical support reaction
Take Moment at hinge D from left end A.
Consider clockwise moment as positive and anticlockwise moment as negative.
Consider the unit load at a variable position x to the right hinge D. (Place portion DG of the beam
Find the vertical support reaction
Take Moment at hinge D from left end A.
Consider clockwise moment as positive and anticlockwise moment as negative.
Thus, the equations of the influence line ordinate for
Equation of influence line ordinate of support E:
Refer Figure 1.
Find the equation of influence line ordinate for the vertical reaction
Apply moment equilibrium at G.
Consider clockwise moment as positive and anticlockwise moment as negative.
Find the influence line ordinate of vertical reaction
Substitute
Find the influence line ordinate of vertical reaction
Substitute
Thus, the equations of the influence line ordinate for
Equation of influence line ordinate of support G:
Refer Figure 1.
Find the equation of influence line ordinate for the vertical reaction
Consider the vertical forces equilibrium condition, take the upward force as positive
Find the influence line ordinate of vertical reaction
Substitute
Find the influence line ordinate of vertical reaction
Substitute
Thus, the equations of the influence line ordinate for
Influence line for shear at point F.
Find the equation of shear at F of portion AG
Sketch the free body diagram of the section AD as shown in Figure 2.
Refer Figure 2.
Apply equilibrium equation of forces.
Consider upward force as positive
Find the influence line ordinate of shear at F
Substitute
Find the influence line ordinate of shear at F
Substitute
Find the equation of shear at F of portion FG
Sketch the free body diagram of the section FG as shown in Figure 3.
Refer Figure 3.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute 0 for
Thus, the equations of the influence line ordinate for
Find the influence line ordinate of
x (ft) | Points | Influence line ordinate of |
0 | A | |
12 | 0 | |
24 | C | 0.25 |
36 | D | 0.5 |
48 | ||
60 | ||
60 | 0.5 | |
72 | G | 0 |
Sketch the influence line diagram for the shear at point F as shown in Figure 4.
Refer Figure 2.
Find the equation of bending moment at E of portion AD
Take moment at F.
Consider clockwise moment as positive and anticlockwise moment as negative.
Find the influence line ordinate of bending moment at F
Substitute
Find the influence line ordinate of bending moment at F
Substitute
Refer Figure 3.
Find the equation of bending moment at F of portion FG
Take moment at F.
Consider clockwise moment as negative and anticlockwise moment as positive.
Substitute
Thus, the equations of the influence line ordinate for
Find the influence line ordinate of
x (ft) | Points | Influence line ordinate of |
0 | A | 3 |
12 | 0 | |
24 | C | |
36 | D | |
48 | E | 0 |
60 | F | 6 |
72 | G | 0 |
Sketch the influence line diagram for the bending moment at point F as shown in Figure 5.
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