Find the maximum positive and negative shears and the maximum positive and negative bending moments at point C.
Answer to Problem 6P
The maximum positive shear at point C is
The maximum negative shear at point C is
The maximum positive moment at point C is
The maximum negative moment at point C is
Explanation of Solution
Given Information:
The concentrated live load (P) is 150 kN.
The uniformly distributed live load
The uniformly distributed dead load
Calculation:
Apply a 1 kN unit moving load at a distance of x from left end A.
Sketch the free body diagram of beam as shown in Figure 1.
Refer Figure 1.
Find the equation of support reaction
Take moment about point D.
Consider moment equilibrium at point D.
Consider clockwise moment as positive and anticlockwise moment as negative.
Sum of moment at point D is zero.
Find the equation of support reaction
Apply vertical equilibrium equation of forces.
Consider upward force as positive
Substitute
Influence line for the shear at point C.
Apply 1 kN load at just left of C.
Find the equation of shear force at C of portion AB
Sketch the free body diagram of the section AC as shown in Figure 2.
Refer Figure 2.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Apply 1 kN load at just right of C.
Find the equation of shear force at C of portion CF
Sketch the free body diagram of the section AC as shown in Figure 3.
Refer Figure 3.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Thus, the equations of the influence line for
Find the value of influence line ordinate of shear force
x | Position | Influence line ordinate of |
0 | A | |
4 | B | |
8 | ||
8 | ||
16 | D | 0 |
19 | E | |
22 | F |
Draw the influence lines for the shear force at point C using Table 1 as shown in Figure 4.
Refer Figure 4.
The maximum positive ILD ordinate at point C is
The maximum negative ILD ordinate at point C is
Find the positive area
Here,
Substitute 4 m for
Find the negative area
Here,
Substitute 4 m for
Find the maximum positive shear at point C using the equation.
Substitute 150 kN for P,
Therefore, the maximum positive shear at point C is
Find the maximum negative shear at point C using the equation.
Substitute 150 kN for P,
Therefore, the maximum negative shear at point C is
Influence line for moment at point C.
Refer Figure 2.
Consider clockwise moment as positive and anticlockwise moment as negative.
Find the equation of moment at C of portion AC
Substitute
Refer Figure 3.
Consider clockwise moment as negative and anticlockwise moment as positive.
Find the equation of moment at C of portion CF
Substitute
Thus, the equations of the influence line for
Find the value of influence line ordinate of moment
x | Position | Influence line ordinate of |
0 | A | |
4 | B | 0 |
8 | ||
16 | D | 0 |
19 | E | |
22 | F |
Draw the influence lines for the moment at point C using Table 2 as shown in Figure 5.
Refer Figure 5.
The maximum positive ILD ordinate of moment at C is
The maximum negative ILD ordinate of moment at C is
Find the positive area
Here,
Substitute 12 m for
Find the negative area
Substitute 4 m for
Find the maximum positive moment at point C using the equation.
Substitute 150 kN for P,
Therefore, the maximum positive moment at point C is
Find the maximum negative moment at point C using the equation.
Substitute 150 kN for P,
Therefore, the maximum negative moment at point C is
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