ahmad alhayek

Question

The boom of a crane has a rectangular box section with the dimensions shown. Determine the principal stresses in the boom at points A and B. Neglect the weight of the boom,

pulley, and cable.

Answer 1

Draw the free body diagram of the beam as follow:

Answer 2

Apply force equilibrium condition along axial direction.

Answer 3

Apply force equilibrium condition along vertical direction.

Answer 4

Apply moment equilibrium condition about node A.

Answer 5

Apply the equation for moment of inertia,

, for a hollow rectangle.

Here, the width of the outer rectangle is

, the height of the outer rectangle is

, the width of the inner rectangle is

, and the height of the inner rectangle is

,

Substitute 100 mm for

, 150 mm for

, 90 mm for

, and 140 mm for

.

Answer 6

Apply the equation for area,

, of the rectangle.

Substitute 100 mm for

, 150 mm for

, 90 mm for

, and 140 mm for

.

Answer 7

Apply the formula for static moment of area,

, at point

.

Substitute

for

,

for

,

for

, and

for

.

Apply the formula for shear stress at point A.

Substitute

for

,

for

,

for

, 90 mm for

, and 100 mm for

.

Answer 8

Apply the formula for the normal stress at point A.

Substitute

for

and

for

.

Answer 9

Apply the formula for the average normal stress at point A.

Substitute

for

.

Answer 10

Apply the formula for the radius of Mohr’s Circle for point A.

Substitute

for

and

for

.

Answer 11

Apply the formula for the principle stresses at point A.

.

Substitute

for

and

for

to calculate the maximum and the minimum principal stresses at point

.

Therefore, the stresses at point

are

and

.

Answer 12

Apply the formula stresses at point B by using Equation 9.10.

Substitute

for

,

for

,

for

,

for

and

for

.

Point

lines on the bottom surface of the boom, so there is no shear stress.

Thus, the stresses at point

are

and

.

ahmad alhayek

Problem

The boom of a crane has a rectangular box section with the dimensions shown. Determine the principal stresses in the boom at points A and B. Neglect the weight of the boom,

Step-by-step solution

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