An aluminum cantilever beam of length L = 0.65 m carries a distributed load, which includes its own weight, of intensity $q / 2 \text { at } A \text { and } q \text { at } B \text {. }$ . The beam cross section has a width of 50 mm and a height of 170 mm.Allowable bending stress is 95 MPa and allowable shear stress $12 \mathrm { MPa }$ The permissible value of load inten-Sity $q$ is approximately:

Question

Quizplus · Accepted Answer

First, we need to find the total load intensity (including the weight of the beam).
Total load = (q/2)x + q(L-x) + (W/L)x, where W is the weight of the beam per unit length.

The weight of the beam per unit length is equal to the density of aluminum (2700 kg/m^3) times the cross sectional area (50 mm x 170 mm = 0.0085 m^2), times the gravitational constant (9.81 m/s^2):
W = 2700 kg/m^3 x 0.0085 m^2 x 9.81 m/s^2 = 22.24 kN/m

Substituting W into the equation for total load and simplifying:
Total load = qx - (q/2)x + qL + 22.24x/L
Total load = (3q/2)x + qL + 22.24x/L

To find the maximum load intensity (q) that the beam can handle, we need to find the maximum bending moment and maximum shear force.

Maximum bending moment occurs at the fixed end (point A):
Mmax = (qL^2)/2 + 22.24L/2
Mmax = (3qL^2 + 44.48L)/4

Allowable bending stress for aluminum is 95 MPa. Using the bending stress formula:
Mmax/I = σmax/y = (bh^2/6)/y
Mmax = σmaxI/y = (bh^2/6)σmax/y
(3qL^2 + 44.48L)/4 = (95 x 10^6)(50 x 170^2)/(6 x 0.17)

Solving for q:
q = 129.68 kN/m

Maximum shear force occurs at point B:
Vmax = qL/2 + 22.24
Vmax = (3qL + 44.48)/4

Allowable shear stress for aluminum is 12 MPa. Using the shear stress formula:
Vmax/A = τmax = VQ/It
Vmax = τmaxIt/Q
(3qL + 44.48)/4 = (12 x 10^6)(50 x 170)/(2 x 170 x (50/2)^2)

Solving for q:
q = 130.12 kN/m

Since the lower value of q is the allowable one, the permissible value of load intensity q is approximately 130 kN/m. Therefore, the answer is C.

An Aluminum Cantilever Beam of Length L = 0 q/2 at A and q at B. q / 2 \text { at } A \text { and } q \text { at } B \text {. }q/2 at A and q at B.

An Aluminum Cantilever Beam of Length L = 0 $q / 2 \text { at } A \text { and } q \text { at } B \text {. }$