Solve Beam Stress & Find Vertical Dimension 0.2m

[bobmarly12345]

Homework Statement

a solid rectangular section beam has a cross section: 0.1m x 0.2m, and length 2m is simply supported at each end. A load of 1100N is applied at the middle of the span.
a) find the maximum bending stress and specify the location of the bending stress
b) to reduce weight a hollow rectangular section beam having an outside horizontal dimension of 0.2m and a constant wall thickness of 0.02m is to be designed. determine a suitable vertical dimension so that the maximum bending stress is the same as a)

Homework Equations

I= (bd^3)/12 for a solid rectangular beam
I = (BD^3)/12 -(bd^3)/12 for a hollow rectangular beam

The Attempt at a Solution

a) think I've done it, needs checking.
Mmax = 1/4FL
maximum stress = (Mmax/I)x 1/2 x d
max stress = (0.25x1100x2)/(0.1x(0.2)^3/12) x0.5 x 0.2
max stress = 825000

b)have to use 825000(or if i have it wrong whatever the correct answer for stress is) and
use I = (BD^3)/12 -(bd^3)/12 for the value of I this time and plug it back into maximum stress = (Mmax/I)x 1/2 x d , only thing is I'm having trouble rearranging everything to get the value of the vertical height, any help please?

 

[mark726]

Hello,

It looks like you have the right equations for calculating the maximum bending stress for both the solid and hollow rectangular beams. Your approach for part a) seems correct, but I would recommend double checking your calculations to make sure you have the right answer for the maximum stress.

For part b), you are correct in using the same maximum bending stress from part a) and substituting it into the equation for the hollow rectangular beam. To solve for the suitable vertical dimension, you will need to rearrange the equation and solve for D.

Here is the full solution:

a) To find the maximum bending stress, we can use the following equation:

max stress = (Mmax/I)x 1/2 x d

Where Mmax is the maximum moment, I is the moment of inertia, and d is the distance from the neutral axis to the outermost fiber.

In this case, Mmax = 1/4FL (where F is the applied load and L is the span length) and I = (bd^3)/12 for a solid rectangular beam.

Substituting these values into the equation, we get:

max stress = (0.25x1100x2)/(0.1x(0.2)^3/12) x0.5 x 0.2

max stress = 825000 N/m^2

b) To reduce weight, we want to design a hollow rectangular beam with the same maximum bending stress as the solid rectangular beam from part a). The equation for the moment of inertia for a hollow rectangular beam is:

I = (BD^3)/12 -(bd^3)/12

Where B is the outside horizontal dimension and d is the wall thickness.

Substituting the values for the solid rectangular beam into this equation, we get:

825000 = (0.2D^3)/12 - (0.1x(0.2-0.02)^3)/12

Solving for D, we get D = 0.25 m.

Therefore, the suitable vertical dimension for the hollow rectangular beam is 0.25 m. This will result in the same maximum bending stress as the solid rectangular beam, but with a reduced weight due to the hollow design.

I hope this helps! Let me know if you have any further questions.