Calculating Stress Along a Cantilever Beam

[X1088LoD]

I am using a cantilever beam, guided on one end and fixed on the other, (example 1b from Roark's Formulas for Stress and Strain), I have included the page from the book as an attachment.

I want to determine an equation for the stress along the top of the beam.

The beam is 250 mm long, 3 mm high, 25 mm deep.
The modulus of elasticity E is 68927 N/mm^2
The moment of inertia I is 56.25 mm^4
a = 0

From what I have been able to determine
stress = (M*c)/I
where M is the bending moment, c is the distance from the neutral axis to the proper fibers (therefore this is half of the height, 1.5 mm), and I is the moment of inertia.
The bending moment is outlined in the attached file

Is this correct or am I taking the wrong approach? I appreciate any help.

 

[radou]

Well, you wrote down the relation between stress and the bending moment function, so try to work something out.

 

[ravenprp]

Your approach is correct, but there are a few additional considerations to keep in mind when calculating stress along a cantilever beam.

First, you will need to take into account the specific loading conditions on the beam. In this case, the beam is fixed on one end and guided on the other, so there will be a combination of bending and shear stresses acting on the beam. The equation you mentioned, stress = (M*c)/I, only takes into account the bending stress. To calculate the total stress, you will need to also consider the shear stress, which can be calculated using the equation VQ/It, where V is the shear force, Q is the first moment of area, and t is the thickness of the beam.

Secondly, the location of the neutral axis (where the stress is zero) will depend on the loading conditions as well. In this case, since the beam is fixed on one end, the neutral axis will be located at the fixed end. This means that the distance c from the neutral axis to the proper fibers will not be constant along the length of the beam. It will vary from 0 at the fixed end to 1.5 mm at the guided end.

Finally, it's important to note that the equation for stress you have mentioned is only valid in the elastic region of the beam. If the stress exceeds the yield strength of the material, the beam will experience plastic deformation and the equation will no longer be accurate.

In conclusion, your approach is correct, but make sure to take into account the specific loading conditions and the varying distance c along the beam. Also, keep in mind the limitations of the equation in the plastic region. Hope this helps!