Maximizing Beam Design: Balancing Bending and Shear Forces

[rock.freak667]

Homework Statement

This is not a homework problem but a general situation where I'd like to know which is better to do.

Say for example I have a cantilever beam of length L and some moment of inertia I, first moment of area Q and thickness t with a force P applied at the free end.

Homework Equations

σ = My/I ...(1)

τ = VQ/It...(2)

The Attempt at a Solution

So at the fixed end the maximum bending moment is PL and the reaction force is P.

Normally to get the bending stress to see if it less than the allowable stress I'd normally just use equation 1 and check if it is less than the allowable bending stress.

However due to the shearing force P at the fixed end, should I also now use

τ_xy=PQ/It

then combine to get the maximum shear stress:

$$\tau_{max} = \frac{1}{2} \sqrt{\sigma_{bend}^2 +4 \tau_{xy}^2}$$

or does the reaction force count as something else? I ask because normally I was taught to just assume pure bending but in a real life application the presence of the reaction force would mean an assumption of pure bending will be violated I believe. (The assumption that plane sections remain plane before and after bending).

 

[Chestermiller]

When you solve a problem using "Strength of Materials" approximations, you are not getting the exact solution that the Theory of Elasticity would give you. However, it is still an excellent approximation. For beam bending, Strength of Materials assumes pure bending locally along the beam. So estimating the state of stress as purely tensile at the outside of the bend is pretty accurate.

 

[rock.freak667]

Then when designing using hand calculations it is best to just assume pure bending then and ignore the shear force based on the maximum bending moment?

I doubt in practicality I would ever find a stress function to get the stresses.

 

[Chestermiller]

Then when designing using hand calculations it is best to just assume pure bending then and ignore the shear force based on the maximum bending moment?

Yes. Just design the thing with a reasonable factor of safety.

 

[rock.freak667]

Yes. Just design the thing with a reasonable factor of safety.

Nice. Thanks man!

 

[pongo38]

In the case of the cantilever, the max bending stress occurs at the extreme fibres, whereas the maximum shear stress is at the neutral axis, So, the question now becomes: what are the situations where bending and shear stresses should be combined? I expect there are many answers to this, but certainly an example arises in plastic design where the whole of the section is subject to (uniform) yield stress from bending, but the yield stress in the area of maximum shear is reduced by the presence of shear stress, Similarly for the presence of co-existent axial stress, but in that case, the combination is for normal stresses not both bending and shear. In soil mechanics there are procedures using Mohr's circle for combined shear and normal stresses to determine their overall effect.

 

[rock.freak667]

Ah I completely forgot to consider the axis for the shear stresses. The only time I've ever had to consider bending and shear is if you had a beam subjected to both torsion and bending in which I'd use the equation for Tmax above (which is the radius of the Mohr's Circle).

Though since it is best to design using maximum shear stress theory I should in fact not really consider bending stress and consider shear stress as in equation 2?

 

[pongo38]

The assumption that plane sections remain plane before and after bending is valid when there is no shear force present. When it is present, how important that is depends on the ratio of bending modulus to shear modulus. In most common construction materials, the error involved in ignoring this assumption (that sections remain plain...) is outweighed by other considerations and would not make a significant difference to any design decision based on it. I can see situations arising with new materials such as carbon fibre composites combined with the need for optimal very light design, as in aircraft design, where you might need more than one mathematical model to explore the solutions and the range of their errors.