# Bending shear stress distribution

• piygar
In summary, the cantilever beam model showed that the shear stress at the center is 6.25, but the shear stress across the width is varying. The reason for this is that the shear stress is not uniform across the width.
piygar
I modeled a simple cantilever rectangular beam in Patran of (L,W,H = 100, 6, 4) with Hexa8 elements and applied a vertical shear load (upwards) of some magnitude at the free end.
Ran in Nastran and plotted the vertical shear stress distribution on a cross-section in middle of length (in order to avoid any boundary condition effects). which is parabolic as expected being zero at top and bottom edges and maximum at center.
But shear stress distribution is not uniform across width. I was expecting rectangular bands parallel to horizontal edges but i get curved bands across widths. what could be the reason of that?
Have attached the fringe plot of vertical shear stress distribution.

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The parabolic shear stress formula VQ/It assumes a uniform shear stress distribution along the width. The computer shows otherwise. What sort of a range of shear stresses do you get along the width at say the neutral axis? How do the values compare to the hand calculation?

Shear stress at neutral axis along the width varies from 5.15 at the center to 8.07 at the either end of the width.
Have inserted fringe plot with spectrum.
I applied 100 unit shear load at the cantilever tip so according to formula max shear stress at center should be 1.5* 100/(6*4) = 6.25

piygar said:
Shear stress at neutral axis along the width varies from 5.15 at the center to 8.07 at the either end of the width.
Have inserted fringe plot with spectrum.
I applied 100 unit shear load at the cantilever tip so according to formula max shear stress at center should be 1.5* 100/(6*4) = 6.25

View attachment 98605
Thanks for the data. Funny things happen at edges and corners and when beams are not narrow. Anyway, the 6.25 seems like a good weighted average. With the addition of a good catch-all safety factor, of course! Unless the cantilever is of short length, shear doesn't add much in comparison to bending stresses, and I've always ignored it in long cantilever designs using steel.

Thanks for reply. Wanted to understand the corners/edges effect clearly.
Also plotted the horizontal shear stress (due to vertical shear load!) on the same cross-section and got this.

Horizontal shear is small but not numeric zero.
Discussed with a colleague and suspicion pointed to Poisson's ratio. Ran another simulation with Poisson ratio = zero and voila, Vertical shear stress is constant across width, and zero horizontal shear stress due to vertical shear load.
Since width of beam is not small,came to understanding that cross-section behaving like a plane strain condition instead of plane stress.
Though plane strain condition should produce secondary normal stress, not shear stress.

When you set Poisson's ratio equal to 0 and got uniform stress distribution across the width, how did the result compare to the VQ/It shear stress 6.25 value at the neutral axis?

It comes 6.12 at the neutral axis with zero Poisson ratio.
(Tried with higher order Hexa, Hexa20 elements and got same result, as i had shear locking effect in mind)

Thanks for info!

## 1. What is bending shear stress distribution?

Bending shear stress distribution is the variation of shear stress along the cross-section of a structural member subjected to bending. It is caused by the internal forces and moments acting on the member, and it is an important factor to consider in the design of structural elements.

## 2. What factors affect the bending shear stress distribution?

The bending shear stress distribution is affected by the magnitude and location of the applied loads, the geometry of the structural member, and the material properties of the member, such as its cross-sectional shape and shear modulus.

## 3. How is bending shear stress distribution calculated?

Bending shear stress distribution can be calculated using the principles of mechanics and the equations of equilibrium. The shear stress at any point along the cross-section can be determined by considering the internal forces and moments acting on that point.

## 4. Why is it important to consider bending shear stress distribution in structural design?

Bending shear stress distribution is important in structural design because it affects the strength and stability of a structural member. If the shear stresses exceed the strength of the material, it can lead to failure or deformation of the member. Therefore, it is necessary to consider bending shear stress distribution to ensure the safety and durability of the structure.

## 5. How can bending shear stress distribution be optimized in structural design?

Bending shear stress distribution can be optimized by choosing appropriate cross-sectional shapes and dimensions, as well as by redistributing the loads and moments to minimize the shear stresses. Advanced design techniques, such as finite element analysis, can also be used to optimize the bending shear stress distribution in complex structural systems.

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