The discussion centers on the applicability of plane stress theory in scenarios where one end of a 2D model is fixed. It is established that under fixed boundary conditions, the z-direction stress component cannot be zero, which contradicts the requirements of plane stress. The participants conclude that for plane stress to be valid when one end is fixed, shear must be applied at that boundary. They also provide numerical examples comparing stress results from 3D, 2D plane stress, and analytical models, demonstrating minimal differences in stress values.
PREREQUISITESStructural engineers, finite element analysts, and students studying mechanics of materials will benefit from this discussion, particularly those interested in the implications of boundary conditions on stress analysis.
is plane stress never applicable here under this BC?Chestermiller said:Sure.
How can it be? It requires the z principal stress and one other principal stress to be equal.feynman1 said:is plane stress never applicable here under this BC?
i don't get you by 'equal'Chestermiller said:How can it be? It requires the z principal stress and one other principal stress to be equal.
If the stretching were in the x-direction (i.e., BC applied at x = 0), the strains in the y- and z-directions would be zero. That would mean that the stresses in the y and z directions would have to be equal.feynman1 said:i don't get you by 'equal'
'the strains in the y- and z-directions would be zero' for x=0 or for all x?Chestermiller said:If the stretching were in the x-direction (i.e., BC applied at x = 0), the strains in the y- and z-directions would be zero. That would mean that the stresses in the y and z directions would have to be equal.
At x = 0.feynman1 said:'the strains in the y- and z-directions would be zero' for x=0 or for all x?
at x=0, stress y=stress z not equal to 0, because strain x isn't 0. but plane stress requires stress z to be 0.Chestermiller said:At x = 0.
Rightfeynman1 said:at x=0, stress y=stress z not equal to 0, because strain x isn't 0. but plane stress requires stress z to be 0.
but you seemed to disagree with plane stress being not applicable here.Chestermiller said:Right
Sure. So...?feynman1 said:but you seemed to disagree with plane stress being not applicable here.
we have been discussing the plane stress theory. of course these results differ little as you go away from the fixed end.FEAnalyst said:Let's consider an example - cantilever beam 1.5 m long, rectangular section 0.1x0.2 m. Loaded with 16000 N at the free end. Here are the results (maximum stress):
- 3D model: 36,98 MPa
- 2D plane stress model: 36,88 MPa
- beam model: 36,1 MPa
- analytical: 36 MPa
The only problem is with stress singularity caused by the unrealistic assumption of fixed constraint itself (both in case of 3D and 2D model) but it can be ignored and the results read slightly away from the very end of the beam.
plane stress says stress z=0.Chestermiller said:Sure. So...?
As I said, it certainly applies to shear at the boundary you described.feynman1 said:plane stress says stress z=0.
so is plane stress applicable here?
but principal stress z is already not 0, violating plane stress requirements, why do we mention shear anyway?Chestermiller said:As I said, it certainly applies to shear at the boundary you described.
It is zero if we apply shear at your boundary.feynman1 said:but principal stress z is already not 0, violating plane stress requirements, why do we mention shear anyway?
to fix the end with shear is really a brilliant idea of rendering the shaky plane stress applicable. but on 'the principal in-plane stresses are not zero, even at the boundary.', aren't the principal in-plane stresses 0 as you wrote in your equation?Chestermiller said:If the boundary is at x = 0, and u = w = 0, and ##v=\gamma x##, then, at the boundary, u = v = w = 0 and $$\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=0$$ and $$\sigma_{xy}=G\gamma$$where G is the shear modulus. So we have plane strain throughout for this homogeneous deformation even though the principal in-plane stresses are not zero, even at the boundary.
Do you know how to determine the principal stresses for this specific state of stress? They are not zero. They are at two different angles to the boundary, offset from one another by 90 degrees.feynman1 said:to fix the end with shear is really a brilliant idea of rendering the shaky plane stress applicable. but on 'the principal in-plane stresses are not zero, even at the boundary.', aren't the principal in-plane stresses 0 as you wrote in your equation?
sorry i got it wrong and the principal stresses aren't 0. then we can come to the conclusion that for plane stress to be applicable when 1 end is fixed, that fixed end has to be sheared.Chestermiller said:Do you know how to determine the principal stresses for this specific state of stress? They are not zero. They are at two different angles to the boundary, offset from one another by 90 degrees.
In-plane shear parallel to the boundary was the only way I was able to think of.feynman1 said:sorry i got it wrong and the principal stresses aren't 0. then we can come to the conclusion that for plane stress to be applicable when 1 end is fixed, that fixed end has to be sheared.