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(Update a new question) Need Help! Bending beam under follower axial force
New question!
I want to explore the same problem in two-dimensional structure (e.g. thin plate).
According to Love simplification, the cross section rotates but the shear
deflection is neglected. This means the edge cross section is not
vertical actually. Can I still assume the shear force on the edge cross
section is always vertical? e.g. free boundary condition on a circular plate under a concentrated in-plane edge force.
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Hi everybody,
I have some questions about the cantilever beam bending boundary condition for follower and conservative forces.
Ff: follower force (its direction is always tangential to the deformed axis)
Fc: conservative force (its direction remains horizontal)
V: shear force at the end section
According to previous reference
Equation:
EI*d4y(x,t)/d x4 = m*d2y(x,t)/d t2
Boundary Condition
For fixed end at x=0
y = dy/dx =0
For free end at x=L
EI* d2y/d x2=0
EI* d3y/d x3=0 under follower force
or
EI* d3y/d x3+ Fc*dy/dx =0 under conservative force
(dy/dx at x=L is the angle between the follower force and the conservative force)
I cannot understand why the conservative force affects the fourth boundary condition.
First, I want to ask which direction of the shear force V is: vertical (along y axis) or parallel to beam end section?
Second, I think the whole problem is in vertical dimension (y axis). The follower force should have a component along y axis but the conservative doesn’t have, so why does the conservative force appear in the shear force boundary condition?
Any idea and suggestion are welcome. Appreciate any help.
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Take a look at Timoshenko's book
You should take a look at Timoshenko and Gere's book 'Theory of elastic stability'. It talks about relevant things at the very beginning: Sec 1.2.
For the first question,
For the first question, though the beam has bending and the surface of the right end is not in a real vertical direction any more, but for elastic beam and small deformation, the shear force is always assumed in the vertical direction. Of course, for large deformation, such a statement doesn't stand.
For the second question, the boundary conditions at the right end can be obtained from the force balance at a finite small element at the end, thus the sum of forces applied on the element is zero respectively in the horizontal and vertical directions, and the sum of the moments is zero. Therefore, you need to double check those boundary conditions at the right end.
Hi, Thanks! Yi, I want to
Hi, Thanks! Yi,
I want to explore the same problem in two-dimensional structure (e.g. thin plate).
According
to Love simplification, the cross section rotates but the shear
deflection is neglected. This means the edge cross section is not
vertical actually. Do I still assume the shear force on the edge cross
section is always vertical: e.g. free boundary condition on a circular plate under a concentrated in-plane edge force.
Many thanks to both of you.
Many thanks to both of you. I have figured out.