Timoshenko - Cantilever with point load at a distance 'x' 1

[GB1000]

I'm after some experts to assist with Timoshenko. I'm trying to calculate the deflection at the tip due to shear deformation and loading. I've found many formulas showing deflection at tip when the load is at the end, however none where the point load is at 'x'

I know of Pa^2/6EI x (3L-a) which we generally use when designing say steel beams.

However this is for a composite material where we specifically need to be taking into account the shear deformation as well given the low modulus of elasticity. I know timoshenko is the answer but I have no idea.

My only solution was to use an equivalent UDL and use those formulas but wanted something accurate

Any assistance would be great.

 

[GB1000]

Update: Found a formula relating to Timoshenko. is this the answer or is this purely for the deflection at the tip

 

[3DDave]

Might want to lead off with "Composite Material Shear Deflection Analysis" because a formula for an isotropic material is unlikely to be suitable for an anisotropic one.

 

[GB1000]

@3DDave, you mean change the title? You are correct, Euler-Bernoulli is ineffective for that reason.

 

[271828]

Virtual Work, more specifically the principle of virtual forces, is an easy way to solve this problem. Any basic structural analysis textbook, Hibbeler for example, will have a chapter on that.

Edit: I was referring to a regular beam subject to shear. I'm not as sure what to do with composite. I'd have to see more info on the beam.

 

[Denial]

Do your sums for a cantilever whose length is only x, calculating both the deflection and slope at its end point.[ ] The unmodelled section of beam from x to the actual end will be straight and at a constant slope.[ ] Problem solved.

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[GregLocock]

Aye, Denial nailed this structures 101 problem. Rotation, and deflection, at the point of application of the load is a standard result.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[GB1000]

I’m not sure this is correct with rotation and deflection? How does this take into account shear deformation? This is a composite material where shear deformation will account for some of the deflection. I’m tendering to agree with 3DDave. This is a anisotropic material.

 

[rb1957]

would the unloaded tip (beyond the load) have any shear deflection ?

is weight a consideration ? (then the unloaded overhang wouldn't be "unloaded")

what material is the beam ??

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[ENGIRL]

I don't think you need any complicated formulas to find deflection. just use the transform section method to go from composite to one homogeneous beam then use static equilibrium to find deflection. Find the rotation at point x : bending moment at point x divide it by E.I (I is the moment of inertia of transformed beam). Use the rotation to find deflection.

 

[ENGIRL]

if the beam is long and thin, shear deflection can be ignored

 

[GB1000]

The material is a FRP e-glass. My understanding is with these products because of the low modulus shear deflection will occur

 

[rb1957]

then I think denial's idea works for you

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[SWComposites]

GB - its not just the material, but the beam cross section shape and the length that will affect the amount of shear deformation vs bending deformaton.

Post dimensions of your beam.

 

[GB1000]

Thanks all. I have come across a formula in a FRP composite book. This takes into account both shear deformation and bending deformation. I'd be interested to compare this answer with denial idea (when I get time).