deflection of short beams 5

[gio1]

Can anybody please suggest a formula for the deflection of cantilever beams with very short spans (L/D around 1) subject to concentrated end load?. This would contain a shear term (the highest contribution?) and a bending term (negligible?)

I struggle to find such a formula in literature

Thanks

 

[JAE]

Here's a formula that I use for deck diaphragms - contains both a flexural and shear component:

[Δ]_c = [Δ]_m + [Δ]_v

[Δ]_m = Ph³ / (3E_mI)

[Δ]_v = 1.2Ph / (AE_v)

P is the end of cantilever load
E_m is the modulus of elasticity (your typ. E)
E_v is the shear modulus (sometimes referred to as G)
h is the cantilever length
I is the moment of inertia of the member
A is the area

 

[Bagman2524]

What's wrong what Pl3/3EI. (P x l cubed / 3 x E x I).

 

[gio1]

Thanks for your answers

PI^3/3EI gives the bending deflection, but I think this formula is only valid for slender beams (L/D>10) (Saint Venant's assumptions)

As for the shear deflection I will use JAE's formula

 

[JAE]

If you notice, the flexural portion of [Δ] in the above equation is PL³/3EI, the same as what Bagman2524 presents. You just need to add the shear deflection for short conditions to be more accurate.

 

[gio1]

JAE

I did notice that Bagman's formula was the flexural term in yours. I think this formula is applicable only under the Saint Venant assumptions, and among them is slenderness of the beam (L/D>10)

I don't know how inaccurate this formula would be for short beams

 

[miecz]

The 1.2 Factor in the above formula for shear deflection applies to rectangular shapes. Article 7.10 of Roarkes Formulas for Stress and Strain shows different factors for round, hollow, or I shapes.

 

[prex]

All the above is correct, but quite theoretical. And gio1, the bending formula is not incorrect at short lengths, it simply doesn't account for local effects that may be more important than the pure bending deflection.
That formula is anyway the only one available. And consider that the ratio of bending deflection to shear deflection for a cantilever is of the order of (L/D)² so that for L/D[≈]1 the two contributions are of the same order of magnitude.
What I wonder on a more practical basis is why you could be interested in the deflection of a so rigid thing.
Also consider, again on a practical basis, that boundary conditions with their local effects (how and how much the beam is fixed, how is the load applied) may contribute more than the theoretical values.

prex

http://www.xcalcs.com

Online tools for structural design

 

[gio1]

Thanks for your feedback. I am trying to maximise the stiffness of a piston pin of given mass and length. Increasing the diameter results in lower bending but higher shear deflection (because of the reduction in cross section area). The detailed analysis will be done by FEA but I need a starting point close to the optimum.

 

[gio1]

Sorry, I take that back - shear deflection does not depend on diameter as cross section area remains constant

 

[miecz]

For a solid circular section, The shear factor is (10/9), so the shear deflection is (10/9)PL/AG.

 

[gio1]

thanks jmiec, do you also have the factor for hollow circular sections?

 

[miecz]

For thin walled hollow circular sections, the shear factor is 2.0.

 

[gio1]

In my case the section is definitely thick-walled...

 

[miecz]

Nothing is said about thick walled circular sections. Rourke references a report by Newlin and Thayer, "Deflection of Beams with Special Reference to Shear Deformation," Natl. Adv. Comm. Aeron., Rept. 180, 1924.