Deflection in pure bending beam 1

[tjbd32]

I am trying to calculate the deflection in an I-beam that is in pure bending. Basically I have a beam with a cylinder attached at on end that is pushing on a plate at the other end (kind of like a bow with the tension string). I am looking for an equation to calculate the deflection in the center of the beam.

 

[BAretired]

Hokie93, I don't have that book, but I am wondering why P_cr enters into it when the axial force in the beam is tension, not compression.

BA

 

[Denial]

Hokie93.[ ] Thanks for that result.[ ] I have seen it before, but could not find it when I went a-hunting yesterday.

BA.[ ] I derived my result assuming a sinusoidal shape, versus your parabolic shape.[ ] There is only a 3% difference between our simplified results.[ ] FWIW I suspect your assumed shape will give a slightly better approximation for axial tension, mine for axial compression.[ ] Regarding your question on why Pcr appears, the simplest answer is that it doesn't appear.[ ] What appears (in my derivation at least) is pi².EI/L², which we choose to call Pcr in some contexts.[ ] Think of it instead as some measure of the sensitivity of the column to axial force, a sensitivity that happens to be measured in units of force.

 

[rb1957]

the block on the LH side looks like an actuator ... that'll deflect in an almost linear fashion, no?

or at least will have a much higher I than the RH slender strut, yes?

Quando Omni Flunkus Moritati

 

[BAretired]

Denial, Okay, I think I understand your method now.

rb1957, The way the rod or actuator deflects is of no importance insofar as beam deformation is concerned. The cylinder and rod deliver an equal and opposite force to a stub column at each end of the beam. Each force is parallel and eccentric to the beam's neutral axis.

BA

 

[Hokie93]

BA - The definition of P_cr (pi²EI/L²) referenced by Denial matches that used in the Timoshenko equation I cited. It also matches that used in Chapter H1.2 of the 2005 AISC Specification for the capacity increase (a multiplier on C_b) permitted for doubly symmetric members subject to axial tension and flexure.

tjbd32 - You can also derive the maximum deflection value for this case by integrating d²y/dx² = M/EI twice (result: EIy = Mx²/2 + C1x + C2), solving for constants C1 and C2 (C1 = -ML/2 and C2 = 0), and evaluating the resulting equation at x = L/2. It is simple and direct.