Deflection in pure bending beam
Deflection in pure bending beam
(OP)
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.






RE: Deflection in pure bending beam
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Deflection in pure bending beam
RE: Deflection in pure bending beam
Have a look at problem 9.5-9 at the link below:-
http://bjalmonte.files.wordpress.com/2012/09/chapt...
RE: Deflection in pure bending beam
The beam shown has constant moment over its full length combined with tension T applied at each end where T is numerically equal to P. As the beam deflects, the eccentricity decreases at every point along the beam except at the two ends. This reduces the deflection below that calculated for pure bending.
I don't have a closed form solution for the magnitude of the decrease, but it could be found by numerical integration over the span.
BA
RE: Deflection in pure bending beam
RE: Deflection in pure bending beam
Otherwise this was a classical problem in my first class session in structures. So basic, I don't think it is in a structures book.
RE: Deflection in pure bending beam
It is a second order effect which would not have a profound affect on deflection. A conservative estimate of deflection in this case would be P*e*L2/8EI.
BA
RE: Deflection in pure bending beam
Analysis and Design of arbitrary cross sections
Reinforcement design to all major codes
Moment Curvature analysis
http://www.engissol.com/cross-section-analysis-des...
RE: Deflection in pure bending beam
BA
RE: Deflection in pure bending beam
RE: Deflection in pure bending beam
Neglecting the tensile force in your beam, the bending moment is uniform over the span, namely P*e. The curvature at every point on the span is M/EI, so you have cylindrical bending.
Using the Conjugate Beam Method which is a particular adaptation of Moment Area Principles, the conjugate beam is an imaginary beam loaded with the M/EI diagram which is also the curvature diagram. The shear at any point on the conjugate beam is equal to the slope at the same point on the real beam. The bending moment at any point on the conjugate beam is equal to the deflection at the same point on the real beam.
M = P*e is constant over the span, so curvature is P*e/EI at all points in the span. The area under the curvature diagram is P*e*L/EI. The end reactions of the conjugate beam are equal to the end slopes of the real beam, namely P*e*L/2EI. The bending moment of the conjugate beam is maximum at midspan and has the value P*e*L2/8EI which is also the maximum deflection of the real beam.
BA
RE: Deflection in pure bending beam
I’d use Newmark’s numerical integration techniques on this problem, which I assume is really where you where going with your explanation.
RE: Deflection in pure bending beam
RE: Deflection in pure bending beam
is the cylinder a spring, an actuator ?? is the arm extending under hydraulic pressure ?
is the plate rigid ? are the endplates ???
looks like there's a pin joint at one end, and some moment capability at the other ??
lots of questions ...
Quando Omni Flunkus Moritati
RE: Deflection in pure bending beam
D = D0 / (1 - P/Pcr)
where D0 is the calculated deflection ignoring axial force and
Pcr is the Euler buckling compression for the member
to calculate the increased deflection caused by the axial force.
I believe that for small values of tension it is still valid to use this formula (but with a plus sign rather than a minus sign). Just don't ask me to define "small".
RE: Deflection in pure bending beam
If I wanted to include the effect of variable eccentricity due to beam deflection, I would probably have chosen Newmark's procedure as well, but if the initial eccentricity is as large as it appears on the sketch, the small variation in eccentricity could reasonably be neglected.
BA
RE: Deflection in pure bending beam
what's bending what ? Compression in the rod is pushing against a stub column each end of the beam. where are the reactions ? Anywhere you want. It doesn't matter as it is a self contained system which means there are no reactions. draw a free body diagram. Maybe another time.
is the cylinder a spring, an actuator ?? is the arm extending under hydraulic pressure ? Don't know and it doesn't matter.
is the plate rigid ? are the endplates ??? What endplates?
looks like there's a pin joint at one end, and some moment capability at the other ?? My interpretation is pin joint both ends.
The OP can correct me if I'm wrong.
BA
RE: Deflection in pure bending beam
The approximate equation mentioned by Denial (using a plus sign, of course) should be plenty accurate for most cases.
RE: Deflection in pure bending beam
If a tension T, numerically equal to P is applied at each end of the neutral axis of the deflected shape, a secondary deflection Δ2 is found to be 5*Δ1*TL2/48EI and is opposite in direction to Δ1.
Combining these results, Δ = Δ1 - Δ2.
So Δ = (P*eL2/8EI)*(1 - 5TL2/48EI)
This is the first iteration. I have over-corrected by using Δ1 to calculate Δ2 but another iteration is not necessary as Δ2 is very small.
BA
RE: Deflection in pure bending beam
BA
RE: Deflection in pure bending beam
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.
RE: Deflection in pure bending beam
or at least will have a much higher I than the RH slender strut, yes?
Quando Omni Flunkus Moritati
RE: Deflection in pure bending beam
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
RE: Deflection in pure bending beam
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.