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.

 

[paddingtongreen]

We need a better description. I don't understand the cylinder bit but is this force couple along one flange of the beam? that sounds like compression as well as bending. A drawing would help.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[tjbd32]

Something like this.

 

[desertfox]

Hi

Have a look at problem 9.5-9 at the link below:-

http://bjalmonte.files.wordpress.com/2012/09/chapter-9-mecmat-gere7chambauer.pdf

 

[BAretired]

First of all, the beam is not in pure bending. If it were, the solution is very easy. A beam with constant moment, M over its entire span has a deflection of ML²/8EI at midspan. M = P*e where P is the compressive force between cylinder and end plate and e is the eccentricity of P from the neutral axis of the 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

 

[XR250]

@BA - Isn't that a second order effect that would not have a significant affect on the deflection?

 

[oldestguy]

If you are making a wood splitter, pay attention to those parts sitting on the beam ends. In the one I built, those were the critical places as to possible failure.

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.

 

[BAretired]

@ExcelEngineering,
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

 

[johnbridge231]

Deflection value is related to the beam theory you are considering. Euler theory neglects shear deformation effects, which may be important in case of small-deep beams, where as Timoshenko theory considers such effects.

Analysis and Design of arbitrary cross sections
Reinforcement design to all major codes
Moment Curvature analysis

http://www.engissol.com/cross-section-analysis-design.html

 

[BAretired]

For the beam in this thread, shear deformation is not a consideration as shear is zero over the entire span.

BA

 

[tjbd32]

How do you get to pel^2/8ei from the strain energy of bending equations?

 

[BAretired]

tjbd32,

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

 

[dhengr]

BA:
I’d use Newmark’s numerical integration techniques on this problem, which I assume is really where you where going with your explanation.

 

[Denial]

Once you know the value of the constant curvature (M/EI by definition) and the secant length (L), the deflection can very easily be deduced from the geometric properties of the circle (plus an assumption that the deflection is small relative to the radius of the circle).

 

[rb1957]

what's bending what ? where are the reactions ? draw a free body diagram.

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

 

[Denial]

As commented by others above, the presence of the axial tension (T) reduces the deflection, albeit probably only by a small amount.[ ] Were the axial force a compression (P), one would quite happily use the simple moment magnifier formula
D = D₀ / (1 - P/P_cr)
where D₀ is the calculated deflection ignoring axial force and
P_cr 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".

 

[BAretired]

dhengr,
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

 

[BAretired]

rb1957 writes, BAretired responds:
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

 

[Hokie93]

A closed form solution to this problem can be found in "Strength of Materials" by Timoshenko. In my second edition, the problem is discussed in Chapter 1 of Volume 2. The muliplier on the deflection due to the bending moments (M_oL²/8EI) is: [COSH(u) - 1]/[1/2*u²*COSH(u)]. The variable u is defined as pi/2*(P/P_cr)^0.5. If P/P_cr = 0.10, the deflection is reduced by approximately 9%. If P/P_cr = 0.50, the deflection is reduced by approximately 34%.

The approximate equation mentioned by Denial (using a plus sign, of course) should be plenty accurate for most cases.

 

[BAretired]

If the beam has an equal and opposite moment M applied at each end, the deflection Δ₁ at midspan is M*L²/8EI and the deflection diagram is parabolic in shape. In this problem, M = P*e.

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