Verifying Beam Reactions: A Chemical Engineer's Challenge

[danago]

Hey.
Im currently doing a course in numerical analysis, and while most of the focus is on the mathematics side of things, the assignments are set in an engineering context.

One of my assignments describes a beam that is "pin jointed" at both ends, is 1 meter long, and is subject to a 0.5 N/m uniform load upwards from one end to 0.3m into the beam, and then for the remaining 0.7m is subject to a 1 N/m uniform load downwards.

Here is the diagram I've drawn:

http://img16.imageshack.us/img16/1130/beamu.th.gif

I am required to calculate the reaction forces.

I actually study chemical engineering, so i haven't really dealt much with beam mechanics since my first year introductory courses. I think i have solved it, but i'd really appreciate it if somebody could verify whether my solution is correct or not.

http://img11.imageshack.us/img11/1289/30115449.th.gif

So both my reactions are vertical with the magnitudes as specified above.

Have i done that correctly?

Thanks in advance, any help is greatly appreciated.
Dan.

 

[danago]

Also, would i be correct in saying that the bending moment at any point along the beam is independent of the flexural rigidity of the beam?

 

[PhanthomJay]

Your reactions are correct. Your statement regarding the independence of beam rigidity and bending moment is also correct. Not bad for a ChemE!

 

[danago]

Your reactions are correct. Your statement regarding the independence of beam rigidity and bending moment is also correct. Not bad for a ChemE!

Haha thank you!

For that second statement about the bending moment being independent of the rigidity, will that apply to non-uniform beams also? Non-uniform in the sense that the product of modulus and moment of inertia (E*I) will change along the beam?

 

[PhanthomJay]

Haha thank you!

For that second statement about the bending moment being independent of the rigidity, will that apply to non-uniform beams also? Non-uniform in the sense that the product of modulus and moment of inertia (E*I) will change along the beam?

Have you considered changing your major? Anyway, I should clarify my statement. Moments about a point of a member in equilibrium always satisfy the the equilibrium equation "sum of moments about a point= 0", independent of EI. It is also true that internal bending moments are independent of EI, provided deflections are relatively small, but that's more than you (or I) may need to know. But generally speaking, EI rigidity of non uniform beams doesn't matter. For example, you might have a cantilevered beam that is tapered in cross section: fat with high EI at the fixed end, and skinny with small EI at the free end, with high moment at the fixed end and no moment at the free end , but independent of EI in any case. The bending stresses and deflections may vary as a function of E or I, but the moments vary with distance and force, not with rigidity. BTW, you don't have to resort to calculus to find moments when you have uniform distributed loading. The moments can be determined by taking the load distribution over the given length and finding its total resultant force, then apply that force at the centroid of the load distribution and determine the moment about a point by using "force times perpendicular distance to that point". For example, in your first term where you have 'integral of x dx' , it is simpler to use ' moment = .5(.3)* (.15) '. Change in moment is actually the integral of the shear times dx. The calculus may get you into trouble, which is why I seldom use it!

 

[danago]

Ah ok that makes sense. Thanks again for the reply.

The main reason i asked was because my lecturer gave us a MATLAB script that numerically computes the bending moment, shear force, deflection and slope of the beam and then plots them as function of distance along the beam.

For one particular scenario, we are required to alter the end conditions so that script models a beam fixed at both ends with a uniform rigidity EI=0.2 along the whole beam. We then need to change the script so that EI is 0.06 for the first 0.17m of the beam and also the last 0.17m, but still 0.2 for the interior 0.67m. In comparing the bending moment and deflection plots for each, i find that the bending moment plots are not the same. Going by what you have explained to me, should i consider this discrepancy as an error in my alteration of the script?

PS. I am quite content with ChemE thanks haha

 

[PhanthomJay]

When you alter the end conditions (the number of support points, and whether fixed or pinned), you dramatically change everything...moments, shear, deflection, slope...if that's what you mean by altering end conditions. This is a boundary condition, not a beam rigidity condition. But for beams with the same support end conditions, the moments and shears are independent of EI rigidity (witin limits of certain asumptions), but the deflections and slopes of the deflection do change as a function of the flexural rigidity. For this case, I don't know why you'd be getting significantly different moment and shear diagrams for differnt EI conditions.

 

[danago]

Hmm ok ill double check the script to see if i can find a reason why different EI properties give rise to different bending moment profiles.

Thanks again for your help. I really appreciate it