Statically indeterminate beam problem
Statically indeterminate beam problem
(OP)
I have a beam 85.4 inches long that is fixed at one end. There is a load of 600 lbs at the other end. It is supported by a pin support (0 defelection) at 6.8 inches from the fixed end. I need the reaction forces at the fixed end, at the support and the moment at the fixed end. I have done the calculations and gotten the following:
Reaction at fixed end: -10,404 lb
Reaction at support: 11,004 lb
Moment at fixed end: 23,587.2 inlbs counterclockwise
It's been a long time since I've done this and want to check my work.
Thanks,
Seth
Reaction at fixed end: -10,404 lb
Reaction at support: 11,004 lb
Moment at fixed end: 23,587.2 inlbs counterclockwise
It's been a long time since I've done this and want to check my work.
Thanks,
Seth





RE: Statically indeterminate beam problem
Actual loading would be highly dependent upon the vertical placement of that support at 6.8".
Assuming your numbers are correct, the maximum moment occurs at that support, not at the fixed end.
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
i'd try two different methods. i prefer unit force method and 3 moment equation.
i don't know how you calc'd the problem, but unit force method is easy to apply. count the pin support as redundant; solve the cantilever, determine the deflection at the pin support position, then apply a unit force on the cantilever at the pin support position, calculate the deflection (this is simply end deflection of a cantilever 6.8" long !); then the pin support reaction is the ratio of these deflections (how many unit forces do you have to apply to restore the statically determinate deflection to zero).
RE: Statically indeterminate beam problem
As far as needing the modulus of elasticity and the moment of inertia, I didn't think those were required to solve for the reaction forces and reaction moment at the fixed end. I would need those if I were to calculate the moment at some other point on the beam, the beam's deflection, or it's slope.
The method I used was to create the moment equation for the system using singularity functions. I did this by writing the load equation and integrating twice. I then integrated the moment equation twice and used my boundary conditions (zero deflection at x=0 and x=6.8, zero slope at x=0) to solve my constants of integration and one of the unknowns in terms of another. In all cases, EI ended up being multiplied by zero so did not factor into the results. I then used the equilibrium equations for force and moment combined with the deflection equation to solve the three unknowns. Was this the incorrect approach or did I not apply it correctly?
RE: Statically indeterminate beam problem
The support force is actually being used to represent the resultant of a reactionary load being applied on the beam by a triangular support gusset. I assumed the load from the gusset would be linearly applied to the beam with a triangular area (not sure how accurate that is, but didn't know how else to do it) and calculated it's position (6.8" from the fixed end) that way.
The methodolgy I used is listed in the post above. I'll take a look at your recommended technique.
RE: Statically indeterminate beam problem
i'm not 100% sure about your method (it's been too long since i heard "singularity equations", though maybe i'd recognise one if i saw it !) ... i can see the moment equation of the beam is two spans, the connection between them is slope continuity and a zero deflection, but i don't see immediately how you can express the moment due to the simple support. try the unit force approach i outlined above.
RE: Statically indeterminate beam problem
i quickly solved for the simply support load; i got 11,003,
i take it that you applied the load in the -ve direction
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
In developing my load equation using singularity functions, I treated the moment at the fixed end as a moment acting at zero: -M*<x-0>^-2. Integration brought me to -M*<x-0>^0 as the term representing that moment in my moment equation. Since x will always be greater than 0, the singularity goes to 1, leaving me with -M. I'm trying your method shortly.
It's been over 4 years since I graduated and I didn't have to use any of this stuff in my previous job. I'm the only one at this job that has any experience with it, so I don't really have any back-up. I'm trying to rmemeber and relearn enough from textbooks to get the results.
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
However, the span between your fixed end and roller support is short. So the answer to your problem appears to be highly sensitive to the cross section dimensions and elastic modulus of your beam, such that the above results might not even be in the correct ball park. Assuming your supports are truly rigid, you would need to specify cross section dimensions and tensile modulus of elasticity.
RE: Statically indeterminate beam problem
As to whether you need the E & I- I would probably go ahead and calculate a number for that deflection, so I would use it. Depends on what you need, I guess. As the formulas become more complex, it becomes easier to work with a specific deflection rather than a term. And then, I'm thinking in terms of checking allowable stresses along the beam, which assumes you have beam properties anyway.
Anyway, using your method, you should be able to check quite easily by looking up those same formuals, calculate deflection due to both loads, and see if it's zero.
RE: Statically indeterminate beam problem
The beam is actually a manipulator arm with 3 segments which I am treating as a single 'system'. The first two segments are 4X4 inch tube with a wall thickness of 3/16ths (hot rolled steel). My goal is to calculate the stress experienced by the first segment. I know there are more accurate methods to use in solving this problem but my experience with this sort of problem is limited.
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
4"x4"x0.19" square tube,
I = 4*0.19*1.905^2*2+0.19*3.625^3/12*2
= 7in4
bending stress = 51600*2/7 = 15ksi ("squat", and that's the technical term !)
but i think the issue is going to be the welds. at the end of the beam, assume the moment is carried by 24" welds, 4" apart. this means that each weld carries 51600/4 = 12900 lbs, or 3175 lbs/in (in tension). not being a "weld" guy, this looks like a lot
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
RE: Statically indeterminate beam problem
in my business i only need a safetey factor of 1.5, so 15ksi is no problem.
back to your problem, this is a simple linear bending anaylsis. You could try a plastic bending analysis, this'll increase the allowable alot.
a couple of design ideas ...
a) you could extend the top surface of the tube back, so that it covers the top of the vertical post. this would give you twice the weld area (compared with the end edge of the tube), and the welds would be in shear.
b) are you married to the weld concept ? given your forces, i'd have thought a fastener group would accomplish the end joint more effectively.
c) is weight an issue for you ? add lightening holes in the webs.
good luck
RE: Statically indeterminate beam problem