Question on general stiffness method

[awa5114]

I am trying to develop a simple program that does linear analysis of beams in bending and shear. I am trying to have as user input standard stuctural information such as nodes, elements, modulus of elasticity, area etc... I have a decent grasp on what needs to go into the program however one mystery remains:

How do structural analysis programs typically generate shear and moment diagrams for each element. Usually the elements are far and few between, and nodes only occur to mark a change in geometry or material. I undersstand the concept of equivalent nodal loads but this only gives you the internal forces/deflections at the nodes. What is a generic way to find the inter-nodal values within the element, without having to break it up into an unreasonable amount of sub elements. For example, a simply supported beam with a distributed load should in my opinion be comprised of only 1 element. How do these programs generate a complete parabolic moment and linear shear diagram for a case like this? I understand that one can use shape functions to generate the complete deflection diagram. Does something similar exist for shear and moment diagrams?

 

[IDS]

You know the shear force and bending moment at each end, and the applied loads along the beam, so starting from one end, you just integrate the applied pressure to get the shear force, and the shear force to get the bending moment (and then check the shear and moment at the other end are what they should be).

For a continuous uniform load it is straightforward. For partial and/or non-uniform loads Macaulay's method is convenient way to do the analysis in a way that can be simply programmed:
Beam actions and deflections by Macaulay’s Method

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

"Usually the elements are far and few between, and nodes only occur to mark a change in geometry or material." ... really ?? not in my FEMs. maybe if I was looking at a finite difference model.

"What is a generic way to find the inter-nodal values within the element," ... have you heard of the method of sections (for a beam). Your finite elements should be balanced, so you can determine the intermediate values.

You're right, a simply supported beam with a UDL can be one element. Your FEM will calculate the end reactions. Then you can use the method of sections to determine the internal loads (balancing the external reactions and the applied UDL).

Isn't this something done in 2nd or 3rd year ??

another day in paradise, or is paradise one day closer ?

 

[awa5114]

I know how to do the method of sections. THe issue here is that I'm trying to get my program to output shear and moment diagrams for the whole beam and not just the nodal forces/diplacements. As far as I know, there is no generic numerical way to do the method of sections for generic loading.

IDS, your comment is interesting, and might be a solution. Of course, analytically (and in general terms) we differentiate the displacement to get the rotation, then the rotation to get the curvature, then the curvature to get the internal moment then the internal moment to get the shear and then the internal shear to get the loading. Although a standard FEM program using the stiffness method will only output nodal displacements/forces, we can use hermite cubics (at the very least) to get intermediate displacements. This has me thinking of two possible solutions:

1) There may be some "hermite cubic" equivalent for nodal forces that there is for displacements to find intermediate internal shear and moment values, and thus generate the full shear and moment diagram
2) The only way to generate shear and moment diagrams (aside from breaking up a simply supported beam into 100 elements) is to go the route of solving the Euler-Bernoulli beam equation numerically via numerical integration. that is numerically integrate displacements to get rotations... etc until you get to moments and shears.

I'm not too optimistic about the first option though the second might work, albeit presenting a lot of additional complexity. Please let me know if I'm going in the right direction.

 

[IDS]

awa5114 - You are working in the wrong direction.

From a frame analysis you get the shear, moment, curvature, slope and deflection at one end. Knowing the applied load distribution it is straightforward to get the shear distribution along the beam, then integrate that to get the moment and curvature, integrate again to get the slope, and again to get the deflection. The actions slope and deflection at the other end should agree exactly with the frame analysis results, but the intermediate actions and deflections will be different if there are any intermediate applied loads.

You need to subdivide the beam for each change in the applied load, but no further.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/