6. Stiffness matrix for a beam element - part 1

Direct Stiffness Method - Beam and Frame Analysis
1

Questions and discussion for this lecture live here. Fire away by hitting Reply below :fire:

2

From what I understand, shear deformations are neglected in this course?

3

Hi,
Yes indeed - that’s correct.
Seán

4

Thanks for your answer. Do you cover this in any of your other courses? Or will it be added in the future?

5

No I don’t have anything on this and I’m afraid I don’t have any immediate plans to add anything - the pipeline of production is already pretty full for the next 8-12 months. Just not enough time to fit everyone’s requests in unfortunately.
S

6

In minute 5:39, why did you choose different polynomials for displacements and rotations?. For the first you used grade 3 polynomial, and for the second one grade 2 polynomial ?

7

When you say the interpolation function is for linear displacements, what do you mean?. If th equation is grade 3 polynomial how could the displacemnts be linear?

8

Hi @edwinhenaove, the fact that the rotations are a lower order polynomial is simply a result of the fact that the rotations are obtained as the first derivative of the displacements. By definition, the rotation at a point is the slope of a tangent to the curve passing through that point - we obtain this slope as the first derivative, which results in a lower order polynomial.

9

When I used the term linear, I meant translational, rather than rotational displacements.

10

Hi Sean - at around 11:00 for the stiffness matrix for beam element part 1 video you have the polynomial equation for u_y expressed in vector multiplication form. Why is it the x vector goes first then the A vector? Once the A coefficients are derived and A is expressed as a matrix the order of vector x and matrix A is important. Why choose it this way as opposed to A vector first? My intuition would be that the coefficients A would be expressed first in the product of vectors defining the polynomial. Thanks.

11

Hey @Bryan_Horton

The x-vector is written first mainly because the coefficient vector is being treated as an unknown column vector, which is a fairly typical convention in finite element formulations.

Once the boundary conditions are applied, those coefficients are expressed in terms of the element’s nodal displacement and rotation degrees of freedom, which are also written as a column vector.

So, writing the polynomial this way means the displacement field naturally becomes…

[shape functions] x [nodal displacements]

…matching the usual finite element form.

It is not that the coefficients couldn’t be written first (through working through it this way yourself); that would also work if the whole formulation were transposed. The chosen order is mostly a matter of compatibility with the standard FEM convention.

Hope that helps,
S