Beams

Statical indeterminate beams problem is difficult to solve because the amount of unknowns exceeds the number of available equations. From a time where handwriting was the only options, we have inherited the force- and the deformations method. Both methods give only information about deflection in specific note points. It was not sufficient, as I in 2001 was asked to decide the dimensions of the steel beams in the flat roof over the cinema shown below.

What I really want to do, was to solve the fourth order differential equation, the beam equation the whole way down to the deflection function. My goal was to end up with a single value share force-, moment- and deflection curves, running at the same independent variable x and executed in Mathcad.

Every point- and moment load create a discontinuity in the load function.The challenge was, to deal with an integration of non-continuous and non-differentiable functions. The mathematical tools which are available for the purpose are named Singularity Functions.

The subject is part of every engineering math. textbook. e.g. Advanced Engineering Mathematics by Erwin Kreyszig, or Elastic and inelastic Stress Analysis by Irving H.Shames and Francis. A. Conzzarelli, Appendix II, in the case, that somebody wants to live it out.

To control the deflection in the calculation was significant, as the roof was 16 m wide.

The task triggered a whole range of parametric written beam programs that Mathcad's Solve Function practiced to treat properly. Also when the determinant approach zero in the matrix equation!

You can see a small collection of the beam programs which serves me in nearly 15 years without competition by clicking on them in the drop-down menu. You can get access to my beam programs by request.

Here are two examples of tasks, which I could impossibly have been solved, without my parametric writing beam programs.

A concrete element factory with a problem.

At the crack picture, it is easy to see, that it is a deformations problem. The concrete form was 30 m long and very slack.

The weak points were identified and with the parametric written beam program, it was easy to change the distance at the cantilever end until the deformations crawled back to something acceptable. Just change one variable parameter out of many and keep an eye on the deformation curve.

A long lifting beam.

One 10 m long lifting beam for handling rebar. Not a challenge for an engineer with a parametric beam program!

You have to controle

the sling angle
the hook points positions for the sling
the deformation
and optimize the moment- and shear force and keep an eye on the column force between the hook point simultaneously to be able to balance them all at once.

Surprisingly difficult to do right and impossible without a finite element program or a parametric written beam program.

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