ReyBEng minimal prompt, authored by Grok 18.8.26
6. Practical tables and worked examples
What Lewe supplied for the designer and how the numbers are used
Lewe does not stop at the general method. In the second half of the dissertation he works out complete numerical examples so that a designer can see exactly how the number rectangle is built and used.
He begins with a continuous beam that has four intermediate supports (five spans). The spans have different lengths and different moments of inertia.
He first calculates the four flexibility coefficients for each span,
then forms the two continued fractions that give the end coefficients a11 and ann,
records every intermediate ratio i and k, and fills the complete number rectangle by successive division.
Once the rectangle exists, he places unit loads (or uniform loads) in each span in turn and multiplies the appropriate load terms by the rows of the rectangle. The results are the support moments. From those moments he draws the field-moment diagrams, calculates the shears, and finally obtains the support reactions.
The same process is repeated for frames with rigid joints. Lewe supplies ready-made tables of moments for multi-storey frames in which every storey has the same stiffness ratio. The designer simply looks up the coefficient that belongs to the number of storeys and the position of the storey being examined, multiplies by the appropriate load factor, and obtains the beam and column moments directly.
In the appendix he goes one step further and tabulates the actual ordinates of every influence line (moments, shears and reactions) for the beams he has treated. These ordinates are already multiplied by the necessary constants, so the designer can read a number and multiply it by the real load to obtain the force or moment at once.
With the finished tables in front of him, an engineer in 1915 could design a continuous beam or a multi-leg frame without ever having to solve a fresh set of simultaneous equations. That was the practical goal of the whole dissertation.
ace-consultancy.uk 