## What is Equilibrium Finite Element Analysis?

In the now archaic language of Sir Christopher Wren (1660s) the importance of equilibrium in design is stated thus:

"The design must be regulated by the art of statics, or invention of the centers of gravity, and the duly poising of all parts to equiponderate; without which, a fine design will fail and prove abortive. Hence I conclude, that all designs must, in the first place, be brought to this test, or rejected." Bill Addis, Building: 3000 years of Design Engineering & Construction, Phaidon, 2007

A more modern and succinct interpretation is provided by Ed Wilson:

"Equilibrium is Essential, Compatibility is Optional".

These statements beg the questions: Why is equilibrium so important and why might compatibility be ignored? The answer to these questions lie in the lower bound theorem of plasticity:

"The only reason why structural designers sleep soundly is the second [lower bound] theorem of plasticity theory. This theorem says that no matter how I designed my structures, they are safe because everything was in equilibrium, nowhere the stresses were too large and I used ductile components and joints."

Thus, if the engineer can invoke the lower bound theorem of plasticity he/she can be satisfied that the design has sufficient strength to take the imposed loads. This requires the engineer's numerical solution to be in equilibrium with the loads and with conventional conforming finite elements (CFE) this is not usually the case. This article, based on, presents a different approach which, through an equilibrium finite element (EFE) formulation, offers the engineer an approach which may be used to satisfy the lower bound plasticity theorem.

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Reference BM_Jan_17_9 Ramsay. A Maunder. E Moitinho de Almeida. J English Analyst Magazine Article 1st January 2017 Ramsay Maunder Associates University of Lisbon Global