Static Collapse of Elastic Circular Arches

freedom. These shape functions are derived to be consistent with the approach present by Allman 3 for drilling freedoms. The stress e eld is derived within the natural coordinate system of the element by considering the equilibrium conditions for a dif- ferential beam segment. The element stress e eld is dee ned as beam resultant forces and moments over the cross section. The vector of six resultant forces and moments fae ¤ g are related to the unknown stress parameters fg by a matrix of polynomial approximations along the element ( P)I that is, fae ¤ g6 £ 1 D (P.»/)6 £6fg6 £1 .2/ where(P) is a matrix of polynomial approximation functions along the beam element length and » is the natural coordinate of the ele- ment. The present element has six unknown stress parameters that are used to describe a constant state of axial, torsional, and trans- verse shear stress resultants and a linear distribution of the bending moments along the beam element length. The assumed-stress e eld satise es the equilibrium conditions in a variational sense through the Hellinger -Reissner principle. The formulation ofthe beamis based on thefollowing variational statement: