Elastic-Plastic and Ultimate State of Perfect Plastic Bodies

Abstract

Although exact elastic solutions at torsion are known only for some cross-sections the ultimate state can be found for any problem because in this case we should consider only two equations for two unknowns (Fig. 2.8) – condition τ_e = τ_yi together with static law (2.48). It can be satisfied if we take

$$\tau_{\rm x} = \partial {\rm w}/\partial{\rm y},\tau_{\rm y} = -\partial {\rm w}/\partial{\rm x}$$

(4.1)

and from 2.52 we find

$$(\partial {\rm w}/\partial{\rm x})^2 + (\partial {\rm w}/\partial{\rm y})^2 = (\tau_{\rm yi})^2$$

or

$$/{\rm gradw}/ = \tau_{\rm yi} = {\rm constant}.$$

(4.2)

Here the gradient w(x, y) is the maximum slope of that function which can be interpreted as a sand heap with angle of repose equal to $\tan ^{-1} \tau_{\rm yi}.$. Expression (4.2) means that the distance between lines in which acts are constant and it allows to compute an elementary moment of torsion as (Fig. 4.1)

$$\rm dM_* = \tau_{yi}pdps = \tau_{yi}2dpdA $$

where A is an area under curve w = constant and p–perpendicular to it from a pole. Summarizing we find the ultimate moment as

$$\rm M_* = 2V.$$

(4.3)

Here V is volume of the heap.