Tension field theory, a new approach which shows its duality with inextensional theory

Abstract

Tension field theory describes the highly buckled (wrinkled) state of membranes or very thin plates whose boundaries are subjected to certain planar displacements well in excess of those necessary to initiate buckling. The present interest in tension field theory is because lightweight structures with stretched membrane components have potential applications in space. In addition, membrane structures which are pretensioned by internal pressure have application to lightweight portable bridges, protective coverings and various air cushion devices. The theory was conceived by Wagner (1929) [1] whose primary concern was to explain the behaviour of thin metal webs in beams and spars carrying a shear load well in excess of the initial buckling value. Such webs offer little resistance to the compressive strain component of the shear and the spar flanges must be held apart by struts to prevent collapse. In the simple case of rigid spar flanges and rigid perpendicular struts the stress field in the web in the highly buckled state is primarily that of tension at 45°. As the shear load increases so does the magnitude of this tensile stress field and, just as a taut string resists a kinking action, so too does this tensile stress field resist the out-of-plane displacements engendered by the buckling action of the compressive stresses; these opposing actions result in a decreasing wavelength along the compressive buckles which form at right angles to the tension field. Strictly speaking such problems are non-linear and their exact analysis presents formidable difficulties. However, within the framework of large-deflexion plate theory it may be shown that for large values of the ratio (applied shear strain)/(shear strain at initial buckling) the relation between applied loads and planar displacements and stresses again approaches linearity, and it is this asymptotic regime for which tension field theory is applicable. In this regime the flexural stresses and the planar compressive (post-buckling) stresses are negligible compared with the tensile stresses; the assumption that their magnitude is zero is physically equivalent to the assumption of zero flexural membrane stiffness, and it is this which characterises tension field theory: the membrane is envisaged as being finely wrinkled at right angles to the lines of tension. In general these “tension rays” are not necessarily parallel and the boundary conditions need not be those of pure shear, as in our previous example, but shear must play a dominant role in the boundary deformation because of the requirement that the principal strains at any point are of opposite sign. This requirement will be considered in greater detail later but it is clear that if the principal strains are both positive so too are the principal stresses, and if the principal strains are both negative the membrane is ineffective in carrying load.