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G. Hofstetter, H.A. Mang:
"Collapse Load Analysis of PC Surface Structures with Unbonded Post-Tensioned Tendons by the Finite Element Method";
Finite Elements in Analysis and Design, 5 (1989), 141 - 165.

The middle surface of curved surface structures (shells) and the axes of the tendons embedded between the two surfaces of a shell are assumed to be analytically describable. Essential features of the mathematical formulation are the exact consideration of the geometric form of the middle surface of a given shell by means of thin-shell finite elements and of the axes of the tendons. Prestress losses occuring during the tensioning operation as well as changes of the prestress forces resulting from loads after prestressing are taken into account. For prestressed concrete (PC) surface strucutres (panels, slabs and shells) with unbonded tendons, determination of these changes of forces is difficult if friction between the tendon and the duct is considered. The basis of the theoretical investigation is a formulation of the principle of virtual work which is suitable for incremental-ierative analysis of PC structures by the finite element method (FEM). It contains the expression for the virtual work of forces exerted by the tendons on the remaining part of the surface structure, treted as a free body. Geometric nonlinearity is considered on the basis of Koiter's shell theory of small displacements and moderately large rotations. Physical nonlinearity is taken into account by means of appropriate constitutive equations for intact and cracked concrete, respectively, reinforcing steel and prestressing steel. The numerical investigation consists of collapse load analysis of a PC slab with unbonded tendons. It demonstrates the usefulness of the theoretical concept.

http://dx.doi.org/10.1016/0168-874X(89)90019-X