'-~ 2~:. :i~;" 1 ELSEVIER Theoretical and Applied Fracture Mechanics 22 (1995) 63-70 theoretical and applied fracture mechanics Torsional damage of concrete beams with softening behaviour C.G. Karayannis Department of Civil Engineering, Democritus University of Thrace, Xanthi 67100, Greece Abstract Analysed in this paper is the torsional damage of concrete beam with softening behaviour. Change in the local stiffness and dissipated strain energy density are determined as the torsional load or rotation is increased. The idealized stress-strain curve is bilinear with a positive and negative slope. Use is made of the equations of elasticity for torsion and isoparameric mapping with finite difference. Numerical results are obtained for the pure torsion of a rectangular beam and combined torsion/compression of an I-beam. Determined are the critical torques which tend to agree well with the test data. 1. Introduction Material damage is a gradual process where its original homogeneous state is disturbed with in- creasing load. A reduction in the local stiffness could occur in a homogeneous fashion. For con- crete the uniaxial stress and strain response need not be monotonic even though the load is in- creased monotonically. That is, the concrete could continue to absorb energy even after the equiva- lent uniaxial stress has reached the maximum. The strain would continue to increase with de- creasing stress. Such a behaviour has been re- ferred to as softening. Stress and failure analyses of concrete with softening can be found in [1-3]. The method in [4] accounts for change in the local moduli of a material as it is damaged with each loading step. This work further explores the damage of con- crete beams subjected to torsional loads. Gradual damage is associated with changes of the local moduli and the dissipated strain energy density. Results of the two examples showed that the critical torques are in good agreement with the experiments. 2. Concrete behaviour Recent experiments have shown that concrete under tension acquired a postcracking softening behaviour. Even though the concrete has cracked, the concrete can still function while softening takes place. Determination of its remaining strength involves the assessment of damage due to deflection and cracking [1-7]. 2.1. Bilinear stress-strain curve The traditional design approach is to termi- nate the use of a concrete structure when the maximum tensile stress, say O'm~,, reaches the tensile strength of the concrete O'er It is now well recognized that this would be evenly conservative 0167-8442/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-8442(94)00049-2