Membrane Behavior of R/C Shell Element and Limits on Reinforcement

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Abstract
The behavior of reinforced concrete (R/C) shell elements subjected to membrane stresses is examined. The governing equations are derived from equilibrium, compatibility, and the constitutive law of reinforced concrete under biaxial stress states. Using these equations, load‐deformation relations of membrane elements are traced up to and past the ultimate load. There are three types of failure modes: (1) Yielding of reinforcement in both directions; (2) yielding of reinforcement in only one direction combined with the compressional failure of concrete; and (3) the failure of concrete without the yielding of any reinforcement. Using an analogy with beams in flexure, it is shown that the element behavior is ductile for the first two modes and brittle for the last mode. One of the major factors that influences the failure mode is the amount of reinforcement. Two sets of boundaries that separate the three different failure modes are identified in terms of reinforcement ratios. One set of boundaries separates the first two failure modes. This set defines the limit of reinforcement beyond which a commonly used strength formula based on the yielding of reinforcement alone is not valid. The other set, which separates the last two failure modes, defines the limit beyond which the element behavior is brittle. These boundaries are presented in a form suitable for use as design guides.

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