chapter  21
Cyclic behavior of reinforced concrete beams: different responses depending on mechanical and geometrical properties
Pages 10

As well known, the stress intensity factor Kj is connected with the strain energy release rate Qj through the following expression:

e, - % m in which E is the Young's Modulus of concrete.

Furthermore, by means of an energy balance approach, one can demonstrate the equivalence between Qj and the total potential energy W released during the formation of a unitary increment of the crack surface A:

By applying Betti's theorem (from which \PM — ^ M P ) a n d Clapeyron's theorem, the total potential energy of the beam element takes the form:

W = - \ \ p P P 2 + \ P M P M - l - \ M M M 2 (9)

Finally, the expressions of the localized compliances are achieved through the integration of eq.(8) and by equating the result to the right-hand side of eq.(9):

2bE

Hence, by making use of a proper congruence condition, one can obtain the unknown reaction P in the different situations of loading and unloading, ac­ companied by crack propagation, steel yielding and/or slippage between steel and concrete.