determined from Figure B-7. The elastic limit envelope in general 

 stress space was obtained by scaling the fracture envelope down to a 

 size where uniaxial yield point corresponded to about 43% of the uniaxial 

 strength . 



For plastic concrete, a strain-hardening plasticity model as pro- 

 posed previously in Reference 21 was used to describe the nonlinear 

 irreversible stress-strain response of concrete material. The plastic 

 incremental stress-strain relationship based on the normality flow rule 

 in the theory of plasticity are developed in detail in Reference 22. 



For fracture, the concrete failed when the state of stress reached 

 a certain critical value. Two different types of fracture mode are 

 defined here. 



(1) "Cracking" Type - When the principal stresses are 

 either in the tension-tension state or tension- 

 compression state and their values exceed the limit 

 values. 



(2) "Crushing" Type - When the principal stresses are in 

 the compression-compression state and their values 

 exceed the limit values . When concrete cracks , the 

 material is assumed to lose only its tensile strength 

 normal to the crack direction but to retain its strength 

 parallel to the crack direction. On the other hand, 

 when concrete crushes, the material element loses its 

 strength completely. 



In the present analysis, a dual representation of fracture criterion 

 was expressed in terms of both stresses and strain and specifies .the 

 limit value under multiaxial state of stresses or strains in the following 

 forms : 



71 



