Mander et al. nonlinear concrete model - con_ma
This is a uniaxial nonlinear constant confinement model, initially programmed by Madas [1993], that follows the constitutive relationship proposed by Mander et al. [1988] and the cyclic rules proposed by Martinez-Rueda and Elnashai [1997]. The confinement effects provided by the lateral transverse reinforcement are incorporated through the rules proposed by Mander et al. [1988] whereby constant confining pressure is assumed throughout the entire stress-strain range.
Five model calibrating parameters must be defined in order to fully describe the mechanical characteristics of the material:
Compressive strength - fc
This is the cylinder (100x200 mm) compressive stress capacity of the material. Its value typically varies from 15 MPa up to 45 MPa. The default value is 28 MPa.
Tensile strength - ft
This is the tensile stress capacity of the material. It can usually be estimated as 
, where ft varies from 2 MPa (concrete in direct tension) to 3 MPa (concrete in flexural tension), as suggested by Priestley et al. [1996]. When this value is reached, the concrete is assumed to abruptly loose its tensile resistance, without any sort of tension softening. The default value is 2.2 MPa (see note below).
Modulus of elasticity - Ec
This is the initial elastic stiffness of the material. Its value usually oscillates between 18000 and 30000 MPa. The default value is 24870 MPa.
Strain at peak stress - 
This is the strain corresponding to the point of unconfined peak compressive stress (fc). For normal strength plain concrete, this value is usually considered to lie within the range of 0.002 to 0.0022. The default value is 0.002 mm/mm.
Specific weight - 
This is the specific weight of the material. The default value is 24 kN/m3.
Notes
- Values of compressive strength capacity obtained through testing of concrete cubes are usually 25 to 10 percent higher than their cylinder counterparts, for cylinder concrete strengths of 15 to 50 MPa, respectively.
- Some researchers [e.g. Scott et al., 1982] have suggested that the influence of the high strain rates expected under seismic loading (0.0167/sec) on the stress-strain behaviour of the core concrete can be accounted for by adjusting the results of tests conducted at usual strain rates (0.0000033/sec); the adjustment could consist simply of applying a multiplying factor of 1.25 to the peak stress, the strain at the peak stress, and the slope of the post-yield falling branch. Mander et al. [1989] also present methods by which strain rate effects can be incorporated into the model, although the basic formulae, implemented here, do not include the effect.
- On occasions, depending on the structural model and applied loading, crack opening may introduce numerical instabilities in the analyses. If, on some of those instances, the user is interested in predicting, for example, the top displacement of a building (i.e. global response) rather than accurately reproducing the local response of elements and sections (e.g. section curvatures), then tensile resistance may be simply ignored altogether (i.e. ft=0 MPa), and in this way stability of the analysis will most certainly be achieved in easier fashion.
- The confinement factor employed by this material type is a constant confinement factor. It is defined as the ratio between the confined and unconfined compressive stress of the concrete, and used to scale up the stress strain relationship throughout the entire strain range. Although it may be computed through the use of any confinement model available in the literature [e.g. Ahmad and Shah, 1982; Sheikh and Uzumeri, 1982; Eurocode 8, 2004; Penelis and Kappos, 1997], the Mander et al. [1989] is used y the program both in the Sections and in the Confinement Factor Calculation module. Its value usually fluctuates between the values of 1.0 and 2.0 for reinforced concrete members and between 1.5 and 4.0 for steel concrete composite members.