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ConcreteCM

Chang-Mander Concrete Model

Syntax

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material ConcreteCM (1) (2) (3) (4) (5) (6) [7] [8] [9] [10]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, compression strength, should be negative but sign insensitive
# (4) double, tension strength, should be positive but sign insensitive
# (5) double, NC
# (6) double, NT
# [7] double, strain at compression strength, default: -2E-3
# [8] double, strain at tension strength, default: 1E-4
# [9] bool string, linear transition switch, default: false
# [10] double, density, default: 0.0

Remarks

  1. The Chang-Mander concrete model uses Tsai's equation as backbone curves for both tension and compression.
  2. Parameters NC and NT control the shapes of backbone curves. A detailed explanation is presented later.
  3. In the original model, the transition between compression and tension could sometimes have larger stiffness than initial stiffness. This is unlikely to be true in reality.
  4. A linear behaviour of part of hysteresis behaviour can be applied by turning on the linear transition switch. It shall be noted that a linear hysteresis rule is much more stable than the original version.
  5. The original CM model has some undefined behaviour which may cause stability issues. This is a simplified model.

History Variable Layout

location value
initial_history(0) unload_c_strain
initial_history(1) unload_c_stress
initial_history(2) reverse_c_strain
initial_history(3) reverse_c_stress
initial_history(4) residual_c_strain
initial_history(5) residual_c_stiffness
initial_history(6) unload_t_strain
initial_history(7) unload_t_stress
initial_history(8) reverse_t_strain
initial_history(9) reverse_t_stress
initial_history(10) residual_t_strain
initial_history(11) residual_t_stiffness
initial_history(12) connect_c_stress
initial_history(13) connect_c_stiffness
initial_history(14) connect_t_stress
initial_history(15) connect_t_stiffness
initial_history(16) inter_strain
initial_history(17) inter_stress
initial_history(18) reload_c_stiffness
initial_history(19) reload_t_stiffness

General Description of the Model

Unload from backbone

example one

Reload from unload branch

Reload before residual

example two

Reload between two residuals

example three

Reload after the opposite residual

example four

Small Cycle

example five

example six

Determination of Parameters

The Tsai's equation (Tsai, 1988) can be expressed as

\[ y=\dfrac{mx}{1+(m-\dfrac{n}{n-1})x+\dfrac{x^n}{n-1}}. \]

where \(x=\varepsilon/\varepsilon_c\) or \(x=\varepsilon/\varepsilon_t\) is normalized strain and \(y=f/f_c\) and \(y=f/f_t\) are normalized stress. The stress decreases to zero if \(m(n-1)>n\) and \(n>1\). The initial stiffness is related to \(m_t\) and \(m_c\) by

\[ E_0=mE_s=m_t\dfrac{f_t}{\varepsilon_t}=m_c\dfrac{f_c}{\varepsilon_c}. \]

Once \(E_0\) is given, \(m_c\) and \(m_t\) are determined automatically.

The parameter \(n\) controls the slope of descending branch, normally \(n_t>n_c\). Some empirical expressions are available to determine both \(m\) and \(n\). But most are unit dependent. Users shall do manual conversion. For example,

\[ m_c=1+\dfrac{17.9}{f_c}\quad{}f_c\text{ in MPa}, \]
\[ m_c=1+\dfrac{2600}{f_c}\quad{}f_c\text{ in psi}, \]
\[ n_c=\dfrac{f_c}{6.68}-1.85>1\quad{}f_c\text{ in MPa}, \]
\[ n_c=\dfrac{f_c}{970}-1.85>1\quad{}f_c\text{ in psi}. \]