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NM2D2

\(N\)-\(M\) Interaction Inelastic Section (Linear Hardening)

Reference

  1. 10.1061/JSENDH.STENG-12176

Syntax

Option One

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section NM2D2 (1) (2...9)
# (1) int, unique section tag
# (2) double, EA
# (3) double, EI
# (4) double, yielding axial force
# (5) double, yielding strong axis moment
# (6) double, c
# (7) double, isotropic hardening parameter H
# (8) double, kinematic hardening parameter K
# (9) double, linear density

If the NM2D2 section is defined by using the above command, it is assumed that the \(N\)-\(M\) interaction surface is defined as follows.

\[ f=1.15p^2+m_s^2+3.67p^2m_s^2-c \]

where \(p\) and \(m_s\) are normalised axial force and moment (about strong axis). The surface is suitable for I-sections.

Option Two

One may wish to customise the surface by assigning different weights and orders, it is possible by using the following syntax.

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section NM2D2 (1) (2...9) [(10) (11) (12) ...]
# (1) int, unique section tag
# (2) double, EA
# (3) double, EI
# (4) double, yielding axial force
# (5) double, yielding strong axis moment
# (6) double, d
# (7) double, isotropic hardening parameter H
# (8) double, kinematic hardening parameter K
# (9) double, linear density
# (10) double, a_i
# (11) double, b_i
# (12) double, c_i

In the above command, parameters (10), (11) and (12) form a triplet and can be appended as many groups as analyst wishes. The surface is assumed to possess the following form,

\[ f=\sum_{i=1}^na_ip^{b_i}m_s^{c_i}-d. \]

For example, the previous surface \(f=1.15p^2+m_s^2+3.67p^2m_s^2-c\) can be equivalently expressed with the second syntax as follows.

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section NM2D2 (1) (2...9) 1.15 2. 0. 1. 0. 2. 3.67 2. 2.

The only validation implemented is the number of triplets. The command takes \(3n\) parameters and interprets them accordingly. Please make sure the definition is correct.

Remarks

The true hardening ratio is defined as

\[ \dfrac{H+K}{1+H+K}, \]

given that \(H\) and \(K\) are hardening ratios based on plastic strain.

Output Type

This model supports the following additional history variables to be recorded.

variable label physical meaning
YF yielding flag (vector of size 2)