NM3D2

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

References

1. 10.1061/JSENDH.STENG-12176

Syntax

Option One

section NM3D2 (1) (2...11)
# (1) int, unique section tag
# (2) double, EA
# (3) double, strong axis EI
# (4) double, weak axis EI
# (5) double, yielding axial force
# (6) double, yielding strong axis moment
# (7) double, yielding weak axis moment
# (8) double, c
# (9) double, isotropic hardening parameter H
# (10) double, kinematic hardening parameter K
# (11) double, linear density

The $N$-$M$ interaction surface is defined as follows.

$$f=1.15p^2+m_s^2+m_w^4+3.67p^2m_s^2+3p^6m_w^2+4.65m_s^4m_w^2-c$$

where $p$, $m_s$ and $m_w$ are normalised axial force and moments about the strong and weak axes. This 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.

section NM3D2 (1) (2...11) [(12) (13) (14) (15)...]
# (1) int, unique section tag
# (2) double, EA
# (3) double, strong axis EI
# (4) double, weak axis EI
# (5) double, yielding axial force
# (6) double, yielding strong axis moment
# (7) double, yielding weak axis moment
# (8) double, e
# (9) double, isotropic hardening parameter H
# (10) double, kinematic hardening parameter K
# (11) double, linear density
# (12) double, a_i
# (13) double, b_i
# (14) double, c_i
# (15) double, d_i

In the above command, parameters (12), (13), (14) and (15) form a set of parameters and can be appended as many groups as the analyst wishes. The surface is assumed to possess the following form,

$$f=\sum_{i=1}^na_ip^{b_i}m_s^{c_i}m_w^{d_i}-e.$$

For example, the previous surface $f=1.15p^2+m_s^2+m_w^4+3.67p^2m_s^2+3p^6m_w^2+4.65m_s^4m_w^2-c$ can be equivalently expressed with the second syntax as follows.

section NM3D2 (1) (2...11) \
1.15 2. 0. 0. \
1. 0. 2. 0. \
1. 0. 0. 4. \
3.67 2. 2. 0. \
3. 6. 0. 2. \
4.65 0. 4. 2.

The only validation implemented is the number of triplets. The command takes $4n$ parameters and interprets them accordingly. Please make sure the definition is correct.

Remarks

The true hardening ratio is defined as

$$\dfrac{H+K}{1+H+K},$$

since $H$ 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)