MultiPhysicsVault/.raw/AbaqusTheoriesManual/AbaqusTheoriesManual_052.md

Elements

\pmb {\varepsilon} ^ {i n i t i a l} = \mathbf {D} ^ {- 1} \cdot \mathbf {F} ^ {i n i t i a l},

where D denotes the 4 £ 4 material matrix evaluated at the nodal temperature \theta \mathrm { : }

\left[ \begin{array}{c c c c} A E (\theta) & 0 & 0 & 0 \\ 0 & I _ {2 2} E (\theta) & - I _ {1 2} E (\theta) & 0 \\ 0 & - I _ {1 2} E (\theta) & I _ {1 1} E (\theta) & 0 \\ 0 & 0 & 0 & J _ {p} G (\theta) \end{array} \right],

where A is the cross-section area; E is Young's modulus; G is the shear modulus; I _ { 1 1 } , I _ { 1 2 } , and I _ { 2 2 } are cross-section moments of inertia; and J _ { p } G is the torsional stiffness. The vector of generalized initial forces includes the following components:

\mathbf {F} ^ {i n i t i a l} = \left\{N _ {x}, M _ {z}, M _ {y}, M _ {x} \right\} ^ {T}.

Initial strains, when needed, are interpolated from the nodal values to the integration points using appropriate interpolators: linear for the axial component, quadratic for the bending components, and constant for the torsional component.

ABAQUS integrates the elastic stiffness matrix numerically:

\mathbf {K} ^ {e} = \int_ {- 1} ^ {1} \mathbf {B} ^ {T} \mathbf {D} (\theta) \mathbf {B} \frac {L}{2} d \xi ,

where temperature-dependent material properties are evaluated at the integration points, assuming a linear variation of temperature along the element axis.

For the simplest case of a temperature-independent material and a pipe cross-section, the elastic stiffness matrix can be integrated analytically to give:

Twisting moments at the end nodes:

\left\{ \begin{array}{l} M _ {x 1} \\ M _ {x 2} \end{array} \right\} = \frac {G J _ {p}}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} \phi_ {x 1} \\ \phi_ {x 2} \end{array} \right\}.

Axial forces at the nodes:

\left\{ \begin{array}{l} N _ {x 1} \\ N _ {x 2} \\ N _ {x 3} \end{array} \right\} = \frac {A E}{3 L} \left[ \begin{array}{c c c} 7 & 1 & - 8 \\ 1 & 7 & - 8 \\ - 8 & - 8 & 1 6 \end{array} \right] \left\{ \begin{array}{l} u _ {x 1} \\ u _ {x 2} \\ u _ {x 3} \end{array} \right\}.

Bending moments at the end nodes and transverse forces for all three nodes:

Elements

\left\{ \begin{array}{l} N _ {y 1} \\ N _ {z 1} \\ M _ {y 1} \\ M _ {z 1} \\ N _ {y 2} \\ N _ {z 2} \\ M _ {y 2} \\ M _ {z 2} \\ N _ {y 3} \\ N _ {z 3} \end{array} \right\} = \frac {8 E I}{1 0 L ^ {3}} \left[ \mathbf {K} _ {\mathbf {m}} ^ {\mathbf {e}} \right] \left\{ \begin{array}{l} u _ {y 1} \\ u _ {z 1} \\ \phi_ {y 1} \\ \phi_ {z 1} \\ u _ {y 2} \\ u _ {z 2} \\ \phi_ {y 2} \\ \phi_ {z 2} \\ u _ {y 3} \\ u _ {z 3} \end{array} \right\},

where { \bf K } _ { \mathbf { m } } ^ { \mathrm { e } } is the bending part of the elastic stiffness matrix and takes the following form:

\left[ \begin{array}{cccccccccc} 7 9 & 0 & 0 & 4 7 \frac {L}{2} & 7 9 & 0 & 0 & - 1 7 \frac {L}{2} & - 1 2 8 & 0 \\  & 7 9 & - 4 7 \frac {L}{2} & 0 & 0 & 7 9 & 1 7 \frac {L}{2} & 0 & 0 & - 1 2 8 \\  &  & 9 L ^ {2} & 0 & 0 & - 1 7 \frac {L}{2} & - 3 \frac {L ^ {2}}{2} & 0 & 0 & 3 2 L \\  &  &  & 9 L ^ {2} & 1 7 \frac {L}{2} & 0 & 0 & - 3 \frac {L ^ {2}}{2} & - 3 2 L & 0 \\  &  &  &  & 7 9 & 0 & 0 & - 4 7 \frac {L}{2} & - 1 2 8 & 0 \\  &  &  &  &  & 7 9 & 4 7 \frac {L}{2} & 0 & 0 & - 1 2 8 \\  &  &  &  &  &  & 9 L ^ {2} & 0 & 0 & - 3 2 L \\  &  &  &  &  &  &  & 9 L ^ {2} & 3 2 L & 0 \\  &  &  & s y m &  &  &  &  & 2 5 6 & 0 \\  &  &  &  &  &  &  &  &  & 2 5 6 \\ \end{array} \right].

Lumped plasticity model

We assume that the displacement and rotation increments admit an additive decomposition into elastic and plastic parts. Hence,

\Delta \mathbf {q} = \Delta \mathbf {q} ^ {e} + \Delta \mathbf {q} ^ {p l}.

The total forces and moments result from the elastic constitutive relation

\mathbf {F} = \mathbf {K} ^ {e} \cdot \left(\mathbf {q} - \mathbf {q} ^ {p l}\right).

Introduce the lumped plasticity concept such that plastic deformation can develop at the beam external (end) nodes only and develops through plastic rotations (hinges) and plastic axial displacement at one or both nodes. Further assume that the plastic deformation at the external nodes can be caused by the interaction of all three moments and the axial force. Therefore, the vector of plastic deformation at those end nodes has the following form:

\mathbf {q} ^ {\mathbf {p l}} = \left\{\mathbf {q} _ {1} ^ {p l}, \mathbf {q} _ {2} ^ {p l}, \mathbf {0} \right\} ^ {T},

Elements

\mathbf {q} _ {1} ^ {p l} = \left\{u _ {x 1} ^ {p l}, 0, 0, \phi_ {x 1} ^ {p l}, \phi_ {y 1} ^ {p l}, \phi_ {z 1} ^ {p l} \right\} ^ {T},

\mathbf {q} _ {2} ^ {p l} = \left\{u _ {x 2} ^ {p l}, 0, 0, \phi_ {x 2} ^ {p l}, \phi_ {y 2} ^ {p l}, \phi_ {z 2} ^ {p l} \right\} ^ {T}.

The yield function ©, called here the plastic interaction surface, is written in terms of the nodal forces and moments. To calculate the increment of plastic deformation, the plastic interaction surface, ©, is checked at each of the two ends of the frame element during the loading history. In general, the plastic interaction surface is a function of the sectional forces, its plastic cross-sectional capacities, and the hardening parameters. The frame element is elastic if the following conditions are fulfilled at both frame ends:

\Phi_ {1} <   0 \quad \text { and } \quad \Phi_ {2} <   0.

The frame element is elastic-plastic if the plastic interaction surface is exceeded at one or both frame ends:

\left(\Phi_ {1} <   0 \quad \text { and } \quad \Phi_ {2} \geq 0\right)

\text { or } \quad (\Phi_ {1} \geq 0 \quad \text { and } \quad \Phi_ {2} <   0)

\text { or } \quad (\Phi_ {1} \geq 0 \quad \text { and } \quad \Phi_ {2} \geq 0).

Assuming associated plasticity with the direction of the increment of plastic deformation along the outward normal to the plastic interaction surface, the following relationship holds:

\Delta \mathbf {q} _ {I} ^ {p l} = \Delta \lambda_ {I} \mathbf {V} _ {I},

where \Delta \lambda _ { I } denotes the magnitude of the plastic deformation at end I and \mathbf { V } _ { I } denotes the direction of the plastic flow at that end.

The hardening model

The hardening model follows a nonlinear kinematic hardening rule generalized from the linear Ziegler hardening law and the relaxation term (the recall term), which introduces the nonlinearity. For details on the hardening model, see ``Models for metals subjected to cyclic loading,'' Section 4.3.5. Now introduce the generalized backstress vector, \pmb { \alpha } _ { I } . , which defines the origin of the moving plastic interaction surface, and define it as

\pmb {\alpha} _ {I} = \left\{\alpha_ {N _ {x I}}, 0, 0, \alpha_ {M _ {x I}}, \alpha_ {M _ {y I}}, \alpha_ {M _ {z I}} \right\} ^ {T}, \qquad I = 1, 2.

Elements

Thus, the plastic interaction surface can be expressed as a function of the generalized force \mathbf { S } _ { I } , \Phi _ { I } = \Phi _ { I } ( { \bf S } _ { I } ) at the end I, where for each component i,

S _ {i I} = F _ {i I} - \alpha_ {i I}.

The nonlinear kinematic hardening evolution law for the increment of the backstress takes the form

Equation 3.9.3-1

\Delta \alpha_ {i I} = (C _ {i} V _ {i I} - \gamma_ {i} \alpha_ {i I}) \Delta \lambda_ {I}, i = 1, 4, I = 1, 2,

where

V _ {i I} = \frac {\partial \Phi_ {I}}{\partial S _ {i I}}.

C _ { i } and \gamma _ { i } are sectional parameters for the ith plastic component, which must be calibrated from the test data defining the hardening response of the cross-section. The parameters C _ { i } are the initial kinematic hardening moduli, and the parameters \gamma _ { i } determine the rate at which the kinematic hardening modulus decreases with increasing plastic deformation. The test data required for this implementation are the values of the sectional force and moment components as a function of generalized plastic displacements. This will be either the axial force versus plastic axial displacement or a moment versus plastic rotation of a hinge. The data are given as pairs of values: generalized sectional force versus conjugate generalized plastic displacement. The data must be supplied on at least one of the plastic options (*PLASTIC AXIAL, *PLASTIC M1, *PLASTIC M2, *PLASTIC TORQUE) under the *FRAME SECTION option or by specifying a yield stress value with the YIELD STRESS parameter on the *FRAME SECTION option. The curve fitting algorithm will determine the parameters C _ { i } and \gamma _ { i } for each component i , since the hardening law is written separately for each sectional force component, F _ { i } , where the range of i depends on the number of forces and moments entering the plastic interaction surface. Integrating the hardening rule, Equation 3.9.3-1, the following evolution law for the backstress is obtained:

\alpha_ {i I} = \frac {C _ {i}}{\gamma_ {i}} V _ {i I} \left(1 - e ^ {- \gamma_ {i} \Delta \lambda_ {I}}\right) + \alpha_ {i I} ^ {0} e ^ {- \gamma_ {i} \Delta \lambda_ {I}},

where the backstress \alpha _ { i I } ^ { 0 } indicates the value of the backstress at the beginning of the increment.

Plastic interaction surface

Plastic interaction surfaces formulated in the generalized sectional variables depend on the cross-section profile. Frame elements with lumped plasticity are valid for tubular cross-sections only, and in the simplest form the interaction surface can be expressed as an ellipsoid in a space of four sectional components: axial force and three moments. Normalized with the ultimate forces and moments for each of the sectional components, the plastic interaction surface, \Phi _ { I } , can be written for each end I of the frame element as

Elements

Equation 3.9.3-2

\left(\frac {N _ {x I} - \alpha_ {N _ {x I}}}{N _ {x 0}}\right) ^ {2} + \left(\frac {M _ {x I} - \alpha_ {M _ {x I}}}{M _ {x 0}}\right) ^ {2} + \left(\frac {M _ {y I} - \alpha_ {M _ {y I}}}{M _ {y 0}}\right) ^ {2} + \left(\frac {M _ {z I} - \alpha_ {M _ {z I}}}{M _ {z 0}}\right) ^ {2} - 1 = 0,

where N _ { x 0 } , M _ { x 0 } , M _ { y 0 } , and M _ { z 0 } represent the cross-sectional capacities at initial yield: the axial force and three moments, respectively. Any other cross-sectional profile for which plastic interaction can be approximated well enough by the above ellipsoidal surface can be used within the lumped plasticity concept for frame elements. For two-dimensional problems modeled with frame elements the plastic interaction condition becomes an ellipsoid in the axial force and bending moment plane. By checking the plastic interaction condition at any time of the deformation at both frame element ends, it is determined that if

1. \Phi _ { 1 } < 0 and \Phi _ { 2 } < 0 . , the frame element stays elastic.
2. © \nu _ { I } < 0 and \Phi _ { J } \geq 0 \left( I \neq J \right) , the frame element is elastic-plastic. If the plastic condition at the end J is exceeded, an iterative procedure is needed to find the final deformation state at the end of the increment. Either end I stays elastic and end J becomes plastic, or both ends become plastic.
3. \Phi _ { I } \geq 0 and \Phi _ { J } \geq 0 _ { : } , the frame element is elastic-plastic. If the plastic condition is exceeded at one or both nodes, an iterative procedure is needed to find the final deformation state at the end of the increment. Depending upon the ratio of plastic deformation at both ends, one or both ends will become plastic.

The integration of the plasticity model for frame elements follows the same general rule as described in ``Integration of plasticity models,'' Section 4.2.2.

To solve for the value of the deformation and sectional forces at the end of the increment for an arbitrary load increment, an iterative process is required. To set up an appropriate Newton loop, the following relationships are used, with some of them linearized:

Elastic equilibrium equation:

F _ {i I} = \sum_ {J = 1} ^ {2} \sum_ {j = 1} ^ {6} K _ {i I j J} ^ {e} \left(q _ {j J} - q _ {j J} ^ {p l}\right),

where F stands for the generalized force at the end of the increment.

The associated flow rule:

Equation 3.9.3-3

\Delta \mathbf {q} _ {I} ^ {p l} = \Delta \lambda_ {I} \frac {\partial \Phi_ {I}}{\partial \mathbf {s} _ {I}}.

The hardening evolution law:

\alpha_ {i I} = \frac {C _ {i}}{\gamma_ {i}} V _ {i I} \left(1 - e ^ {- \gamma_ {i} \Delta \lambda_ {I}}\right) + \alpha_ {i I} ^ {0} e ^ {- \gamma_ {i} \Delta \lambda_ {I}}.

Equation 3.9.3-4

The backstress definition:

Equation 3.9.3-5

S _ {i I} = F _ {i I} - \alpha_ {i I}.

The plastic interaction surfaces at both frame ends:

Equation 3.9.3-6

\Phi_ {I} \left(\mathbf {S} _ {\mathbf {I}}\right) = 0.

Large-displacement and large-rotation formulation

The frame elements admit large overall displacements and rotations; however, it is assumed that the strains are small. Accordingly, the nonlinear geometric formulation corresponds to Euler-Bernoulli beam theory superposed on a rotating reference frame. An Euler-Bernoulli displacement field, \boldsymbol { \mathbf { q } } ^ { e b } , is defined relative to this rotation reference configuration, which causes straining. The Euler-Bernoulli displacement field is defined as follows.

Let x¹ and \bar { \phi } be the average position and average rotation of the frame element in the deformed configuration:

\bar {\mathbf {x}} \stackrel {\mathrm{def}} {=} \frac {1}{2} (\mathbf {x} _ {1} + \mathbf {x} _ {2}) \qquad \mathrm{and} \qquad \bar {\boldsymbol {\phi}} = \frac {1}{2} (\boldsymbol {\phi} _ {1} + \boldsymbol {\phi} _ {2}).

The motion of the rotating reference system is defined by the rigid body motion, where the translational part of the motion is the displacement vector \left( { \bar { \bf x } } - { \bf X } _ { 3 } \right) , since { \bf X } _ { 3 } initially corresponds to the element centroid. The rotation part of the motion is the rotation matrix created from the average rotation vector:

\bar {\mathbf {R}} = \exp [ \hat {\bar {\boldsymbol {\phi}}} ].

The average rotation defines the rotation of the element's local directions from the reference values ( \mathbf { T } , \mathbf { N } _ { 1 } , \mathbf { N } _ { 2 } ) to the current values \mathbf { ( t , n _ { 1 } , n _ { 2 } ) } through

\mathbf {t} = \bar {\mathbf {R}} \cdot \mathbf {T} \quad \text { axial   direction   and }

\mathbf {n} _ {\alpha} = \bar {\mathbf {R}} \cdot \mathbf {N} _ {\alpha} \quad \text { cross - section   directions }.

To define the strain-inducing rotation contributions, we multiplicatively decompose the rotation at each node,

Equation 3.9.3-7

Elements

\mathbf {R} _ {I} \stackrel {\mathrm{def}} {=} \exp [ \hat {\pmb {\phi}} _ {I} ] = \bar {\mathbf {R}} \cdot \mathbf {r} _ {I},

where \mathbf { r } _ { I } is the strain-inducing part of the nodal rotation. Since the strains are assumed small, define the Euler-Bernoulli rotations at node I as the axial vector \hat { \pmb { \phi } } _ { I } ^ { e b } , such that

Equation 3.9.3-8

\mathbf {r} _ {I} = \exp [ \hat {\boldsymbol {\phi}} _ {I} ^ {e b} ].

Using Equation 3.9.3-8 in Equation 3.9.3-7, we can solve for \hat { \phi } _ { I } ^ { e b } by quaternion extraction. In components relative to the reference element coordinate directions

\phi_ {y I} ^ {e b} = \hat {\pmb {\phi}} _ {I} ^ {e b} \cdot \mathbf {N} _ {1},

\phi_ {z I} ^ {e b} = \hat {\phi} _ {I} ^ {e b} \cdot \mathbf {N} _ {2},

\phi_ {x I} ^ {e b} = \hat {\phi} _ {I} ^ {e b} \cdot \mathbf {T}.

The Euler-Bernoulli displacement field is the difference between the position of the node relative to the element center and the reference position of the node relative to the reference center rotated to the current configuration by the average rotation:

\mathbf {u} _ {I} ^ {e b} = (\mathbf {x} _ {I} - \bar {\mathbf {x}}) - \bar {\mathbf {R}} \cdot (\mathbf {X} _ {I} - \bar {\mathbf {X}}),

or in components relative to the rotated local element coordinate directions

u _ {x I} ^ {e b} = \mathbf {u} _ {I} ^ {e b} \cdot \mathbf {t},

u _ {y I} ^ {e b} = \mathbf {u} _ {I} ^ {e b} \cdot \mathbf {n} _ {1},

u _ {z I} ^ {e b} = \mathbf {u} _ {I} ^ {e b} \cdot \mathbf {n} _ {2}.

Once the equivalent Euler-Bernoulli displacements and rotations are determined from the nonlinear displacements and rotations, standard expressions following Euler-Bernoulli beam theory are used. The element interpolations are

Elements

\begin{array}{l} u _ {y} ^ {e b} = \frac {1}{4} (- 3 \xi + 4 \xi^ {2} + \xi^ {3} - 2 \xi^ {4}) u _ {y 1} ^ {e b} + \frac {1}{4} (3 \xi + 4 \xi^ {2} - \xi^ {3} - 2 \xi^ {4}) u _ {y 2} ^ {e b} + (1 - 2 \xi^ {2} + \xi^ {4}) u _ {y 3} ^ {e b} \\ + \frac {L}{8} (- \xi + \xi^ {2} + \xi^ {3} - \xi^ {4}) \phi_ {z 1} ^ {e b} + \frac {L}{8} (- \xi - \xi^ {2} + \xi^ {3} + \xi^ {4}) \phi_ {z 2} ^ {e b}, \\ \end{array}

u _ {z} ^ {e b} = \frac {1}{4} (- 3 \xi + 4 \xi^ {2} + \xi^ {3} - 2 \xi^ {4}) u _ {z 1} ^ {e b} + \frac {1}{4} (3 \xi + 4 \xi^ {2} - \xi^ {3} - 2 \xi^ {4}) u _ {z 2} ^ {e b} + (1 - 2 \xi^ {2} + \xi^ {4}) u _ {z 3} ^ {e b}

- \frac {L}{8} (- \xi + \xi^ {2} + \xi^ {3} - \xi^ {4}) \phi_ {y 1} ^ {e b} - \frac {L}{8} (- \xi - \xi^ {2} + \xi^ {3} + \xi^ {4}) \phi_ {y 2} ^ {e b}, _ {x} ^ {e b}

= \frac {1}{2} (- \xi + \xi^ {2}) u _ {x 1} ^ {e b} + \frac {1}{2} (\xi + \xi^ {2}) u _ {x 2} ^ {e b} + (1 - \xi^ {2}) u _ {x 3} ^ {e b},

\phi_ {x} ^ {e b} = \frac {1}{2} (1 - \xi) \phi_ {x 1} ^ {e b} + \frac {1}{2} (1 + \xi) \phi_ {x 2} ^ {e b}.

The strain increments, following Euler-Bernoulli beam theory, are

\epsilon_ {x} = \frac {2}{L} \frac {\mathrm{d} u _ {x} ^ {e b}}{\mathrm{d} \xi}, \quad \kappa_ {z} = \frac {4}{L ^ {2}} \frac {\mathrm{d} ^ {2} u _ {y} ^ {e b}}{\mathrm{d} \xi^ {2}}, \quad \kappa_ {y} = - \frac {4}{L ^ {2}} \frac {\mathrm{d} ^ {2} u _ {z} ^ {e b}}{\mathrm{d} \xi^ {2}}, \quad \epsilon_ {t} = \frac {2}{L} \frac {\mathrm{d} \phi_ {x} ^ {e b}}{\mathrm{d} \xi},

where \epsilon _ { x } is the axial strain, \kappa _ { z } and \kappa _ { y } are the bending strains, and \epsilon _ { t } is the twist strain.

Additional data

To supply hardening data for sectional forces, the following options can be used in conjunction with the *FRAME SECTION option:

*PLASTIC AXIAL
*PLASTIC M1
*PLASTIC M2
*PLASTIC TORQUE

Each option (designed similarly to *AXIAL, *M1, *M2, and *TORQUE) will be followed by a pair of data on each line relating the nodal force or moments with the plastic extension or plastic rotations. If both these data and the YIELD STRESS parameter on the *FRAME SECTION option are omitted, ABAQUS assumes that the frame element will remain elastic.

The curve fitting algorithm is used for the evolution hardening equation. At least three pairs of data are required for ABAQUS to fit the curve and solve for the constants C _ { i } and \gamma _ { i } for each generalized force component, i, as shown in Figure 3.9.3-2.

Figure 3.9.3-2 Hardening model for a frame element.

text_image

F F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞ F∞

The parameters \gamma _ { i } are scaled according to the following formula:

\gamma_ {i} = \bar {\gamma} _ {i} V _ {i} ^ {0},

where V _ { i } ^ { 0 } denotes the component i of the plastic direction at the beginning of plastic deformation. For the plastic interaction surface in the form of Equation 3.9.3-2, the scaling factor is equal to

V _ {i} ^ {0} = 2 / S _ {i} ^ {0}.

3.9.4 Buckling strut response for frame elements

Frame elements admit an optional force response in which axial force only is supported by the element. Furthermore, the axial force is constant along the element; all transverse forces and all moments in the element are zero. The axial forces in the element may be linear elastic or may admit a buckling strut response where the force versus axial strain is characterized by a buckling envelope with hysteresis, as described below. For details on the standard frame element response, see ``Frame elements with lumped plasticity,'' Section 3.9.3.

In compression the buckling strut response models, in a simple way, the highly nonlinear buckling and postbuckling damage of slender members when loaded monotonically or cyclically. In tension the response is modeled by isotropic hardening plasticity. The buckling strut envelope is phenomenological, derived from experiments with pipe-like members. Since the description of the buckling envelope includes the outer pipe diameter and the pipe thickness, only PIPE cross-section types are permitted with buckling strut response.

The buckling strut response is linear elastic until the compressive loading exceeds P _ { c r , } the critical load to cause buckling. The value of P _ { c r } is determined with the ISO (International Organization for Standardization) equation, as described below.

Buckling prediction and the ISO equation

The ISO equation is used to predict the onset of buckling in slender members with pipe-like cross-sections. All quantities with dimensions have dimensions of stress. We define I, which is a

Elements

function of the axial compressive stress, f _ { c } , and the maximum bending stresses about the local 1 and 2 axes, f _ { b 1 } and f _ { b 2 } , by the expression

I (f _ {c}, f _ {b 1}, f _ {b 2}) = \frac {f _ {c}}{F _ {c}} + \frac {1}{F _ {b}} \sqrt {\left[ \frac {c _ {m 1} f _ {b 1}}{1 - \frac {f _ {c}}{F _ {e 1}}} \right] ^ {2} + \left[ \frac {c _ {m 2} f _ {b 2}}{1 - \frac {f _ {c}}{F _ {e 2}}} \right] ^ {2}}.

Here, F _ { c } is a characteristic axial compressive stress, F _ { b } is a characteristic bending stress, c _ { m 1 } and c _ { m 2 } are reduction factors corresponding to the cross-section directions 1 and 2, and F _ { e 1 } and F _ { e 2 } are the Euler buckling stresses corresponding to the 1- and 2-directions. The ISO equation states that buckling does not occur as long as

I (f _ {c}, f _ {b 1}, f _ {b 2}) <   1. 0.

To define the terms in I, we use the following notation:

\sigma^0 is the yield stress, E is Young's modulus of elasticity, A is the cross-sectional area, k_{1}, k_{2} are the effective length factors in the 1- and 2-directions (user-defined), L_{1}, L_{2} are the unbraced lengths for the 1- and 2-directions (user-defined), I_{11}, I_{22} are the bending moments of inertia about the local cross-section directions, Z_{e} is the elastic section modulus, Z_{p} is the plastic section modulus, r is the radius of gyration.

For PIPE sections if D is the outside diameter and t is the pipe wall thickness,

\begin{array}{l} I _ {1 1} \\ = I _ {2 2} = \frac {\pi}{6 4} (D ^ {4} - (D - 2 t) ^ {4}), \\ \end{array}