The elastic response of the frame elements is formulated in terms of Young's modulus, ; the torsional shear modulus, ; coefficient of thermal expansion, ; and cross-section shape. Geometric properties such as the cross-sectional area, , or bending moments of inertia are constant along the element and during the analysis.

If present, thermal strains are constant over the cross-section, which is equivalent to assuming that the temperature does not vary in the cross-section. As a result of this assumption only the axial force, , depends on the thermal strain

*FRAME SECTION, ELSET=name

*FRAME SECTION, SECTION=library_section, ELSET=name

*FRAME SECTION, SECTION=GENERAL, ELSET=name

*FRAME SECTION, SECTION=section_type, ELSET=name
first_data_line
second_data_line
elastic_modulus, torsional_shear_modulus, coefficient_of_thermal_expansion, temperature, fv_1, fv_2, etc.

To include elastic-plastic response, specify , , , and directly as functions of their conjugate plastic deformation variables or use the default plastic response for , , , and based on the material yield stress. ABAQUS/Standard uses the specified or default values to define a nonlinear kinematic hardening model that is “lumped” into plastic hinges at the element ends. Since the plasticity is lumped at the element ends, no length dimension is associated with the hinge. Generalized forces are related to generalized plastic displacements, not strains. In reality, the plastic hinge will have a finite size determined by the structural member's length and the loading, which will affect the hardening rate but not the ultimate load. For example, yielding under pure bending (a constant moment over the member) will produce a hinge length equal to the member length, whereas yielding of a cantilever with transverse tip load (a linearly varying moment over the member) will produce a much more localized hinge. Hence, if the rate of hardening and, thus, the plastic deformation at a given load are of importance, you should calibrate the plastic response appropriately for different lengths and different loading situations.

In the plastic range the only plastic surface available is an ellipsoid. This yield surface is only reasonably accurate for the pipe cross-section. Box, circular, I, and rectangular cross-sections can be used at your discretion with the understanding that the elliptic yield surface may not approximate the elastic-plastic response accurately. The general cross-section type cannot be used with plasticity.

*FRAME SECTION, SECTION=PIPE, ELSET=name
*PLASTIC AXIAL for  
*PLASTIC M1 for  
*PLASTIC M2 for  
*PLASTIC TORQUE for 

*FRAME SECTION, SECTION=PIPE, ELSET=name, 
PLASTIC DEFAULTS, YIELD STRESS= 
plastic options if user-defined values are necessary for a particular generalized force

Elastic-plastic data curve fit and calculation of default values

The elastic-plastic response is a nonlinear kinematic hardening plasticity model. See “Models for metals subjected to cyclic loading,” Section 11.2.2, for a discussion of the nonlinear kinematic hardening formulation.

Nonlinear kinematic hardening with N, M₁, M₂, and T defined directly

For each of the four plastic material variables ABAQUS/Standard uses an exponential curve fit of the user-supplied generalized force versus generalized plastic displacement to define the limits on the elastic range. The curve-fit procedure generates a hardening curve from the user-supplied data. It requires at least three data pairs.

The nonlinear kinematic hardening model describes the translation of the yield surface in generalized force space through a generalized backstress, . The kinematic hardening is defined to be an additive combination of a purely kinematic linear hardening term and a relaxation (recall) term such that the backstress evolution is defined by

where is a component of generalized force, and and are material parameters that are calibrated based on the user-defined or default hardening data. is the initial hardening modulus, and determines the rate at which the kinematic hardening modulus decreases with increasing backstress, . The saturation value of (), called , is

See Figure 15.4.2–1 for an illustration of the elastic range for the nonlinear kinematic hardening model.

Figure 15.4.2–1 Nonlinear kinematic hardening model: yield surface for positive loading and the center of the yield surface, .

Allowing ABAQUS/Standard to generate the default nonlinear kinematic hardening model

To define the default plastic response, three data points are generated from the yield stress value and the cross-section shape. These three data points relate generalized force to generalized plastic displacement per unit length of the element. Since the model is calibrated per unit element length, the generated default plastic response is different for different element lengths. The generalized force levels for these three points are , , and . is the generalized force at zero plastic generalized displacement. and are generalized force magnitudes that characterize the ultimate load-carrying capacity. The slopes between the data points (i.e., the generalized plastic moduli and ) characterize the hardening response. See Figure 15.4.2–2 for an illustration of the default nonlinear kinematic hardening model.

Figure 15.4.2–2 Data points generated for the default nonlinear kinematic hardening model.

For the plastic axial force, is the axial force that causes initial yielding. For the plastic bending moments about the first and second axes, is the moment about the first and second cross-sectional directions, respectively, that produces first fiber yielding. For the plastic torsional moment, is the torque about the axis that produces first fiber yielding. The generalized force levels and , along with the connecting slopes and , are chosen to approximate the response of a pipe cross-section made of a typical structural steel, with mild work hardening, from initial yielding to the development of a fully plastic hinge. The work hardening of the material corresponds to the default hardening of the section during axial loading. For different loading situations the size of the plastic hinge will vary; hence, the default model should be checked for validity against all anticipated loading situations. Default values for , , , and corresponding to each plastic variable are listed in Table 15.4.2–1.

Table 15.4.2–1 Default values for generalized forces and connecting slopes for corresponding plastic variables.

Plastic axial force
First plastic bending moment
Second plastic bending moment
Plastic torsional moment (for box and pipe sections)
Plastic torsional moment (for I-sections)

These default values are available for pipe, box, and I cross-section types with the values for the coefficients , , and as shown in Table 15.4.2–2.

Table 15.4.2–2 Coefficients , , and .

Cross-section type
Pipe	0.30	0.07	1.35
Box	0.17	0.02	1.20
I (strong)	0.10	0.02	1.12
I (weak)	0.43	0.10	1.50

Frame elements optionally allow only uniaxial response (strut behavior). In this case neither end of the element supports moments or forces transverse to the axis; hence, only a force along the axis of the element exists. Furthermore, this axial force is constant along the length of the element, even if a distributed load is applied tangentially to the element axis. The uniaxial response of the element is linear elastic or nonlinear, in which case it includes buckling and postbuckling in compression and isotropic hardening plasticity in tension.

*FRAME SECTION, SECTION=library_section, ELSET=name, PINNED

Defining buckling, postbuckling, and plastic uniaxial behavior: buckling strut response

If uniaxial buckling and postbuckling in compression and isotropic hardening plasticity in tension are modeled (buckling strut response), the buckling envelope must be defined. The buckling envelope defines the force versus axial strain (change in length divided by the original length) response of the element. It is illustrated in Figure 15.4.2–3.

Figure 15.4.2–3 Buckling envelope for uniaxial buckling response.

The buckling envelope derives from Marshall Strut theory, which is developed for pipe cross-section profiles only. No other cross-section types are permitted with buckling strut response.

Seven coefficients determine the buckling envelope as follows (the default values are listed, where is the pipe outer diameter and is the pipe wall thickness):

Elastic limit force (). is the yield stress.

Isotropic hardening slope ().

Critical compressive buckling force predicted by the ISO equation, defined in “Buckling strut response for frame elements,” Section 3.9.3 of the ABAQUS Theory Manual.

Slope of a segment on the buckling envelope, ( and ).

Corner on the buckling envelope ().

Slope of a segment on the buckling envelope ().

Corner on the buckling envelope ().

The axial force in the element is required to stay inside or on the buckling envelope. When tension yielding occurs, the enclosed part of the envelope translates along the strain axis by an amount equal to the plastic strain. When reverse loading occurs for points on the boundary of the enclosed part of the envelope, the strut exhibits “damaged elastic” behavior. This damaged elastic response is determined by drawing a line from the point on the envelope to the tension yield point (force value ). As long as the force and axial strain remain inside the enclosed part of the envelope, the force response is linear elastic with a modulus equal to the damaged elastic modulus. At any time that the compressive strain is greater in magnitude than the negative extreme strain point of the envelope, the force is constant with a value of zero.

The value of is a function of an element's geometrical and material properties, including the yield stress value.

Buckling strut response cannot be used with elastic-plastic frame section behavior; the strut's plastic behavior is defined by and the isotropic hardening slope .

Defining the buckling envelope

You can specify that the default buckling envelope should be used, or you can define the buckling envelope. If you define the buckling envelope directly and specify that the default envelope should be used, the values defined by you will take precedence.

In either case you must provide the yield stress value, which will be used to determine the yield force in tension and the critical compressive buckling load (through the ISO equation described later in this section).

Input File Usage:	To specify the default buckling envelope, use the following option:
	*FRAME SECTION, SECTION=PIPE, ELSET=name, BUCKLING, PINNED, YIELD STRESS= To specify a user-defined buckling envelope, use both of the following options: *FRAME SECTION, SECTION=PIPE, ELSET=name, PINNED, YIELD STRESS= *BUCKLING ENVELOPE

Defining the critical buckling load

The critical buckling load, , is determined by the ISO equation, which is an empirical relationship determined by the International Organization for Standardization based on experimental results for pipe-like or tubular structural members. Within the ISO equation, four variables can be changed from their default values: the effective length factors, and , in the first and second sectional directions (the default values are 1.0) and the added length, and in the first and second sectional directions (the default values are 0). These variables account for the buckling member's end connectivity. The effective element length in the transverse direction () is . For details on the ISO equation, see “Buckling strut response for frame elements,” Section 3.9.3 of the ABAQUS Theory Manual.

Input File Usage:	To define nondefault coefficients for the ISO equation with the default buckling envelope, use both of the following options:
	*FRAME SECTION, SECTION=PIPE, ELSET=name, BUCKLING, PINNED, YIELD STRESS= *BUCKLING LENGTH To define nondefault coefficients for the ISO equation with a user-defined buckling envelope, use all of the following options: *FRAME SECTION, SECTION=PIPE, ELSET=name, PINNED, YIELD STRESS= *BUCKLING ENVELOPE *BUCKLING LENGTH

Frame elements allow switching to uniaxial buckling strut response during the analysis. The criterion for switching is the “ISO” equation together with the “strength” equation (see “Buckling strut response for frame elements,” Section 3.9.3 of the ABAQUS Theory Manual). When the ISO equation is satisfied, the elastic or elastic-plastic frame element undergoes a one-time-only switch in behavior to buckling strut response. The strength equation is introduced to prevent switching in the absence of significant axial forces.

When the frame element switches to buckling strut response, a dramatic loss of structural stiffness occurs. The switched element no longer supports bending, torsion, or shear loading. If the global structure is unstable as a result of the switch (that is, the structure would collapse under the applied loading), the analysis may fail to converge.

To permit switching of the element response, use the default buckling envelope or define a buckling envelope and provide a yield stress, but do not activate linear elastic uniaxial behavior for the frame element.

The ISO equation is an empirical relationship based on experiments with slender, pipe-like (tubular) members. Since the equation is written explicitly in terms of the pipe outer diameter and thickness, only pipe sections are permitted with buckling strut response. The ISO equation incorporates several factors that you can define. Effective and added length factors account for element end fixity, and buckling reduction factors account for bending moment influence on buckling. You can define nondefault values for these factors in each local cross-section direction.

*FRAME SECTION, SECTION=PIPE, ELSET=name, BUCKLING,   
YIELD STRESS=

*FRAME SECTION, SECTION=PIPE, ELSET=name, BUCKLING,   
YIELD STRESS= 
*BUCKLING LENGTH
*BUCKLING REDUCTION FACTORS

*FRAME SECTION, SECTION=PIPE, ELSET=name, YIELD STRESS= 
*BUCKLING ENVELOPE
*BUCKLING LENGTH
*BUCKLING REDUCTION FACTORS

You can define a thermal expansion coefficient for the frame section. The thermal expansion coefficient may be temperature dependent. In this case you must define the reference temperature for thermal expansion, .

*FRAME SECTION, ZERO= 
*THERMAL EXPANSION

Define temperatures and field variables by giving the value at the origin of the cross-section (i.e., only one temperature or field-variable value is given).

*TEMPERATURE
*FIELD
*INITIAL CONDITIONS, TYPE=TEMPERATURE
*INITIAL CONDITIONS, TYPE=FIELD