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 $P _ { y } )$ . 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 $P _ { c r }$ 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 $P _ { y }$ and the isotropic hardening slope $\gamma E A$ . # 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, $P _ { c r }$ , is determined by the ISO equation, which is an empirical relationship determined by the International Organization for Standardization based on experimental results for pipelike or tubular structural members. Within the ISO equation, four variables can be changed from their default values: the effective length factors, $k _ { 1 }$ and $k _ { 2 } .$ , in the first and second sectional directions (the default values are 1.0) and the added length, $\Delta L _ { 1 }$ and $\Delta L _ { 2 }$ , 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 $i ( i = 1 , 2 )$ is $\bar { L } _ { i } = k _ { i } ( L + \Delta L _ { i } )$ . For details on the ISO equation, see “Buckling strut response for frame elements,” Section 3.9.3 of the Abaqus Theory Guide. # 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 # Switching to optional uniaxial strut behavior during an analysis 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 Guide). 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. # Input File Usage: To allow switching to buckling strut response with default coefficients for the ISO equation and the default buckling envelope, use the following option: \*FRAME SECTION, SECTION=PIPE, ELSET=name, BUCKLING, YIELD STRESS= To allow switching to buckling strut response with nondefault coefficients for the ISO equation and the default buckling envelope, use all of the following options: $* \mathrm { F R A M E ~ S E C T I O N } , \mathrm { S E C T I O N } = \mathrm { P I P E } , \mathrm { E L S E T } = n a m e , \mathrm { B U C K L I N G } ,$ YIELD STRESS= $\sigma^{0}$ *BUCKLING LENGTH *BUCKLING REDUCTION FACTORS To allow switching to buckling strut response with nondefault coefficients for the ISO equation and a user-defined buckling envelope, use all of the following options: *FRAME SECTION, SECTION=PIPE, ELSET=name, YIELD STRESS= $\sigma^{0}$ *BUCKLING ENVELOPE *BUCKLING LENGTH *BUCKLING REDUCTION FACTORS # Defining the reference temperature for thermal expansion 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, $\theta ^ { 0 }$ . Input File Usage: Use both of the following options: *FRAME SECTION, ZERO= $\theta^{0}$ *THERMAL EXPANSION # Specifying temperature and field variables 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). Input File Usage: Use one or more of the following options: ```c *TEMPERATURE *FIELD *INITIAL CONDITIONS, TYPE=TEMPERATURE *INITIAL CONDITIONS, TYPE=FIELD ``` # 29.4.3 FRAME ELEMENT LIBRARY Product: Abaqus/Standard # References • “Frame elements,” Section 29.4.1 • \*FRAME SECTION # Overview This section provides a reference to the frame elements available in Abaqus/Standard. # Element types # Frame in a plane FRAME2D 2-node straight frame element Active degrees of freedom 1, 2, 6 Additional solution variables Two additional variables relating to the axial and lateral displacements. # Frame in space FRAME3D 2-node straight frame element Active degrees of freedom 1, 2, 3, 4, 5, 6 Additional solution variables Three additional variables relating to the axial and lateral displacements. # Nodal coordinates required Frame in a plane: X, Y (Direction cosines of the normal are not used; any values given are ignored.) Frame in space: X, Y, Z (Direction cosines of the normal are not used; any values given are ignored.) # Element property definition Local orientations defined as described in “Orientations,” Section 2.2.5, cannot be used with frame elements to define local material directions. The orientation of the local section axes in space is discussed in “Frame elements,” Section 29.4.1. Input File Usage: \*FRAME SECTION # Distributed loads Distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

Load ID (*DLOAD)	Units	Description
GRAV	$LT^{-2}$	Gravity loading in a specified direction (magnitude is input as acceleration).
PX	$FL^{-1}$	Force per unit length in global X-direction.
PY	$FL^{-1}$	Force per unit length in global Y-direction.
PZ	$FL^{-1}$	Force per unit length in global Z-direction (only for frames in space).
P1	$FL^{-1}$	Force per unit length in frame local 1-direction (only for frames in space).
P2	$FL^{-1}$	Force per unit length in frame local 2-direction.

# Abaqus/Aqua loads Abaqus/Aqua loads are specified as described in “Abaqus/Aqua analysis,” Section 6.11.1.

Load ID(*CLOAD/*DLOAD)	Units	Description
$FDD^{(A)}$	$FL^{-1}$	Transverse fluid drag load.
$FD1^{(A)}$	F	Fluid drag force on the first end of the frame (node 1).
$FD2^{(A)}$	F	Fluid drag force on the second end of the frame (node 2).
$FDT^{(A)}$	$FL^{-1}$	Tangential fluid drag load.
$FI^{(A)}$	$FL^{-1}$	Transverse fluid inertia load.
$FI1^{(A)}$	F	Fluid inertia force on the first end of the frame (node 1).
$FI2^{(A)}$	F	Fluid inertia force on the second end of the frame (node 2).

Load ID(*CLOAD/*DLOAD)	Units	Description
PB(A)	$FL^{-1}$	Buoyancy load (closed-end condition).
WDD(A)	$FL^{-1}$	Transverse wind drag load.
WD1(A)	F	Wind drag force on the first end of the frame (node 1).
WD2(A)	F	Wind drag force on the second end of the frame (node 2).

# Foundations Foundations are specified as described in “Element foundations,” Section 2.2.2.

Load ID(*FOUNDATION)	Units	Description
FX	$FL^{-2}$	Stiffness per unit length in global X-direction.
FY	$FL^{-2}$	Stiffness per unit length in global Y-direction.
FZ	$FL^{-2}$	Stiffness per unit length in global Z-direction (only for frames in space).
F1	$FL^{-2}$	Stiffness per unit length in frame local 1-direction (only for frames in space).
F2	$FL^{-2}$	Stiffness per unit length in frame local 2-direction.

# Element output All element output variables are given at the element ends (nodes 1 and 2) and midpoint (node 3). # Section forces and moments

SF1	Axial force.
SF2	Transverse shear force in the local 2-direction.
SF3	Transverse shear force in the local 1-direction (only available for frames in space).
SM1	Bending moment about the local 1-axis.
SM2	Bending moment about the local 2-axis (only available for frames in space).
SM3	Twisting moment about the frame axis (only available for frames in space).

See “Frame elements with lumped plasticity,” Section 3.9.2 of the Abaqus Theory Guide, for a discussion of the section forces and moments. Section elastic strains and curvatures

SEE1	Elastic axial strain.
SKE1	Elastic curvature change about the local 1-axis.
SKE2	Elastic curvature change about the local 2-axis (only available for frames in space).
SKE3	Elastic twist of the beam (only available for frames in space).

Plastic displacements and rotations in the element coordinate system

SEP1	Plastic axial displacement.
SKP1	Plastic rotation about the local 1-axis.
SKP2	Plastic rotation about the local 2-axis (only available for frames in space).
SKP3	Plastic rotation about the beam axis (only available for frames in space).

Section force and moment backstresses

SALPHA1	Axial force backstress.
SALPHA2	Bending moment backstress about the local 1-axis.
SALPHA3	Bending moment backstress about the local 2-axis (only available for frames in space).
SALPHA4	Twisting moment backstress about the beam axis (only available for frames in space).

# Node ordering on elements 

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1 end 1 2 end 2

2 - node element For frames in space an additional node may be given after a frame element’s connectivity (in the element definition—see “Element definition,” Section 2.2.1) to define the approximate direction of the first crosssection axis, . See “Frame elements,” Section 29.4.1, for details. # 29.5 Elbow elements • “Pipes and pipebends with deforming cross-sections: elbow elements,” Section 29.5.1 • “Elbow element library,” Section 29.5.2