The nonlinear general beam section (SECTYPE,,GENB) is an abstract cross section type that enables you to define axial, flexural, torsional, and transverse shear behavior as a function of axial strain, bending curvature, twist, and transverse shear strains.
The generalized section form of input does not require cross section geometry data or material data independently. For evaluating mass matrices, the program assumes a unit area of cross section. This form of data is useful for including an experimentally measured nonlinear response of a beam-like structural component, or for including complex behavior such as cross section distortion (not possible when using normal beam sections).
Nonlinear general beam sections also allow a nonlinear relationship of transverse shear forces to the corresponding transverse shear strains. Often, the input of generalized beam sections may be a result of a prior detailed slice analysis (for example, a segment of pipe analyzed using generalized plane strain elements).
The behavior of beam elements is governed by the generalized-stress/generalized-strain relationship of the form:
where:
N = Axial force |
M
1 = Bending moment in plane XZ |
M
2 = Bending moment in plane XY |
| τ = Torque |
S
1 = Transverse shear force in plane XZ |
S
2 = Transverse shear force in plane XY |
| ε = Axial strain |
| κ1 = Curvature in plane XZ |
| κ2 = Curvature in plane XY |
| χ = Twist of the cross section |
| γ1 = Transverse shear strain in plane XZ |
| γ2 = Transverse shear strain in plane XY |
A
E
(ε,T) = Axial stiffness as a function
of axial strain and temperature |
I
1
E
(κ1,T) =
Flexural rigidity as a function of curvature and temperature in plane XZ |
I
2
E
(κ2,T) =
Flexural rigidity as a function of curvature and temperature in plane XY |
J
G
(χ,T) = Torsional rigidity, as a function
of torsion and temperature |
A
1
G
(γ1,T) =
Transverse shear stiffness as a function of shear strain and temperature in plane
XZ |
A
2
G
(γ2,T) =
Transverse shear stiffness as a function of shear strain and temperature in plane
XY |
T is the current temperature |
Thermal expansion coefficients and mass density for the section as a function of temperature complete the definition of a generalized cross section.
Each of the following commands specifies a particular component quantity necessary for defining a nonlinear general beam section:
Table 12.2: Commands for Specifying Nonlinear General Beam Section Data
| Command | Quantity Defined and Data Specified |
|---|---|
| BSAX[a] |
Axial strain and force ε,N, T
|
| BSM1[a] |
Bending curvature and moment in plane XZ κ1,M
1, T
|
| BSM2[a] |
Bending curvature and moment in plane XY κ2,M
2, T
|
| BSTQ[a] |
Cross section twist and torque χ, τ,T
|
| BSS1[a] |
Transverse shear strain and force in plane XZ γ1,S
1, T
|
| BSS2[a] |
Transverse shear strain and force in plane XY γ2,S
2, T
|
| BSMD[b] |
Mass density of the beam section (assuming a unit area) DENS
|
| BSTE[b] |
Thermal expansion coefficient |
You can define each of the generalized section data components as
temperature-dependent. It is possible to specify up to six temperatures
(T) by reissuing any command as necessary. If you
issue a command for a temperature specified earlier, the most recent data
supersedes the previous value.
Unspecified data values default to zero except for the temperature. If the temperature is unspecified, the section data becomes temperature-independent.
To define temperature-dependent section data, specify all temperatures with numerical values including zero temperature. The section data outside of the temperature range cannot be used. If an out-of-range temperature is detected, the program is terminated. If section data is specified at single temperature, the data is effective at the temperature only.
Each component of a nonlinear beam section definition (axial, bending, torque, and transverse shear) can be a nonlinear function of the corresponding strain. The terms generalized stress and generalized strain describe the data defined via the BSAX, BSM1, BSM2, BSTQ, BSS1, and BSS2 commands.
The generalized stress to generalized strain relationship can be nonlinear. The nonlinear response can be either purely elastic--that is, no permanent deformation, and fully recoverable deformation even though the behavior is nonlinear--or elastoplastic. The option of a purely elastic or elastoplastic response (SECTYPE) applies to all components of a beam section definition.
The following input illustrates a typical (temperature-independent) nonlinear axial behavior:
sectype,1,genb,plastic ! Elastoplastic response bsax ,0.0008,200 ! Axial strain 0.0008, Force of 200 bsax ,0.001 ,240 ! Axial strain 0.001 , Force of 240 bsax ,0.0014,300 ! Axial Strain 0.0014, Force of 300
The range of strain values must cover the anticipated maximum deformation of the structure. For all nonlinear beam section specifications, the stiffness disappears if the structural response is beyond the maximum strain value of the section data. You can define a maximum of 20 generalized stress-strain points at each temperature value, and the successive slopes must be smaller than the first slope. Isotropic hardening is assumed for the material response.
For a plastic nonlinear beam section subtype (SECTYPE,,GENB,PLASTIC), you must define all section components with two or more stress-strain points. However, you can define a linear behavior for any section component by specifying a larger maximum strain value. Following is an example where axial behavior is nonlinear, but bending response is linear:
sectype,1,genb,plastic ! Elastoplastic response bsax,0.0008,200 bsax,0.001 ,240 bsax,0.0014,300 ! bending bsm1,0.1,10000 bsm1,1,100000
For an elastic nonlinear beam section subtype (SECTYPE,,GENB,ELASTIC), a single stress-strain point is adequate for defining linear behavior.
You can define nonlinear general beam sections only when using element BEAM188 or BEAM189. When using nonlinear beam section data, the following conditions apply:
The section data defined by each command listed in Table 12.2: Commands for Specifying Nonlinear General Beam Section Data is associated with the section most recently defined via the SECTYPE command.
Beam stresses are not available for output; however, the stress resultants are available as ETABLE quantities.
Section offsetting (SECOFFSET) is not available.
General beam sections do not support linear perturbation.
Only the temperature of the beam axis is relevant.
Beam section rotary inertia is calculated internally based on axial and bending stiffnesses provided. If beam section rotary inertia of a more general form is available, use MASS21 elements (rather than the BSMD command) to define beam section mass and rotary inertia at the beam nodes.