| Matrix or Vector | Option | Shape Functions | Integration Points |
|---|---|---|---|
| Stiffness and Stress Stiffness Matrices, and Thermal and Newton-Raphson Load Vectors | Linear (KEYOPT(3) = 0) | Equation 11–6, Equation 11–7, Equation 11–8, Equation 11–9, Equation 11–10, and Equation 11–11 |
Along the length: 1 Across the section: see text below |
| Quadratic (KEYOPT(3) = 2) | Equation 11–19, Equation 11–20, Equation 11–21, Equation 11–22, Equation 11–23, and Equation 11–24 |
Along the length: 2 Across the section: see text below. | |
| Cubic (KEYOPT(3) = 3) | Equation 11–26, Equation 11–27, Equation 11–28, Equation 11–29, Equation 11–30, and Equation 11–31 |
Along the length: 3 Across the section: see text below. | |
| Consistent Mass Matrix and Pressure, Hydrostatic, and Hydrodynamic Load Vectors | Linear (KEYOPT(3) = 0) | Equation 11–6, Equation 11–7, Equation 11–8, Equation 11–9, Equation 11–10, and Equation 11–11 |
Along the length: 2 [1] Across the section: 1 |
| Quadratic (KEYOPT(3) = 2) | Equation 11–19, Equation 11–20, Equation 11–21, Equation 11–22, Equation 11–23, and Equation 11–24 |
Along the length: 3 [1] Across the section: 1 | |
| Cubic (KEYOPT(3) = 3) | Equation 11–26, Equation 11–27, Equation 11–28, Equation 11–29, Equation 11–30, and Equation 11–31 |
Along the length: 4 [1] Across the section: 1 | |
| Lumped Mass Matrix | Linear (KEYOPT(3) = 0) | Equation 11–6, Equation 11–7, and Equation 11–8 |
Along the length: 2 Across the section: 1 |
| Quadratic (KEYOPT(3) = 2) | Equation 11–19, Equation 11–20, and Equation 11–21 |
Along the length: 3 Across the section: 1 | |
| Cubic (KEYOPT(3) = 3) | Equation 11–26, Equation 11–27, and Equation 11–28 |
Along the length: 4 Across the section: 1 |
| Load Type | Distribution |
|---|---|
| Element Temperature | Bilinear across cross-section and linear along length |
| Nodal Temperature | Constant across cross-section, linear along length |
| Pressure | Linear along length. The pressure is assumed to act along the element x-axis. |
References: Simo and Vu-Quoc([238]), Ibrahimbegovic([239]).
The element is based on Timoshenko beam theory; therefore, shear deformation effects are included. It uses three components of strain, one (axial) direct strain and two (transverse) shear strains. The element is well-suited for linear, large rotation, and/or large strain nonlinear applications. If KEYOPT(2) = 0, the cross-sectional dimensions are scaled uniformly as a function of axial strain in nonlinear analysis such that the volume of the element is preserved.
The element includes stress stiffness terms, by default, in any analysis using large deformation (NLGEOM,ON). The stress stiffness terms provided enable the elements to analyze flexural, lateral and torsional stability problems (using eigenvalue buckling or collapse studies with arc length methods). Pressure load stiffness (Pressure Load Stiffness) is included.
Transverse-shear strain is constant through cross-section; that is, cross sections remain plane and undistorted after deformation. Higher-order theories are not used to account for variation in distribution of shear stresses. A shear-correction factor is calculated in accordance with in the following references:
Schramm, U., L. Kitis, W. Kang, and W.D. Pilkey. “On the Shear Deformation Coefficient in Beam Theory.” [Finite Elements in Analysis and Design, The International Journal of Applied Finite Elements and Computer Aided Engineering]. 16 (1994): 141-162.
Pilkey, Walter D. [Formulas for Stress, Strain, and Structural Matrices]. New Jersey: Wiley, 1994.
The element can be used for slender or stout beams. Due to the limitations of first order shear deformation theory, only moderately “thick” beams may be analyzed. Slenderness ratio of a beam structure may be used in judging the applicability of the element. It is important to note that this ratio should be calculated using some global distance measures, and not based on individual element dimensions. A slenderness ratio greater than 30 is recommended.
These elements support only elastic relationships between transverse-shear forces and transverse-shear strains. Orthotropic elastic material properties with bilinear and multilinear isotropic hardening plasticity options (BISO, MISO) may be used. Transverse-shear stiffnesses can be specified using real constants.
The St. Venant warping functions for torsional behavior is determined in the undeformed state, and is used to define shear strain even after yielding. The element does not provide options to recalculate the torsional shear distribution on cross sections during the analysis and possible partial plastic yielding of cross section. As such, large inelastic deformation due to torsional loading should be treated with caution and carefully verified.
The elements are provided with section relevant quantities (area of integration, position, Poisson function, function derivatives, etc.) automatically at a number of section points by the use of section commands. Each section is assumed to be an assembly of predetermined number of nine-node cells which illustrates a section model of a rectangular section. Each cell has four integration points.
When the material has inelastic behavior or the temperature varies across the section, constitutive calculations are performed at each of the section integration points. For all other cases, the element uses the precalculated properties of the section at each element integration point along the length. The restrained warping formulation used may be found in Timoshenko and Gere([247]) and Schulz and Fillippou([248]).
The element has the same ocean effects as described in section Ocean Effects, specialized to other cross sections, but with the following limitations:
Sections with internal voids (such as circular tubes [SECTYPE,,BEAM,CTUBE]) cannot have a separate input for internal pressure; rather, the external pressure is used (similar to setting
KFLOOD= 1 on the OCDATA command).Hydrostatic crushing of the cross-section cannot be considered directly as no strains exists in the element y and z directions. To ensure that the linear elastic materials are handled correctly, axial loads are adjusted by:
where:
= adjustment to the axial term of the element
load vector
= external pressure 
= Poisson’s ratio (input
via PRXY on the MP command)
= cross-section areaCompared to the working stress in the material,
is normally
very small. In deep-water applications, however, the approximation
is more significant.
Several stress-evaluation options exist. The section strains and generalized stresses are evaluated at element integration points and then linearly extrapolated to the nodes of the element.
If the material is elastic, stresses and strains are available after extrapolation in cross-section at the nodes of section mesh. If the material is plastic, stresses and strains are moved without extrapolation to the section nodes (from section integration points).