13.189. BEAM189 - 3-D 3-Node Beam

Matrix or Vector	Shape Functions	Integration Points
Stiffness and Stress Stiffness Matrices; and Thermal and Newton-Raphson Load Vectors	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 BEAM188 - 3-D 2-Node Beam
Consistent Mass Matrix and Pressure, Hydrostatic, and Hydrodynamic Load Vectors	Same as stiffness matrix	Along the length: 3 [1] Across the section: 1
Lumped Mass Matrix	Equation 11–19, Equation 11–20, and Equation 11–21	Along the length: 3 Across the section: 1

Matrix or Vector

Shape Functions

Stiffness and Stress Stiffness Matrices; and Thermal and Newton-Raphson Load Vectors

Along the length: 2

Consistent Mass Matrix and Pressure, Hydrostatic, and Hydrodynamic Load Vectors

Same as stiffness matrix

Along the length: 3 [1]

Across the section: 1

Lumped Mass Matrix

Along the length: 3

Across the section: 1

For partially submerged elements, the hydrostatic/hydrodynamic load vector integration point(s) is/are centered on the wetted portion.

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 theory for this element is identical to that of BEAM188 - 3-D 2-Node Beam, except that it is a 3-node beam element.