Interpolation
The interpolation for the 2-node element is
and for the 3-node element,
where , , and are the values of a variable at the nodes and is the interpolated value of this variable.
ProductsAbaqus/StandardAbaqus/Explicit There are two truss elements in Abaqus: a 2-node linear interpolation truss and a 3-node quadratic interpolation truss. The quadratic interpolation version is in the library mainly for compatibility with the quadratic interpolation elements of other types, such as shell element S8R5. The same interpolation functions are used for both the Cartesian displacement components and for the Cartesian components of the initial position vector, so these elements are the simplest form of isoparametric elements. The elements are one-dimensional: a single material (isoparametric) coordinate, g, is defined along the element, with in the element. In a 2-node element node 1 is at and node 2 is at . In the 3-node version node 1 is at , node 2 is at , and node 3 is at . InterpolationThe interpolation for the 2-node element is and for the 3-node element, where , , and are the values of a variable at the nodes and is the interpolated value of this variable. Strain measureThese are one-dimensional elements, and the only strain considered is that along the axis of the element. The stretch ratio along the axis is where l measures length along the truss axis in the current configuration: and measures length along the axis in the original configuration. For geometrically nonlinear analysis we use a logarithmic strain measure: First variation of strainThe first variation of strain is where is a unit tangent along the truss axis. Second variation of strainThe second variation of strain is Integration
Virtual work contributionThe virtual work contribution from the stress in a truss element is where a is the current cross-sectional area of the truss, is the “true” (Cauchy) stress along the truss, is the logarithmic strain, and l is the length of the element. Since we assume the truss is incompressible, , where A is the original area and L the original length of the truss. So, This is the form in which the internal virtual work contribution is used for truss elements. Mixed (hybrid) forms“Hybrid” truss elements are also available in Abaqus/Standard. In those elements the axial force at the integration points is taken as an additional variable, with the compatibility condition introduced to define these variables. The formulation is identical to that used for the hybrid beam elements (Hybrid beam elements), without the bending terms. |