Product: ABAQUS/Standard
The elastic behavior of frame elements with different cross-sections (BOX, CIRC, GENERAL, I, PIPE, RECT) is tested under the following loads: *CLOAD, GRAV, PX, PY, PZ, P1, P2, *TEMPERATURE, FDD, FD1, FD2, FDT, PB, WDD, WD1, WD2, FX, FY, FZ, F1, F2. These loads are considered to act either individually or in combination. Both regular static steps and linear perturbation steps are considered.
The *TRANSFORM option is also tested. Temperature dependence of frame element properties is tested under thermal loading. The *INITIAL CONDITIONS, TYPE=STRESS and *INITIAL CONDITIONS, TYPE=TEMPERATURE options are also verified.
The problem consists of a cantilever with a length of 75.0 units made of five frame elements. Various orientations of the cantilever in space are considered. The cross-sectional dimensions shown in “Verification of beam elements and section types,” Section 1.3.22, are used for the five section types (BOX, CIRC, PIPE, RECT, and I).
The cantilever is subjected to concentrated tip loading that leads to both flexure and torsion. The wind loads, WD1 and WD2, and the Aqua loads, FD1 and FD2, also apply concentrated forces at the nodes. The remaining loads cause uniformly distributed loading on the cantilever. Under thermal loading the free end of the cantilever is fixed. The wind velocity profile is made nearly uniform with the height by setting the exponent 
to 1 × 10.0–9 on the *WIND option. The fluid velocity in the Aqua loading is constant with height. With *FOUNDATION loads the boundary conditions of the cantilever are changed to simple supports, and the cantilever is pressed uniformly into the foundation using distributed loads.
| Young's modulus at temperature –10.0 units: | 3 × 106 |
| Poisson's ratio at temperature –10.0 units: | 0.3 |
| Young's modulus at temperature 90.0 units: | 1.5 × 106 |
| Poisson's ratio at temperature 90.0 units: | 0.3 |
| Reference temperature for definition of thermal expansion coefficient: | –10.0 |
| Thermal expansion coefficient at –10.0 temperature: | 0.001 |
| Thermal expansion coefficient at 90.0 temperature: | 0.002 |
| Initial temperature: | –10.0 |
| Material density: | 0.8 |
| Gravitational constant: | 10.0 |
| Density of air for wind loads: | 0.008 |
| Density of fluid for Aqua loads: | 0.008 |
| Seabed level: | –100.0 |
| Still fluid level: | 50.0 |
| Foundation stiffness: | 1500.0 |
The problem is statically determinate. The section forces and section strains match the analytical values.
Box section with thermal loading.
Circular section with wind loading and *TRANSFORM.
General section with *FOUNDATION loading.
General section with initial stress, perturbation step with *LOAD CASE.
I-section with Aqua fluid loading.
Pipe section with initial stress.
Rectangular section with Aqua fluid loading.
Rectangular section with Aqua fluid loading and *TRANSFORM.
Rectangular section with *FOUNDATION loading.
Box section with wind loading.
Circular section with *FOUNDATION loading.
Circular section with *TRANSFORM.
General section with initial stress.
I-section with Aqua fluid loading.
Pipe section with *FOUNDATION loading.
Pipe section with thermal loading.
Rectangular section with initial stress and *TRANSFORM.
The linear elastic uniaxial behavior of frame elements under a concentrated load is tested.
The use of the PINNED parameter on the *FRAME SECTION option is required in this case indicating that the element's ends are pinned. In this example the frame element behaves as an axial spring with constant stiffness. In small-displacement analysis the element can be compared with truss or spring elements. The model and geometry used are the same as in the verification problem “Three-bar truss,” Section 1.3.32.
All tests match the exact solution; for details, see “Three-bar truss,” Section 1.3.32.
Rectangular section with *CLOAD loading.
Rectangular section with *CLOAD loading.
The uniaxial buckling strut behavior of frame elements with both ends pinned is tested.
The buckling strut envelope corresponds to Marshall Strut theory. The tests consist of one frame element fixed at one end and subjected to a prescribed displacement on the other. The value of the prescribed displacement changes according to an amplitude definition. The variation of the amplitude is chosen in such a way that the buckling strut envelope is traced for the compressive as well as for the tensile behavior up to and beyond the yield stress value. The PINNED, BUCKLING, and YIELD STRESS parameters on the *FRAME SECTION option are required for this case.
Model: Material:The uniaxial buckling and postbuckling behavior in compression and isotropic hardening behavior in tension can be seen by plotting the axial force in the element against the prescribed displacement; see Figure 1.3.30–1.
Pipe section with prescribed displacement.
Pipe section with prescribed displacement.
The scaffold is made of three pinned frame elements with pipe cross-sections. The buckling strut envelope corresponds to Marshall Strut theory. The collapse occurs under a force-controlled loading.
Model: Material:The snap-through character of the response requires the Riks analysis procedure. Figure 1.3.30–2 plots the section force in each element versus the load factor from the Riks analysis. The buckling of frame elements 2 and 3 changes the force distribution of the entire structure. After element 3 buckles, it remains buckled throughout the loading process; element 2 buckles, then regains stiffness and develops tensile force, as seen in Figure 1.3.30–2.
A collapsing scaffold with geometry and material properties as described in “Elastic frame element with buckling strut response for nonlinear geometry” in “Verification of the elastic behavior of frame elements,” Section 1.3.30 is investigated using frame elements with the switching algorithm.
The BUCKLING parameter is used on the *FRAME SECTION option to switch from frame element to buckling strut response. The ISO equation is used as a criterion for the switching algorithm, and the default buckling envelope governs the postbuckling behavior.
Two types of problems are tested here: an in-plane scaffold structure modeled with FRAME2D and FRAME3D elements and a three-dimensional scaffold supported by an additional out-of-plane element. The default buckling envelope is used for the in-plane scaffold problems, and a nondefault buckling envelope is used in the three-dimensional scaffold. In all problems the buckling reduction factors are 1.0 in both directions. All end points of the scaffold structure are fixed, and a prescribed displacement is applied to node 2. The value of the displacement is chosen such that elements 1, 3, and 4 in the three-dimensional scaffold will violate the ISO equation and, therefore, will cause a switch to strut response.
Figure 1.3.30–3 plots the axial force in elements 1 and 3 versus the time for the scaffold in plane. Element 3 buckles at the value of critical compressive force 
–56.75 and loses its stiffness at 58% of the prescribed displacement values; element 1 buckles next and retains a small stiffness through the loading history.
The behavior of the three-dimensional scaffold is different. The first element that switches to the strut response is element 4, followed by elements 3 and 1. At 72.5% of the prescribed displacement values, elements 3 and 4 have already lost their stiffness.
FRAME2D element with switching algorithm.
FRAME3D element with switching algorithm.
FRAME3D element with switching algorithm.
