Verification of the elastic behavior of frame elements

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

ProductsAbaqus/Standard

Simple load tests

Elements tested

FRAME2D

FRAME3D

Features tested

The elastic behavior of frame elements is tested. The different cross-sections considered for the frame elements are:

  • rectangular hollow box section

  • solid circular section

  • general cross-section

  • I-beam section

  • hollow circular section

  • solid rectangular section

The tests are performed for the following load conditions:

  • concentrated loads

  • gravity loads

  • force per unit length in the global and the local frame directions

  • thermal loads as nodal temperatures

  • transverse fluid drag load

  • fluid drag force on the ends of the frame

  • tangential fluid drag load

  • buoyancy load (closed-end condition)

  • transverse wind drag load

  • wind drag force on the ends of the frame

  • foundation loads in the global and local frame directions

These loads are considered to act either individually or in combination. Both regular static steps and linear perturbation steps are considered.

The local coordinate system is also tested. Temperature dependence of frame element properties is tested under thermal loading. The initial stress conditions and the initial temperature at the nodes are also verified.

Problem description

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 are used for the five section types (rectangular hollow box, solid circular, hollow circular, rectangular, and I-beam).

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 α to 1 × 10.0−9. 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.

Material:

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

Results and discussion

The problem is statically determinate. The section forces and section strains match the analytical values.

Input files

frame2d_bs_thermal.inp

Box section with thermal loading.

frame2d_cs_wind_transform.inp

Circular section with wind loading and TRANSFORM.

frame2d_gs_foundation.inp

General section with FOUNDATION loading.

frame2d_gs_sig0.inp

General section with initial stress, perturbation step with LOAD CASE.

frame2d_is_aqua.inp

I-section with Aqua fluid loading.

frame2d_ps_sig0.inp

Pipe section with initial stress.

frame2d_rs_aqua.inp

Rectangular section with Aqua fluid loading.

frame2d_rs_aqua_transform.inp

Rectangular section with Aqua fluid loading and TRANSFORM.

frame2d_rs_foundation.inp

Rectangular section with FOUNDATION loading.

frame3d_bs_wind.inp

Box section with wind loading.

frame3d_cs_foundation.inp

Circular section with FOUNDATION loading.

frame3d_cs_transform.inp

Circular section with TRANSFORM.

frame3d_gs_sig0_transform.inp

General section with initial stress.

frame3d_is_aqua.inp

I-section with Aqua fluid loading.

frame3d_ps_foundation.inp

Pipe section with FOUNDATION loading.

frame3d_ps_thermal.inp

Pipe section with thermal loading.

frame3d_rs_sig0_transform.inp

Rectangular section with initial stress and TRANSFORM.

Elastic frame elements with pinned ends

Elements tested

FRAME2D

FRAME3D

Features tested

The linear elastic uniaxial behavior of frame elements under a concentrated load is tested.

Problem description

Pinned connections are specified for the ends of frame elements by declaring the relevant parameter for the frame section. 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.

Results and discussion

All tests match the exact solution; for details, see Three-bar truss.

Elastic frame elements with buckling strut response

Elements tested

FRAME2D

FRAME3D

Features tested

The uniaxial buckling strut behavior of frame elements with both ends pinned is tested.

Problem description

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. Uniaxial response, buckling strut response, and yield stress are considered for the cross-section for frame elements.

Model:

Pipe's radius: 2.
Pipe's thickness: 0.08122693
Cross-sectional area: 1.

Material:

Young's modulus: 30 × 106
Shear modulus: 10 × 106
Yield stress: 1 × 106

Results and discussion

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.

Figures

Figure 1. Buckling response of FRAME2D element.

Elastic frame element with buckling strut response for nonlinear geometry

Elements tested

FRAME2D

Features tested

A collapsing scaffold is investigated in a geometrically nonlinear analysis.

Problem description

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:

Pipe's radius: 0.2
Pipe's thickness: 0.01

Material:

Young's modulus: 3. × 106
Shear modulus: 1.5 × 106
Yield stress: 51.9 × 103

Results and discussion

The snap-through character of the response requires the Riks analysis procedure. Figure 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 2.

Figures

Figure 2. Buckling response of scaffold with FRAME2D pipe elements.

Elastic frame elements with switching algorithm for nonlinear geometry

Elements tested

FRAME2D

FRAME3D

Problem description

The buckling strut response is enabled for the elastic frame elements. The ISO equation is used as a criterion for the switching algorithm, and the default buckling envelope governs the postbuckling behavior.

Results and discussion

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 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 Pc⁢r=​ −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.

Figures

Figure 3. Buckling response of scaffold with FRAME2D and switching algorithm.