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2024-09-03

VE0217 | Bending with Imperfection and Warping

Description

A structure consists of I-profile simply supported beam. The axial rotation φx is restricted on the both ends but the cross-section is free to warp (fork support). The beam has an initial imperfection in Y-direction defined as a parabolic curve with maximum displacement 30 mm in the middle. Uniform loading is applied in the middle of the top flange of I-profile. The problem is described by the following set of parameters. The verification example is based on the example introduced by Gensichen and Lumpe see the reference.

Material Steel S235 Modulus of Elasticity E 210000.000 MPa
Shear Modulus G 81000.000 MPa
Geometry Structure Length L 6.000 m
Imperfection Maximum Imperfection imax 30.000 mm
I-profile Height h 400.000 mm
Width b 180.000 mm
Web Thickness s 10.000 mm
Flange Thickness t1 14.000 mm
Load Continuous Load q 30.000 kN/m
Eccentricity ez -200.000 mm
Bending with Imperfection and Warping
Bending with Imperfection and Warping

Analytical Solution

Analytical solution is not available. Results from software S3D are taken as reference.

RFEM and RSTAB Settings

  • Modeled in RFEM 6.06 and RSTAB 9.06
  • The element size is lFE= 0.010 m
  • Isotropic linear elastic material model is used
  • The number of increments is 10
  • Second-Order and Large Deformation Analysis are used
  • Torsional Warping (7DOF) Add-on is used
  • The problem is modeled both by members and a combination of members and surface elements
  • Stiffness is reduced by means of Partial Safety Factor γM=1.1

Results

Two modeling techniques are used in RFEM 6. At first, the I-section is modeled as a beam with given imperfection (parabolic shape). Next, the I-profile is modeled by means of surface elements (plates). In this case the boundary conditions are modeled as close as possible to the beam case, but the results can be influenced by the differencies in the modeling style. In RSTAB 9 the imperfection is modeled by means of the set of short beams with given imperfection in nodes.

RSTAB 9 results:

Quantity S3D RSTAB 9 - Second-Order Analysys Ratio RSTAB 9 - Large Deformation Analysys Ratio
uy(x=3 m) [mm] 24.2 31.041 1.283 30.182 1.247
uz(x=3 m) [mm] 18.8 16.772 0.892 22.644 1.204
φx(x=3 m) [mrad] 152 186.528 1.227 194.596 1.280
My(x=3 m) [kNm] 134 134.738 1.006 135.550 1.012
Mz(x=3 m) [kNm] -20.5 -24.875 1.213 -26.716 1.303
Mω(x=3 m) [kNm2] 4.02 5.053 1.257 5.276 1.312
MTpri(x=0 m) [kNm] 2.91 3.165 1.088 3.301 1.134
MTsec(x=3 m) [kNm] 1.78 2.307 1.296 2.410 1.354

RFEM 6 results:

Quantity S3D RFEM 6 - Second-Order Analysys Ratio RFEM 6 - Large Deformation Analysys Ratio RFEM 6 - Plates - Large Deformation Analysys Ratio
uy(x=3 m) [mm] 24.2 14.476 0.598 26.962 1.114 26.339 1.088
uz(x=3 m) [mm] 18.8 14.022 0.746 20.213 1.075 20.159 1.072
φx(x=3 m) [mrad] 152 86.937 0.572 175.234 1.153 172.512 1.135
My(x=3 m) [kNm] 134 133.477 0.996 132.992 0.992 - -
Mz(x=3 m) [kNm] -20.5 -17.476 0.852 -23.546 1.149 - -
Mω(x=3 m) [kNm2] 4.02 2.335 0.581 4.716 1.173 - -
MTpri(x=0 m) [kNm] 2.91 1.490 0.512 3.002 1.032 - -
MTsec(x=3 m) [kNm] 1.78 1.160 0.652 2.300 1.292 - -

Model 005052 | Verification Example 0217 | 1
30x
7x
Verification Example 0217 | 1
References
  1. LUMPE, G. and GENSITEN, V. Evaluation of Linear and Nonlinear Member Analysis in Theory and Software: Test examples, causes of failure, detailed theory. Ernest.
  2. LUMPE, G. S3D (Vers. 09/25/2011). Biberach University of Applied Sciences, 2011.
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