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2024-08-02

VE0217 | Bending with Imperfection and Warping

Description

The structure consists of a simply supported I-profile beam. The axial rotation φx is restricted on both ends, but the cross-section is free to warp (fork support). The beam has an initial imperfection in the 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 the 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

An analytical solution is not available. The results from the S3D software are taken as reference.

RFEM and RSTAB Settings

  • Modeled in RFEM 6.15 and RSTAB 9.14
  • 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 by surface elements
  • Stiffness is reduced by means of the Partial Safety Factor γM=1.1

Results

Two modeling techniques are used in RFEM 6. First, the I-section is modeled as a beam with the 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 closely as possible to the beam case, but the results can be influenced by the differences in the modeling style. In RSTAB 9 the imperfection is modeled by means of the set of short beams with the given node imperfections.

RFEM 6 results:

Quantity S3D RFEM 6 - Second-Order Analysis Ratio RFEM 6 - Large Deformation Analysis Ratio RFEM 6 - Plates - Large Deformation Analysis Ratio
uy(x=3 m) [mm] 24.2 27.431 1.134 26.868 1.110 26.339 1.088
uz(x=3 m) [mm] 18.8 15.734 0.837 20.173 1.073 20.159 1.072
φx(x=3 m) [mrad] 152 167.493 1.102 174.656 1.150 172.512 1.135
My(x=3 m) [kNm] 134 133.090 0.993 132.990 0.992 - -
Mz(x=3 m) [kNm] -20.5 -22.496 1.097 -23.461 1.145 - -
Mω(x=3 m) [kNm2] 4.02 4.503 1.120 4.701 1.170 - -
MTpri(x=0 m) [kNm] 2.91 2.877 0.989 2.998 1.031 - -
MTsec(x=0 m) [kNm] 1.78 2.204 1.238 2.292 1.288 - -

RSTAB 9 results:

Quantity S3D RSTAB 9 - Second-Order Analysis Ratio RSTAB 9 - Large Deformation Analysis Ratio
uy(x=3 m) [mm] 24.2 30.995 1.281 30.178 1.247
uz(x=3 m) [mm] 18.8 16.766 0.892 22.621 1.203
φx(x=3 m) [mrad] 152 186.237 1.225 194.295 1.278
My(x=3 m) [kNm] 134 134.748 1.006 135.545 1.012
Mz(x=3 m) [kNm] -20.5 -24.839 1.212 -26.671 1.301
Mω(x=3 m) [kNm2] 4.02 5.041 1.254 5.265 1.310
MTpri(x=0 m) [kNm] 2.91 3.165 1.088 3.301 1.134
MTsec(x=0 m) [kNm] 1.78 2.301 1.293 2.404 1.351

Model 005052 | Verification Example 0217 | 1
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Verification Example 0217 | 1
References
  1. Lumpe, G., & Gensichen, V. (2014). Evaluierung der linearen und nichtlinearen Stabstatik in Theorie und Software: Prüfbeispiele, Fehlerursachen, genaue Theorie. Berlin, Ernst & Sohn.
  2. LUMPE, G. S3D (Vers. 09/25/2011). Biberach University of Applied Sciences, 2011.
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