Design of a Steel I-section for Bending and Shear

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1 Benchmark Example No. 13 Design of a Steel I-section for Bending and Shear SOFiSTiK 2018

2 VERiFiCATiON MANUAL DCE-EN13: Design of a Steel I-section for Bending and Shear VERiFiCATiON MANUAL, Version Software Version: SOFiSTiK 2018 Copyright 2018 by SOFiSTiK AG, Oberschleissheim, Germany. SOFiSTiK AG HQ Oberschleissheim Office Nuremberg Bruckmannring 38 Burgschmietstr Oberschleissheim Nuremberg Germany Germany T +49 (0) T +49 (0) F +49 (0) F +49(0) info@sofistik.com This manual is protected by copyright laws. No part of it may be translated, copied or reproduced, in any form or by any means, without written permission from SOFiSTiK AG. SOFiSTiK reserves the right to modify or to release new editions of this manual. The manual and the program have been thoroughly checked for errors. However, SOFiSTiK does not claim that either one is completely error free. Errors and omissions are corrected as soon as they are detected. The user of the program is solely responsible for the applications. We strongly encourage the user to test the correctness of all calculations at least by random sampling. Front Cover Project: New SOFiSTiK Office, Nuremberg Contractor: WOLFF & MLLER, Stuttgart Architecture: WABE-PLAN ARCHITEKTUR, Stuttgart Structural Engineer: Boll und Partner. Beratende Ingenieure VBI, Stuttgart MEP: GM Planen + Beraten, Griesheim Lead Architect: Gerhard P. Wirth gpwirtharchitekten, Nuremberg Vizualisation: Armin Dariz, BiMOTiON GmbH

3 Overview Design Code Family(s): EN Design Code(s): EN Module(s): AQB, AQUA Input file(s): usage steel.dat 1 Problem Description The problem consists of a steel I- section, as shown in Fig. 1. The cross-section is designed for bending and shear. b t ƒ t r h V y y z Figure 1: Problem Description 2 Reference Solution This example is concerned with the resistance of steel cross-sections for bending and shear. The content of this problem is covered by the following parts of EN :2005 [1]: Structural steel (Section 3.2 ) Resistance of cross-sections (Section 6.2) 3 Model and Results The I-section, a HEA 200, with properties as defined in Table 1, is to be designed for an ultimate moment and a shear force V y, with respect to EN :2005 [1]. The calculation steps are presented below and the results are given in Table 2. The utilisation level of allowable plastic forces are calculated and compared to SOFiSTiK results. SOFiSTiK 2018 Benchmark No. 13 3

4 Table 1: Model Properties Material Properties Geometric Properties Loading S 235 HEA 200 V y = 200 or 300 kn b = 20.0 cm = 20 knm h = 19.0 cm t ƒ = 1.0 cm, t = 0.65 cm r = 1.8 cm Table 2: Results Case Result SOF. (EC3 Tables) SOF. (FEM) Ref. V p,rd,y [kn] V p,rd,z [kn] Util. level V y Util. level Util. level V y Util. level Benchmark No. 13 SOFiSTiK 2018

5 4 Design Process 1 Material: Structural Steel S 235 ƒ y = 235 N/mm 2 Tab. 3.1 : Nominal values of yield strength ƒ y and ultimate tensile strength ƒ for hot rolled structural steel. Cross-sectional properties: W p,y = cm 3 Plastic section modulus of HEA 200 w.r.t. y and z axis W p,z = cm 3 Where the shear force is present allowance should be made for its effect on the moment resistance. This effect may be neglected, where the shear force is less than half the plastic shear resistance : Bending and shear V Ed 0.5 V p,rd (2) V p,rd = A V ƒ y / 3 γ M0 A Vy = 2 A ƒ nge (in the y direction only contribution the two flanges ) A Vy = 2 t ƒ b = = 40 cm 2 A Vz = A 2 b t ƒ + (t + 2 r) t ƒ but not less than η h t (2): Eq. 6.18: V p,rd the design plastic shear resistance A V the shear area w.r.t. y and z axis, respectively A Vz = ( ) (3): The shear area A V may be taken as follows for rolled I-sections with load parallel to the web h = h 2 t ƒ = 17 cm A Vz = cm 2 > = / 3 V p,rd,y = 1.00 = kn / 3 V p,rd,z = 1.00 = kn (3): η may be conservatively taken equal to (1): Partial factor γ M0 = 1.00 is recommended a) Finite Element Method: Case : According to FEM analysis: V p,rd,y = kn V p,rd,z = kn we have: V Ed = V y = 200 kn, = 20 knm 1 The sections mentioned in the margins refer to EN :2005 [1] unless otherwise specified. SOFiSTiK 2018 Benchmark No. 13 5

6 V Ed = 200 = < 0.5 V p,rd,y (5): Eq. 6.30: The reduced design plastic resistance moment (for ρ = 0) no reduction of moment resistance due to shear,v,rd = W p,z ƒ y = = 4789 kncm = knm γ M0 1 = 20,V,Rd = Case : V Ed = V y = 300 kn, = 20 knm V Ed = 300 = < 0.5 V p,rd,y no reduction of moment resistance due to shear,v,rd = W p,z ƒ y γ M0,V,Rd = 47.89kNm = 20,V,Rd = b) EC3 Tables: V p,rd,y = kn V p,rd,z = kn Case : V Ed = V y = 200 kn, = 20 knm V Ed = 200 = < 0.5 V p,rd,y (5): Eq. 6.30: The reduced design plastic resistance moment (for ρ = 0) no reduction of moment resistance due to shear,v,rd = W p,z ƒ y = = 4789 kncm = knm γ M0 1 = 20,V,Rd = Case : V Ed = V y = 300 kn, = 20 knm V Ed = 300 = > 0.5 V p,rd,y (3): The reduced moment resistance should be taken as the design re- reduction of moment resistance due to shear sistance of the cross-section, calculated using a reduced yield strength (1 ρ) ƒ y for the shear area 6 Benchmark No. 13 SOFiSTiK 2018

7 2 VEd ρ = 1 V p,rd,y = = ,V,Rd = (1 ρ) Wp,z ƒ y γ M0,V,Rd = ( ) = knm = 20,V,Rd = SOFiSTiK 2018 Benchmark No. 13 7

8 5 Conclusion This example shows the calculation of the resistance of steel cross-section for bending and shear. It has been shown that the results are reproduced with excellent accuracy. y V y V y z a) FEM Analysis (SOFiSTiK) a) Reference Figure 2: Visualisation of areas and approach for calculating the V p,rd,y V z V z y z a) FEM Analysis (SOFiSTiK) a) Reference Figure 3: Visualisation of areas and approach for calculating the V p,rd,z By using FEM analysis the V p,rd,y and V p,rd,z values are deviating from the reference values. The reason behind this difference is shown in Fig. 2, 3, 4 and 5. FEM analysis represents a real physical behaviour, because as shown in Fig. 2 and 3 the calculated area is taken approximately for the reference V p,rd,y and V p,rd,z values. The results by using FEM analysis are more accurate in reality than the reference example. 8 Benchmark No. 13 SOFiSTiK 2018

9 SOFiSTiK AG * Bruckmannring 38 * Oberschleißheim SOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00) SOFiSTiK AG * Bruckmannring 38 * Oberschleißheim Page 1 SOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00) Page SOFiSTiK AG m Y X Z Figure 4: V p,rd,y - SOFiSTiK Results (FEM) Shear stress of the Cross section in Element, Loadcase 2 Vy = 1.0, 1 cm 3D = MPa (Max=0.377) Shear stress of the Cross section from middle of element, Loadcase M 21 Vy : 1.23 = 1.0, from e-04 to step MPa Y X Z M 1 : 1.23 m SOFiSTiK AG * Bruckmannring 38 * Oberschleißheim SOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00) SOFiSTiK AG * Bruckmannring 38 * Oberschleißheim Page 1 SOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00) Page SOFiSTiK AG m Y X Z Figure 5: V p,rd,z - SOFiSTiK Results (FEM) Shear stress of the Cross section in Element, Loadcase 3 Vz = 1.0, 1 cm 3D = 2.16 MPa (Max=0.895) Shear stress of the Cross section from middle of element, Loadcase M 3 Vz 1 : = , from to step MPa Y X Z M 1 : 1.23 m 6 Literature [1] EN : Eurocode 3: Design of concrete structures, Part 1-1: General rules and rules for buildings. CEN [2] Schneider. Bautabellen für Ingenieure. 21th. Bundesanzeiger Verlag, SOFiSTiK 2018 Benchmark No. 13 9

Design of a Steel I-section for Bending, Shear and Axial Force

Benchmark Example No. 20 Design of a Steel I-section for Bending, Shear and Axial Force SOFiSTiK 2018 VERiFiCATiON MANUAL DCE-EN20: Design of a Steel I-section for Bending, Shear and Axial Force VERiFiCATiON