Design of a Rectangular CS for Bending

Transcription

1 Benchmark Example No. 2 SOFiSTiK 2018

2 VERiFiCATiON MANUAL DCE-EN2: VERiFiCATiON MANUAL, Version Software Version: SOFiSTiK 2018 Copyright 2019 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 protecte by copyright laws. No part of it may be translate, copie or reprouce, in any form or by any means, without written permission from SOFiSTiK AG. SOFiSTiK reserves the right to moify or to release new eitions of this manual. The manual an the program have been thoroughly checke for errors. However, SOFiSTiK oes not claim that either one is completely error free. Errors an omissions are correcte as soon as they are etecte. 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 ranom sampling. Front Cover Project: New SOFiSTiK Office, Nuremberg Contractor: WOLFF & MLLER, Stuttgart Architecture: WABE-PLAN ARCHITEKTUR, Stuttgart Structural Engineer: Boll un Partner. Beratene Ingenieure VBI, Stuttgart MEP: GM Planen + Beraten, Griesheim Lea Architect: Gerhar P. Wirth gpwirtharchitekten, Nuremberg Vizualisation: Armin Dariz, BiMOTiON GmbH

3 Overview Design Coe Family(s): DIN Design Coe(s): DIN EN Moule(s): AQB Input file(s): rectangular bening.at 1 Problem Description The problem consists of a rectangular section, as shown in Fig. 1. The cross-section is esigne for an ultimate moment M E an the require reinforcement is etermine. b A s2 h 2 M E A s1 Figure 1: Problem Description 2 Reference Solution This example is concerne with the esign of oubly reinforce sections for ULS, subject to pure flexure, such as beams. The content of this problem is covere by the following parts of DIN EN :2004 [1]: Design stress-strain curves for concrete an reinforcement (Section 3.1.7, 3.2.7) Basic assumptions for section esign (Section 6.1) Reinforcement (Section , ) ε c σ c A s2 2 z s2 ε s2 σ s2 F s2 F c z z s1 A s1 1 ε s1 σ s1 F s1 Figure 2: Stress an Strain Distributions in the Design of Doubly Reinforce Cross-sections In oubly reinforce rectangular beams, the conitions in the cross-section at the ultimate limit state, are assume to be as shown in Fig. 2. The esign stress-strain iagram for reinforcing steel consiere in this example, consists of an incline top branch, as presente in Fig. 3 an as efine in DIN EN :2004 [1] (Section 3.2.7). SOFiSTiK 2018 Benchmark No. 2 3

4 σ A ƒ tk,c = 525N/mm 2 ƒ yk ƒ y = ƒ yk /γ s B A B Iealise Design ε ε = 25 0 / 00 Figure 3: Iealise an Design Stress-Strain Diagram for Reinforcing Steel 3 Moel an Results The rectangular cross- section, with properties as efine in Table 1, is to be esigne, with respect to DIN EN :2004 (German National Annex) [1], [2], to carry an ultimate moment of 135 knm. The calculation steps with ifferent esign methos [3] [4] [5] are presente below an the results are given in Table 2. Here, it has to be mentione that these stanar methos employe in orer to calculate the reinforcement are approximate, an therefore eviations often occur. Table 1: Moel Properties Material Properties Geometric Properties Loaing C 20/25 h = 40.0 cm M E = 135 knm B 500A = 35.0 cm 2 = 5.0 cm b = 25 cm Table 2: Results SOF. General Chart [3] ω Table [3] k Table [3] A s1 [cm 2 /m] A s2 [cm 2 /m] Benchmark No. 2 SOFiSTiK 2018

5 4 Design Process 1 Design with respect to DIN EN :2004 (NA) [1] [2]: 2 Material: Concrete: γ c = 1.50 Steel: γ s = 1.15 ƒ ck = 20 MP ƒ c = cc ƒ ck /γ c = /1.5 = MP (NDP) : (1), Tab. 2.1DE: Partial factors for materials Tab. 3.1: Strength for concrete 3.1.6: (1)P, Eq. (3.15): cc = 0.85 consiering long term effects ƒ yk = 500 MP 3.2.2: (3)P: yiel strength ƒ yk = 500 ƒ y = ƒ yk /γ s = 500/1.15 = MP MP Design Loa: 3.2.7: (2), Fig. 3.8 N E = 0 M Es = M E N E z s1 = 135 knm Design with respect to General Design Chart Bening with axial force for rectangular cross-sections: μ Es = M Es b 2 ƒ c = μ Es > μ Es, m = = compression reinforcement require 5.4: (NA.5): Linear elastic analysis ξ = height of comression zone / 0.45 for C12/15 C50/60 from esign chart for μ Es, m = an 2 / = : Tab. 9.1 [3]: General Chart for up to ε s1 = ; ε s2 = ; ζ = z/ = for ε s1 = σ s1 = MP for ε s2 = σ s1 = MP M Es, m = μ Es, m b 2 ƒ c = knm ΔM Es = M Es M Es, m = = 32.3 knm A s1 = 1 MEs, m σ s1 ζ + ΔM Es + N E = cm 2 2 A s2 = 1 σ s2 ΔM Es 2 = 2.47 cm 2 C50/60 - Section with compression reinforcement Design with respect to ω (or μ s )Table for rectangular crosssections: μ Es = M Es b 2 ƒ c = = : (NA.5): Linear elastic analysis Because the internal force etermination is one on the basis of a linear ξ = height of comression zone / 0.45 for C12/15 C50/60 1 The tools use in the esign process are base on steel stress-strain iagrams, as efine in [1] 3.2.7:(2), Fig. 3.8, which can be seen in Fig The sections mentione in the margins refer to DIN EN :2004 (German National Annex) [1], [2], unless otherwise specifie. SOFiSTiK 2018 Benchmark No. 2 5

6 elastic calculation, then ξ m = 0.45 is chosen. Referring to the esign Tab. 9.2 [3]: ω Table for up to C50/60 table with compression reinforcement an for 2 / = 0.15: - Rectangular section with compression reinforcement ω 1 = ; ω 1 = A s1 = 1 ƒ y (ω 1 b ƒ c + N E ) = cm 2 A s2 = ƒ c ƒ y (ω 2 b ) = 2.52 cm 2 Tab. 9.3 [3]: k Table for up to C50/60 - Rectangular section with compression reinforcement 5.4: (NA.5): Linear elastic analysis Design with respect to k Design Table for rectangular crosssections: k = MEs /b = 35 = /0.25 Not able to rea values from k table for simply reinforce rectangular cross-sections compression reinforcement is require ξ = height of comression zone / 0.45 for C12/15 C50/60 Because the internal force etermination is one on the basis of a linear elastic calculation, then ξ m = 0.45 is chosen. Referring to the k table with compression reinforcement: k s1 = 2.740; k s2 = (interpolate values for k = 1.51) ρ 1 = 1.021; ρ 2 = (interpolate values for 2 / = an k s1 = 2.740) A s1 = ρ 1 k s1 MEs A s2 = ρ 2 k s2 MEs + N E σ s1 = cm 2 = 2.43 cm2 6 Benchmark No. 2 SOFiSTiK 2018

7 5 Conclusion This example shows the calculation of the require reinforcement for a rectangular beam cross-section uner bening. Various ifferent reference solutions are employe in orer to compare the SOFiSTiK results to. It has been shown that the results are reprouce with excellent accuracy. 6 Literature [1] DIN EN /NA: Eurocoe 2: Design of concrete structures, Part 1-1/NA: General rules an rules for builings - German version EN :2005 (D), Nationaler Anhang Deutschlan - Stan Februar CEN [2] F. Fingerloos, J. Hegger, an K. Zilch. DIN EN Bemessung un Konstruktion von Stahlbeton- un Spannbetontragwerken - Teil 1-1: Allgemeine Bemessungsregeln un Regeln für en Hochbau. BVPI, DBV, ISB, VBI. Ernst & Sohn, Beuth, [3] K. Holschemacher, T. Müller, an F. Lobisch. Bemessungshilfsmittel für Betonbauteile nach Eurocoe 2 Bauingenieure. 3r. Ernst & Sohn, [4] Beispiele zur Bemessung nach Eurocoe 2 - Ban 1: Hochbau. Ernst & Sohn. Deutschen Betonun Bautechnik-Verein E.V [5] R. S. Narayanan an A. W. Beeby. Designers Guie to EN an EN Eurocoe 2: Design of Concrete Structures. Thomas Telfor, SOFiSTiK 2018 Benchmark No. 2 7