Creep and Shrinkage Calculation of a Rectangular Prestressed Concrete CS

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1 Benchmark Example No. 18 SOFiSTiK 2018

2 VERiFiCATiON MANUAL DCE-EN18: 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 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): DIN Design Code(s): DIN EN Module(s): AQB, CSM Input file(s): creep shrinkage.dat 1 Problem Description The problem consists of a simply supported beam with a rectangular cross-section of prestressed concrete, as shown in Fig. 1. The time dependent losses are calculated, considering the reduction of stress caused by the deformation of concrete due to creep and shrinkage, under the permanent loads. h M y N p A p z p Figure 1: Problem Description 2 Reference Solution This example is concerned with the calculation of creep and shrinkage on a prestressed concrete cs, subject to bending and prestress force. The content of this problem is covered by the following parts of DIN EN :2004 [1]: Creep and Shrinkage (Section 3.1.4) Annex B: Creep and Shrinkage (Section B.1, B.2) Time dependent losses of prestress for pre- and post-tensioning (Section ) The time dependant losses may be calculated by considering the following two reductions of stress [1]: due to the reduction of strain, caused by the deformation of concrete due to creep and shrinkage, under the permanent loads the reduction of stress in the steel due to the relaxation under tension. In this Benchmark the stress loss due to creep and shrinkage will be examined. 3 Model and Results Benchmark 17 is here extended for the case of creep and shrinkage developing on a prestressed concrete simply supported beam. The analysed system can be seen in Fig. 2, with properties as defined in Table 1. Further information about the tendon geometry and prestressing can be found in Benchmark 17. The beam consists of a rectangular cs and is prestressed and loaded with its own weight. A calculation of the creep and shrinkage is performed in the middle of the span with respect to DIN EN :2004 SOFiSTiK 2018 Benchmark No. 18 3

4 (German National Annex) [1], [2]. The calculation steps [3] are presented below and the results are given in Table 2 for the calculation with AQB. For CSM only the results of the creep coefficients and the final losses are given, since the calculation is performed in steps. Table 1: Model Properties Material Properties Geometric Properties Loading (at = 10 m) Time C 35/45 h = cm M g = 1250 knm t 0 = 28 dys Y 1770 b = cm N p = kn t s = 0 dys RH = 80 L = 20.0 m t eƒ ƒ = dys A p = 28.5 cm 2 L Figure 2: Simply Supported Beam Table 2: Results Result AQB CSM+AQB Ref. ε cs ε ϕ ϕ(t, t 0 ) Δσ p,c+s [MP] ΔP c+s [kn] Benchmark No. 18 SOFiSTiK 2018

5 4 Design Process 1 Design with respect to DIN EN :2004 (NA) [1] [2]: 2 Material: Concrete: C 35/ 45 E cm = N/mm 2 ƒ ck = 35 N/mm 2 3.1: Concrete 3.1.2: Tab. 3.1: E cm, ƒ ck and ƒ cm for C 35/45 ƒ cm = 43 N/mm 2 Prestressing Steel: Y 1770 E p = N/mm 2 3.3: Prestressing Steel (3): E p for wires ƒ pk = 1770 N/mm , 3.3.3: ƒ pk Characteristic tensile strength of prestressing steel Prestressing system: BBV L mm 2 19 wires with area of 150 mm 2 each, giving a total of A p = 28.5 cm 2 Cross-section: A c = = 1 m 2 Diameter of duct ϕ dct = 97 mm Ratio α E,p = E p / E cm = / = A c,netto = A c π (ϕ dct /2) 2 = m 2 A de = A c + A p α E,p = m 2 Load Actions: Self weight per length: γ = 25 kn/m At = 10.0 m middle of the span: M g = g 1 L 2 / 8 = 1250 knm N p = P m0 ( = 10.0 m) = kn (from SOFiSTiK) Calculation of stresses at = 10.0 m midspan: Position of the tendon: z cp = 0, 3901 m Prestress and self-weight at con. stage sect. 0 (P+G cs0) N p = kn and M g = 1250 knm 1 The tools used in the design process are based on steel stress-strain diagrams, as defined in [1] 3.3.6: Fig The sections mentioned in the margins refer to DIN EN :2004 (German National Annex) [1], [2], unless otherwise specified. SOFiSTiK 2018 Benchmark No. 18 5

6 σ c M g + M p N p z p P m0,=10 +σ c z s the new position of the center of the cross-section for cs0 z p = z cp + z s M p bending moment caused by prestressing σ c,qp stress in concrete M p1 = N P z cp = = knm M p2 = N P z s = = knm M p = = knm M y = = knm σ c,qp = = 4.82 MP Calculation of creep and shrinkage at = 10.0 m midspan: t 0 minimun age of concrete for loading t s age of concrete at start of drying shrinkage t age of concrete at the moment considered t 0 = 28 days t s = 0 days t = t eƒ ƒ + t 0 = = days (6): Eq. 3.8: ε cs total shrinkage strain (6): Eq. 3.9: ε cd drying shrinkage strain (6): Eq. 3.10: β ds (6): h 0 the notional size (mm) of the cs h 0 = 2A c / = 500 mm ε cs = ε cd + ε c ε cd (t) = β ds (t, ts) k h ε cd,0 The development of the drying shrinkage strain in time is strongly depends on β ds (t, ts) factor. SOFiSTiK accounts not only for the age at start of drying t s but also for the influence of the age of the prestressing t 0. Therefore, the calculation of factor β ds reads: β ds = β ds (t, t s ) β ds (t 0, t s ) (t t s ) β ds = (t t s ) h 3 0 (t 0 t s ) (t 0 t s ) h 3 0 ( ) β ds = ( ) β ds = = (6): Tab. 3.3: k h coefficient depending k h = 0.70 for h mm on h 0 ε cd,0 = 0.85 ( α ds1 ) exp Annex B.2 (1): Eq. B.11: ε cd,0 basic drying shrinkage strain Annex B.2 (1): Eq. B.12: β RH RH the ambient relative humidity (%) Annex B.2 (1): α ds1, α ds1 coefficients depending on type of cement. For class N α ds1 = 4, α ds2 = 0.12 (28 0) (28 0) α ds2 ƒcm 10 ƒ 6 β cmo RH RH β RH = = = RH ε cd,0 = 0.85 ( ) exp Benchmark No. 18 SOFiSTiK 2018

7 ε cd,0 = ε cd = = ε cd = = / ε c (t) = β s (t) ε c ( ) (6): Eq. 3.11: ε c autogenous shrinkage strain ε c ( ) = 2.5 (ƒ ck 10) 10 6 = 2.5 (35 10) (6): Eq. 3.12: ε c ( ) ε c ( ) = = / Proportionally to β ds (t, ts), SOFiSTiK calculates factor β s as follows: β s = β s (t) β s (t 0 ) β s = 1 e 0.2 t t 0.2 t e 0 = e 0 e 0.2 t (6): Eq. 3.13: β s β s = ε = ε cd,0 + ε c ( ) = ε = ε c = = = / ε cs = = ε absolute shrinkage strain negative sign to declare losses negative sign to declare losses ϕ(t, t 0 ) = ϕ 0 β c (t, t 0 ) Annex B.1 (1): Eq. B.1: ϕ(t, t 0 ) creep coefficient ϕ 0 = ϕ RH β(ƒ cm ) β(t 0 ) Annex B.1 (1): Eq. B.2: ϕ 0 notional creep coefficient ϕ RH = RH/ α 1 α 2 Annex B.1 (1): Eq. B.3: ϕ RH factor for h 0 effect of relative humidity on creep β(ƒ cm ) = 16.8 ƒcm = 16.8/ 43 = α 1 = α 2 = = ƒ cm = ƒ cm Annex B.1 (1): Eq. B.4: β(ƒ cm ) factor for effect of concrete strength on creep Annex B.1 (1): Eq. B.8c: α1, α2, α3 coefficients to consider influence of concrete strength α 3 = ϕ RH = = ƒ cm β(t 0 ) = t 0 = t 0,T / t t 1.2 0,T = Annex B.1 (1): Eq. B.5: β(t 0 ) factor for effect of concrete age at loading on creep + 1 α 0.5 Annex B.1 (2): Eq. B.9: t 0,T temperature adjusted age of concrete at loading adjusted according to expression B.10 SOFiSTiK 2018 Benchmark No. 18 7

8 Annex B.1 (3): Eq. B.10: t T temperature adjusted concrete age which replaces t in the corresponding equations Annex B.1 (2): Eq. B.9: α a power which depends on type of cement For class N α = 0 t T = n =1 e (4000/[273+T(Δt )] 13.65) Δt t 0,T = 28 e (4000/[273+20] 13.65) = = t 0 = = 28.0 β(t 0 ) = 1 = Annex B.1 (1): Eq. B.7: β c (t, t 0 ) coefficient to describe the development of creep with time after loading β c (t, t 0 ) = (t t 0 ) (β H + t t 0 ) β H = (0.012 RH) 18 Annex B.1 (1): Eq. B.8: βh coefficient h α α 3 depending on relative humidity and notional member size β H = ( ) β H = = β c (t, t 0 ) = ϕ 0 = = Annex B.1 (3): The values of ϕ(t, t 0 ) given above should be associated with the tangent modulus E c (2): The values of the creep coefficient, ϕ(t, t 0 ) is related to E c, the tangent modulus, which may be taken as 1.05 E cm ϕ(t, t 0 ) = /1.05 = According to EN, the creep value is related to the tangent Young s modulus E c, where E c being defined as 1.05 E cm. To account for this, SOFiSTiK adopts this scaling for the computed creep coefficient (in SOFiSTiK, all computations are consistently based on E cm ) (2): Eq. 5.46: ΔP c+s+r time dependent losses of prestress ε cs E p + 0.8Δσ pr + ϕ(t, t 0 ) σ c,qp E ΔP c+s+r = A p Δσ p,c+s+r = A cm p 1 + E p A p 1 + A c z 2 [ ϕ(t, t E cm A c cp 0 )] c In this example only the losses due to creep and shrinkage are taken into account, the reduction of stress due to relaxation (Δσ pr ) is ignored. E p Δσ p,c+s variation of stress in tendons Δσ p,c+s = due to creep and shrinkage at location, at time t Δσ p,c+s = MP ( 4.82) [ ] ΔP c+s = A p Δσ p,c+s = = kn 8 Benchmark No. 18 SOFiSTiK 2018

9 5 Conclusion This example shows the calculation of the time dependent losses due to creep and shrinkage. It has been shown that the results are in very good agreement with the reference solution. 6 Literature [1] DIN EN /NA: Eurocode 2: Design of concrete structures, Part 1-1/NA: General rules and rules for buildings - German version EN :2005 (D), Nationaler Anhang Deutschland - Stand Februar CEN [2] F. Fingerloos, J. Hegger, and K. Zilch. DIN EN Bemessung und Konstruktion von Stahlbeton- und Spannbetontragwerken - Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. BVPI, DBV, ISB, VBI. Ernst & Sohn, Beuth, [3] Beispiele zur Bemessung nach Eurocode 2 - Band 1: Hochbau. Ernst & Sohn. Deutschen Betonund Bautechnik-Verein E.V SOFiSTiK 2018 Benchmark No. 18 9

Shear at the interface between concrete cast

Benchmark Example No. 11 Shear at the interface between concrete cast SOFiSTiK 2018 VERiFiCATiON MANUAL DCE-EN11: Shear at the interface between concrete cast VERiFiCATiON MANUAL, Version 2018-7 Software