How To Model The Non Linear Behaviour Of A Concrete Beam

Transcription

1 Stiness-oriented numerical model for non-linear reinforced concrete beam systems Master Thesis presentation SEng Agnieszka KNOPPIKWRÓBEL Supervised by: PhD SEng Grzegorz WANDZIK Silesian University of Technology Faculty of Civil Engineering Gliwice, 9 Feb 2011

2 Presentation outline Introduction 1 Introduction Objective of thesis Range of thesis 2 Structural analysis Problems in standards 3 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM 4 5 Objective of thesis Range of thesis

3 Objective of thesis Range of thesis Objective of thesis To derive a numerical model for designing of exural reinforced concrete beams taking into consideration non-linear behaviour of reinforced concrete and redistribution of internal forces as a result of stiness degradation of the elements due to crack formation at exure, thus providing a unied algorithm for static calculations and dimensioning.

4 Objective of thesis Range of thesis Range of thesis 1 methods of non-linear analysis 2 theory of stiness degradation Practical solution 1 numerical model

5 Structural analysis Problems in standards Linear-elastic analysis The most simplied method, but with inconsistencies: Ultimate Limit State linear analysis in static calculations M Sd plastic properties considered in cross-section resistance M Rd concrete section only Serviceability Limit State modied stiness values B I and B II cracking moment M cr creep and shrinkage (in parametric form)

6 Structural analysis Problems in standards Linear-elastic analysis with limited redistribution Bending moments distribution resulting from linear-elastic analysis redistributed: δ = M red M el 100% Yielding of steel ensured by a specic value of reinforcement strain or a depth of neutral axis limitation. Resulting allowable redistribution: European standards δ 30% American standards δ 20%

7 Structural analysis Problems in standards Plastic analysis Introduction to plastic properties of concrete: plastic hinge, rotational capacity, ductility conditions. Method taking into account plastic properties of concrete for beam elements: limit equilibrium method.

8 Structural analysis Problems in standards Non-linear analysis Applicable for both ULS and SLS, provided that equilibrium and compatibility are satised and an adequate non-linear behaviour for materials is assumed. No consistent design procedure is provided except for some design requirements which must be satised.

9 Structural analysis Problems in standards Problems with non-linear analysis in standards little or no information; poor and extensively simplied representation of material properties, applicability to a very narrow range of engineering problems, need of reference to literature with no available consistent solution, only numerical analysis possible, great designer experience needed.

10 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Geometry Arbitrary statically-indeterminate beam with known geometry, reinforcement, material properties and loading pattern. Figure 1: Exemplary continuous beam

11 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Discretisation Division into segments of the same cross-section type. Assignment of initial stiness. Figure 2: Discretisation of the beam

12 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Models for concrete and steel CONCRETE: MC2010 model for compression and tension: STEEL: EC2 model, steel as isotropic material: Figure 3: Material model for concrete Figure 4: Material model for steel

13 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Curvature Curvatures κ k taken at k equal intervals for α k ( 90, 90 ): Boundary values for x k = 0 and x k = h: (a) κ > 0 (b) κ < 0 Figure 5: Geometrical interpretation of curvature Figure 6: N 0k and N hk

14 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Bending moment Axial force determined in each sub-step for x kf until for x kf N k = 0. Then values of bending moments M k in pure bending in each k step for κ k calculated. (a) cross-section (b) strains (c) stresses (d) internal forces Figure 7: Determination of internal forces in section with numerical integration

15 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Momentcurvature relationship Momentcurvature relationship derived for each cross-section type j in both positive and negative range of bending moments (curvatures). Figure 8: Hypothetical Mκ diagram

16 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Stiness of cross-section Stiness must be determined in every cross-section after each r th loading step. Figure 9: Geometrical interpretation of section stiness Methods to determine the value of stiness: 1 nite dierence methods 2 curve tting interpolation approximation spline

17 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Stiness of segment Stiness of each segment i of a given cross-section type j = t(i) is taken as a mean value calculated for the maximum bending moment M i,r acting in that segment after r th loading step, taking into account the tension stiening eect coecient ξ i,r and stiness of a single cross-section B t(i),r : B mean,t(i),r = (1 ξ i,r )B t(i),r Stiness is changed within a segment when cracking occurs. Stiness cannot be regained, cracking cannot be reversed

18 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations General assumptions displacement nal element method used incremental method used for small increments to avoid necessity of iterations linearisation of K(u)u = P problem into Ku = P Calculations constant increments of load q bending moments determined in each node the maximum value of bending moment taken for the segment modication of stiness after each loading step

19 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 10: Step 1. Initial loading

20 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 11: Step R 1. Formation of the rst plastic hinge

21 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 12: Step R 2. Formation of the last plastic hinge

22 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Bending moment distribution 1 determination of stiness matrix and vector of nodal loads of each segment 2 formation of global stiness matrix and global vector of nodal loads (aggregation) global stiness matrix is a band matrix only entries on diagonal are aggregated 3 denition of vector of nodal displacements 4 introduction of boundary (support) conditions 5 solution u = K 1 P equation 6 calculation M and V forces

23 General conclusions design process requires idealisations model proposed approaches too simplied, inconsistent, with little information provided non-linear analysis must be used but it requires experience and numerical calculations numerical simulation requires development of numerical model and co-operation between scientist/engineer and software developer

24 Summary and future prospects objective of this thesis was achieved procedure was described in a very detail mathematical and geometrical interpretations were provided graphical representations were shown model is ready for implementation by a software expert not necessarily familiar with the discussed physical problem validation and evaluation of redistribution as a result of stiness degradation will be possible

25 Books 1 Gaw cki A.: Mechanika materiaªów i konstrukcji pr towych. Archiwum AlmaMater (2003) 2 Godycki wirko T.: Mechanika betonu. Wydawnictwo Arkady, Warszawa (1982) 3 Jirásek M., Baºant Z.: Inelastic analysis of structures. John Wiley & Sons, Chichester (2002) 4 Klemczak B.: Modelowanie efektów termiczno-wilgotno±ciowych i mechanicznych w betonowych konstrukcjach masywnych. Wydawnictwo Politechniki l skiej, Gliwice (2008) 5 Kanu M., ed.: Podstawy projektowania konstrukcji»elbetowych i spr»onych wedªug Eurokodu 2. Dolno±l skie Wydawnictwo Edukacyjne, Wrocªaw (2006) 6 Klamka J., Paweªczyk M., Wyrwaª J.: Numerical methods. Wydawnictwo Politechniki l skiej, Gliwice (2001) 7 Kuczy«ski W.: Konstrukcje betonowe. Kontynualna teoria zginania»elbetu. Pa«stwowe Wydawnictwo Naukowe, Šód¹ (1971) 8 Majewski S.: Mechanika betonu konstrukcyjnego w uj ciu spr»ysto-plastycznym. Wydawnictwo Politechniki l skiej, Gliwice (2003) 9 Mosley B., Bungey J., Hulse R.: Reinforced concrete design to Eurocode 2, 6 th ed. Palgrave MacMillan. New York (2007) 10 Pietrzak J., Rakowski G., Wrze±niowski K.: Macierzowa analiza konstrukcji, Wyd. 2 zm. Pa«stwowe Wydawnictwo Naukowe, Warszawa (1986) 11 Starosolski W.: Konstrukcje»elbetowe. Tom I, wyd. 8 zm. Wydawnictwo Naukowe PWN, Warszawa (2003) 12 Tichý M., Rákosník J.: Obliczanie ramowych konstrukcji»elbetowych z uwzgl dnieniem odksztaªce«plastycznych. Wydawnictwo Arkady, Warszawa (1971)

26 Articles 1 Bathe K.J., Walczak J., Welch A., Mistry N.: Nonlinear analysis of concrete structures. Computers & Structures vol. 32, no. 3/4, p (1989) 2 Bondy K.B.: Moment Redistribution: Principles and Practice Using ACI PTI Journal, p. 321 (2003) 3 D browski K.: Kilka uwag na temat metody plastycznego wyrównania momentów stosowanej w konstrukcjach»elbetowych. In»ynieria i Budownictwo nr 8/2002, str (2002) 4 J drzejczak M., Knau M.: Redystrybucja momentów zginaj cych w»elbetowych belkach ci gªych zasady polskiej normy na tle Eurokodu. In»ynieria i Budownictwo nr 8/2002, str (2002) 5 Noakowski P., Moncarz P.: Stiness-Oriented Design of R.C. Structures. Close-to-Reality and Practicable Computations. Procedures and Their Applications. Ksi ga jubileuszowa z okazji 70-lecia profesora Tadeusza Godyckiego- wirko, Politechnika Gda«ska, p (1998) 6 Scott R.H., Whittle R.T.: Moment redistribution eects in beams, Magazine of Concrete Research, 57, No. 1, p. 920 (2005)

27 Standards 1 ACI 318:2002 Building Code Requirements for Structural Concrete 2 BS 8110:1997 Structural Use of Concrete Part 1: Code of Practice for Design and Construction 3 CEB-FIP Model Code CEB-FIP Model Code 2010 (nal draft) 5 EN :2008 Eurocode 2: Design of Concrete Structures. Part 1-1: General Rules and Rules for Buildings 6 PN-B-03264:2002: Konstrukcje betonowe,»elbetowe i spr»one. Obliczenia statyczne i projektowanie

28 THANK YOU