LOAD CARRYING BEHAVIOR OF PRESTRESSED BOLTED STEEL FLANGES CONSIDERING RANDOM GEOMETRICAL IMPERFECTIONS

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1 8 th ASCESpecialtyConferenceonProbabilisticMechanicsandStructuralReliability PMC LOAD CARRYING BEHAVIOR OF PRESTRESSED BOLTED STEEL FLANGES CONSIDERING RANDOM GEOMETRICAL IMPERFECTIONS C. Bucher and M. Ebert Bauhaus-University, Weimar, Germany Abstract Prestressed ring flanges are frequently used in slender structures subjected to wind loads such as chimneys or wind turbine towers. The alternating forces due to wind can cause both fatigue failure and ultimate load failure of the bolts. For the design of such connections it is therefore important to take into account possible variations of the bolt forces. Due to manufacturing tolerances the contact surfaces are not ideally plane. While the initial forces (under dead load only) in the bolts can be considered deterministic and uniformly distributed along the circumference of the tube, the forces will be redistributed non-uniformly in the presence of lateral loads. This is a consequence of the non-uniform prestressing state of the flange material. A random field model is assumed to describe the geometry of the contact surface. The mean surface is assumed to be the ideal contact plane, deviations are assumed to be log-normally distributed with an exponential spatial correlation function. A Stochastic Finite Element Model with 13,000 DOF and 1500 random variables in conjunction with Monte-Carlo methods is then utilized to derive the statistical properties of the random ultimate load of the joint. It is shown that the effect of imperfect contact surfaces can considerably deteriorate structural performance. Introduction Recent investigations of prestressed flange connections focussed on fatigue-relevant tension force amplitudes in the screws and on failure analysis. These investigations were based on numerous experiments, such as those by Petersen (1990a, 1990b, 1998). Schmidt & Neuper (1997) and Ebert & Bucher (1998) analyzed a segment by means of finite elements and found results similar to those reported by Petersen (1990a). A simplified model for the full flange was developed by Ebert & Bucher (1998). This model showed a considerable re-distribution of tension forces which leads to a reduction of stress amplitudes in the screws. An open problem is the question of geometrical imperfections of the contact surface in the flange. This leads to non-uniform distribution of contact pressures and screw forces, and finally to an increase in tension force amplitudes in the screws. Petersen (1990b) performed an experimental study on these imperfections and derived recommendations for the design. Based on a segment model, his conclusions are that the fatigue life is not substantially influenced by imperfections. In contrast, Schmidt et al. (1999) based their analysis on quarter- and half-ring flange models. Assuming deterministic imperfection shapes they show a significant increase in fatigue-relevant stress amplitudes. Bucher, Ebert 1

2 In reality, imperfections are random in space, extension, and magnitude. This leads to a loss of structural symmetry. Consequently, a full ring flange model must be investigated. This paper describes an approach to analyze the effect of random imperfection based on the Monte-Carlo method. Modeling Finite-Element-Model of the flange An interior flange connection with 66 screws (M36) is investigated. The outer diameter of the flange is 3.20m, the flange thickness is 65mm and the flange width is 130mm. Thewall thickness of the cylinder is 14mm.The screws are pre-stressed with a force of F v = 510kN. Figure 1. Finite Element Model of the Flange. The FE-models of the full ring (cf. Fig. 1) was modeled within the software package SLang (2000). In view of the required stochastic analysis the model was chosen in a rather simple way. Details are also shown in Fig. 1. The model consists of 3,500 elements with 13,000 degrees of freedom. 1,500 non-linear spring elements represent the contact in the flange. These springs have a high stiffness in compression and a low stiffness in tension. Additionally, these elements can have an offset H as shown in Fig. 2. Force H Displacement (Compression) Offset Figure 2. Force-Displacement Relation of Contact Springs. In a first step, an analysis with perfect geometry is carried out. The bending moment (e.g. Bucher, Ebert 2

3 caused by wind) is increased from 0 to 30MN. The resulting maximum screw force in the tension zone is shown in Fig. 3. The results compare well to those obtained by Schmidt et al. (1999). 10.0E5 Screw Force [N] Half Ring (Schmidt et al., 1999) Present Model Applied Moment [MNm] Figure 3. Maximum Screw Force vs. Bending Moment. Stochastic Model of the Contact Surface A hypothetic stochastic model for the surface assuming a homogeneous and isotropic lognormally distributed random H(x) field is chosen. In the computational procedure, the random field is represented by offsets H i of the contact springs (cf. Fig. 2). The random field H has a standard deviation of σ H = 0.2mm and an autocorrelation function R HH (x,y)=σ 2 x y Hexp( L ) with a correlation length L = 200mm. x and y are the Cartesian coordinates in the contact surface. The simulation model is based on a spectral decomposition of the random field (Ghanem & Spanos, 1991). Specific details regarding the treatment of geometrical imperfections are given e.g. by Schorling & Bucher, Realizations of the random field are obtained by linear combination of the eigenvectors of the covariance matrix with random amplitudes. Three selected eigenvectors (random field mode shapes) are shown in Fig. 4. Since the model contains 1500 contact elements, the spectral decomposition yields 1500 random field mode shapes. For computational reasons it is useful to reduce this number. A comparative anaylsis taking into account 800 and 128 random variables was performed. There were virtually no differences in the results. A realisation of the contact surface (based on 128 random variables, magnified) is shown in Fig. 5. For the stochastic analysis it is assumed that the prestressing forces in the screws are deterministic. The initial deformation state for one sample is shown in Fig.5. Due to the random pressure distribution in the contact zone the application of a bending moment causes random changes in the screw forces. Based on the Monte-Carlo method the statistics of the screw forces are determined. This requires a nonlinear contact analysis in each simulation. The computational effort can be reduced by applying the Latin-Hypercube sampling technique. This method is particularly useful for estimation of response variability from a very small number of random samples. A recent study by Novák et al. (2000) showed the excellent applicability of the method for linear and nonlinear random field problems. Bucher, Ebert 3

4 Figure 4. 2 nd 4 th,and6 th Mode Shapes of Random Field. Results Figure 5. Realization of Random Field, a) initial; b) after prestressing. In a first step, the evolution of the screw forces with increasing bending moment is monitored in the range of linear material behavior. The results from 300 simulations are summarized in Figs. 6. Screw Force [kn] 10.0E E Deterministic Bending Moment M = 18 MNm Deterministic Geometry Bending Moment [MN E2 Screw Force [kn] Figure 6. Random Screw Forces on Tension Side. Movie 1 shows the evolution of the stresses in a section of the flange (perfect vs. imperfect) with increasing bending moment. Movie 1. Stresses in Bolts and Flange on Tension Side, a) perfect; b) imperfect. Bucher, Ebert 4

5 For the load level 12, Fig. 6 shows the histogram of the screw force on the tension side of the flange ring. Selected results are also summarized in Table 1. Bending Moment [MNm] Mean Value [kn] Coeff. of Var. [%] Deterministic [kn] Table 1: Statistics of Screw Forces The comparison to Latin Hypercube Sampling results from 32 simulations (using 128 random variables) indicate a highly favorable performance of LHS. Fig. 7 compares the mean values and standard deviations of the screw force on the tension side from plain MC and LHS. The results are in very good agreement. Mean Values LHS PMC Coefficient of Variation LHS PMC 6.5E E7 9.0E E7 Figure 7. Statistics of Random Screw Forces on Tension Side vs. Bending Moment. In a second step, the effect of brittle failure of the screws on the load carrying capacity of the flange connection was analyzed. Redistribution of the internal forces was taken into account. The ultimate bending moment leading to structural failure showed an significant level of uncertainty with a coefficient of variation of about 8%. Again, the results from LHS are very good as indicated in Table 2. Method Mean Value [MNm] Coeff. of Variation [%] LHS (32 samples) MCS (200 samples) Table 2: Statistics of Ultimate Bending Moment Bucher, Ebert 5

6 Conclusions The results indicate the significance of geometrical imperfections in terms of fatigue life of prestressed flange connections. In addition, also the ultimate bending moment of such a flange is considerably influenced by imperfections. The results are based on a full FEmodel of the flange which takes into account the non-uniform initial distributions of stresses in the structure as well as the redistribution due to increased loading. The assumptions on the underlying random field are hypothetical as yet. They will be substantiated by means of experimental investigations to be carried out in the future. Application of advanced Monte-Carlo simulation techniques such as Latin Hypercube Sampling can lead to a substantial reduction of numerical efforts and thus make a stochastic analysis of fairly complex structural systems feasible. Acknowledgements The theoretical background of this work is based on results obtained within the research project Bu-987/5-1 Experimental and numerical investigation of the dynamic behavior of beams taking into account stochastic material properties supported by the German Research Foundation. The authors gladly acknowledge this support. The Latin Hypercube samples were provided by D. Novák and W. Lawanwisut of the Technical University of Brno, Czech Republic, which is gratefully acknowledged. References Ebert, M., Bucher, C. (1998), Stochastische nichtlineare Untersuchung vorgespannter Schraubenverbindungen unter Windeinwirkung in:baukonstruktionen unter Windeinwirkung, WTG-Berichte Nr.5, Hrsg. Udo Peil, Windtechnologische Gesellschaft e.v., Aachen Braunschweig, 1998, Ghanem, R.G., Spanos, P.D. (1991), Stochastic Finite Elements: A Spectral Approach. Springer, Berlin. Novák, D.; Lawanwisut, W.; Bucher, C. (2000), Simulation of random fields based on orthogonal transformation of covariance matrix and Latin Hypercube Sampling for SFEM, paper accepted for presentation, Monte Carlo Petersen, C. (1990a) Stahlbau, F. Vieweg, Braunschweig, Wiesbaden. Petersen, C. (1990b), Tragfähigkeit imperfektionsbehafteter geschraubter Ringflansch-Verbindungen,Stahlbau, 59, Petersen, C. (1998), Nachweis der Betriebsfestigkeit exzentrisch beanspruchter Ringflanschverbindungen, Stahlbau, 67, Schmidt, H; Neuper, M. (1997), Zum elastostatischen Tragverhalten exzentrisch gezogener L-Stöße mit vorgespannten Schrauben, Stahlbau, 66, Schmidt H., Winterstetter Th. A., Kramer M. (1999), Nonlinear Elastic Behaviour of Imperfect, Eccentrically Tensioned L-Flange Ring Joints with Prestressed Bolts as Basis for the Fatigue Design, CD ROM Proceedings of the European Conference on Computational Mechanics, Munich, Germany. Schorling, Y.; Bucher, C. (1999), Stochastic Stability of Structures with Random Imperfections. In: B.F: Spencer Jr. and E.A. Johnson (eds.): Stochastic Structural Dynamics, A.A. Balkema, Rotterdam/Brookfield, pp SLang (2000), SLang - the Structural Language Version 4.0, User s manual, Institut für Strukturmechanik, Bauhaus-Universität Weimar. Bucher, Ebert 6