Extended options and improved accuracy for determining of buckling load with Southwell plot method

Transcription

1 Abstract Extended options and improved accuracy for determining of buckling load with Southwell plot method Boris Blostotsky, Ph.D., Elia Efraim, Ph.D., Oleg Stanevsky, Ph.D., Leonid Kucherov, Ph.D., Alexander Zakrassov, Ph.D. Ariel University Southwell method consists of measuring some characteristic displacement or deformation in compressed structure as a function of the compression load. Buckling load is determined by approximating and extrapolating experimental data presented via specific combinations of measurements and experimental parameters. Theoretical analysis and experiments carried out in the civil engineering laboratory of the Ariel University comprise a research aiming at enhancement of the buckling load assessment method and at improvement of its accuracy. In this article the following is presented: - An analysis of the methods for creating a pre-strained state and the methodological basis for determining the required shape; - The analysis of experimental data approximation and extrapolation schemes which are used to determine the critical load; - The objectives and the methodology of experiment design, methods of estimation and improvement of the accuracy. The findings of this work can be used in the practice of laboratory and industrial stability tests. Keywords: buckling load, Southwell plot, laboratory testing 1. Introduction Non-destructive experimental approach for determining the critical load has been proposed by Ayrton and Perry [1], Southwell [2], Donnell [3] to test the theory of the stability of the Euler rods. It was found that the critical load can be determined by measuring lateral deformation of the rod subjected to a compressive force. This deformation is caused by initial imperfection of some kind ( form ) - curved bar, eccentric application of the compressive load, the heterogeneity of mechanical properties, etc. The maximum load during the test is limited by the mechanical strength, which is smaller than the critical strength. Therefore, the critical load is determined via approximation of the compressive load and lateral strain measurement presented in certain combinations proposed by Ayrton and Perry, Southwell, Donnell. To improve test accuracy, it has been proposed to create imperfection artificially by applying lateral force [4]. A method incorporating the lateral load for determination of the critical load for the frame in sway mode was proposed in [5]. A method of creating imperfection by bending moment with consequent measurement of the 1-10

2 rotation angle was proposed [6]. The method for the rod with an additional intermediate elastic support has been proposed in [7]. In this case, the critical load for a bar with hinged end supports was measured for given intermediate support stiffness. Numerous examples of use of the Southwell plot for determining the critical load of rods and plates, and also lateral buckling of beams are given in [8]. In all cases, the method hinges on the possibility of expressing the relationship between compression load and displacement in the special form [8]. Significant experience of laboratory testing of rods and frames has been accumulated in the Ariel University laboratory [9-13]. In this paper, the analysis and generalization of experience with Southwell plot shows options for method enrichments and the possibility to improve the accuracy of the experiment. 2. The test parameters and variables The deformed shape of the rod in the absence of axial compression force may be created by applying lateral force F e or bending moment M e. The rod may have a straight shape distortion due to manufacturing (imperfection). As measured parameters for the test lateral displacement of any section of the rod, rotation angle and the bending moment can be adopted. The moment can be measured by two strain gauges on both sides of a symmetrical cross-section of the column, connected by a bridge circuit with a common point (half-bridge). For the deformed shape due to the loading by F e or M e and axial compression force N, displacement, rotation angle ϕ and moment M are measured. In the case of imperfection a change in shape is due to compressive force. I this case displacement δ, rotation angle ϕ, and moment M are measured. In the first case, any measured value can be the result of the loading force or moment, as well as the result of loading axial force. Accordingly, the measured values can be F, M, ϕ F, ϕ M, M F, M M. Naturally, the loading section and the measurement can be different. 3. Mathematical models of approximation and variables of extrapolation The simplest model of an approximation and extrapolation is the linear model [14]. Analytical base for application of Southwell plot is a widely used formula P [15]: = 0, (1) 1 N N cr where - deformation due to application of an external lateral load ( F, M ) or deformation due to imperfection 0 and compression load, 0 - deformation or imperfection before the application of compression load, N cr - critical load for stability. 1-11

3 Structure of the expression (1) is similar when measuring the rotation angle ϕ F, ϕ M or moment M F, M M. The advantage of the mathematical model (1) is that when using the imperfection in the experiment, an unknown quantity 0 is not required. A similar advantage occurs when using a non-linear model in the form of = 0 n, (2) N 1 N cr However, its use for approximation and extrapolation requires information about the exponent n. All known ways to determine the critical load from the test follow from (1). For example, consider the case when the measured quantity is the displacement (in general) or δ (in the case of axial force). Thus, if the displacement is the result of the loading F e, then lateral stiffness at the adopted point of displacement measurement. From (1) follows k = k N 0 1, (3) Ncr F 0 = e k0, where k0 - where F k = e is the lateral stiffness conditioned by the compressive force. It follows from (3) that it is necessary to apply a linear model for approximation and extrapolation k = b + rn, (4) where b and r are the parameters determined by using the least squares method from the experimental values N and k. The critical load is determined from (4) when k = 0 N cr = b. r A method of determining similar. N cr by measuring of F, M, ϕ F, ϕ M, M F, M M is If the measured value is δ, then it follows from (1), that 1-12

4 δ = 0, (5) N cr 1 N which may be presented in the following form or δ = N δ 0 N cr N (6) cr = N N (7). δ N cr 0 Expression (6) which corresponds to the representation of Southwell, requires approximation in the form of δ = δ * r b. (8) N Critical force N cr = 1 is defined by the parameter r, obtained via least squares on r the test results. Expression (7) corresponds to the representation of Donnell, it requires approximation in the form of N = b + r * N. (9) δ The critical load N cr = b, where b - the parameter determined from the test results by using the least squares method. 4. Experiment design 4.1. Objectives. The aim of the design of experiment is developing a scheme that allows determining by the non-destructive method (Southwell plot) a critical stability load of an element or a structure for given error level and prescribed costs. Design tasks: - Adopt a method of creating a pre-deformed shape, natural or artificial type of loading for its creation; - Adopt the deformation parameter to measure (displacement, rotation angle, bending or torque moment); 1-13

5 - Adopt the type of variables for approximation and extrapolation (stiffness, variables of Southwell or Donnell) Methods of implementation The criterion for the validity of the initial deformed shape is its proximity to the buckling form (stability loss shape). This shape is determined by integrating the equations of deformation in a state of post buckling. The type and method of force loading is adjusted in accordance with the established shape. In the case of the imperfection the initial shape is set by measurement taking into account imperfection due the inaccuracy of installation in the preparation of the test. Measured variable and variables type for approximation and extrapolation are established by using numerical simulation of tests. The initial data for the modeling are the initial shape, the maximum compressive load N max defined by strength or permissible deformation, and the permissible error in determining the critical load. In the interval N ( 0, Nmax ) values N i are set, at which the strain parameter is measured. The most common strategy is uniformly distributed points test. Since the lateral stiffness of the test column or system is reduced significantly with the compressive force approaching to the critical value, the load F e and moment M e can be reduced during test process. It was established experimentally that testing columns and frames requires at least 5 measurements at different values of N i. Since the growth in the number of values N i increases the effect of measurement errors, the appropriate number of N i is established upon statistical process simulation [16]. The linear approximation with respect to the measured parameters appearing in models (4), (8), (9) is carried out and experimental value of the critical load P crm is determined. The approximation and extrapolation error P % crm P D = cr %, (10) Pcr where P cr - the estimated value of the critical load. If one need to take into account the measurement errors of the compressive force and the deformation parameter, the total error is determined by the use of statistical simulation [16]. Increased accuracy can be achieved: - By repeating the tests followed by corresponding statistical processing of the results; - By selecting the proper approximation and extrapolation variables; - By changing the pre-deformation shape and the pre-load. The theoretical analysis has been confirmed by the following stability tests: 1-14

6 - Columns with boundary conditions of rigid-rigid, pinned-pinned, rigid-pinned; - Central loaded and eccentrically loaded frames; - System, including compressive rod with an end sliding along a circular guide. The results can be used in the practice of industrial and laboratory test of element and structural stability. 5. Conclusions In order to enhance and improve the accuracy of determining the buckling load by Southwell plot: - Analysis of the methods for creating a pre-deformed state has been performed; - The mathematical models of approximation and extrapolation of measurements used to determine the critical load have been analyzed on the basis of a common methodology; - The purpose and problems of experimental design have been formulated. The method has been proposed for selection of the measured experimental parameters and of the approximation and extrapolation variables, and for estimation of the accuracy of the experiment. The results can be used in the practice of industrial and laboratory test of element and structural stability. References 1. Ayrton, W.E., and Perry, J.: ON STRUTS, The Engineer, Vol. 62, December 10, 1886, pp , and December 24, 1886, p Southwell R.V. (1932) On the analysis of experimental observations in problems of elastic stability. Proc Royal Soc, Ser A 135: Donnell L.H. (1938) On the application of Southwell's method for the analysis of buckling tests. In: Stephen Timoshenko 60th Anniversary Volume, the Macmillan Co., NY, 1938, pp Lokkas P. (2002) A Search on the Instability of Frame Structures Tested to Buckling Under Side Sway, Proceedings of the 4th GRACM Congress on Computational Mechanics GRACM 2002, Patra, Greece, June 2002, Vol. 1, pp B.Blostotsky, E.Efraim and Y.Ribakov. Improving the reliability of measuring critical buckling load in sway mode frames. Experimental Mechanics 56, 2, (2015). 1-15

7 6. Vaswani H.P. Model Analysis Method for Determining Buckling Load of Rectangular Frames. Experimental Mechanics 1, 8, (1961). 7. Hayashi, T., and Kihira, M.: ON A METHOD OF EXPERIMENTAL DETER IINATION OF THE BUCKLING LOAD OF AN ELASTICALLY SUPPORTED COLUMN, Paper presented at Japan Congress on Testing Materials, Kyoto, Japan, September Applicability of the southwell plot to the interpretation of test data obtained from stability studie of elastic column and plate structures, w.n.norton, f.l.cudari, r.w.johnson, november B.Blostotsky, E.Efraim. Dynamic method for measuring critical buckling load of sway frames. 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal, 30 June - 2 July B.Blostotsky, E.Efraim. Frames with tensioned and compressed columns. Stability in sidesway mode, EUROSTEEL 2014, September 10-12, 2014, Naples, Italy. 11. B.Blostotsky, Y.Ribakov, E.Efraim Methodology of measuring critical buckling load for sway mode frames, The 10th Conference on Contemporary Issues in Higher Education: The Ethos of the Academe-Standing the Test of Time, Ariel, Israel, September 10, E.Efraim, B.Blostotsky Enhanced Learning of Buckling Effect in Columns by Laboratory Test in Civil Engineering Courses, 5th International Conference on Education and New Learning Technologies EDULEARN 13, Barcelona, Spain, July 1-3, B.Blostotsky, E.Efraim, Y.Dachkovsky Dynamic method test of buckling load of one-storey sideway permitted frame, 5th International Mechanical Engineering Forum 2012, June 20-22, 2012, Prague, Czech Republic, pp Rao C.R., Toutenburg H. (1999) Linear Models: Least Squares and Alternatives. 2nd edn., Springer Series in Statistics, Springer-Verlag, New York. 15. Eurocode 3: Design of steel structures, Fahrmeir L, Tutz G. (2010) Multivariate statistical modeling based on generalized linear models, 2nd edn., Springer, New York. 1-16