- Email: [email protected]
Finding the buckling load of non-uniform columns using the iteration perturbation method
THEORETICAL & APPLIED MECHANICS LETTERS 4, 041011 (2014)
Finding the buckling load of non-uniform columns using the iteration perturbation method Aref Afsharfard,a) Anooshiravan Farshidianfar Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran (Received 2 February 2014; revised 3 April 2014; accepted 26 May 2014)
Abstract The aim of this study is to calculate the critical load of variable inertia columns. The example studied in this paper can be used as a paradigm for other non-uniform columns. The wavelength of equivalent vibratory system is used to calculate the critical load of the trigonometrically varied inertia column. In doing so, the equilibrium equation of the column is theoretically studied using the perturbation method. Accuracy of the calculated results is evaluated by comparing the solution with numerical results. Effect of improving the initial guess on the solution accuracy is investigated. Effects of varying parameters of the trigonometrically varied inertia and the uniformly tapered columns on their stability behavior are studied. Finally, using the so-called “perfectibility” parameter, two design goals, i.e., being lightweight and being strong, are studied for the discussed columns. c 2014 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1404111] ⃝ Keywords perturbation method, critical load, non-uniform column, perfectibility
Nowadays, interests of researchers for using analytical techniques to solve nonlinear problems are increased. The reason lies in the fact that every physical process is really a nonlinear process, and should be described using nonlinear equations. Kerschen et al.1 studied the sources of nonlinearity and classified them into five categories: (1) geometry nonlinearity, which is the results of large displacements in structures, (2) inertia nonlinearity, which is derived from nonlinear terms in the equation of motion, (3) nonlinear material behavior, which may be observed when the constitutive law is nonlinear, (4) damping dissipation being essentially a nonlinear phenomenon, and (5) boundary conditions or external nonlinear body forces resulting in nonlinearity. Many efforts are made to develop methods for investigating the nonlinear problems.2–4 One of the best tools to study nonlinear problems is the perturbation method.5–7 This method is widely used to investigate practical nonlinear problems.8–10 Nayfeh11,12 used the perturbation method to study the weakly nonlinear problems. Hajj et al.13 combined the perturbation techniques and spectral moments to characterize the nonlinear parameters of vibratory systems. Chen and Cheung14 extended the classical perturbation techniques to study the strongly nonlinear problems. Moeenfard et al.15 used the perturbation method to analyze the vibratory behavior of micro-beams considering the effects of rotary inertia and shear deformation. One of the most important branches of widely studied mechanical engineering is the stability of structures. Li16 investigated the buckling load of non-uniform columns subjected to axial a) Corresponding
author. Email: [email protected].
041011-2 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
concentrated and distributed loading. He presented a class of exact functional solutions for the buckling problem of non-uniform columns. Iremonger17 determined buckling load of tapered and stepped columns using finite difference method. Lacarbonara18 used the perturbation approach to obtain the post-buckling solutions of non-uniform linearly and nonlinearly elastic rods. Singh and Li19 presented a low-dimensional mathematical model to calculate the buckling load of uniform and non-uniform functionally graded column the axial direction. Whereas finding the critical load of columns is widely considered by researchers,20,21 there is little attention toward analytically calculating the critical load of non-uniform columns. The main goal of the present study is to find the critical load of variable inertia columns. In doing so, the equilibrium equation of a trigonometrically varied inertia column is investigated using the perturbation method. Using an admissible initial guess, the critical load of column is calculated. To evaluate the accuracy of solution, the analytical solutions are compared with the finite element method (FEM) results. Effects of improving initial guess in the iteration perturbation method are analyzed. Finally, the trigonometrically varied inertia column is compared with uniformly tapered columns. A parameter named “perfectibility” is used to combine two design goals for the discussed columns. Non-uniform column A simply supported ideal column of length L under the action of axial load P is shown in Fig. 1. It is assumed that the axial load is quasi-statically applied on the column. Based on the Euler–Bernoulli assumption, the equilibrium equation for the column can be given as EI(x) y,xx + Py = 0, where x is an independent variable, y denotes the lateral displacement of the column and I(x) is inertia of the column, which is variable. In the present study the inertia of column trigonometrically changes along the column. Inertia of the trigonometrically varied inertia columns is as I(x) = I0 (1 + sin(πx/L)), where I0 is the column inertia at both ends of the column and L is the column length. y
P
P
x
L
Fig. 1. Schematic of the trigonometrically varied inertia column.
Finding the critical load The critical load of the trigonometrically varied inertia columns is theoretically obtained, using the iteration perturbation method. Note that, the iteration perturbation is a technique, which is coupled to the iteration method and can be used to solve strongly nonlinear problems.9,10 Governing equation of lateral deformation of the trigonometrically varied inertia column can be written as y′′ + sin (kx) y′′ + [P/(EI0 )]y = 0,
(1)
where y′′ = y,xx and k is equal to π/L. An iteration formula for Eq. (1) can obviously be constructed as y′′n + y′′n+1 [(yn+1 /yn ) sin(kx)] + [P/(EI0 )]yn+1 = 0, n = 0, 1, 2, · · · .
(2)
041011-3 Finding the buckling load of non-uniform columns
The above equation is a Mathieu type of nonlinear equations. The parameter yn describes n-th approximate solution of nonlinear oscillation. Suppose that the first iterative solution is equal to y0 = sin(ω x). Note that, the first solution is a geometrically admissible trial function. In this assumption parameter ω is the angular frequency of the equation. Regarding the first iterative solution, Eq. (2) can be re-written as y′′1 + [P/(EI0 )]y1 − εω 2 y1 sin (kx) = 0,
(3)
where ε is an artificial parameter. Solution of the above equation (y) can be expanded into a series in artificial parameter.11 In other words, y can be written as y = y0 + ε y1 + ε 2 y2 + · · · .
(4)
Similarly, P/(EI0 ) can be expressed as P/(EI0 ) = ω 2 + ε C1 + ε 2C2 + · · · .
(5)
Substituting Eqs. (4) and (5) into Eq. (3) results in y′′1 + ω 2 y1 = −C1 sin(ω x) + ω 2 sin (ω x) sin(kx).
(6) ∫ π/2ω
sin(ω x)(−C1 sin(ω x) Avoiding the presence of secular terms requires it to satisfy the relation 0 +ω 2 sin (ω x) sin(kx)) dx = 0. Therefore, parameter C1 satisfies the relation C1 = (4k2 ω 3 − 8ω 5 ) cos(πk/2ω ) + 8ω 5 /[kπ(4ω 2 − k2 )]. In the present study, the longitudinal deformation of the column under the axial load is considered negligibly small. But, it should be noted that increasing the axial load leads to lateral deformation of the column when the axial load becomes greater than the buckling load. Therefore, before reaching the critical load and consequently curving the column, the first mode shape wavelength of the column λ is equal to 2L. The wavelength of the first mode shape of column is clearly shown in Fig. 2. y
Column
P
P L λ/2
x λ/2
Fig. 2. Relation between the column first mode shape and the wavelength.
Subsequently, before the column buckles, it can be concluded that ω = π/L, therefore, C1 = 8ω 2 /(3π). Regarding Eq. (5), the critical load is equal to Pcr = (π2 /L2 )(1 + 8/3π)EI0 ≈ 18.247 2EI0 /L2 . Using the variational iteration method, y1 can be calculated from Eq. (6) to be y1 = [2/3 − C1 x/(2ω )] cos(ω x) + C1 sin(ω x)/(2ω 2 ) − 1/2 − cos(2ω x)/6. New iteration can be performed by substituting y1 into Eq. (4). Therefore the new iteration can be considered as y = sin(ω x) + [2/3 − C1 x/(2ω )] cos(ω x) + C1 sin(ω x)/(2ω 2 ) − 1/2 − cos(2ω x)/6. Substituting the new iteration into Eq. (2) and avoiding the presence of secular terms results in the following relation: C1 = [(128π2 − 15π3 )π2 ]/[(9πL2C1 + 48π3 − 80π2 )L2 ]. The coefficient C1 can be ob-
041011-4 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
tained by solving the above equation using Newton’s method. Therefore the critical load, in the new iteration, can be obtained as Pcr = 18.281 2EI0 /L2 . This paper focuses on finding the critical load of non-uniform columns. For this reason the critical load of a trigonometrically varied inertia column is theoretically calculated. In doing so, the relation between the critical load and the wavelength of the first mode shape of column is used. To evaluate the result accuracy, the theoretical answers are compared with the numerical result obtained by using the ANSYS software. The element used to obtain buckling load of the column is solid 95. This element is defined by twenty nodes and it has three degrees of freedom per node. Numbers of elements in the coarse and the refined model were respectively equal to 960 and 1 440. The buckling loads determined from eigenvalue analysis for the two meshes differed by less than one percent. The comparison between analytical and numerical results is presented in Table 1. As shown in this table, the iteration perturbation method provides high accurate results even in the first iteration. Furthermore, it is shown that improving the initial guess in iteration perturbation technique leads to more accurate results. Table 1. Comparing FEM results with analytical solution for trigonometrically varied inertia column. Method Pcr Error /%
Numerical method 18.351 6EI0 /L2 –
Perturbation method (1st iteration) 18.247 2EI0 /L2 0.56
Perturbation method (2nd iteration) 18.281 2EI0 /L2 0.38
Note that, the discussed technique can be used to calculate the critical load of other nonuniform columns. For example, consider the inertia of a trigonometrically varied inertia column and a uniformly tapered column as { Tri I(x,a)
= I0 (1 + a sin(πx/L)),
Uni I(x,a)
=
I0 (1 + 2ax/L),
0 6 x 6 L/2,
I0 [1 + 2a(1 − x/L)], L/2 6 x 6 L,
where a is a constant showing the rate of changing inertia along the column. Furthermore, superscripts “Tri” and “Uni” show that the values are respectively related to the trigonometrically varied inertia column and the uniformly tapered column. Using the first iteration of the presented perturbation method, the critical load of columns can be obtained as PcrTri = (9.869 6 + 8.377 6a)(EI0 /L2 ) and PcrUni = (9.869 6 + 6.934 8a)(EI0 /L2 ). Furthermore, volume of the trigonometrically varied inertia column and the uniformly tapered ∫ √ √ √ column can be calculated as V Tri = 2L πI0 ( 0L 1 + a sin(πx/L) dx) and V Uni = 2L πI0 {[2(1 + a)3/2 − 2]/(3a)}. Variation of the dimensionless radius of the uniformly tapered and the trigonometrically varied inertia columns versus their dimensionless length are shown in Fig. 3. Relation between the volumes of the discussed columns is shown in Fig. 4. Furthermore, in this figure, the relation between the critical loads of the columns is depicted. As shown in this figure, rate of increasing the critical load ratio is larger than that of the volume ratio. For example, if the volume of the trigonometrically varied inertia column is 1.097 times larger than the volume of the uniformly tapered column, the critical load of the trigonometrically varied inertia column
041011-5 Finding the buckling load of non-uniform columns 1.5
1.20 1.15 1.10
Uni
1.0
Pcr / Pcr
r/r0
& V Tri/V Uni
r0 = (4I0/π)1/4
a=1
0.5 0
0.2
0.4
0.6
0.8
Tri
Uniform column Uniform tapered column Trigonometrically variable inertia column
Tri
Uni
V Tri /V Uni
1.05
Pcr / Pcr 1.00
1.0
0
1
2
3
4
5
a
x/L
Fig. 4. Variations of the volume and the critical load of columns versus the parameter a.
Fig. 3. Variation of the column dimensionless radius versus the dimensionless length.
will be 1.162 times larger than the critical load of the uniformly tapered column. Note that it is desirable to have a column, which buckles at higher axial loads. Furthermore, this column should not be heavy. In other words, a perfectly designed column not only can support heavy axial loads, but also is lightweight itself. In doing so, in the present study, to satisfy both of the design goals, a parameter named “Perfectibility” is introduced as Pper = (V0 /V )W F1 + (1 − Pcr0 /Pcr )W F2 , where V0 and Pcr0 are respectively the volume V0 = 2L(πI0 )1/2 and the critical load Pcr0 = π2 EI0 /L2 of the uniform column. Furthermore, W F is the weight factor of the column volume and critical load (note that W F1 + W F2 = 1). Variation of the perfectibility with the parameter a is shown in Fig. 5. As shown in this figure, when higher critical load is more important than smaller volume of columns (W F1 < 0.5), the trigonometrically varied inertia columns are better choices. The uniformly tapered columns are perfect choices, if the weight of column is the more important parameter (W F1 > 0.5). (a) 1.0
(b) 1.0 WF1 = 0
0.8
0.6
Pper
Pper
0.8
0.4
0.6 WF1 = 1.0 0.4
Tri
0.2
Tri
0.2
Uni
Uni
0 0
2
4 a
0 0
2
4 a
Fig. 5. Variation of the Pper versus the parameter a for (a) W F1 = 0 and (b) W F1 = 1.0.
Conclusion A novel technique is developed to find the critical load of non-uniform columns. In doing so, the critical load of a trigonometrically varied inertia column is theoretically calculated using the perturbation method. It is shown that, using the presented technique, accurate results can be calculated when an admissible initial guess is initially used. Furthermore, it is shown that improving the initial guess can reduce the error of solution more than 32%. The critical load and volume of the trigonometric variable inertia column and the uniformly
041011-6 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
tapered column are combined into the so-called perfectibility parameter. Using this parameter, it is shown that the trigonometric varied inertia columns are good choices, if high critical load is demanded. 1. G. Kerschen, K. Worden, A. F. Vakakis, et al. Past, present and future of nonlinear system identification in structural dynamics. Mechanical Systems and Signal Processing 20, 505–592 (2006). 2. A. H. Nayfeh, D. T. Mook. Nonlinear Oscillations. Pure and Applied Mathematics. Wiley, New York (1979). 3. M. H. Kahrobaiyan, M. Asghari, M. Rahaeifard, et al. A nonlinear strain gradient beam formulation. International Journal of Engineering Science 49, 1256–1267 (2011). 4. I. Tawfiq, T. Vinh. Nonlinear behaviour of structures using the volterra series—signal processing and testing methods. Nonlinear Dynamics 37, 129–149 (2004). 5. J. Awrejcewicz, I. V. Andrianov, L. I. Manevich. Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications, Springer Series in Synergetics. Springer, Berlin (1998). 6. M. Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics, 4th edn. SpringerVerlag, New York (1993). 7. J. A. Murdock. Perturbations Theory and Methods. John Wiley & Sons, New York (1991). 8. S. B. Li, W. Zhang, L. J. Gao. Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities. Nonlinear Dynamics 71, 175–185 (2013). 9. H. H. Khodaparast, J. E. Mottershead, M. I. Friswell. Perturbation methods for the estimation of parameter variability in stochastic model updating. Mechanical Systems and Signal Processing 22, 1751–1773 (2008). 10. A. Afsharfard, A. Farshidianfar. Free vibration analysis of nonlinear resilient impact dampers. Nonlinear Dynamics 73, 155–166 (2013). 11. A. H. Nayfeh. Introduction to Perturbation Techniques. Wiley, New York (1981). 12. A. H. Nayfeh. Perturbation Methods, Wiley Classics Library edn. John Wiley & Sons, New York (2000). 13. M. R. Hajj, J. Fung, A. H. Nayfeh, et al. Damping identification using perturbation techniques and higher-order spectra. Nonlinear Dynamics 23, 189–203 (2000). 14. S. H. Chen, Y. K. Cheung. A modified Lindstedt–Poincare method for a strongly non-linear two degree-of-freedom system. Journal of Sound and Vibration 193, 751–762 (1996). 15. H. Moeenfard, M. Mojahedi, M. Ahmadian. A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams. J. Mech. Sci. Technol. 25, 557–565 (2011). 16. Q. S. Li. Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading. European Journal of Mechanics - A/Solids 20, 485–500 (2001). 17. M. J. Iremonger. Finite difference buckling analysis of non-uniform columns. Computers & Structures 12, 741–748 (1980). 18. W. Lacarbonara. Buckling and post-buckling of non-uniform non-linearly elastic rods. International Journal of Mechanical Sciences 50, 1316–1325 (2008). 19. K. V. Singh, G. Li. Buckling of functionally graded and elastically restrained non-uniform columns. Composites Part B: Engineering 40, 393–403 (2009). 20. R. Lagrange. Compression-induced stiffness in the buckling of a one fiber composite. Theoretical and Applied Mechanics Letters 3, 061001 (2013). 21. C. Chen, W. Tao, Z. J. Liu, et al. Controlled buckling of thin film on elastomeric substrate in large deformation. Theoretical and Applied Mechanics Letters 1, 021001 (2011).
Finding the buckling load of non-uniform columns using the iteration perturbation method Aref Afsharfard,a) Anooshiravan Farshidianfar Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran (Received 2 February 2014; revised 3 April 2014; accepted 26 May 2014)
Abstract The aim of this study is to calculate the critical load of variable inertia columns. The example studied in this paper can be used as a paradigm for other non-uniform columns. The wavelength of equivalent vibratory system is used to calculate the critical load of the trigonometrically varied inertia column. In doing so, the equilibrium equation of the column is theoretically studied using the perturbation method. Accuracy of the calculated results is evaluated by comparing the solution with numerical results. Effect of improving the initial guess on the solution accuracy is investigated. Effects of varying parameters of the trigonometrically varied inertia and the uniformly tapered columns on their stability behavior are studied. Finally, using the so-called “perfectibility” parameter, two design goals, i.e., being lightweight and being strong, are studied for the discussed columns. c 2014 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1404111] ⃝ Keywords perturbation method, critical load, non-uniform column, perfectibility
Nowadays, interests of researchers for using analytical techniques to solve nonlinear problems are increased. The reason lies in the fact that every physical process is really a nonlinear process, and should be described using nonlinear equations. Kerschen et al.1 studied the sources of nonlinearity and classified them into five categories: (1) geometry nonlinearity, which is the results of large displacements in structures, (2) inertia nonlinearity, which is derived from nonlinear terms in the equation of motion, (3) nonlinear material behavior, which may be observed when the constitutive law is nonlinear, (4) damping dissipation being essentially a nonlinear phenomenon, and (5) boundary conditions or external nonlinear body forces resulting in nonlinearity. Many efforts are made to develop methods for investigating the nonlinear problems.2–4 One of the best tools to study nonlinear problems is the perturbation method.5–7 This method is widely used to investigate practical nonlinear problems.8–10 Nayfeh11,12 used the perturbation method to study the weakly nonlinear problems. Hajj et al.13 combined the perturbation techniques and spectral moments to characterize the nonlinear parameters of vibratory systems. Chen and Cheung14 extended the classical perturbation techniques to study the strongly nonlinear problems. Moeenfard et al.15 used the perturbation method to analyze the vibratory behavior of micro-beams considering the effects of rotary inertia and shear deformation. One of the most important branches of widely studied mechanical engineering is the stability of structures. Li16 investigated the buckling load of non-uniform columns subjected to axial a) Corresponding
author. Email: [email protected].
041011-2 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
concentrated and distributed loading. He presented a class of exact functional solutions for the buckling problem of non-uniform columns. Iremonger17 determined buckling load of tapered and stepped columns using finite difference method. Lacarbonara18 used the perturbation approach to obtain the post-buckling solutions of non-uniform linearly and nonlinearly elastic rods. Singh and Li19 presented a low-dimensional mathematical model to calculate the buckling load of uniform and non-uniform functionally graded column the axial direction. Whereas finding the critical load of columns is widely considered by researchers,20,21 there is little attention toward analytically calculating the critical load of non-uniform columns. The main goal of the present study is to find the critical load of variable inertia columns. In doing so, the equilibrium equation of a trigonometrically varied inertia column is investigated using the perturbation method. Using an admissible initial guess, the critical load of column is calculated. To evaluate the accuracy of solution, the analytical solutions are compared with the finite element method (FEM) results. Effects of improving initial guess in the iteration perturbation method are analyzed. Finally, the trigonometrically varied inertia column is compared with uniformly tapered columns. A parameter named “perfectibility” is used to combine two design goals for the discussed columns. Non-uniform column A simply supported ideal column of length L under the action of axial load P is shown in Fig. 1. It is assumed that the axial load is quasi-statically applied on the column. Based on the Euler–Bernoulli assumption, the equilibrium equation for the column can be given as EI(x) y,xx + Py = 0, where x is an independent variable, y denotes the lateral displacement of the column and I(x) is inertia of the column, which is variable. In the present study the inertia of column trigonometrically changes along the column. Inertia of the trigonometrically varied inertia columns is as I(x) = I0 (1 + sin(πx/L)), where I0 is the column inertia at both ends of the column and L is the column length. y
P
P
x
L
Fig. 1. Schematic of the trigonometrically varied inertia column.
Finding the critical load The critical load of the trigonometrically varied inertia columns is theoretically obtained, using the iteration perturbation method. Note that, the iteration perturbation is a technique, which is coupled to the iteration method and can be used to solve strongly nonlinear problems.9,10 Governing equation of lateral deformation of the trigonometrically varied inertia column can be written as y′′ + sin (kx) y′′ + [P/(EI0 )]y = 0,
(1)
where y′′ = y,xx and k is equal to π/L. An iteration formula for Eq. (1) can obviously be constructed as y′′n + y′′n+1 [(yn+1 /yn ) sin(kx)] + [P/(EI0 )]yn+1 = 0, n = 0, 1, 2, · · · .
(2)
041011-3 Finding the buckling load of non-uniform columns
The above equation is a Mathieu type of nonlinear equations. The parameter yn describes n-th approximate solution of nonlinear oscillation. Suppose that the first iterative solution is equal to y0 = sin(ω x). Note that, the first solution is a geometrically admissible trial function. In this assumption parameter ω is the angular frequency of the equation. Regarding the first iterative solution, Eq. (2) can be re-written as y′′1 + [P/(EI0 )]y1 − εω 2 y1 sin (kx) = 0,
(3)
where ε is an artificial parameter. Solution of the above equation (y) can be expanded into a series in artificial parameter.11 In other words, y can be written as y = y0 + ε y1 + ε 2 y2 + · · · .
(4)
Similarly, P/(EI0 ) can be expressed as P/(EI0 ) = ω 2 + ε C1 + ε 2C2 + · · · .
(5)
Substituting Eqs. (4) and (5) into Eq. (3) results in y′′1 + ω 2 y1 = −C1 sin(ω x) + ω 2 sin (ω x) sin(kx).
(6) ∫ π/2ω
sin(ω x)(−C1 sin(ω x) Avoiding the presence of secular terms requires it to satisfy the relation 0 +ω 2 sin (ω x) sin(kx)) dx = 0. Therefore, parameter C1 satisfies the relation C1 = (4k2 ω 3 − 8ω 5 ) cos(πk/2ω ) + 8ω 5 /[kπ(4ω 2 − k2 )]. In the present study, the longitudinal deformation of the column under the axial load is considered negligibly small. But, it should be noted that increasing the axial load leads to lateral deformation of the column when the axial load becomes greater than the buckling load. Therefore, before reaching the critical load and consequently curving the column, the first mode shape wavelength of the column λ is equal to 2L. The wavelength of the first mode shape of column is clearly shown in Fig. 2. y
Column
P
P L λ/2
x λ/2
Fig. 2. Relation between the column first mode shape and the wavelength.
Subsequently, before the column buckles, it can be concluded that ω = π/L, therefore, C1 = 8ω 2 /(3π). Regarding Eq. (5), the critical load is equal to Pcr = (π2 /L2 )(1 + 8/3π)EI0 ≈ 18.247 2EI0 /L2 . Using the variational iteration method, y1 can be calculated from Eq. (6) to be y1 = [2/3 − C1 x/(2ω )] cos(ω x) + C1 sin(ω x)/(2ω 2 ) − 1/2 − cos(2ω x)/6. New iteration can be performed by substituting y1 into Eq. (4). Therefore the new iteration can be considered as y = sin(ω x) + [2/3 − C1 x/(2ω )] cos(ω x) + C1 sin(ω x)/(2ω 2 ) − 1/2 − cos(2ω x)/6. Substituting the new iteration into Eq. (2) and avoiding the presence of secular terms results in the following relation: C1 = [(128π2 − 15π3 )π2 ]/[(9πL2C1 + 48π3 − 80π2 )L2 ]. The coefficient C1 can be ob-
041011-4 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
tained by solving the above equation using Newton’s method. Therefore the critical load, in the new iteration, can be obtained as Pcr = 18.281 2EI0 /L2 . This paper focuses on finding the critical load of non-uniform columns. For this reason the critical load of a trigonometrically varied inertia column is theoretically calculated. In doing so, the relation between the critical load and the wavelength of the first mode shape of column is used. To evaluate the result accuracy, the theoretical answers are compared with the numerical result obtained by using the ANSYS software. The element used to obtain buckling load of the column is solid 95. This element is defined by twenty nodes and it has three degrees of freedom per node. Numbers of elements in the coarse and the refined model were respectively equal to 960 and 1 440. The buckling loads determined from eigenvalue analysis for the two meshes differed by less than one percent. The comparison between analytical and numerical results is presented in Table 1. As shown in this table, the iteration perturbation method provides high accurate results even in the first iteration. Furthermore, it is shown that improving the initial guess in iteration perturbation technique leads to more accurate results. Table 1. Comparing FEM results with analytical solution for trigonometrically varied inertia column. Method Pcr Error /%
Numerical method 18.351 6EI0 /L2 –
Perturbation method (1st iteration) 18.247 2EI0 /L2 0.56
Perturbation method (2nd iteration) 18.281 2EI0 /L2 0.38
Note that, the discussed technique can be used to calculate the critical load of other nonuniform columns. For example, consider the inertia of a trigonometrically varied inertia column and a uniformly tapered column as { Tri I(x,a)
= I0 (1 + a sin(πx/L)),
Uni I(x,a)
=
I0 (1 + 2ax/L),
0 6 x 6 L/2,
I0 [1 + 2a(1 − x/L)], L/2 6 x 6 L,
where a is a constant showing the rate of changing inertia along the column. Furthermore, superscripts “Tri” and “Uni” show that the values are respectively related to the trigonometrically varied inertia column and the uniformly tapered column. Using the first iteration of the presented perturbation method, the critical load of columns can be obtained as PcrTri = (9.869 6 + 8.377 6a)(EI0 /L2 ) and PcrUni = (9.869 6 + 6.934 8a)(EI0 /L2 ). Furthermore, volume of the trigonometrically varied inertia column and the uniformly tapered ∫ √ √ √ column can be calculated as V Tri = 2L πI0 ( 0L 1 + a sin(πx/L) dx) and V Uni = 2L πI0 {[2(1 + a)3/2 − 2]/(3a)}. Variation of the dimensionless radius of the uniformly tapered and the trigonometrically varied inertia columns versus their dimensionless length are shown in Fig. 3. Relation between the volumes of the discussed columns is shown in Fig. 4. Furthermore, in this figure, the relation between the critical loads of the columns is depicted. As shown in this figure, rate of increasing the critical load ratio is larger than that of the volume ratio. For example, if the volume of the trigonometrically varied inertia column is 1.097 times larger than the volume of the uniformly tapered column, the critical load of the trigonometrically varied inertia column
041011-5 Finding the buckling load of non-uniform columns 1.5
1.20 1.15 1.10
Uni
1.0
Pcr / Pcr
r/r0
& V Tri/V Uni
r0 = (4I0/π)1/4
a=1
0.5 0
0.2
0.4
0.6
0.8
Tri
Uniform column Uniform tapered column Trigonometrically variable inertia column
Tri
Uni
V Tri /V Uni
1.05
Pcr / Pcr 1.00
1.0
0
1
2
3
4
5
a
x/L
Fig. 4. Variations of the volume and the critical load of columns versus the parameter a.
Fig. 3. Variation of the column dimensionless radius versus the dimensionless length.
will be 1.162 times larger than the critical load of the uniformly tapered column. Note that it is desirable to have a column, which buckles at higher axial loads. Furthermore, this column should not be heavy. In other words, a perfectly designed column not only can support heavy axial loads, but also is lightweight itself. In doing so, in the present study, to satisfy both of the design goals, a parameter named “Perfectibility” is introduced as Pper = (V0 /V )W F1 + (1 − Pcr0 /Pcr )W F2 , where V0 and Pcr0 are respectively the volume V0 = 2L(πI0 )1/2 and the critical load Pcr0 = π2 EI0 /L2 of the uniform column. Furthermore, W F is the weight factor of the column volume and critical load (note that W F1 + W F2 = 1). Variation of the perfectibility with the parameter a is shown in Fig. 5. As shown in this figure, when higher critical load is more important than smaller volume of columns (W F1 < 0.5), the trigonometrically varied inertia columns are better choices. The uniformly tapered columns are perfect choices, if the weight of column is the more important parameter (W F1 > 0.5). (a) 1.0
(b) 1.0 WF1 = 0
0.8
0.6
Pper
Pper
0.8
0.4
0.6 WF1 = 1.0 0.4
Tri
0.2
Tri
0.2
Uni
Uni
0 0
2
4 a
0 0
2
4 a
Fig. 5. Variation of the Pper versus the parameter a for (a) W F1 = 0 and (b) W F1 = 1.0.
Conclusion A novel technique is developed to find the critical load of non-uniform columns. In doing so, the critical load of a trigonometrically varied inertia column is theoretically calculated using the perturbation method. It is shown that, using the presented technique, accurate results can be calculated when an admissible initial guess is initially used. Furthermore, it is shown that improving the initial guess can reduce the error of solution more than 32%. The critical load and volume of the trigonometric variable inertia column and the uniformly
041011-6 A. Afsharfard, A. Farshidianfar
Theor. Appl. Mech. Lett. 4, 041011 (2014)
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