Acta Mech DOI 10.1007/s00707-013-0828-z M. Baghani B. Fereidoonnezhad Limit analysis of FGM circular plates subjected to arbitrary rotational symmetric loads using von-mises yield criterion Received: 7 December 2012 Springer-Verlag Wien 2013 Abstract In this paper, employing the limit analysis theorem, critical loading on functionally graded (FG) circular plate with simply supported boundary conditions and subjected to an arbitrary rotationally symmetric loading is determined. The material behavior follows a rigid-perfectly plastic model and yielding obeys the von-mises criterion. In the homogeneous case, the highly nonlinear ordinary differential equation governing the problem is analytically solved using a variational iteration method. In other cases, numerical results are reported. Finally, the results are compared with those of the FG plate with Tresca yield criterion and also in the homogeneous case with those of employing the von-mises yield criterion. A good correspondence is observed between the calculated results and those available in the literature. 1 Introduction In many practical situations, accurate prediction of the load carrying capacity of structures is highly important. In general, traditional elastic design cannot present realistic estimates for the actual load carrying capacity. Therefore, the structures manufactured according to elastic design principles are quite heavy and expensive. One of the most efficient methods of overcoming this difficulty, which is applicable to the evaluation of the behavior of structures in the plastic region, is the limit analysis technique. Application of the limit analysis approach results in the design of lighter structures in comparison with those obtained through the elastic design. Such structures are commonly used, especially in aerospace transportation devices. Limit analysis is a structural analysis field which is devoted to the development of efficient methods to directly determine estimate values of the collapse load of a given structural model without using iterative or incremental analysis methods. The limit analysis approach is based on lower and upper bound theorems, which results in corresponding estimate values for the critical load (the limit load of the structure). If these two bounds coincide, the exact solution of the given problem is obtained. These bounds relate the load carrying capacity of the structure to factors like material behavior, governing yield criterion, and imposed boundary conditions. Among the most fundamental works on this subject, the contributions of Prager [1], Horne [2], Mansfield [3], and Hodge [4] are remarkable. In the last decades, limit analysis is used to analyze the plastic behavior of plates under rotational symmetric loading. Kargarnovin and Ghorashi [5] considered circular plates obeying the Square yield criterion and obtained upper and lower bound solutions for the critical load factor of arbitrary M. Baghani (B) School of Mechanical Engineering, College of Engineering, University of Tehran, P. O. Box 11155-4563, Tehran, Iran E-mail: baghani@gmail.com Tel.:+98-21-66163614 Fax:+98-21-66000021 B. Fereidoonnezhad Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
M. Baghani, B. Fereidoonnezhad rotational symmetric loading. In another attempt, Kargarnovin et al. [6] presented the analytical solution for evaluating the critical load factor of simply supported homogeneous circular plates subjected to arbitrary rotationally symmetric loadings assuming the von-mises yield criterion. Several attempts are also made to use the limit analysis concept in design and analysis of different structures (see e.g. [7 10] among others). Functionally graded materials (FGMs) have several applications in aerospace, transportation, energy, electronics, and biomedical engineering [11]. In applications involving severe thermal gradients (e.g., thermal protection systems), FGMs exploit the heat, oxidation, and corrosion resistance typical of ceramics and the strength, ductility, and toughness typical of metals. Recently, Kargarnovin et al. [12] derived the limit load of functionally graded circular plates, using the Tresca and the Square yield criterion. Nevertheless, no works on the limit analysis of the FG plates has been reported based on the von-mises yield criterion. It is noted that the governing equations of such systems are essentially nonlinear. Generally in a nonlinear problem, it is hard to arrive at an analytical solution unless a number of different simplifying assumptions is considered. Otherwise, application of different numerical techniques is unavoidable [13,14]. Among different analytical methods, the variational iteration method (VIM) is one of the most accurate and efficient methods for studying nonlinear systems [15]. The basic idea of VIM is to construct a correction functional with a general Lagrange multiplier which can be identified optimally via variational theory [15]. In this paper, the governing equations for simply supported FGM circular plates subjected to arbitrary rotationally symmetric loadings assuming the von-mises yield criterion are derived. In the homogeneous case, employing the VIM method an analytical solution is presented in order to evaluate the critical load factor of the circular plate. In general cases, the nonlinear governing equations are solved numerically. Finally, the results are compared with those of the FG plate with Tresca yield criterion and also in the homogeneous case with those employing the von-mises yield criterion available in the literature. 2 FGM plastic parameters Functionally gradient materials (FGMs) utilized in, e.g., space devices are high-performance heat-resistant materials which are able to tolerate extremely large temperature gradients [16]. Meanwhile, FGM concepts have triggered worldwide research activities which are applied to metals, ceramics, and organic composites to produce modified components with superior physical and mechanical properties [17]. In this research, for a simply supported FGM plate, it is assumed that the compositional variation of ceramic and metal phases is approximated via an idealized power-law expression as ϕ c = r n, (1) where ϕ c denotes the volume fraction of the ceramic phase, r is the dimensionless radial coordinate ( r = R r ), R is the plate radius, and n is the power exponent. In fact, the core of the plate is pure metal and the edge is fully ceramic. Employing the mixture rule, for the effective elastic modulus, it can be easily found that [12,18] E ef f = [ ϕ m E m q + E c q + E m + ϕ c E c ]/ [ ϕ m q + E c q + E m + ϕ c ], (2) where E c and E m are the elastic modulus of ceramic and metallic phases, respectively. The normalized ratio of the stress to strain transfer is also defined by a positive parameter q to describe the ratio of stress to strain transfer, q = σ c σ m ε c ε m. For applications involving plastic deformation of ceramic/metal (brittle/ductile) composites, it is assumed that the composite yields once the metal constituent yields. With these assumptions, the yield stress, σr y,of the composite may be obtained as follows [12]: σ y r = σ y m [ ϕ m + q + E m q + E c ] E c ϕ c, (3) E m where σm y denotes the yield stress of the metal. Equation (3) indicates that the yield stress of the material depends on the yield stress of the metal, the volume fraction of the metal, elastic modulus of the constituent phases, and the parameter q.inviewof(1), we may write [12]: σ y r ( r) = σ y m [ 1 + q (E c E m ) (q + E c ) E m r n c ] σm y + A 1 r c n, (4)
Limit analysis of FGM circular plates subjected Fig. 1 a von-mises (solid lines) and Tresca (dotted line) yield criteria, b rotational symmetric loading on a radially FG circular plate where A 1 is an FG material parameter. For example, the following material parameters are reported where the metal is Ti and the ceramic part is TiB [19]: E m = 107 GPa, E c = 375 GPa, q = 4.5GPa,σ y m = 450 MPa, A 1 = 13.365 MPa. 3 Governing equations Consider a circular plate of radius R and thickness t (as shown in Fig. 1b) which is subjected to an arbitrary rotationally symmetric loading f (r) per unit area. Following other works in limit analysis [6,20 23], we assume that all deflections are neglected until the collapse mechanism is fully activated. Recasting the loading function as μf (r), inwhichμ is the load factor, the main goal of this paper is calculating the critical value of μ, μ cr, which is responsible for the plate collapse. It is assumed that the FGM properties of the plate in its undeformed shape are only a function of the radius. It is also assumed that the plate is made of a rigid-perfectly plastic material which obeys the von-mises yield criterion as demonstrated in Fig. 1a. In this figure, m r, m θ, and m u are the radial, tangential, and ultimate (fully plastic) bending moments per unit length, respectively. Considering relation (4), the ultimate bending moment per unit length for a plate with constant thickness t and yield stress σ y was obtained as: m u = t 2 t 2 σ y r z dz = t2 4 σ y r A ( 1 + B r n), (5) in which A and B are material parameters ( / y ) B = A 1 σm. The collapse mechanism consists of a few sets of yield lines along which the plate becomes fully plastic through the thickness. The formation of the collapse mechanism is usually considered as the last step of load carrying function. Thus, evaluating μ cr,theload carrying capacity of the plate could be expressed as μ cr f (r) [6]. The governing equations on the classic limit analysis problem are equilibrium equations and the yield stress surface. For the problem under consideration, the equilibrium equations are: d ( rq) + μ r f( r) = 0, d r (6) d ( rm r ) m θ rq = 0, d r where Q is the shearing force per unit length. One may combine Eq. (6) and arrive at: d ( rm r ) d r r m θ = μ 0 ξ f (ξ) dξ. (7) For the von-mises criterion, it can be shown that [6] m θ = 1 ( ) m r + 4m 2 2 u 3m2 r. (8)
M. Baghani, B. Fereidoonnezhad Substituting (8)into(7), we obtain 2 r dm r r d r + m r 4m 2 u 3m2 r = 2μ For a simply supported plate, the boundary condition on the edge of the plate is [6] Moreover, in case of μ = μ cr, corresponding to point A in Fig. 1a, we have 0 ξ f (ξ) dξ. (9) m r (1) = 0. (10) m r (0) = m θ (0) = m u. (11) / Employing the change of parameter U = m r mu and considering (5), Eq. (9) may be recast as U + (1 + B r n (1 + 2n)) 4 3U 2 2 r (1 + B r n U + λ r r f( r) d r = 0, (12) ) 2 r in which λ = μ / m u. The boundary conditions (10)and(11) are also rewritten as U (0) = 1, U (1) = 0. (13) Considering (12)and(13), we have a nonlinear first-order differential equation in which an unknown parameter λ must also be identified. Having two boundary conditions lets us find λ and U( r) simultaneously. It is noteworthy to mention that the function f ( r) must always have the same sign. In this paper, a quadratic format is chosen for f ( r): f ( r) = a + b r + c r 2. (14) In the general case (n = 0), we solve this problem using numerical methods such as the fourth-order Runge- Kutta method coupled with a shooting method (required for finding λ). However, in next section, we present a semi-analytical solution for the case n = 0. 4 Solution method: variational iteration method (n = 0) In this method, the problems are initially approximated with possible unknowns. Then, a corrected functional is constructed using a general Lagrange multiplier which can be identified optimally via the variational theory [13 15,24]. To explain the basic idea of the method, consider the following general nonlinear system: L [U ( r)] + N [U ( r)] = 0, (15) where L [U ( r)] and N [U ( r)] are linear and nonlinear differential operators, respectively. The basic character of the method is to construct a correction functional for the system as follows: r U i+1 ( r) = U i ( r) + 0 { λ (τ) L [U i (τ)] + N [Ũi (τ)]} dτ, (16) where (τ) is a general Lagrange multiplier which can be identified optimally via the variational theory. Also, U i is the i-th approximate solutions, and Ũ i represents a restricted variation, i.e., δũ i = 0. In the special case n = 0, we recast the problem described in Eqs. (12) (13) in the following form: { L [U ( r)] = U N [U ( r)] = ( 2 r U + U + 2λ r f( r) d r ) 2 4 + 3U 2 U. (17)
Limit analysis of FGM circular plates subjected Table 1 Comparison of the present work results with those available in the literature at case n = 0 Load a b c B λ[numeric] λ[vim-1] von-mises yield-criterion [6] Tresca yieldcriterion [12] Constant 1 0 0 6.515 6.460 6.382 6 Linear 1 1 0 4.329 4.216 4 Linear 1 2 0 3.2395 3.120 3 Linear 1 1 0 13.075 12.963 13.487 12 Parabolic 1 0 1 4.998 4.836 4.615 Parabolic 1 1 1 3.600 3.445 3.333 Parabolic 1 1 1 8.1605 8.064 7.5 Parabolic 1 2 1 3.818 3.735 3.529 Calculating the variation of Eq. (16) and noting that δũ(τ) = 0, the Lagrange multiplier is produced as (τ) = 1[25,26]. Substituting (17)and(13) 1 in (16) yields: r U i+1 ( r) = 1 0 { ( 2 r U i + U i + 2λ ) } 2 r f( r) d r 4 + 3Ui 2 U i dτ. (18) Selecting the initial guess in the form of U 0 ( r) = 1 r 2, the following expression is obtained for the next iteration: U 1 ( r) = 1 1 36 λ2 c 2 r 9 1 12 λ2 cb r 8 + 1 (5c 7 λ λca 49 ) λb2 r 7 + 2 6 bλ (5 λa) r 9 ( + 2λa 1 5 λc 1 5 λ2 a 2 28 ) r 5 1 5 3 λb r 4 + 2 3 (8 λa) r 3 r 2. (19) Applying the boundary condition (13) 2 on the obtained solution gives the following value for the critical load factor: λ cr = 420a + 245b + 162c + 155232a 2 + 53305b 2 + 23304c 2 + 182280ab + 120960ac + 70560bc 126a 2 + 40b 2 + 17.5c 2. + 140ab + 90ac + 52.5bc (20) The procedure explained in this section could be followed up to higher orders. For the sake of brevity, we only report the construction of U 1 ( r). 5 Results and discussion In thissection,numerical resultsfor differentloadingconditionsare reported.in Table1, the analytical results as well as the numerical results for different loading conditions are reported. The obtained results are also compared with those available in the literature for some cases. As it is known, the von-mises yield criterion is a less conservative yield model compared to the Tresca model [27,28]. Along this line, as observed from Table 1, the present work results predict higher values for the critical load factor. For instance, for constant (a = 1, b = 0, c = 0), linear (a = 1, b = 1, c = 0), and parabolic (a = 1, b = 1, c = 1) loadings, von Mises model (compared to the Tresca model) predicts 8.58, 8.96, and 8.8 % higher values, respectively. In Figs. 2, 3,and4, the effects of the FG exponent n on the critical load factor λ cr for constant (a = 1, b = 0, c = 0), linear (a = 1, b = 1, c = 0), and parabolic (a = 1, b = 1, c = 1) loadings are shown, respectively. As observed from these figures, in all cases increasing the FG exponent n results in a decrease in the value of λ cr. Moreover, in larger values of B,λ cr becomes smaller. In fact, in these cases, the volume fraction of the ceramic part is much higher in comparison with the metallic part; thus, smaller values for λ cr are expected. To compare the results of the present work with those of the Tresca model, in these figures, the Tresca model results are also plotted. Similar to the results reported in Table 1, in these cases, the von-mises model predicts larger values for λ cr. The non-dimensional radial moment U versus r is illustrated in Figs. 5 (constant load) and 6 (linear load). From these figures, it is concluded that changes in n may lead to a considerable change in the form of non-dimensional radial moment distribution along the radius of the FG plate.
M. Baghani, B. Fereidoonnezhad Fig. 2 The critical load factor, λ cr for different values of exponent n under constant load (a = 1, b = 0, c = 0). Thicker lines denote the von-mises yield criterion results, while thinner ones stand for the Tresca yield criterion results Fig. 3 The critical load factor, λ cr for different values of exponent n under linear load (a = 1, b = 1, c = 0). Thicker lines denote the von-mises yield criterion results, while thinner ones stand for the Tresca yield criterion results Fig. 4 The critical load factor, λ cr for different values of exponent n under parabolic load (a = 1, b = 1, c = 1). Thicker lines denote the von-mises yield criterion results, while thinner ones stand for the Tresca yield criterion results
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