Paper received: 1.8.2006 Paper accepted: 28.9.2007 Investigation of Stress Concentration Factor for Finite-Width Orthotropic Rectangular Plates with a Circular Opening Using Three-Dimensional Finite Element Model Kambiz Bakhshandeh* - Iraj Rajabi - Farhad Rahimi MUT University, Air Naval Research Center, Iran In this paper, theoretical stress concentration factor for finite-width orthotropic plates with centered circular opening is investigated by use of three-dimensional finite element analysis. In the first part, the finite element (FE) model is validated by comparison analytic equations for infinite orthotropic and isotropic plates. With verified model, the stress concentration factor is calculated for orthotropic rectangular plates with centered circular opening subject to uniform tension. The effect of opening radius to width ofplate ratio for various orthotropic plates with different orthotropy ratia is investigated. This study shows the accuracy of analytical method is dependent on orthotropy ratio and a/W ratio. In high opening radius to plate width ratio, the orthotropy ratio has a little effect on stress concentration factor. © 2008 Journal of Mechanical Engineering. All rights reserved. Keywords: stress concentration factor, finite element methods, orthotropic plates 0 INTRODUCTION Stress concentrations around cutouts have great practical importance because they are normally the cause of failure. For most materials, the failure strengths of the materials are strongly notch (or hole) sensitive. The net failure stress, taking into account the reduction in cross-section area, is typically much less than the ultimate tensile strength of the same materials without the notch or hole. Until now, the stress concentration factor for various isotropic structures is investigated. Heywood [1], Peterson [2] and Pilkey [3] have investigated various isotropic shape with wide range of holes. Howland [4] determined the solution of the problem of a long isotropic rectangular plate with a centered hole subject to a tension load. Peterson and Heywood introduced various equations for long length and finite width plate with different opening shapes. Lekhnitskii [5] and Tan [6] introduced various formulations for stress concentration in orthotropic materials. Lekhnitskii derives equations for infinite plate with circular holes. For finite-width orthotropic plate Tan introduces various equations. He derives these equations by use of equilibrium relations, but he did not investigate the effect of orthotropy ratio. The orthotropy ratio is defined as the ratio of two elastic moduli in both main directions. The analysis presented here suggested, that Tan's equations are applicable only for a specific range of orthotropy ratio. In this paper, the effect of orthotropy ratio on the stress concentration factor of rectangular finite-width plates with circular opening is investigated. These plates are under action of unidirectional tension loads. In the first part, the finite element model is validated for infinite orthotropic and isotropic rectangular plate with circular opening. With validated model, the effect of orthotropy ratio on stress concentration factor is investigated. 1 THEORETICAL STRESS CONCENTRATION FACTOR FOR FINITE WIDTH PLATE As is well known the theoretical stress concentration factor (TSCF) for normal stress is defined according to [2] and [7]: snom where the stress a represents the maximum stress max A to be expected in the member under the actual loads and the normal stress a is reference normal *Corr. Author's Address: MUT University, Air Naval Research Center, Shiraz, Postal Code: 7194915685, Iran, 140 kambiz@bakhshandeh.ir stress. In a rectangular plate under action of unidirectional tension load S (Fig. 1), a is V ^ / ? nom calculated based on the whole cross-section of the plate (gross area) as [2]: S 2 Wh (2), where: S - traction force, uniform on edges of plate, W - plate half-width, h - plate thickness. In this way, defined reference stress is in fact the far away field stress. TSCF calculated from Equations (1) and (2) is denoted by KTg and is very useful when investigating wide plates with small openings, i.e. when approaching infinite domain problems. Some values of the TSCF have been compiled by some major references Peterson, Pilkey, Tan and Lekhnitskii, and are reproduced in a number of mechanical design texts used worldwide. For a finite width plate, the stress concentration factor can be calculated by use of finite-width correction (FWC) factor [6] and [8]. FWC factor is a scale factor defined as the stress concentration factor of an infinite to a finite-width plate. Thus, the stress concentration of finite-width plate can be calculated with specification of FWC factor and stress concentration of infinite plate with similar opening and material (orthotropic or isotropic). FWC factor for isotropic material is independent of material properties thus can be determined accurately by use of curve fitting technique. Whereas in orthotropic material, this factor must be obtained with stress analysis by use Fig. 1. A finite-width plate containing a central elliptical opening of elasticity equations or mainly using finite element methods [6]. According to investigation of Tan on rectangular orthotropic plate with elliptical opening under action of unidirectional axial load (Fig. 1), the FWC factor is introduced in the form of following equation: (1- 21) K Tg l2 K Tg (1-1)2 (1-1)2 1 + (12 -1) (1-1) W -M a — M W -1/2 1 + (12 -1) W M a M W K % 1-I 1 (3), 1 + (12-1) W M -5/2 W M -7/2 1 + (12 -1) W M where: 1 - 8 M2 = - i - — W 2 + / \3 i - — w -1 -1 V (4), X = a/b, Kg - stress concentration factor in infinite plate, KTg - stress concentration factor in finite plate. The magnification factor M, is only a function of a/W and is independent of material properties. For a circular opening X = a/b = 1 with using the L'Hospital's rule twice w.r.t X [6], Equation (3) becomes: 11 a Tg . +1 (—M 1 (( - 3)1 -{—M 2 {W ) T ' \ W 2'\ KTg 2+fj - a, I W J (5). This equation can be used for orthotropic and isotropic finite-width plates with circular opening under action of unidirectional tension load for a/W up to 0.9 [6]. For calculation of stress concentration factor in a finite-width plate besides of FWC 2 7 factor the stress concentration factor of infinite plate with similar opening must be calculated. For orthotopic plate, the stress concentration factor of finite-width plate is complicated with respect to isotropic plates. The Lekhnitskii [5] introduces stress concentration factor for orthotropic infinite plate, KTgo as below: Krg.o = — ([- cos2 j + (k + n) sin2 J k cos2 6 + En + [(1 + n) cos2 j — ksin 2 j sin 2 6 — (6), —n(l + k + n)sin j cos j sin 6 cos 6) where: n = V lElL - u12 + -E1_ E22 2G12 k = E11 E22 (7). l sin4 q ^2 2u 12 G 12 Eli .In 2 n COS 0 sin2 q cos2 q +-- E E22 Ejj and E22 are elasticity moduli in main directions, G12 is modulus of rigidity in shear plane, u12 is Poisson ratio, 9 is polar angle measured from X axis (main direction 1) and ç is the angle of acting force with respect to main direction 1 (Fig. 2). For an infinite plate with circular opening under action of unidirectional tension, with direction of the applied force aligned with the main direction 1, ç is 0 and polar angle 9 at points 'c' and 'd' on Figure 2 is 0 and 90 degree, respectively. The stress concentration factors for these two points are: KTgJ _ 1 + n Tg ,o K7l _ -k (8), where: KTT'O - stress concentration factor of point 'c' for orthotopic infinite plate, KTT ,2 - stress concentration factor of point 'd' for orthotopic infinite plate. In isotropic material E = E22 and G = E/ 2(1+v), and according to Equations (6) and (7), the stress concentration factor for point 'c' and 'd' are 3 and -1, respectively. Substituting, K^g = 3 in the Equation (5), it is simplified to: Kg _ 3(1 - a/W) K (9). T 2 +(1 - a/W) This is familiar Heywood equation for isotropic infinite-plate with circular opening. Also, Howland determined the solution of the problem of a long rectangular plate with a central hole subject to tension in an open series form [4]. Another way to approaching the stress concentration factor and FWC factor is the application of the average normal stress across the net section (2(W-a)h) instead of the whole width of the plate (gross area). This stress concentration factor is known as net stress concentration factor, KTn and can be related to the previous stress concentration factor, K by: KTb _ Kg (1 - a/W) (10). In the common engineering sense it is more convenient to give stress concentration factors using reference stress based on the net area rather than the gross area [9]. Thus in this study the net stress concentration factor is used. 1.1 Finite Element Model Validation The stress concentration of an isotropic and orthotropic plate with circular opening under action of unidirectional tension is calculated by use of a finite element software MSC.Nastran [10]. Due to symmetry and using the appropriate displacement boundary conditions only one quarter of the plate was used as occupied by the shaded area indicated in Figure 3. The utilized restrictions Fig. 2. An infinite-plate containinga central circular opening Fig. 3. Finite element model region are zero displacement in the X direction for all the mesh nodal points located on the line X = 0, and zero displacement in the Y direction for all mesh nodal points, located on the line Y = 0. The finite element model is discretized with twenty-node solid brick elements (Fig. 4). Various mesh densities for each of plates are used to achieve a good mesh density for improving accuracy of results. The mesh density is changed from coarser to finer. On the average and roughly the fine mesh have ten times more nodes and elements than the coarse mesh. Densifying is iteratively repeated until the difference between the consecutive results becomes smaller than 0.35%. In Figure 5 the final discrepancy of the TSCF for various opening ratia is shown. This figure reflects the absolute value of the difference (percent) in the magnitude of the TSCF due to the mesh refinement. Between 67171 to 70501 elements are used in the study. Because the isotropic material properties are not different in main directions, this is used for Fig. 4. A typical finite element model improving the finite element model accuracy besides of orthotropic material. The finite element model is created for various opening ratia, on plates with dimensions: half-length L = 1000 mm, half width W = 100 mm and thickness 10 mm for isotropic and orthotopic material. The opening radius to half-length of plate ratio a/W is changed from 0.1 to 0.8. The isotropic plate has elastic modulus E = 210 GPa, Poisson ratio v = 0.3 and mass density p = 7800 kg/m3. The net stress concentration factor is calculated for isotropic plates by use of Equations (9) and (10) and compared with FE method results. Table 1 shows the net stress concentration factors that are concluded by FE and analytical method for isotropic plates. The error e in this table is defined as: TSCFTheory — TSCFFEM TSCF, Theory 100 where TSCFTheory and TSCFfem are, respectively, the o 5 0.25 ll" E 13 Opening radius to plate width [a/W] Fig. 5. Convergence study results Table 1. Net stress concentration factor for isotropic plate Net Stress Concentration Factor EEOR*** a/W Heywood* Howland* FE e % 0.1 2.72 2.72 2.73 0.37 0.2 2.61 2.62 2.59 0.61 0.3 2.34 2.36 2.35 0.31 0.4 2.22 2.24 2.23 0.35 0.5 2.12 2.16 2.11 0.41 0.6 2.06 2.09 2.05 0.29 0.7 2.03 2.05 2.04 0.24 0.8 2.01 2.03 2.02 0.40 * Eq. (9) **Ref. [4] *** Error (Eq. 11) of FE results with respect to Heywood results stress concentration factors calculated with the analytic and FE method. As it can be seen from Table 1 the agreement between the finite element method and theoretical solution for isotropic plate is excellent. The distribution of net stress concentration factor in a vicinity of circular opening for isotropic plate with a/W = 0.1 is shown in Figure 6. For orthotopic plate KTn is also calculated by use of FE and analytical method (Eqs. (5), (8) and (10)). The properties of the orthotopic material are: Corresponding to a typical orthotropic material [11]. In Table 2, the calculated net stress concentration factor is compared. The error that is shown in this table is calculated by Equation (11). Similar to isotropic plate, the FE method has a good accuracy for orthotropic materials. The distribution of net stress concentration factor in a vicinity of circular opening for orthotropic plate with a/W = 0.1 is shown in Figure 7 . E11 = 44.7 GPa, G23 = 3.45 GPa, v* = 0.34, E22 = E33 = 17.9 GPa, v12 = 0.25,