XI CONGRESO NACIONAL DE MATERIALES COMPUESTOS. Celebrado en Móstoles los días 6, 7 y 8 de julio de 2015The aim of this article is to provide an analytical tool to estímate the stress
concentration in anisotropic plates weakened by a circular or elliptical hole; it is
achieved by developing Lekhnitskii formalism in order to allow fínite boundary effects
to be added (original formulation considers infinite plates). Only membrane problem is
herein considered, particularly when prescribed displacements are applied at externa!
boundaries, which could simulate boundary conditions of a manhole structure between
wing spars. A boundary collocation method in conjunction with a least squares
approach is used to solve the problem.Ministerio de Economía y Competitividad de España DPT2012-3718

Blázquez Gámez, Antonio

Correa Montoto, Elena

López Romano, Bernardo

Montes, A.

París Carballo, Federico

Pastorino Junquero, Daniel

English

idUS. Depósito de Investigación Universidad de Sevilla

ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïë çèíStress analysis of finite anisotropic plates with cutouts under displacement boundary conditions D. Pastorinol, A. Blázquez2, B. Lópezl, E. Correa2, F. París2, A. Montes3 1 Advanced Materials Department, FlDAMC, Getafe (Madrid), Spain. 2Group ofElasticity and Strength ofMaterials, ESI, University of Seville, Spain. 3 Airbus Operations, Getafe (Madrid), Spain ABSTRACT The aim of this article is to provide an analytical tool to estímate the stress concentration in anisotropic plates weakened by a circular or elliptical hole; it is achieved by developing Lekhnitskii formalism in order to allow fínite boundary effects to be added (original formulation considers infinite plates). Only membrane problem is herein considered, particularly when prescribed displacements are applied at externa! boundaries, which could simulate boundary conditions of a manhole structure between wing spars. A boundary collocation method in conjunction with a least squares approach is used to solve the problem. KEYWORDS: Analytical procedure, cutout, open-hole, stress concentration, composite structures, anisotropic plates l.INTRODUCTION Composite structures are widely u sed in aeronautical industry, mostly due to the lightweight requirements of this sector. It is common to have structures of aircrafts weakened by cutouts dueto maintenance and accessibility requirements. For instance, the so-called manholes are cutouts in the structure which provide human access to internal systems or structures. Cutouts are commonly circular or elliptical and cause a stress concentration around them, which have to be quantified for a safe design of the structure. Commercial FE software packages are useful to quantify this stress concentration. Nevertheless, this way of estimation requires to solve the problem many times (most involving remeshing) in order to satisfy the parametric studies required in the design phase. A complex variable formulation, based on Lekhnilskii [1] formalism for anisotropic plates, seems a better choice for a most versatile resolution that could be useful for parametric studies, sin ce remeshing is not necessary. The objective of present article is to develop an analytical tool (based on Lekhnitskii formali sm) to estimate the stress concentration produced by cutouts in anisotropic finite plates. Only membrane forces are considered in the model, since in-plane displacements are applied at the outer boundaries. ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïëçèì--------------~ Finite plate theories have been developed by several authors, mostly considering outer load boundary conditions. There are two main methods which can be found in literature; on one hand there is a method based on the use of Boundary Collocation Points in conjunction with the Least Squares Method, e.g.: Lin and Ko [2}, Xu et al. {3 J and Hufenbach el al. [4]. On the other hand, there is an energetic procedure based on the use of Minimum Total Potential Energy Theorem, e.g.: Xiong [5]. The Boundary Collocation Method Ieads to a simpler solution but is sensitive to point selection; even so this method will be used in what follows. Methodology herein presented is implemented into mathematical software. The results obtained from the method show satisfactory agreement with respect to those obtained with FE, while reducing computational time and avoiding the need of renew the model every time that data change. 1.1 Geometry and boundary conditions The problem herein considered is geometrically defined by the external dimensions of the rectangular plate and the open-hole geometry, as shown in Figure l. As mentioned befare, the main interest of this article will be to apply prescribed displacements to external boundaries. Thus, the following configuration is chosen: • The open-hole boundary is considered unloaded. • Prescribed uniform displacements will be applied at the externa! boundaries, defined by fi and v (Figure 1 ). ,.. L -~ '""1 Ay ~ cb(. ~ w :1 a ~ ~ ,..1 , 2. THEORETTCAL 1\'TODEL 2.1 Lekhnitskii formalism J.ekhnitskii r !J proposed a complex variab le formalism for the anal y ti cal estimation of the stress distribution near a cutout structure in infinite plates. It is based on the accomplishment of equilibrium equations by using an Airy stress function <p, i.e.: - jj 2 cp axay (1) ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïë çèëFollowing this formalism, the derivative <1>¡ = acp¡ is taken as main variable, where z¡ = íJ z¡ x + ll¡Y and (x, y) are the coordinates. As well as J.1j are the characteristic eigenvalues of laminate, i. e. the roots of the characteristic equation: (2) Where aij = Ai/ are the components of the inverse of the elastic stiffness matrix for generalized plain stress. Characteristic eigenvalues are conjugated two-by-two; hence it is possible to write: (3) The variable <l>j (zj) is expanded in series making use of Cnj unknown coefficients: (4) (j is the conforma! mapping of Zj; (5) It transforms the externa! region of the elliptical hole into the externa! region of an unitary radius circle, and it is defined below, i.e. At the hole boundary ( 1 = ( 2 = a = eil/J. A sign selection algorithm for the square root sign could be found in Koussios f6j. Load boundary conditions in Lekhnitskii formalism are developed as: + J; Ynds = 2Re{ <1>1 (z1) + <1>2 (z2 )} = a0 + L~=1 (ancrn + an a - n) (6a) ± J; Xnds = 2Re{f.1.1<1>1(z1) + f.lzcf> z(Zz)} = f3o + L~= 1(f3ncrn + f3n CT-n) (6b) Where the right-hand side term is a Fourier series expansion that could be used when conditions are applied at the cutout boundary. Similarly for applied displacements: .- = 2Re{p1 <1>1 (Z¡) + Pz<l>z (Zz)} = ao + ¿~=1 CanCTn + an CT-n) ~- = 2Re{q1<1>1(z1) + qzcf>z(Zz)} = f3o + L~=1(f3nan + f3n CT- n) Where Pj and qj are: 2.2 Application to finite plates (7a) (7b) (7) The problem of finite plates is solved on the basis of Lekhnitskii formalism by introducing a new term in the expression of cf> j ( Zj) presented in ( 4): ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïëçèê--------------~ (8) On the basis of Lekhnitskii formalism, the actual problem could be reduced by simplifying with a set of two sub-problems, i.e.: l. A fínite plate without cutout, with displacement boundary conditions applied to the external boundaries equal to those applied to the original problem. 2. A finite plate with the cutout and undisplaced external boundaries. A load is applied to the internal contour, in such a way that the sum of the first and second problem gives the solution ofthe original problem. Solution to the first sub-problem is elementary and will not be discussed. The second one is solved by substituting boundary conditions into external and internal boundaries. On the first place, conditions at internal boundary have to satisfy conditions (9). Note that <1>1(z1) and <1> 2 (z2 ) will be replaced by h1((1) and h2 ((2 ) respectively according to the conformal mapping presented in (5). h1(a) + h1C~-) + hz(a) + hz(j-) = J;vn ds J11h1 (a)+ fl1h1 (a) + Jlzhz (a)+ flzhz (a) = - J; Xn ds (9a) (9b) Substituting (8) into expression (9) it is easy to obtain an expression which relates h 2 ((2 ) and h1 ((1) . Hence, unknowns are reduced to half since Cn2 and c;2 coefficients are related to Cn1 and C~1 coefficients. Additional equations are needed to solve the remaining coefficients; it can be achieved by collocation of points at the externa) boundaries, where null displacement conditions must be satisfied: P1h1((1) + P1h1(~) + Pzhz((z) + Pzhz((z) = O q1h1((1) + lhh1(~) + qzhz((z) + Cfzhz((z ) =O (lOa) (lOb) For each collocation point a set of two equations is obtained. Hence, at least 2N + 1 points are required to sol ve real and imaginary parts of remaining coefficients. A linear system of equations will be obtained, which is solved by the least squares method. 3. RESULTS AND COMPARISON The methodology presented above is applied to a particular case in arder to be able to validate it by comparing with respect to a FE analysis. Considered material is AS4/8552, its properties as well as boundary conditions and geometry are found in Table 1 and 2 respectively. a e . es e T bl 1 T t d compos1 e proper Ies. 't f Fiber percentage Thickness (mm) Eu/Ezz Eu/G12 0% +45 % 90% v12 25 60 15 23 .92 15.55 30.1 0.3 ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïë çèéT bl 2 G a e . t eometry an db o un d ary con d"f 1 IOllS. L(m) W(m) a(m) b(m) ü(mm) v(mm) 1.5 1.2 0.3 0.2 2 o Present methodology results are compared against FE results. Resultant forces (through-thickness stress) and displacements at the cutout boundary are shown in Figure 2 and 3 respectively. A great agreement between analytical and finite element model is observed, since relative errors of maximum values remain below 3 per cent (see Table 3). The approximation order N for the calculus has been taken equal to 1 O. And 40 collocation points have been used. 1 1 1 1 1'. , / ,.-.., ""---..... ,. ~ Fo~• around hokl boundary a Presen1Nx o Presen1Ny • Preseru Nxy ---FEMN• - FEI.4Ny -- FEI.INKY "3o~---=~l:----:-,o-!-:-o -----,.,-!-:-~ -----=2~oo------=2~------=3.¿-,oo,....-----=3~5o,-J !Ingle ~ (<119) Fig. 2. Resultant forces com parison of the present methodology with respect to FE results s o.s ~ ~ 3 o .. ~ ·0.5 . - 1 " "" . " Displacemenl:s around OOkt boundary " Pre:!!lenlu o Pre.:senl~ ---FEMu - FEM • " / . ..... '•1"" . .. . .. .. '? ' , "-, // ' ~ ......... . ,.,,.. -........ / · l .so_L----~.1-----,...L.oo----l,:;..,...o ~-..o.L:2oL-o-----.,.25.L.o---3-ooL---3-l5-lo Angle ~ (degl Fig. 3. Displacements comparison ofthe present methodology with respect to FE results T bl 3 R 1 a e . es u ts com panson. Resultant Force Max. Present (MPa· m) Max. FE (MPa·m) Relative Error (%) Nx 6.074 6.129 0.93 N y 2.111 2.167 2.69 Nxy 2.428 2.451 0.93 3. SUMMARY AND CONCLUSIONS An analytical method for the estimation of stress field of finite anisotropic plates with cutouts under uniform displacement boundary conditions has been presented. ÓßÌÛÎ×ßÔÛÍ ÝÑÓÐËÛÍÌÑÍ ïëçèè--------------~ Once the program is implemented, it could be executed on a standard PC leading to accurate results which are obtained in a few seconds ( ~2 sec). Results have shown excellent agreement with respect to finite element analysis, with the main benefit of having a simplified method for stress estimation, which could be easily repeated with different data for iterative design. Further developments of the present methodology should be made to take into account more complex loading cases, such as mixed boundary conditions or complicated displacement distributions at the external edges. Similarly, bending problem of finite plates could be developed following Lekhnitskii [1] formalism for bending plates. ACKNOWLEDGEMENTS Authors wish to express their gratitude to the funding through DPT2012-37187 of Ministerio de Economía y Competitividad de España. REFERENCES l. LEKHNITSKII, S.G. (1 968). Anisotropic Plates. Gordon and Breac/1 Science Publishers. 2. LIN, C.C. AND KO C.C (1988). Stress and strength analysis of finite composite laminates with elliptical holes . .Journal o.fComposite Materials, 22(4), pp. 804-81 O. 3. XU, X.W.; SUN, L, AND FAN, X (1995). The stress concentration of fínite composite laminates with elliptical hole. Computers & Stn.tctures, 57(1), pp. 29- 34. 4. HUFENBACH, W.; GRUBER, B.; GOTTWALD, R.; LEPPER, M. AND ZHOU B. (201 0). Analytical and experimental analysis of stress concentration in notched multilayered composites with finite outer boundaries. Mechanics o.f Composite Materials, 46(5):531-538. 5. XIONG, Y. (1996). An analytical method for failure prediction of multi-fastener composite joints. International Journal ofSolids and Stntctures, 33(29), pp. 4395-4409. 6. KOUSSIOS, S (2008) . Sign selection strategies. American Sociely jór Composites, 1:289-307. 

Stress analysis of finite anisotropic plates with cutouts under displacement boundary conditions

https://idus.us.es/bitstream/handle/11441/81050/Stress%20analysis%20of%20finite%20anisotropic%20plates%20with%20cutouts%20under%20displacement%20boundary%20conditions.pdf?sequence=1&isAllowed=y

Stress analysis of finite anisotropic plates with cutouts under displacement boundary conditions

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