Journal of Mechanical Engineering Research Vol. 4(4), pp. 148-162, April 2012 Available online at http://www.academicjournals.org/jmer DOI: 10.5897/JMER11.088 ISSN 2141 - 2383 © 2012 Academic Journals
Full Length Research Paper
Static analysis of an isotropic rectangular plate using finite element analysis (FEA) Vanam B. C. L.1*, Rajyalakshmi M.1 and Inala R.2 1
Department of Mechanical Engineering, Sir CRR College of Engineering, Eluru, Andhra Pradesh-534007, India. 2 Department of Mechanical Engineering, National Institute of Technology -Rourkela, Odissa-769008, India. Accepted 16 January, 2012
This research work aims to analyze the static analysis of an isotropic rectangular plate with various boundary conditions and various types of load applications. In this paper, finite element analysis has been carried out for an isotropic rectangular plate by considering the master element as a four noded quadrilateral element. Numerical analysis (finite element analysis, FEA) has been carried out by developing programming in mathematical software MATLAB and the results obtained from MATLAB are giving good agreement with the results obtained by classical method - exact solutions. Later, for the same structure, analysis has been carried out using finite element analysis software ANSYS. This job is helpful for obtaining the results not only at node points but also the entire surface of the rectangular plate. Finally, comparison has been done between the results obtained from FEA numerical analysis, and ANSYS results with classical method - exact solutions. Numerical results showed that, the results obtained by finite element analysis and ANSYS simulation results are in close agreement with the results obtained from exact solutions from classical method. During this analysis, the optimal thickness of the plate has been obtained when the plate is subjected to different loading and boundary conditions. Key words: Finite element analysis (FEA), isotropic rectangular plate, ANSYS, static analysis.
INTRODUCTION Finite element analysis (FEA) is a powerful computational technique used for solving engineering problems having complex geometries that are subjected to general boundary conditions. While the analysis is being carried out, the field variables are varied from point to point, thus, possessing an infinite number of solutions in the domain. So, the problem is quite complex. To overcome this difficulty FEA is used; the system is discretized into a finite number of parts known as elements by expressing the unknown field variable in terms of the assumed approximating functions within each element. For each element, systematic approximate solution is constructed by applying the variational or weighted residual methods. These functions (also called interpolation functions) are,
*Corresponding author. E-mail:
[email protected]
included in terms of field variables at specific points referred to as nodes. Nodes are usually located along the element boundaries, and they connect adjacent elements. Because of its flexibility in ability to discretize the irregular domains with finite elements, this method has been used as a practical analysis tool for solving problems in various engineering disciplines. FEA is used in new product design, and existing product refinement. Because of its characteristics, researchers are able to verify a proposed design to the user’s specifications before manufacturing or construction. In case of structural failure, FEA may be used to determine the design modifications to meet the required conditions. Structural analysis consists of linear and non-linear models. Linear models consider simple parameters and assume that the material is not plastically deformed. Non-linear models consider that the structure is pre-stressed and is plastically deformed. Generally,
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three types of analysis are used in FEA: 1-D modelling is useful for solving beam, bar and truss elements; 2-D modelling is useful for solving plane stress and plane strain problems and 3-D modelling is useful for solving complex solid structures. In the earlier period, some of the researchers have been dealt exact solutions of plate deflections with different boundary conditions. Bhattacharya (1986) investigated the deflection of plates under static and dynamic loads by using a new finite difference analysis. This approach gives the forth order bi-harmonic equation which varies from node to node and found the true mode shape of the plate at each node. Defu and sheikh (2005) have presented the mathematical approach for large deflection of rectangular plates. Their analysis, based on the two forth order and second degree partial differential Von karman equations, found lateral deflection to applied load. This solution can be used to direct practical analysis of plates with different boundary conditions. Bakker et al. (2008) have studied the approximate analysis method for large deflection of rectangular thin plate with simply supported boundary condition under action of transverse loads. This approach gives the shape of initial and total deflection of plates. From this analysis, the large deflection behaviour of plate under transverse load can be expressed as a function of the pre to post buckling inplane stiffness of plate. Liew et al. (2001) developed the differential quadrature method and harmonic differential quadrature method for static analysis of three dimensional rectangular plates. This methodology can be used to found the bending and buckling of plates, which are simply supported and clamped boundary conditions only. In the past, some researchers utilized FEA in solving problem plates with holes. Jain (2009) recently analyzed the effect of D/A ratio (where D is hole diameter and A is plate width) upon stress concentration factor and deflecttion in isotropic and orthotropic plates under transverse static loading. Chaudhuri (1987) presented a theory for stress analysis by using rhombic array of alternating method for multiple circular holes. He worked on effects of stress concentration on a laminated plate with hole by finite element method. Paul and Rao (1989) presented a theory for evaluation of stress concentration factor of thick and FRP laminated plate with the help of LoChristensen-Wu higher order bending theory under transverse loading. In the last few decades, authors have considered plates with stiffener. Although the stiffened rectangular plates have been thoroughly studied, the application of stiffeners to circular plates is not so popular. Steen and Byklum (2004) presented an analysis for reducing deflection in isotropic circular plates under the action of static pressure by considering two stiffening rings. In their methodology, they conducted clamped edge boundary conditions throughout the analysis. Troipsky (1976) carried out his work for stiffened plates under bending, stability and vibration. Pape and Fox (2006) presented an infinite series approach for verity of plate aspect ratio by
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using stiffened elastic beam. For smoothen the elements, Xuan et al. (2007) have used boundary integral method. Das et al. (2009) have developed a quite general method which can be applied to any classical boundary conditions. Jain (2009) presented analysis of stress concentration and deflection in isotropic and orthotropic rectangular plates with central circular hole under transverse static loading. He considered three types of elements to solve square plate problems with various boundary conditions and loadings. Shaiov and Vorus (1986) developed an integral equation formulation for an elasto-plastic plate bending. This technique is used for finding plasticity characteristics as well as the external lateral loading. Paiva and Aliabadi (2004) presented a formulation for the analysis of Kirchhoff plates with sub-regions by varying the thickness for boundary element method. This can be used for analyzing building floors such that the boundary integral equations of curvatures of points located at the zone’s interfaces are deduced in a very easy way that makes it possible to get the bending moments. The present work deals with the analysis of an isotropic rectangular element being considered as a plane stress condition. This paper deals with FEA of isotropic rectangular plates under various boundary conditions and loadings. Throughout the analysis, the master element which is four noded quadrilateral elements, are used. Later, experiments have been conducted for same, using ANSYS. Finally, results have been checked with exact results obtained from Kirchhoff plate theory. MATHEMATICAL MODELLING The use of mathematics is one of the many approaches to solving real-world problems. Others include experimentation either with scaled physical models or with the real world directly. Mathematical modelling is the process by which a problem as it appears in the real world is interpreted and represented in terms of abstract symbols. This makes mathematical modelling challenging and at the same time demanding since the use of mathematics and computers for solving real-world problems is very widespread and has an impact in all branches of learning.
Kirchhoff plate theory The basic assumptions being considered under classical Kirchhoff’s plate bending theory are identical to the Euler-Bernoulli beam theory assumptions. Consider the differential segment from the plate by planes perpendicular to the x axis as shown in Figure 1. Load ‘q’ causes the plate to deform in the z direction and the deflection ‘w’ of point ‘P’ is assumed to be a function of x and y only, that is, w=w(x,y) and the plate does not stretch in the z direction. The line connecting points a and b drawn perpendicular to the plate surface remain perpendicular to the surface before and after loading. The following assumptions are considered with the Kirchhoff theory: i. Normal remains normal: This implies that transverse shears strains
and
. However
does not equal to
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Figure 1. Basic view of plate segment.
Figure 2. Shear stress in plane normal.
zero. Right angles in the plane of the plate may not remain right angles after loading. The plate may twist in the plane. ii. Change in plate thickness can be neglected and normal undergo no extension. It means: . iii. Normal stress
has no effect on in-plane strains
and
in
the stress-strain equations and is negligible. iv. The plate is initially flat. Therefore, the in-plane deflections in x and y directions at the mid-surface are assumed to be zero; U(x, y, 0) =0 and V(x, y, 0) =0. v. The material of the plate is elastic, homogeneous and isotropic. These assumptions result in the reduction of a three-dimensional plate problem into a two dimensional one. Consequently, the governing equation of plate can be derived in a simple and straightforward manner.
Governing equation for deflection The aforementioned assumptions of Kirchhoff plate theory makes it easy to drive the basic equations for thin plates. The plate can be considered by planes perpendicular to the x axis as shown in Figure 2, to drive the governing equation. Based on Kirchhoff assumptions, at any point P, due to a small rotation, displacement in the x direction is given as: U=
(1)
At the same point, the displacement in the y direction is: V=
=
(2)
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The curvatures (rate of change of the angular displacements) of the plate are: (13)
(3)
(4)
Using the curvature relationships, the moments become:
and (14)
(5) Using the definitions of in-plane strains, the in-plane strain/ displacement equations are: Where x=
is called the bending rigidity of the plate.
(6) The governing differential equations are:
y=
(7) (15)
and xy=
(8)
The two points of Equations 1 and 2 are used in beam theory. The remaining equations (Equations 3 to 8) are new to plate theory. According to Kirchhoff theory, the plane stress equations for an isotropic material are:
Where q is the transverse distributed loading and
and
are
the transverse shear loads as shown in Figure 3. Substituting the moment/curvature expressions in Equation 15, and substituting the results into Equation 13, the final form of governing partial differential equation for isotropic thin-plate in bending is:
(9) (16) (10) From Equation 16, the solution of thin-plate bending is a function of the transverse displacement w. (11) Analytical solution for rectangular thin plate Where
)
The in-plane normal stresses and shear stress are acting on the edges of the plate as shown in Figure 2. The stresses are varying linearly in the Z-direction from the mid surface of the plate. Although transverse shear deformation is neglected, transverse shear stresses
yz
and
Navier’s method or double series solution is used to determine unknowns for a simple supported rectangular thin plate when a concentrated and distributed force are applied. Figure 4 shows a plate of side a and b. For a simple supported rectangular plate, the boundary conditions are:
are present. Through the plate thickness,
these stresses are varying quadratically. The bending moments acting along the edge of the plate related to the stresses are as follows:
(12)
w=0 and
for x=0 and x=a
(17)
w=0 and
for y=0 and x=b
(18)
An infinite Fourier series for the deflection and distributed load can be defined as follows: (19)
Substituting strains for stresses gives:
(20)
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Figure 3. Transverse distributed load.
Figure 4. A plate of side a and b.
Where
and
are coefficients to be determined.
Equations 17 and 18 assumed for the deflection can easily be showed to satisfy the prescribed boundary conditions. To determine the
Fourier
coefficients
one
can
multiply
by
and integrated twice between the limits 0, a and 0, b:
(21) It can be shown by direct integration that: (22)
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(31)
(23)
Using the property of Equations 22 and 23, one can get a general expression for the coefficient of distributed load as shown thus:
When the boundary condition changes to clamped or fixed. The detailed calculation of maximum deflection with distributed load can be as follows:
(24) Using the assumed Equation 24 and inserting them in the governing equation, one can obtain the following expression:
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(32) The value of α depends on the boundary condition of rectangular plate as in Equation 32, where p is the value of distributed load. As a result, α will be 0.00516 for simple supported and 0.00148 for fixed or clamped edges.
(25) FEA formulation for 4-noded quadrilateral element After simplifying and rearranging coefficient of the deflection is as follows:
Isotropic four node quadrilateral element is having one node at each corner as shown in Figure 5. There are three degrees of freedom at each node, the displacement component along the thickness (w), and two rotations along X and Y directions in terms of the (
, ) coordinates:
(26) , (33) Inserting the expression obtained for coefficients of deflection and distributed load in Equation 16 to get a solution for the governing equation for a general distributed load, we get:
Therefore, the element has twelve degrees of freedom and the displacement function of the element can be represented by a polynomial having twelve terms as shown in Equation 34:
(27) (34) For a uniform distributed load, the expression in Equation 27 can be reduced further by specifying the general coefficient of distributed load:
This function is a complete cubic to which have been added two quadratic terms
and
which are symmetrically placed in
Pascal's triangle. This will ensure that the element is geometrically invariant: (28) (35) (m,n=1,3,5, …) (29)
(36)
The following expression is a solution of a simple supported rectangular plate: (37)
(30) A detailed calculation to reduce Equation 30 is a special case where the concentrated load is applied at the canter of a simple supported square plate. Accordingly, the maximum deflection for a square plate is given thus:
(38)
(39)
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(44)
(45) Figure 5. Rectangular element geometry.
Substituting the functions N (
) from Equation 42 and
integrating, gives, for the isotropic case:
(46) Where
(40)
Substituting Equations 40 and 35, it becomes Equation 41:
(41)
(42)
(43) In deriving this result, it is simpler to use the expression in Equation 42 for w and substitute for {a} after performing the integration. A typical integral is then the element stiffness matrix, and thus;
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Where
(47) Figure 6. Geometry of shell63 element. As in the case of the inertia matrix, it is simpler to use the expression in Equation 42 for w and substitute for {a} after performing the integration using Equation 47. This procedure has been generalised for a number of plate elements with isotropic material properties in reference is the element equivalent nodal force matrix. Assuming pz to be constant substituting for [N] from Equation 44 and integrating gives:
of “compatibility” at the element nodes. According to law of compatibility, the values of the field variables are same for all the elements joining at that node. In structural analysis, generalized displacements are the nodal variables, which can be translations, rotations, curvatures, or other spatial derivatives of translations. Let ‘nel’ and ‘sdof’ denote the total number of elements and nodal degree of freedom respectively. Then the order of assembled stiffness matrix [K] is sdof x sdof and P is the characteristic load vector having sdof x1 dimension. Thus, the global stiffness matrix and the global load vector can be obtained by algebraic sum as:
(48) [k]= [P]=
(51)
After assembling the element stiffness matrixes, global stiffness matrix is obtained. Here, the analysis carried out 2X2 mesh and 4X4 mesh sizes for a given rectangular isotropic elements. ANSYS analysis
The stresses at any point in the plate are given by:
Finite element analysis software ANSYS is a capable way to analyze a wide range of different problems. ANSYS can solve various problems such as elasticity, fluid flow, heat transfer, and electro-magnetism. Beside those, it can also do nonlinear and transient analysis. ANSYS analysis has the following steps for problem solving:
(49)
Substituting for the strains {e} from Equation 49 gives: (50) Where {X} is defined in Equation 35 and is substituted for w in Equation 50.
Assembly of element stiffness matrices Once the calculation of element stiffness matrices and element vectors in global coordinate system is done, the next step is to construct the assemble stiffness matrix. The assembling process of element stiffness matrices and vectors is based on the requirement
i. Modelling: includes the system geometry definition and material property selection. In this step user can draw either 2D or 3D representation of the problem. ii. Meshing: this step involves discritizing the model according to predefined geometric element. iii. Solution: this step involves applying boundary conditions and loads to the system and solves the problem. iv. Post processing: this involves plotting nodal solutions (unknown parameters), which may be of displacements/stresses/reactive forces etc. In this analysis, shell63 element is used with four nodes as shown in Figure 6. Shell63 element type is having both bending and
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A
B
Figure 7. Boundary condition: All sides clamped and simply supported.
membrane capabilities and these elements permit both in-plane and normal loads. At each node, this element is having six degrees of freedom: three nodal translations and three rotations about x, y, and z directions. This element includes Stress stiffening and large deflection capabilities. For finite rotation (large deflection) analysis, a consistent tangent stiffness matrix option is available in this element.
Problem specification Clamped edge and simple support boundary conditions are applied for the considered rectangular plate. The detailed description of the problem is described in Figure 7. Uniform distributed load on thin plate
RESULTS AND DISSCUSSION From the ongoing discussion, the analyses have been carried out as follows: i. Formulation has been presented for exact solution using classical method (Kirchhoff’s plate theory) for isotropic rectangular plate with different boundary conditions and loadings. ii. Finite element formulation for 4-noded quadrilateral element has been presented to calculate the field variables (deflection and slope). MATLAB programming has been developed for isotropic rectangular plate by considering master element as a 4-noded quadrilateral element. iii. Finally, analysis has been carried out for the same structure (rectangular plate) with analysis software ANSYS. In this analysis, a four node quadrilateral element is considered for mesh generation. The aluminium rectangular plate with dimension 1 x 1 m is considered. The following are the material properties for rectangular plate: Young’s modulus E = 70 x109 Pa Poisson’s ratio ν= 0.33
Throughout the analysis, uniformly distributed load (pressure) of 500 Pa is applied on an isotropic square plate as shown in Figure 8. The variations in deflections of the plate with boundary conditions clamped and simply supported are illustrated in Tables 1 and 2, when the plate thickness is varying in the range of 0.01 to 0.18 m. Simulation results are presented in Figures 9 to 13. Concentrated load on thin plate Point load or concentrated load of 5000 N is applied for a simply supported plate with same dimension using the afore stated case and the deflection of plate by varying thickness of plate is calculated. The simulation result is shown in Figures 14 and 15. The deflection values are tabulated as shown in Table 3. The comparisons are made by varying the thickness of the plate in order to obtain the optimum thickness where the thin plate theory works best and also varying different boundary conditions. Also, analysis software ANSYS is considered to verify the results obtained from the developed numerical method using MATLAB. In the finite element method, by increasing the mesh size, the results give good convergence with exact solutions. Here, the 2 × 2 and 4 × 4 mesh size analysis has been carried out.
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B
A
Figure 8. Load applying: UDL and point load on plate surface.
Table 1. All sides fixed or clamped plate subjected to uniform distributed load (pressure).
S/No.
Thickness of the plate
Exact solution deflection
1 2 3 4 5 6 7
0.01 0.03 0.06 0.09 0.12 0.15 0.18
9.86*10 3.65*10-4 -5 4.56*10 1.35*10-5 -6 5.70*10 -6 2.92*10 -6 1.69*10
-3
From ANSYS deflection (m) -3 9.69*10 -4 3.5*10 -5 4.47*10 1.33*10-5 -6 5.559*10 -6 2.86*10 -6 1.66*10
From FEA with 4 × 4mesh size(MATLAB) deflection (m) -3
9.53*10 3.56*10-4 -5 4.43*10 1.29*10-5 -6 5.62*10 -6 2.74*10 -6 1.54*10
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Table 2. Deflection of plate: All sides simply supported to uniform distributed load (pressure).
S/No.
Thickness of the plate
Exact solution deflection
1 2 3 4 5 6 7
0.01 0.03 0.06 0.09 0.12 0.15 0.18
3.15*10-5 -5 1.15*10 1.43*10-6 4.25*10-7 -7 1.79*10 9.18*10-8 -8 5.3*10
From ANSYS deflection (m) 3.1*10-5 -5 1.15*10 1.44*10-6 4.26*10-7 -7 1.8*10 9.19*10-8 -8 5.32*10
Figure 9. Deflection of plate: All sides clamped or fixed supported subjected to UDL pressure.
Figure 10. Deflection of plate: All sides clamped or fixed supported under UDL pressure
From FEA with 4 × 4 mesh size (MATLAB) deflection (m) 3.0089*10-5 -5 1.135*10 1.394*10-6 4.28*10-7 -7 1.84*10 9.173*10-8 -8 5.24*10
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Figure 11. Deflection vs. thickness graph of fixed plate subjected to uniform distributed load.
Figure 12. Deflection of plate: All sides simply supported subjected to uniform distributed load (pressure).
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Figure 13. Deflection vs. thickness graph for all sides simply supported subjected to UDL.
Figure 14. All sides simply supported plate and concentrated load at centre of the plate.
Finally, the 4 × 4 size gives the approximate results. By observing the results from Figures 11, 13 and 15, the optimum thickness of plate is obtained as shown in Table 4
Conclusion This paper mainly focused on the finite element model for finding field variables of an isotropic rectangular plate.
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Figure 15. Deflection vs. thickness graph for a simply supported plate concentrated load.
Table 3. Deflection of plate: All sides simply supported subjected to concentrated load.
S/No.
Thickness of the plate
Exact solution deflection in m
1 2 3 4 5 6 7
0.01 0.03 0.06 0.09 0.12 0.15 0.18
7.84*10-3 2.90*10-4 3.63*10-5 1.03*10-5 4.54*10-6 -6 2.32*10 1.34*10-6
From ANSYS deflection (m) 8.8*10-3 3.3*10-4 4.1*10-5 1.22*10-5 5.1*10-6 -6 2.6*10 1.5*10-6
From FEA with 4 × 4 mesh size (MATLAB) deflection (m) 8.15*10-3 3.16*10-4 3.864*10-5 1.182*10-5 4.68*10-6 -6 2.74*10 1.68*10-6
Table 4. Result of optimum thickness of plate.
All sides fixed plate with UDL load (m) 0.05
All sides simply supported plate UDL load (m) 0.08
The analysis has been performed by considering a four noded rectangular element as a basic geometric shape. During the analysis, plate thick varies from 0.01 to 0.18 2 m, under different load conditions (UDL of 500 N/m and concentrated point load of 50 KN) and different boundary conditions (simply supported and clamped). Later, for the
All sides simply supported plate with concentrated load (m) 0.06
same structure and load/boundary conditions, analysis has been performed using analysis software ANSYS. Finally, the results obtained from FEA and ANSYS have been compared with exact solutions, which are calculated from Kirchhoff plate theory. From the numerical results, it is observed that structure obtains better results at the
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thickness of 0.065 m. Analysis results showed that the results obtained from FEA and ANSYS are closely converging to the results obtained from exact solutions. Experimental analysis for the same structure has to be performed as a future work and the proposed methodology results has to be compared with the results obtained from experimental work. REFERENCES Bakker MCM, Rosmanit M, Hofmeyer H (2008). “approximate large deflection analysis of simply supported rectangular plates under transverse loading using plate post-buckling solutions”, Struct. Design, 46(11): 1224-1235. Bhattacharya MC (1986). “Static and Dynamic Deflections of Plates of Arbitrary Geometry by a New Finite Difference Approach”. J. Sound. Vib., 107(3): 507-521. Chaudhuri RA (1987). “Stress concentration around a part through hole weakening laminated plate”. Comput. Struct., 27(5): 601-609. Das D, Prasanth S, Saha K (2009). “a variational analysis for large deflection of skew plates under uniformly distributed load through domain mapping technique”. Int. J. Eng. Sci. Technol., 1(1): 16-32. Defu W, sheikh E (2005). “Large deflection mathematical analysis of rectangular plates”, journal of engineering mechanics, 131(8): 809821. Jain NK (2009). “Analysis of Stress Concentration and Deflection in Isotropic and Orthotropic Rectangular Plates with Central Circular Hole under Transverse Static Loading”, 52th international meet World Academy of Science, Engineering and Technology, Bangalore.
Liew KM, Teo TM, Han JB (1999). “Three-dimensional static solutions of rectangular plates by variant differential quadrature method”. Int. J. Mech. Sci., 43: 1611-1628. Paul TK, Rao KM (1989). “Finite element evaluation of stress concentration factor of thick laminated plates under transverse loading”. Comput. Struct., 48(2): 311-317. Paiva JB, Aliabadi MH (2004). Bending moments at interfaces of thin zoned plates with discrete thickness by the boundary element method. Eng. Anal. Boundary Elements, 28(7): 747-751. Pape D, Fox AJ (2006). Fox, Deflection Solutions for Edge Stiffened Plates, Proceedings of the 2006 IJME - INTERTECH Conference, Session: Eng, pp. 203-091. Shaiov MA, Vorus WS (1986). “Elasto-plastic plate bending analysis by a boundary element method with initial plastic moments”. Int. J. Solid. Struct., 22(2): 1213-1229. Steen E, Byklum E (2004). “Approximate buckling strength analysis of plates with arbitrarily oriented stiffeners”, 17TH Nordic seminar on computational mechanics, Stockholm, pp. 50-53. Troipsky MS (1976)."Stiffened Plates, Bending, Stability, and Vibration", journal of applied mechanics, 44(3),p. 516. Xuan HN, Rabczuk T, Alain SPA, Dedongnie JF (2007). “a smoothed finite element method for plate analysis”, Computer Methods Appl. Mech. Eng., 197: 13-16, 1184-1203.
