1 Session: ENG Deflection Solutions for Edge Stiffened Plates David A. Pape Department of Engineering and Technology Central Michigan University Mt Pleasant, Michigan Angela J. Fox Department of Engineering and Technology Central Michigan University Mt Pleasant, Michigan Astract The deflection of thin rectangular plates loaded y point loads and stiffened y elastic eams is determined using an infinite series approach. Both ending and torsional eam stiffness is included in the formulations. Solutions are otained and results presented for a variety of plate aspect ratios and eam stiffness values. Comparisons of the results otained with availale solutions demonstrate good agreement, and parametric studies show the utility of this approach for various ratios of eam stiffness (ending and torisonal) to plate stiffness. Introduction Thin plates with eams affixed to the edges with the purpose of stiffening the assemly are commonly used in automotive, aerospace, marine, civil, and other practical applications. Theoretical expressions for the deflection of such plates with edge stiffening can e found in many classical textooks such as Timoshenko  on plate theory. In general, solutions for thin elastic plates are given in terms of infinite series solutions of the governing differential equation. The solutions for edge stiffened plates are otained y comining the differential equation governing the plate deflection with appropriate differential equation governing the eam deflection and enforcing compatiility conditions. In this way, oth the ending and torsional stiffness of the eams can e included. Many researchers have investigated stiffened plates, using a variety of approaches. A recent review y Satsangi and Mukhopaadhyay  of developments in static analysis of stiffened plates listed 104 papers in this area. They grouped the idealizations into three general classes as orthotropic plate theory, grillage theory, and plate eam idealization, and reviewed solution procedures. Concentrically and eccentrically stiffened plates have also een studied using a finite element approaches [3,4,5], and y oundary element techniques [6,7,8].
2 Although the equations are well known, the actual analytical solution of practical prolems remains a challenge, due to the complexity of the equations involved. As such, to the knowledge of the authors the details of a complete general solution have not een pulished. Therefore, in the current paper we present the general solution for the displacement of thin elastic plates with aritrary values for oth flexural and torsional edge stiffeness under various loading conditions. In addition, numerical solutions are given for edge stiffness values ranging from zero (free) to infinity (completely rigid). The symolic math toolox in the computer application Matla is used to facilitate the calculations. The analytical solutions presented in this paper allow calculation of deflection of aritrarily loaded and supported rectangular plates for a wide variety of design and analysis situations. Further, since the limited numer of currently availale solutions for edge stiffened plates have een used in the development of finite element and oundary element methods, the solutions presented herein may also e useful in the further development of these formulations. Rectangular Plate Equations The governing differential equation for the middle surface deflection of a thin, homogeneous, isotropic plate is: where: w w w q + + = equation (1) 4 4 x x y y D w=w(x,y) q=q(x,y) is the plate deflection as a function of x and y is the applied distriuted load, also as a function of x and y 3 Eh D = is the plate stiffness, with 1(1 v ) E v h the modulus of elasticity of the plate material Poisson s ratio for the plate material the plate thickness This equation is valid for flat plate structures with thickness that is small in relation to the width and length dimensions. In addition, the applied loads and resulting deflection of the plate are oth assumed to e perpendicular to the plane of the plate. There are several other assumptions underlying equation (1), detailed the reference y Timoshenko and Woinowsky-Krieger: 1. Deflections are small compared to the thickness of the plate.
3 . The edges of the plate are free to move in the plane of the plate. 3. The midplane of the plate is a plane of zero strain (neutral axis). 4. The loads are such that the stresses are elow the yield strength of the plate material, so that the material ehaves elastically. Edge Stiffened Plates In this paper deflection solutions for rectangular plates with concentric edge stiffening eams on two opposite sides, loaded y a concentrated point load are found. Figure 1 shows the geometry under consideration. a ζ P y x Figure 1. Edge Stiffened Plate The oundary conditions on the sides of the plate given y x = 0 and x = a are assumed to e simply supported. This condition requires that the vertical deflection and the ending moment at any point along these two edges equal zero. The oundary conditions for the plate edges at y = -/ and y = / with attached stiffeners are more complex. The deflection along this edge is otained from the compatiility requirement etween the stiffener and the plate. In other words, the deflection of the eam must equal the deflection of the plate, resulting in the following oundary condition: ( EI ) 4 w 4 x y = / = D w w + ( v) y y x y = / equation ()
4 with: (EI) equal to the ending stiffness of the eam, given as the product of E, the modulus of elasticity of the cur material, and I, the second moment of inertia of the eam cross section. In equation, the term of the left-hand side represents the eam deflection (from elementary eam theory), and the right hand term is the deflection of the plate at the edges where y = /. The second oundary condition is otained y considering the twisting of the stiffened edge. Again, compatiility requires that the rotation of the plate equal the twisting of the eam. The resulting oundary condition is: ( GJ ) w x x y y = / w w = D + v y x y = / equation (3) with: (GJ) equal to the torsional stiffness of the eam, given as the product of G, the shear modulus of the eam material, and J, the polar moment of inertia of the eam cross section. Note that the shear modulus can e expressed in terms of the modulus of elasticity and Poisson s ratio as: G = E (1 + v) In equation 3, the term of the left-hand side represents the eam rotation (from elementary eam theory), and the right hand term is the rotation of the plate at the edges where y = /. The oundary conditions on the simply supported edge are represented y: w = 0 equation (4) x= 0, a w w + = 0 ν equation (5) x y x= 0, a
5 General Solution We seek a general solution of equation (1) suject to the oundary conditions given in equations (-5). This solution can most easily e accomplished y constructing the total deflection w as the sum of two parts: w = w 1 + w equation (6) Where w 1 is the particular solution, which depends upon the location and distriution of the loads applied to the plate, while w is the homogeneous solution, selected to satisfy the oundary conditions. Symmetric Case To illustrate this approach, we consider a rectangular plate with length a, width, and thickness h as shown in Figure. The two edges (at x = 0 and x = a) are simply supported and the other two edges (at y = 0 and y = ) are supported y elastic eams. For the case of a concentrated load P located y the coordinate ξ on the axis of symmetry x as shown in Figure, the deflection can e expressed y a single infinite series solution as: where: Y w Pa mπx = [ Y1 ] sin equation (7) 3 π D a 1 m= 1 α m m m 1 = ( + α tanhα ) sinh ( y) ( y) cosh ( y) with 1 m m 3 α α mπξ sin a m coshα m α = m mπ a This displacement solution is valid for y 0, and will e symmetric aout the x-axis. This expression satisfies the governing differential equation (1), as well as the simply supported oundary conditions at x = 0, and x = a given y equations 4 and 5.
6 a ζ P x Figure y The solution w must satisfy the homogeneous equation: 4 w + 4 x w x y 4 w + = 0 y 4 4 we choose a solution of the form: Pa mπx w = [ Y ] sin 3 π D m a = 1 equation (8) with: Y = A mπy cosh a m + B m mπy mπy sinh a a The function w evidently satisfies the oundary conditions at x = 0 and x = a. We must find suitale coefficients A m and B m such that the oundary conditions on edges at
7 y = -/ and y = / given y equations () and (3) are satisfied y the total solution, w = w 1 + w, given y: w = Pa mπx [ Y1 + Y ] sin 3 π D m= 1 a equation (9) The solution procedure is as follows: 1. Calculate appropriate derivatives of the assumed deflection solution (equation 9) and sustitute these derivatives into oundary condition equations () and (3).. Solve equations () and (3) simultaneously for the constants A m and B m. 3. Insert A m and B m ack into equation (9) to otain the infinite series expression for plate deflection. 4. Sum an appropriate numer of terms of the series to calculate the desired plate displacement. The aove steps represent a significant analytical challenge. Therefore, the Matla software package, which has the capaility to perform symolic computations, was used to assist with the rather tedious calculations in step one and two. Complete solutions for oth A m and B m are provided in Appendix A. Although equation (9) represents an infinite series solution, in practice it converges very rapidly so that only the first five or six terms are required for convergence. Again, software was written using the Matla programming language to perform the calculations. In this way any variety of plate dimensions, eam properties, and loads can e investigated. Examples By appropriate selection of eam stiffness parameters, the aove solution can e used for a wide variety of practical situations. The following two cases, for which estalished solutions exist, are considered: 1. Plate simply supported on all four sides.. Plate simply supported on two parallel sides and fixed on the other two sides. By comparing existing solutions for these two cases to those otained from equation 9 aove, the solutions otained in this work can e verified. Case 1: Plate Simply Supported on all Four Sides The deflection given y equation 9 is required y virtue of the selected functions to have zero deflection on the sides given y x = 0 and x = a, representing simple supports. By setting the ratio of eam to plate stiffness as an aritrarily large numer, comined with a eam torsional stiffness of zero, the plate effectively ecomes simply supported on the eam sides as well. Accordingly, we set the non-dimensional ending and torsional stiffness ratios to the following values:
8 E I Da 6 = 10 G J Da = 0 Software was written to use equation 9 to calculate the plate deflection over a numer of points across the plate. Figure 3 shows the displaced shape otained a square plate suject to a center load. Figure 3. Square plate, center load, GJ/Da=0, EI/Da= The solution for a simply supported rectangular plate suject to a central point load is well known. The maximum deflection at the center of the plate is given in  as: w max Pa = α equation (10) D For a square plate, the /a ratio is 1 and the maximum deflection from equation 10 is identical to that otained through the series solution of equation 9. To further validate the solution otained in this work, the coefficient α is calculated from equation 9 aove and compared for to the accepted solution from  for various values of the ratio /a in Tale 1. In general there is excellent agreement etween the two solutions, with maximum differences less than 0. percent.
9 Tale 1 Maximum non-dimensional Deflection for Plate Simply Supported on all Four Sides. /a Ref  Eqn Case : Plate simply supported on two parallel sides, fixed on the other two sides By setting oth the ending and torsional stiffness of the eam to high values, the eams in effect ecome fixed supports. Thus, setting E I Da G J Da 6 = 10 6 = 10 The maximum deflection of a rectangular plate fixed at y=+/-/ and simply supported on the other two sides and loaded in the center is given in  as: w max P = α equation (11) 3 π D Figure 4 shows the deflected shape otained from equation 9. Figure 4. Square plate, center load, GJ/Da=, EI/Da=
10 Rearranging equation 11 so that α represents the non-dimensional displacement for the present case: α = w 3 max π P D Tale shows the results for the calculation of the coefficient α from equation 9 for various values of the ratio /a. Once again, excellent agreement with the accepted solution is seen. Tale /a Ref  Eqn Maximum non-dimensional Deflection for Plate simply supported on two parallel sides, fixed on the other two sides. Two other example cases are also illustrated. Figures 5 shows the displaced shape for a square plate with oth ending and torsional eam stiffness set to zero, simulating free edges. Figure 6 shows the displaced shape with high torsional stiffness and zero ending stiffness, simulating a guided support (no rotation). Figure 5. Square plate, center load, GJ/Da=0, EI/Da=0
11 Figure 6. Square plate, center load, GJ/Da=, EI/Da=0 Parametric Studies The dependence of the deflection on the ending and torsional stiffness of the eams is demonstrated through a parametric study in which the ratios of the eam (ending and torsional) stiffness to the plate stiffness is varied. A square plate, 60 inches on a side, 0.30 inches thick, with a modulus of elasticity of 10e6 psi and poisson s ratio of 0.33, and loaded with a single load in the center, was used for this study. The twenty two cases studied are listed in Tale 3, along with the maximum deflection otained. Figures 7 through 14 illustrate the dependence of deflection on eam stiffness for this parametric study.
12 Tale 3 Results for Parametric Study of Stiffened Plate. Case Bending Ratio (r) Torsion Ratio (tr) w max EI/Da GJ/Da
13 Figure 7. Centerline deflection (y=0), square plate, center load, GJ/Da=0, various EI/Da ratios. Figure 8. Centerline deflection (x=a/), square plate, center load, GJ/Da=0, various EI/Da ratios.
14 Figure 9. Centerline deflection (y=0), square plate, center load, GJ/Da=, various EI/Da ratios. Figure 10. Centerline deflection (x=a/), square plate, center load, GJ/Da=, various EI/Da ratios.
15 Figure 11. Centerline deflection (y=0), square plate, center load, EI/Da=0, various GJ/Da ratios. Figure 1. Centerline deflection (x=a/), square plate, center load, EI/Da=0, various GJ/Da ratios.
16 Figure 13. Centerline deflection (y=0), square plate, center load, EI/Da=, various GJ/Da ratios. Figure 14. Centerline deflection (x=a/), square plate, center load, EI/Da =, various GJ/Da ratios.
17 Conclusions Deflection solutions for edge stiffened plates using an infinite series approach have een otained. Although previous authors have outlined a solution procedure, in this work complete expressions for all terms have een found. Comparison with availale analytical solutions show good agreement, and parametric studies reveal the utility of the method for aritrary eam ending and torsional stiffness values. References 1. Timoshenko, S. P., and Woinowsky-Krieger, Theory of Plates and Shells, nd Edition, McGraw-Hill, Satsangi, S.K., and Mukhopadhyay, M., A Review of Static Analysis of Stiffened Plates, Journal or Structural Engineering, Vol 15, No 4, Jan. 1989, pp Rossow, M.P., and Irahimkhail, A.K., Constraint Method Analysis of Stiffened Plates, Computers and Structures, Vol 8, pp Linderg, G.M., Accurate Finite Element Modeling of Flat and Curved Stiffened Panels, AGARD Conference Proc. No 113, Sept McBean, R.P., Analysis of Stiffened Plates y the Finite Element Method, PhD Thesis, Stanford University, Tanaka, M., and Bercin, A.N., Static Bending Analysis of Stiffened Plates Using the Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 1, 1998, pp Hu, C. and Hartley, G.A., Elastic Analysis of Thin Plates with Beam Supports, Engineering Analysis with Boundary Elements, Vol. 13, 1994, pp de Paiva, J.B., Boundary Element Formulation of Building Slas, Engineering Analysis with Boundary Elements, Vol. 17, 1996, pp Matla, The Mathworks Inc., Natick, Massachusetts.
18 Biographies DAVID A. PAPE is a Professor of Mechanical Engineering in the Department of Engineering and Technology at Central Michigan University. From 1998 to 004 he was professor and Chair of the Mechanical Engineering Department at Saginaw Valley State University. From 1989 to 1998 he taught at Alfred University, where he served as Department Chair from Dr. Pape earned a B.S. degree with distinction in Civil Engineering from Clarkson University in 1980, a M.S. from the University of Akron, and a Ph.D. in Engineering Mechanics from the State University of New York at Buffalo in He has een elected to the Chi Epsilon, Tau Beta Pi, and Phi Kappa Phi honor societies. ANGELA J. FOX is an Undergraduate Research Assistant in the Department of Engineering and Technology at Central Michigan University. Ms. Fox is an officer in the student section of the American Society of Mechanical Engineers at CMU. She anticipates earning a B.S. in Mechanical Engineering in May, 008. Her professional interest is in mechanical design and analysis.
19 Appendix A Expressions for A m and B m alm=(m*pi*)/(*a); c1=1+alm*tanh(alm); c=alm/; c3=(sin((m*pi*z)/a)/(m^3*cosh(alm))); c6=(m*pi)/a; am=*(-4*c6^*d*c1*c^*g*j*exp(c6*)*+1*c6*exp(c6*)**d^*c^- 4*c6*exp(c6*)**D^*c1*c^... +4*c6**D^*c1*c^+4*c6*exp(c6*)**D^*c9*c1*c^- 1*c6*exp(c6*)**D^*c9*c^-4*c6**D^*c9*c1*c^... +EB*I*c6^5**exp(c6*)*G*J-EB*I*c6^5**exp(c6*)*G*J*c1-6*G*J*exp(c6*)*c6^3*D+6*G*J*c1*exp(c6*)*c6^3*D *D^*exp(c6*)*c1*c^+4*D^*exp(c6*)*c8*c6^*c1+6*G*J*c6^3*D- 6*G*J*c1*c6^3*D-16*D^*c1*c^+48*D^*c^... +4*D^*c8*c6^*c1+8*G*J*c6*D*c1*c^-4*G*J*c6*D*c^- 4*D^*c8*c6^-8*G*J*c6*exp(c6*)*D*c1*c^+4*G*J*c6*exp(c6*)*D*c^... -4*D^*exp(c6*)*c8*c6^+c6^4*D**exp(c6*)*G*J*c1- c6^4*d**exp(c6*)*g*j-c6^4*d**g*j+c6^4*d**g*j*c1-c6^3**d^*c8*c c6^3**d^*c9*c8+c6^3**d^*c8+c6^3**d^*c9*c8*c1+1*c6^**g*j*d*c^ -4*c6^**G*J*D*c1*c^-1*c6**D^*c^... +1*c6**D^*c9*c^... -EB*I*c6^5**G*J+EB*I*c6^5**G*J*c1-c6^3*exp(c6*)**D^*c9*c8*c1- c6^3*exp(c6*)**d^*c8+c6^3*exp(c6*)**d^*c8*c1... +c6^3*exp(c6*)**d^*c9*c8+48*d^*exp(c6*)*c^+1*c6^*d*c^*g*j*ex p(c6*)*)*c3*c*exp(1/*c6*)/c6^3/(*d*c6*exp(*c6*)*eb*i... +*D*c6*EB*I- G*J*c6^*EB*I+G*J*c6^*exp(*c6*)*EB*I+D^*exp(*c6*)- *D^*exp(c6*)*c8*c6*+D^*exp(*c6*)*c8... +*D^*exp(c6*)*c6*+*G*J*c6*exp(*c6*)*D-3*D^*c9*exp(*c6*)- D^*c8-D^+*G*J*c6^3*exp(c6*)*EB*I*+*G*J*c6*D+3*D^*c *G*J*c6*exp(c6*)*D+*D^*c9*exp(c6*)*c8*c6*+D^*c9*exp(*c6*)*c8- *D^*c9*exp(c6*)*c6*-D^*c9*c8+4*D*c6*exp(c6*)*EB*I); m=-4*c3*c*exp(1/*c6*)*(- G*J*c1*exp(c6*)*c6^4*EB*I+G*J*exp(c6*)*c6^4*EB*I- G*J*exp(c6*)*c6^3*D+D^*c9*exp(c6*)*c8*c6^... -1*D^*c9*exp(c6*)*c^- D^*c9*exp(c6*)*c8*c6^*c1+G*J*c1*exp(c6*)*c6^3*D- 4*D^*exp(c6*)*c1*c^+4*D^*c9*exp(c6*)*c1*c^... +D^*exp(c6*)*c8*c6^*c1+G*J*c6^3*D+G*J*c6^4*EB*I-G*J*c1*c6^3*D- G*J*c1*c6^4*EB*I-4*D^*c1*c^+1*D^*c^+D^*c8*c6^*c *G*J*c6*D*c1*c^-1*G*J*c6*D*c^-D^*c8*c6^- 1*D^*c9*c^+D^*c9*c8*c6^+4*D^*c9*c1*c^-D^*c9*c8*c6^*c1...
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