MHAVIOR CH3 ViSCGELAE-Tlc PHOTES UNDER “WE ACTWN OF INuPLANE FORGE Thesis far the Dawn 0? 53h: D. MECHIGAN $TATE UNNERQTY Sammi L. De Laauw 119-61 This is to certify that the thesis entitled BEHAVIOR OF VISCOELASTIC PLATES UNDER THE ACTION OF IN-PLANE FORCES presented by SAMUEL. I... DEL EEUW has been accepted towards fulfillment of the requirements for DOCTOR 0““ PHILOSOEH! . _degree inWECHANICS @/%/Z¢c flajor professor Date Aug. 101 1961 0-169 LIBRARY Michigan State University ABSTRACT BEHAVIOR OF VISCOBLASTIC HATES MR m ACTION OF IN—PIANB FORCES by Sanuel L. DeLeeuw This thesis is concerned with the behavior of viscoelastic plates acted upon by in-plane forces. The theory used in this thesis is sub- Jected to the sane liuitations. as the classical .all deflection theory for elastic plates. Thus, the thickness of the plate must be shall in comparison with the other dimensions of the piste and the deflections nust not exceed four-tenths of the thickness. The visco- elastic stress-strain law is seemed to be expressed in a linear operator forts. Two approaches are nade upon the problem. The first approach considers the problen fro- the traditional buckling view- point. Tuo types of buckling loads are defined and examples of plates coqiosed of the Marvell and Kelvin uaterials are analyzed. The second approach considers a viscoelastic plate with an initial cur- vature. Expressions for the deflection of the plate in terns of the space coordinates and tine are determined. Soue numerical emples of a‘si-ply supported square plate for both the Maxwell and Kelvin materials are included. when considering the buckling of viscoelastic plates. a sane- what different idea of stability, than that used for elastic plates, nust be considered. In viscoelasticity, two buckling loads are de- fined in contrast to one buckling load defined in elasticity. The two Samuel L. DeLeeuw buckling loads are designated as the lower critical load and the upper critical load. As in the usual procedure for analyzing buckling, the plate is assuued to be given a slight deformation. The results of this thesis then show the following. If the applied load is less than the lower critical load, the deflection will decrease toward the flat for- of equilibriun of the plate as tine increases. If the applied load is between the lower and upper critical loads, the deflection will increase udthout liuit as tine increases. .Finally, if the applied load equals the upper critical load, the deflection immedi- ately becomes very large. In the case of a plate composed of a Maxwell material the lower critical load is zero and the upper criti- cal load is equal to the buckling load of an elastic plate. In the case of a plate composed of a Kelvin material, the lower critical load is equal to the buckling load of an elastic plate and the upper critical load is equal to infinity. 1 The governing equation for viscoelastic plates with initial cur- vature is obtained using a correspondence principle. The equation includes in-plane loads, lateral loads, and vertical inertia. ‘The . nain euphasis here is placed upon in-plane loads, but since super- position,is not valid, plates which are siuultaneously acted upon by both lateral and in-plane loads are also considered. In the case of rectangular plates solutions are expressed in a double series of characteristic functions for vibrating beans. Boundary conditions considered are the fixed edge, the simple support and the free edge. The tine dependent functions are obtained from a set of ordinary, linear, simultaneous. differential equations. In sone special cases these equations are not simultaneous and nuuerical exaaples are Sanuel L. DeLeeuw presented in these cases. The solution of a syuetrically loaded, clanped, circular plate is also presented. BEHAVIOR OF VISCCEIASTIC PLATES UNDER THE ACTION OF III-PLANE FORCES A THESIS Suinitted to Michigan State University in partial fulfill-ant of the requir-aents for the degree of mos mnmomr Depart-ant of Applied Mechanics 1961 (‘2 (ENC-7 é/7/é: ACMLEDGE‘HENTS I wish to express w gratitude to Professor George E. Ruse. l-iis aid and encourageuent were invaluable to me during my investigation. ii TABLE OF CONTENTS I. Introduction A. B. Background and General Discussion Scope of Thesis II. A.General Theory of Viscoelastic Plates A. B. C. D. G. Viscoelastic Stress-Strain Laws General Viscoelastic Plate Equation A Buckling Criterion A Method of Solving the General Equation for Rectangular Plates Equationsfor Circular Plate under Syunetrical Loading Boundary Conditions Initial Conditions III. Special Cases and Examples A. Simply Supported Rectangular Plates 1. In—Plane Forces in Che Direction Only 2. In-Plane Forces in‘Two Directions 3. Iinlane Shear h. .A Consideration of Inertia Plates Other than Simply Supported Rectangular Plates 1. Rectangular Plates with Other Boundary Conditions 2. Fixed Edge Circular Plate with Symmetrical Edge Thrust IV. Discussion V. Bibliography iii. \Oflflw 11! 18 21 L2 311 315 56 63 71 74 78 1. 2. ll. 5. 6. 7. 8. LIST OF TABLES A table illustrating critical loads Critical buckling loads for three viscoelastic models Deflection coefficients for a simply supported square Maxwell plate Deflection of a Maxwell plate under various loads based on a first tern approximation Deflection of a Kelvin plate under various loads based on a first term approximation Removal of load from a Maxwell plate Removal of load from a Kelvin plate Response to loads in two directions applied at different times to a Maxwell plate Response to loads in two directions applied at different times to a Kelvin plate iv. 16 18 39 Al All 43 50 I 0a. 6b. 7a. 10.— 11. 12a. 12b . 13. 11' . LIST OF FIGURES Maxwell Model Kelvin Model A Plate Element A Rectangular Plate A Circular Plate Initial Conditions of Load Initial Conditions of Response Gradual Load Gradual Response A Step Load Deflection Curves for a Sinply Supported Square Maxwell Plate Deflection of a Simply Supported Square Maxwell Plate under Various Loadings Deflection of a Sinply Sapported Square Kelvin Plate under Various Loadings Removal of Load Response to Renoval of Load Deflection of a Sinply Supported Square Maxwell Plate mac: the Load ctz Loads in Two Directions Applied at Different Tines V. 10 19 22 31 31 32 32 36 [:0 42 115 46 46 53 58 LIST OF APPENDICES A Discussion of Simultaneous Differential Equations Vic 81 I INTRODUCTION A. Background and General Discussion In recent years there has been a steadily increasing interest in the field of viscoelasticity. Many books written in the field such as {2}”, [3], £4] and [22] have appeared. Also, numerous papers con- cerning the general viscoelastic theory and the solution of viscoelastic boundary value problems can be found in the various engineering, math- ematical and scientific journals. There are practical reasons for the growing interest in viscoelasticity outside of pure academic interest. One reason is that the co-on engineering materials which normally behave elastically exhibit properties of creep and relaxation at extreme temperatures. Another reason is the development of many new engineering materials some of which may exhibit viscoelastic properties. As in elasticity, an interesting phenomena in viscoelasticity is buckling. In elasticity the buckling load for a given structure is a fairly well established quantity. However, in viscoelasticity deflec- tions vary with time and to define a buckling load is not quite as straightforward. In some cases if a ”critical load” is applied to a viscoelastic structure the deflections may i-ediately become very large. In other cases no such ”critical load" may occur. Also, if the load applied is less than the above-mentioned "critical load” the deflection still may get very large'after'a certain length of time. *Ntaabers in square brackets refer to the Biblim at the end of the paper. -2- Kempner in [18] considers the buckling of a viscoelastic column. In this paper the column is asaned to have an initial curvature and buckling is based upon the condition of the deflection approaching infinity. In this paper only one particular viscoelastic model is considered and a general viscoelastic stress-strain law is not used. Hilton in [13] defined a critical time for the buckling of a visco- elastic column with initial curvature. This critical time occurs when the stress in the colt-n reaches the yield point. Lin in [23] con- siders buckling of plates in a manner similiar to Kempner's method for colt-us. A more general viscoelastic stress-strain law is used in this paper, however. Biot in L9] , [10], [11] and [12] presents a somewhat different approach to buckling. In these papers a layered viscoelastic medium under comression is considered. The application of this method to plates will be considered later in this thesis. Several papers have appeared concerning viscoelastic plates. Mase in [5] discusses viscoelastic plates under the action of lateral loads, and the free vibrations of viscoelastic plates. Further dis- cussion of vibrations of viscoelastic plates occur in [6] , [24] and [25] . In [26] a viscoelastic plate on a viscoelastic foundation is discussed. ‘ In [23] Lin considers a viscoelastic plate with an initial curvature loaded by an in-plane force in one direction only. The force is considered constant throughout the plate and constant with respect to time. The viscoelastic plate equation under such conditions is derived and a method of solving this equation for a simply-supported plate is presented. A mnerical example is presented for a square plate made of a Maxwell material. B. Seas of Thesis It is the purpose of this thesis to investigate the problem of viscoelastic plates subjected to in-plane forces. Two general ideas are discussed in this dissertation. The first idea concerns the estab- lishent of a buckling criterion for viscoelastic plates. The second idea presents a method. of solving the problem of a viscoelastic plate with initial curvature. The concept of buckling in viscoelasticity is somesdlat different than elastic buckling. (he of the purposes of this thesis is to pre- sent a buckling criterion for viscoelastic plates. Instead of one critical buckling load being defined as in elasticity, two critical buckling loads are defined for viscoelastic plates. These two loads will be called the upper critical load and the lower critical load. The physical representation of these two loads is as follows. If the actual applied load is less than the lower critical load, the plate deflection will be bounded for all tine including infinity. If the actual applied load is between the lower and upper critical loads, the ' plate deflection will be finite for any finite time, but will increase without limit as time approaches infinity. If the actual applied load reaches the upper critical load, the 'plate deflections will inediately become large without limit. ' The second purpose of this thesis is to present methods of determining deflections of viscoelastic plates with initial curvature and loaded by in-plane loads. A general viscoelastic plate equation is considered 'which includes lateral loads, but the main emphasis is placed upon in—plane loads. The theory used is subjected to the same -4- limitations as the classical small deflection theory for elastic plates discussed in [-7] and [8]. Thus, h, the thickness of the plate, nust be small in comparison with the other dimensions of the plate and the deflection must not exceed 0.11 b. The method of solution employs Galerkin's method (See [14]). In this method the solution is assued to be approximated by a finite series of functions which satisfy the boundary conditions. This assured solution is substituted into the governing differential equation. Multiplication of the equation by a typical one of the previously mentioned functions and integration over the linits of the plate produces a set of equations for the coefficients of the series solution. For viscoelastic plates these coefficients are time dependent. In the case of rectangular plates, characteristic functions for vibrating beams are used. In the case of the clamped circular plate, Bessel functions are used. The plate material is assured to follow a linear viscoelastic stress-strain law which can be elqlressed in an operator form. Such viscoelastic stress-strain laws are discussed in [:1], [2], [3] and [A]. Since linear operators will be used often in the thesis, the following notation will be used. An operator will be expressed in the form A(p), where 'p' is a symbol representing 3/8t . The function upon which the operator is operating wdll be enclosed in the following type of brackets, { 3. Thus, if the linear operator A(p) is operating on the function ¢ (t) the expression will be A00) {Ma} The usual convention of placing dots over symbols to represent differ- entiation with respect to time will also be used. -5- A general viscoelastic plate equation is considered, but solutions are given only in the case of rectangular plates and a special case of a clamped circular plate. Boundary conditions considered for rec- tangular plates are the single support, the fixed support and the free edge. In general, solutions will involve simultaneous differ- ential equations. However, in some cases where a rectangular plate is airply supported on all four sides the simultaneity of the equations is removed, thereby simplifying the method of solution considerably. Some numuerical examples are given for simply supported plates. Since the deflection of viscoelastic plates is dependent on time, initial conditions must be considered. A very practical case occurs udlen a plate thich is at rest, free from stress and with no loads acting upon it, is at some time, say t a 0, suddenly acted upon by a load or several loads, Thus, there must be sue discontinuities in the loading function and/or its derivatives, and it seems reasonable to expect similar discontinuities in the deflection and/or its derivatives. The usual practice in viscoelasticity is to apply the Laplace transform to the governing differential equations using zero initial conditions. The correct discontinuities will then automat- ically appear in the solution (See [3]). This is usually a very con- venient procedure, but not in the case of viscoelastic plates subjected to edge loads. One of the terms of the viscoelastic plate equation contains the product of the applied load, N, and the deflection, w. i This product is then acted upon by an operator of the form Mp). Thus, this term has the form A (p) [/Wt) Wm] - 6.. Taking the Laplace transform of the above expression is not practical except in the special case where N equals a constant. Lin's paper [2;] considers this case where N is a constant. Another disadvantage of the Laplace transform occurs when inverting to determine the solutions. If an elastic plate solution involves complicated functions of the elastic plate constants, D and H, the corresponding viscoelastic solution will require a very complicated Laplace transform inversion. Thus, in this thesis another method of handling initial conditions is used. Boley and Meiner in [1927 present the method employed here. II A GENERAL THERY 0F VISmBIASTIC PIATBS A. Viscoelastic Stress-Strain Laws are useful representation of viscoelastic stress-strain laws in- volves the use of linear differential viscoelastic operators. Such representations are discussed in [1:], E2], E3] and[ 1|]. According to this theory, the stress-strain law for a viscoelastic material can be expressed by the following equations PM) {5.1} = 2 Q(P){Eu} (1) P '0») I 0:5} = 3 am z 525} m- where 5U and EU are the cowonents of the deviatoric stress and strain tensors, respectively, 07:} and 6U are the components of the stress and strain tensors, and P, Q, P' and Q' are linear differential operators of the form Po»): ind—En mp): 312533— 5—,. 2%) Sci?- 07p): sag—g, $5”) The coefficients "n' bn' cn, and tin are constants which represent the the physical properties of the material. For many materials P' a l and Q' = K, a constant. This represents a material which behaves elastically under hydrostatic pressure with I being the bulk modulus. In the remainder of this thesis this assumtion shall be employed. An incompressible material is obtained by letting x-rco . Substi- tuting P' a l and Q' = K into Equation (2) produces -7- 02'; :BKEZL'. (3) In tin subsequent development eqmtion (3) will be used in place of Equation (2). Equation (1) can be visualized as a group of springs and dashpots arranged into a form called a viscoelastic model. Two well known models are shown in Figures 1 and 2. In both figures 7i represents the coefficient of viscosity. 5a j T 5U G G 1:21 7\ L1” = 5U I, 54'} Figure l. Maxwell Model Figure 2. Kelvin Model The stress-strain law for the Maxwell model is . . .5. .. .. ' . . 5U + 2' ~54} -- 2 G 94.; ) (4) where the dot represents differentiation with respect to time and 2‘:- h/G. Thus, the viscoelastic operators would be PM =(p+;_—1-), mp) = 6p , ‘5’ where p 8 3/3t . A Maxwell material, then, is a material which obeys the stress-strain law corresponding to the Maxwell Model. Equations .. 9 - for a Kelvin naterial are obtained in an analogous nanner to be 5/3 = 266g +2czé£j (6) Pan) = /) Q 11' G{/+ Z'p), (7) Examination of Equation (1) shows that an elastic material is a secial caselpf the general viscoelastic material and is obtained by setting P = l and Q = G where G is the elastic shear modulus. B. General Viscoelastic Pla_t_e_Equation. Many viscoelastic problems are solved by using a correspondence principle. There are several forms of the corremondence principle and these are discussed by Bland in [A]. The form of the correspon- dence principle ewloyed in the thesis is as follows. Upon emination of Equation (1) it is noted the Q]? plays the sane role in visco- elasticity as G does in elasticity. Thus, the governing differential equations in viscoelasticity are obtained by substituting Q]? for G in the corresponding governing differential equations in elasticity. To apply the above correspondence principle, consider the following equation for elastic plates (See [7:], page 327) subjected to both lateral and in-plane loads. 4 .. 37mm) a‘w w, amw 0V w, wing, 3x2 +Ny-g-fi-1-P2nyaxay (8) In this equation N3, fly, and ny are forces in the plane of the plate ‘3 shown in Figure 3 on a plate element. Also, q is the lateral load ~10- T N), X A/xy L___. [4 «<————)} ‘ -———-—-‘>—/Vx :7} ny Y Lg Figure 3. ‘A Plate Ble-ent per unit area, '0 is the initial deflection of the plate, and W1 is the additional deflection of the plate caused by the loads. The total deflection of the plate is then '0 + '1' As usual V 4 2 .39.: + 2 2:...— + 3—4.. 9 X 4 9 x‘ay 2- 3y 4 ' In order to obtain the governing differential equation for a visco- elastic plate, the elastic constant D is replaced by a viscoelastic operator 81(p). This procedure is illustrated very well in [:5]. For an elastic plate 5 A3 _ NEG/(+6) /2//—£/’-) " 3(3K+4G) where h is the plate thickness and E,/J , G, and K are the usual D: elastic constants. To obtain 81(p), replace G by M and obtain [130(3KP7‘0) (9) 3P(3KP+4Q§ ' 8,09) = By substituting Equations (5) into Equation (9), the viscoelastic plate operator for a Maxwell naterial is found to be (10) 5 ( > .. [I Gp[(3K+G)p+ 3V7] ’ P " 3(,o+//g)[(3k+4e)p+3/%j ’ and by substituting Equations (7) into Equation (9), the viscoelastic plate operator for a Kelvin plate is found to be __ ABG(/+Z'P)E3K+G)+GZ’P—7 < ) 8’60) ’ 3[(3K+46)+4Grpj ' n Note that Equation (8) does not include any inertia effects. If the vertical inertia of the plate eleaents are included, Equation (8) becomes 0 74W + Pzflsé'm (12) _. fifww‘ a’fm+w) ;‘/u4,+m) - i "IA/X ”9;?424-M 9y; +ZMr/fi‘éj“ where e is tie plate density. Replacing the elastic constant D in Equation (12) by the viscoelastic plate operator 81(p), a general viscoelastic plate equation is obtained and expressed in the fora B, [/3) {V7 4W,} + plé’FA m} (13) .. ‘ gig/fax, QCWMEW 9.1qu “Z + M," J+N 4'2ny -x~&y . This equation is, of course, subject to the linitation discussed in the introduction. Also, as was pointed out in the introduction, this thesis is primarily concerned with deflections caused by the in—plane forces. Solutions to plates with lateral forces are found in E5]. -12- It must be pointed out, however, that if a plate is acted upon by both lateral and ill-plane loads, the principle of superposition does not hold. Thus, a procedure to solve the general Equation (13) will be discussed. In addition to the deflection of the plate, the detenination of the stresses in the plate are also of prinary importance. In order to determine the stresses in the plate, it is first necessary to deter- mine the moments and shear forces in the plate. Equations for visco- elastic plate nonents are presented in [5]. The procedure here is to use the aouent curvature relations for an elastic plate, and replace the elastic constants by the corresponding viscoelastic operators. Thus, the correspondence principle is again utilized. The moment curvature relations for an elastic plate are (See [7], page 88) __ ‘92 W’ a" __ 92W1+d2144 Nix - " D ax”- +0 Mfg-2)) ///y-— ’0 WW 7:21 (11;) 0 2M4 The additional deflection, '1, is used since moments are dependent upon the change in curvature. For a viscoelastic plate the elastic constants D andp are replaced by viscoelastic operators. D is re- placed by 31(p), which is found in Equation (9). The elastic constant p can be expressed in terns of tin elastic constants K and G and replacing G by M, the corresponding viscoelastic operator, U(p), is found tube (3KP-2Ql 14%) 2B«P+Q) (15) -13- By substituting Equations (9) and (15) into Equation (11:), the visco- elastic plate nonents are found to be M “1.3 QGKPW) 923+ GBA/PJQ) 32m 7 ><" 3 P(3KP+4Q) 3x7- “ fl 2P(3KP+4Q) M '_ __A___3 (BF-Pv‘Q; {93:1wa glam-:29 QMJ‘W X 3P(3¢ 0 53p“ Critical Load Increases with time p 900 Load—p Ibper Critical Load 9.00 inediately -17.. Upon examination of Table 1 it is noticed that if the applied load is between the lower and upper critical loads, the deflection is . finite for any finite time. Thus, one might say that buckling only occurs as time approaches infinity. However, at a finite time several other modes of failure might occur. For example, the stresses in the plate might exceed a yield point of the material. Even if this did not occur, at some finite time the deflection would become so large that the mall deflection theory would not give accurate results. Thus, a "critical time” could be defined as the time when the maximal stress in the plate became equal to the stress at the yield point of the material or the time udlen the maximum deflection of the plate reached a certain value. A ”critical time” or ”ultimate time” for a visco- elastic column is discussed in [13:]. Now, consider the buckling criterion applied to sure examples. First consider the elastic case. The viscoelastic operator Bl(p) equals D, the elastic plate constant, and from Equation (23) C equals D. Thus, Equation (22) becomes the elastic buckling equation. For the Maxwell material Bl(p) is defined by Equation (10). Substitution of p = 0 into Equation (10) gives 81(0) = 0. Thus, from Equations (22) and (23) the lower critical load for a Maxwell material is zero. Substitution of p am into Equation (10) gives 31(0) = D, the elastic constant corresponding to the elastic spring in the Maxwell model (See Figure 1). Thus, the upper critical load is equal to the elastic critical load for a Maxwell material. Next consider the Kelvin material. Substitution of p = 0 into Equation (11) results in 81(0) = D, the elastic constant corresponding to the elastic spring in the Kelvin model (See Figure 2). Thus, the lower critical load for a Kelvin - 18 - material is equal to the elastic critical load. 81(00) am for a Kelvin material and thus, the upper critical load is infinite. These results are marked in Table 2. Table 2. Critical buckling loads for three viscoelastic models Material Lower Critical Load : Upper Critical Load Elastic Standard Elastic Critical Load Maxwell 0 . 1 Elastic Critical Load Kelvin “Elastic Critical Load (D Generalizing the above analysis, some conclusions can be made. If the viscoelastic model contains a free dashpot, the lower critical load will be zero. If the viscoelastic model contains a free spring, then the upper critical load equals the elastic critical load corre- sponding to the free spring. D. A Method of 80va the General Euation for Rectggglar Plates. Equation (13) represents a general viscoelastic plate equation. A general method of solving this equation will now be presented. Galerkin's method (See [143, page 162) shall be used. The solution of Equation (13) is assuased to be approximated by the finite series A z , w, (x, y, t ) = E 5 (A... (‘2‘) X. (x; Y. 0/) PM h-ro I120 where the ¢mn{£)’$ represent functions of time, the X.(x)'s represent functions of x and the Yn(y)'s represent functions of y. X.(x) and - 19 - Yn(y) are now chosen to be the characteristic functions for vibrating beams. Use of these functions for plates has been previously used in the study of vibration of rectangular elastic plates in [15]. The functions X-(x) and Yn(y) are required to satisfy the ”artificial boundary conditions” as mentioned in [15:] . Boundary conditions will be discussed in more detail later. These functions satisfy the following relations (See [15] and [16]). 4!: u 4 (j m X): kijmKX) 0/ Y“ :kfl:Y/7(//V) “(X4 60/4 (25) q A ,.. x r 0 #5 ~ . _. '0 "45 J£Xr(X/X5tX/\5/X: C? r:.5' L X(‘,/)YS//"Jo//'/-Zb rs: In the above relations a and b are the dimensions of the plate as shown in Figure l; and kuand kny are constants depending upon only the corresponding integers. Y Figure I]. A Rectangqu Plate Further, these constants are determined when the functions X‘(x) and Yn(y) are chosen (See [15] and [16]). The next step in the procedure is to expand the lateral load, q, ' and the initial deflection, wo, into a double series of the orthogonal characteristic functions X-(x) and Yn(y). These representations would take the form 10/ y) r)’ - é é)” mn/i/Xm/X/‘flz’fl 4/ M-‘Ohso We fly) :g :5”) AM anx) \IQ/V) The V..." RV: and the A-n's are determined in the usual manner of Fourier analysis. Upon substitution of the above expressions for '0' u1 and q into Equation (13) the following equation is obtained. ééB(P/\Z(¢,m/ W/iifidyfy) gal/Yn/WO/ZLZII’LX 600/ ;J:/—___/_3:)'7 07:0"0 dxl (26) :2 o éégfnn/t/X. (UM/HM” /t)+4.,)(x Wéfi/‘gé'ZM/J 0’ /) 4204’ (“Ll {) ”LA/X MM) 02:; +2AX/ a’x %&)] Galerkin' a method is now formally applied by multiplying the equuation by Xr(x)Y8(y)dxdy and integrating over the plate. The relations (25) are also used here. The result is B/(P) {¢r:“)}[ab (Ar)! 4 +k54)] +2 E} 5, (nfa/t/MZZ X /x2-—-—-— éi—“er‘ZX/WW J K” 4d»? 206%n{0+é 0%[QM E/tflmjgf(x—Md ix 2 flufi/X/XJY/Jn . \ +4/yX,,, /x)/( a)”, €359)“ K/y) +Z/ny 6%“02 a? MAW/:7. The in-plane forces, Ni, N, and Mi, are functions of x, y and t. Carrying out the indicated integrations, there results a series of simultaneous,linear, ordinary, differential equations for the unknown time functions, qqhnl’t) . References [15] and [16] aid in the deter- mination of the given integrals. If the in-plane forces are constant with respect to time, the simultaneous equations become equations with constant coefficients. In this case the solutions will involve ex- ponential functions plus the particular integrals. If the in-plane forces vary with time, the simultaneous equations become linear differential equations with variable coefficients. In general the solutions could be expressed as power series. Standard methods of solving both types of problems are discussed in [17] . Although the method of solving the general Equation (13) is straightforward, obtaining the actual solutions would be very tedious. Section III solves some special cases of the general equation, and although these special cases are much simpler than the general case, the special cases studied are not trivial, but do represent practical applications. E. Equations for Circular Plate under Symmetrical Loading; when dealing with circular plates, polar coordinates are usually more convenient than rectangular coordinates. Therefore, the obvious procedure would be to make a variable transformation on the general Equation (13) in rectangular coordinates. However, for circular plates under the action of symmetrical loading, a more convenient method can be employed. -22- The equation for bending of circular, elastic plates is (See [7], page 57) 2. o’z¢ o’é . 0 r2' r + r - ¢ "- (23) c/r'“ 4/!“ 'D where all of the quantities in the equation are illustrated in Figure 5. \. \ I \ O we Figure 5. A Circular Plate The angle ¢ is the angle between the axis of revolution of the plate and any normal to the plate, r is the distance of any point measured from the center of the plate, Q is the shearing force per unit length and D is the elastic plate constant. The special case where the only load applied is a compressive load, Mr, uniformly distributed around the edge of the plate will be considered. In addition, it will be assumed that the plate has an initial curvature, A, . The additional curvature caused by the load will then be expressed by ¢ . with these assumptions plus the equation Q =3 Nr¢ , Equation (28) becomes -23.. Using the correspondence principle the corresponding equation for a viscoelastic plate would be r1 A»- ($0 + fig (30) B,(p){rljrz 45 +r where Bl(p) is defined by Equation (9). Solutions for a particular case of Equation (30) are presented in Section 111. As might be expected solutions take the form of a series of Bessel functions. Once the angle 4) has been determined, moments and stresses are determined from the viscoelastic moment-curvature relations in the same manner as described previously (See [7], page 56). If the additional deflection, wl, caused by the load is desired, the equation (See [:7], page 56) im— ;_- _ 45 (31) a r~ ’ is employed. Thus, if 4t, , is determined from Equuation (30), one integration will produce the deflection "1‘ The integration also produces an arbitrary function of time which can be determined from the boundary conditions at the edge of the plate (r = a). A few words should be mentioned concerning the buckling of circular, viscoelastic plates. The same buckling criterion previously used can be applied to Equation (30). In Equation (30), $0 would be set equal to zero and "r would be considered a constant. Upon application of this procedure, the results obtained for an elastic -24.. material, a Maxwell material, and a Kelvin material will be exactly the same as the results found in Table 2. F. Boundary Conditions. In Section 11-13 the method of solving the governing differential equation for viscoelastic plates was discussed. The method of solution involved the use of the characteristic functions for vibrating beams. The question naturally arises as to whether these functions satisfy the boundary conditions of a plate. Three canon types of supports: the built-in edge, the simply supported edge, and the free edge will be analyzed. These boundary conditions cause the characteristic functions to be orthogoml (See [16‘], page 32?). Consider first the built-in edge and assume that this edge coin- cides with the xpaxis. The boundary conditions along this edge would be (WA/:0 3' é; (%)y=o : 0, (32) The normal functions Yn(y), used in this case (See [16], page 337) satisfy {Y'VZya-o :0; jjijso :.0 ' From Equation (24) and the fact that w = wo + '1' the total deflection is expressed as A a way, r) zhéo £54,... + ¢,,.,./J]X./x)>€,/y/ Thus, in the case of a built-in edge, the boundary conditions are - 25 - satisfied by each tern in the series solution. Consider next the case of a sinply supported edge and again assume this edge to be coincident with the xpaxis. The boundary conditions for such an edge would be that the deflection and the nonent, My, along this edge are zero. Using Equation (16) for the moment, the boundary conditions becone (W)y:o 2 O; 3LP(3KP+4Q) 3/- 2P{3KP+4Q)¢C;XZ 0' O x. , <33) ._ £[QGK P+QL 2.391, QLBKP-QQ) 31mg: 7 The normal functions Yh(y) used in the case (See [id], page 331) satisfy _ J2)? _' {fix/:0 - 0; (04/2 yzo — 0' Again w'has the forn - P 3 w (x, y) 2f) :: Ago :54”, + ¢m~/tflXM m); (y) . Substituting this into the first of the two boundary conditions, it is noticed that each tern of the series for w satisfies this condition. Substituting the expression for w1 into the second of the boundary conditions égives are/«om d1)”; \ .- KP+4QijMMf y,“ {/Y) a/yZ/é:0 mam-2a) flme‘ y :).. +2P(3KP+4Q)Z{¢W/U (JV-=0 "‘ once again the boundary conditions are satisfied term by tern. ‘Thus, ‘the boundary conditions are satisfied at a simple support. -26.. The third type of support to be considered is the free edge. The boundary conditions for a plate in this case is that the monent must be zero and a combination of the twisting monent and shear force must be zero along such an edge. Expressed nathenatically for the edge 1 a a, this becules (MX>X:a : 0; (ax ‘ 354;”),(26, : O ' Using Equations (16) and (18), these boundary conditions can be ex- pressed in terns of the deflection by __[f[0(3/nn(t)'s, this procedure will give simultaneous algebraic equations for the initial conditions. If the equation is of higher order, the integration procedure can be continued until there are enough conditions to obtain all of the re- quired initial values. III SPECIAL CASES AND EXAMPLES A. Simply Sggported Rectanggalr Plates. In this section plates which are simply supported on all four sides will be considered. Solutions (24) in this case take the form I: . I/V/ (X) y) f) I 0% hi ¢/7:n(£)5/bh‘%y5/”§1;X o (40) General approaches are considered for in-plane forces in one direction only, inpplane forces in two directions, in plane shear, and the effect of inertia. Numerical examples for a square plate are included. 1. In-Plane Forces in One Direction Only The equation for a plate with in-plane forces in the x-direction / is obtained from Equation (13). The resulting equation is BMIV‘W = M Qfizw The plate is assumed to have no lateral loads acting upon it and in- ertia is neglected. The initial deflection is wb and the additional deflection caused by the loads is '1' ‘Thus, the total deflection is '6'+ w . The initial deflection, '6 is assumed to be expressed in l the form A % my .. - &.,- Mr WW) ego-m . we. Substitution of wband w1 into the above plate equation yields the following expression. -35- E Ema-I1 +L21L 23,{p)[¢,,,,,/t)]5m—§~ wjinfl-L’Q’ ”13/ 02/ (41) 7"" g. g”: a1 IV KEAMn+%n{tJ-f’”flgfl “fin-Tl Inf/0:1 First, consider Nx as a compressive force which is constant through- out the plate but is allowed to vary with time. Satisfying Equation (41) term by term gives the following equation for each ¢mn(:)° [(4% £42)1:-3(p) NW] [(¢»..{i)}= --’-"—-—A,,,,, a. ”IN/t) The sign change on N; is due to the fact that Nx is now assumed to be a compressive force. Taking Bl(p) from Equation (9) gives C... A30 (3KP+ ah)”. (3)} — 3POK P + 40) [MM {I‘m/£1} (42) = 3 P(3KP+ 4o)u4m[/1§ m} where Lm’ 1'2" ’72- 71 1 In this equation if Ni(t) is constant, solutions take the form of ex- ponentials plus a particular solution. If N*(t) is a function of time. other types of solutions occur. If N;(t) can be expressed in a power series, a series solution can be utilized. .As an example, let N&(t) be the step function shown in Figure 8. Solutions for the elastic, Maxwell, and Kelvin models will be deter- mined o 36- )V{£) 0 Figure 8. A Step Load For the elastic model, the viscoelastic operators P and Q become 1 and G respectively. Substituting these values into Equation (£32) and solving for ¢m(t) yields é {é} : BNAAIL @le4G) hm‘ EC Noting that mlqu ”171. 2. C .. A 07: 4' 23f") hm " b.7771 a and that the elastic plate constant D 2 5’6 (3K+ 6) 3(3/x’+ 45) the above expression for ¢m(t) can be expressed as I , «(j/rag ”3/ M) = #6: (m 4.71)-” M513 GUM G) —-3N(3K+ 45):] (i>o). which agrees with the elastic plate solution (See [83, page 321). For the Maxwell model, the viscoelastic operators for P and Q are defined by Equations (5). Substitution of Equations (5) into Equation (42) and noting that Nx(t) = N, fix“) = ii;(t) = 0, yields the following differential eqmtion for ¢m(t). III II I'll. ll! I'll Ii] lunllllr xlxl III-Ila; l I ,‘llallal. If, v 1f l «Ill iia II‘ \‘.II. 1" v I [ j I . .‘ -375- Emzficemc) - 3/3/(+46)A/j gs,” m + _§_[&'M:/,3:: - 2(3K+26)/V] dim/i) Using the notation z ZEN/.36 (2w 5) - 3 (3m 4524/? d : ’rnn och..." = BTW/236K - 2 (smacmj (X3 1' 7”“ I r. — film” +‘/q/Zin + 4a/m" d3”)? I ’hm -— 2 09m“ _. —' C(32)”! — V/O‘lzfm + 4dlmnq3mn 7 I2 — a hm é d‘mh the solution of this equation then takes the form 05W) :- Bm e” + 0,”, 5’5“ — AM must be determined from initial conditions. The where B and D an mn initial conditions are obtained from Equations (38) and (39) which its this case are [empwma —3(3K+4a)/vj éM/a), — 2(3 mum] ¢,,,,,/o)+ : ~5- (yew/v.4,” + g-[CMPGK -38- and [EM/Emma) —— 3(3/(+46)A_/7Qm (0,; = gammy/1,." Using these expressions for the initial conditions, the constants an (t) . and ”mu can be determined and the solution for (25“ is then .. 31%... In... Tame/{mgpzamzcfl + WM” tymt ’ QM /Z) " 8 can» nnn(/7n - ’3'”) 3NA,..,. $13,,Er5.,(3k+46)’+2(3k+2d+7/VKAM ’5”? ’ 8 - Am, a’m" Gm" (,2... -— Gun) If N is less than the upper critical load shown in Table 2, an exam— ination of rm and r2mn will show that rh.m>0 and r2“: 0. At t s O, ¢m(t) becomes equal to the corresponding term for an elastic plate. As a numerical example, a square plate (b = a) is considered and values for the material constants taken are G = 2 x 106 p.s.i., K = 10/3 3: 106 p.s.i. and T: 2 hours. The value of N is taken to be one-half the upper critical load. Calculations were made for m and n ranging over the odd numbers from 1 to 5, .Two hour intervals were used for 0 to 10 hours or from O to 52‘. The results are presented in Table 3 and Figure 9 which relates mum to 12/1 for Z’equal to two hours. It is noticed that ¢n/An is quite predominate. The closest term to ¢11/‘ll is ¢13/A13 which is 22% of 11/All at ti; = 0 and 2.5% of ¢11IA11 at t/t 8 5. It seems reasonable to some that A 13 will be less than an. If A13 is taken as 22% of an. ¢13 becomes -39- Table 3. Deflection coefficients for a simply supported square Maxwell plate “._ ' fi—4---T--::‘~.._ ¢mnlAm : I T g . I I | "i 2 I I lg is l 1 "1/“ V730 *ZEE.1J'.-E{E=ZI*/9=3 t/2=4't/t=5__ -_l/_1___..._1.: W/(XJ Y) i) "" ¢ll{t) 5’” a 5M 5 ' In subsequent numerical examples this approximation will be made in ‘Ccordance with the above discussion. The accuracy of this approxi- mation largely depends upon the dominance of All in relation to the Other A '3. mn . Additional calculations were made of ¢11IA11 for a Maxwell plate with the value of N taken as one-fourth and three-fourths of the upper Gritical load. Results of these calculations are presented in Table ll -130- 60' 50. {14*}!1/1/t4 hug 40 - Rd 7‘ [<9 J 30 20' [) («f/e.) C700” /0' f M“ {if/‘6"; O / a 3 4 5 77/778 Raf/'0) t/Z’ Figure 9. Deflection Curves for a Simply Supported Square Maxwell Plate (Note: 4513, $15, ¢33, $35, ¢S3 and ¢55 are too small to include.) “d Figure 100 1.1- It is noted that as the load approaches the upper critical load, deflections becoue very large in a relatively short period of time. In all cases the initial deflection equals the de- flection of an. elastic plate corresponding to the spring element of the Maxwell model and irregardless of the load, the deflection will. eventually become very large. T‘ble a s Deflection of a Maxwell plate under various loads based on a first term approximation 4311/511 -_....- N 22:29 V... t/r'z t/r'j mafia-5-242.: AN : 0.333 0.637 1.122 1.663 2.340 3.188 a) 91222... 1.900 3.958-..- -7933- 45.259. 231.991- 93.531 0" 199....--3-099 _. 32-822 299.93. 2.32:7. .99 .559... 192.839 .- 33...... The viscoelastic operators for the Kelvin model are defined by Equations (7) and substitution of these into Equation 1&2) produces CMABGZ'rlém (.2) + thCmn/Pemze) - MA] (A. (t) +[ZWA36(3K+G)'3(3K+4G)A_I]¢m,,(2) 2' 3NA,,,,, (3K+4G). using the notation -42- go. / A O r Def/eCf/on Ra1‘l'o/ ¢”/.4// N in a) 3 Removal Of/oaa’ A/ t’x ~ I r I-J Time Robb,- 1“/2' Figure 10. Deflection of a Simply Supported Square Maxwell Plate under Various Loadings 2pm .- 2 ”In”, the solution of this equation then takes the form I: 1'- ¢mn(£): B r t 'lrn 3NA»0(3K2:46) ’Ian 0 ’Mnr where an and Din must be determined from initial conditions. The Inn initial conditions are obtained from Equations (38) and (39) which in this case are 3 1 " . Chnl’ 6 Z"l¢’mn/O)+ + Gf‘gNNG/{di‘r—/2/J¢,n(b). 2’ MAME 27V and . 3 r‘ 2 2V 7 X. . ‘. Chen A U L f}I9/~4 r /’+ "' O 4 Thus, )0 I , I f ‘ um. 1 [-1‘7’,‘ I. 1“,:‘ hymn 2'5 ’4. - » I CynnAJG T ‘ Using these expressions for the initial conditions, the constants E.“ and D can be determined and the solution for g5 (t) is then mn mu .5 \ f - an.L-/ 9" I'szzAflfimmwrrn ) I p at q‘hhr‘ I," ”(r/m." 25M; [3N AfinlBKM 4'4” I I: 4': M29]? SI‘V/A ”n (Bk‘f 452,) d o a!!!» Ch". (I; In" '- ’"nnnn rip-n -44- If N is less than the lover critical load found in Table 2, an exam-- ination of than and rm will show that rhn< 0 and rhn< 0. Thus, ‘C t-a-CD ¢ (,0) __ 3mm (3K+4G} _ 3,023,.” (re/(+45) m" 0’1... f“ l3,” LCM/23¢; (3;{+GI"——3(3K+4G)A/_J ’mn which equals the corresponding tern for an elastic plate. The sane values for the constants and loads given for the Maxwell plate are used in a m-erical ens-ole for the Kelvin plate. Results corresponding to those in Table l; and Figure 10 for a Maxwell plate are given in Table 5 and Figure 11 for a Kelvin plate. Table 5. Deflection of 0. Kelvin plate under various loads based on a first tern approximation .__....II I____ ”It (fill/AL]. N t/c=0 t/csl t/r=2 t/c'3 t/call t/z=5 V69, his“. 0.000 0.223 0.285 0.312 0.324 0.329 0-333 was“, 0.000 70.537 0.744 0.858 0.922 0.957 1.000 i using,“ 0.000 ‘o.993 1.545 ‘ 1.9M 2.234 2.444 3.000 4 Another interesting aspect of the problem occurs if the load is removed at sone tine, t1. The governing differential equation in this case for t>t1 is obtained by letting Nx(t) 8 0 in Equation (112). This produces . 7 Q<3KP+ Q)[¢M/t/j :: 0 (a3) -45.. I—-—— ——._.... _ - __._._ ___ ._ V...‘ —— ~.._—- -————-— m 2.5L 3 I] : 4 A/L‘f' yl/’// “i Q'Sb' // 7;} /’ / / .0 / #3 ° 5 0 I' // Q / k / .') 'L; // x) ,. \q; /.u _.___.. /1 .__.. ..___ .. \ I , - K~k /’/ )VI : .2— jl‘v'lc I“. ”'5‘ ‘/ Q, /' /// P.\' 1 l" I- .I' K; // I,/"' K, / --/ A dd N o, 5 » / y 0 // //’ ‘ J 7“) // Y __ ____~...::.’::...:_:::" w";- / /" [98 I770 V0 / \_ N: 71'1"": r. " 0 1C Z. 0 6d 1 1 1 1 “‘MH *‘fi L 9 I 2 3 4 5' 77/» e Raf/'0’ ‘5/33/ Figure 11. Deflection of 3 Simply Supported Square Kelvin Plate under Various Loadings -46- For the elastic case 1’ and Q are constants so that the only possible solution is 45...“) -—‘= O {i7é’24 For the viscoelastic cases, Equation (43) is a homogenous equation and thus the solution depends entirely upon the initial conditions at Since there is a discontinuity in the loading function, discon- tinuities also night occur in the response function, ¢mn(t)' The method of deternining these discontinuities is similar to the nethod of determining initial conditions discussed in Section 11. Figures 12a and 12b correspond to Figures 7a and 7b. l¢(£) (Maw N K . ' V” ”(é fIii: \ \JW‘J o J t, t; t Figure 12a.. Removal of Load A Wt) dj/tJth’) W. ,5 \ wan cab/0K ‘ “‘ <> t, 2. t Figure 12b. Response to Removal of Load If the material is either a Maxwell aaterial or a Kelvin material the differential equation of ¢m(t) for t1< t fd The solution of this equation takes the form (31%) g-a) .615 fix.) ', 1J- '— — - t? ‘” ‘, $17M" ‘“ Bin!) 8 U + Dmn e (i 7147/. Equations (41;) for the Kelvin model become .3 1 \ Chm/7 G 2’ $1,"(r,,‘+ +£M/I36'r (BMZG, £55, ._ /- - 6mm}? 32.6 [ zénnfr ll.- ,4- .,r\ .7 ‘ ‘fJZ :Jmph (aflfsia/‘I/Z/Y'I/ /ir/_ -/26'z"4mn/«'/ In» and 3 1 fl, . f .1 / ‘ Chm/1 G L 2 fine» (fl)+ -[),,,1‘ AU 3 1? Zylzflnn{ty These produce élm (4)4. : ¢nm ([7)— ,Z , __ ' /2/ (Ant +M +17”: (Ki/)4— "' ¢ma(£l)- —. E€n63g% A j . The solution is now written {fa-{1" - “‘1’”- l 4?, (f):§;;“‘!T 325,, (f)++ 4“” 9210‘] ..£15_t.'~_l;) (75:11,} _ - 351.. 9,12: Q4”: (t )+ + {fmn ‘ 4“")? (:3 fl: 7‘5“". -50- with ¢m(t1)+ and ¢mn(tl)+ defined above. Again t1 was taken as 22' in the previous case of a Kelvin model for the load equal to one-half the upper critical load. Table 7 gives the nunerical results for ¢lllA11 up to 10 hours and these results are plotted in Figure 11. It is noticed for a Kelvin naterial that Table 7. Removal of load from a Kelvin plate t/t 0 1 2(-) 200') 3 A 5 w {IL/A, 0.000 0.537 0.744 0.741; 0,169 0.062 0.023 0.000 there is no sudden decrease in the deflection, like in the Maxwell material, but a more gradual decrease. Further, it is observed tut as tine increases the deflection approaches zero in spite of the fact there is a dissipative element in the Kelvin Model. This is caused by the action of the spring and dashpot in parallel. lb to this point in the examples discussed the load has been con- stant. As an example of a load varying with tine, consider Equation ([12) with Nx = ct2 were c is a constant. Equation (42) then becomes 3 s. 7’ thA Q (3K ””0442,” (a); (45) —3P(3m 4a}[czzgn[a§5 :3P8Kp+4o)[ct74m. - 51 - If the plate material is elastic, the solution to Equation (AS) is 7) (f) ._ 3/379 4.4),4M a?” "M E‘MA3C;(3K+6J - 3(3),; 45,1 157;] ' If the plate material corresponds to the Maxwell model, substi- tution of Equations (5) into Equation (45) produces o \ . 2:: O . , ' . ,_ (0’!an din/i ’3 ¢mn (é) + (OCBMn- 4% : -'C\/4 ”162/, an I: am he '(Za +7c/ f ’d- f2)¢ {33:4 {20" 4'2 ’ it“ 752) 2m ‘ ‘4pm 7 5m, ’ 1M ” ’W’ ‘2m1‘ 0‘4!“ 5n» where = [Mn/736 {Wm-6) : 36. (35+46) lmn zwnn as” = {—CMAJGK «W = ~2-é—c(3K+26) “5;... 3 '5‘? C K Power series solutions to differential equations are found in many textbooks such as [:20]. An examination of the above eqmtion will show tint the only singular point of the equation is when the load, Hg, equals the upper critical buckling load. Thus, it is assumed that the load is applied during a time which is shorter than the time re- quired to reach the upper critical load. The homogeneous solution for the above equation takes the form ¢mn (75) a: 8 firm .. nan mn‘o -52- The particular solution is ¢mn (i) 2: — Ann. Thus, the complete solution can be put in the form ¢Mn (t) :1 (BOW, -' AIM/r) + 523/ Bk'mtkm’. Since Nxfo) 213(0) 2, O, the initial conditions on 4).“ (t) are ¢m(0) = $5.,“ (0) 8 0. _ This means that ohm L: A"), B/l‘hn : O a The remaining coefficients, 83.“, are obtained by substituting the homogeneous solution into the huogeneous equation and won doing this the following recurrance fornula is obtained. a“- 4 +~4,.,.(k,.. 95w +or ,mkmm— 08,...2- as ”MM—123, Dunk Fm (Kn-u " I) This formula is valid for km>l if Eda“ is considered zero. BkMF-h: As a numerical example the constant c is taken as 1/200 of the upper critical load and calculations are made for (pm/An up to 10 hours when the load reaches ore-half of the critical load. The other constants are taken to be the same as before. The results are plotted in Figure 13. If the plate material corresponds to the Kelvin model, substi- tution of Equations (7) into Equation (45) produces U 0 2 . dim" ¢Mn (t) + [quhh’ qJMntj ¢Mn (f) flgém'zqémt ’ ”5' is] 47’1" (‘1) .-.— AWEWBM 7“ *°97- i] -53- Time I. a Deflection (moans/a2; Ratio Ratio 0 0 O 1 0.351 0.025 .A . $3.0 2 1.404 0.109 *3. 3 3-15'8 0-404 A? A ,, _ ’5 3'0 a 5.615 1.126 {E .g 5 8.773 3.661 \ ‘1 *h = plate thickness 6 2-0 ‘ a a plate dimension _\ N. <3 /.0 - L" _ 7).)” e, IQJ fie} t/Z' Figure 13. Deflection of a Simply Supported Square Namll Plate under the Load ct2 -54- where [M : Chm/1362 2'2 a,” : thjGrO/wzé) 0.53M : £2ch 00.. = Chm/>36 (3K+6) 09—” : 3(35+ 4€)c. There are no singular points in this differential equation for finite time. The homogeneous solution of the above equation takes the form Ch... ft) == 1 E M ii“ Ah”: 0 and the corresponding recurrance formula is . -. I , g ”SHE/(mi 4. +qjmn{km-/)Bknz3-q4pn8kn;2 me’I'g, DB I'm-I 5 = - - . 1 A'P‘” win,” knm (knm'l) (1.20. Again 8.x“ must be considered zero. Bonn and Blmn are undetermined. The particular solution is not as simple as in the case of a Maxwell plate. The particular solution is assumed to be a series of the form (29 , ¢nm(t) 3 E D}, tkM+1, kwn:o Mn Substitution of this series into the non-homogeneous equation yields Dom 3 O ._ 6(3mn Am” D/nm 3 qua" -- ”DIM. " 4 + %Mn(k"‘+ [)0kng- q/‘flmD/Xm; Z - q2h»(kM0+/) DKnm” ) .. "—— 00” (kmi-Z) (k~+]) ”“ """ (kmfl . . III... illllllll...|:|.1|.ll 1 ill Ills! 1|» '1 , ‘. .lurlllnu. ' UT ‘J‘ | The initial conditions are that 43mm) = 40mm) = 0. Since the lowest power of the particular solution is three, the initial condi- tions cause the homogeneous solution to be zero. Thus, the total ex- pression for ¢m(t) becomes ¢Mn(t) :éo DA,” tkmfz where DonnD’mn and kan are given above and it is understood that D4.“ must be taken as zero in the recurrance formula. In the discussion of simly supported plates up to this point, the load was assmed constant throughout the plate. Next consider the problem where the load varies with the space variables, but remains constant with time. The procedure in this case is to multiply Equation (£11) by sin r‘n‘x/a sin s'rry/b dxdy and integrate over the limits of the plate. The result of this is 2. ' .1 '94..- b r 42:12 +fl3— \" .1 ,6 9 .m22 %¥[AMN + @Mliu (1'6) #1:] A=I [la/N {X man-"3335111 Laws/ngzsin—Ela/xd/j If a finite number of the terms ¢m(t) and Mn are taken as an approxi- mation, there results a set of simultaneous differential equations with constant coefficients for the ¢m(t)'s. An example of a load would be the application of a canbined axial and bending load (See [8], page 350). If the axial load is compressive, the form of NJ: would be -56.. A4, :—/V0(/"‘%'7) where N0 and O( are constants. Substituting this expression into Equation (46) and performing the indicated integrations produce the following set of simultaneous differential equations for the ¢mn (t)'s. [NEE/(P) [V0 0. 1)?{¢r5({)]' 8 AC] 7:35); {”7.h57‘1¢rn(£/ ()7 i 5) odd) 9 0315 Arr: Maw-g» + g;,-:-—~_;,~;~ (n :t 5, odd) As a first approximation only one tern of the series could be used. In this case the above equation becomes ENE/(P) "' No(/’%y {¢,,/f)} 3' AHA/00" it). which is exactly like Equation (42) with n = l, n = l, and Nx(t) = HOG-g). Thus, previous results of a plate under constant load could be used with the constant load N replaced by No(l - 3-). A one tern approxi- mation would probably only be accurate for null values of O( and then this would only be accurate if a one term series is accurate for a plate loaded with a constant compression. For more accuracy, more terms should be used and this leads to the problem of solving simul- taneous differential equations. 2. In-Plane Forces in Two Directions Following the same procedure used in obtaining Equation ([11) for inaplane forces in the x-direction, the corresponding equation for -57.. in-plane forces in two directions is ’5 é[m‘1fl+h;,, a - :l a (In) # 1 - 2 T-Ig—ifiA/X 54mm + ¢,n{£jflhw5/hm 072/ B, (pJZIéM {H} :m—- “a” --—-5m-”—;—’-’ h: A Q 22”""’N[ flu” 'W W n 52 y A... +4§Jt 5m 5. 5m 77—. If Nx and Ny are functions of time alone, term by tern satisfaction of this equation gives [CM B (p)-- -(N (f) +fifT—a b; "E’Nyaj {an (0} (48) :4,” [man +,;’, 4-1;. JAM] Comparing this equation with Equation (42) shows that the solutions of Equation ([12) can be used with Rx“) replaced by Dix“) + (n2a2/m2b2)hly(tfl as long as the discontinuities in Nx(t) and Ny(t) occur at the same tine. An interesting exception to this case occurs when step functions which are applied at different times, as illus- trated in Figure 111, are acting on the plate. The a represents any numerical factor. In the elastic case. the time when the loads are applied has no affect and the solution of Equation ((38) is (t >t1) C? (t) :: 3N0 +m2“? )Amn(3K+4G) M [:CM"A3G(3K+G) ’3N0 +m1b‘M%K*4GZI Al O( N _ __..._.r._.‘ .4, .---_ _. o t, Figure la. Loads in Two Directions Applied at Different Times For a plate of viscoelastic material being subjected to the above loading Equation (48) becomes [CM/)8 07)" N0 +m::‘qfl{¢°m(fl} - -—A,..,.N(/+ 57—me) a»). Let C ’ __ Crown. ’"" " (1+ 5%) Then the above. eqmtion becomes [(1:27 8,60) " A17{¢Mn(t)} : Aha/V (g 7.4} (249) which is exactly like the equation for a constant force in one direction with Cm replaced by th' Thus, previous solutions can be used if the discontinuities at tine t1 are taken into accotmt. These discontinu- ities are determined using the method for initial conditions discussed in Section II. -59.. The solution of Equation (1:9) for a Manell model is taken from the corresponding solution of eqmtion (112) for a Maxwell model with qu) :- N and cum s cm’. This solution is ,l . r; ,f C,” e . , chm/U = am a:- + D 8 -- 4M (am For convenience this solution is put in the form (..-2’ 9km“) : m» C '7,mt& This expression along with the previous equations for Q5.“ (t1) and $1.” (t1) is the solution to Equation (£19) for a plate made of a Maxwell material. Cowputations for $11] [All were nude with N equal to one-quarter of the upper critical load for a Maxwell plate loaded in one direction and Of equal to unity. The plate was considered square and values for the other constants were taken to be the sane as used previously. The results are tabulated in Table 8 and plotted in Pime 10. Table 8. Response to loads in two directions applied at different tines to a Maxwell plate t V: o 1 _2_(-) 2(+) 3 z; 5 co 4911/11110333 0.687 1.122 2.183 5.419 11.837 24.657 co The solution of Equation (49) for a Kelvin model is taken fraa the corresponding solution of Equation (1:2) for a Kelvin model with Nx(t) = N and can In Cm'. This solution is QM“) 3‘5 Bum enmt 1-D”: 8G“: + Bxflmrgjki‘qd (i 7i.) 1,... I an 2pm -61.. For convenience the solution is put in the form I n t' ti) rmn(t-t’.)3 QMn(f/’ ‘ BM 8 I“, + DH” 8 2 + gflmn(3k+46/ Kt>t\ ’mnr [”102 PM Again, the solution for t < t1 is already know. The discontinuities at time t1 are found in the same manner as previously used and the following two equations are obtained. ¢mn (£12,. : (22,0{f/L. ' .. ' I I2 N£§£rzdflm./£.J. +4.4 gxtd" ’ ¢"'(£’)’ +. , Chm/.362“ The left hand limits are known quantities determined from the previous solution of a Kelvin plate loaded in one direction only. The solution for the Kelvin plate can then be put in the form (t >t1) 4> (t)- 12"” w(¢~~W+ Ewan/t, .)+ 3N4...(3k+45)]€ 1-.) m” ‘ qjmnr IMn (nn,,,"'r‘ 2m”) ‘ .—l + Inm Jinn + 3 V ”n?“ . Ya, Main"? GMn(/nm: glut-1,; J8 “In"! finncmn This expression along with the previous equations for ¢m(t1) and O ¢nn Then repeating this step again produces another equation of the form 4 2. 3% - A {045, + A (t)¢ + A3 (0% 54/0934 d'y‘fiZ-(h) -85.. Equations (f) and (g) can be treated as simultaneous equations and solved algebraically for the quantities $2 and ¢ 11' Substituting these quantities into Equation (h) and using the first of Equations (c) gives a single equation for 4’1 in the form if: *k-‘(fl 753 7‘k2 (A) :5"? ”((05% {/raahfirga). (i) Using the sane procedure on the second part of Equations (e) will produce an equation for ¢2 similiar to Equation (i). Equation (i) is then solved using the standard methods for ordinary linear differ- ential equations with variable coefficients. The integration constants can be determined from the initial conditions on ¢1(t) and ¢2(t) and by requiring that Equations (b) must be identically satisfied when the solutions 491“) and 452“) are substituted into them. The initial conditions for ¢1(t) and ¢ 2(t) are obtained by applying the method of determining initial conditions discussed in Section II to Equations (b). This will yield simultaneous algebraic equations for the initial conditions on ¢1(t) and $2“). The discussion so far has considered a two-term approximation. If approximations with more terms are desired, the procedure would be the same. However, it is noted that for a two-term approximation the calculations become quite involved even though the procedure is fairly straightforward. In the first case where the coefficients of the differential equations were constants, the main problem is the deter- mination of roots of a high ordered algebraic equation. Once these roots are determined the solutions are easily obtained. In the second case where the differential equations have variable coefficients, - 86 - the solutions would probably be in the form of power series. Thus, the k(t)'s in equation (i) must be expressed in power series. Perhaps with modern computers these tedious calculations may be feasible. "TIT/'ITIEIITJLEJMMfilfllflfflfijflfllflflifflflmfl