1、NASA TECHNICAL NOTE 0 b T ta I + 4 v3 4 NASA TN D-3010 - NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. SEPTEMBER 1965 I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM I111111 11ll1 11111 11111 lllll11111 lllll
2、 Ill 1111 ON FREE VIBRATIONS OF ECCENTRICALLY STIFFENED CYLINDRICAL SHELLS AND FLAT PLATES By Martin M. Mikulas, Jr., and John A. McElman Langley Research Center Langley Station, Hampton, Va. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Far sale by the Clearinghouse for Federal Scientific and Techn
3、ical Information Springfield, Virginia 22151 - Price $1.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ON FREE VIBRATIONS OF ECCENTRICALLY STIFlFENED CYLINDRICAL SHELLS AND FLAT PLATES By Martin M. Mikulas, Jr., and John A. McElman Langley Resear
4、ch Center SUMMARY Dynamic equilibrium equations and boundary conditions are derived from energy principles for eccentrically stiffened cylinders and flat plates. Inplane inertias are neglected and frequency expressions are obtained for simple-support boundary conditions for both the cylinder and the
5、 plate. in the form of plots of frequencies as a function of mode shape illustrate the effects of eccentricities. It is found that theseeccentricities can have a significant effect on natural frequencies and should be investigated in any dynamic analysis of stiffened structural members. Results INTR
6、ODUCTION The effects of stiffener eccentricities on the buckling characteristics of stiffened circular cylindrical shells are being given a great deal of con- sideration in the design of aerospace structures. In references 1 to 5, the effects of eccentricities on the buckling of stiffened cylinders
7、have been treated analytically. An externally stiffened cylinder under axial compres- sion has been shown experimentally to carry over twice the load sustained by its internally stiffened counterpart (ref. 6). It should be expected, therefore, that substantial eccentricity effects would be found in
8、the vibration characteristics of stiffened cylinders. A survey of the present literature (for example, refs. 7 and 8) reveals that stiffener eccentricity generally has been neglected in studying the vibrations of stiffened cylinders. In the present paper, the differential equations of dynamic equili
9、brium are derived from energy considerations for the free vibrations of ring- and stringer-stiffened cylinders. Donnell-type strain-displacement relations for the cylinder and beam-type strain-displacement relations for the stiffeners. The stiffeners are not con- sidered as discrete elements, but th
10、eir effects are averaged or “smeared out.“ However, the location of the resulting equivalent orthotropic layers relative to the shell middle surface is carefully maintained; that is, the common assumption that the equivalent orthotropic shell is homogeneous through the The derivation is accomplished
11、 by utilizing Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-thickness with a single neutral surface is not made. neglected and the differential equations of dynamic equilibrium and appropriate boundary conditions are found by variational techniques
12、. equations are solved to obtain a closed-form frequency expression for ring- and stringer-stiffened cylinders for the case of simple-support boundary conditions. Results from this expression are presented in the form of plots of natural fre- quencies as a function of mode shape for several practica
13、l configurations. These plots illustrate the effects of stiffener eccentricity. Inplane inertias are The differential A comparable analysis for the free vibrations of stiffened flat plates is presented and again it is shown that eccentricity effects can be important. SYMBOLS The units used for the p
14、hysical quantities defined in this report are given both in the U.S. Customary Units and in the International System of Units, SI (ref. 9). The appendix presents factors relating these two systems of units. A cross-sectional area of stiffener C defined by equation (47) D flexural stiffness of isotro
15、pic plate or isotropic cylinder wall, Et3 E Young s modulus G shear modulus I moment of inertia of stiffener about its centroid moment of inertia of stiffener about middle surface of plate or cylinder IO J torsional constant for stiffener M mass per unit area of cylinder or plate %, My, Mxy, Myx mom
16、ent resultants N number of stringers Nx,Ny,Nw stress resultants R radius to middle surface of isotropic cylinder (see sketch a) 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- ErAr R nondimensional parameter, - Et1 %As - S nondimensional parameter
17、, - Etd Z a2 1/2 curvature parameter, -(I - p) Rt a length of cylindrical shell or plate b width of plate d stringer spacing (see sketch a) Lu f frequency, - 2fl 1 ring spacing (see sketch a) m,n integers t thickness of cylinder or plate u, v,w displacements in x-, y-, and z-directions, respectively
18、 E, 7, fi displacement amplitudes X, Y, orthogonal coordinates defined in sketch a (x and y lie in middle surface of cylinder or plate) - z distance from middle surface of plate or cylinder to centroid of stiffener a, P wavelength parameters E y,7w middle-surface normal and shearing strains E:,E, to
19、tal noma1 and shearing strains (see eqs. (2) to (4) A defined by equations (44a), (44b), (44c), and (44d) CI Poissons ratio II potential energy P mass density w circular frequency 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-v“ = $V2 where V2 is
20、 the Laplacian operator in two dimensions Subscripts : C cylinder r stiffening in y-direction S stiffening in x-direction P plat e 0 inertial load A subscript preceded by a comma indicates partial differentiation with respect to the subscript. DERIVATION OF BASIC EQUATIONS The problem considered is
21、the free vibration of a thin-walled circular stringers. (See sketch a.) Inplane inertias are neglected, and it is assumed that the stiffener spacing is small cam- length so that its effect on the behavior of the cylinder may be averaged (smeared out). The strain energies of the cyl- inder and stiffe
22、ners are pre- sented and the displacements of the stiffeners and the cyl- inder are required to be com- patible. After formulating the potential energy of inertial loading, the equations of dynamic equilibrium and con- sistent boundary conditions are obtained by applying the method of minimum potent
23、ial energy to the total energy of the system. The differential equations of dynamic equilibrium and consist- cylindrical shell which is stiffened by evenlx spaced uniform rings and/or - A -A . ,/ pared with the vibration wave- (I Sketch a.- Geometry of eccentrically stiffened cylinder. ent boundary
24、conditions are then obtained in a similar fashion for a stiffened flat plate. 4 I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Strain Energy of Isotropic Cylinder The strain energy of the unstiffened thin-walled isotropic cylinder is The linear Do
25、nnell-type strain-displacement relations are EXT = EX - Zwyn Efl= Ey - ZWJyy - - 2zw, xY 7m - 7xY where the middle-surface strains are defined as EX = UYX Ey - Vy + TXy = UYy + VYX W - Substitution of equations (2), (3), and (4) into equation (1) and integration with respect to z yields the followin
26、g expression for cylinder strain energy: In this equation, is the flexural stiffness of the cylinder. 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Strain Energy of Stiffeners The strain energy of the stiffeners is derived on the basis that the d
27、is- placements in the cylinder and stiffeners are equal at the point of attachment and stiffener twisting is accounted for in an approximate manner. where both rings and stringers are attached to the same surface of the shell, the effect of joints in the stiffener framework is ignored. In cases Stri
28、nger energy.- The total strain energy is written as E2 as dx + 2 xT j =1 where the first term inside the parentheses of of N stringers on the cylinder equation (6) is the strain energy of bending and extension in the stringer, and the second term is the strain energy involved in twisting of the stri
29、ngers. The quantity dAs is an element of the cross-sectional area of the stringer and GsJs ness of the stringer section. term inside the parentheses of equation (6) can be written as follows: is the twisting stiff- After substitution from equation (2), the first Inspection of these terms reveals tha
30、t the first integral inside the parentheses is the area of the stri er cross section A, the second integral is the first moment of the area (?yXx=o 3 d or w=O ( 35b 1 or v=O (3P) In addition to these boundary conditions the following relationship must be satisfied at free corners: w,xy = 0 (39) It s
31、hould be noted that, even though this theory is linear, the inplane displacements u and v are involved in the boundary conditions for one- sided stiffened plates. For the case of symmetric stiffening zs = Zr = 0, the boundary conditions given in expressions (3l), (32), (35), and (36) uncouple from t
32、he inplane displacements and the other boundary conditions, expres- sions (33), (34), (37), and (38), need not be considered. - - The flat-plate equilibrium equations and boundary conditions may also be written in terms of stress and moment resultants as was done for the stiffened cylinder in the pr
33、evious section. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SOLUTIONS FOR SIMPLY SUPPORTED CYLINDERS AND PLATES Solutions are presented for the natural frequencies of vibration of simply supported cylinders and plates with eccentric stiffenin;. T
34、hese solutions illustrate in a straightforward manner some of the significant effects of stiff- ener eccentricity on the vibration behavior.of such structures. Stiffened Cylinder The coordinate system chosen has its origin located at one end of the cylinder. x = 0,a are The simple-support boundary c
35、onditions to be satisfied at each end w = MX = v = NX = 0 (40) The expressions for the displacements u, v, and w, which satisfy these boundary conditions, are given as Y - m?tX nY u = u cos - cos - a R nY sin - R v = T sin - IMX a IMX nY w = w sin - cos - a R d where m is the number of axial half wa
36、ves and n is the number of circumfer- ential fuU waves. equations (eqs. (ll), (12), and (13) the following equation is obtained after some manipulation: After substitution of equations (41) into the equilibrium 14 Provided by IHSNot for ResaleNo reproduction or networking permitted without license f
37、rom IHS-,-,-where and the following nondimensional parameters are defined: na p =- mnR - To obtain a nontrivial solution, the determinant of the coefficients of u, v, and G is set equal to zero. After more manipulation, the following nondimen- sional frequency equation is obtained: 2 1 + 3, + EAr +
38、sqs ) 122 ( A +- n4 where (43) (44a) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- p2) has been defined. a4( 1 and another nondimensional parameter 22 = R2t2 In equation (43), the effect of eccentricity of stiffening is reflected by - - the terms
39、containing E, and zr. The quantities zs and Er are positive when the stiffeners are located on the external surface of the cylinder and negative when the stiffeners are on the internal surface so that sign changes can occur in equation (43). quantity (1 - p2p) in A, geometry and vibration mode shape
40、. should be exercised in drawing general conclusions as to the influence of eccen- tricity of stiffening on the vibration behavior of stiffened cylinders. Notice that the quantity (p2 - p) in As and the can also change signs depending upon the cylinder These facts suggest that some caution Stiffened
41、 Flat Plates A coordinate system is chosen having its origin at one corner of a plate of length a and width b. The simple-support boundary conditions which must be satisfied are w(0,y) = w(a,y) = w(x,O) = w(x,b) = 0 Expressions for the displacements u, v, and w which satisfy these boundary condition
42、s are 16 I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-mrX - u = u cos - sin a IMX cos - - v = v sin - a (45) w = w sin - mrX sin “J a b where for flat plates m and n are the numbers of half waves in the x- and y-dire ctions, respectively . Follo
43、wing a procedure similar to that used in the previous section, the following nondimensional frequency equation is obtained: where and the following nondimensional parameters are defined: - - In equation (46), the terms which involve zs and Zr are present because of eccentric stiffening. If the plate
44、 is stiffened by only longitudinal or transverse stiffeners, all terms involving Zr or zs are squared and hence the surface on which the stiffener is attached is unimportant. However, if both longitudinal and transverse stiffeners are present, the coupling term C defined in equation (47) has a term
45、with the coefficient negative if the longitudinal and transverse stiffeners are on opposite sides of the plate. - - ZrZs which can be 17 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . , -. . . . RESULTS AND DISCUSSION I Because of the large numb
46、er of parameters appearing in the frequency expressions, it is impractical to present results of a general nature. fore, computed results for stiffened cylinders and plates with proportions of contemporary interest are presented in order to illustrate the magnitude of eccentricity effects. frequenci
47、es as a function of mode shape; for each configuration considered the physical properties of the structure are given in the figure. There- Results are presented in the form of plots of natural Stiffened Cylinders Stringer-stiffened cylinders.- In figure 1, the natural frequencies as obtained from eq
48、uation (43) are given for a stringer-stiffened cylinder (cyl- inder 1) with physical properties similar to one of the integrally stiffened cylinders considered in reference 6. attached to the outside occurs at a higher mode number n and is approximately 35 percent higher than the lowest natural frequency for the same stringers attached to the inside. The lowest natural frequency for stringers As a matter of interest, natural frequencies were calculated for a cylinder with the same physical properties as those given in figur
