1、i b- IF: 3 I NASA TN D-402 TECHNICAL NOTE 0-402 ANALYSIS OF FRAME-REINFORCED CYLINDRlCAL SHELLS PART III - APPLICATIONS By Richard H. MacNeal and John A. Bailie Lockhe ed Aircraft C o rpor at ion California Division Burbank, California WASHINGTON May I360 NATIONAL AERONAUTICS AND SPACE ADMINISTRATIO
2、N N89-7C766 rhes8-5K-9-!ior) ALELYSlS CF EABE-iEILFCRCEC CYLIBCKICAL Et.L1$. PAEZ 3: A E L1C A 1 IC N E 137 F Unclas (Lo ckkf E d Airc ref t COEP. ) 00/39 0199044 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-% , c NATIONAL AERONAUTICS AND SPACE AD
3、MINISTRATION C . TECHNICAt NOTE D-402 ANALYSIS OF FRAME-REINFORCED CYLINDRICAL SHELLS PART 111 - APPLICATIONS1 By Richard H. MacNeal and John A. Bailie ABSTRACT Tables are presented giving the loads and displacements in a flexible frame supported by a circular cylindrical shell and subjected to conc
4、entrated radial, tangential, and moment loads. Additional tables give the loads in the shell. presented in terms of two basic parameters, one of which is of second-order importance. Procedures for modifying the important parameter to account for certain non-uniform properties of the structure are pr
5、esented. This enables the one set of tables to be used for the solution of a wide variety of shell-frame problems, some of which have not been solved previously. The solutions are The parameters of the two companion publications are computed on a more rational basis than previously. This increases t
6、he confidence in, and range of application of, the charts in these publications. NOTATION 2.25 4 - parameter of references 4 and 5 A Y influence coefficient il a B - parameter of references 4 and 5 d eL - parameter of reference 3 2 E Youngs modulus - lbs/in Originally prepared as IMSD 49734, Lockhee
7、d. Missiles and Space Division, Sunnyvale, California, and reproduced in original form by NASA, by agreement with Lockheed Aircraft Corporation, to increase availability. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 Ef EO ESK e F G IO i Kn L LC
8、Lr M 2 Youngs modulus of unloaded frames - lbs/in Youngs modulus of loaded frame - lbs/in 2 Youngs modulus of skin - lbs/in base of natural logarithms axial force in loaded frame - lbs 2 shear modulus - lbs/in: eccentricity between neutral axis of loaded frame and median plane of skin - in. 4 moment
9、 of inertia of a typical unloaded frame - in moment of inertia of an unloaded frame, distant .e from the loaded frame - in4 moment of inertia the loaded frame - in 4 /io -in 3 distance from loaded frame to undistorted shell section - in. 1/4 characteristic length (see Glossary) = $1 - in. fi charact
10、eristic length (see Glossary) = fi - in. frame spacing - in. bending moment in loaded frame - in lbs I- I I- * c Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-3 c 0 M 0 4 r; MO m n P 9 r S S TO t tl te U V W X e externally applied concentrated mome
11、nt - in. lbs bending moment per inch in shell - in lbs/in. index of harmonic dependence in the 6 direction externally applied radial load - lbs axial load per in inch in the shell - lbs/in. shear flow in shell - lbs/in. radius of skin line - in. transverse shear force in loaded frame - lbs transvers
12、e shear per inch in shell - lbs/in. externally applied tangential load - lbs skin panel thickness - in. effective skin panel thickness for axial loads - in. weighted average of all the bending material (skin and stiffeners) adjacent to the loaded frame, assumed uniformly distributed around the perim
13、eter - ins. axial displacement of shell - in. tangential displacement of shell - in. radial displacement of shell - in. axial co-ordinate of shell, in. “beef-up“ parameter Io/2iLc y for a nearby heavy frame polar co-ordinate of frames and shell g/r - eccentricity parameter rotational displacement; r
14、adians Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 GLOSSARY OF TERMINOLOGY The terms Tnput Impedance, “Transmission Matrix,“ and “Characteristic Length? are used in this report and are defined as follows: Input Impedance : The relationship betw
15、een the tangential displacement and shear flow harmonic coefficients of the shell at the section of the loaded frame. Transmission Matrix: The forces and displacements at one end of a finite length of unloaded shell can be written in terms of their values at the other end; the square matrix defining
16、 these relationships is the transmission matrix. Characteristic Length: In this report there are two characteristic lengths, defined as follows. the lowest order self-equilibrating stress system to decay to l/e of its value at x = 0 , provided that the skin panels are rigid in shear. Lr is the dista
17、nce required for the envelope of the lowest order self-equilibrating stress system to decay to l/e of its value at x = 0 , provided that the frames are rigid in bend- ing. Lc is the distance required for the exponential envelope of INTRODUCTION There are basically two approaches to determining stres
18、ses that are due to concentrated loads applied to flexible frames which are supported by cylindrical shells. One is to make a complete redundant analysis of each problem, utilizing a large digital computer. The second, which is adopted here, is to devise simple approximate procedures. Such procedure
19、s should promote a better understanding of the problem, and provide tables or charts of results as functions of as few para- meters as practicable. The analysis is undertaken in references 1 and 2. The results obtained from that analysis are summarized in this report which is intended to be a refere
20、nce for persons interested in using the results, but who are not concerned with the mathematical derivations. In any theoretical analysis in the field of mechanics, the first step IS to set up a mathematical model that contains the essential physical characteristics of the system, and yet is amenabl
21、e to known mathematical techniques. analysis is described in detail in next section. Tables for the load in the shell and the externally-loaded frame, derived for the model, are included in this report. In practice, many shells deviate markedly from any such simple model, and in refer- ence 2 a cons
22、iderable effort is devated to the derivation of simple corrections to the basic parameter to account for such deviations. This enables the tables to be utilized in the solution of a much wider range of shell-frame problems. The results of the investigation, while initially intended mainly for airpla
23、ne fuselage analysis, have been successfully applied to a ballistic missile body and an airplane landing- gear strut. These uses suggest wide applications of the techniques in cylindrical shells whose skin thickness is small compared to their radius. The model for this 2 C I I2 P Provided by IHSNot
24、for ResaleNo reproduction or networking permitted without license from IHS-,-,-5 8 r; 0 I4 18 I i BASIC ASSUMPTIONS AND COMPARISON WITH ASSUMPTIONS OF PREVIOUS ANALYTICAL METHODS It is necessary to be aware of the basic assumptions made in this analysis in order to make effective use of the results.
25、 A comparison with the assumptions of references 3 and 4 is made to indicate differences in the solutions of the three methods. In the method of attack with which this report is mainly concerned, a simplified structural model (fig. 1) is used to obtain a solution for a uniform shell stretching to in
26、finity on both sides of the loaded frame. Clearly the effect of any frame can be propagated only a finite distance along the shell. In practice, the perturbations from “elementary beam theory“ are, at worst, negligible at Lc inches away from the loaded frame. Procedures for modifying the solution to
27、 account for discontinuities and non-uniform properties are discussed in the following sections. For the model used.the following assumptions are made: Concentrated loads are applied to the loaded frame and are reacted an infinite distance away on either one or both sides. The shell extends to infin
28、ity on both sides. The loaded frame has in-plane bending flexibility. It is free to warp out of its plane and to twist. It has no axial or shearing fiexibilities. Its moment of inertia for circumferential bending is constant. The effect of the eccentricity of the skin attachment with respect to the
29、frame neutral axis is ignored for both the loaded and unloaded frames. The shell consists of skin. longerons, and frames similar to the loaded frame, but possibly with different moments of inertia. The skin and longerons have no bending stiffness. All properties of the shell are uniform. The longero
30、ns are “smeared out! over the circumference giving an equivalent constant thickness, t , (including effective skin), for axial loads. The shell frames, but not the loaded frame, are “smeared out“ in the direction of the shell axis, giving an equivalent moment of inertia per inch, i , for circumferen
31、tial bending loads. The model and the assumptions employed in this report differ from those used by other authors. In reference 3, the model shown in figure 2 is used. The assump- tions regarding the loaded frame are identical to those employed in this report. The assumptions regarding the idealized
32、 shell are: Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(1) It contains no frames. (2) Longerons are infinitely rigid for axial loads, infinitely flexible for bending, and continuously “smeared out. (3) The panels have uniform shear stiffness. Th
33、e length to the undistorted section, L , is chosen on an empirical basis. Nevertheless, as will be shown, the length L can be chosen such that the results agree reasonably well with those given by the present theory. In references 4 and 5, models are used similar to the model employed in this report
34、. The main differences are that the shell frames are concentrated, rather than “smeared out, equal to the moment of inertia of the loaded frame. Charts, calculated from the equations in reference 4, are given in reference 5. are equally spaced, and have moments of inertia The assumption of concentra
35、ted frames is more realistic than the assump- tion of “smeared outf7 frames. On the other hand, the following advantages are claimed for the latter assumption: (1) Simplified mathematical processes in getting a solution. This simplification of the theory promotes clearer insight into the processes i
36、nvolved. It also permits easier extension to more involved problems. (2) Eliminatingoneparameter in plotting solutions. In reference 5, two parameters were required to describe the properties of a uniform shell. In the present analysis, variations in one of the two parameters produce little change i
37、n the results and it is possible to compensate for these variations by modifying the other parameter, (3) Greater accuracy in the analysis of a heavily reinforced frame. The simplified structural models previously described bear only slight resemblance to practical aircraft shells. The differences m
38、ight be described as the omission of the following effects: Finite frame spacing Nonuniformity of longeron area, skin thickness and frame moment of inertia both in the circumferential and axial directions Local reinforcement of the loaded frame near concentrated loads Eccentricity of the skin connec
39、tion to frames Internal bracing of the frames Nearby discontinuities in the shell, such as free ends, rigid bulkheads, etc. s r Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I F 7 t (7) Elliptical cross-sections (8) (9) Presence of cut-outs Changes
40、 in radius of shell in the axial direction (10) Finite axial and shear stiffness of the frames 0 K 9 4 (11) Bending stiffness of longerons (12) Assumptions of linearity, etc. A considerable effort is expended in references 1 and 2 in attempts to account for some of these effects. The main objective
41、of this effort is to express the influence of the above effects as modifications to the basic parameter so that the tables are still applicable. A detailed discussion of some of the above effects is given in the following sections, where it is shown that items 1 to 6 inclusive can be handled quite s
42、imply. Reference 6 presents solutions accounting for items 4, 10,and 11 above, together with experimental results for shells representative of airplane construction. It is shown that in such shells the high stear stresses adjacent to the loaded frame predicted by references 1, 3,and 5 do not occur i
43、n practice due to the neglect in the analyses of effects such as items 4, l0,and 11 above. EVALUATION OF PAMETERS Lr , Lo and y Case of Uniform Shell In cases where the shell happens to satisfy all of the assumptions listed in the previous section, and in particular, if the skin thickness, stringer
44、area, and shell-frame moment of inertia are uniform in both the axial and circumferential directions, the following formulas may be used: L r =:Jig Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 Youngs modulus for skin. stiffeners and all frames i
45、s assumed equal. Using these parameters and interpolating in the tables give coefficients which, when substituted into the equations on pages 14 and 15 , yield the required loads and deformations. In non-uniform shells, use the modified parameters indicated in the following equations : Case of Non-U
46、niform Shell (1) In case that the shell properties, i , t , and t , vary over the surface of the shell to a moderate degree, the following formulas and definitions are appropriate: Lr = 3 r /y Eo Io = 2Ef i Lc The stiffness factors, Gt , ESK te , and Ef i , must be averaged in the neighborhood of th
47、e loaded frame. The factors Gt and ESK i should be averaged over a length of shell extending approximately one-half of a characteristic length from the loaded frame in both directions. (2) When unloaded frames have unequal moments of inertia or are unequally spaced, the following weighting factor me
48、thod may be used for computing Ef i : E f i = (Ef i)fwd + (Ef Qaft (7) =1 2 (WEf If) (Ef i)fwd L_ Y I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-9 where C VJ = 1 - X/L, for x L C (x is measured forward and aft of the loaded frame). The summation
49、s in equations (8) and (9) are to be extended over all frames except the loaded frame. The method of calculation gives greater importance to frames closest to the loaded frame and less importance to those farther way. For the case of a single, particularly heavy, neighboring frame, or for other neighbor- ing
