JOURNAL OF THE AERONAUTICAL SCIENCES VOLUME 23 SEPTEMBER, 1956 NUMBER 9 Stiffness and Deflection Analysis of Complex Structures M. J. TURNER,* R. W. CLOUGH,t H. C. MARTIN,t AND L. J. TOPP** ABSTRACT tion on static air loads, and theoretical analysis of aero- A method is developed for calculating stiffness influence co- elastic effects on stability and control. This is a prob- efficients of complex shell-type structures. The object is to pro- lem of exceptional difficulty when thin wings and tail vide a method that will jdeld structural data of sufficient accuracy surfaces of low aspect ratio, either swept or unswept, to be adequate for subsequent dynamic and aeroelastic analyses. are involved. Stiffness of the complete structure is obtained by summing It is recognized that camber bending (or rib bending) stiffnesses of individual units. Stiffnesses of typical structural components are derived in the paper. Basic conditions of con- is a significant feature of the vibration modes of the tinuity and equilibrium are established at selected points (nodes) newer configurations, even of the low-order modes; in the structure. Increasing the number of nodes increases the in order to encompass these characteristics it seems accuracy of results. Any physically possible support conditions likely that the load-deflection relations of a practical can be taken into account. Details in setting up the analysis can structure must be expressed in the form of either de- be performed by nonengineering trained personnel; calculations are conveniently carried out on automatic digital computing flection or stiffness influence coefficients. One ap- equipment. proach is to employ structural models and to determine Method is illustrated by application to a simple truss, a flat the influence coefficients experimentally; it is antici- plate, and a box beam. Due to shear lag and spar web deflection, pated that the experimental method will be employed the box beam has a 25 per cent greater deflection than predicted extensively in the future, either in lieu of or as a final from beam theory. It is shown that the proposed method cor- rectly accounts for these effects. check on the result of analysis. However, elaborate Considerable extension of the material presented in the paper models are expensive, they take a long time to build, is possible. and tend to become obsolete because of design changes ; for these reasons it is considered essential that a con- (I) INTRODUCTION tinuing research effort should be applied to the devel- RESENT CONFIGURATION TRENDS in the design of opment of analytical methods. It is to be expected that modern developments in high-speed digital com- Downloaded by VIRGINIA TECHNICAL UNIVERSITY on September 12, 2014 | http://arc.aiaa.org DOI: 10.2514/8.3664 Phigh-speed aircraft have created a number of difficult, fundamental structural problems for the puting machines will make possible a more fundamental worker in aeroelasticity and structural dynamics. The approach to the problems of structural analysis; we chief problem in this category is to predict, for a given shall expect to base our analysis on a more realistic elastic structure, a comprehensive set of load-deflection and detailed conceptual model of the real structure relations which can serve as structural basis for dynamic than has been used in the past. As indicated by the load calculations, theoretical vibration and flutter title, the present paper is exclusively concerned with analyses, estimation of the effects of structural deflec- methods of theoretical analysis; also it is our object to outline the development of a method that is well Received June 29, 1955. This paper is based on a paper adapted to the use of high-speed digital computing presented at the Aeroelasticity Session, Twenty-Second Annual machinery, Meeting, IAS, New York, January 25-29, 1954. * Structural Dynamics Unit Chief, Boeing Airplane Company, Seattle Division. (II) REVIEW OF EXISTING METHODS OF STRUCTURAL f Associate Professor of Civil Engineering, University of Cali- ANALYSIS fornia, Berkeley. J Professor of Aeronautical Engineering, University of Wash- (1) Elementary Theories of Flexure and Torsion ington, Seattle. ** Structures Engineer, Structural Dynamics Unit, Boeing Air- The limitations of these venerable theories are too plane Company, Wichita Division. well known to justify extensive comment. They are 805 806 JOURNAL OF THE AERONAUTICAL SCIENCES — S E P T E M B E R , 1956 lQ n adequate only for low-order modes of elongated struc- (5) Direct Stiffness Calculation: Levy, Schuerch > tures. When the loading is complex (as in the case In a recent paper Levy has presented a method of of inertia loading associated with a mode of high order) analysis for highly redundant structures which is par- refinements are required to account for secondary ticularly suited to the use of high-speed digital com- effects such as shear lag and torsion-bending. puting machines. The structure is regarded as an assemblage of beams (ribs and spars) and interspar (2) Wide Beam Theory: Schuerch1 torque cells. The stiffness matrix for the entire struc- ture is computed by simple summation of the stiff- Schuerch has devised a generalized theory of com- ness matrices of the elements of the structure. Fi- bined flexure and torsion which is applicable to multi- nally, the matrix of deflection influence coefficients is spar wide beams having essentially rigid ribs. Torsion- obtained by inversion of the stiffness matrix. Schuerch bending effects are included but not shear lag. It is has also presented a discussion of the problem from the expected that wide beam theory will be used extensively point of view of determining the stiffness coefficients. in the solution of static aeroelastic problems (effect of air-frame flexibility on steady air loads, stability, etc.). However, the rigid rib assumption appears to limit its (Ill) S O M E U N S O L V E D P R O B L E M S utility rather severely for vibration and flutter anal- At the present time, it is believed that the greatest ysis of thin low aspect ratio wings. need is to derive a numerical method of analysis for a class of structures intermediate between the thin (3) Method of Redundant Forces: Levy, Bisplinghoff and 2 stiffened shell and the solid plate. These are hollow Lang, Langefors, Rand, Wehle and Lansing ^ structures having a rather large share of the bending These writers have contributed the basic papers material located in the skin, which is relatively thick leading to the present widespread use of energy prin- but still thin enough so that we may safely neglect ciples, matrix algebra, and influence coefficients in the its plate bending stiffness. In order to cope with this solution of structural deflection problems. Redundant class of structures successfully, we must base our internal loads are determined by the principle of least analysis upon a structural idealization that is suffi- work, and deflections are obtained by application of ciently realistic to encompass a fairly general two- Castigliano's theorem. The method is, of course, dimensional stress distribution in the cover plates; perfectly general. However, the computational diffi- and our method of analysis must yield the load-deflec- culties become severe if the structure is highly re- tion relations associated with such stresses. It is char- dundant, and the method is not particularly well acteristic of these problems that the directions of prin- adapted to the use of high-speed computing machines. cipal stresses in certain critical parts of the structure Rand has suggested a method of solution for stresses cannot be determined by inspection. Hence, the in highly redundant structures which might also be familiar methods of structural analysis based upon the used for calculating deflections. Instead of using concepts of axial load carrying members, joined by member loads as redundants, he proposes to employ membranes carrying pure shear, are not satisfactory, systems of self-equilibrating internal stresses. These even if we employ effective width concepts to account redundant stresses may be regarded as perturbations for the bending resistance of the skin. We should like of a primary stress distribution that is in equilibrium to include shear lag, torsion-bending, and Poisson's with the external loads (but does not generally satisfy ratio effects to a sufficient approximation for reliable compatibility conditions). The number of properly prediction of vibration modes and natural frequencies chosen redundants required to obtain a satisfactory of moderate order. Also, we should like to avoid any solution may be considerably less than the "degree of assumptions of closely spaced rigid diaphragms or of Downloaded by VIRGINIA TECHNICAL UNIVERSITY on September 12, 2014 | http://arc.aiaa.org DOI: 10.2514/8.3664 redundancy." Successful application of this method orthotropic cover plates, which have been introduced requires a high degree of engineering judgment, and in many papers on advanced structural analysis. The the accuracy of the results is very difficult to evaluate. actual rib spacing and finite rib stiffnesses should be accounted for in a realistic fashion. In summary, what (4) Plate Methods: Fung, Reissner, Bens cot er, and is required is an approximate numerical method of MacNeaV* analysis which avoids drastic modification of the As the trend toward thinner sections approaches the geometry of the structure or artificial constraints of its ultimate limit, we enter first a regime of very thick elastic