Shear Flow

     This subsection contains an Example Problem for finding the internal shear loads in a bulkhead web.   Equilibrium requirements necessitates adding to the initial free body diagram -- additional internal balancing end-moments. It also illustrates the deficiency when compared to the exact shear flow solution, of the FEM in regard to web shear flow. (Actually, stress per se doesn't flow.) Especially see ROBINSON's paper referenced directly below too.      This model is being revaluated as of June 2023, but will be soon updated.
     This subsection also contains GRAVERY's famous "The Quadrilateral 'Shear' Panel " paper. He accounts for both axial and shear energy in a panel, although the panel example itself is loaded only with shearing forces. GARVEY's formula could have been used to develop a stiffness matrix for rectangular shear panels, however, this was not done.
     Panel buckling per se has not been effected, but to this purpose use George S. JOHNTON's Stringer Panel Analysis Methods.
     This analyses is an effort to go beyond ROBINSON's (and BOEING's) approach. To this purpose see BOROWIEC's analyses.
     An aside: when creating a FEM it is important to include CRODs around the periphery of a shear web panel (the cap loads); and at each and every corner grid of that shear web panel's CQUAD4 install a local coordinate system -- in-plane with the shear web panel -- with its out-of-plane axis zeroed out: this will cancel the DOF of the out-of-plane parameter associated (by default) with the shear web (as the isoperimetric math model for the shear web panel can't support out-of-plane forces or moments; only in-plane bending and shear). If this is not done than the FEM is garbage.
     Please report any errors (or opinions).
     Plus References at the end.
Read first:

RIVELLO, Robert M.: Theory and Analysis of Flight Structures, McGRAW-HILL, hardcover, 1969. This is perhaps the best general textbook into this subject. He is weak on Z-stringer panel analyses, so see George JOHNSON's book.
MEGSON, T. H. G.: Aircraft Structures for Engineering Students, Edward Arnold, hardcover, Index, 1972.
JOHNSON, George S.: Stringer Panel Analysis Methods (in BASIC For PC's), 1st Edition, Copyright 1985 by Optimization Inc., paperback. References listed below were mainly extracted from George's book. George's book is difficult to understand but he does a comprehensive analyses of Z-stringer panels, and other type panel configurations as well -- a revised and expanded edition, 1 copy, has been written by R. K. M. See his "Buckling of Tapered Columns" (in BASIC for IBM PCs) too. George's approach is really an echo of LOCKHEED's approach to stringer panel analyses; which was originally partly invented by George.
PEERY, David J, Ph. D: Aircraft Structures, McGRAW-HILL, hardcover, 566 pages, indexed, 1950. (Don't consult the revised edition.) The original textbook is a good general presentation of some the older analytical methods, with most still relevant.
Marshall Space flight Center, Anon: Astronautic Structures, Vol. 1-3, NASA TM X-73305, 06, 07, 8-1/2'' by 11'', in binders, on microfiche, 1975. Their three volumes covers all the main subject of aerospace analyses and as such is a standard reference. All three volumes may be downloaded from the Internet.
ESDU's: The many (really excellent) bulletins of the English aerospace stress manual represent the current methodologies.
ROBINSON, John: "The Mode-Amplitude Technique and Hierarchy Stress Elements -- A Future Technology," Finite Elements in Analysis and Design, Elsevier Science Publishers, Revised May 1989. Abstract: "A general summary is given and key items are demonstrated using examples representing thin walled structures." Especially see his reference list too. His paper doesn't check for equilibrium directly.
MacNEAL, Richard H.: "Toward a Defect-free Four-Noded Membrane Element," Finite Elements in Analysis and Design, 5(1989) 31-37, Elsevier Science Publishers B.V. Amsterdam.
BOROWIEC, Dr. Zygmunt: "Rectification of MSC/NASTRAN Output of Skin to Stringer Interface Loads," MSC World Users Conference, Los Angles, Mar. 11-15, 1991, 18 pages. Dr. Zygmunt was (an associate) from Poland, who later in life worked for Canadair. This rather obscure paper, as well as his others, are soon to be uploaded onto this webpage.

Shear Bulkhead Web Example Problem

Background Material

Basic Shear: Quadrilateral Shear Panel Free Body Diagram.
GRAVEY's Paper

The three figures above are not part of GRAVERY's Paper below; they are shown only to illustrate how the C.G. is determined.

References

These older papers serve as the foundations for later analyses and the reason they should be read. Blue colored books, etc., are main references. General References

BIJLAARD, P. P. and JOHNSTON, George S.: “Compressive Buckling Of Plates Due To Forced Cripping Of Stiffeners,” IAS paper SMF FF8, Jan. 1953.
BIJLAARD, P. P.: “Analysis Of The Elastic And Plastic Stability Of Sandwich Plates By The Method Of Split Rigidities, Parts I And II,” Journal of the Aeronautical Sciences, pp. 339, 790 respectively, 1951. This is difficult to understand.
BIJLAARD, P. P.: “Method Of Split Rigidities and Its Application To Various Buckling Problems,” NACA TN 4085.
BIJLAARD, P. P.: “Interaction Of Column And Local Buckling In Compression Members,” NACA TN 2640, March 1952.
BIJLAARD, P. P.: “Analysis Of The Elastic And Plastic Stability Of Sandwich Plates By The Method Of Split Rigidities - I,” Journal Of The Aeronautical Sciences, May. 1951.
BIJLAARD, P. P.: “Theory And Tests On The Plastic Stability Of Plates And Shells,” Journal Of The Aeronautical Sciences, Sept. 1949.
HICKMAN, W. and DOW, N.: “Data On The Compressive Strength Of 75S-T6 Aluminum Alloy Flat Panels With Longitudinal Extruded Z-section Stiffeners,” NASA TN 1829, March 1949.
HICKMAN, W. and DOW, N.: “Data On The Compressive Strength Of 75S-T6 Aluminum Alloy Flat Panels Having Small, Thin, Widely Spaced, Longitudinal Extruded Z-section Stiffeners,” NASA TN 1978, Nov. 1949.
DOMAN, J. and SCHWARTZ, E.: “Study of Size Effect In Sheet-Stringer Panels,” NACA TN 3756, July 1956.
RAMBERG and OSGOOD: “Description of Stress-strain Curves By Three Parameters,” NACA TN 902, 1942.
MENDELSON, A. and MANSON, S. S.: “Practical Solution Of Plastic Deformation Problems In The Elastic-plastic Range,” NASA TR R-28, 1959.
STOWELL, E. Z. and PRIDE, R. A.: “The Effect Of Compressibility Of Material On Plastic Buckling Stress,” Journal of the Aeronautical Sciences, pg. 773, 1951.
KRIVETSKY, A.: “Plasticity Coefficients For The Plastic Buckling Of Plates and Shells,” Journal of the Aeronautical Sciences, pg. 432, 1955.
GERARD, G.: Handbook Of Structural Stability, Part 1 - Buckling Of Flat Plates, NACA TN 3781, July 1957. There are many more Parts to this set.
TIMOSHENKO, S. P.: THEORY OF ELASTICITY, McGraw-Hill Book Co., Inc., 1934.
TIMOSHENKO, S. P. and GERE, J. M.: Theory Of Elastic Stability, Second Ed., McGraw-Hill Book Co., Inc., 1961.
ROARK, R. J.: Formulas For Stress and Strain, Third Ed., McGraw-Hill Book Co., Inc., 1954; Cases 18 and 19, pg. 178.
BRUHN, E. F.: Analysis And Design Of Flight Vehicle Structures, S. R. Jacobs and Associates, 1973. Out-of-print. A well known textbook but some of the analyses methodologies are wanting.
WAGNER, H. W.: “Torsion and Bending Of Open Sections,” NACA TM 807, Oct. 1936.
SIEDE, P. and STEIN, M.: “Compressive Buckling Of Simply Supported Plates With Longitudinal Stiffeners,” NACA TN 1825, 1949.
SANDORF, P. E.: “Notes On Columns,” Journal of the Aeronautical Sciences, pp. 1-12, Jan. 1944. Was classified.
Anon.: Handbook Of Aeronautics, No. 1, Structural Principles And Data, Fourth Edition, The New Era Publishing Co., Ltd., pp. 126-170, no date, about 1954.
KROLL, W. D.: “Tables Of Stiffness And Plates Under Compression,” NACA ARR 3K27, Nov. 1943. Was classified.
COX, H. L.: “Computation Of The Initial Buckling Stress For Sheet Stiffener Combinations,” Journal Of The Royal Aeronautical Society, Sept. 1954.
VISWANATHAN, A. V. and TAMEKUNI, M.: “Elastic Buckling Analysis For Composite Stiffened Panels and Other Structures Subjected To Biaxial Inplane Loads,” NASA CR-2216, 1973.
ANDERSON, M. S.: “Local Instability Of Elements Of A Truss Core Sandwich Plate,” NACA TN 4292, July 1958.
ROSSMAN, C. A.; BARTONE, L. M., and DOBROWSKI, C. V.: “Compressive Strength Of Flat Panels With Z Section Stiffeners,” NACA ARR 4B03, Feb. 1944. Was classified.
Anon.: “Summary Report On The Influence Of Fillet Radius On The Instability Of Compression Panels Having Integrally Machined Unflanged Stiffeners,” British S & T Memo 9/61, Sept. 1961.
JOHNSTON, G. S.: “Buckling Of Orthotropic Plates Due To Biaxial In-plane Loads Taking Rotational Restraints Into Account,” Fibre Science And Technology, pp. 435-443, Dec. 1979.
BATDORF, S. B. and HOUBOLT, J. C.: “Critical Combinations Of Shear And Transverse Direct Stress For An Infinitely Long Flat Plate With Edges Elastically Restrained Against Rotation,” NACA Report 847, 1946.
STOWELL, E. Z. and SCHWARTZ, E. B.: “Critical Stress For Infinitely Long Flat Plate With Elastically Restrained Edges Under Combined Shear And Direct Stress,” NACA ARR 3K13, Nov. 1943. Was classified.
KROMM, A. and MARGUERRE, K.: “Behavior Of A Plate Strip Under Shear And Compressive Stresses Beyond The Buckling Limit," NACA TM 870, July 1938.
KOITER, W. T.: “Theoretical Investigation Of The Diagonal Tension Field Of Flat Plates,” National Luchvaart-laboratorium Report S. 295, 23 Nov. 1946.
PRIDE, R. A.; ROYSTER, D. M.; and GARDNER, J. E.: “Influence Of Various Fabrication Methods On The Compressive Strength Of Titanium Skin-Stringer Panels,” NASA TN D-5389, Aug. 1969.
PETERSON, J. P. and CARD, M. F.: “Investigation Of The Buckling Strength Of Corrugated Webs In Shear,” NASA TN D-424, 1960.
BRUSH, D. O.; and ALMROTH, B. O.: BUCKLING OF BARS, PLATES AND SHELLS, McGraw-Hill, 1975.
BUSHNELL, D.: “BUCKLING OF SHELLS - PITFALL FOR DESIGNERS,” AIAA J., 18, pp. 1183-1226 (1981).
MURRAY, Noel W.: INTRODUCTION TO THE THEORY OF THIN-WALLED STRUCTURES, Oxford Engineering Science Series 13, Oxford Science Publications, 1986. Cover six types of diagonal tension.
YUNG-KUANG Chen; KUTT, Lembit M.; Christopher M. PIASZCZYK; and Maciej P. BIENIEK: “ULTIMATE STRENGTH OF SHIP STRUCTURES,” Trans. SNAME, Vol. 91, 1983.

V. I. WEINGARTEN and P. SEIDE:

“BUCKLING OF THIN-WALLED CIRCULAR CYLINDERS,” NASA SP-8007, August 1968.
“BUCKLING OF THIN-WALLED, TRUNCATED CONES,” NASA SP-8019, Sept. 1968.
“BUCKLING OF THIN-WALLED, DOUBLY CURVED SHELLS,” NASA SP-8032, August 1969.
BARUCH, M.; SINGER, J.; and WELLER, T.: “Effect Of Eccentricity Of Stiffeners On The General Instability Of Cylindrical Shells Under Torsion,” Israel Journal Of Technology, Vol. 4, No. 1, pp. 144-154, Feb. 1966.
BLOCK, D. L.; CARD, M. F., and MIKULAS, M. M. Jr.: “BUCKLING OF ECCENTRICALLY STIFFENED ORTHOTROPIC CYLINDERS,” NASA TN D-2960, Aug. 1965.
CHAJES, Alexander and WINTER, George: “TORSIONAL-FLEXURAL BUCKLING OF THIN-WALLED MEMBERS,” Journal of the Structural Div., ASCE, pp. 103-124, August 1965.
R. FRISCH-FAY: FLEXIBLE BARS, Butterworths & Co. publishers, LTD., 1962. His solutions involve solving elliptical integrals for large deflections.

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