# Shear Flow

This subsection contains an

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*for finding the internal shear loads in a bulkhead web***Example Problem****.**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 subsection also contains****GRAVERY**'s famous "**The Quadrilat****er****al '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**.****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 "**Bu**" (in**ckling of Tapered Columns****BASIC**for IBM PCs) too**.****PEERY**, David J, Ph.D**:****Aircraft Structures**,*McGRAW-HILL*, hardcover, 566 pages, indexed, 1950. (Don't consult the revised edition**.**)**.****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 bulletins of the English aerospace stress manual represent the current methodologies**.**-
**ROBINSON**, John**:**"," Revised May 1989. Abstract: "**The Mode-Amplitude Technique and Hierarchy Stress Elements -- A Future Technology***A general summary is given and key items are demonstrated using examples representing thin walled structures*." Especially see his reference list too. In this paper he doesn't check for equilibrium directly**.**

**Shear Bulkhead Web Example Problem**

**Shear Bulkhead Web Example Problem**

Questions in regards to this solution may be addressed to rudy.k.mckee@gmail.com. Any error are those of the authors.

**Background Material**

**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**

**References**

These older papers serve as the foundations for later analyses and the reason they should be read

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**Blue colored books, etc., are main references**

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**General References**

**BIJLAARD**, P. P. and**JOHNSTON**, George S.**: “**,”**Compressive Buckling Of Plates Due To Forced Cripping Of Stiffeners**paper**IAS****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 Combination***s*,”*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 Load***s*,”**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 Stres***s*,”**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.,*, pp. 103-124, August 1965**ASCE****.****R. FRISCH-FAY:****FLEXIBLE BARS****,****Butterworths & Co****. publishers,****LTD****., 1962.**His solutions involve solving elliptical integrals for large deflections**.**