1、NATIONAL,i !ADVISORY COMMITTEETECHNICAL NOTE 2661A SUMMARY OF DIAGONAL TENSIONPART I - METHODS OF ANALYSISBy Paul Kuhn, James p. peterson,and L. Ross LevinLangley Aeronautical LaboratoryLangley Field, Va.WashingtonMay 1952,_e,ELI!U liX,.iGEReploducod b,fNATIONAL TECHNICALINFORMATION SERVICESptrngfie
2、ld, Va. 22151Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A SUMMARY OF DIAGONAL TENSIONPART I - METHOD OF ANALYSISPLANE-WEB SYSTEMSI. Theory of the “Shear-Resistant“ Beam2. Theory of Pure Diagonal Tension3. Engineering Theory of Incomplete Diagona
3、l Tension4. Formulas and Graphs for Strength Analysis of Flat-Web Beams5. Structural Efficiency of Plane-Web Systems6. Design Procedure7. Numerical ExamplesCURVED-WEB SYSTEMS8. Theory of Pure Diagonal Tension9. Engineering Theory of Incomplete Diagonal TensionlO. Formulas and Graphs for Strength Ana
4、lysis of Curved-Web Systemsll. Combined Loading12. General Applications13. Numerical ExamplesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2661CONTENTSSUMMARY 1INTRODUCTION 1FREQUENTLY USED SYMBOLS . 2PLANE-WEB SYSTEMS . 9I. Theory of the “
5、Shear-Resistant“ Beam 92. Theory of Pure Diagonal Tension . 62.1. Basic concepts . 72.2. Theory of primary stresses 72.3. Secondary stresses . ii2.4. Behavior of uprights 122.9. Shear deformation of diagonal-tenslon web . 143. Engineering Theory of Incomplete Diagonal Tension 153.1. General consider
6、ations . 163.2. Basic stress theory 173.3. Remarks on accuracy of basic stress theory . 223.4. Comparison with analytical theories . 233.5. Amplification of theory of upright stresses 3.6. Calculation of web buckling stress . 263.7. Failure of the web . 273.8. Upright failure by column action . . .
7、313.9. Upright failure by forced crippling 323.10. Interaction between column and forced-cripplingfailure . 333.ii. Web attachments . 343.12. Remarks on reliability of strength formulas . . . 363.13. Yielding . 384. Formulas and Graphs for Strength Analysis of Flat-Web Beams . 414.1. Effective area
8、of upright . 414.2. Critical shear stress . 424.3. Nominal web shear stress . 434.4. Diagonal-tension factor . 434.5. Stresses in uprights . 434.6. Angle of diagonal tension 444.7. Maximum web stress 4344.8. Allowable web stresses . 454.9. Effective column length of uprights 464.10. Allowable stress
9、es for double uprights . 464.11. Allowable stresses for single uprights 474.12. Web-to-flange rivets 48Precedingpageblank iiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A;.jNICA TN 2661D4.13. Upright-to-flange rivets . 484.14. Upright-to-web rive
10、ts . 494.15. Effective shear modulus 504.16. Secondary stresses in flanges . 505. Structural Efficiency of Plane-Web Systems . 516. Design Procedure . 557. Numerical Examples . 56Example i. Thin-web beam . 56Example 2. Thick-web beam 60CURVED-WEB SYSTEMS 638. Theory of Pure Diagonal Tension . 639. E
11、ngineering Theory of Incomplete Diagonal Tension 689.1. Calculation of web buckling stress 689.2. Basic stress theory 689.3. Accuracy of basic stress theory 719.4. Secondary stresses . 719.5. Failure of the web 729.6. General instability 739.7. Strength of stringers . 739.8. Strength of rings . 749.
12、9- Web attachments 759.10. Repeated buckling 76i0. Formulas and Graphs for Strength Analysis of Curved-WebSystems 78i0. I. Critical shear stress . 7810.2. Nominal shear stress . 7810.3. Diagonal-tension factor 78i0.4. Stresses, strains, and angle of diagonal tension 7910.5. Bending moments in string
13、ers . 7910.6. Bending moment in floating ring 8010.7. Strength of web 8010.8. Strength check, stringers and rings 8010.9. Riveting . 8182Ii. Combined Loading 12. General Applications 86ivProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 266113.
14、Numerical Examples 86Example 1. Pure torsion . 86Example 2. Combined loading . 91Example 3. Angle of twist . 9_APPENDIX - PORTAL-FRAME EFFECT 97REFERENCES . 99FIGURES 102VProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-! .pNATIONAL ADVISORY COMMITTEE
15、 FOR AERONAUTICSTECHNICAL NOTE 2661A SUMMARY OF DIAGONAL TENSIONPART I - METHODS OF ANALYSISBy Paul Kuhn, James P. Paterson,and L. Ross LevinSUMMARYPreviously published methods for stress and strength analysis ofplane and curved shear webs working in diagonal tension are presentedas a unified method
16、. The treatment is sufficiently comprehensive anddetailed to make the paper self-contained. Part I discusses the theoryand methods for calculating the stresses and shear deflections of websystems as well as the strengths of the web, the stiffeners, and theriveting. Part II, published separately, pre
17、sents the experimentalevidence.INTRODUCTIONJThe development of diagonal-tension webs is one of the most out-standing examples of departures of aeronautical design from the beatenpaths of structural engineering. Standard structural practice had beento assume that the load-bearing capacity of a shear
18、web was exhaustedwhen the web buckled; stiffeners were employed to raise the bucklingstress unless the web was very thick. Wagner demonstrated (reference l)that a thin web with transverse stiffeners does not “fail“ when itbuckles; it merely forms diagonal folds and f_nctions as a series oftension di
19、agonals, while the stiffeners act es compression posts. Theweb-stlffener system thus functions like a tr_ss and is capable ofcarrying loads many times greater than those producing buckling of theweb.For s number of years, it was customary to consider webs either as“shear-resistant“ webs, in which no
20、 buckling takes place before failure,or else as diagonal-tension webs obeying the laws of “pure“ diagonaltension. As a matter of fact, the state of pure diagonal tension is anideal one that is only approached asymptotically. Truly shear-resistantwebs are possible but rare in aeronautical practice. P
21、ractically, allwebs fall into the intermediate region of “incomplete diagonal tension.“An engineering theory of incomplete diagonal tension is presented hereinwhich msy be regarded as a method for interpolating between the twoProvided by IHSNot for ResaleNo reproduction or networking permitted witho
22、ut license from IHS-,-,-2 NACATN 2661limiting cases of pure-diagonal-tension and “shear-resistant“ webs, thelimiting cases being included. A single unified method of design thusreplaces the two separate methods formerly used. Plane webs as well ascurved webs are considered.All the formulas and graph
23、s necessary for practical use are collectedin two sections, one dealing with plane webs and one with curved webs.However, competent design work, and especially refinement of designs,requires not only familiarity with the routine application of formulasbut also an understanding of the basis on which
24、the methods rest, theirreliability, and their accuracy. The method of diagonal-tension analysispresented herein is a compoundof simple theory and empiricism. Both con-stituents sre discussed to the extent deemeduseful in aiding the readerto develop an adequate understanding. The detailed presentatio
25、n of theexperimental evidence, however, is made separately in Part II (refer-ence 2); a study of this evidence is not considered necessary forengineers interested only in application of the methods.FREQUENTLYUSEDSYMBOLSAEGGeHIJLLeMPPucross-sectional area, square inchesYoungs modulus, ksishear modulu
26、s, ksieffective shear modulus (includes effects of diagonaltension and of plasticity), ksiforce in beam flange due to horizontal component ofdiagonal tension, kipsmoment of inertia, inches 4torsion constant, imches 4length of beam, incheseffective column length of upright, inchesbending moment, inch
27、-klpsforce, kipsinternal force in upright, kipsiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-oiNACA TN 2661QRR vtRRSTddcehhchehRhukkssqt3static moment about neutral axis of parts of cross sectionas specified by subscript or in text, inches 3total
28、shear strength (in single shear) of all upright-to-webrivets in one upright, kipsshear force on rivets per inch run, kips per inchvalue of R required by formula (40)restraint coefficients for shear buckling of web (seeequation (32)transverse shear force, kipstorque, inch-kipsspacing of uprights, inc
29、hesclear upright spacing, measured as shown in figure 12(a)distance from median plane of web to centroid of (single)upright, inchesdepth of beam, inchesclear depth of web, measured as shown in figure 12(a)effective depth of beam measured between centroids offlanges, inchesdepth of beam measured betw
30、een centroids of web-to-flangerivet patterns, incheslength of upright measured between centroids of upright-to-flange rivet patterns, inchesdiagonal-tension factortheoretical buckling coefficient for plates with simplysupported edgesshear flow (shear force per inch), kips per inchthickness, inches (
31、when used without subscript, signifiesthickness of web)angle between neutral axis of beams and direction ofdiagonal tension, degreesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACATN 2661i_,8EP(I(_03“T all_dSubscripts:DY diagonal tens ionIDT in
32、complete diagonal tensionPDT pure diagonal tensionF flangeS shearU uprightW weball allowableav averagecr criticalcy compressive yielde effe ct ivedeflection of beam, inchesnormal strainPoissons ratiocentroidal radius of gyration of cross section of uprightabout axis parallel to web, inches (no sheet
33、 should beincluded)normal stress, ksi“basic allowable“ stress for forced crippling of uprightsdefined by formulas (37)_ ksishear stress, ksi“basic allowable“ value of web shear stress given by fig-ure 19, ksiflange flexibility factor, defined by expression (19a)Provided by IHSNot for ResaleNo reprod
34、uction or networking permitted without license from IHS-,-,-NACATN 2661 5m_xultmaximumultimateRZdhSubscripts:RGSTSymbols Used Only for Curved-Web Systemsradius of curvature, inchescurvature parameter, defined in figure 30spacing of rings, incheslength of arc between stringers, inchesringstringerPLAN
35、E-WEB SYSTEMSi. Theory of the “Shear-Resistant“ BeamTypicsl cross sections of built-up beams are shown in figure I.When the web is sufficiently thick to resist buckling up to the failingload (without or with the aid of stiffeners), the beam is called “shear-buckling resistant“ or, for the sake of br
36、evity, “shear resistant.“ Webstiffeners, if employed, are usually arranged normal to the longitudinalaxis of the beam and have then no direct influence on the stressdistribution.If the web-to-flange connections are adequately stiff, the stressesin built-up beams follow fairly well the formulas of th
37、e engineeringtheory of bendingMz (i)Iq = (2)IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FNACA TN 2661with the understanding that the shear flow in outstanding legs of flangeangles and similar sections is computed by taking sections such as A-Ain
38、 figure l(a). As is well-known, the distribution of the shear flowover the depth of the web follows a parabolic law. Usually, the dif-ference between the highest shear flow in the web (along the neutralaxis) and the lowest value (along the rivet line) is rather small, andthe design of the web may be
39、 based on the average shear flowwhere QF is the static moment about the neutral axis of the flangearea and QW, the static moment of the web material above the neutralaxis. When the depth of the flange is small compared with the depthof the beam (fig. l(c) and the bending stresses in the web are neg-
40、lected, the formulas are simplified to the so-called “plate-girderformulas“(3)M (4)s (5)q = -which imply the idealized structure shown on the right in figure l(c).When the proportions of the cross section are extreme, as in fig-ures l(a) and l(b), formulas (i) and (2) should be used, because theuse
41、of formulas (3) to (9) may result in large errors. In such cases,the web-to-flange connection, particularly if riveted, is often over-loaded and yields at low loads. The beam then no longer acts as anintegr81 unit, the two flanges tend to act as individual beams restrainedby the web, and the calcula
42、tion of the stresses becomes very difficultand inaccurate.2. Theory of Pure Diagonal TensionThe theory of pure diagonal tension was developed by Wagner inreference 1. The following presentation is confined to those resultsthat are considered to be of practical usefulness, and the method ofpresentati
43、on of some items is changed considerably. Mathematical com-plexities have been omitted, and an empirical formula is introduced forone important item where Wagners theory appears to be unconservative.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NAC
44、A_ 2661 72.1. Basic concepts.- A diagonal-tension beam is defined as abuilt-up beam similar in construction to a plate girder but with a webso thin that it buckles into diagonal folds at a load well below thedesign load (fig. 2). A pure-diagonal-tension beam is the theoreticallimiting case in which
45、the buckling of the web takes place at an infini-tesimally small load. Although practical structures are not likely toapproach this limiting condition closely, the theory of pure diagonaltension is of importance because it forms the basis of the engineeringtheory of diagonal tension presented in sec
46、tion 3.The action of a diagonal-tension web may be explained with the aidof the simple structure shown in figure 3(a), consisting of a parallelo-gram frame of stiff bars, hinged at the corners and braced internallyby two slender diagonals of equal size. As long as the applied load Pis very small, th
47、e two diagonals will carry equal and opposite stresses.At a certain value of P, the compression diagonal will buckle (fig. 3(b)and thus lose its ability to take additional large increments of stress.Consequently, if P is increased further by large amounts, the additionaldiagonal bracing force must b
48、e furnished mostly by the tension diagonal;at very high applied loads, the stress in the tension diagonal will beso large that the stress in the compression diagonal is negligible bycomparison.An analogous change in the state of stress will occur in a similarframe in which the internal bracing consists of a thin sheet (fig. 3(c).At low values of the applied load, the sheet is (practically) in a stateof pure shear, which is statically equivalent to equal tensile and com-pressive stresses at 45 to the frame axes, as indicated on the insetsketch. At a certain crit
