A Theoretical Investigation of the Distribution of Maximum Shearing Stresses in a Stiffened Flat Panel Citation Wang, Tsun-Kuei (1939) A Theoretical Investigation of the Distribution of Maximum Shearing Stresses in a Stiffened Flat Panel. Master's thesis, California Institute of Technology. doi:10.7907/A1JY-DH94. https://resolver.caltech.edu/CaltechETD:etd-04142008-083306 Abstract No abstract submitted. Item Type: Thesis (Master's thesis) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1939 Record Number: CaltechETD:etd-04142008-083306 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04142008-083306 DOI: 10.7907/A1JY-DH94 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1385 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 21 Apr 2008 Last Modified: 17 Aug 2023 00:29 Thesis Files Preview PDF (Wang_tk_1939.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

A Theorecticel Investigation of the Distribution of Maximum Shearing Stresses ina Stiffened Flat Panel. In Partial Fulfillment of the Requirements of the Degree of Macter of Feience in Aeroneutical Fnoineering, by T. KR. Wang California Institute of Technelcsy Pesedena, California ACKNOWLEDGHENT . Grateful aclmowledgment is hereby made to Dr, Th. von Karman for the suggestion of this research subject and the assistance which he has given, The author also wishes to express his appreciaticn to Dr. E. E. Sechler and Dr. T. 2. Sears who gave criticisms and help in preparing this paper. INTRODUCTION, In studying stress distribution several methods have been used, The purely experimental method is one in which local stresses are measured directly by means of sensitive strein gages. The photoelastic method is the second one which gives a definite visual stress pattern to each stress system, from which the shearing stress ean be read di- rectly. There is a third one which is a purely a mathematical method. This method is based upon the assumption of continuity and the law of equilibrium for establishing:the stress functions and equetions derived by logical processes. The solution is then obtained by the use of boundary conditions. At the California Institute of Technology the shear oharacteristics of thin plates reinforced by one centrally locat- ed stiffener was investigated by the photoelastic method. The third m:thod has been used by the author for studying the seme naturel ¢ro- blem in hopes of checking the result already obtained. It is common practice to use jenels of thin plates reinforced by longitudinal stiffeners in the construction of modern metal airplanes. So far as the strength characteristics are concerned, it is necessary to Imow the stress distribution of the sheets in order to obtain the strength Limits of the structures. This thin plate with a stiffener is assumed to be leaded in to different ways. They are : first, uniform normal loads are applied to the stiffener at both ends; second, uniform normal load is applied to the stiffener at both end with another load uniformly distributed along the opposite edge of the slate and stiffen- ele By utilizing Airy's fumcticn and Fourier's series, the suthor derived the stress components 6x, Oy and Gg for the two cases above. Then from these expressions the stress components were obtained for the principal stresses and the miximum shearing stress and their directions at any point on the thin plate were determined. The curves of the max- imum shearing stress distributions were then drawn for the particular specimens tested by the yhotoelastic methods. Tho material cf the plete studied is Bakelite. The value of Foisson's ratio for it is 0.36, which was aupplisd by Mr. 7, A. Zinzow > E'S > Rssearch.Physicist at the Bakelite Corperstion, Bloomfield, N. J.. Contents. Introduction Symbols used, Dimensions of the plate and stiffener. Part One 3 (a). General assumptions and method, (b). Boundary conditions. (c), Determination of f,(y) and By (4). Result (e) Discussion, Part Two : (a) General assumptions . (b) General methods (c) Boundery conditions (a) Trensformation of coordinates (e) Result (f) Discussion. Reference Appendix, x, J V¢ 6x54 Tc Tax, ee Vey Gy, Sg Ow a tA 4 a a Symbols. Used. Rectangular coordinates Polar coordinates Components of the displacement parallel to the x end y axes in two dinensional doordinetes,. Components of the normal stress carellel to the x end y axes, The principal stresses, Shearing stress in rectangular coordinates Mieximum shearing stress, Unit elongetion in x and y directions. Shearing strain in rectangular coordinates, Redial and tangential normal stresses in ¢olar coordinates, . shearing stress in .olar coordinates Stress function Modulus of elasticity in tension and compression Modulus of rigidit Modulus of Elasticity in shear ). g v Poisson's ratio, Intensity of a unifcraily distributed load, Dimensions of the Plate and Stiffener, — —| | W—5/5 | —.257 ty = 0.355 in, P = 2,000 lbs, £ = 6.0 in, a = 3 1/32 in, b=1 in, te = 0,515 in, ray PART ONE Maximum shearing stress distribution in a rectangular plate re- enforced by a stiffener with uniform load applied at the twoiands of the stiffener, -l- (a) General Assumptions, Consider the compression of a beam of constant rectangular cross section by forces applied at the ends and parallel to one of the principal axes of the cross section, Then the shape of the compression will be as- shown in the figure l. ql FIGURE /. Hii If a plate is added ,without fastening it will be found that the plate is longer than the top of the beam in figure l. Thus ,for the case in which the beam and. plate are fastened toge- ther , there must be a shearing force applied. This shearing forcem varies from zero in the center of the beam to a maximum value at the ends, This shear is between the beam and plate. @o Now we consider ons half of the plate as OPQR in figure 2. The edge OP is fixed to the beam, edge QR is free and the edge OR and PQ may be either fixed or fres, d4 Q ~2- Assumptions : For convenience we assume that 1. The lateral contraction of the beam can be neglected | without much error. 2. The deformation of the elastic plate takes plave in the xy plane only thus it is two-dimensional. The displacements of the particles of the plate can then be resolved into components u, v parallel to the coordinatesaxes xX, y respectively. In elasticaLy it is well known that the stress function has to satisfy the compatibility equation : 4 ot 4 vy*é = 8 2 fab y + 24 = 0 (1) The stress components are _ eg _2g _ __ daz 6, = $58 ’ & = Sx , c* 7 Sh Provided that there is no body force. Since any stress distribution can be expressed by a Fourier's series and the accuracy of the calculations only depends upon the rapidity of the convergence of the series and the number of the terms taken, thus the assumption can be legitimately made that sine or cosine functions may be used for expressing this distribu- tion. Furthermore we assume that the plate concerned is at rest under the action of external force as in the usual case. The re- sulting problem is then static, that is, independent of time. “Assume.a: stress functicn % such that, B=f (y) sin (2) where n's are integers and f ts are functions of y only. This stress function means thet the distribution in ell sections along x is similar and only differs in amount. By substitution yohe stress com;onents are then : _ affn . n7yx 6, = dy * sin £ 645 ~ (yy* f, sin ye (5) ny te F) Cia oor 7 A (b) | Boundary Conditions. Since the beam and plate are fastened together the displacements o&atiees over the line of ccntact should be the same or, The ether boundary conditicns which are self evident are : v=0 at y=o (&) 6y = 0 at yra (6) @w=0 at yrua. (7) From (3) and (6) : f, (y) =0 at ysa (8) Again from elasticity we know that Yxy = Ou + oy y —°d ve (9) €x= oY OX From (5) and (9) we have 2& = 26-2 2b =0 ars Ls Substituting (3) into (10) and using the relation G=_E we have (10) 2(1+y/) ei, - (ax) 22 +y) df, = 0 (11) dy3 oh From (7) we have dg =0 at yea (12) (c) (i), To find the expression:of fr (y) From the com,atibility equation (1) and the stress function (2), -we obtain ;: ¢ dfn -2 az )* dtp (a fn = 0 (13) dy* j This differential equation is a linear one with constant coeffi- cients, So there is” a first solution f(y) = ey. Substitution _ gives. m=+ny +n7 3 that is, it has two pairs of re speated roots, hence its kL ~— & complete solution is ; nly - nity BAY — DAY _ £. & fy) = Ae | +Boe ft + Cy ee +Dye Where An , Bn, Cn,Dn are constants which are nov to be determined by the boundary conditions. First assume an infinite lerge plate,i.e. y=a St oo af = OQ Hence Then from boundary conditions y =a, f,(y) =0 & ay _ 2 ~i fy) = Be va + Dy oe Substituting the last expression into (11) we then have : Di = F a, a and hence, f= Bie - (1 - a ) (14) All the scluticns of the type “Dy an be the sclutions of (11) & : (15), since the ejustions are linear, therefore 6, = XEN Bae De St OTP) in a (I-p” @ 1-y 2 Ps, OF cn Bye) nh ny Atv (ii). To determine tne coefficients B, by using condition (4), From the reletions in (9) we have u_l 2 =F (6, VG ) Hence the disylacemcnt u, of either the slate or beam is : x for ul lke (6 IG) ae : oO Evaluating it at y = 0 for the ,late by (3') we have then : Wists = 2S DE f, Sata. cos 2kxtD By See y-y* | TTY Now we ekaluate u for the beam as follows : (4) From the equilibrium condition we know that (Fara y C6 y¥( plate ) eam) ‘Assume that the normal stresses in the x direction are uniformly distributed over every cross-section of the beam and their valuelis then, Gx = rral te a where P is the total load on the end of the beam, A the area of the cross section. Evaluating at y= 0 then we have’ u= -Px+ 2 St, By _2 cosnJx -1 Yt, BR 2 -lLty(MDB ycos Hm i ows ER BA > 1-y & BA = 1 1-y Ete g 4 tlt y( az )ya <—— 4 It is well known mathematically that xh -9 > 1 cos nx ond yL= TT 2 7 Spe g Using this mathematicel expansion we get the foltowing: exvression’. for u = AtiB, + 4VP - nT ) Mean S Cp) Rye i ig ye } ly (33) + aLP, or) ; E A(l-y Ayén® - t) (n7Z )pBy cos n7 x a — Ne y aE* (47 Because of the displacements of the beam and plate are always equal, so the , coefficients of the corresponding terms of expressions (@) and (as should be identically equal, Then we get: * - £p _(1-») L+Ay- Z42V—yR --— Yr pe jn yp WASNT Sores | 4ti 2 4 ta 9 Put Bn as (£)° 0-0) Cm nT ~T=— = > e fre Ferg h as eran oe nye =-T en e % _ 2 ItLgf Coe TA 4 “FE + Am (SBE? -¥S_ y-v (a) GAA! The result, It is well kmown that Mchr's circle can be used for determining the principal stresses and the meximum shearing stress if any set of the stress components Gy, 6, y? xy are given, or by the formulae : Tmax =7 (6-6) =/| “5 2) cy For the problem concerned we have: “Cmax “ale begets e + Sy ‘f2-7 eas @ +65 f + 3r eo tess fe gene; Boies ge nity © WX a = Fo; oF +y4 FTV Aden a (3) oy where : Cc. = / n= (+ An (Sstev-y' _ yv- vs " orr. tH, L )n The directions of the principal stresses can be found for any particular points ( x,y ) by the formula : tan 20 = 2 Lavy Cy 76 } The direction of the maximuj shearing stress for the same point lies on the plane passing this point, whtich ‘makes an angle of 7/4 with re- spect to the principal stresses, that is, on the plane bisecting the angle between the two principal stresses. : i: wether “ semen n8 Pee Peet + eaeaees + rose PS hr cae Sues oi ches —~ ORIGINAL CURVE, w-0+ CORRECTED CURVE, — BP =K000 Pounas L 4 4 Car /4B SH LBS Se if fo Any j0 $ dik a 7 r 7 a BISTRIBUTICN OF MAXIMUM SHEARING 2TRESS (FROM REFERENCE 8 ) (e} Discussion comparing the theoretical curves of the maximum shearing stress distribution with those obtained by photoelastic method.°® we Tind that : 1. The curvatures of the latter ones are bigser; thet is the magnitudes of the maximum shearing stress distribution of the plate near the two ends of the beam decvesase a Little more rapidly at the points farther from the beam, 2. The curvatures of the curves obteined by the theoretical method are smaller comparatively. However the curvature increases with an increase in Poisson's ratio. 4. The magnitudes of the maximum shearing stresa in the nlddile part of the plate obtained by these two methods closely asree, 4. On the two edses ( x = © yk? of the olate the veneral solution shows that Tis different from zero, but alone these edges for a certain distance from the beam, C is very emall. In calculatine & it has been assumed thet on the sdres Tequals maxX. Zero. 5. The equation of the T, Sor Sy breaks down when it con- -2T -on 5 nx tains no factors e £” and cos “F or e ZI Pcin z simul- on taneously. This is the case at points y=O and x=0O or y=0 and x=R - Since then the series which represents the T, Sy oF Sy is diver- °See reference No.8 - Ja - gent its value becomes infinite. 6. By an inspection of these expreesion (3") we can find that if ~@ aoproaches infinity all the values of t » Oy or Sy are identically eaual to zero. These are correct because at infinite distance from the points where the force is anvlied there is no disrtibution of stresses at all. PART TiO ~*~ Meximua shearing stress distribution in » rectansuler vlete re- enforced by a stiffener with uniform losd esvolied «at the tro ends of the beam and .om one sdge of the slate, (a) General Assumstions, ‘Let « uniform norm: ] pressure be applied elons 2. finite length of one edge of the plete and tro ends of the boam as shown in firure 4. ALJJJJSLASIANIAAJLU TT Figure 4, The gssunptions of other things end the initiel shesring force applied in tue section between ths beam end the plete bear the sane charac (b) Genorel Method The : Toblem stated A above is dealt with by eunerimposing two cs Case (1). The uniform normal pressure acts at tro ends of the beam investizeted in Part One « fo ct oO fe oO oO for Q @ Oo a) ct i O bey Lo " ra tA Case (2), The uniform normal pressure ects will now be studied, La F ‘on 7) Q ir] @ oO 3 9 ™m We can assume that RC is merely. c line in cn infinite slat figure 5, bh -ji- etresses at any point of the glate is with the srozer units, eiven Me e by the stress function £-=8 (¢- rtéy,’) . (1s) / : Lt +s where ( r,¢), (rt@) sre poler coordinates .t tho crisins 0 ond R! re- Figure 6, apres we gee thet «t ony point EH of the plete the first term cf tre stress eat function #, thet is 2) = 6 or yb st tre orisin © gives : 2 * The solution cf this problem is due to J.¥,. ichell, Proce. London wnth. oy = + 3 * 1 sh = 2 w= - (fF 38) = Similarly Pf, =~ = y* “7 at the origin Q' gives Cy _ z , TCyyre \¢ ~~ == # 27 The directions of the principal stresses bisect the angle between the radii r and r', Ths megnitudes of the totel principal stresses at any point Mare 3: 6, . | ~ B(x + her = (x + ox ) [ 6. =- 2(a _ dr x) where K = Yl £ . Hence Tmay= 3 (6 - 6) (c) Boundary Conditicns _ (1) The uniform normal lozd along OR is c, that is , S45 =2(¢-4¢)=2 It is reduced to 3 ¢ ~ yp’ = 7 (2) The rest of the boundary is free from the stress, (d) Trencformetion of Coornin:tes. ing ensec (1) end (©) ve should hove trom in ths same coordinate systen, It is evident thst this can be easily dene Dy us- ing = rectanguler coordinete systen, We trerefore choose the rigin ss in - a7 Pisore 7, From this fisure we have, ra sabsti tno 4 int pe oteeee Pouention (15) am than he Substitcutine them inte thc stress function (L&E), "ec tron neve 3: eile +(b- DY tao a8 4 - [x 4(arb- el ake ~/ ath Put b= 6 for setiefying ovr cose, then cn Eta ~4e Xf x la~f) ] ANY RE ee (a- yy S} fa! 4% AA, x4 4 Xx Ca-Z) ai aH NE Kit (a - g)* ae lgs and smile pert of the ~ BY tas xX" yap * age a ~ (a aH Tmaxs ~ AL OTE OPE ¢ mo cpecoe In superim, cei ss (1) end (2), we heave the Oma x. ( part - & = , (Part) 1/224 fae 7 (a phe (f) >: soussion. 1, As sa whole the = @ 4 is ‘snes ering strs Ss obtain- lier then thet of the snximun css obtained by the photo- elastic method.* a os 7 ~, . t2¢ weaximun sheering stres: - . CN es rn fae pe A 4 NM] Uber part of the ginte ae droavn in the curves is ls uarger mesr the cdg rer Me OvrsY wiate sround ths bean m, then thet obtained expsrimentally, eer ste by i See Heference @ Ve a1S - a+ (atb-4)° | a+ (6-4)? _ Av + (ath~4)* z+ (b-4p ks _ ta th (6-4) gee (ath~-ty® t+ (b-JP) ete (a thf" xe 4 (6-ppe Henee there ere three forms for expressing the scearing stress cf toe second probici, . { tay = Baw b-pe : é _ (a +b-) + (6 ~ 4)” a | Tc ——— *¢ zi +(atb~-p) 2+ Cb- gy and. _ v4 t (atb- 2 a ~ e Tx = eset sae Se Theodor von Merman: Lectures of Blasticity epzlied to Asronsutics 1930 - 1936. ( given at Caltech ) S. Timcshenko : Thecry of Elasticity. Aug, Fogpl und Ludwig Feppl : Drang und 7eng. ‘Technicel tiemorendums, Neticnal Advisory Committes of Asroneutics. No. 769, iarch 1938. Technicel Semorendums, Ne. 4. 0. £. No. 793, April 1258, : GC, & Inglis : Two dimensional stresses in elastic pistes, "ngineer- ing 112, (1921). C. B. inglis : Stress distribution in 2 rectangular olate h-ving two opsosing edzes sheared in opposing directions. Proc. Roy. Soc. London, A 103 (1923), A ghotoelastic investigation of the distribution of shesring stress in s stiffened flet panel. By “. N. Bell and J. K. Bussey, Californic Institute of Technology, 1958,