TWO PROBLEMS OF THERMAL STRESS IN THE INFINITE SOLID A Thesis Presented to the Faculty of the Graduate School of Cornell Unlrersity for the Degree of Dootor of Philosophy By Nils Otto Myklestad June, 1940 ProQuest N um ber: 10834652 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is d e p e n d e n t upon the quality of the copy subm itted. In the unlikely e v e n t that the a u thor did not send a c o m p le te m anuscript and there are missing pages, these will be noted. Also, if m aterial had to be rem oved, a n o te will ind ica te the deletion. uest ProQuest 10834652 Published by ProQuest LLC(2018). C opyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 (1) Blographloal Sketoh of the Author# The author was born In Wllliston, North Dakota on March 24th, 1909, but reoelved moot of hia education in Nomay and Danmark and graduated from junior college (gymnasium) in Oslo, Norway In 1926# In the fall of the same year he entered the Royal Technical College in Copenhagen9 Denmarkt graduating from this school in Marcht 1932* During this period he also worked one year, 1928*29, at the South Philadelphia plant of the Westlaghouse Electric and Manufaturing Company, and again went to work for this company in August, 1932* He was then continuously engaged in engineering work, with various companies, until August, 1937* In 1937*38 he was teaching assistant and graduate student in Mechanical Engineering at the University of California and, since 1938 instructor and graduate student in Mechanics of Engineering at Cornell university* (2 ) History of the Thermal StressProblem, The general equations for determining thermal stress were probably first derived by J.r.C.^uhamel and published in Paris in 1838 (r,l). He develops the general thermo-elastic equations by considering a solid body as a system of particles and, as the sphere and the cylinder, with radial stress variation only, are the simplest cases of thermal stress, he derives the expressions for the stresses in these two oases. For the sphere he arrives at the very interesting result that whatever the law of radial temperature variation may be, the extreme fibres have the same length as If the whole sphere were at the average temperature• Independently, and at about the same time as Duhamel, Franz Neumann also arrived at the thermoelastic equations. However, he was primarily interested In the effect of uneven temperature distribution upon the double refraction of light, and he worked out the strains for the cases of a sphere and a long circular eyllnder. His work on this subject was published in Berlin in 1841 (r,2). (3) Since Duhamel and Neumann very little work haa been done on the development of the theory; moat papera on the subject have dealt with the aame problems in different ways or developed the theory more fully for epeelal oases* Almost the only problems dealt with to date are the sphere, the oyllnder and plates* The following brief abstraots of some of the papers pub lished will perhaps Illustrate this* C*W*Borchardt (r*3) develops a method of solving the thermo-elastlo equations by means of potential theory* He arrives at rather cumbersome expressions for the displacements u f v, w In terms of potential fnnotions, and he treats particularly the sphere and the olroular disc with arbitrary temperature distribution* J* Hopklnson (r*4) brings the temperature term into the equation of equilibrium and applies the equations thus found to the problem of a sphere with nonuniform radial temperature distribution* Rayleigh (r*5) is mostly interested in the double refraction of glass due to temperature stresses; and he treats the sphere, cylinder and flat plate* He states that the investigation of temperature stresses was first attaoked by J* Hopklnson (r*4), while in re ality all conclusions reached by Hopklnson were to (4 ) he found in the muoh earlier paper by Duhamel (r.l). It seems that neither Rayleigh nor Hopklnson knew that was being done on the oontlnent. Alfons Leon (r.6) develops the general equations for a body of revolution, and proeeeds from these to treat the sphere and the cylinder for the oase of linear temperature variation with the radius. Lorenz (r#7) develops the equations for displace ments and stresses for an infinite hollow eyllnder with only radial variation in temperature. For a logarithmic temperature distribution he determines the maximum stresses at the inside and outside of the cylinder. Lees (r.8) treats a hollow circular eyllnder in a more general way and gives a simple graphical method for determining the stresses for any temperature dis tribution and any law of expansion. Barker (r.9) develops charts for the rapid cal culation of temperature stresses in tubes. Kent (r.10) treats a finite, thin walled, hollow eyllnder, with free and damped end oondltlons, by the method of assuming an element, out from the cyl inder wall along Its entire length, to be a beam on am elastic foundation. He finds the stresses and def lections due to variations of temperature both radially and axially. (5) Kent (r*ll) also determines the distribution of stresses in spheres and long oiroular oylinders, both solid and hollow, when the temperature varies with time aooording to the law of heat conduction* He starts with the well known formulas for the stresses in spheres and cylinders when the temperature varies with the radius only* Then he determines the temperature distribution under certain conditions of cooling and heating of the outer surfaoe, and substitutes these in the formulas for stress* He also plots numerous ourves for the variation af temperature and stress with the radius* Biot (r*12 and r*15) treats, as part of a some what more general theory, the temperature stresses set up in a cylinder of arbitrary oross seotion and with a steady state temperature variation* lhe surfaoe temperature varies in any manner along the boundary of a oross seotion, but is constant along a generator* He ehows that, for this case, a solid oyllnder has only axial stress| but a hollow oyllnder will, in general, also have radial and tangential stresses, unless the oyllnder can be slit lengthwise without producing any relative motion between the two sides of the slit* (6 ) Ueulbetsoh (r*14) develops the formulae for def» I m U q m tad momenta in triangular end square plates, which are simply supported at the edges and subjeoted to eonstant| but different, temperatures on the upper end lover sides* Be shows, that for these oases, the stresses are Independent of the thickness of the plate* Goodler (r*15) investigates the thermal stress set up by unequal heating of a flat strip, the tem perature varying along the length and through the thiokness, but not aoross the width * He treats par ticularly a strip that is heated uniformly through the thiokness, but only along a narrow transverse bandj and a strip that is heated uniformly along a narrow transverse band, but only on one side, the temperature variation through the thiokness being linear* Dsn HOrtog (r*16) treats the following oases, all of vhldh to a certain degree resemble welding processes t 1* A reotangular plate heated along a center line; £•A large flat plate with a central 1 circular hot spot; Z. A tube with a hot olrole around its middle end oool at the ends; 4* A tube with a hot generating line* Starting with a sinusoidal tsmperature distribution he derives sens interesting results. The maximum (*l i l i m to « m l| I toi 4 m m h t o n t o hot toad to m y W 9 N » omd to t o n t o t |«r somewhat larger)# la m m • t o m>iU— t o n i t o M m i u m at the edge of toa hat elreular ration, and la £ s « t regardless of atoa of plate or alaa of hot spot* Caaaa 1 and 4 f i n riaa to aaoh aoro aarloua atroaaaa than oaaa Z for eempareble dtoanalona of the hot band* Ooodlar (r*17) gives a sunmary of tha formulas for thasaal atraaa in Zlatas, rods and oyllnders arranoad in tha form of design data* Ooodlar (r*18) also investigates the thermal atraaa In thin vailed oyllnders of any oross seotion, for internal and external temperatures varying In any ■aimer round tha olroumferanoa, but not along a gen erator* ferenoe* Tha thiokness also may vary round the ciroumThe olroular oyllnder is treated in detail* Ooodlar (r*19) makes the most Important contri bution to the theory of thermo-elastio stress slnoe Duhamel and Neumann* He shows how all stresses and displacements In a heated body can be derived in a vary simple manner from a single function ^ 9 vhloh ha oalls tha thermo-elastic "Displacement Potential"* Tha aquation for f is of the same form as Polsson9s aquation and its solution is facilitated by the vail known theory of tha Newtonian Potential of a distri- (8) button of natter* He also treats the particular eases of plain stress and plain strain, and finally applies the method to the two problems of a plate with cun elllptlo hot spot and a plate with a reotangular hot spot* The shore theory by Dr* Goodler forms the basis for the two thermal stress problems of this thesis, and It will therefore be dealt with In more detail under "General Theory"* Porltsky (r*20) treats a cylindrical tube by means of complex variables, very nruoh like Blot (r*13)* Of books that oontain material on thermal stress Timoshenko's "Theory of Elasticity" (r*28) Is the most oomplete* It contains the fundamental thermo-elastlo equations and applications to plates, dlsos, spheres and cylinders. Almost the same material Is treated In F^ppl's *Vorle8ungen iiber Teohnlsche Meohanlk” (r*29)* Neumann (r*22) derives the thermo-elastic equations, but gives no useful applications* Love (r*27) barely mentions the thermal stress problem, and the other books referred to only treat the circular oyllnder and elementary problems* (9) General Theory* When an elaatlo solid, which Is unstressed at a uniform temperature, Is given a nonuniform temperature T(x,y,z) that Is not linear In (x,y9z), the natural expansion of a volume element will he restricted by the surrounding material and a state of stress will ensue* The difference between the actual strain and the natural expansion 01T of a volume element Is related to the stress through Hooke’s lawj oi being the linear eoeffiolent of thermal expansion* For simplicity It Is assumed that the original uniform temperature Is zero* This makes no difference In the theory because a uniform change of temperature creates no stress In an Isotroplo solid* It Is also assumed that the elastlo constants of the material do not vary consider ably over the temperature range In question, although a variation of oc with temperature oan be taken oare of by carrying « T through as a single variable quan tity* let u,v,w be the total displacements in the x,y,z directions respectively; then the components of normal and shear strain are (r*28, p*7), Also tha volume expansion e » . ii a uniform change of temperature of a small Tolume element does not oreate any angular distortion of the element the shear stresses vlll be unaffected by the term otT or9 etc., where G« 18 the modulus of elasticity in shear , B the modulus of elas ticity in tension and compression (assumed equal) and 1/ Poison's ratio# The normal stresses, however, are deter mined by the following equations (r«28, PP*11 and SOS) C* * 777“|/P57 e * * * ~ F I T * T] and two more by interchanging x with y and z« The three equations of equilibrium (r*£8, p»£08) take the form ( * + * ) & + e v ' u - 1% r £ L = o and two more by interchanging x with y and z9 and u with v and w. Also (U) Substituting for A and G these equations become +(l-iv)V*u f eto%* Equations (1) and (2) are the fundamental equations of the theory, but the boundary conditions and the con ditions of compatibility between the strain components must also be satisfied* The compatibility equations In terms of strain components (r*28, p*196) are seen to be satisfied automatically when the strains are ex pressed in terms of the displacements u,v,w acoordlng to (a). This is also quite obvious from a physloal standpoint as there Is no restriction upon the displace ment of a point of the solid as long as the material Is not being stretched beyond the elastic limit* The boundary conditions are mostly given as freedom from surface forces* Should any external forces aot on the surface of the solid the stresses resulting from these can be determined separately and, by the principle of superposition, added to the stresses already found* In order to obtain particular Integrals of (2), a funotion ip is introduced and defined in such a manner that 41 4% i (12) By m n i of (a) the atrains oan be expressed in terms of the funotion f as follows: (b) e» v*r. (c) Equations (2) oan now be written £ eto.. [ (l + v ) e t T ] = 0 It Is seen that equations (2) are all satisfied when (3) Evidently any funotion f , which satisfies (3), pro* Tides a particular integral of (2). The state of stress and strain represented by this funotion will ordinarily require oertain surface foroes at the boundary of the solid. These surfaoe foroes oan be determined from the conditions of equilibrium at the boundary and the stresses there, expressed in terms of f • To oomplete the problem It Is necessary to apply equal and opposite foroes to the surfaoe of the solid and determine the stresses from these* as in an ordinary (isothermal) boundary value problem. (13) By superposition the complete stress resulting from unequal heating of the solid oan then he determined* The funotion V is oalled the "Displacement Po tential" (r*£7, p.38). By means of (3) equations (1) oan now be written in the form (ia) eto.* It Is seen that only the seoond derivatives of ¥ are necessary to determine the stress components* The equation (3) is of the same form as Polsson's equation 7*V*-4Tff (r*299 p*148)9 and a particular Integral is given by the Newtonian Potential of a distribution of matter of density f per unit of volume9 In this equation V is the potential at the point (x9y9z)9 the coordinates £ refer to a point in the solid where the density is f 9and the inte gration is throughout the solid* The distance r9 between the points (x9y 9z) and (tf by the equation r * ^ (x-jf f + Putting ccT » -4Tff £!▼» (y-f )z + (z-JT) • the following equation for (14) ip Is obtained: W M , *) =- £ 7 / f f T (* -y<-f ) d S d J d t . When the Inequalities of temperature are of a purely looal nature In an infinite solld9 T = 0 outside the hot part of the solid 9 beoause £ 9y , will vanish at infinity are bounded 9 and consequently r goes to infinity vhen the point (z9y 9z) goes to infinity. As T (<$%y 9{f ) is independent of (z9y 9z)9 differentiation under the integral sign shows that the displacements will vanish as r~* and the stresses as r’’5 9 when the point (z9y 9z) goes to infinity (r.30v p.28). As this satisfies the boundary conditions for an infinite solid9 the solution in this case is represented by (4). Discontinuous temperature distribution. The general equations (2) imply the ezlstenoe of the first derivatives of the temperature9 and the validity of any solution at a surface S of temperature discontinuity therefore requires examination. Consider first the displacement potential the infinite solid9 equation (4). of Let the subscript i refer to the limit of a funotion when the surfaoe S is approached from its interior9 and the subscript (15) • refer to the limit when s ie approaohed from it* exterior* The temperature discontinuity on s may then he written T|-Tt , whioh is a funtion of position on 8* At any point F on S take right handed reetangular coordinates $jr suoh that ct lie in the tangent plane and it along the outward normal* The oondltlons that must be satisfied if (4) is to he valid, in spite of the temperature discontinuityt are that, on s, the displacement must he continuous and the surfaoe foroes must be continuous* These require that the first derivatives of ¥ must be continuous and the stress components 05,9 Zm* end Zp* must be continuous* Now 9 9 as given by (4)9 is the potential of a distribution of matter f •- 9 and it is shown in works on the theory of the potential that a distribution f 9 whioh is bounded and lntegrable9 has a potential suoh that its first derivatives are continuous everywhere9 Including surfaces there f is discontinuous, and its second derivatives are continuous exoept at surfaces where f is discontinuous (r*30, paragraphs 24,25,29,61,73,91,92,93,94)* In a coordinate system or,^ ,^ , as defined above, all second derivatives are continuous on a surfaoe where f Is discontinuous, exoept * The discontinuity ** of this last second derivative is given by the equation (r.30, p«175) ( & ) The displacements (first derivatives of ^ ) derived from (4) will therefore be continuous* From (la) the normal stress components in the new coordinate system will be etc.* The stress components <7^ and <7^ are not continuous* By means of (lb) the discontinuity may be written as °*f" <r^ ‘ ^ 6 [ (4c^Ji ~ ( l ^ ) e ~ f t “ ft 'T«)] . As ~ is continuous, ( 0 ) . - ( 4 - f j ^ o , and Similarly the same expression is obtained for the discontinuity of (fy • The stress component ls9 howevert oontinuous* 9 normal to the surface S, From (lb) (17) Also and consequently <r*< - * 0* tinuous on the surfaoe d* The shear stresses on the That is, <Tr is con surface s are immediately seen to be oontinuous, and the solution (4) is therofore valid when T has surface discontinuities* It follows that any component of normal stress, noting on an element of area normal to, and at, the surfaoe of discontinuity is discontinuous by an amount while the component of noxmal stress acting on an element of the surface s Is continuous. All these conclusions remain true even when the solid has finite boundaries* The boundary foroes, coresponding to (4), are to be cancelled by superpos ing an isothermal stress distribution having equal and opposite boundary values* But any distribution of boundary foroes, whether continuous or not, produces a stress that is oontinuous in the interior* The die* (IB) continuity of stress duo to a temperature dlaeontlsulty remains unaffooted9 although the absolute values of the stresses on either side of s will In general be modified* It Is not neoessary that S lie entirely within the boundary* It may out It in one or more ourves* then It doest the complementary surfaoe foroes on the boun dary are discontinuous at these ourves• When it does not| these foroes are oontinuous all over the boundary* This is part of the general theory developed by Professor j.N.Goodler (r*19), and the original part of this thesis oonslsts of its application to two new problems* In these problems the temperature Is con sidered equal to zero throughout the infinite solid exoept within a oertaln region* The temperature in this region is uniform* but higher than in the rest of the solid* There is then a temperature discontin uity at the boundary of the hot region* In the first problem the hot region is an ellipsoid of revolution* In the seoond problem the hot region is a semi-infinite oireular oyllnder, and here the effect of the sharp corners are particularly interesting* (19) Problem 1, Stresses aat up in an Iaotroplo Infinite solid when a part of it, bounded by an Ellipsoid of Revolution, la at a higher uniform temperature than the rest of the solid* Let the ellipsoid be formed by rotating an ellipse, with semi-axes a and o, about the z axis* The equation for the ellipsoid of revolution then becomes c* • In the oold exterior of the infinite solid it is shown under "General Theory" that the stresses are found immediately from the potential af an ellipsoid of uniform mass density p a -J ± !L £ il T h* 41T . The temperature in the ellipsoid is considered equal to T and in the oold exterior equal to zero* The normal stresses are given by equations (la), and as T * 0 for the oold exterior, the stresses in this part of the infinite solid will be to ) t*' The shear stresses are T = -L - — 11 " !♦*/ r - J - lll L»* “ 14* JgiZ , . First oonslder tha aasa of an Oblato Spharold. *>o, for whioh tha potantlal la (r.30, p.68) and k is determined as heing the positive root in the equation (£1) ***** . j1 *ci«7 f | | By introducing tha eooentrlolty e« V '£ C " » o oan ha eliminated* This gives o*» a*(l-a*) and a*-o*« 0*0*f where e la a dlmenslonless quantity. Also put y** **• r* and (7) gives, by solving for *, (a) c * * x = j [ r a- * * * * * e b *)*+ 4 e2a * z 2 ] In differentiating (6) with respeot to x, y and z, X oan be treated as a oonstant beoause gjj- * 0 (r.30, p.63). This is easily verified when it is reoalled that from (7) ***** C*+K z ± _ *I. U0 whioh is a faotor in the above derivative. The partial derivatives of (6) then beoome 4 * , <1Trailhe* f 'jc**X 1Z e» L *•** I r!rt~f a t at *,n J* , 4V M A i r f a f c P t I r?r,-< a e 4z e* L«« 1„ 1~ » ) ia**x P I k J la different lating equations (9), however, X oust be treated as a funotion of (x,y,z), whioh funotion la Immediately given by either of equations (8), The second partial derivatives of (6) become d*v £vri>4i/i-e* Jk* e* • -L. la*+x a* 4 *y ■ £WPaJi-ei \ y/c*+K L > y* ’ e* L a * + x ~ ae _ * ir w E P fz 4z*~ In whioh e* eiS 'ae ♦ ***** J* 7 \fa*+x (am+*} \/c*4>x d* «*, in- t a e . tfr*7x z . ela * y J* ] (ox* * H c * t x 4<i J, e*a*z 4*1 f i X S i F * ! * ' * J» * - f i - 4% L dx _ r». rV * * ‘ h e*4*)*+ 4e*a*z* J ' /■*♦«*«* _ i_ . Jl l1 h r ± ' W . 4 ' & ? r ’ and d*V tTTPat \/I-ei V . £ 5w * tdz rf*y fi. r z-e *a * g jffa H h ^ _ f , A___ r ‘ * e*a* ' (a***)*\fc*** L' \i(r*-e»a*),*4etaizt tir r ^ /F e * f , r dydz~ (a*+x)*)/c*-tx L r * f e *a * y fr^ ^ *a *)*+ 4 e *a *z * ~|... 1„ , J ' 1 .. - By substituting for f aooording to (e)9 In equations (10) and (11) the seoond partial derlvatlYes of will be obtained. Then(5) will yield the following expressions for the stresses in the infinite solid outside of the hot ellipsoid (64) tr -* W * I'V #r. a\fbt* [<fc**K g P ~ l**+ x f"T tfl/Te* * ” i-v i ri„-i ae . ***** ** I(5*7* ( a * * * ) * V c ‘ *M cbl I S *1 I (<**■*k )\/c M*it Jy J / . ^ - I ge . L a *** _ Va*4k -ft1w «yg c**~T7~~ZF~L7e * 2 1Z*T* W T * e *a *9 . e *a *z a-i rf>| <**♦*)£'♦*)**i\ and E *T s r _ a * / h F _ f,. /■*-*' h* 2(a*+x) xF7c*n<L ^(r*- e*a*)*+ 4e*a*Z * F*T LM - ~ X - 1*1 a V f-e * ]x</ f > .______ r * *+e*a* e *a * 1 .,, \j(r*.e *a *)**+ e *a *z *J 2(S** * ! * & * * L' ^/Te5 f. r**e *a * f [ l 9) _ 1 „, Nov consider the case of a Prolate Spheroid, o>a, for vhloh the potential Is (r«30, p.63) #§<*vi ‘#£1 This time a will be eliminated by bringing in the eooentrlolty e« - - Lt-- ^ c t o*-a*« e*o* and a2= o*(l-e*)» (25) The formulas for a*+x and o* + k then take the fora x = j [ r l - el c2+ \J(r*+e*c*)e- 4 e * c * z * J (U) c z* ) k- j[ r * + e * c * + { ( r * 7 e ° P ) ^ - 4 e * c * P J Also <&. s [ , . r * t e Mc M ]r L /(#•*+ e3^ ) * - 4 f * c V ‘ J ’ — -/"/+ * " L r £+ e * c * ~| . *>* r ,. r*-e V 1^ L‘ )((r*+e£c * )* -4 e * c *Z g J The normal stresses at exterior points are rr ~-i*l *“ >-•» <r - d L £ ! 2 f - L -I'rh** e c _ >£*♦* * « * C * x £ * ] Lee V^*+K c r/-« *;f I .j-L-1ec /-.> *e4 Lecswh& I ^ t r - - f * T cO~e,) [ * £ *' * e r l \ ( F Z r “ «c «a+K y/c^K . «>»1 I /.,% a *>* * < * • * & & . * «tyj f U 5) ec e*c*y e *c *z ^*1 («*+*)(c*+*)* and the shear stresses are I - -M c •((•**) f,. r**e *c* 1 *» i-»» < ^ k j 1 r c * » K I )J(rt te *c */i - 4e'ctz t J » - t«r fV /-**) **' r -_gw cVz-eff f.. r*-e*c* i... L hr^e'c*)*- 4e*c*zal r,. i-v 2 ( a * + * ) * \ f F Z i l ' « r*-e*c* \fr r * * e W -4 e 1 e fr l All stresses considered above are for the oold exterior of the infinite solid* The stresses inside the hot ellipsoid of revolution oan easily be found from the potential at Interior points of an ellipsoid of uniform mass density* This is done on pages 50-52, and all interior stresses are shown to be constant for any one ellipsoid of revolution* The most interesting stresses in the cold ex terior are the stresses along the axis of the ellipsoid and along a line, through the centroid, perpendicular to the axis* These stresses, along the z and x axes respectively, have been calculated for different radii of ourvature at the ends of the semi-axes of the ellipse Xa 2* I The results are plotted in figure 1, page 49* Appendix to Problem 1* A* Stresses along the x axis. Put x * an* Then, for an Oblate Spheroid* € t*+ x *a *n * # s* c** jc= a*(n*~ e * \ * un > A, Jz ° From (12) the normal stresses become <T = B* T f 1 s in '(e \ 1- 1/ [< >fi) na ?/<-«* <2 e 4 rt- - *«»T f • f . 6 '/r7*-e4 ) VT-e4 °i - — [ e s 'rt 7T TP- zr . ««T f Cr* a — L / t f ^ I - T «7 4 l-e * s,n n j — The radius of curvature at the end of the semi-axis a9 of the ellipse v2 .2 — 1*— = I ** c* 9 is f*a(l-e*)f giving e*= ! " • For an Indefinitely flat ellipsoidt .p*0f e * 1# all stresses are immediately seen to be zero for all (sa) values of n ozoept n»l, when <T - - (T - O (T s MSS. For a sphere, p* a 9 e*0, the terms in equations (17) will be expressed in the form of a series* nils gives , -i« * i && $to zr s it + 7“— i * terms of higher order ft n 6 n* s terms of higher order * n ( h j* * t0rms of higher order)* Substituting these expressions In (17)f multiplying out the equations, and then dropping terms of higher order In e, the following formulas for normal stress will be found <rx = Q- _ 2 EotT 3 n * 1- 1/ / feT *~3n* I (T = * iSil J/;* /-* . These agree with the known results, (22), page 35, / ( f8 ) at p For a Prolate spheroid put cr» af*#» a#n* and o*f* » s then y ~ j )• From (15) the normal stresses beeome e« fr t fa T r * *'17 * -« * -» > e ® n* r — _ f « T r / ' ' * * * n* •»-— 1— / n//-e*j2e* - « 5-------- — t e * « 1 I 5"* ov = J t t f i ' - W ' - T i . Also /* 7 i . '-«* rtit-e *. 14 1 4 J e* * which gives e*« For a spheret /» a, e* 0, the terms in equations (19) may be expressed in the form of the following series g £ ~ h - e * + 4 / e3 7 n > o -e * )& = if/1** = / - ^ e * f Substitution in (19) yields + . ' .............. ' which oheoks with (18) • For an indefinitely thin ellipsoid« p = * o » eel. In equations (19) it is then necessary to find the Uniting value of as e approaches 1* This can be written in the form of a fraotlon as I /-«* the numerator and denominator of which both approaoh Infinity when e approaches 1« Considering the numer ator and denominator of the above fraotlon as funotlons of e, it is shown in books on the calculus that the Uniting value of the fraotlon will be the same as that of another fraotlon whose numerator and denomi nator have been derived from those of the original fraotlon by differentiation with respeot to e. (91) Differentiating both numerator and denominator with reapeot to e glrea I 11( b p * V/-e* i ^ r i nll- ‘) . ,-e* This is easily seen to approaoh zero ae e approaohes 1, Then Llm(l-» e*)sinh * ■ ulti s 0 e+l nil-e* and equations (19) immediately give —. x Bo I E ar 'I7Ta ~ rr , I E ar ^ ’ Z J ^ T T , ~ _ G i-O . Stresses along the z axls> Put z » on* Then, for an Oblate Spheroid, put j a » ■~ -y • and obtain 1-9* **+* s £*(”*+jz§k), i t H Sy =. 0 * c*+x = c*n*t * £ .S £ cn . ( ft From (12) the normal stresses become The radius of curvature at the end of the sen! axle o9 of the ellipse X* , z * _ » a* * c* " 1 la / = , giTing e*» 1-y>. For an Indefinitely flat ellipsoid* p = c o 9 e » l # all stresses are Immediately seen to he aero* For a Prolate Spheroid, *C — ** ■- — >= 0 *9 •. > C*+ k s c*n* > 4Z s 2CR TToa (15) the normal stresses became (as) <Tu* <rt Uf) i - ^ e* Also f t o d - e 1) and e1* l-~* For an infinitely thin ellipsoid, f * 0 9 e*l, all etreaeea are seen to be zero ezoept for n > l f when (T S(T - J. I S il 2 1-* , For a sphere, f * o 9 e« 0, the stresses for both the oblate and the prolate spheroid are found to oheok with the ones previously found along the z axis, equations (18). However 9 a wholly Independent oheok on the validity of the above developed formulas oan be had by oaloulatlng the radial and tangential stresses, for the spherloal oase, by means of the ordinary theory of elasticity. The radial and tangential stresses at the surface (84) Of a sphereI of radius 0| duo to on external pressure P O N (*»M» P«9*5)§ At the surface of o spherioal oavity, of radius a, lnoldo on lnflnlto solid, tho corresponding strossos duo to on Internal pressure p are If the sphere fitted snugly In the spherical cavity at temperature zero the pressure p will be due solely to the uniform heating of the sphere and the tangential strains of the two surfaoes considered ■not differ by an amount ce T* For the surface of the sphere this tangential strain is and for tho spherical cavity it is Dlls gives (35) Oil the outside of the oomon spherioal surface , then t r * - -S- SSS ^ <T = - Ssu 3H, * J I-* and In the Infinite solid, at a distance from the oenter of the sphere of n times the radius, the stresses w l U be These stresses oheok with (18) when It Is remembered that In (18) <TX corresponds to the radial stress and both dj and OJ correspond to the tangential stress. A further oheok on the calculations for exterior points can be obtained from (3) and (5) when It Is remembered that TsO. (T* ♦ ( f y ♦ This gives - Oj that Is Immediately seen to oheok with equations (19) and (20) and Is a very useful check on all numerical ealdilations. (36) Numerical Calculations for Problem 1. A. Stresses along the i axis. In the following calculations the radius of curvature f refers to the end of the semi-axis a of the ellipse a * and e is the eccentricity of the same ellipse. When f is smaller than a, equations (17)t fbr an oblate spheroid,are referred to, and when p is larger than a, reference is made to equations (19), for a prolate spheroid. n is the distance from the oenter of the ellipsoid in terms of the length of the semi-axis a. p« .la e*= .9 1 2 n J. n 3 4 -I—e sin e = .949 4 £.(») 5 n*+ e* 6 n*-e* 1 •949 1.250 1.318 1.9 •1 1.1 •864 1.043 1.100 2.11 •31 1.25 .759 .862 .908 2.462 .662 1.6 •655 .685 .721 3.15 1.35 2 .475 .495 .521 4.9 3.1 (37) 7 {w Q 9 n*-(7) (5) (8) 10 (4)-(9) — 11 12 Ox (7) n* .316 •316 6.01 -4.69 -.823 .361 .557 .674 3.13 -2.03 -.356 .460 •814 1.270 1.937 -1.029 -.180 •522 1.161 2.61 1.205 -.484 -.085 •516 1.760 7.04 .697 -.176 -.031 .440 13 14 (4)-<12) E«T ° * 15 1 (7) 16 17 18 n* (15)—(4) .645 1 .697 .245 1.21 1.229 .321 •113 1.562 •036 .861 •140 •049 2.25 .014 .568 .047 .016 4 1.002 .176 3.16 .640 .112 1.797 .386 .068 .205 .081 1.84 (38) P*#85a e*s *75 e « *866 1 1 11 2 n 3 4 -l c sin | £•<») 5 6 na* e* al n •o 1.75 •25 •81 1 •866 1.047 1.25 •695 •765 •883 2.31 1.5 •577 •615 .710 3.00 1*5 2 .483 •448 .518 4.75 3.25 8 9 10 11 12 7 (7) (S) is) •5 •5 •90 1.405 1.645 1.224 2,75 1.804 7.23 13 14 LJL ( f (4)-(12] E«7UU 3.5 1*21 (4)-(9) h !L (y BUT u * -2.29 (7) n* -.763 •5 -.762 -.254 .577 1.090 -.380 -.127 .544 .657 -.139 -.046 .451 16 17 15 1 (7) 18 na (15)-(4) i^T °Z .71 .237 2.000 .790 .527 1 .306 .102 1.111 .228 •152 1.56 .166 .055 .816 .106 .071 2.25 .067 •022 .554 .036 •024 4 (39) Q» »8a eas .5 1 2 n n 3 sln-'f • s .707 4 i'W 5 n ▼ e 6 n e 1 •707 •785 1.110 1.5 1*25 .565 •600 ••649 2.062 1.062 1.5 .471 •490 •694 2.75 1.75 2 .353 •361 •511 4.5 3.5 8 9 10 11 12 7 fit) (4)—(9) n**(7) •707 .707 1.030 1.610 1.322 2.12 -1.01 4T«T *5 •5 (7^ /t* -.714 .707 1.280 -.431 -.305 .659 2.97 .926 -.232 -.164 •583 1.870 7.48 •602 -.091 -.064 .467 13 14 16 17 (4)-(12) ilH <7£*7 15 / (V (15) — (4) <r 18 n* •403 .285 1.414 .304 .430 1 •190 .134 .971 •122 .173 1.562 •106 •075 .757 .063 .089 2.25 •044 •031 .535 .024 .034 4 (40) e * r .5 Prn 2* 1 2 n e* n* 3 e » .7 0 7 4 l-e*-(2) l-e**U) 5 ^(4) 6 n<(5) 1 •500 .000 1.000 1.000 1.000 1*25 •320 .180 •820 •907 1.134 1.5 •222 .278 .722 •850 1.275 2 •125 .375 •625 •790 1.580 7 8 9 10 11 (3) (6) e slnh f(8) n ft-e * 12 (7)-{10) £«T *** * .000 1.000 •882 •623 -.623 -.623 •159 •800 .733 •518 -•359 -•359 •218 .667 •625 •442 -•224 -•224 •237 •500 •481 •340 -.103 -.103 13 14 15 16 (S) n (13)-(10) iz ± * 17 18 (10)-(16) (6) 1.000 .377 .377 •500 •123 •246 •750 •207 •207 •441 .077 •154 •567 •125 •125 •392 •050 •100 •595 •055 •055 •317 •023 •046 (41) .« 4* 8*. .75 1 2 n 6* H* 3 e = .866 4 5 l-e*-(2) l-e*t (2) m 6 n*(5) 1 .750 -.500 1.000 1.000 1.000 1.25 .480 -.230 .730 •855 1.070 1.5 .535 -.083 •583 .763 1.145 2 .188 •062 •438 .662 1.324 7 8 9 10 11 (3) (0) e 12 (7)—(10) Id L r r slnh f(8) £«T * -.500 1.732 1.317 •381 -.881 -.587 -.215 1.385 1.129 •326 -.541 -.361 -.072 1.155 .987 .287 -.359 -.239 .047 •866 .783 .227 -.180 -.120 13 14 15 16 (£> if (13M10) t L fr f*r v* /-•« (*) 17 18 (10)—(16) h ± r r 1*000 *019 •413 •250 •131 •175 •500 *807 *286 •233 •093 •124 •500 *221 *147 •219 •066 •091 #801 *104 *009 •189 •038 •051 (40 p » 10a e * = .9 1 2 n e* n* 3 « 4 b . 949 5 1—8^—(2) l-e*+(2) V u T 6 n*(5) 1 .900 -.800 1.000 1.000 1.000 1.25 •576 -.476 .676 •822 1.026 1.5 •400 -•300 •500 •707 1.060 2 •225 -•125 •325 •570 1.140 10 11 7 W (6) 8 e 9 slnh '(8) ^ 4 9 ) 12 (7)-(10) I S -.800 3.000 1.819 •192 -.992 -.551 -.464 2.400 1.609 .170 -.634 -.352 -.283 2.000 1.444 •152 -.435 -.242 -•110 1.500 1.195 •126 -.236 -.131 13 14 15 (£) n <TL (13M10) — SctT 7 16 (6) 17 18 (lo)-(ie) ilt i r £«T z 1.000 •808 .449 •100 •092 .102 •657 •487 .270 •097 .073 •081 •472 •320 .178 •093 •059 •066 •285 •159 •088 .088 •038 •042 (48) Tho maximum shear stress along the x axis Is determined by the formula ^ T*Mtfff * J { p n * m x " Grnirt) # 0g and 0J are the prinolple stresses along the As x axis the maximum shear stress is found by subtract ing the smallest of these normal stresses from the largest one* The result, for different values of n and f , are shown In the following table. fm 0 n p * ola f s .25a P * .5a .734 .645 .572 1*85 0 .147 •203 •239 1.5 0 .067 •099 •127 8 0 .084 •035 •049 /■ In r*4 . fa 10* • 1 ■ 1 •to o •M O •M O •800 •M O .8 8 4 •M B •BOO •811 •810 •1 4 . < in •IB B •H O •888 •M l •O ff •OBB •110 •188 All n l m to «!>• i M n tahl. «r» to b« Multiplied by t o o b ta in th o M t W i i l m r t t r o o s . (44) B. Stresses along the z axis. In the following calculations the radius of ourrature p refers to the end of the semi-axis o of the ellipse 1 « . f i. 3 a * * c* I • Alien p Is larger than o reference Is made to equations (£0), for an oblate spheroid, and when p is smaller than of equations (£1), for a prolate spheroid, are referred to. n Is now the dlstanoe from the center of the ellipsoid In terms of the length of the semi-axis o* For p *00 all stresses are zero along the z axis* p « 10c e*« *9 1 2 n 1-e1* i^»* 5 es *949 4 n» (5) 5 e (4) 6 sln"*(5) 1.000 1*000 1.000 •949 1*250 1*2$ .676 •822 1*026 .925 1.181 1*0 •500 .707 1*060 *895 1*108 2 •525 .570 1*140 .855 .985 1 (45) 7 8 9 n* (2 ) /-e * w 10 (7 )-(9 ) 11 £<*r * 12 *■«/ — fir * .417 1.000 •100 .317 .176 - .3 5 2 •392 • 845 .11 8 • 276 •153 - .3 0 6 .369 .750 •133 •236 •131 - .2 6 2 •328 • 650 .154 .174 •097 - .1 9 4 P * 40 e2 = *75 e « .866 1 2 3 4 n l-e% —* r»a VTil n«(3) 5 e (4) 6 sin^iS) 1.000 1.000 1.000 .866 1.047 1*25 .730 .855 1.070 •809 •942 1.5 •583 .763 1.145 .756 .857 2 .438 .662 1.324 .654 .713 8 9 11 12 n « (2 ) i-e * IS ) 1 7 S .I.) 10 (7)-(9) S 7 ,r- •605 1.000 • 250 .355 .237 -.4 7 4 •545 • 912 .274 .271 .180 - .3 6 0 •496 .875 .286 .210 •140 - .2 8 0 •412 .876 .28 5 .127 .085 -.1 9 0 (46) P» e*= .5 80 1 n 2 1-e** e s .707 3 4 VT2) n*(3) 5 e (4) 6 8liT*(5) 1,000 1.000 1.000 .707 .765 1*25 •820 •907 1.154 .622 .671 1.5 •722 •850 1.275 •552 •586 2 •625 .790 1.580 .447 •462 7 8 9 11 12 £ ^ ( 6) n*(2) hJL1 (*) 1 10 (7)-(9) — <T. L i a~ £«T .785 1.000 .500 •285 .285 -.570 .671 1.025 .488 •183 .183 -.366 .586 1.082 .462 .124 •124 -.248 •463 1.250 •400 .063 .063 -.126 p » .5 0 • « .707 e*s .5 1 2 3 4 5 6 n n* (2)-e* (1) (1) \[(3) C (57 1 1.000 .500 2.000 .707 1.000 1*25 1.562 1.062 1.176 1.030 .686 1#5 2.250 1.750 .857 1.322 .534 2 4.000 3.500 1.870 .373 .571 (47) 7 slnh J(6) 8 9 ±*(7) (4)-(8) 10 11 — 12 n £ *T ** •882 1.248 .752 .376 -.752 1 •641 .907 .269 •135 -.269 1.25 •511 .723 .134 •067 -.134 1.5 .370 .524 .047 •024 -.047 2 JE>»_j »25o © s #75 1 2 3 n n* (2)-e* 6 s #866 4 5 6 e (S) (1) (3) 1 1.000 •250 4.000 .500 1.731 1.1 1.210 •460 2.390 •679 1.275 1.25 1.562 •812 1.540 .901 .960 1.5 2.250 1.500 1.000 1.224 .707 2 4.000 3.250 •615 1.804 .480 7 8 9 (4)-(8) slnh /(6) 11 10 f«r 2 12 n 1*316 1.520 2.480 .413 -.825 1 1*063 1.228 1.162 •194 -.387 1.1 *863 .985 .555 .092 -.184 1.25 *658 •761 .239 •040 -.080 1.5 *463 *535 .080 .013 -.027 2 (48) P» »lo e*s .9 1 2 3 n n* (2)-e* e s .949 4 5 ill (3) /(ST 6 € (S I 1 1.000 •100 10.000 •316 3.000 1.1 1.210 •310 3.550 •557 1,704 1*25 1.562 •662 1.888 •814 1.234 1.5 2.250 1.350 1.111 1«161 •817 2 4.000 3.100 .645 1.760 .549 7 8 10 9 (4)-(8) slnh *(6) 11 /T- £«T 12 n 1.819 1.915 8.085 •448 ••898 1 1.303 1.373 2.177 •121 ••242 1.1 1.038 1.094 .794 .043 ••088 1.25 .746 .786 •325 .018 -.036 1.5 •525 .553 .092 •005 ••010 2 The following table gives the maximum shear stress# n 100 fm 4C ps2o pm 0 .5c .25c •lc 0 1 .264 •356 .428 .500 •564 •619 •673 .75 1.25 .230 .270 .275 .256 .202 .138 •066 0 1.5 •197 •210 •186 .148 •101 •060 •027 0 2 •146 •138 .095 .062 •036 •020 •008 0 (49) Stresses along XAxis* .6 CTX .4 P*44 2 I 1.8 1.6 1.4 U 1.2 6. 4 * 4 P*4 ~ 1.6 1.6 I ' * *1 j t ■P« 44 ^ max. rt t r----Z A 1.4 A|k O ims .-j XA kis Radius of Cmvotuf, Stresses <aton4 Z Axis SSI I-* O 2 14 1.6 1.4 14 14 119J- 14 1.6 LS I PMN 2 (50) Stresses at Interior Points* The potential of a homogeneous solid ellipsoid for interior points is (r*S09 p.49) a** s b +s %j <*s /(a*« s)(b** s )(c** s ) U i ) in which a 9 b and o are the semi-axes of the ellipsoid* Equation (23) is seen to he the same as the equation for the potential at exterior points (r*309 p*56) exoept that the lower limit is zero Instead of Jc • The normal stresses are then immediately derived from (12) and (15) by putting and subtracting y according to (la)* For an Oblate Spheroid these stresses beoome <ry S <rM (24) (51) and for a Prolate Spheroid ' e i | i-e *l j l-e * ~ i - e * J 2 e * l ^ 5 = (Tx (ssi It Is Interesting to note that the normal stresses are all oonstant and independent of the size of the ellipsoid* These stresses do9 however, depend upon the shape of the ellipsoid of revolution, in whioh the semi-axes a and b are equal* The shear stresses are immediately seen to be zero, as the first derivatives of the potetlal in this ease are functions of one variable only, see (9), page 22* By putting e = 0 in equations (24) and (25) the normal stresses inside a hot sphere are determined* In both oases it is easily verified that the stresses are 0^ s 0^ fhloh oheoks with what was previously found in this oast, see page 34• (52) For an Indefinitely flat ellipsoid, e= 1, equations (84) Immediately yield O .’ ■ <r, * - ° i ‘ °, , whioh oheoks with what was previously found on pages 28 and 32. For an Indefinitely thin ellipsoid, e»l, and from page 51 Lim(l- e2 )sinh *= = *+• \[h e* =0* Then equations (25) give _ I E<*T ^.-oi’- F — , £<xT _ < 5 - — , which oheoks with what was previously found on pages 51 and 33• A check on the formulas from (3) and (la). (24) and (25) canbe had If the three normal stresses are added together and (la) eliminated by means of (3) $ the following result is obtained ^ _ o E<XT (Tx + (Ty + <T2 = ~ c 777 - If the three normal stresses from either (24) or (25) are added together it is easily verified that the result oheoks with (26)• (2 6 ) (53) Remarks on Problem 1, The maximum stress that occurs anywhere is -65LE i-\/ i and this stress occurs only when the ellipsoid is Indefinitely flat (oblate spheroid) or indefinitely thin (prolate spheroid). In these two cases the e^yy maximum stress inside the hot ellipsoid is j everywhere, but in the cold exterior it reaches this value only at points of the surface of the ellipsoid where p is zero, and the stress then immediately drops to zero outside of these isolated points. To keep the boundaries of the ellipsoid finite, it is assumed that a beoomes zero for an indefinitely thin ellipsoid and that o beoomes zero for an indefin itely flat ellipsoid. At points where p is zero infinite stress might be expeoted, but as this only ooours when the ellipsoid is either indefinitely flat (o = 0 ) or Indefinitely thin (a m 0 ) the amount of material that tries to expand in these oases approaches zero, and the result Is a finite stress of focJ- as previously i*t/ OSjUlMd* N r ■ sphere the maximum stress Is j -J7 7 both ioslde M M outside of the hot reclon. (M) II It tl« 0 U N w i l i k r to not* that ror i prolot* •yfceroid Ibo i i l w o oboor stress at the and of tna • w l < u 1 i a lo eonotont B rorardless of the ?alua of p at tblo point. for othor potato of tha x axis, however, tblo otrooo differs with f , being largest for an indefinitely thin allipoold ( p * * o ) and soallest for o opbero ( p « a). In general, ao la seen from figure 1, page 49. the otreas In the oold exterior is a maximum at the ourfbee of the hot ellipsoid of revolution and falls off rather rapidly for points farther out in the exteriorj according to Saint-Venant*s principle. If the indefinitely thin ellipsoid is considered to have a finite thickness ( a finite), but Infinite length (o Infinite), the stresses along the x axis will he the same as for an infinite oircular cylinder. This case oan be oheoked by the* ordinary theory of alastloity, and this is done in problem 2 as a check an the oaloulatlons for the semi-infinite oircular cylinder. The results oheok with those on page 31. (5b) Not all the numerical calculations are plotted In figure 1* However, enough curves have been plotted to Indicate the trend of the stress variation as p varies from zero to infinity, and any more curves would olutter the figure up so as to reduce its usefulness. It might have been slightly preferable to use cylindrical coordinates (r,d,z) instead of the rectangular coordinates used* To obtain the stresses in these coordinates it is only neoessary to put x= r and y = 0 in the expressions for stress* <T# would then become and (Ty would become <TJ . r„z would become 7>z and rrt and Tj* would be zero, where t is the direction tangent to the olrole r equal to a constant* (56) Problem 2 * Stresses set up In an Isotropic Infinite solid when part of it, bounded by a Semi-Infinite Circular Cylinder, is at a higher uniform temperature than the rest of the solid* As far as the author has been able to asoertaln the potential of a semi-infinite circular cylinder is not known. However, it is the seoond derivatives of this potential that are required in order to find the stress9 and the determination of these second derivatives constitutes the major part of this problem* The first derivatives of the potential of a homo geneous body can be found in terms of a surface potential by means of Gauss* theorem (r*31, p*Q), see also (r*SO, p.110). Then the attraction at a point F of a solid body of any form, when resolved parallel to any straight line, (taken as the axis of x )9 is given as Ui) Where f is the constant density of the body, r the distance from P to any element da/ of the area of the surface and $ is the angle the normal at dia, drawn Inwards* makes with the positive direction of x* (57) This Is true both for external and Internal points. The ooordlnate system is now located in such a manner that the origin is at the center of the base olrole^and the positive z axis forms the axis of the cylinder. The base of the semi-infinite cylinder, then, lies in the xy plane. For the attraction in the z direction the angle # is zero for all elements of the base of the oyllnder and 90° for all elements of the cylindrical surface. The attraction in this direction, then, is equal to which is the potential of the surface of the base of the cylinder with a mass density p per unit of area. This result could also have been obtained directly by considering the difference between the potentials of two homogeneous cylinders a distance dz apart. The difference in these potentials is obviously equal to the potential of a circular disc at the base of the oyllnder and with a surface density of pdz. Dividing this by dz the above result for the attraction is obtain#** (58) The potential of a circular disc of radius a and mass density p at per unit of area Is (r*21,p*398) i j 1' ^ ~ T where the dlso lies In the zy plane with center at the origin* and Thenr is defined by the equation r** x*4 y* is the algebraically larger (positive) root of the equation -J± a **X X X I (i9) ' ' ' the smaller (negative) root of which is /Ij* Equation (29) gives (30) * also £[r‘tI*- a*-f(rl*z*-a*)z* 4a*z* ] ♦ f a r*4 a* and ^ s - a* z*, which gives (*-*.)(*-**) ■ r*-(r*+z*-a*)t The partial derivative of (28) with respect to - 2 Pa* J h - r - - -?-7 — (3l) is I----- Whioh Is Immediately seen to be zero, by means of (29). (59) The partial derivatives with respeot to x 9 y and z of (28) can therefore be obtained simply by differentiating under the integral sign with respect to these quantities. Doing this, and using (51), the following results are obtained i t _ M l ' !* ± a (T t-* Z'n i J _____ J t/ + jLf _ 211 * i a gu f°° iyii ~ I-* 2 ™ — “J .oo i t - « T „ &y f _______ d ± ______ in which equations p has been put equal to ~' /-o 4 Tf . Evaluation of the elliptic integrals in (52) and substitution into the formulas for stress (see appendix) yields 7 - ill a *X E ** ' C a '+ A jifP Z , /-W TT T - z**' TT \ (33 ) A-f (a *.At ) \ f t t t everywhere in the infinite solid, and (60) °*S w { k ^ n f ^ k n L ) *(t -KE)%]-k J for exterior points only. (34) For interior points it is only necessary to subtract from (34), according to (la), to obtain the normal stress in the direction of the axis of the cylinder. The functions entering into the above formulas for stress are defined in the following manner: (35) which quantity is always smaller them 1 , as smaller than • le Also k is always taken as positive. By means of (30) equation (35) can be written in the following two new forms k ________________________________________________( 3 f a ) ♦ ^(r*+2*-a*)*+ + a zz * and k 2 ra Then JT K . (* <** is the complete elliptic integral of the first kind, and (61) B » J |fhUsiit1? df la the complete elliptic integral of the second kind, With k* * \f 1 -k 2 the following quantities are defined ir K,a f * 4+ , y/-A '*sinef jr e B * * J ]} l-k '* S in * V d V o /' !. — SW f r*___________ _ E(T,k») = (f h df ■n(T,k')« sin 'f on(v,k*) * oo» /’ » / l-sn*(v,k') dn(v,k')« ^l-k'*sn*(T,k*) (62) The function dn(Tfk 9) is found from the following equation (36) a and from this all the other elliptic functions, entering into equation (34), can be found* From (35a) it is easily seen that k is the same for z«f m as for z> -m, which then also holds for the elliptic functions and integrals* It follows immediately that the two shear stresses (33) are symmetrical with respect to the xy plane and that the normal stress <JJ » according to (34), has equal, but opposite, stress for z > « m and z * -m* For the purpose of numerical calculation the equations for stress oan be greatly simplified by means of (30) and (35a and b), and also by putting r x pa and z s na • Then it is easily verified that rn __ p+ k ~ k (63) and (a****)\/aV3, s (ral^/ks . The two shear stresses (32) oan immediately be reduced to one by introducing cylindrical coordinates r, 0 ,z« Putting xa r * pa, y * 0 and z r na gives (K-e) r n ' 0 The elliptic functions entering into equation (34) can now be reduced to the following: A 1 i 3 on (y,kf)• 1-sn (vfk*)* k and finally /-k * P-Ar (64) dn(v,k*) '^sntVjk* )on(v,k') I p If (1 -pk) (p-k) IT^l Then equation (34) can be written (KiM*n.-KCf.)-K] (**•) It is desirable to eliminate n in equation (34a). This can be done by expressing n in terms of k and p by means of the equation k 9 - L £ p * + f r * + / - ^ ( p * + n * - if + 4 n e J (see 35b) solving for n gives n *t)j (+\) where the plus sign is to be used when z is positive and the minus sign when z is negative. The normal stress (J“L can then finally be written <5 . : ♦ i z] where the plus sign is to be used for positive z and «> (6b) the minus sign for negative z. Equation (42) is good for all external points except the z axis (p=0)* For internal points it is EotT only neoessary to subtract from the right hand side of equation (42) to obtain 0J . Along the z axis (see appendix) (43) where n is positive when z is positive and negative when z is negative, and (43) is good for both external and internal points* Radial and tangential stresses* The radial and tangential stresses can also be found by means of Gaussf theorem* The attraction of the semi-infinite cylinder in the radial direction is the same as the attraction in the x direction for a point in the xz plane, and is equal to the potential of a cylindrical shell with a surface distribution of matter of - p c o s 0 (see sketch)* This result could also have been obtained by considering the difference between the potentials of two semi- (66) Infinite homogeneous circular cylinders a dlstanoe dz appart, just as was done for the attraction in the z direotion. In order to find the radial to find the potential of the above cylindrical shell9 but f ^ v<\ — IILyc-k— k— I • i* - stress it is not neoessary ■— s. V / J only its attraction in the x direction upon a point in ■ the zz plane* This is done by splitting the surface up into semiinfinite rods of width add and mass per unit of length p cos* add • *' cost a d * x where dtl/i^fa2- 2axoosd (67) Substituting r for x gives 277 ^„ f r~ 4 Cos6 /, . ‘" ’I ' - . S - t . r c * . d*V 7P z \ < **> where r* in general, is equal to x^* y*. A*yj it is only necessary to turn the 91 To find -777 cylindrical shell 90 about its axis and find its attraction in the y direction upon a point in the xz plane* This gives (r.31,p.4) 27T Jp s~ (f+Si»c()sin'faJ* /ft where sin ^ ■ Substituting the previous values of d and sin of gives 271 j7 = Jt* a f t f in 0 — (l* Jo r +a -2arco»0 \ 1 \lr*+z 1 — \d s .2 *r c» » f Now put z * na and for r larger than a put r* pa, but for r smaller than a put a*p*r. (44) and (45) will reduce to the form Then equations (45) (68) iV £ ± - o + * f f e,J f z £ £ £ * l Jt p + 1 -2 pcose / / . ______ ( H \ ^ { P .n ^ - e p c ^ r in A_ r * * ' y)t p * + w > * \ ! /-£*s*0 n \ /„ (46) Zpc* 9 r ° when r is larger than a and to the form &ir c p ' p f cosf-pjcos** J P Z+ I- 2 p'cosi /, ( »P‘ ) ^ '}p 't+ » >P % h 2 p c o s » ) (■ » ) in 4 jt - - b '* p I * • - p rl !-*< > *** ( u P '^ - ip c o s * V ______ 2£|______ iFZmZpZZl when r Is smaller than a , Integrating (46) an (47) and proceeding in the same manner as with the equation for <rz $ (see appendix), the following results are obtained <rr m- — - r —* i v * n \( ( p * i) * * » * ) ( * ' £ ) (*•) Hr'**M)k} * fp.'frtotf-epr.glj (69) (49 ) tor r larger than a and, by putting pci-( P (4 S a ) ~ (p **2 p and (49‘) *<p- 0 ‘k ] - # [ * ( ( « ) - ■ & ' & ] } for r smaller than a. These are the stresses at exterior points« (70) For Interior points the stresses are (4 8 b ) -(r**ip-l)k\-Zpr[KCM')-*|j.Jj and (*n) ■ & [*" < *)-■ *? * & ] j In formulas (48) to (49b) the elliptic integrals are determined by (p+t)*+n* k.*. !- k *a < £ 0 f * «f (p *0 *+ n * £ «n (T,k«)« sin*/'« n The above stress formulas do not hold when p equals zero, or for points along the z axis. In this oase, however, it is quite simple to find the attractions of the cylindrical shell at points along Its axis* This gives (see appendix) for external points (n negative), and W - 7 & ( for internal points (n positive)* From symmetry it is Immediately seen that Tr ^ - 0 . The most Interesting stresses are those near the •ltd of the oyllnder, and particularly at the sharp (* ‘> (72) olroular edge. Here there la a discontinuity in the normal stresses and the shear stress becomes infinite, as shown on figures 2 to 5. Figure 2 shows the normal stresses along a radial line in the xy plane (just below the base of the oylln* der nsO-, and just above the base n=0+), along a radial line at a distance of 0.1a below the base ( m -0.1 ) and along a radial line at a distance of 0,1a above the base (n« 0.1 ). Figure s gives the normal stresses along a generator on the outside of the cylindrical surface (p*l+) and along a line parallel to the generator at a distance of 1.1a from the axis of the oyllnder (p»l.l). Figure 4 gives the normal stresses along a generator on the Inside of the cylindrical surfaoe (p»l-) and along the axis of the cylinder. Figure 5 gives the shear stress along a radial line in the xy plane and along a generator. With reference to figures 2 to 5, a is the axis of the cylinder, the longitudinal distance z from the base of the oyllnder is put equal to na and the radial distance r from the axis of the cylinder is put equal to pa# Appendix to Problem 2 A« Evaluation of the elliptic Integrals» To evaluate the elliptic Integrals entering Into equations (32), page 59, consider first the Integral 40 I ( &) i 't w + J X * - * # * - * . ) In vhioh A 24 04 * , 4 ^ 4 Nov put (Si) a *+ * vhere 0 < W4 I Then dw. or» dV'= - 2 ^ (o *Hr) also, from (52), dw* w (74) , * **Mi where k * ■ r-- (34 -) and 0 4 k 4 1 • Expression (51) oan then be written w cfw____ / ( V M fa**A, J. NOW put Is sinf , whioh gives dw« cos / d/ and If the Integral In (Sla) is put equal to ut then fw Z I - 4 ” , '9 i(h w * )(l-k * w*) f r* d * K fl-k *s m *Y there * » ' i m f • snu • l/ ■ » • • • f a f * </ a f b $ H * U m »/ 2 3 f **♦», m f/y**) (V$) <7b| •aft A t*fhk*sm*t * th v •\fl-k‘sn*u = | / p ^ F ( / » k ) I s tlM •lliptio Integral of the first kind. Then 00 du i* (ss) + *, and the first Integral in equations (32) becomes ’X ” <***•)* I * * udtt ^ a* u^, Is obtained by putting V' = />i in (54), making S 0 s S and </. = f - = ft share K Is the eomplete elliptic integral of the first kind* From (53) and (54q) it is immediately seen that an. d l 4f (? 7 ) (76) and an This gives f* I sn*u du s 1 ) where E s f ^l-k*sin*]f d ^ o and is called the complete elliptic integral of the second kind* Then (56) gives f ,('t* * * W ('t* 4 * )(* -U )(+ -X,) <a * + h ) \ ^ and the shear stresses (55) follow immediately from (52) and (58), and the formulas (d)9 page 12* From (52) and (54) a ,, a ***, - ^ s n ' u sn *u (77) Using (55) and the above equation, the expression under the Integral sign in the last of equations (52) oan be written sn*u n, Putting a **2 ,-a *s n *u 0 ^ — 7- 2 2 s k sn of gives (S9J a**?, *> r H / _________ _ 2 I k s n ^ s n ^ u d tt (6o) 2 sn« 0*Ha*+1kt CHd dHot 7 T (« ,< *) in Jaoobl's notation, where //(k 9ei) is the eomplete elliptic Integral of the third kind (r«339 P.420) and (r«349 pages 15 and 142)• 7Toy,A) * J, In general / - k sn « sn u where u Is defined by (53)* A1so9 see (r«339 p.421) f(K,«)* K2(«) d (78) S (6 /) where Z(u)*E(u) - — uf see (r*349 p«15)« From equations (35) and (59) is obtained (6Z) a *+ *a and it is easily seen that sn* * 4 JH (43) 2 as sn 0< is larger than l f oi must be a complex quantity, and due to the periodicity of sn u there are Infinitely many values of or that satisfy (62)• The perlodloity of sn(u9k) is (4K, 21K*)» see (r*329 p.35), and along a line K+iv the function sn(K+lv) varies from 1 to 4- when v varies from zero to K99 k (r#32, P«36)• But 9 from (63), 1 ^ sn «c and one particular value of e€ that satisfies (62) oan be obtained by putting o( s K* iv9 where O ^ v ^ K 9, and v is always real* Then, (r#33, p*422)9 (79) Z(K + l v ) * - [ Z(v ♦ K ’, lc*) + ^ K K 7 J and, by the addition theorem, Z( v * K ’f k*) « Z(v, k')f Z|K», M M - k ^ s n f v . k M s n f K S k M s n f v f But, from (r.34, p.15), Z(Kv,k')s O« Also, sn(K* ,k*) = sin = 1, and, see (r*32, p,33) sntv^KSk*) = £ hSv»k ') ctn(V'k') This gives Zlv+K1,**)- Z(v#k f)-k,2sn(v,k*) and finally, by means of (61), *j£]‘ Row, see (r.32, p.32), in* c»*4n* m sn(K+iv) dnfckj cn(K*'tv)dn(K*iv) ~ V * snW.V)cn(v,k') and equation (60) beoomes . (80) , f *—-_ [ f - \ * * i <»*+*, d n (*k ') [tfp /v L ) ft'*Mfoh')cnW)l ■j (6 ]-Kj From (5)t (32) and (64) equation (34) oan be written down at once. Evaluation of the Integrals appearing in equations (46) and (47)♦ Consider first r rw I tfcose- e.cos*e JB 0 p * + l- 2 p c o s e __/_r *ir I dcoso- * c a s 9 ^ ~ P *H J /-c ,c a s fi where c = * P p *+ l and is always smaller than 1 for p4 ^t for p equal to 1 # The above Integral oan be written oqual to 1 (81) and is equal to 2W For r larger than a d * p and e * 1, and for r smaller than a d « 1 and e * p *• In the xy plane, then, ^-7 7 = r* jr* = ^p* (f \>or r>*)• Us) (forr<a) Which is just half of the corresponding seoond deriva tives of the logarithmic potential for an infinite eireular cylinder, see (r*30, p*72)» The radial stress at points in the xy plane will then be <Tr -f— r* - 47^-j p * /-v (rya) (66) J- (r<a) (82) These stresses should be just half of those for z « whloh oheeks vlth what is later found for this ease* Now oonslder I (d c o s * - eco s*e)d * _ 2 J0 (p*H -2pco$6)fp*+nz+l-2pcos* f JJ^p*+rf+^2pm 0 7T (6 7) / (hC,cos*)Jp**ff+t^2pcot0 where P *HL 0 & * - £». = -C— Ci Zp Cf \ C» ' e a - b and f * &P *p p *+ / J ' f - </ *P lb* right hand side of equation (67) can now be written a (0 j f [(*£ *** gpeest ] (63) To integrate (68) put <Ps ■■ £ Then cos0 * 2sin -1 d0= - 2d* and the limits are 0 « O f and 0 * 7 T» 0. The first integral in (68) then becomes ft ~ 4 = — r ^ ftp + ! ) * + » * j where k*s (p+l)‘+ n e . r<#; ' (70) The integral in (69) is easily seen to be which is equal to j y [ ( ( P * 0 * + » * ) e ( K ‘ F ) + ((f>-0ee - 2 P < t ) K \ Using the same transformation on the last of the integrals in (68) gives (7/) (84) Og s with ZC' = !*c, * p (pH) this beoomes 2 £ _____ a ft e f(P*+l) f (P*0‘*)!(ph )*+ n* J, £ ' * /-.» (t- c&sin*</)ff-/'*sin*v The integral in (72) can be written Jr J K du J f-Cg£tra ti share fr * _ sn** = £ V ______ I- k * s n * « t $tr‘i = !£+?)*+»* k* (p + ,)* This last integral is easily Identified with Jaoobi9s elliptic integral of the third kind, and as 14 ^ ka it oan be solved in exactly the same manner as the Integral of the same type in equation (60). This gives ( du_ [l-Wu , dm*') fun../.•)-](£!!'i .ILL 1 M m W c W n *' **'* (85) With the values of the elllptlo functions given on pages 70 and 71, this becomes I ^ (»> By oombining (69), (71), (72) and (73) the expression (6 8 ) oan be written \(74) - [<A - H f c * * ^ ♦#] When r is larger than a d • pn and e= n. Then, by combining (65) and (74), 0 = - p { ir' -) Whenever a double sign ooours in any of these equations the upper sign is to be used for positive z, or n, end the lower sign for negative z9 or n* (7f) (06) When r is smaller than a dcn'p9 and a * n ,p ^ l vher* n*« np*# Then, (74) v TTv ZK In equation (76) 4 p - k*= P but, on putting p f« -L , this is seen to be identioal vlth equation (70), and (76) takes the font £ Jr* . r ; & I k c m i • - * & * & -i) ] ) Equation (77) is only good for r smaller than a* From equations (75) and (77) it is now easy to arrive at the expressions for radial stress, by means (77) of (Xa)9 when it is reoalled that T» 0 for exterior points. The tangential stresses are derived in exaotly the same manner from the lower equations in (46) and (47). Thus equations (48) to (49b) are obtained. B» Special oases. At an infinite distance above the plane of the base of the semi-infinite cylinder the stress distri bution will be the same as for an infinite oylinder. This stress distribution oan easily be oaloulated by means of the ordinary theory of elasticity, as it oan be considered one of plane strain (r.36). The boundary conditions at infinity will then only require that uniform pressure be applied to the ends of the infinite oylinder. The effect of removing this local pressure is negligible in the interior of the infinite solid. When a uniform pressure p aots on the surfaoe of an in finite cylindrical hole9 of radius a9 cut through an infinite solid, the stresses in the solid are (r#28,p.57), (88) Tbo elongation of the radius due to the pressure p will then be which for r s a becomes For a solid infinite oyllnder of radius a and outside pressure p the stresses are (r«£8, p# 56) (j't = <rt = - p . The radial dlsplaoement at the outside of the oyllnder then Is **)) = - f j M f ♦ •'‘S) . The axial strain Is j - f c - u ( < r r t <rt ) J = y (<rz +2>>p). If the pressure p is solely due to the temperature expansion of the hot cylinder (89) or The condition of plane strain In the oyllnder gives Summing from the last two equations gives » - 2p, whloh Then, in the infinite solid, far above the plane of the base of the semi-infinite circular oyllnder, the stresses are (76 ) <rt*o (80) outside the hot cylinder, and <T -*c er - — ISLL Qr r t -- - J (79) 1 /-I/ inside the hot oyllnder. From the symmetry of two seal* infinite oyllnders put end to end (see sketch) it is easily seen that for r > a the stresses <Tr and CTt are just one half as large for the semi- Infinite oyllnder as for the Infinite oyllnder, when nsO. For r < a the stresses fTr and <Tt for the infinite oyllnder will obviously be equal to the stun of the corresponding stresses just above and just below the xy plane for the semi-Infinite cylinder, as there is a discontinuity of stress in this case. The stress <JJ however, is continuous across the xy plane (except for p= 1 ), and it is easily seen to be (91) just one half as large for the semi-infinite ae for the Infinite cylinder everywhere in the xy plane9 exeept for p*l# As the stress for the infinite oyllnder is the same as that for the semi-infinite oyllnder at n * e »9 this consideration will serve as a oheok on the formulas developed# The stresses trz and <T» # From (35a) it is easily seen that k becomes zero when n becomes infinite# Also9 k 9»l and K a E s ^ # From (38) it is seen that it is neoessary to evaluate the expression K- X & of whloh both the numerator and denominator are zero# To do this put From this equation it is immediately seen that (so) K-E Also Llm — •0 k -*0 k (si) (92) K-E and Llm k-*0 k« ' 4 • (3 2 ) From (80) It follows that TrL = 0 for n:tf, which it should be. For the stress C \ equation (42) will be used. From page 63 sn*(v,k»)« sin*JP« 1, whloh gives E(v,kf)s Ef* 1. v Also tsK's co %and — *1. Kf Then (42) yields <T2 s 0 whloh cheeks with (78). <TZ - fP« 90# and for exterior points, For interior points , whloh oheekswlth (79). In the xy plane n s O and, from (35b) k « *P ( p *+i - )/(p*-0e ). In using this equation for k it must be remembered (93) Then, fop p>l, k » ^ and for p<l, k*p* Equation (38) then gives T - +*7 * by For p * 1, K -E k ir (S3) k * 1 and Ztz « e© . For p * 0, k * 0 , and equation (61)Immediately gives Tt l s o, whloh Itshould be* As p-ksO for p <l and 1-pkaO for p > l 9 equation yields From page 63 it Is then seen that s i n * ^ ^ 0 for p > l and sin*J^#= 1 for p< 1 * .Then p is larger than 1, then, E(vfk*)a va 0 and 0 * * 0 in the xy plane* . ’/hen p Is smaller than 1 , however, E(v,k,) * E f and v » K f, which gives v Strik1) * -* E*« 0, and K* (48) (94) (84 ) just below the base of the oyllnder (n«0-). Just above the base (n * 0+) It Is easily seen that the axial stress has the same value as riven by (94), whloh It should, as <J^ Is continuous* These results oheok with (78) and (79) In oonneotlon with the dlsousslon of these formulas* Along the axis of the oyllnder (peO) It Is obvious that r 1-2* 0. From (35a) k = Using this, and letting p-*0, gives (P -h )U -P k ) m {n * Also, it is immediately seen that k * 0* p In terms of k and n gives p . l+k* ik - i ( * £ - ) * - ( i , n l solving for (95) where the minus sign Is taken because p is smaller 1. When k-tO, and for finite values of n9 this oan be written „ - h i l ~ [ u jl . i V ~ £k •*** Z £*$[• I tk . ,+ -*)L ~ " J1 ~. r\ t4,t I* . i* k-fO Then pk • (1 ♦ n*)k*and equations (81) and (38) immediately give T/-23 09 which it should be* From page 63 it is seen that sin* / « l 9 whloh v*K'«*> v , — • I, Kf IT and 3(v9kf)l Kvv « Kay gives 1* Then (42) gives <rz = 1 1 1*1 •-* * 1 (43) fiZ p ! both for positive and negative values of n* Equation (43) oan easily be oheoked by finding y v “JJj direotly, for points along the axis of the semi* infinite circular oyllnder* Along any line p a l there is no difficulty in applying formulas (38) and (42), except for the case of n » 0* Then k a 1 and K * ae 9 whloh immediately gives t r t s d? (96) This oheoks with what was found in the xy plane. Also, sin J - , whloh gives EfT,**)* v s and S ’s K's £ . 9 - 45*, and Equation (42) then gives Writing it is seen that each term in this series is finite, and if the series therefore is multiplied by either Ef EjVfk1) - - v or 1 -k, each of whioh becomes zero K* when ksi , the terms all become zero. The series thus obtained will satisfy the conditions of convergence and approach zero as k approaches 1 , Then it is easily seen that (97) and there la a discontinuity lr <rs of ^ £ £ aa the xy plane la orossed* The atreaaea Or and Oj > in equations (48) to (49b) theModulus k oftheelllptlo integrals la determined by theequation .* ka 4P —- t (P+0 * + if * When n Is infinite the tern examination* ((p+l)*+ n*)(K«S) requires In this oaae k » 0 and, from (8b), m 4p (p 4 1)a♦ n*»-^ , which gives k ((P ♦ 1 K*5 n4 )(K-E)* 4p — 7 • This, by k* virtueof(88), is seen to approaoh piT when k approaches zero* n Also, - p s s s s s n approaches 1 as n approaches infinity y(P 4 l )*4 n* and, from page 70, sin^ / s !• K*y , E(v,kf)S 1, With k** 1 this gives v* K*« ee and — * 1, (98) and aquation (48) baeonaa, for eztarior points, <Tt * - I 2p* CttT /-* , whloh oheoks with (78). For Interior points equation (48b) glwes t 1 £* T * r - - J — whloh oheoks with (79)• n For n » - «o t ; — T" y f V(p*l)«* n* 1 and both equations (48) and (48a) give <7J.a 0. In the same manner <Jf Is seen to oheok with (78) and (79) for n • so and to be zero for n* -ee f whloh are the proper boundary conditions in these two oases* In the xy plane (n«0) / * 0, see page 70. It is then Immediately seen that equations (48) to (49b) take the form (99) 01s- I E«T 4p*~t~v (p/i) cr.- 4 pi * But /-* to; oj.o-.^fsr r f 4 /-«/ (p< 1, » » 0»1 (p<l» n « 0+) This oheoks with (66) and also with (70) and (79) In oonneotlon with the disoussion of thoaa formulas* Along the axis of the cylinder (p*0) <7> and dj are equal everywhere* Equations (46a) to (49h) then beeoas very difficult to apply on aeoount of the term */p* , However, in this case a staple expression for the stress perpendicular to the axis of the cylinder can be derived very easily, by considering the potential at points along its axis of the semi-infinite cylindrical shell previously used, see pages 65, 66 and 71. (100) Then» for the solid semiinfInite oiroular cylinder of uniform mass density p lit j p = - J (/- sin*)cos 6 ad& where sin et s -Z ll a**, This gives fj =** 7 T p (l+ -= = ) 1 fh n *' i which is good for both positive and negative values of n. For n < 0 (external points) equation (50) is obtained and for n >0 (internal points) equation (50a) Is obtained. 2 2 * P For p* 1 it is convenient to put (p+1) ♦ n * --y f from (86)f in equations (48) t (49) and (49b). n Then V ( P + 1 ) S n* (101) and with p «l this becomes ♦ \/l-k* = Ik*, plus or minus aocording to whether n is positive or negative* Also, sin*J^=l, which gives v a K 1 and E(v,kf)sE*, It is then easily seen that for n > 0 — £ « rf /- u L < (61) and for n < 0 * r - f S 4 ( ¥ - i « ) (87 a) «?■ Whan n a t c c , k* 0 and kf* 1* By means of (82) it if than immediately seen that (87) checks with (78) and (79)* for n a * oo f and that (87a) gives a o for n ■ • oo , (102) NUmerloal Calculations for Problem 2 , A. The stresses X r z and (Xj . n *0. 1 3 4 *'(<) 0- sm'(t) K £ 1.571 1.596 1.571 1.54 6 1.468 1.320 1.171 1.029 n 2 0 •25 .50 .75 •90 .99 1.00 0 .785 1.57 2.35 2.82 3.11 3.14 14° °30f 30* 48# 30* 64* 10* 81* 50* 90* r>a r <a /-* T T TZt tUL it -M “M 0 •064 IT 0 4.0U0 2.000 .016 .069 •139 •251 1.333 .188 .380 •746 1.111 .353 1.010 .740 CO 1.000 1.686 1.910 2.280 3.346 0O 5 1.000 6 0 .050 .218 .590 1.109 2.317 oo (102) 1 3 2 4 5 6 (2) n n 2 0 .01 .10 .20 •50 T +l 0 .0001 .01 1.00 1 .00 •04 •25 1.01 1.0025 1.00 2.00 1.00 1.0625 1.25 4.00 2.00 7 8 (S) 4 (6) 1.00 1.01 1.105 1.221 1.641 2.618 5.828 13 9 \I(T) 1.00 1.005 1.050 1.105 1.282 1.617 2.41 IT A = 15 16 .516 1.118 2.828 •990 •905 .820 .610 .382 .172 CO .719 .382 .271 .140 •060 .018 ' m 77 ? * ■ l.Ou 1.00 1.005 1.020 1.125 1.500 3.000 12 &K 90 0 82° 65 • 55# 37# 30* 22* 30* 9* 50* 17 ▼ ILL * (MU) •122 0 .01 .100 .201 tt)-(s) £ •024 0)^(4) 11 1.00 /-* 2.242 1.145 .77 .343 1.005 1.033 1.118 1.414 .318 .320 .334 .352 .408 .515 .767 14 CO 1.00 1.00 1.001 10 18) MOT 1.00 1.028 1.164 1.26 1.413 1.512 1.559 \fl3) eo 3.370 2.309 2.03 1.756 1.634 1.583 18 / l+ (l°) CO 2 .00 1.074 .735 .647 .558 •520 .504 1.990 1.905 1.820 1.610 1.382 1.171 Ul) .500 .503 .525 .549 .621 .723 .854 (104) 19 20 21 22 23 2A e'* if(755 $##T/09) .707 .709 .785 .742 .788 .855 •925 25 E l* k ') .7854 .7852 .797 .809 •836 .876 •934 45* 45* 46° 48° 52 # 58 # 67° 0*)* 1.00 10* 30* 10* 40* 26 .998 .820 .672 .372 .142 .029 27 cc ' 1.571 1.570 1.498 1.432 1.280 1.137 1.040 /-te/J 0 .002 .180 .328 .628 .858 .971 28 29 .7854 .7856 .826 .868 .990 1.198 1.600 p« 1+ 31 (ishtoht*) 0 •0012 •048 •089 •182 •307 •422 32 (K )* (U ) 0 •0013 •035 •058 .1017 •1596 •213 33 / (* * ) i'zr •25 •2499 •250 •251 •249 •248 •254 0 .0447 .424 .573 .793 .926 .985 V 1.571 1.572 1.649 1.731 1.983 2.415 3.153 /(ID 34 L i <Tm **rz L (27j .7854 .7854 .750 .718 •638 •565 •528 p« 1 35 U S i* s(S2)+Ctt) 04)-/ •250 •2486 •215 •193 •147 •088 •041 -•750 -.751 -•785 -•807 -•853 -•912 -•959 sin'7*5) 0 2° 34* 25* 35* 52* 30* 67* 50* 80* 30 /-(/*) 0 .001 .095 •180 •380 •618 •828 (105) nt io.l 1 p .500 .800 •950 1.000 1.050 1.200 1.500 2.000 7 2 p 5 * •25 •64 •9025 1.00 1.1025 1.44 2.25 4.00 /.OI+(2) 1.26 1.65 1.9125 2.01 2.1125 2.45 5.26 5.01 6 9 (3 )* (7) .766 •403 •218 •2005 •229 •493 1.276 3.02 13 2.026 2.053 2.130 2.210 2.341 2.943 4.536 8.03 14 5 ff (*h •74 •55 •0875 •01 •1125 •45 1.26 3.01 10 d (S) .493 .780 •893 •906 •898 •815 •662 •498 15 sin"1(9) 29° 51* 63* 65 # 63* 54* 41* 29* 30f 20* 20f 50* 40' 30* 50* 16 (J3) /- ( i * ) •753 •376 •152 •094 •057 •022 •007 •004 w * •243 •608 •797 •820 •807 •665 •438 •248 ------■ 6 (4 )* •547 •1225 •0076 •0001 •0126 •205 1.588 9.07 11 •587 •1625 •0476 •0401 •0526 •245 1.628 9«11 12 05 A* m 4 Os) •757 •392 •203 •180 •193 •535 •562 •752 •995 •959 .749 •522 •295 •066 •0125 .0053 K 1.6818 1.9602 2.2541 2.3088 2.2702 2.0209 1.7057 1.6847 17 jW ) •998 .978 •865 .723 •543 •257 •112 •073 « )*(9 ) •247 •624 •848 •906 •943 •978 •993 •996 18 $inm,07) 86* 20* 78* 59* 50* 46* 20* 32* 50* 14* 50* 6* 30* 4* 10* - (106) 19 20 21 22 24 23 a '= m V V £ ' 1.176 1.240 1.015 .789 .570 •261 .110 .070 2.02 1.510 1.060 .817 •582 •263 .110 .070 2.17 1.775 1.660 1.649 1.656 1.735 1.910 2.156 1.210 1.401 1.490 1.498 1.492 1.430 1.320 1.210 27 28 29 30 s in '1'(19) .870 .626 .450 .424 .439 .579 .750 .867 25 (22 (23) 1.13 1.20 .970 .742 .525 .217 .076 .040 31 OQ*(30) V -.030 -.036 -.036 -.035 -.033 -.026 -.015 -.013 60# 38* 26“ 25“ 26' 35“ 48“ 60“ 30* 50* 40’ 10* 20* 40' 10* 26 0 )-(9 ) .007 .020 .057 •094 .152 .385 .838 1.502 32 (26)* (*3) (D .0105 .0094 .0091 .0083 .0083 .0071 .0039 .0030 IT *-.I 33 £ £ 0%s ^ (23) .465 .426 .325 .248 .176 .076 .029 .016 (u v •1025 •0970 .0955 .0939 .0912 .0843 .0625 .0548 1.232 1.297 1.065 •836 •616 .301 •138 .095 ft-*./ 34 35 J ^ < rz <y £<*T ( * <l ) -.435 -.390 -.289 -.213 -.143 -.050 -.014 -.003 (2 *)+(**) (29)-(19) -.565 -.610 -.711 -.787 (PH) .215 .145 .05 .014 .005 -.056 -.057 -.050 -.047 -.046 -.040 -.028 -.025 | ' (107) P»l.l 1 2 3 n /?* 2.21+(*) 0 •l •2 .3 .4 •5 1.0 2.0 0 .01 •04 •09 .16 •25 1.00 4.00 2.21 2.22 2.25 2.30 2.37 2.46 3.21 6.21 •21 •22 •25 •30 •37 •46 1.21 4.21 •044 •048 •062 •09 •137 •212 1.46 17.7 0 •04 •16 •36 •64 1.00 4.00 16.00 7 8 9 10 11 12 •044 •0885 .222 •45 .777 1.212 5.46 33.7 13 (3) ♦M •21 .297 .472 .671 .882 1.100 2.33 5.80 14 2.42 2.517 2.722 2.97 3.25 3.56 5.54 12.01 5 6 * (S)+(6) 4 4-W ft* Li ( t) •908 .875 •808 •741 •677 •618 •397 •183 16 (i*)* A I**) « K 65* 61* 53* 47* 42* 38* 23* 10* 10* 50* 50* 40v 10* 20* 30* 17 1.000 •963 •889 •815 •744 •660 •437 •201 A in) 0 •037 •111 •185 •256 •320 •563 •799 •824 •765 •653 •550 •456 •382 •158 •0335 •176 •235 .347 •450 •542 •618 •842 •966 SIS /f*W) 2.31 2.18 2.01 1.89 1.82 1.763 1«638 1.582 18 i 15 e 0 .1575 •320 .411 •473 •517 •668 •827 m 0 •397 •566 •641 •688 •719 •816 •910 (108 ) 19 20 I I 22 23 24 e '= ^' = if o r , 0 22° 34° 39° 43° 46° 54° 65° 21 20* 30* 50* 30* 40* 30* 25 £ ' 1.499 1.474 1.425 1.376 1.33 1.287 1.150 1.043 •419 .485 •588 .671 .737 .786 .918 .983 26 (23) *(2S) (24) 0 .360 .502 .550 .560 .560 .529 .503 s i n ' ,Uo) 24° 29° 36° 42° 47° 51° 66° 79° 50* 10* 30* 50* 40* 25* 27 U -O *) .192 .225 .292 .359 .423 .482 .703 .917 £ ( * , A ’J 0 •399 .590 .672 .725 .754 .839 .920 28 Q4) *(27) t.t 0 .0076 .0295 .0604 .0985 .1401 .360 .667 t V K 0 .410 .614 .723 .795 .859 1.090 1.50 1.649 1.678 1.742 1.81 1.89 1.972 2.27 29 30 {(28) 2.100 (26) + ( * 0 .087 .172 .24 6 .314 .374 .600 .817 j 0 .447 .67-1 .796 .874 .934 1.129 1.320 (109 ) The stresses <Tt jijid <T* . B. 1 2 3 4 5 6 k - 1 1 n2 \»\ 4 + (2) 2 ffi HJ 5.0 2.0 1.0 ..5 .2 .1 0 25.00 4.00 1.00 .25 .04 .01 29.00 8.00 5.00 4.25 4.04 4.01 5 •38 2 .ci3 2.24 2.06 2.01 2.00 7 a 9 10 K-E (3) * ( 8 ) JO <4J .372 .707 .895 .970 .995 .999 .92 • > .707 .4'7 .24; .0991 .049 j 11 12 <9r s i n ' 9(s) 21° 45 * 63° 76° 84° 87° 50* .1148 .5035 1.0816 1.7641 2.700 3.446 30* 20* 19* 3.33 4.03 5.41 7.50 10.91 13.80 2 H (V -<'•> Ifei 477 .07 3.26 3.70 4.52 5.66 7.42 8.90 .072 i .056; .Otl'5Li .0192 .0079 .00'0 .89 1.84 3.49 4.90 1 14 tetr £ r (n <o) L* <rr e a r —£ on i 13 .005 •019 .032 .035 .028 .019 0 (9)*02) i -N (al *(**) (/»>o) 15 -.505 -.519 -.532 -.535 -.528 -.519 -.500 .246 .227 .193 .145 .086 .055 16 <«**•) . 2 $ -US) .004 .023 .057 .105 .164 .195 .250 17 18 /-«/ i-0 rr JSTrZl tar ft (»*• 2 S ♦ fIS) ( 9 f ) - . 7 S -.504 .496 •> .4 77 -.557 .442 -.60: .395 -.6t .326 • -.69. .305 -•71 .250 — . * *7 (110) p«l.l 1 2 5 4 5 {<7) ** 4.49*M 2.00 1.00 •50 •20 •10 0 4*00 1*00 •25 •04 •01 0 0.41 5.41 4«66 4*45 4.42 £•90 £•55 £•16 £•11 £•10 7 8 9 10 l"l I * A* ft1 w * •690 •450 •£51 •095 •040 i p : • m •902 •992 •995 •9V9 . 12 0 * 46* 20 * 64* 259 76 # 25f 04* 169 86 * £5 9 K-£ £ Si*miM 1.8744 2*2895 2.8612 3.6968 4.1574 1.3388 1.1693 1«0654 1.0160 1.0072 •5556 1.1200 1.7950 £.6800 5.1502 "■ 13 14 15 ■ 1■ i 16 4.61 6.06 0.56 11.95 14.24 It 4.52 5.52 6.09 0.91 10.00 10 4 #*» 00-(u) -.011 .54 1.47 3.02 4.24 M -.001 .232 •340 •286 .202 4.01 1.01 •26 •05 •02 fur) 2.00 1.005 •510 •224 •1414 00 •690 •451 •256 •106 •067 «* C9* U* •* 88 * »• Z9* 6* (Ill) 19 20 21 K' E9 ill VQ 1*884 1 *6 5 2 1*595 1*575 1*5 7 3 1 *3 4 3 1 *4 9 5 1 *5 4 9 1 *5 6 6 1 *5 6 9 1 *0 0 0 *995 •980 *895 *707 £8 26 27 2*56 3*20 3*86 4*11 3*27 31 1 *5 7 1 *4 6 1 *3 7 1 *1 0 *785 4*18 4 *6 6 5*23 5*21 4*05 n>o 33 32 22 y v 90# 84* 7 8 * 80* 83 * 80* 48* ~ 3*48 3*52 3*35 2*85 1*95 0 6*62 6 *6 6 6 *4 9 5 *9 9 5 *0 9 3*14 28 i.;u 1*40 1*88 l.3b 1.10 *786 1.11 •vat> —, 29 ------ V M M im (t’h W 2*55 5*17 8*87 4*05 3*26 1*56 1*49 1*36 1*16 *79 8*49 3.29 3*01 2*55 1*76 34 35 36 5>f> .*t*M - m m -*4 3 6 -*4 3 8 -*4 2 7 -•3 9 4 -*3 3 5 -* 2 0 7 1*854 1*55 ■■ — - -i tSLsr - ¥ B h T V itf i» r £ r (14)+(3*) 28 -.84 — *88 -*a *29 1*19 3*14 *022 *025 *014 -* 0 1 9 -* 0 7 8 -•2 0 7 •019 #02 •00 *04 •04 1 (112) 37 38 39 0 0+ (n) ($)t (37) .£/ * fe* 3.12 2.62 1.94 1.132 •680 •332 •313 •286 .244 •166 4.52 6.08 8.39 11.97 14.28 40 n to 42 41 — <r* (38)-(39) 3M +(40) #wr«iL (41) iS.8 2.79 2.31 1.65 •888 •514 5.93 5.45 4.79 4.03 3.66 .390 •358 •315 •265 •240 •207 n <0 43 44 £ £ rr ttcr 3/4-(44) (43) AT.2 •35 •83 1.49 2.25 2.63 n •023 •055 •098 .148 .173 •207 mt*1 1 P 2*0 1*5 1.2 1.1 .9 #8 .5 #2 2 3 .01 +(3) PH 3.00 2.50 2.20 2.10 1.90 1.80 1.50 1.20 4 9.00 6.25 4.84 4.41 3.61 3.24 2.25 1.44 9.01 6.26 4.85 4.42 3.62 3.25 2.26 1.45 5 m 3.00 2.50 2.20 2.10 1.90 1.80 1.502 1.204 6 (S) •0333 •0400 •0454 •0476 •0526 .0555 •0666 •0831 (113) 7 H 4^1) (4) 8 9 10 r ii 12 8= K £ 2.53 2.98 3.59 4.16 3.96 3.45 2.50 1*90 1.11 1.05 1.02 1.01 1.01 1.02 1.12 1.32 1.42 1.93 2.57 3.15 2.95 2.43 1.38 •58 16 17 18 K -E •888 •958 •989 •996 •994 •985 •886 •552 •942 •979 •994 •998 .997 •992 •941 •743 13 14 (4) * 0 2 ) (D * 2 *0) 12.93 12.08 12.47 13.91 10.67 7.88 3.12 •84 4.00 2.25 1.44 1.21 .81 .64 .25 .04 4.00 3.00 2.40 2.20 1.80 1.60 1.00 .40 8.00 5.25 3.84 3.41 2.61 2.24 1.25 .44 7.00 4.25 2.84 2.41 1.61 1.24 .25 -.56 17.72 12.67 10.20 10.02 6.38 4.28 .62 -1.06 19 20 21 22 23 24 -4*79 ••59 2.27 3.89 4#29 3.60 2.50 1.90 70° 78* 83* 86* 85° 82° 70° 48* 20* 10' 40* 20* 30' 40’ 10' 15 (14)* (/S) 0)*04) (14) f / 6' = 90*- (4) • •160 -.024 •103 •185 •225 •200 •167 •158 5.00 3.25 2.44 2.21 1.81 1.64 1.25 1.04 19# 40* 11* 50' 6 • 20* 3 # 40* 4* 30* 7* 20* 19* 50* 42 * ( i*)- / (/7)*00) $ K £ ' 1.62 1.59 1.58 1.57 1.57 1.58 1.6S 1.81 1.52 1.55 1.57 1.57 1.57 1.57 1.52 1 *38 (11 4 ) 25 (0 -1 26 27 Us)* •Ot+fel) 28 - .. . 31 V 1.00 .25 .04 .01 .01 .04 .25 •64 32 1.01 .26 .05 .02 .02 .05 .26 .65 .099 .198 .462 .785 .785 .462 .198 .125 /•* l/itea) .010 .038 .200 .500 .500 .200 .015 .100 .196 .447 .707 .707 .447 .196 .124 34 35 .038 i l * 11* 26* 45* 45* 26* 11* 7* 40* 20* 30* 30* 20* 10* 1 33 *) £ ( * , k ') .099 .198 .463 .786 .786 .463 .198 .125 30 • Of u v 1.00 .50 .20 .10 -.10 -.20 -.50 -.80 29 .250 .590 1.655 3.27 3.11 1.593 .495 .237 36 (*» (3 3)+ (34) '■(/»)( 31)(£ 4) / S 7 (83) .096 .195 .460 .786 .786 .460 .192 .108 \ ♦ .346 < .785 2.115 4.06 3.90 2.053 .687 .345 (23) Co .575 1.650 3.27 3.11 1.590 .465 .181 [ 1 37 ___ 1 38 39 40 42 41 ■ . \ (35)-(H) ! | ! | .111 .210 .465 .79 .79 .463 .222 .164 (*0 *M .553 .683 1.135 1.755 1.430 .760 .278 .171 (eo)i (38) .395 .659 1.238 1.940 1.655 .960 .^45 .329 3.14-(31) 2.74 2.48 1.90 1.20 1.48 2.18 2.70 2.81 TS-rlL ( 3 8 ) - ( 2«) -.055 -.088 -.105 -.080 1.205 .360 .111 .01: (115) !»•-.# 48 (42) (14) 1 .4 9 .8 8 .4 4 .5 2 /!« ♦ ./ 49 /- * — i«? £ r 44 45 MJ — 1W? * £ 3J4-(43) *44) /I.I4 1 .6 5 2 .2 6 2 .7 0 2 .8 2 50 ww^ — 4? 46 SwZt 3 .5 4 3 .8 0 4 .3 7 5 .0 8 .131 •180 •215 .224 51 & * ) « ( /» ) 48 -•0 7 0 -•1 3 4 -•2 4 1 - .3 3 5 -▼•98 -6.54 -8 * 9 8 - 9 .1 0 52 (S*)*M 53 64 (Mhl /«.«» - .6 8 2 - .6 8 0 - .7 1 5 - .7 2 5 2 .5 8 .7 4 .1 4 .0 4 •04 •14 .6 2 1 .2 2 55 56 1 5 .4 6 1 2 .8 2 1 2 .6 1 1 8 .9 5 1 0 .7 1 8 .0 2 8 .7 4 2 .0 6 57 (£ih(34) SH-(SS) m l * •515 •518 •573 •664 •563 •445 •249 .171 [ 58 is&m 3 .0 0 1 .2 5 •44 •21 -•1 9 —•36 -•▼5 -.9 6 •382 .262 •606 •166 -.100 -•169 -.167 -.166 59 60 (SB) 04) 3.t4-(fH 1 • •182 •251 •36 8 .4 9 8 2 .9 6 2 .6 9 2 .7 7 2 .6 4 .0 5 9 .1 0 2 .1 5 3 a73 .4 1 3 .2 7 8 .0 8 2 .0 1 3 .5 1 .4 8 •3 3 .3 2 | — 2 . 69 2*71 2*81 2*82 ! (U0| 61 9 mf*9 62 63 66 #•* &** ism . 3 • 3.92 5.99 3.51 9.64 i •066 •100 •194 •040 •209 •016 .204 •225 -•n o -e.n ••no <*•00 -•too <*•09 -9.10 -•▼06 Stroiaoi along till (p>0)> 1*/ H* !+ (*) 0 •1 •0 •5 1.0 2«0 0 •01 •04 •25 1.00 4.00 & £ * & £ L 5 -•500 -•550 -•598 -•710 -•854 -•948 3.00 0*90 2 .eo 2.58 2.29 0.10 »«# U 10 9 0 ) - « 0 •100 •196 .425 .707 .895 1.000 1.005 1.020 1016 1.414 2.255 i&2k / • W -•800 -.450 —•400 -•750 -•725 -•TOO -•645 -.579 -•525 • j r~*4 # 12 & & ~ S -.*•04 .**•04 -•m - 8 1.00 1.01 1*04 1.25 2.00 5«00 ; <11 0 ) L it>© 7 -. 5 4 - 5 m m 1H H H H H H I ’ JI 0 1 i c. 64 -•14T -.069 .280 .225 .201 •158 •074 1 •026 j (117) Normal Stress**. n«0». 5 - 4=1 J * l-* ||. 0.1 ■■■ ■ (118) Normal Stresses. p» I ♦. (119) Normal Stresses. £«T . w-i4 |-tf P* • " • AjciSOfCyl -.3 Axis ofCul (120) Shear Stress Tr z . p» i. (121) Remarks on Problem 2« On aooount of the discontinuity of stress at the surfa06 of the semi—InfInite oiroular oylinder separate formulas had to he derived for the oases p> 1 and p< 1* Also, for p < 1 separate formulas had to be derived for the oases n > 0 and n < 0* Hie shear stress Is oon- tinuous everywhere exoept at the oorner (p»l, n«0), where It beoomes infinite, but of the normal stresses only the radial stress is oontlnuous when the oyllndrloal part of the surface is orossed and only the axial stress is oontlnuous when the base of the oylinder is orossed* The stresses obtained from the different formulas when p is put equal to one and n is larger than zero, or n is put equal to zero and p is smaller than one, must therefore be considered to be either for points im mediately inside the cylinder or immediately outside* When the oorner is approached from different directions different limiting values of stress are obtained, all normal stresses being discontinuous and the shear stress becoming infinite* This is shown on figures 2-5, and it is also shown how the stress along a line that passes near the oorner has a rapid variation as the oorner is passed, although it is not discontinuous* The greatest normal stress that occurs anywhere (122) la the axial stress Inside the oylinder and far rroa Its base* This Is a ooapreaslTa stress of »•# The greatest radial and tangential stresses are both the same and ooeur Inside the oylinder9 but Infinitely close to its base* Their value is *75 ,and the stress Is compressive* Outside the oylinder the largest stresses are at the surfaoe and drops off rather rapidly for points farther out in the exterior, in aoeordanoe with saint* Venant*s principle* In this case the greatest axial stress occurs just below the base of the oylinder* is a compressive stress of magnltute *5 ^ It * The greatest radial stress occurs at the oylindrloal surfaoe and at a distance of about *5a from Its base* It Is a compressive stress of magnitude *535 TTJr • Th* greatest tangential stress is tensile and of magnitude *5 -y— . xt ooours at the surfaoe of the oylinder and far from its base* _ / ^ I t e numerical calculations were all checked by the faot that for exterior points the sum of the three normal stresses is zero, and for interior points -2 SSI ^ # / / This Is also a oheok on the method of solution, as the / axial stress was oaloulated from an entirely different surfaoe potential than the radial and tangential stresses* ..... — •— (123) Notations. E Modulus of elastloity Coefficient of thermal expansion T t x 9y,z Rectangular coordinates u,v,w Components of displacement p €z Normal strain components ¥ *it ¥yz (Tjc Temperature Shear strain components e Unit volume expansion G Modulus of elastloity in shear \) Poissonvs ratio ^ Displacement potential cTz tty, T T i f Z Normal stress components Shear stress components V Potential at point (x,y,z) p Density of matter S surfaoe of discontinuity Rectangular coordinates tangential to and perpendicular to the surfaoe s a fo semi-axes of generating ellipse r* e Eooentrioity of ellipse n Ratio of x to semi-axis a, or z to semi-axis o (124) p (Tf.f G~t Badius of curvature Radial and tangential stresses r x y a radius of semi-infinite circular cylinder K Complete elliptio integral of the first kind E Complete elliptic integral of the second kind k p Modulus of elliptic integrals Ratio of radial distance r to radius a of semi-infinite cylinder n Ratio of axial distance z to radius a of semi-infinite cylinder T f0 9z t Cylindrical coordinates tangent to oircle r equal constant p f Reciprocal of p TTtKf0*) Jaoobifs complete elliptic integral of the third kind Z(u) Jaoobi*s Zeta funotion (125) References* Parentheses containing the letter r followed by a number refer to this list. Papers. 1. "Memoirs sur le Calcul des Actions Moleculaires develop pees par les Changements de Temperature dans les Corps Sol ides"; by j. M. C. Duhamel, Aoademie Royale des Sciences de L vinstitute de France, vol. 5 1838, p. 440. 2. "Die Gesetze der Doppelbreohung des Llchts in oomprimirten Oder ungleichformlg erwarmten unkrystal linisohen Korpern"; by F# Neumann, Abhandlungen der Konlglisohen Akademie der Wissensohaften zu Berlin, 1841, p. 3. 3. "TJhtersuchungen iiber die 31asticitat fester iso- troper Korper unter Beriicksiohtigung der Warme"; by C. W. Borohardt, Monatsberichte der Konigllch Preussischen Akademie der Wissensohaften zu Berlin, 1873, p. 9* 4* "On the stresses caused in an Elastic solid by (126) Inequalities of Tnetntnrv"; by j# jnpliasu^ z^es* ••near of Uathenatles, vol. e9 1879, p. 16a# 5* "On the stresses in Solid Bodies due to unequal Heating, and on the Double Refraction resulting there* from"; by Lord Rayleigh, Fhilosophloal Megaslne, 1601, p# 169* 6* "Spannungen und Foraanderungen elnes Hohlsyllnders und elner Hohlkugel, die ton lnnen ersarat verden, unter der Annahme elnes llnearen Toaperaturvertellungs* geaetzea"; by a, V. Leon, Zeltsohrlft fur lfetheaatlk und Physlk, vol. 52, 1905, p# 174, 7. "Temperaturepannung in Hohlzyllnder”; by r # Lorens, 7. D. I., vol. 51, 1907, p# 745. 8. "The Thermal Stresses in solid and in Hollos Clroular Cylinders oonoentrloally Heated”; by C. H. toes, Proceedings Royal Society of London, rol# Id, 1922, p«411* 9. "The Calculation of Temperature Stresses in Tubes"; ty L# H. Barker, Engineering, rol# 124, 1927, p. 445# 10# "Thermal stresses in Thin walled Cylinders"; hy (127) C. H. Kent, Transactions A.S.IT.E., vol. 55, 1951, p.167. 11. "Thermal stresses In Spheres and Cylinders Produced by Temperatures Varying with Time"; by c. H. rent, Transactions A.S.M.E., 1932. 12. "Distributed Gravity and Temperature Loading la Two-Dimensional Elasticity Replaoed by Boundary Pres sures and Dislocations"; by r. a . Blot, Transactions A.S.M.E. , vol. 57, 1935, p. A—41. 13. "A General Property of Two-Dimensional Thermal stress Distribution"; by ??. a . niot, Philosophical Magazine, 1935, p. 540, 14. "Thermal stresses in Plates"; by J. L. raulbetseh. Transactions A.S.M.E., vol. 57, 1935, p. A-141. 15. "The Thermal stress in a strip due to Variation of Temperature Along the Length and Through the Thickness"; by j. n . Goodler, Physics, vol.7, April 1936, p. 156. 16. "Temperature Stresses in Flat Rectangular Plates and in Thin Cylindrical Tubes"; by j. p. Den Harteg, Journal of the Franklin Institute, vol. 222, ffo. 2, August 1936, p, 149. (128) 17* "Thermal Stress"* by j. jy. Goodier, Transactions A.S.M.E., March 1937, p. A-33. 18. "Thermal stress in Long Cylindrical Shells due to Temperature Variation Round the Circumference, and Through the Wall"; by j. n . Goodier, Canadian Journal of Research, vol. 15, April 1937. 19. "On the Integration of the Thermo-Elastic Equations” by J. N. Goodier, Philosophical Magazine, vol. 23, 1937, p. 1017. 20. "Thermal Stresses in Cylindrical Pipes"; by H. Poritsky, Philosophical Magazine, vol. 24, p. 209. 21. "The Stress produced in a Semi-Infinite Solid by Pressure on Part of the Boundary"; by a . E. H. Love, Philosophical Transactions of the Royal Society of London, Series A, vol. 228, 1929, p. 398. Books containing articles on thermal stress. 22. "Vorlesungen liber die Theorie der Elasticitat"; by Franz Neumann, Leipzig, 1885, p# 107. 23# "Vorlesungen uber Teohnisohe Mechanik"; by Aug. (129) Foppl, Leipzig and Berlin, 1922, 4th ed., vol. 5, p.238. 24. "Strength of Materials"; by j. cose, New York and London, 1925, p. 456. 25. "Applied Elasticity"; by s. Timoshenko and j. r. Leeeelle, East Pittsburgh, 1925, pp. 146 and 278. 26. "Strength of Materials"; by s. Timoshenko, M i York, 1930, part 2, pp. 467 and 550. 27. "A Treatise on the IGathsnatloal Theory of Elasticity"; by A. 15, H. Lore, Cambridge, 1927, 4th ed., p. 106. 28. "Theory of Elastlolty"; by S. Timoshenko, M r York and London, 1934, pp. 203, 364, 366 and 276. Other refereneee. 29. "spherioal Harmonies"; by T. X. Use Robert, New York, 1927. 30. "The Theory of the Potential"; by *># ?*e Milan, New York, 1930. 31. "A Treatise on Analytical Statics"; vol. 2, by (130) E. J. south, Cambridge, 1892. 32. "Elliptic Integrals"; by h . Hancock, New York, 1917. 33. "Theory of Elliptic Functions"; by h . Hancock, New York, 1910. 34. "Elllptlo Functions"; by A. Cayley, London, 1895. 35. "Funotlonentafeln mlt Formeln und Kurven"; by E. jahnke und F. Emde, Leipzig and Berlin, 1909. 36. "On Frictional Effects in Shrink Fits"; by J. N. Goodier, Stephen Timoshenko 60th anniversary volume, UaoMillan, 1938, p. 51. I (131) Table of Contenta. Blograpbloal sketch of the author 1 History of the thermal stress problem 2 General theory 9 Discontinuous temperature distribution 14 Problem 1 19 Appendix to problem 1 27 Numerical calculations for problem 1 56 Figure 1 (stresses pertaining to problem 1) 49 Stresses at interior points 60 Remarks on problem 1 65 Problem 2 66 Appendix to problem 2 75 Evaluation of the elliptic Integrals 75 Special oases 07 Numerical calculations for problem 2 102 Figures 2-5 (stresses pertaining to problem 2) 117 Remarks on problem 2 121 Notations 125 References 125 Ac k n o w l e d g emen t s The author wishes to express his gratefulness to Dr, J. N. goodier, Professor of Mechanics of Engineering, Cornell University, for having suggested the two problems of this thesis and for invaluable aid in their solution* He also wishes to thank Dr. H* poritsky, Engineering General Department, General Electric Company, for a valuable suggestion in the solution of the second problem.

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