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Reprinted from
A D V A N C E S I N A P P L I E D M E C H A N I C S , Vol. I , 1948
Survey of Papers on Elasticity
Published in Holland
1940—1946
by C. B. B i B Z E N O Copyright by A C A D E M I C PRESS I N C . , PUBLISHERS N E W Y O R K 10, N . Y .A D V A N C E S I N
A P P L I E D MECHANICS
EDITED BY
R I C H A R D VON M I S E S T H E O D O R E VON K A R M A N
Harvard University California Institute of Technology Cambridge, Massachusetts Pasadena, California
V O L U M E I
1948
A C A D E M I C P R E S S I N C . , P U B L I S H E R S N E W Y O R K , N . Y .
B Y C . B . B I E Z B N O I Technical University, Delft, Holland
C O N T E N T S
Page
I . Publications of the Laboratory for Applied Mechanics of the Technische
Hoogesohool at D e l f t 108 A. Elastic Stability 108
1. On the Stability of Elastic Equilibrium . 108 j 2. On the Buckling of a Thin-Walled Circular Tube Loaded b y Pure
; Bending 110 3. The Generalized Buckling Problem of the Circular R i n g . . . . 1 1 4
( 4. The Circular Ring Under the Combined Action of Compressive
and Bending Loads 117 ! 5. On the Nonhnear Deflection of a Semicircular Ring, Clamped at
Both Ends. . . . ' 118 6. On the Nonlinear Deflection of a Semicircular Ring, Clamped at
Both Ends 118 7. Large Distortions of Circular Rings and Straight Rods 119
I 8. On the Buckhng and the Lateral Rigidity of Helical Compression
I Springs 119 i B. Plates and Shells 121
1. On the State of Stress i n Perforated Plates 121 2. On the State of Stress i n Perforated Strips and Plates 125
3. Some Explicit Formulas of Use i n the Calculation of Arbitrary
Loaded Thin-Walled Cyhnders 128 4. The Effective W i d t h of Cylinders, Periodically Stiffened b y
Circular Rings 129 5. On Circular Plates, Supported i n a Number of Points 3)
Regularly Distributed along I t s Boundary and
Rotatory-I Symmetrically Loaded 131 I C. Miscellaneous Papers 133
1. On Elastically Supported Beams 138 2. On a Special Case of Bending 133 3. Critical Speeds of Rotating Shafts 134 4. On the Elastic Behavior of the So-Called " B o u r d o n " Pressure
Gage 135 5. On the Torsion of Prismatic Beams the Cross Section of Which
is Bounded by Two Pairs of Orthogonal Circular Arcs 136 6. Some Applications of Fourier Integrals w i t h Respect to Elastic
Problems 137 105
Page
7. The Convergence of a Specialized Iterative Process i n Use i n
Structural Analysis 141 8. A D r a f t Nomenclature for Deformations 143
D . Experimental Work 144 1. The "Reduced" Length of (Cylindrical) Twisted Shafts of
Variable Cross-Section 144 2. Some Experimental Data Concerning Flange Couplings . . . . 145
3. Strength and Stiffness of a Twisted Cross Section Weakened
b y a Deep Key Way w i t h Sharp Corners 145 4. Experimental Determination of the Stresses Occurring i n a Ship
Propeller ' 146 5. Optical Determination of Stress Concentrations i n Fillets of Flat
Bars of Constant Thickness and Sharply Varying W i d t h . . . . 147 6. - Effect of Change i n Section of Rotating Shafts on Fatigue
Properties 147 7. A n Abbreviated Method for the Determination of the Factor /3
Mentioned i n the Preceding Section as Well as for Bending as
for Torsional Stress Cycles 148 8. A Highly Sensitive Electrical Apparatus for the, Magnification,
Reproduction, and Measurement of Mechanical Quantities
Occurring w i t h Vibratory Phenomena 148 I I . Publications of the Nederlandsche Centrale Organisatie voor Toegepast
Natuurwetensohappehjk Onderzoek (T.N.O.), the Hague 149 1. Experimental Determination of the Time Integral of an Impact w i t h
the A i d of a Vibrating System 149 2. The Infinite, Elastically Supported Beam, Subject to an Impact Load 150
3. The Smallest Characteristic Number of Certain Eigenwert Problems . 151 4. A Comparative Study about the Main Photoelastic Properties of
Different Materials 152 5. On the Stress Concentration i n the Corners of a Gross, the Rectangular
Branches of Which Have Equal W i d t h 153 6. Photoelastic Determination of the Stresses Occurring i n a Tunnel,
Projected between the Banks of the "Noordzee K a n a a l " Connecting
Amsterdam w i t h the North Sea 153 7. Stress Concentrations i n Corner Joints 153 I I I . Publications of the National Luchtvaartlaboratorium at Amsterdam . . . 154
1. Stresses of the Second Order i n the Skin and Riba of Wings i n Bending 154
2. Stress and Strain i n Shell Wings w i t h Two Spars 154
3. Effective W i d t h 157 4. The Tension Field for Stresses Beyond the Buckling Stress 159
5. The Effect of the Flexibility of Fuselage Frames on the Stress
Dis-tribution i n Fuselage Shells 160 6. Qualitative Pictures of Stresses i n Wings and Fuselages 160
7. General Instability of Longitudinally and Laterally Stiffened
Cylin-drical Shells under Axial Compression 161 8. The Stability of Sandwich Strips and Plates 163
IV. Books 164 1. Technische Dynamik by 0 . B . Biezeno and R. Grammel 164
Page
2. Plasticiteitsleer by F. K . T h . van Iterson 165
Bibliography 167 The i n v i t a t i o n of Professor v o n Mises to contribute to these " A d
-vances" a survey of the worlc done i n H o l l a n d during the war i n the field of appUed mechanics, i n particular i n the field of elasticity, has been met b y the author w i t h the greatest pleasure. A f t e r the sad years i n which Holland suffered f r o m the monstrous brutalities to which a l l western European peoples were subjected b y a merciless occupant, i t is generally felt as a d u t y to re-establish as soon as possible the manifold scientific connections b y which our country was bound t o its aUies.
Since he has devoted a great deal of his endeavour to the international exchange of scientific advancement, the author enjoys the possibility of re-establishing contact w i t h his American colleagues and hopes to p r o f i t as soon as possible f r o m the enormous amount of knowledge and experi-ence t h a t has been heaped up during the past few years i n the U n i t e d States of America where the war d i d not stop—but even gave rise to—the creation of different scientific institutions, as for example, the Society f o r Experimental Stress Analysis.
I t w i l l not surprise the reader that—given the prevailing circumstances — H o l l a n d cannot claim to have contributed a great deal to the t o t a l number of newly issued pubhcations. As a matter of f a c t i n this country only t w o or three centers of specialized "mechanical" a c t i v i t y do exist: the Technische Hoogeschool (Technical University) at D e l f t , the recently founded department f o r experimental research of constructions of the Centrale Organisatie voor Toegepaste Natuurwetenschappehjk Onderzoek at the Hague (Central I n s t i t u t e f o r Technical Scientific Research), and the Nationaal Luchtvaartlaboratorium (the N a t i o n a l Aeronautical laboratory) at Amsterdam. I t is mainly the work of these three insti-tutes t h a t w i l l be reviewed here.
Chronological order has been neglected i n favor of a methodical treat-ment, b u t i t must be emphasized that, i n consequence of the diversity of the different subjects, coherence of this monograph as a whole could not be reached. Matters are divided according to their theoretical or experi-mental character, whereas publications i n book f o r m have been treated separately. The reader is referred to the index f o r f u r t h e r i n f o r m a t i o n about the topics of research, i
' Of nearly all treatises mentioned i n this monograph, reprints are still available, and i t w i l l be a pleasure to the author to send them on application to the reader who may be interested i n any of them; requests may be directed to h i m personally at the address; Laboratorium voor Toegepaste Mechanica, Nieuwe Laan 76, Delft, Holland.
I . PUBLICATIONS O F T H E LABORATORY F O R APPLIED MECHANICS O P T H E TECHNISCHE HOOGESCHOOL A T D E L F T
A. Elastic Stabiliiy
1. On the Stability of Elastic Equilibrium. H i t h e r t o the general theories of stability as developed b y B r y a n , Southwell, Biezeno and Hencky, T r e f f t z , Marguerre, B i o t and others have been restricted t o the investigation of neutral equilibrium; they aim particularly at the deter-mination of the stability l i m i t . The phenomena occurring on reaching and possibly on surpassing this l i m i t were l e f t out of account. This restriction as to the extent of the investigations is caused b y t w o cir-cumstances. First of all there must be mentioned the great mathematic'al difficulties t h a t obstruct the theoretical treatment of elastic behavior after the stability l i m i t is surpassed. Whereas the investigation of states of neutral equilibrium is still possible b y means of linear differential equa-tions, the equations describing elastic behavior after the stability l i m i t is surpassed are no longer linear. Moreover engineering has long been satisfied w i t h the knowledge of the stability l i m i t (critical or buckhng load). The recognized principle based on considerations of safety was t h a t the load on a structure should always be kept below this l i m i t , so t h a t an investigation of the phenomena occurring above this l i m i t seemed superfluous. However, i t has been k n o w n f o r a long time t h a t some structures {e.g. f l a t plates) are capable of sustaining considerably larger loads than the buckhng one w i t h o u t exceeding the elastic l i m i t at any point of the structure; i n modern engineering, especially i n aircraft engineering, where saving on structural weight is of the greatest impor-tance, these higher loads are actually allowed. The theoretical treatment of this problem has been given b y Marguerre and T r e f f t z , among others. On the other hand i t has been noted t h a t the experimentally determined buckling loads of some shell structures, e.g., axially compressed t h i n -walled cylinders, lie considerably below the theoretical stability l i m i t . Moreover the experimental results are m d e l y divergent. (Cf. investiga-tions b y Flügge, Donnell, Cox, v o n Kd,rman and Tsien.)
F r o m all this i t appears t h a t the general theories of stabiUty framed so f a r do not suffice. They have to be completed i n such a way t h a t the divergent behavior of various structures i n the case of loads i n the neighborhood of the theoretical buckling load can be described as well. The present treatise aims at such an extension.
Chapter 1 gives a brief survey of the theory of elasticity f o r finite deformations, whereas the general equations of motion are derived b y means of Hamilton's principle, using the potential or strainenergy f u n c
-t i o n -to describe -the elas-tici-ty of -the body. The equa-tions of equihbrium are obtained b y p u t t i n g equal to zero all inertia forces.
I n Chapter 2 the general theory of stability is dealt w i t h . A precise definition is given b y means of the energy criterion. I n accordance w i t h T r e f f t z i t leads to t w o conditions of stability. The first condition is t h a t the first variation of the potential energy is zero f o r any Idnematically possible variation of displacement. The second condition requires t h a t the second variation of the energy cannot be negative f o r any such pos-sible v a r i a t i o n of displacement. The stabiUty at the stability h m i t is studied at f u l l length.
I n the next chapter the states of equilibrium at loads i n the neighbor-hood of the buckling load are investigated. The approximate method used f o r this purpose naturally gives better results the less the load differs f r o m the critical one. The character of these states actually appears to be governed b y the stability at the buckling load. I t must be said explicitly t h a t the method exclusively enables to deal w i t h buckling problems w i t h a so-called point of b i f u r c a t i o n ; so-called " o i l c a n n i n g " problems are not taken i n t o account. The most i m p o r t a n t result pre-sented i n Chapter 3 is t h a t w i t h stability of the equilibrium at the critical load neighboring states of equilibrium exist only f o r larger loads; these states are stable. Therefore larger loads than the buckhng can be sus-tained. W i t h an unstable buckhng load, on the contrary, neighboring states of equilibrium do occur at loads smaller than the buckhng load; these states are unstable.
The theory treated up to now does not give an explanation of the f a c t t h a t f o r some structures the experimental buckling loads are con-siderably smaller t h a n the theoretical buckhng load. To arrive at such an explanation the influence of small deviations of a real structure f r o m the simphfied model, designed to represent the structure, is considered i n Chapter 4. The most i m p o r t a n t result t h a t this investigation shows is t h a t w i t h an unstable buckling state of the model the buckhng load of the structure may be considerably lower i n consequence of very small differences between structure and model. Hence the discrepancy between theoretical and experimental critical loads can be explained purely elastically b y assuming small deviations of such a structure f r o m the corresponding model; moreover the great sensibility of the buckling load of the structure f o r small variations i n the magnitude of the devia-tions explains too the wide divergence of experimental results.
The most interesting example of the application of the theory devel-oped here is the application to the axially compressed thin-walled cylin-der; f o r i n this technically i m p o r t a n t case the great discrepancy between the theoretical and experimental buckhng loads has up to now not been
accounted f o r satisfactorily. T o apply the general theory to this problem i t is necessary to dispose of the knowledge of the elastic energy of the thin-walled cylindrical shell f o r finite displacements. W i t h a view to the possibility of application to other shell structures as well, a general theory of t h i n shells f o r finite displacements is given i n Chapter 5. Before passing to the rather complicated theory of the thin-walled cyhnder i t seemed advisable to deal first w i t h some simpler applications, i n order t o elucidate the general theory (Chapter 6). The well-known problem of the elastica, Cox's problem, and the problem of equivalent w i d t h of compressed flat rectangular plates are considered i n this connection.
Finally, Chapter 7 is concerned w i t h the axially compressed cylindrical shell. Neglect of boundary conditions leads to the same result f o r the buckling load as known f r o m existing hterature. The same neglect leads to the conclusion t h a t equilibritim i n the buckling state is unstable. As t o the neighboring states of equilibrium at loads i n the neighborhood of the critical load, i t is f o u n d t h a t all existing neighboring states of equilibrium are unstable. The results are compared w i t h those obtained b y v o n K4rmd.n and Tsien, and seem to be more satisfactory, at least f o r loads i n the neighborhood of the critical load. Thereupon the i n f l u -ence of small deviations f r o m the true cylindrical f o r m is discussed. As the investigation is rather complicated the detailed calculations are restricted to one f o r m of deviations. I n this case a very marked decrease of buckling load is f o u n d already w i t h v e i y small deviations. This result is i n striking contrast to t h a t of Donnell and i t may be stated indeed t h a t the theory given here supphes an explanation of the large discrep-ancy between theoretical and experimental critical loads. The wide divergence of experimental results is Ukewise satisfactorily accounted for by the extreme sensibility of the critical load f o r small variations i n the magnitude of the deviations.
2. On the Buckling of a Thin-Walled Circular Tube Loaded by Pure
Bending. I f a thin-walled circular tube (length U, radius a, thickness h) is loaded b y two equal and opposite terminal bending moments M,
its cross section alters its circular shape i n t o an oval one, owing to the fact t h a t apart f r o m the normal bending stresses i n the cross section of the tube, tangential bending stresses arise i n i t s meridional sections. A closer examination of this fact reveals t h a t the curvatute of the " a x i s " of the tube does not increase proportionally w i t h the loading moment M. The phenomenon has been studied at great length b y Brazier. I n the present paper another phenomenon is studied, which is characterized b y the simultaneous appearance of longitudinal and circumferential waves i n the cylindrical shape of the tube. I t is assumed that—^if unloaded—the tube possesses such i n i t i a l curvature t h a t under the action of the buckling
moment i t is straight and—^in cross section—circular and of constant thickness h. The coordinates used are cylinder coordinates r, <p, z; the corresponding displacements are denoted b y u, v, w; the surface stresses b y R, $, Z. I t is well k n o w n t h a t w i t h the load system
z z R = üpq COS p<p sin X -•; # = bp^ sin p<p sin X - ; Z = 0 (1)
a a
(where p and q represent arbitrary positive, integers and X stands f o r X = TT qa/l), the following displacements are compatible:
. . 2 . 2 z
u = Upq cos p<p sm X - ; v = Vpq sm X - ; w = Wp„ cos vtp cos X —
a a a
(2) As f a r as u and v are concerned Upq and Vpq are represented b y
Upq = apqttpq " j - Ppqbpq', Vpq = fipqapq -\- "Ypqbpq (3) Ppq and jpq standing f o r
« - _ / ^ A ^ - / ^ ^ ^
(4) Tl, Ti, Ts, N themselves standing f o r
= (1 + k)p^ + [2X^ + 2(1 - v)m]p^ + (1 + 3fc)X* T2 = p { [ l + ki2\' + l)]p' + {v + 2)X2 + 2/cX*} l + ' ^ k X ^ 2(2 — v) Ti = Ap2(p2 _ 1)2 + ^ .A 1 X V + 1 — I' 2{v - v"" - 2 ) + ' kX' + k I — V + 2(1 + v)X' + k{X' - 2vX^) I — V N = kp« + k[iX' - 2]p' + klQX' - 2(4 - v)X' + l ] p * + k[iX^ - QX' + 2(2 - v)X']p^ + (1 - v')X' + kX' - 2vX'k + (4 - dv')X%. I t can be proved t h a t t w o systems of so-called "elementary charac-teristic loads" B, w i t h corresponding "elementary characcharac-teristic deforma-t i o n s " D, exisdeforma-t, f o r which deforma-the load componendeforma-ts Rpq, ^pq are propordeforma-tional to the corresponding displacements Upq, Vpq, such t h a t
C. B . B I E Z E N O
These systems are defined except f o r a factor of proportionahty, and possess certain properties of orthogonahty. The factor of proportionahty can be fixed b y a suitable condition of standardization.
Returning to the buckling problem under consideration, we refer t o the f a c t (cf., f o r instance, W. Flügge, Statik und Dynamik der Sciialen, or C. B . Biezeno u n d R. Grammel, Teciinische Dynamik) t h a t the differen-t i a l equadifferen-tions f o r differen-the supplemendifferen-tary displacemendifferen-ts u, v, w (which differen-transfer the cylinder f r o m its state of " n e u t r a l " equilibrium i n t o an infinitesimally neighboring state of equihbrium) agree w i t h the differential equations for the displacements u, v, w oi a, cylinder, the loads R, Z of which are represented b y
where Q = Ma cos (p/ira^ stands f o r the normal load per u n i t of circum-ferential length i n the terminal cross sections of the cylinder, as exerted b y the bending moment M.
L e t (u,v,w) occurring under the critical load i l f = fiM {M = u n i t moment) be named the " t o t a l " characteristic deformation T, Then T can be decomposed into a (infinite) series of elementary normal deforma-tions D
00
T = 2 diDi. (8)
i = l
I f the load system, derived f r o m D, w i t h the aid of (7) be called J5,-, then the load system belonging to the " t o t a l characteristic d e f o r m a t i o n " T may be w r i t t e n as S diBi.
Each separate system 5,- can be expanded into a (finite or infinite) series of "elementary characteristic loads" Bi, so t h a t the load system corresponding w i t h the " t o t a l characteristic d e f o r m a t i o n " finally can be looked upon as the sum of an infinite number of groups of a (finite or infinite) number of ' ' elementary characteristic loads " Bi. I n this reason-ing i t has been assumed t h a t all deformations and load systems are connected w i t h the same parameter X, characteristic f o r the " t o t a l deformation" under consideration, so t h a t all loads and deformations have the same number of longitudinal waves. A l l functions T, B, D can therefore be denoted b y a single subscript p.
I f artificially the tube be given the "elementary characteristic defor-m a t i o n " Dj, then we can, b y the aid of (7), fordefor-maUy calculate the corre-sponding load system {R,'è,Z) produced b y two u n i t terminal moments
infinite) series of "elementary characteristic" functions B. The coeffi-cient ffii,-, which i n this expansion belongs to the elementary characteristic f u n c t i o n Bi, is called the "influence number of the elementary charac-teristic deformation Dj w i t h respect to the elementary characcharac-teristic load Bi."
This f o r m a l definition provides us w i t h an expedient f o r obtaining a system of homogeneous linear equations f o r the coefficients di i n the expansion (8), viz.,
' di = J^ ixdjaa (t = 1, 2 • • • ) (9) j
and therefore the problem of the buckling cylinder under bending is f o r m a l l y reduced to the solution of the equation
«11 «12 «21 «22
= 0. (10)
I t can be proved, that—^if only the elementary characteristic functions are suitably numbered—an = an, so t h a t equation (10) is a secular one possessing only real roots.
For the computation of the smallest root n of this equation recourse is made t o a generalization of an iterative method, which b y the second author of this treatise has been established i n his doctor's thesis, and which consists i n deriving f r o m an arbitrary set of quantities d-u another set t?2t, defined b y
#2i = ^aijêu etc. (11) Two consecutive iterations êsi and ^s+i,i tend w i t h increasing s t o
proportionality, the factor of which represents the smallest characteristic n u m -ber. I t s value fii is f a i r l y approximated b y the expression given b y K o c h :
SO t h a t only one iteration needs to be performed. The iteration process relates to a, fixed value of X, viz., to a fixed number of longitudinal waves i n the buckling deformation. Therefore the iteration process has t o be repeated f o r a great number of values X i f the absolute m i n i m u m of the critical buckling moment is required. This seems to be a rather tire-some technique, b u t means are provided to abbreviate considerably the numerical calculations.
3. The Generalized Buckling Problem of the Circular Ring. A circular ring, subjected to a u n i f o r m radial pressure q per u n i t of circumferential length, is apt to buckle under the action of one of the so-called " c r i t i c a l " loads g = (n^ — l)EI/r^ (n integer and ^ 2), i?7 representing the flexural r i g i d i t y of the ring and r its radius. This case of buckling is analogous to the buckling of a straight rod under the action of t w o compressive forces, as far as the cross sections of b o t h ring and rod are loaded b y a normal force of constant magnitude. I f the straight rod is loaded b y a prescribed system of axial forces, so t h a t the noi'mal force of the cross section varies w i t h its coordinate, proportional increase of the load-system, say t o the multiple X, leads as well to critical b u d d i n g loads. The (positive or negative) value of the factor of magnification X can best be f o u n d b y a method of iteration. Obviously the analogous problem exists f o r the circular ring, i f only i t is subjected t o an external load-system such t h a t i n every cross section the bending moment M and the shearing force D are zero, whereas the normal force of the section varies w i t h its coordinate. E v i d e n t l y the first of these conditions w i l l not be f u l f i l l e d if the ring is loaded b y an arbitrary system of radial and tangen-t i a l forces, b u tangen-t i tangen-t can be shown tangen-t h a tangen-t every systangen-tem can be sphtangen-t up i n tangen-t o
two components A and B, the first of which is called the "compressive" ^ system because i t is characterized b y i l f = Z) = 0, whereas the second
one is called the " b e n d i n g " system characterized b y = 0.
The first system A, i f suitably magnified, leads to the generalized buckling problem of the circular ring, which is the object of this treatise, but i t is seen at once t h a t an arbitrary load-system, consisting of b o t h components A and B, gives rise to a problem, which again is analogous to a weU-known problem, viz. the straight rod subjected to axial thrust
and transverse bending loads. Just as w i t h the straight rod the axial Here only
forces tend to increase the deflections caused b y the transverse loads, the presumed A-load-system of the circular ring w i l l affect the deflections due to the to the
B-system (cf. Section 4). of the I f the arbitrary load-system {q,t)—g = radial load, t = tangential
load per u n i t of circumferential length—of the ring is expanded i n t o Fourier series
CO eo • 00
g = ao + ^ «4 cos /cp + y &j sin fc^s; i = co + ^ c;j cos A;^
1 1 1 + ^ 4 sin (1) the 4 - and 5-coniponents are represented b y
g* = ao + i «1 cos ¥5 + ^ + - 6i sin ^5 +
Ï
1 , k _ -y-, r a* + — r fc^ — 1 /c^ — 1 1 Oft - 7^^ 7 fc2 - 1 " /c2 _ 1 COS fcv? COS fc^ sin k(p 1 r i * = -2-6.cos^ + 2^^ — 5 . + — + ^ Ol sin ^ + ^ '** = ^ Ctl cos ¥3 + ^ sin fc^ fc2 _ 1 '^'^ _ 1 + ^ 6i sin V + ^ K I fe h + T-o 7 Ci k ' - l k ' - l fe 7 1 fc2 - 1 " fc2 _ 1 + - ai sin 9? +t
fe 1 , k' + 1 fc^ - 1 COS fc(p sin k(p j cos fc^ sin k(p. (2) (3)Here only attention w i l l be paid to A-load systems f r o m which i t is presumed t h a t the g* and t* components keep their directions w i t h respect to the element of the ring on which they act. The differential equation of the problem is then given b y
(U = u" + u,—u representing tlie increase of tlie radius vector r— N = prescribed normal force acting on the cross section, C = constant
of integration to be f o u n d i n the course of the investigation). B o t h N and U can be expanded into Fourier series
N = \No = X Ao + Ak oos kip + sin ktp
U = cto + ^ cos l<p + bl sin Icp
(5)
1 = 1 1=1
and i t can be proved t h a t buckling of the ring can occur only i f all m u l -tiples of (p i n the Fourier series of N have a factor i n common.
I f the greatest factor i n common be called p, all terms ao, ai cos <p,
bl sin (p, a„p±i cos (ap ± l)<p, and sin {ap + l)<p, {a representing
any positive integer 5^ 0) have to be excluded f r o m U.
Substitution of N and Ü i n (4) would lead to an infinite system of recurrent relations between the coefhcients ai and bi, and i t would be seen t h a t the system Avould break up i n t o a number of minor systems, each of which is related to a distinct class of coefhcients ai, bi, defined b y
Z = + g, (mod p) g = 0, 2, 3 g = 0, 2, 3 P - 1 {p = odd) (6) {p = even).
W i t h any prescribed p the buckhng problem therefore is split up i n t o
p — 1 p
— ~ — or - cases, i n every one of which U is composed exclusively of
2 2
terms relating to one of the congruences (6).
W i t h respect t o the constant C i n (4) i t must be observed t h a t i f the product NU in the right-hand side of the equations happens t o miss a constant t e r m after having been w r i t t e n as an ordinary Fourier series, then C must be suppressed; if not so, C serves t o annul this t e r m .
The practical solution of the problem is given i n an iterative way, the justification of which is given b y expressing the problem i n terms of an integral equation.
Starting w i t h an arbitrary f u n c t i o n Vi{(p), containing only cosine and sine terms of suitable multiples of (p, another f u n c t i o n V2{<p) is derived, formally defined b y the differential equation,
v." +V, = - ^ V i + C 7 sin <p (C hi Y% another f i I t can be mated b y on the second practical itei i n fact, canr treatise itsel The case \ ( 1 - i - 2 cos: istic values normal force expected. ï stretched pa therefore ca example i t ( cos 1^) the - 2 . 1 1 3 3 — ^2 0.6275 — • ,.2 4. TU C Bending Loc already bee A - and 5-sy guarantee t l tions (and i B-system, w action of thi The influent B-system c£ the correspc end the so-c tem has t o
7 sin (p (C being again a constant t o be determined a posteriori). F r o m
V2 anotlier f u n c t i o n Vs is deducted i n tlie same way, etc.
I t can be proved t h a t the smallest characteristic value Xi is approxi-mated b y one of the following formulas:
the second of which exceeds i n general the first i n accuracy. The practical iteration scheme, along which the iteration has to be performed i n fact, cannot be described here, b u t is elucidated at f u l l length i n the treatise itself.
The cases N = \No = X ( l + 2 cos M ( A ; = 2, 3, • • • 12) and N = \ ( 1 -|- 2 cos 2^3 - f cos 4<p) are worked out numerically. Higher character-istic values can be calculated as well. I n all such cases where the normal force N changes its sign, negative characteristic values are t o be expected. A negative value of X interchanges the compressed and the stretched parts of the construction, and a sufficiently great negative X therefore causes the buckling of the i n i t i a l l y stretched parts. As an example i t can be stated t h a t w i t h a compressive force N = X ( l - f 4 cos 2(p) the second characteristic number proves t o be negative X2 =
3i?/
— 2.1133 > whereas the first characteristic number is equal to Xi =
4. The Circular Ring under the Combined Action of Compressive and
Bending Loads. I n the previous section the problem treated here has
already been announced. The ring is simultaneously subjected t o an
A- and B-system, b u t i t is supposed t h a t the A-system is small enough to
guarantee the elastic stabihty of the ring. I t is obvious t h a t the deflec-tions (and internal stresses) of the ring, due to the single action of the B-system, w i l l be affected i n a rather complicated way b y the simultaneous action of the A-system, which, alone, would produce no deflections at all. The influence of the A-system upon the bending effect produced b y the B-system can be expressed i n terms of the characteristic numbers X and the corresponding characteristic functions U of the A-system. T o this end the so-called "reduced moment," — •—- Mb, belonging t o the B-sys-tern has t o be expanded i n t o a series
of the characteristic functions Uk of the A-system. The resultant bend-ing moment M, resp. the resultant f u n c t i o n U, due to the j o i n t systems
A and B, is then represented b y
The'resultant distortion of the ring is governed b y the equation
A numerical application is given for the ring compressed b y t w o diametral forces P.
5. On the Nonlinear Deflection of a Semicircular Ring, Clamped at
Both Ends. The aim of this paper is t o deduce i n a theoretical way the
force-deflection relation (P,iO of a semicircular ring clamped at b o t h ends and loaded (in its midpoint characterized b y cp = 0) b y a radial force P. The treatment of the problem (which finds its origin i n a remarkable paper on the buckling of cylindrical shells b y T h . von K a r m a n , L . G. D u n n , and Hue-Shen Tsien) is essentially based upon the methods developed i n the t w o preceding papers. Instead of the semiring clamped at b o t h ends, first a complete ring is considered loaded b y t w o pairs of diametral forces
P and Q, placed at right angled (P at .p = 0 and TT, Q at 7r/2 and 37r/2. For
a number of values of the ratio P/Q, viz. P/Q = 1, 1,05, 1,10 and 1,2 and their reciprocals, the radial deflections u are calculated at ^ = 0 and tp = 7r/2. F r o m each of these results (relating t o a distinct value of PQ), the absolute value of P can be deduced, f o r which the deflection u&t tp =
Tv/2 and 37r/2 becomes zero, and b y substituting the smallest of these
values i n the corresponding expression of u for ^ = 0, the deflection of the point <p = Q can be calculated. I n a graph - — is plotted against u/r and the results are compared w i t h the experimental ones of von K a r m a n
6. On the Nonlinear Deflection of a Semicircular Ring, Clamped at
Both Ends. The method discussed i n the preceding paper requires a
con-siderable amount of computation. I t therefore seemed desirable t o investigate whether the results could be obtained i n a simpler (be i t approximative) way. Again the problem is attacked b y considering a
complete ring, loaded b y t w o pairs of diametral forces P and Q. The
simplification of the problem consists i n replacing the rather complicated A-load-system (connected w i t h these forces) b y a radial constant pressure
(P - j - Q); 7rr equally distributed along the circumference of the ring, giv-(2)
u" -{-u = U. (3)
ing rise t o a constant normal force (P + Q): TT i n the cross section. The approximative results obtained i n this way are quite satisfactory.
7. Large Distortions of Circular Rings and Straight Rods. The present paper is closely-connected w i t h the foregoing three papers, and provides the reader w i t h a valuable extension. Indeed the authors of these previous treatises use the differential equation of the shghtly curved beam to describe the distortions, an approximation, which can be made only if the difference i n slope of the elastic hne i n its deformed and its undeformed shape is small. So their solution is limited to distortions not too large. Moreover, the distortions, as calculated b y their method, appear as an infinite series of terms. As the calculation of the separate terms is rather tedious, one has for practical reasons to confine oneself t o a small number of terms. The authors, for instance, use only t w o terms.
I t therefore is of interest to give an exact treatment of the same prob-lem, p a r t l y t o have the exact results for themselves, p a r t l y to provide a check to the approximate results. The method used i n this paper is principally based on the ideas used b y some authors i n the classical period of elasticity and consists i n the use of a system of natural coordinates
{s,<p) where s is the length of the arc measured f r o m an arbitrary zero
point, and p the angle included b y the normal of the distorted ring and a fixed direction. The method is not restricted to closed circular rings: many other problems of large distortions (as for instance the one referred to i n section C2) can be solved b y the same means. As may be expected, i t calls f o r a knowledge of eUiptic integrals. Four particular problems, one dealing w i t h a closed circular ring, one w i t h a semicircular ring, and two w i t h straight beams a d m i t t i n g large deflections, have been numer-ically calculated, and the results have been plotted i n diagrams. Com-parisons are made w i t h approximate results of other authors and w i t h own experiments.
8. On the Buckling and the Lateral Rigidity of Helical Compression
Springs. The buckhng of hehcal springs was first studied b y R. Grammel
i n a paper read before the first International Congress f o r Applied Mechanics. Later on C. B . Biezeno and J. J. K o c h pointed out t h a t the results obtained d i d not.agree w i t h the experiments and t h a t the reason for this discrepancy was to be sought i n the neglect of the "shear elastic-i t y " of the sprelastic-ing. U n f o r t u n a t e l y a partelastic-ial melastic-iselastic-interpretatelastic-ion of thelastic-is shear effect leads Biezeno and K o c h to an overestimation, which makes itself clearly perceptible i n the end results. I n this paper the correction is given, and i t is proved t h a t now f u l l agreement between theory and experi-ment exists. The theory is n o t restricted to the determination of the buckling load of hehcal springs under compression b u t can be applied as weh t o springs subjected t o an axial force combined w i t h a bending
moment and a lateral force, as t o springs under the combined action of com-pression and torsion. A p a r t f r o m the fact t h a t the normal and shear elasticity of the spring have to be brought i n t o account i f the latter (as a whole) is expected to behave like a very elastical rod, i t must be remem-bered t h a t the compression of the spring results i n an increase of the number of coils per u n i t of axial length, and therefore leads t o a decrease of all coefficients of rigidity as compared w i t h those i n the unloaded state. I f n is the number of active coils, D the mean spring diameter, / the linear moment of inertia of the circular wire section, lg the length of the unloaded spring, I the length of the loaded spring, E Young's modulus of elasticity, m Poisson's ratio, then the various coefficients of elastic r i g i d i t y per u n i t of length of the unloaded spring are (in succession of bending, shear, compression, and torsion)
1 _ nTD2m+ 1 1 nwD^m + 1
ao ~ UEI 2m ' Po" loSEI' 70 ~ To EI
(1)
6o lo EI
those defined per u n i t of length of the loaded spring are represented b y
I I I I
a = - a o ; /3 = - / 3 o ; 7 = 7 7 0 ; S = - 80. (2)
lo Iq lo Iq
The differential equation of the problem is
'" + ~{' + ^)'">
(in contrast t o the equation
(3)
given b y Biezeno and Koch), and i t is an easy matter t o deduce f r o m this equation the required buckhng load, taking due regard t o the boundary conditions. The relative compressions ^ [defined b y Z = (1 — OU] at which the spring buckles can be plotted against the ratio U/D and be compared w i t h experimental results; this leads t o the agreement already mentioned. M o s t interesting is the behavior of a spring the length of which is about 2.7 times its diameter. I f the length exceeds this amount by some 5 per cent, the spring buckles under a compression of about SO per cent, whereas no instability at all occurs i f the length is some 5 per cent smaller. Similar results hold f o r other end conditions of the spring.
W i t h the aim i n view t o use his results i n the construction of so-called " v i b r a t i o n " tables, the author pursues his treatise i n examining a spring clamped at one end and loaded at its free end b y an axial force P, a lateral force L , and a bending moment ilfo, and calculates the lateral displace-ment ui and the slope of the free end i n terms of these loads. P being prescribed, the lateral force and the moment required to produce a given yi and xj/i can be w r i t t e n as
The coefficients ci, ct, ca, which are of the greatest importance to the designer, can be derived f r o m graphs (for different values of ^ resp. P) as f u n c t i o n f r o m U/D.
F i n a l l y the author considers the buckling of the spring under com-bined compression (P) and torsion {W), b u t the treatment of this problem is f a r more intricate so t h a t no details can be given here. The results are given i n graphs, i n which f o r various values of the "angle of t w i s t " curves are drawn the ordinate of which represents the critical relative compression ^ of the spring as f u n c t i o n of the ratio 7o/D. Theoretical and experimental data are again i n f u l l agreement.
1. On the State of Stress in Perforated Plates. This important mono-graph deals w i t h the stress distribution occurring i n plates perforated b y an arbitrary number of circular holes of arbitrary size and arbitrary situation (though the author restricts himself to holes of a fixed radius a). The starting point of all calculations is the well-known stress f u n c t i o n
F of A i r y , satisfying the equation:
E v e r y solution of this equation—if only satisfying certain conditions of continuity—gives rise t o a realizable distribution of stress {<Ty,ffz,ry^ resp. (o-r,o-^,Tr,,), represented b y
L = ciyi — C2i/i; Mo = — C22/i + C3\pl. (5)
B. Plates and Shells
/ 52 32 52 1 a 1 5 AAP = 0 ( A = — • + — resp. \ h - — \ dy'' dz^ ^ r dr r^ d<f d^F dW dW (2) dydz resp. 1 dF 1 dW dW d / I dF'
I f we restrict ourselves to stress functions the corresponding displace-ments of which are zero at i n f i n i t y and unambiguous throughout the field the most general F useful i n our problem is given b y
oo F = AoPo + A i P i + I i F i + 2 A + BJ'^* -h Ajn + BnFn*) (4) w i t h Fo = Inr, Fl = r " ' cos (f>, Fl = r~^ sin <p, F„ = r^" cos ( jP„* = 7--n+2 cos n<p, F„ = sin n(p, F^* = r-"+2 sin n,p (n è 2) (5)
E v i d e n t l y every f u n c t i o n F^, Fa*, F^, F^*, gives rise t o stresses cxr, <r^ and i n the points of the circle r = a (afterwards t o be considered as the boundary of a circular hole I ) . B u t i t is an easy matter t o construct such linear combinations F^s, Frs, F„s, F^s of these elementary functions t h a t apart f r o m tangential stresses cr^ all along the circle r = a only cosinusoidal or sinusoidal radial stresses ar resp. only cosinusoidal or sinusoidal shearing stresses tr^ occur. I t can easily be checked, f o r instance, t h a t w i t h the stress f u n c t i o n
fjS+2 „3
the stresses o-, = cos s<p and r^^ = 0 do correspond i n points of the circle r = o. I n many cases i t is recommended t h a t the f u n c t i o n F be developed i n t o a series of the arguments F^s, Frs, F^s, Frs. I t takes the f o r m
F = CoF^o + Ci{F„i + Fri) + Gi{F,i - Fri) A ^ (C.F., Z ) A «
s = 2
+ GsF.s + DAS). (7)
I f now a second circle (radius a) is considered (later on t o be regarded as the boundary of a second hole I I ) , a second system of polar coordinates (rVO may be introduced i n the center of this circle. The polar axis coincides w i t h the polar axis of the first system b u t has the inverse posi-tive direction, whereas the azimuthal posiposi-tive directions are opposite as well. Under these conditions the stresses (Xr and T , ^ occurring i n the points of circle I as produced b y the stress functions F'„s, F'rs, F^s, F'rs connected w i t h the center of circle I I can be expanded i n t o a Fourier series of the argument (p b y the aid of a simple complex transformation.
T h i s being stated we draw our attention t o an infinite plate w i t h t w o circular holes I and I I (of equal radius) at i n f i n i t y subjected to u n i f o r m tension perpendicularly directed t o the centerline of the holes. - I f f o r a m o m e n t the plate is thought to be unperforated, the stresses ar° and Tr^" occurring along the boundaries I and I I can be calculated and brought into the f o r m
COS TKp,
n = l
-Tr^ = ^ D„° sin n,p. (8)
Thereupon a stress function F is introduced into each of the centers I and I I , represented b y
F = CoF.o + CiiF.i + Fri) + 2 + ^"^™)- [(9)
The stress f u n c t i o n F connected w i t h the center I gives rise to stresses along the boundary I of the magnitude
o-ri = Co + ^ C„ COS n<p, Tr^i = ^ D„ si sm n<p. '(10)
The stress f u n c t i o n F connected w i t h the center I I gives rise t o stresses along the boundary I , which can be w r i t t e n as
eo 0 0 — 0 0
s = l (1 = 1 L s = l
+ Dsin') COS n(p ^ (11) 00 _ oo
sm n<p.
The requirement t h a t ffr and Tr<f, shall be zero along the boundary I (and consequently along the boundary I I also) leads t o the following infinite system of linear equations f o r the constants Cn and D„:
Cn — Cn" •
Dn = Dn"
CoA/ + '2 iCshn' + Dsin')
s = l
Cojn'+ ^ iCsÜ + D,kn')
( n = 1,2, • • • ) j
(n = 1, 2, • • • )• (12)
The solution of this system can be f o u n d b y iteration. B y p u t t i n g CJ = - Co"-'^hJ + + D.<*-"t„0 s = l r ^ DJ = - C o ( - « i „ ° + ^ ( C . f - i ' i / + D,^'-»K') s = l
Cn and D„ are represented b y
00 00
= ^ (7/ Dn= 2
1=0 i=0
(13)
(14)
provided t h a t the convergence of the iteration process be guaranteed. The process itself can be interpreted b y the following mechanical direction :
a. Calculate (expressed i n Fourier series) the stresses a/ and occurring along the boundaries I and I I i n the unperforated plate.
b . Introduce i n the centers I and I I (identical) stress functions Fi° and F I I " such t h a t Fi° annuls the stresses ar" and Tr^" along the boundary I and t h a t Fn" annuls the stresses Cr" and Tr,," along the boundary I I .
c. Calculate the stress systems aJ Tr/ (expressed i n Fourier-series) called i n t o existence b y F i " and Fu" along the boundaries of the non-corresponding holes.
d. Introduce i n the centers I and I I new stress functions F^, which annul these stresses along their corresponding boundaries, etc.
The convergence of the iteration process is brought i n t o relation w i t h the " E i g e n w e r t " pi'oblem, defined b y the system
CoK' + 2 iCshn' + Ddn') ; f^Dn = —
s = l
+ Dskn (15) b u t i t is impossible to go i n t o f u r t h e r details.
The exposition given here contains (in principle) all data, required for the treatment of the following cases (all of w i i i c h i n the original treatise are considered i n f u l l detail, and calculated numerically):
Three holes (the centers of which are collinear) A n infinite row of holes
Three holes (the centers of which are not coUinear) T w o infinite parallel rows of holes (shifted or not).
I n the last t w o paragraphs of the treatise the plate w i t h one or t w o rows is reconsidered, now under the assumption t h a t each hole is loaded
b y a resultant force P, adequately distributed along the boundary of the hole. AU forces P are equal and perpendicular to the' row (or rows) as may be expected t o occur i n a riveted joint. The new element i n the treatment of these problems consists i n the starting f u n c t i o n of the itera-t i o n . I itera-t is shown itera-t h a itera-t itera-the m a x i m u m sitera-tress is smallesitera-t f o r itera-t w o nonshifitera-ted rows. I t is proved t h a t the least m a x i m u m stress occurring i n a plate w i t h shifted rows exceeds the m a x i m u m stress occurring i n a plate w i t h one row of holes (the t o t a l load of the plate being equal i n b o t h cases). 2. On the State of Stress in Perforated Strips and Plates. The preceding treatise restricted itseh exclusively t o infinite plates. I n a subsequent series of papers the same author draws attention t o the strip of finite w i d t h (and infinite length) w i t h one or more rows of holes (pitch = b). The calculations are performed only f o r a strip w i t h one row of holes, though an extensive program of f u r t h e r investigation was projected, which unfortunately could not be executed; the Jewish author, who lived i n Holland, was arrested and marched off to Germany or Poland, which meant his destruction.
The system of coordinates 0{yz) used now is rectangular; the y-axis coincides w i t h the center line of the holes. Consequently the stress functions Fo, Fn, Fn, Fn*, Fn* introduced i n the preceding treatise are w r i t t e n now as
Fo = Re I n X, F„ = Re x-'% F„ = -Im .-u"" (n ^ 1) Fn = Rex Fn* = -Imx .'u-'+i (w ^ 2).
Furthermore stress functions Fo'', Fn", FJ, Fn*", Fn*" are introduced, identical w i t h the preceding ones, every set related however t o a point 0' of the y-axis, having the abscissa y = kb. Finally these functions are expressed i n terms of the coordinates (yz) of the point 0, so t h a t , f o r instance, Fn" = Re{x — kb)-". The t o t a l system of identical stress functions Fn, related t o the i n f i n i t y of centers y = kb, can w i t h the aid of this transformation be represented b y one single stress f u n c t i o n Un
= Fn+/ {Fn" + Fn""), related t o the system of coordinates (){yz).
I t can readily be understood t h a t the generahzed stress f u n c t i o n
{x=y + iz) (1)
k
+ BnUn*) (2)
produces the same set of stresses <Tr, rr^ along the boundaries of all holes. Just i n the same way as i n the preceding treatise the functions F^,
F s * , Fs, Fs* were linearly combined, here the functions Us, U,*, Ds, Us*
are combined b y introducing functions U„s, Urs, Ü^s, Ürs, the first of which, f o r instance, is defined b y
Each of these functions produces a radial stress o-p and a shearing stress
Tr^ along the boundary I of all circular holes, on the understanding
how-.ever t h a t these stresses are composed of a main t e r m and an infinite series of minor terms; the main t e r m corresponds w i t h the stress t h a t would have been produced b y the corresponding functions Fes, Frs, F^s,
Frs.
I f restriction is made t o such functions t h a t satisfy the conditions of symmetry inherent to the strip under tension, the most general stress f u n c t i o n that can be brought into play is given b y
F = CoU,o+ ^ (CisU,,2s + DisUr,2s) + ^ ((?2.£^,,2a+l + D,s+1
s = l s=0
Ur.2s+l). (4)
I t produces along the boundaries I I and I I I of the strip ( 2 = ± c ) stresses
(Tz and Tyz, for which a Fourier series (as functions of y, and w i t h the
period h) can be developed.
On the other hand i t is required to construct a stress f u n c t i o n F' having reference t o the infinite half-plane such t h a t along the straight boundary of the stress field prescribed stresses of the type
to iO
= ^ C,/ COS 27r« ^ ^ ^ ^™ ^'^^ 5
Jl = 1 n = 1
come into existence.
This f u n c t i o n has the f o r m
F ' = I {Cs'F.s' + Ds'Frs') (6) s = l
w i t h
= ~ COS sr,, F'rs = ^^^"^ ^'J
I t is defined w i t h respect t o a system of coordinates, the y-a.xis of which coincides w i t h the straight boundary of the half plane, the s-axis of which points t o the inner region of the field). The stresses occurring along the boundaries I of circular holes, the centers of which have a m u t u a l distance
b and a distance c t o the boundary can be expanded i n t o a Fourier series
of the arguments t o cp of the holes.
N o w the unperforated infinite plate may be considered covered w i t h the contours I of the holes afterward to be introduced, and the boundaries I I and I I I of the strip, afterward t o be cut out of the plate; and a state of stress may be introduced as belonging t o :
a. a stress f u n c t i o n F i (4) w i t h respect t o the center of one of the holes
/3. a stress f u n c t i o n F ' n (6) w i t h respect t o the boundary I I
y. a stress f u n c t i o n F ' m (6) w i t h respect t o the boundary I I I 8. a u n i f o r m tension p' i n the direction y.
Then a l l means are available t o calculate the stresses o-j and Tyz along the boundaries I I and I I I and ar and T , ^ along the boundaries I f o r each of the cases separately. The requirement t h a t the t o t a l stresses along these boundaries be zero provides an infinite system of hnear equations for the unloiown constants C^n, Din, Cn', and DJ, v i z . :
Cn'
8 = 1
w ^ 0
D 2n Coi2n» + ^ (C2j2„28 + D^sh^J') +
n ^ 1
CJ CJ". CohJ" + ^ iC^shJ'^ + Duin"')
+ CJpJ + DJqJ n ^ 1
DJ DJ" CojJ" + ^ iC^JJ'^' + DM^^)
+ CJrJ + DJtJ w ^ 1 (8) i n which the quantities h, i, j, k, p, q, r, t, C" etc. are known.
This system is resolved i n an iterative way, f o r which a mechanical interpretation can readily be given.
Three quantities are of outstanding practical importance: (1) The magnitude of the t o t a l tensional force P = 2cp, to which the strip is sub-jected; (2) the maximal stress occurring i n the strip; (3) the mean elonga-t i o n of elonga-the selonga-trip.
The answers to these three questions are:
I I = I I Co + (72 - J'2 p' 2cp' be p'
2. The maximal stress occurs at the boundaries I , and can for different values X = a/b and JJ. = a/c be read f r o m Table 1:
T A B L E 1 Maximum Stress: p \ 0 0.20 0.40 0.60 0 3.0000 0 10 2.7677 3.0905 3.7510 0 20 2.3261 2.7121 3.5210 5.3256* 0 30 1.9956 2.3936 3.0741 4.8310 * Belongs to X = 0.15. 3
^ ^tjL.
' b p 2EcA l l calculations mentioned before are repeated f o r the bent strip and all required numerical data are p u t at the disposal of the reader. The same holds for the problem connected w i t h a row of rivet holes, i n the neighborhood of a straight boundary, which is solved under the condition t h a t the boundary stresses of every hole give rise t o a resultant force
P perpendicular to (and directed toward) the straight boundary, whereas
the reaction of these resultant forces consists of a continuous constant tension p = P/b acting at i n f i n i t y at the "opposite edge" of the infinite plate. The resultant force P is supposed t o be exerted b y a normal pres-sure (Tr = (2P/7rtt) sin (p acting on the upper half of the circular boundary of the hole; no other stresses on this boundary are present.
I t w i l l be clear t h a t only the principal features of the work could be outlined and t h a t a tremendous amount of computation hides itself behind the theoretical deductions.
3. Some Explicit Formulas of Use in the Calculation of Arbitrary
Loaded Thin-Walled CylindersJ
A n arbitrary load system iR,^,Z) acting upon a thin-walled cyhnder can be expanded into a double Fourier series;
CO 00 • M CO ^
R = ^ ^ a'pa cos P ' i ' ^ ^ + ^ ^ ^'^'3 cos p(p cos X ^ p = 0 a = 0 I P = 0 q = 0
M M QO to
"'"XX
^'^'^
sin >^^ + X X*^' ^^"^ ^ ~
(1)p = 0 a = 0 P = 0 g = 0
and t w o other expansions f o r $ and Z, which can be derived f r o m (1) by replacing the coefficients a'b'c'd' b y a", h", c", d", and a'", V", c'",
d'", respectively (p and q integers, X = irqa/l, a = radius, I = length of
the cyhnder).
I t is the aim of this treatise t o give explicit formulas f o r all internal forces and moments k^^, k^„ k,^, k^^, m^^, m,^, m,, occurring i n the cyhnder i f i t is loaded b y any of the elementary loads contained i n series (1) or b y any of the elementary loads of which $ and Z are b u i l t up. The derivation of the formulas caUed for a good deal of laborious computation, which, however, needed only t o be done once forever, and can be avoided now b y others.
4. The Elective Width of Cylinders, Periodically Stiffened by Circular
Rings. The purpose of this treatise is t o provide the reader w i t h a simple
rule f o r the computation of the greatest tangential stress occurring i n a thin-walled cylinder which i n itself possesses only a shght flexural r i g i d i t y against alteration of its circular cross section and which therefore i n con-stant distances ( = 2 0 has been stiffened b y circular rings. The load system of the cylinder is subject t o the condition of periodicity i n the axial direction, the period being identical w i t h t h a t of the rings; moreover the load is supposed t o be symmetrical w i t h respect t o the plane of sym-metry of t w o consecutive rings and t o consist of radial and tangential components only. A specialization of this load system presents itself i f aU loads are concentrated i n the planes of the rings, and as a matter of f a c t i n this paper attention is drawn only to such a special system. The justification of this restriction is mainly t o be sought i n its agreement w i t h actual practice b u t also i n the fact that the stress problem raised b y a general load system can always be decomposed into t w o other problems, one of which refers to a cylinder of length 21 loaded i n the pre-scribed way, b u t clamped at b o t h ends, while the other one relates t o the actual cylinder exclusively loaded i n the planes of its stiffening rings. The orthodox method of solving this latter stress problem consists i n the separate treatment of the rings and the cylinder, the first ones under the
action of unlmown radial and tangential load components, the last one stiffened under the action of their reaction components and the prescribed load AJ cos pi
system. The unknown load components can then be f o u n d b y equalizing system r„ the radial and tangential displacements of the corresponding points of the cients AJ cylindrical shell and the stiffening rings. The authors used this method b o t h conf i n solving a certain mineshaft problem and had thereby f u l l o p p o r t u n i t y rings are to take note of the laborious computation t h a t i t involves. I t was this A l l t h very problem t h a t led t h e m t o the approximate method developed i n this stress occ paper. is loaded
I f the cylinder, stripped of its stiffening rings, is subject to an a r b i t r a r y Fourier £ load system L i n each of its former ring planes, i t shows—if t h i n — a high Ap" sin y degree of flexibility, characterized b y large radial displacements. There ring of coi
exists, however, an infinite class of "characteristic" load systems Lp obtained (p = 1, 2, • • • ) w i t h radial and tangential components f j , = Bp'cosp^s, I t goe
tp = BJ' sin pip {(p designing the azimuthal coordinate), which, i n their rather te(
planes of application produce no radial displacements Mp b u t solely t a n - involved, gential displacements Vp, so t h a t i n these planes the cylinder seems t o and f o r a
lack any flexibility; the amplitudes BJ and BJ' of these characteristic \ graphs, ii components have a well-defined ratio. The circumferential elasticity of 7iV12a^, ' the cylinder can be measured b y the quotient Vp'.tp, which f o r all points case occu of the circumference appears t o be a constant. case the i
We now examine a circular ring of the same radius a and the same interpolal thickness h as the cylinder, and of the w i d t h I' = /xa. This ring too can values of
be loaded b y "characteristic" loads fp = BJ cos pip, ip = BJ' sin pip i n a grapl (but now u n i f o r m l y distributed along the w i d t h I', so t h a t the cross sec- the f o u r
tions of this ring remain plane), such t h a t no radial displacements Up correspon occur and only tangential displacements Vp are present. Again i t can 5. On
be stated t h a t this ring (which does not bend i n its plane) possesses a DistribiiU tangential elasticity t o be measured b y the quotient Vp-.tp, which f o r a l l The prob
points of the circumference proves to be constant. The w i d t h Z = is said t o represent the effective w i d t h Ip = jUj,a of the cylinder (and the
ring is said t o be equivalent t o the cyhnder w i t h respect t o its tangential A A M elasticity), if Sp :ip = Wp: tp. Consequently the cylinder and its equivalent
ring of effective w i d t h ïp = /Xpfl w i l l have the same tangential
displace-ments i f loaded b y characteristic loads irp,tp), (fp,ïp) resp. the i-compo- (u = defl
nents of which are equal. I t oan be shown t h a t the equality t = ip involves displacen — w i t h a high degree of approximation—the equality Vp = fp, and there- u* + u**
fore i t can be said t h a t the cylinder and its equivalent ring of effective (1) «'
w i d t h ïp = Hp-a behave similarly ( w i t h respect t o the displacements b o u n d a r j
Up ( = 0) and Vp occurring i n their midplanes) under the action of equal (2) (
characteristic loads (rp,i5j,). b o u n d a r j B y comparing a section of the stiffened cylinder w i t h an equally Herel
stiffened ring of effective w i d t h ïp = ix^a and loaded b y a system r = Ap' cos p<p, t = Ap" sin ftp, which differs f r o m the characteristic load system Vp = BJ cos p<p, tp = BJ' sin p<p (so t h a t the ratio of the coefh-cients AJ and AJ' differs f r o m the ratio BJ:BJ'), i t can be proved t h a t b o t h constructions are elastically identical, i n so f a r as the transverse rings are similarly loaded at their joints w i t h the cyhnderical sheU.
A l l this taken f o r granted, i t is easily understood t h a t the m a x i m u m stress occurring i n the cylinder, which i n the planes of its stiffening girders is loaded b y an arbitrary system L , can be f o u n d b y expanding L i n t o a Fourier series, computing f o r every component {r = AJ cos pip, t =
AJ' sin pip) the maximal tangential stress t h a t occurs i n the T-shaped
ring of corresponding effective w i d t h ïp = iXpO, and summing up the results obtained i n this way.
I t goes w i t h o u t saying t h a t the computation of the coefhcients np is a rather tedious matter, i n which the theory of cylindrical sheUs is f u h y involved. I t has been the aim of the authors t o do the computation once and f o r all, and the result of their attempt is laid down i n tables and graphs, i n which np can be f o u n d up t o p = 50 f o r f o u r values of /c = ftV12a2, viz. k = 10-*, 10"^ 10-" and l O " ' . Obviously any particular case occurring i n practice is related t o another value of k. I n such a case the procedure recommended is t o calculate (eventually b y suitable interpolation) the required effective w i d t h corresponding t o the f o u r values of & = 10-*,10-^,10-^10-''; t o represent these values as ordinates i n a graph, the abscissa of which indicates lg k, to join b y a smooth curve the four points so obtained and t o read f r o m this curve the ordinate corresponding t o the value of k under consideration.
5. On Circular Plates, Supported in a Number of Points {'^3) Regularly
Distributed Along Its Boundary and Rotatory-Symmetrically Loaded.
The problem is governed b y the differential equation
AAu^Ap; A = - + - - + - — ; A = ^^^^^ (1)
{u = deflection; p = local load; h = thickness of the plate). The
displacement u is decomposed i n t o t w o components u* and u** (u =
u* -\- u**) such t h a t :
(1) u* satisfies the differential equation (1) w i t h o u t satisfying the boundary conditions of the problem.
(2) (u* + u**) satisfies b o t h the differential equation and the boundary conditions.
t h e case u n d e r c o n s i d e r a t i o n is done i n s u c t i a w a y t h a t a t t h e b o u n d a r y r = R-^u* a n d <7r = 0. T h e d i f f e r e n t i a l e q u a t i o n f o r M * * o b v i o u s l y is AAu* 0. (2) T h e b o u n d a r y c o n d i t i o n s a r e : (a) « * * = 0 i n t h e p o i n t s of s u p p o r t , ( b ) ffr = 0 a t r = i ? ; (c) shear f o r c e ( p r o p o r t i o n a l t o u* + zero i n a l l p o i n t s r = R, e x c e p t i n t h e p o i n t s of s u p p o r t , w h e r e i t has a p r e s c r i b e d v a l u e , v i z . one /c**" of t h e t o t a l l o a d , i f k designs t h e t o t a l n u m b e r of s u p p o r t s . T o c o v e r t h e m o s t g e n e r a l case of l o a d i n g , u* is c a l c u l a t e d u n d e r t h e a s s u m p t i o n t h a t t h e p l a t e is l o a d e d o v e r t h e a n n u l a r s u r f a c e 2Tradro o n l y (0 < ro < R) b y a c o n s t a n t pressure p ; i t is f o u n d t h a t t h e ( i n f i n i t e s i m a l ) d e f l e c t i o n is g i v e n b y du* = 3(m2 - 1) proR^ m — 1 2 ( m + 1) ' ^ 2 ^ \R2 ^ I n R i f 0 ^ r ^ ro, a n d 3(m2 - 1) proR2 ( 3 m + 1) / r l _
