On stress calculations in helicoidal shells and propeller blades

(1)

•jf^'i^t iimv'

ON STRESS CALCULATIONS IN HELICOIDAL SHELLS

AND PROPELLER BLADES

(2)

O n stress calculations in helicoidal shells

and propeller blades

PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGE-SCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR O. BOTTEMA, HOOGLERAAR IN DE AFDELING DER ALGE-MENE WETENSCHAPPEN, VOOR EEN COM-MISSIE U I T DE SENAAT TE VERDEDIGEN OP WOENSDAG 6 JULI 1955, DES NAMIDDAGS

TE 4 UUR

DOOR

JACOB WILLEM COHEN

WERKTUIGKUNDIG INGENIEUR G E B O R E N T E L E E U W A R D E N

(3)

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR IR W. T. KOITER

(4)

Het Delftse Hogeschoolfonds ben ik zeer erkentelijk dat het mij als assistent in de gelegenheid heeft gesteld deze studie aan te vangen. Aan het Studiecentrum T.N.O. voor Scheepsbouw en Navigatie betuig ik mijn hartelijke dank voor de verleende hulp bij het uit-voeren van het rekenwerk van hoofdstuk III en het uitgeven van dit proefschritt. Daarnaast dank ik het Mathematisch Centrum te Amsterdani voor het uit-voeren van deze berekeningen.

(5)

Aan mijn Ouders

Aan mijn Vrouw

(6)

(7)

E R R A T A

p. XI line 20, for: Q„ read: Q"

p. 21 fig. (1.12. f 1), the moment vector at/?2 should be reversed. p. 22 form. (1.12.5), for: - K^" H M^" read: K"" M^". p. 25 last sentence, for: R" and /?" read: IjR" and l//^^

p. 33 and 34 form. (2.10.16), (2.10.18), (2.10.19),

a a

for: / read: / , where a„ is a constant and in the 5 «c

calculations equal to 7TJ9.

(10)

XI

NOTATIONS

x,y, z, rectangular coordinates with respect to a left-handed triad of axes. a, p, curvilinear coordinates.

/^— - r ' > Gaussian quantities of the first order.

M = \' KG — F ,j

L, M, N, Gaussian quantities of the second order.

n', [i = 1, 2, 3) direction cosines of the normal.

a' (b'), (i = l, 2, 3) direction cosines of the tangent to the parametric curve /3 (a) of the middle surface.

~D^ \~Dfi)' 'Töfl ( T ^ ) ' curvature and torsion of the parametric curve /3 (a) of the middle surface.

dl" (s) {dl^{s)), length of a linear element at a distance s from the middle surface in the direction of a

parametric curve fi (a).

(p (j), angle between the directions of (//" {s) and dl^ {s).

e" (e^), unit elongation in the direction of a parametric curve /3 (a) of the middle surface. •tp"^, change of angle between the directions of parametric curves /? and a of the middle

surface.

e" (s) (e^ (5)), unit elongation in the direction of a parametric curve /3(a) at a distance s from the

middle surface.

yj"^ (s), change of angle between the directions of parametric curves /3 and a at a distance s

from the middle surface. x", x^, H"^, X^", changes of curvature.

M" (M^), M"^ {M^"), bending and twisting moment per unit length on the lateral side of a shell element. K", K^, K"^, K^", Qa, Q^, normal and shear forces per unit length on the lateral sides of a shell element. l"" (W i"? f^a

s""', s^i

u, V, w, displacements in the directions of a', b' and n', respectively. Eh^ ^ Eh

D = — ^ , C

^a^ ^fia j Stress quantities.

1 2 ( 1 - ^ 2 ) ' \ - v^

E, modulus of elasticity. V, Poisson's ratio. h, thickness of the shell.

p", /»', /)", loads per unit area in the directions of a', b' and n', respectively. H = 2 na, pitch of a helicoid.

D^ = 2 r^, Df = 2 ^2, root and tip diameter of a helicoid. HjDt, face pitch ratio.

r, (p, polar coordinates.

Q = r/ri,

Af, Af', radial and tangential bending moment per unit length in a circular or sectorial plate. Q'', Qf, V, F ' , shear forces and reduced shear forces per unit length in a circular or sectorial plate. a0 (fi 0), real (imaginary) part of a complex quantity 0.

?,j , ( i = 0 , 1,2...,) eigenvalues for a sectorial cantilever plate. Wj, {j=0, 1,2...,) eigenfunctions for a sectorial cantilever plate.

2 0 opening angle of a sectorial plate.

W, section modulus.

S , denotes summation from 2 = 1 to i ^ 3 inclusive. (1.2. f 1), denotes first figure in para. 1.2.

(1.2. t 1), denotes first table in para. 1.2. [ x ] , denotes reference x of bibUography. (a. 10), denotes formula 10 of appendix a.

(11)

(12)

GENERAL INTRODUCTION AND SUMMARY 1

G E N E R A L I N T R O D U C T I O N A N D S U M M A R Y

T h e object of this study is to gain an insight into the stress distribution in statically loaded propeller blades so as to provide a basis for a justified strength calculation for this type of loading.

A method commonly used for strength calculations in ship's propellers is T a y 1 o r's method [ 1 ] , where the maximum stress is determined by the familiar formula ffmax = MjW and where a special interpretation is given to the bending moment M and the section modulus W.

R ö s i n g h [2] raises serious objections against the application of Taylor's method. In order to gain a better insight into the stress distribution, Rösingh has carried out stress measurements. I n this investigation the stresses were measured at the circumference of two plane cross-sections of the propeller blade. From the results of the measurements, Rösingh developed a formula for the maximum stress occurring in a cross-section. This formula proved to yield a somewhat better approximation than Taylor's formula.

A few months after the publication of Rösingh's paper B i e z e n o [3] published the results of stress measurements made at the circumference of four plane cross-sections of a propeller, two of the cross-sections having been chosen at a large distance from the root. The results of Biezeno's measurements for all cross-sections agreed very well with the values obtained for the same propeller by the use of Rösingh's formula. However, we note that Taylor's formula appears to agree equally well with Biezeno's experiments. It also appeared that the stress concentration in the fillet between the blade and the hub could be calculated accurately with the aid of a formula for the stress concentration factor given by V a n M a n s u m , K o c h and B i e z e n o [4].

R o m s o m [5] has also carried out stress measurements on a ships's propeller. He does not apply Rösingh's method, but proposes a stress calculation based on Taylor's method, where the section modulus W is determined empirically.

In the present investigation a theoretical approach to the problem of stress calculations for statically loaded propellers will be used. It will be clear that the theory of elasticity must be applied if this course is taken. Unfortunately, there are only a few cases in which the true solutions of the equations of the theory of elasticity can be found, and the determination of the stress distribution in ship's propellers certainly does not come into this class.

T h e usual way of dealing with problems of this kind is to make assumptions with regard to the nature of the stress distribution. In other words, a model is made of the actual structure and its loading. The requirements on such a model are, firstly, that the stress distribution occurring in it is sufficiently representative of that encountered in the actual structure and, secondly, that it is better suited to a theoretical approach. Before giving a more detailed account of the procedure to be followed, a rough description of the geometrical shape of a propeller blade will be given which will be confined to unraked propeller blades of constant pitch. One of the bounding surfaces of the propeller blade, which is called the face, is part of a right helicoid bounded by two cylinders (radii r^ and r^, r^ > r-^, the axes of which are coincident with that of the helicoid. The ratio of the helicoid's pitch H to the outer diameter Z)( ( = 2 r^ varies firom 0.6 to 1.4 in ship's propellers. For wide-bladed propellers the greatest width is about 0.5 — 0.9 Z)(, the vertical projection of the blade onto a plane perpendicular to the axis of the helicoid having a roughly elliptical outline, with a rapid increase of blade width outwards from the root. T h e plane development of the cross-section of the propeller blade and a coaxial cylinder has the shape of a circular segment near the outer edge, whilst an airfoil section is often used near the root. The thickness of the propeller blade is usually a linear function of the radius, decreasing outwards from the root. T h e thickness of the blade at the root is approximately 0.05 — 0.10 Z),.

(13)

2 GENERAL INTRODUCTION AND SUMMARY

T h e thickness of the propeller blade, therefore, is small as compared to its width and length so that the propeller' blade may be considered as a shell, i.e., a body of which one dimension is small as compared to the other two. When the propeller blade is considered as a shell, it is possible to make a few assumptions on the elastic behaviour of the shell which simplify the problem to a certain degree.

Although the reasoning above has virtually provided a model of the propeller blade, it is hardly suited to a theoretical approach.

A model of the loaded propeller blade by means of which some insight may be gained into its stress distribution is a shell the middle surface of which is a right helicoidal sector, whilst its thickness and loading are functions of the distance to the helicoid axis only. This shell is clamped along the cylindrical helix r = r^o{ the helicoid and free at both radial edges and at the outer edge formed by the cylindrical helix r = rj. T h e loading and the geometrical dimensions of this model can be chosen so as to conform to those of the actual propeller blade.

T h e equations of the theory of shells have been discussed by many writers but their results are nearly always given for the lines of curvature as parametric curves. For helicoidal shells the asymptotic lines are more appropriate and, therefore, we start in chapter I with a systematic derivation of the equations for infinitesimal displacements of homogeneous, isotropic, elastic shells, referred to arbitrary orthogonal parametric curves. The most important results of this theory for practical applications are the relations between the strains and the displacements

u, V and w of points situated on the middle surface of the shell, the stress-strain relations, the

equations of equilibrium for a shell element and the boundary conditions.

In chapter II the theory of shells as developed in chapter I is applied to helicoidal strips, i.e., to shells the middle surface of which is a right helicoid, unbounded in the direction of the cylindrical helices of the middle surface. The strip is clamped along the cylindrical helix r = Tj and free at the outer edge formed by the cylindrical helix r = rj *. The load and the thickness are functions of the radius r only. For the sake of simplicity it is assumed that the load in the direction of the radius is zero. In ship's propellers this load is due to the cen-trifugal force. Since this load has hardly any effect on the stress distribution in unraked pro-peller blades there is no objection against leaving it out of consideration. However, there are no difficult problems involved in determining the effect of such a load.

When establishing the differential equations for the displacements of points of a shell, use has to be made of the stress-strain relations. These relations can usually be reduced to a simpler form by deleting a number of terms. Chapter II gives an examination of both the solutions obtained from the approximate relations and those obtained from the more accurate stress-strain relations; a uniform shell thickness has been assumed in these calculations. O u r cal-culations show that it is not always permissible to apply the approximate stress-strain relations to helicoidal shells. Besides, another approximative method is described in chapter I I . This method, which is called the quasi-statical method, is based exclusively on considerations of the equilibrium of a narrow sector of the shell. For helicoidal shells having a large face pitch ratio the method gives a very good approximation of the distribution of radial moments.

For a helicoidal strip having a face pitch ratio HjDt -^ 1 it appears that the distribution of bending moments has much in common with that in a complete circular plate of the same radial dimensions, except for a small area near the clamped edge.

At the end of chapter II the distribution of bending moments is examined in helicoidal

* Several papers on helicoidal strips have been published recently [24], [25], [26]. However, these papers a p p e a r e d after our investigations, contained in chapter I I , h a d been completed. Solomon's results are in agreement with our a p p r o x i m a t e solution of p a r a . 2.9, Mikhlin's results confirm the general conclusions of p a r a . 2.15,whereas Reissner's results a r e n o t relevant for o u r problem.

(14)

GENERAL INTRODUCTION AND SUMMARY 3

strips the thickness of which is a linear function of the radius. It appears that the distribution of radial moments is affected only slightly by the variation of the thickness.

For the calculation of the stress distribution in a helicoidal cantilever sector the three differential equations for the displacements u, v and w can be set up with the aid of the equations of chapter I. It proves to be virtually impossible to obtain the solution that satisfies these simultaneous partial differential equations and the associated boundary conditions. However, the similarity between the distributions of bending moments in circular plates and those in helicoidal strips having a face pitch ratio HjDf -^ 1, leads to the conjecture that an insight can be gained into the distribution of bending moments in a cantilever helicoidal sector from the distribution of bending moments in a sectorial cantilever plate which is clamped at the inner edge.

In chapter H I the distribution of bending moments in a uniformly loaded sectorial cantilever plate is examined. The solution to this problem is governed by the biharmonic equation in polar coordinates r and q).

Solutions of this equation are constructed of the type w = ƒ (A, r) cos X <p, where 2 is a para-meter, and which satisfy the boundary conditions along the clamped inner edge and the boundary conditions along the free outer edge. A doubly infinite denumerable set of such eigenfunctions is obtained. With the aid of a finite number of these eigensolutions, a solution is constructed which approximates the boundary conditions along the radial edges. Although the approximation of the boundary conditions, especially for the shear forces, is not as good as desired, it is shown that a satisfactory approximation is obtained for the distribution of moments near the axis of symmetry.

From the solution obtained for the distribution of radial moments in the axis of symmetry of the uniformly loaded sectorial cantilever plate it appears that the bending moments are only sHghtly larger than in a uniformly loaded circular plate of corresponding dimensions, except in the area near the clamped edge, where the moment in the sectorial cantilever plate increases much more rapidly towards the root than it does in the circular plate.

T h e distribution of radial moments in the axis of symmetry of the sector is also compared to the distribution obtained when a sectorial element is calculated as a cantilever beam. It appears that, almost along the entire length, the bending moments obtained by the theory of beams are considerably larger than those according to the theory of plates. Only near the clamped edge is the difference small.

In chapter IV the results of the preceding chapters are used to derive tentative upper and lower bounds for the distribution of radial moments in the axis of symmetry of a cantilever helicoidal sector having a face pitch ratio HjDf --^ 1 and a thickness and load which are functions of the radius only. The upper bound is calculated by means of the quasi-statical method. The lower bound is determined by the distribution of bending moments in the helicoidal strip. T h e radial bending moments in the axis of symmetry are the critical moments from the point of view of strength of the sector. Since the upper and lower bounds obtained for these bending moments do not differ too much, it is possible to perform reasonably accurate strength calculations for such helicoidal sectors.

Finally, Taylor's method and Rösingh's method are compared to the results obtained in the present investigation. It appears at once that Taylor's method of calculation is equivalent to the quasi-statical method of chapter I I . Remembering that the quasi-statical method is slightly conservative, Taylor's method may, therefore, be recommended for practical strength calculations. In order to permit a comparison with Rösingh's method of calculation, this method is applied to two helicoidal sectors. It appears that Rösingh's method yields satisfactory results for not too wide hehcoidal sectors, but the stresses obtained for wide helicoidal sectors are somewhat too small.

(15)

1.1 GENERAL THEORY OF SHELLS 5

C H A P T E R I

G E N E R A L T H E O R Y O F SHELLS 1.1 Introduction

The stress and strain distribution in loaded bodies can be determined from the equations of equilibrium for the stresses and the compatibility conditions for the strains if only infinitesimal displacements are considered. However, due to the difficult mathematical problems involved, these equations can be solved rigorously in a few cases only; therefore, it is necessary to find methods of approximation.

In the theory of elasticity for shells for infinitesimal displacements the approximations are based on the characteristic of shells that the thickness is small as compared to the other two dimensions. Various theories of shells have been developed in the course of years. T h e assump-tions on which these theories are based are in better agreement with actual condiassump-tions as the thickness of the shell is smaller compared to its other dimensions.

The theory developed by L o v e [6] is originally based upon the following assumptions. i. Points on a normal to the middle surface of the shell before deformation will all remain

on one normal to the strained middle surface.

ii. A linear element on a normal to the unstrained middle surface will not be elongated or contracted.

iii. The normal stresses on surfaces parallel to the middle surface are neghgible as com-pared to the other normal stresses.

T h e deformation of the immediate vicinity of the shell's middle surface is completely deter-mined by i and ii once the deformation of the middle surface of the shell is known. Due to the introduction of i, this theory neglects the shear deformation between a direction in the middle surface and the normal to the middle surface. Another objection is that assumptions ii and iii are incompatible. T h e latter objection is met by Love by introducing a refinement. Generally speaking, Love's first approximation gives satisfactory results for engineering applications, provided that the displacements are small with respect to the thickness of the shell.

T r e f f t z [7] developed a theory based upon iii and the assumption that the state of stress at any point is essentially plane; the stresses are assumed to be linear functions of the distance s to the middle surface. T h e relations between the strains and the stress resultants are derived from these assumptions by applying Castigliano's theorem. For a first approxima-tion, i.e., neglecting the product of the thickness of the shell and the curvature with respect to unity, the results obtained are in agreement with the corresponding approximation given by Love.

In recent years many attempts have been made to refine shell theory by a more systematic and accurate approximation, e.g., [8], [9], [10], [11], [12].

As stated above. Love's first approximation is sufficiently accurate for engineering applications. Unfortunately, this theory has been formulated only for a special system of curvilinear coor-dinates, i.e., the lines of curvature of the middle surface. More convenient coordinates for helicoidal shells are the generators and the helices, i.e., the asymptotic lines. This chapter, therefore, gives a derivation of the equations of shell theory in Love's first approximation for arbitrary orthogonal curvilinear coordinates on the middle surface.

T h e line of thought to be taken in this development will be briefly sketched before going into detail. If the displacements of points on the middle surface of the shell are known, the

(16)

6 GENERAL THEORY OF SHELLS 1.2 expressions for the strains of the middle surface (extensional strains) may be derived by means of the differential geometry of curved surfaces. T h e relation between the metrics of the im-mediate vicinities of strained and unstrained middle surfaces of the shell is given by the first two assumptions; hence the strain at a generic point at a distance s from the middle surface may be expressed in the strains of the middle surface itself, and in s and the changes of curva-ture of the middle surface. T h e strained state of the shell is now completely expressed as a function of the displacements of the middle surface. Once the stresses on a shell element have been defined, six equations of equilibrium can be established for the ten stress resultants. By means of Hooke's law and the third assumption, the relations between eight stress resultants and the strains are obtained; the differential equations for the components of displace-ment of a point on the middle surface now follow from the equations of equilibrium, of which one appears to be an identity, by elimination of the remaining two stress resultants. T h e boundary conditions for these displacements are obtained by the well-known method of T h o m s o n and T a i t [13].

1.2 Some concepts and formulas from the differential geometry of curved surfaces

When describing the relations between the geometrical quantities of curved surfaces, it is convenient to use Gaussian coordinates. With respect to the left-handed Cartesian system of coordinates OXYZ the curved surface may be defined by the formulas

x = x^{a,P), y=x^a,^), z=x^[a,fi), (1.2.1)

where a and /? are parameters; the functions x'{a, /?) {i = 1, 2, 3) and their derivatives u p to the highest order occurring in the equations are assumed to be continuous. The distance

dl between points P{a, ft) and Cl{a -{- da, ft ^ dft) on the surface is given by

rf/2 = Eda^ + 2Fdadft + G dft\ (1.2.2) where E, F and G are the Gaussian quantities of the first order, defined by *

, £ = 2 ( x g ^ F=^xi,x^„ G = 2 ( ^ V ) ' - (1-2-3) It is convenient to put

A = E'l', B = G'l', H = {EG - F^)'''. (1.2.4)

For a constant value of a(/J), the geometric locus of the points for variable ft{a), by (1.2.1), is a curve on the surface, which will be called the parametric curve a {ft). Since each point P on the surface is the intersection of a parametric curve a and a parametric curve ft, two directions can be defined at each point of the surface with respect to these parametric curves, viz.: the direction of the tangent to the parametric curve ft and the direction of the tangent to the parametric curve a. A third direction at point P is defined by the direction of the normal to the surface at P. T h e cosines of these three directions with respect to OXYZ be indicated by

a\ ¥ and n' [i = 1, 2, 3.) It is easily seen that

a' = 'Ux\, b' = 'lBx',, (1.2.5) n' = 'IH {x'a x% - x^n x\}, (1.2.6)

where i, j , h is a cyclic permutation of 1,2, 3. T h e positive sense of the direction defined by the parametric curve a{ft) is in the sense of increasing ft{a); the positive sense of the normal has been selected in such a way that the system of directions a', ¥ and w' in the order mentioned is left-handed.

*) Differentiations with respect to a or /5 are indicated by a subscript, e.g., x'^ = ^—. All summations are

(17)

From (1.2.3) and (1.2.5) it follows that

F^AB'La'b', (1.2.7)

where "La'b' represents the cosine of the angle between the directions a' and b'; these directions are orthogonal if F = 0.

T h e curvature of the surface is completely described by (1.2.3) and the Gaussian quantities of the second order

L = S x\, n*, M = S x\p n\ jV = 2 x^^ nK (1.2.8) It is convenient to define 1 1 / 1 L 1 'E' R^ ~G 1 M F L 1 M F JV H ^ H E' T"" H H G' (1.2.9) 1 / 1

\vhere-—I —^ I and—j^l—^- will be called "curvature" a n d "torsion" of a parametric curve ft{a), although generally they are not the actual curvature and torsion of the parametric curve ft{a) [14].

1.3 The metrics in the vicinity of the middle surface

Formula (1.2.2) describes the metrics on the curved surface completely. In order to (examine the deformation of a shell, we must also know the metrics in the vicinity of the middle surface.

T h e distance between two points R and R^ (1.3. f 1) in the vicinity of the middle surface is given by

dl^ = 2 {d{xi + s n^Y = 2 {(x*„ + J n\) da + (A:'^+ S n%) dft + n* dsf.

Remembering the definitions oï E, F, G, A, B, a\ b' and n\ and the orthogonality of n' and the directions a*, b\ we may write

dl^ = {E + 2si: Aa*n\ -^ s^I, («'J^j ^„2

+ {2F + 2sl. {Aa' nip + Bb'«'„) + 2 J^ 2 n\ w'^} dadft

(18)

8 G E N E R A L T H E O R Y OF SHELLS _1.4 O n the assumption that the parametric curves a and ft are orthogonal, the expression (1.3.1) may be simplified by means of the formulas (a.21) of appendix a. The result is

+ f^ j (l - :^y + ( y i j j ^/^^ + '^''- (1-3-2)

T h e length dl''{s) of a linear element for which ds = 0, dft = 0 is now given by S \2 1'/,

dr{s)=.E'''}^[i-^j + M I '^^ ^^-^-^^

In a thin shell we may restrict our attention to the immediate vicinity of the middle surface. Hence sjR" {sjR^) and sjT"^ i^lT^") are small as compared to unity; therefore, terms of the second and higher degrees in these quantities will be neglected. We now write

and, likewise,

dl''{s) = A[\-^]da, (1.3.4)

dl^{s)=B{l-^]dft. (1.3.5)

Let (p {s) denote the angle between the directions of the linear elements dl" {s) and dl^ {s); its cosine is given by (cf (1.3.2))

cos9>(j) = - ^ ( ^ - ^ j , (1.3.6) where squares and higher powers of sjR", etc., have again been neglected. Putting

cp{s)=j + r^{s), (1.3.7)

we may write

o^{s)=-s{l,^~^^. (1.3.8)

1.4 The deformation of the middle surface

As soon as the displacement of every point of the middle surface is known, the strained surface is completely defined. Let «', u^, u^ denote the components of the displacement of a point P of the middle surface in the directions of the positive X, Y and Z axes. Since the surface is defined by (1.2.1), it is assumed that u^, «^ and u^ are also given as functions of a and ft, and that these functions are single-valued. It is also assumed that these functions and their first and higher derivatives are continuous.

Let X, y, ~z *) denote the coordinates of point P after deformation; the strained surface in relation to the triad of axes OXYZ is then given by

~x = x^{a,ft) +u^{a,ft), } = xâ,ft)+uâ,ft), I ^ x'^ {a, ft) + uâ, ft). (1.4.1)

(19)

1.4 GENERAL THEORY OF SHELLS 9 T h e distance dl between two points P {a, ft) and Q {a -\- da, ft + dft) on the unstrained middle surface is given by (1.2.2). O n the strained surface the distance dl between these points is given by the corresponding formula

d? = Ëda^ + 2Fdadft + G dft^, (1.4.2)

where

Ë = 2 {(x^- + u'),Y, F = 2 {x' + «'•)„ {x* + u')^, G = 2 {(A:^ + u*)^}^ Remembering (1.2.3) we may put

Ë = £ + 22x^„^^^„ + 2 ( « g ^ (1.4.3)

F = F + ^ {x\ u'l, + x'p u\) + 2 u\ «V > (1.4.4)

G = G + 22xV^/V + S ( « V ) ' - (1-4.5) T h e unit elongation is defined by

dl - dl dl '

the unit elongations e" and e'' of linear elements on the unstrained surface along the parametric curves ft and a are thus given by (cf (1.4.2))

«' = 7^, • (1-4.6)

E'l' ' G'l'

T h e angle between the two directions will also change under deformation. Before deformation, the angle cp between the directions with direction cosines a' and b* was given by

F

cos (p = T;ir-7,ir • _{E'' G''}

T h e angle 95 between the directions of the parametric curves a and ft on the strained surface is, therefore, given by

F

cosw = ——^-. (1.4.7)

T h e change of angle xp"^ is now defined by

ip"^ = cp -'^. (1.4.8)

O n the assumption that the displacements a' are small we omit non-linear terms in the deriva-tives of the displacements. We may now rewrite (1.4.3) . . . (1.4.5)

E-E = 2i:x\u\, F-F=i:{x\u'i, + x*pU':), G - G ' = 2 2 x^ «V • (1-4.9)

The relations between the strains e" and s^ and the displacement components can now be simplified to

E" = — 2 x'„ u',, / = — 2 XV «V- (1.4.10) In order to obtain a simple expression for the change of angle y)"^ (1.4.8) we now assume

that the parametric curves a and ft are orthogonal, i.e., F =0. Omitting again all non-linear terms in the displacement derivatives we obtain

/ ^ = ^ ^ = - - - 2 (x'„ « V + ^V «'.) • (1 •4.11)

AB AB

T h e strains of the middle surface, defined by (1.4.10) and (1.4.11), will be called extensional strains.

(20)

10 GENERAL THEORY OF SHELLS 1.5 1.5 The deformation of the immediate vicinity of the middle surface

T h e analysis of the deformation in the immediate vicinity of the middle surface is based on the two assumptions i and ii of para. 1.1. In the unstrained state we considered two points

R {a, ft, s) and R^ {a + da, ft -\- dft, j + ds) (cf. (1.3.f 1)). O n account of the assumptions i and

ii of para. 1.1. the distance dl between these points in the strained state is (//2 ^ 2 { ö ' ( x ' + sn')Y

= 2 {{x\ + s n\) da + (x'^ + s n*p) dft + n' dsf. (1.5.1)

This expression may be rewritten in a form similar to (1.3.1)

dl"" =^{Ë + 2sAI. ~a'n\ + s^Z {n\)^} da^

+ {2 F + 2 J 2 ( J a' «V + B~bi n\) + 2 J^ 2 n\ ra^} ^« ^^

+ {G + 2 J 5 2 é* Hv + ^^ 2 ( « g ^ i dft^ _^ <^_y2_ ( 1 5 2 ) T h e parametric curves a and ft do not form a system of orthogonal trajectories on the strained

middle surface. Therefore, we cannot expect that (1.5.2) can be simplified in the same way as (1.3.1). We introduce, as in (1.2.9) 1 R" L — — E' 1 N R^ G ' 1 — 7-«/3 M F L 1 H HE 1 M H F N H ;i.5.3) ; 1.5.4) {EG - F^)'l' = AB,

because /^ is a small quantity of the first order in the displacement derivatives (cf (1.4.9)). Introducing (1.5.3) into (a. 12) for the deformed state we obtain

^ 2 a' n\ = B 2 ¥ «V = E R" G Rf B'Lb* n\ = A 2 a' n\ = H F H F T" R"

Disregarding quadratic terms in F, we obtain in a similar way from (a. 15 for the deformed state

1 '^2 2 {n\)' = E

+

{n',Y

'='^F

+

J-/30 2 n\ nK = V2 a 1 1

+ /?J \T»"

i^-'^^l(i

/?" R^I \T^" T"^! ' " \\R By means of (1.5.4) . . . (1.5.6) we can reduce (1.5.2) to

d,^Ë

\(l-^X+l^X\da^-+

1

+

y-/3a j - a ^ :i.5.5) 17) ;i.5.6) s R" s

(21)

T h e new length dl" {s) of the linear element ds = 0, dft = 0 is now

dl^s) =A(\ - ^ ) , (1.5.8)

and, likewise,

d?{s) =B(I - ^ y (1.5.9)

where the same approximations have been made as in (1.3.4). The angle (p {s) between the directions of these linear elements is (cf (1.4.11) and (1.5.7)) to the same approximation given by

cos w {s) = w"^ — s I ^= ^ ^ Hence (cf (1.3.7))

i;.{s) = -rp"^ + s{±-2-^. (1.5.10)

The strains, defined by

a,. . d? {s) - dr {s) d? {s) - dl^ {s) I

' ^'' dl"{s) ' ' ^'' dl^{s) ' (1.5.11)

y"^ (j-) = (p {s) — ^ (j) = w {s) — w {s), I

may now be written down from (1.3.8), (1.4.6), (1.4.11), (1.5.8), (1.5.9) and (1.5.10)

s" {s) =8" ~s x", e^ {s) = e" - J x^, y,"^ {s) = w"^ - ^ {--^J" - ^^Z"). (1.5.12)

where the "changes of curvature" are given by

x'' = l- L, ^^ = i L, xj'^ = 2 -, p./" = i - . (1.5.13)

Da Da R^ R^ T"^ T"^ T^" T^"

The second terms in the right-hand members of (1.5.12) will be called the flexural strains. The expression for ip"^ {s) may be simplified by disregarding all non-linear terms. Remembering (1.4.11) a n d (1.5.3) we rewrite (1.5.13)

R_J^] + fl ,^^-(R_^]_rl (1514)

H HJ^R-' ""^ - \ H Hj RO- ^ ' Defining ; , ' " ' = - „ " ^ = ^ _ . ^ , (1.5.15) H ff we obtain x/" - x:^ = 2 .^" - rp"^ ( i „ + i ^

Neglecting •f (-5^ + ^ with respect to unity, we obtain the final formulas for the strains

e" {s) = e" -sx", y,"^ {s) = y,"!" - 2 s x^^", e^ {s) = e^ - s x^. (1.5.16)

1.6 The strains expressed in the displacements

T h e displacements a' {i = 1,2,3) as defined in para. 1.4 are the displacements of a point on the surface in the directions of the X, Y and Z axes. Let u, v and w denote the displace-ments of the same point in the directions of a', b* and n'-; we may put

(22)

12 GENERAL THEORY OF SHELLS 1.6

:i.6.2)

;i.6.3) Differentiation of (1.6.1) results in

u\ = a% + b% + n'w^ + a\u + b\ v + n\w , u'p = a'Up + bhfi + n'Wp + a'^ M + b*^ v + n*^w,

and

u\a=a'u,„ + ¥v,„ + n'w,, + 2a\u, + 2b\v, + 2n\w„ + a\„u + b\^v + n\^w, u'pf, ^â'Ufiji + b' Vfip + n' W^D + 2 a^ "/J + 2 b*^ y^ + 2 n'f, Wp + a'^f, u-\-b'f,f,v-\- w'^^ w, u'an = a'Kis + b' v^p + n' w^^ + a\ Uf, + b\ Vf, + n\ w^ + a^ «„ + é > â + «V â

Expressions (1.4.10) for the normal strains are now reduced by means of (1.2.4), (1.2.5), (1.6.2), (a. 19) . . . (a. 21) to (1.6.4) (1.6.5) (1.6.6)

'^

A"^ AB'^£^ = "" + ^' . -B ' AB w R"' w

In a similar way we reduce the expression for the shear strain to

^ B ^ A AB " AB' " T^"' U r ^ V — 2

In order to express the changes of curvature x", x^ and x^" in u, v and w, we have to consider the expressions

2 x \ „ n\ 2 x'„^ n* and 2 x'^^ n\ By writing 2 x\^ n* = 2 x\^ w' + 2 M^^ ra% etc., and disregarding all non-linear terms, we have

2 u\a w' = 2 M»„„ n', etc. From (1.6.3) we obtain

2ü«„„ ra< = »„„ + 2 2 (M„ a ^ n ' + y„ è'„;2*) + 2 {ua\, «' + vb\„ n* + w «'„„ n^, etc. Reducing this expression by means of (a. 19), (a. 20), (a. 22) . . . (a. 24) we find

A A

2 u\, n' = w,, + 2u,— + 2v, y^„

A A,

^''

B

T^"

U"jJ

'^^

{ B

R"

A A, 2 u'pn n* = B B ^w + 2 "^ ^ a + 2 z;^ - ^

I n a similar way we have

B_B^ A T^"

w

yl ^ 5 5 2 M'„^ «' = «'a^ + "^ - ^ + »/J 7 ^ + «a ^ + ^'a ^

"I(FI-^1+"I(^1"^1

^ 5 f 1

AV

I A

Y

5 \2 5 «^ 2-/ia I Tja + ^^ ;i.6.7) (1.6.8) ;i.6.9) Before considering the expressions 2 x^^ n', etc., we have to derive the expressions for n\ the direction cosines of the normal to the strained middle surface. According to (1.2.6),

(23)

«* = ^ {(^'a + U\) {X% + U"^) - ( X \ + U\) {X^f, + M^)}, H

where i, j , h is a cyclic permutation of 1, 2 and 3. Disregarding non-linear terms, we get 1

where Hence

n' = ^ {x'„ x"^ - x\ x'p + x\ u% - x'f, u\ + u\ x"^ - u^^ x^J., H

H= {EG - F^)'l' = AB (1 + e" + e")

^1.6.10)

n* = «'•(! — e" — £^) + ^ - {a' «"^ — a" u'^) {b'u^„ — b" u'^), (1.6.11) B A

where i, j , A is a cycHc permutation of 1, 2, 3. The expressions for x'„„, etc., are given by (a. 26). Because 2 x'„„ n' is independent of the choice of our rectangular frame of reference, we calculate this expression, for the sake of simplicity, for the frame of reference coinciding with the directions of a', b' and n\ i.e.

«1 = 1 , b^ = 0, a^ = 0, ^2 = 1,

a^ = 0, b^ = 0,

T h e expressions for n* are now

«1 = 0, n^ = 0, n^ = 1. ;i.6.12) 1 n^ = n'=-^u^,, n^={\-e"-e')+^u-, + ^ u \ , (1.6.13)

and for x'„„ etc.,

B ^0! B X ap — Aii T h e result is X Rfi iJft x\, = B„ X Of} R" ' B^

W

'

AB 7 ^ " (1.6.14) 2 x\^ n* = ^ u\ + --- ^ u\ ^ 1 - e" - / + — — u\ + -—u\ . B B, B. B^ , ,. B B B S ^ V «'• = ^ — «'a - ^ «3, -f ^ (1 - e" - £^) + - ^ . ^ , + "1 /2^ i?" '^ ' J i?^

2 X*„^ ;Ï^- = - - ƒ u\

_{-fu% + ^ { \ - £"-£") +^U^, + ^U\.}

B, AB , A B _J-Pa _j-jia

J-fia

A B„ B

u\ = u„ ^--5^^> U^0 = Vp+-~U w,

A A B B

;i.6.i5)

By eliminating the derivatives of a\ b' and n* from (1.6.2) by means of (a. 19) . . . (a. 21) we obtain for the frame of reference defined by (1.6.12)

(24)

14 GENERAL THEORY OF SHELLS 1.6 T h e final form for the expressions 2x*„„ n\ etc., is

2 x\„ n* A^ R" ' ! - £ " - e") - ^a + A A, Ws-\- A A B B " B R ' A A, A A ^f<+ J^^a A Ap A B, ^

'^"^'BW''^^R"~^\^'"\'JW^'B^ T^"

A V ^ [^ + _{R" R'-R^ ) '} 2x^ £ 2 B B„ ps B B^ B B„ B, B. B B B B, B Af, B B, 'BY B^ R^l R-'R^ AB S^^a,«^ = ^ ( l - £ " - « " w„ A AB I 1 B„ A B - y «^? + j ^ ^^ + 7 ^ «a A, 5„ ABi I I ;i.6.i7)

From the definitions of x", x^, x^" = — x"^ and their relations with the Gaussian quantities we obtain 1 _ _ x" = ^"L x\„ n' A"" 1 2 x\, n\

'''^"F^^'^^"'^ 5^^''''^^"'

X»" = - x"^ = ^ Y . x\, K"- - - - 2 x\, n\

H a

a/3 ' ;i.6.i8)

By means of (1.4.6), (1.5.4), (1.6.7) . . . (1.6.9), (1.6.17) and (a. 25) these expressions for the changes of curvature are reduced to

w„„ A„ Af, ^ v„ _ \ i VV i \ X = A^

z>" + :^"^ + ^ly^ + "l(i"I-(Y^)1

+ "z(^")a+^l:^(Fl-2:4l^M'

;i.6.i9)

^ = # -

F"'^

+

:4^""

+

^^7^"

+

"UFJ

+ 4 ( ^ l + "lT(^)a-22^}'

1 ^2 (1.6.20) ^Pa _ _ ^a^ _ "W _ ^ ^ B„ , «P , ^"

AB A^B " AB^ " BR" AR^

+ u

_{A \ r ^ 7 a ABR"}1 / 1

±\-

+ V 1 / 1 B„

J_j J_ J_

B \T^ii, ABR" I "^ "^ T ^ i F "^ ^

(25)

1.7 GENERAL THEORY OF SHELLS 15 1.7 Stresses on a lateral side of a shell element

A lateral side of a shell element is generated by the normals to the middle surface along a parametric curve (cf. (1.7. f 1)).

1.7.f 1)

T h e curves PR and PT are the parametric curves a and ft on the surface through P at a constant distance s from the middle surface. T h e direction cosines a* {s) and b' {s) of these curves PT and PR are given by (cf (1.2.5))

(x^+jreOa l,,.^ {x' + sn')p aUs) s

+

s ¥ {s) = B s V 2 I '/i ;i.7.i)

T h e stress vector p" {s) at P, acting on a surface element da = 0, may be resolved into three components t"" {s), t^^ {s), t""^ {s) in the directions of a' {s), ¥ {s) and n\ or into three components

s"" {s), / ^ {s) and s"" {s) in the directions of a\ b' and n*. T h e same can be done with the stress

vector p^ {s). The normal ra* to the middle surface is also perpendicular to the directions a' {s) and b* {s). T h e projection of the stress vectors on the tangential plane described by a* and b* is

s .

given in (1.7.f2) on the assumption t h a t — ^ is positive.

« • b '

(26)

16 GENERAL THEORY OF SHELLS 1.8

Using the fundamental theorem of the theory of stress, we obtain

tt"' {s) = ^"(5). ;i.7.2)

T h e relations between the orthogonal components s"" {s), etc., and the oblique components

t'"'{s), etc. are

s"" {s) = f" {s) cos X {s) - t"" {s) sin ^l {s), s^^ {s) = t^^ {s) cos fj, {s) - t^" {s) sin 1 {s), s"» {s) = t"^ {s) cos pc {s) - f" {s) sin A {s), / " (5) = ü^" {s) cos X {s) - t^^ {s) sin [j. {s).

We now derive the expressions for the angles X{s) and pc {s). From (1.7.1) cos — + A (.y) = 2 ¥a' {s).

s s

Using {a. 12) and disregarding squares and higher order terms in —^, —^, etc. we obtain

R T

;i.7.3)

sin X{s) = X{s) = Yfa' cos X {s) = 1, sin/t {s) = pi{s) =—, cosfi{s) = \.

;i.7.4)

1.8 The stress resultants

T h e stresses on a lateral side of an element cut from a shell give rise to a resulting force and a resulting moment (cf. (1.8.fl)). In order to establish expressions for the j^rei'jreja/tonii, i.e., the resulting force and moment per unit length, we shall consider a shell element whose middle surface is bounded by the parametric curves a and a -\- da, and ft and ft -\- dft, the lateral sides being generated by the normals to the middle surface along these curves. We resolve the stress resultants into three components in the directions of n', 0» and b\

(1.8.f 1)

(27)

Denoting

1.9 The eq

the forces and moments per unit length ;

M" M^ M"» M"" K" K^ K"^ jrfla Q" Q.' nations of = = = = = = = = = = equ 'Uh -'Uh 'Uh

-js.

-'Uh 'Uh

- i S

.

-'Uh 'Uh

[s.

-'Uh 'Uh

1 s""

-'Uh _'Uh

1 /"

— 'Uh 'Uh \ S"^ -'Uh 'Uh -'Uh 'Uh

[ s""

-'Uh 'Uh

] S^^

-'Uh ilibrium s"" s^^ s"^ / " {s) (s) is) is) is) is) for

w|(.

« ! ( •

. | ( , . , | ( .

1/

l('

! ( •

10 l('

!(•

a shell

s

Y

Rn

'1

R"l

'1

R'j R"!

s

Y

R^l

sV R")

R^j

R"! R")

s

Y

R" element

+

is indicated in (1.8. f 1),

/ s Yi

l s V \ \T"^I

( H J

1 ' V

[rP") 1

\T"^I \TM \

[T-^I

1 M

(T"^)

1

ds, 'U ds, 'U ds, 'U ds; 'U ds, 'U ds, 'U ds, 'U ds; 'U ds, 'U ds. we have ( i . 8 . n ^ (1.8 (1.8 2) 3)

The stress resultants and the loads per unit area /)", ^ and p" acting on a shell element are shown in (1.9.fl). T h e resulting force and moment acting on a lateral side /3 = constant (a = constant) are decomposed into the directions of the fixed frame of reference OXYZ; these components are given by

parametric curve tx

KldK"

• K "

(28)

-18 GENERAL THEORY OF SHELLS _1.10

dP"' =-{K''a'+K''n' + Q"n')Bdft, dP"' = - {K^" a'+ K^ b* + Q^ n*) Ada, ]

[ ( 1 9 1')

dW' = - (M"^fl' + M"¥) Bdft, dW' = - {M^a' + M^"b') Ada. )

The corresponding components of the forces and moments on the opposite lateral sides are given by

- dP"' - {dP"'),da, - dW' - {dW"'),da and - dP^' - {dP^')f,dft, - dW»' - {dW^')^dft. The resulting force due to the loads is also decomposed into the directions of OXYZ. These components are given by

(/)" a' + p» ¥ + p" n*) AB dadft.

The equations of equilibrium for a shell element are obtained by equating to zero the resulting force on the element and the resulting moment with respect to an arbitrary point in space; for this point we choose the centre R of the shell element (1.9. f 1). Putting the resulting force zero we obtain

- {dP"')Ja - {dP^')pdft + {p'^ai + f¥ + p-ni) AB dadft = 0. (1.9.2) The stress resultants Q", Qf, K"^, K^" contribute to the moment with respect ioR (1.9. f 1) (the

contribution of the other forces being small of higher order). The components of this moment are

{C^a' -^ Q"¥ + {K"^ - K^") n'} AB dadft.

Putting the resulting moment zero, we obtain

- {dW'''),da - {dW^')p dft + {Q^a' - Q"¥ + {K"^ - K^") n*} AB dadft = 0. (1.9.3) Substituting (1.9.1) into (1.9.2) and (1.9.3), eliminating the derivatives of fl% ¥ and ¥ by means of (a. 19) . . . (a. 21) and remembering the linear independence of ¥, A' and ¥, we obtain finally

^ > ^ ( Q " ^ ) + ^ ( Q ^ ^ ) + f ^ " + W"^' + 7 ^ ^ " + ^ a ^ " " + r ^ 5 = 0, (1.9.4)

^ -^Q." - ^ Q ' + 1^{BK''^--B.K'' + ApK"^ +^^(^AK'^'') + p^AB = 0, (1.9.5)

'^ -^.(T-^Qf- ApK"

+ ^[AK')

+

J-^[BK"^)

+ B^Ki^" ^ pf AB = 0, (1.9.6)

M" M" M"'' M""

- • y ^ a + ^ ^ a + ^ + ^ + ^ - ' - ^ ' - ^ O , (1.9.7)

AgM" -\ \AMn + — {BMA - B, M^" + ABQ^ = 0, (1.9.8) dft \ I da \ I

.' — (sAf") + S„ Af^ - Ap M"^ + ~ [AMA - ABQ" = 0. (1.9.9)

Equation (1.9.7) expressing the condition of equilibrium of moments about the normal is an identity. This well-known fact [15] may be proved by expressing the stress resultants of (1.9.7) in the stress components f", etc., using the formulas (1.7.2), (1.7.3) and the rigorous expressions for sin X {s), sin fi {s), etc., which may be derived from (1.7.1).

1.10 The stress-strain relations

The stress-strain relations for infinitesimal deformations of an elastic body are expressed by Hooke's law. In order to apply this law we need expressions for the stresses and strains referred to an orthogonal frame of reference.

(29)

We shall first derive relations between the strains e" {s), s^ {s), y)"^ {s), on the one hand, and the strains e^ {s), s^ {s), yi^^ {s) referred to the orthogonal directions a* and ¥, on the other hand (1.10. f 1).

s ^ ^ s ) *

The sides OB, AB and ^ 0 of the triangle OAB will have unit elongations e^ {s), e^ {s) and e° {s) and the angle ABO will increase by y)^^ {s). Remembering that X{s) is a small quantity (1.7.4), we may write

e" {s) = £i {s) - X {s) v)i2 (^s), AX {s) = X {s) {e^ {s) - s^ {s)},

where AX {s) is the change of the angle AOB. In the same way we have £^ {s) = e' {s) - fl {s) y>^' {s), Ap, {s) =pi{s){ e^ {s) - e^ {s) } . Remembering the definition of y)"^ {s) (1.5.11), we may now write

£i {s) = e" {s) + X {s) v"" is), e' {s) = e^ {s) + /i {s) y,"i> {s),

yl2 (,) = ^a^ (,) _ i^x{s) - A* (^) } {£" {s) - é ( . ) } .

:i.io.i)

We now derive similar relations between the stress components s"" {s), etc., and CT'I {S), a^^ {s),

•^{s) = CT^i (i) (l.lO.fl); these relations are given immediately by means of the fundamental

theorem of stress. If all second order terms in X {s) and pi {s) are disregarded again, we have

S"" {s) = ffll {s) + pi {S) (T12 {S) , ^"/J {s) = (x21 j^s) + pi {s) ff22 {s),

5"^ {S) = ff22 (^) ^ X (j) j;21 (^) ^ / a (^) = 0-12 {S) + A {S) ff" (j) . 1.10.2) T h e stress-strain relations for homogeneous, isotropic, perfectly elastic materials, are according to Hooke's law

e' is)=-^{ ff" is) - V {a^^ {s) + CT" ( . ) ) } , 1

yl2 {s)

2(1 + 1')

E 2{\ + v)

E

(30)

20 G E N E R A L T H E O R Y O F S H E L L S 1.11 where E is Young's modulus * ) , v is Poisson's ratio and CT" {S) is the direct stress on a plane parallel to the middle surface. According to the assumption iii of para. 1.1., ff" (J) is small as compared to a^^ {s) and a'''''' {s); hence, v.'e write

— ^ ^ {£' {s) + V e^ {s) }, a^^ {s) = — ^ { £^ {s) + v e^ {s)},

012 (J-) = CT21 {S) = 2 ( 1

y>i2 (^)

;I.IO.3)

By means of equations (1.10.1) . . . (1.10.3) we now establish the final relations between the oblique components of strain and stress; disregarding again all second order terms in

X {s) and pi {s) (1.7.4), we have

s"" is) = T ^ I ^" is) + '"=" is) + 4fa r ' is) (1V2 + V2 •") 1, 1

"'' (') = T ^ I "' (") + ''" ^') + ^ a ^"' (^) (1'/^ + '/^ ") j '

^"^ (s) s"- [S E W'"is) + 2 (1 H-j') '' '"' ' I -v^ T^'

^

^«z' (,)

+

_ L _

2 (1 +>-) '^ ^ ^ ^ 1 _ ^ 2 7-^^ , { £ " ( . ) + . £ " ( . ) } , r a K ( ^ ) (^)}. : i . i o . t )

1.11 77i^ relations between the strains and the stress resultants

T h e stress resultants (1.8.1) and (1.8.2), expressed in the extensional strains and the changes of curvature, are obtained f-om (1.10.4) if we express the strains e" {s), etc., by means of (1.5.16) and perform the integrations with respect to s. Neglecting all contributions which are small of higher order than the second in hjR", hjT^", etc., we obtain

M" = - D[x'-+ vx^ —- ^ + ^

M" = D U" + vx"

2 Tf"" ' R^ 3 + »> y)"^ £^ + VE"

- ' ^ ' ( - ' " + ^

2 7"^" /JO «"'^ £^ + r£" M^" = - D K" =C (£" + -r- ^„ D j-fia e" + ve'' K^ =C (fi" + ve") + D R^ x^ -\- vx" ^" J-Pa (3 + 1-) J" J-Ha {3 + v[ x'^ -\- vx

^•" = V ^^^"' + ^{(1-")^" T^"

I —v „ \ x''" x" + vx''

K'" = ^ - G^"^ + i ) j ( l - r ) —

7-^a where Z) = Eh^ 12 (1 and C = £A 1

' i . i i . n

; i . i i . 2 )

(31)

1.12 GENERAL THEORY OF SHELLS 2 1

T h e complete set of equations describing the elastic behaviour of the shell consists of (1.6.4) . . . (1.6.6) and (1.6.19) . . . (1.6.21) (expressing the extensional strains and the changes of curvature in the displacements), (1.11.1) and (1.11.2) (expressing the stress resultants in the strains) and, finally, the equations of equilibrium (1.9.4) . . . (1.9.9).

O u r assumptions and approximations find expression only in (1.11.1) and (1.11.2), the other equations being rigorous. It might be argued that a number of terms, viz-, those involving the extensional strains in (1.11.1) and those involving the changes of curvature in (1.11.2), should be disregarded on account of similar approximations which have been made earlier. However, the identity (1.9.7) would be violated if we dropped these terms.

1.12 The boundary conditions for the displacements

We distinguish between two different types of boundary conditions, viz. i. the geometric boundary conditions, ii. the dynamic boundary conditions.

In the case of geometric boundary conditions the displacements along the edge of a shell and the direction of the normal to the middle surface at the edge are imposed, yielding four boundary conditions. If, on the other hand, the forces and moments along the edge are imposed, the resulting relations between the displacements and their derivatives are called the dynamic boundary conditions.

In the case of dynamic boundary conditions three forces and two moments (the third moment being small of higher order) are prescribed along the edge. This larger number of boundary conditions cannot be satisfied generally but the number is reduced to four by an appeal to St. Venant's principle. O n the strength of this theorem we may replace the twisting moment of the load applied to an element dt of the edge by a couple of forces at the ends of the element, the moment of which equals the twisting moment (1.12. f 1).

(1.12.fl) We replace the twisting moment

^M'" ~^u~^dtyt

(32)

22 GENERAL THEORY OF SHELLS 1.12 these forces is the direction of the normal to the middle surface of the shell at P j . T h e intensity of these forces is M"' — 1/2 dt. We apply the same procedure to the twisting moment

!

riAA^ 1

^"^ + V2 dt dt of the load on PP^. Fig. (1.12.fl) also shows the forces Q* dt. A"" dt, K* dt and the moment M^ dt of the load on R^R^. T h e direction of the force M'" — 1/2 dt at

P is given by ra' — 1/2 dt, and the direction of the force Af" + V2 dt at P by

/ dn' \ ^ ' ^^ - ( " ' + V 2 ^ ^ ^ ) .

We decompose the resulting force of the load on R^R2 into three orthogonal components in the directions of the normal ¥ to the middle surface at P, the tangent p' to the edge, and the normal q' to the edge. aA/f"

T h e component of M" ±, ^j^ dt in the direction p' is

dt

(

9K'

I f _ dM'" ] f dM" 1 d¥

1.T P^\¥ ±Û~dt\\M'" ±Û^dt\ = -^M^ ±Û — dt\i:p'^dt;

the components in the directions q' and ¥ are obtained in the same way. Finally, we distribute the resulting force uniformly over the element R1R2; we obtain the forces per unit length

K*-M"''S,pi— in the direction/)% (1.12.1) dt

K" - AP" 2 9' - ^ in the direction q\ (1.12.2) dt

d A/f"

Q* in the direction ¥. (1.12.3)

dt

It should be observed that the replacement of the twisting moment by a couple of forces results in concentrated forces at the ends of the edge. At an angular corner, which is the inter-section of two edges having dynamic boundary conditions, these concentrated forces must be taken into account as an additional dynamic boundary condition.

If the edge where the forces and moments are imposed coincides with one of the parametric curves a or ft, we can reduce the expressions (1.12.1), (1.12.2) and (1.12.3) by using (a. 21). If the edge coincides with the curve a, we have

d¥ 1 p' = ¥, q' = fl' and -r-= -^ n'^. ot a Hence, (1.12.1) . . . (1.12.3) reduce to 1 1 1 dAi"^ K"" + r-^ M"", K" +--^ Ad"", Q"--^ , (1.12.4) ' RK ' T"" ' ^ B dft ^ '

and, likewise, for an edge coinciding with the parametric curve ft, to 1 1 1 dAl""

^fC''" + — M"", K" r ^ " " , Q^ + • (1.12.5) R" ' T"" ' ^ A da ^ '

(33)

2.1/2.2 HELICOIDAL STRIP 23

C H A P T E R I I

SHELLS T H E M I D D L E S U R F A C E O F W H I C H IS AN I N F I N I T E H E L I C O I D A L S T R I P

2.1 Introduction

A right helicoid, referred to a rectangular system of coordinates OXYZ, may be represented by

x = r c o s / 3 , y = rs,inft, z~aft, (2.1.1) where r and ft are parameters. T h e pitch of the helicoid is given by 2 Jt fl.

An infinite helicoidal strip is defined by (2.1.1), where r^ <, r <, r^ and — oo < (9 < oo. In order to simplify the investigation, we assume that the load, the thickness of the shell, and the boundary conditions are independent of the parameter ft. This assumption implies, that the stress and strain distribution depends only on the parameter r.

The boundary conditions for the helicoid are derived from the idealized model of a ship's propeller. T o a first approximation the propeller blade may be considered to be a shell, the middle surface of which is a part of a right helicoid bounded by the parametric curves /9 = i 0 and the parametric curves r = r^ and r = r^. Since the helicoid examined in this chapter is unbounded in the direction of the parametric curves r, we need only consider the boundary conditions at the parametric curves r = rj and r =^ r^; in other words, the effect of the finite width of the propeller is not considered in this chapter. T h e appropriate boundary conditions for the helicoid that is unbounded in the ft direction are a clamped edge at r = r^ and a free edge at r = r^.

2.2 The geometrical quantities of a right helicoid

According to (2.1.1), the right helicoid is given by

x = rcosft, y = rsinft, z = aft, (2.2.1)

where r is the distance between a point on the helicoid and its axis x = 0, j = 0. T h e relations for the geometrical quantities of the helicoid to be derived in this paragraph will obtain a simpler form if we write

r = a t a n a . (2.2.2) In accordance with (1.2.1) we have

X = x'(a,/?) = a tan a cos/?, j E= x^ (a,/?) = a tan a sin/3, z^^x^ {a, ft) ~ aft. (2.2.3) From (2.2.3) we obtain

(2.2.4)

cos ft sin /3 V >f'a = « — 7 ~ , x\ = a — , x\ = 0,

cos^a cos'a I

x^^ = — fl tan a sin ft, x^^ = a tan a cos ft, x^^ = a. > From (2.2.4) and (1.2.4) we obtain

A^ = E = - ^ , B' = G = ^ ^ , F=0, H = ^ ^ . (2.2.5)

(34)

24 H E L I C O I D A L S T R I P 2.3/2.4

T h e parametric curves a and ft are orthogonal. From (2.2.5) we obtain further

x\„ = 2 fl cos ft sin a cos^a x^fj^ = — a tan a cos ft, x^^ sin/9 2fl sin ft sin a 0, cos'a - fl tan a sin |S, x'^^ = 0, cos/? ^=a^ = 0 . C O S ' ' a C O S ' ' a

T h e direction cosines fl', é' and ¥ are calculated from (1.2.5) and (1.2.6)

fli = cos ft, a^ = sin ft, a^ = 0, ^ ^1 = — sin a sin ft, b^ = sin a cos /3, 6' = cos « , [ (2.2.6)

¥ = sin /? cos a, «^ = _ cos /9 cos a, w^ = sin a. i

From the above expressions we can derive the components of curvature; using (1.2.9) we find

w-'-

1 0, 1 _ 1

Y^" ~~ ~ T"" (2.2.7)

2.3 The strains expressed in the displacements

As already stated in para. 2.1., we assume the load, the thickness and the boundary conditions of the shell to be independent of the parameter ft. Hence, all relevant quantities will be functions of a only and all derivatives with respect to ft are zero.

From (1.6.4) . . . (1.6.6), we obtain cos^a K, £^ = U , „ cos a sin a „"f COS' fl a and from (1.6.19) . . . (1.6.21) cos* a

at-

'-'

i

~[v„-vtana'+2w), (2.3.1) fl" cos* a

w„„ — 2 w^ tan a — 2Va — w + 2v tan a),

(w„ tan a — w), x''" = — x"" = 2 cos* a _{u tan a.}

(2.3.2)

fl' «'' With a view to later application, the expression for x" is reduced to the following form

cos* a

(w„a — 2 w„ tan a — 2 v^ — w -^ 2 v tan a)

(»„„ — 2 w„ tan a + 3 w) — 2 — (&„ — v tan a + 2 w)

fl2 cos* a fl2 COS* a (tü„„ - 2 «;„ tan « + 3 w) + 2 „°/5 J-|8a' (2.3.3)

2.4 TAe equations of equilibrium

By substituting the expressions for the geometrical quantities (cf para. 2.2.) in the equations of equilibrium of para. 1.9., we obtain

d

Q"

K"" +

K''"

ap"

da cos a d K" da cos a t a n a

+

cos a

cos-+

+

= 0,

ap"

= 0,

cos a cos a cos'' a

(2.4.1)

(35)

2.5/2.6 HELICOIDAL STRIP 25

d K"i' K"" Q" atf>

+ tan« + ^ ^ + - _ £ _ = o, (2.4.3)

da cos a cos a cos a coS'' a

cos^a fl

{M" + M'') + A-°^ - r^" = 0 , (2.4.4)

rf Af"^ Ai"" a (y

: tan« H — ^ ^ = 0 , (2.4.5) da cos a cos a CDS'* a cos a

^ Af" Af^ fl Q"

-^ h t a n « — - ^ = 0 . (2.4.6)

da cos a cos a CDS'* a cos a

It has been stated in chapter I that (2.4.4) is an identity.

By eliminating Q° and Qf from (2.4.1) . . . (2.4.6), we obtain three equations between the forces and the moments. Elimination of Q" from (2.4.6) and (2.4.3) gives

d Ad" a d K"" I Ad'' a K''"\ a^p

. + ^ - . + t a n a + = ^ . (2.4.7) da cos a cosâ da cos a \cosa cosâ cos a/ cosâ

Elimination of Q^ from (2.4.5) and (2.4.2) gives

d Ad"" a d K" (Ad"" a K" \ a"" p"

^ t a n a = — ^ . (2.4.8)

da cos a cosâ da cos a \cosa cosâ cos a/ cosâ

The third equation is obtained by eliminating Q" from (2.4.3) and (2.4.1). (/2 K"" d I K""\ K"" + K"" ap^ d atP

T T + — t a n a + ^ = _ J L _ _ ^ ^ . (2.4.9) fla'' cos a da \ cos a] cos a cos'^ a da cos^ a

These three equations contain a total of eight unknowns. It should be noted that only two unknowns occur in (2.4.9).

2.5 The boundary conditions

The boundary conditions for the helicoidal strip are a clamped edge at r = rj and a free edge at r = r^.

Let «1 be the value corresponding withfj and aj the value corresponding with r^ (cf (2.2.2)). At the clamped edge a = aj we have

„ = 0, v = ^, w = ^, w„ = 0. (2.5.1) At the free edge a = «a we have from (1.12.4) and (2.2.7)

C O S OL

K"" = 0, K" Ad"" = ^, Q° = 0, Af" = 0. (2.5.2)

2.6 The relations between the strains and the stress resultants

The relations between the strains and the stress resultants are obtained from (1.11.1) and (1.11.2) by putting R" and R" zero

M"=-D{x"-^vx"-'-p:^^, M""= D[{1-V)X""-^'^^

On stress calculations in helicoidal shells and propeller blades

ON STRESS CALCULATIONS IN HELICOIDAL SHELLS

AND PROPELLER BLADES

O n stress calculations in helicoidal shells

and propeller blades

JACOB WILLEM COHEN

Aan mijn Ouders

Aan mijn Vrouw

CONTENTS

E R R A T A

NOTATIONS

+ f^ j (l - :^y + ( y i j j ^/^^ + '^''- (1-3-2)

dr{s)=.E'''}^[i-^j + M I '^^ ^^-^-^^

o^{s)=-s{l,^~^^. (1.3.8)

+

{n',Y

'='^F

+

+ /?J \T»"

i^-'^^l(i

d,^Ë

\(l-^X+l^X\da^-+

+

d?{s) =B(I - ^ y (1.5.9)

i;.{s) = -rp"^ + s{±-2-^. (1.5.10)

x'' = l- L, ^^ = i L, xj'^ = 2 -, p./" = i - . (1.5.13)

R_J^] + fl ,^^-(R_^]_rl (1514)

'^

^''

B

T^"

U"jJ

'^^

{ B

R"

"I(FI-^1+"I(^1"^1

AV

I A

Y

W

'

-fu% + ^ { \ - £"-£") +^U^, + ^U\.

'^"^'BW''^^R"~^\^'"\'JW^'B^ T^"

'''^"F^^'^^"'^ 5^^''''^^"'

H a

z>" + :^"^ + ^ly^ + "l(i"I-(Y^)1

+ "z(^")a+^l:^(Fl-2:4l^M'

^ = # -

F"'^

+

:4^""

+

^^7^"

+

"UFJ

+ 4 ( ^ l + "lT(^)a-22^}'

+ u

±\-

J_j J_ J_

+

-js.

- i S

.

[s.

1 s""

1

/"

[ s""

] S^^

w|(.

« ! ( •

1/

l('

10

l('

!(•

!(•

s

Y

Rn

_{-fu% + ^ { \ - £"-£") +^U^, + ^U\.}