CHAPTER 1
BACKGROUND ANALYSIS AND INTRODUCTION TO SHELL THEORY by John Spence University of Strathclyde, Glasgow
SYNOPSIS Many of the design procedures in BS 5500 and in other standards are based primarily upon linear elastic small displacement stress analysis. Even where inelastic behaviour is allowed to some limited extent the design base frequently remains essentially linear elastic. This section briefly discusses the type of analysis – thin shell analysis – which underpins much of the thinking in pressure vessel codes. Typical equations and solutions are given for spherical and cylindrical shells together with an indication of how junctions between different geometries are analysed. Although this chapter can only provide a limited introduction to shell analysis, the important concepts of membrane and discontinuity effects are emphasised and the inherent limitations of the approach are highlighted.
CONTENTS LIST 1.1
BASIC PRINCIPLES AND THEIR ILLUSTRATION 1.1(a) 1.1(b) 1.1(c)
Introduction Basic concepts Illustration of a beam loaded with direct and bending actions
1.2
SHELLS AND SHELLS OF REVOLUTION
1.3
EQUILIBRIUM EQUATIONS FOR A SHELL ELEMENT 1.3(a) 1.3(b)
Membrane shells Rotationally symmetric load on membrane shells
1.4
STRAIN DISPLACEMENT EQUATIONS
1.5
CONSTITUTIVE RELATIONSHIPS
1.6
METHODS OF SOLUTION 1.6(a) 1.6(b) 1.6(c) 1.6(d) 1.6(e) 1.6(f)
Rotationally symmetric load on a spherical shell Rotationally symmetric load on a cylindrical shell Die out effects A cylindrical vessel with hemispherical ends under internal pressure The cylindrical nozzle in a spherical shell type problem Limitations of shell theory
1.1
BASIC PRINCIPLES AND THEIR ILLUSTRATION
1.1(a) Introduction The term ‘stress analysis’ is frequently used, not only to cover the analysis of stress but the complete behaviour of a body including, on occasions, the distribution of load, stress, displacement, strain and temperature throughout the body when some external actions such as load, displacement or temperature act upon the body. The rather obvious justification for doing stress analysis is that it is one of the main activities that can give a designer a quantitative yard-stick as to the efficiency of a load carrying component and the possibility of its failure in service. It is thus a necessary stage in the rational design of most engineering components but is usually a fairly approximate tool unless the loads and the geometry of the component in question are simple and well defined. In recent times the ability to conduct sophisticated stress analyses has increased dramatically; although the understanding of component behaviour under load has also increased, the design decisions related to these analyses are not necessarily obvious. Thus ‘stress analysis’ needs frequently to be liberally seasoned with engineering judgement. In pressure vessel standards the basic thicknesses of the main shells are derived from calculations based upon rather simple methods of stress analysis in conjunction with the allowable design stress. This may be illustrated as follows. Consider the case of a long, thin cylindrical shell of radius r and thickness t subject to internal pressure loading. As one might expect, forces and stresses will be generated in the cylinder wall in the circumferential direction and if the cylinder has closed ends, in the axial direction. A longitudinal half section across such a cylinder is shown below in Fig 1.1. In this the circumferential stress is denoted by σ θ . The stress is assumed to be uniform across the thickness and along the length, since the cylinder is considered to be “long” and “thin”. By “long” we mean that the ends are sufficiently far away as not to disturb the stresses in the main part of the cylinder. Of course near to the end, as we shall see, the stresses may be quite different but if necessary a different thickness may be employed near to the end. By “thin” we mean that the radius is much greater than the thickness; thus at this stage we do not need to identify the radius as being associated with a particular surface (i.e. internal, external or mid-surface), although strictly speaking for the analysis which follows we should use the inside radius. The forces must be in equilibrium so that 2σ θ tL = p 2rL
i.e.,
σθ =
pr t
FIG 1.1
A half section of a Cylindrical Vessel
Similarly by considering a diametral section and taking longitudinal equilibrium we find the longitudinal stress σ x is pr 2t
σx =
These stresses are frequently referred to as “membrane” stresses since there are no variations in the stress through the thickness i.e. no bending stresses. Notice they are obtained from equilibrium considerations alone. It is useful to consider the stresses as acting at the mid surface of the shell. Similar arguments applied to a thin sphere (Fig 1.2) result in uniform and equal stresses in mutually perpendicular directions.
FIG 1.2 σθ = σ φ =
A half section of a Spherical Vessel
pr 2t
Real pressure vessels, of course in practice, often have a significant thickness compared with the radius. The complexity of dealing with “thick” shells is usually considered too involved for pressure vessel design standards. Consequently in almost all cases the significance of thickness is ignored and the theory related to thin shells is employed. In fact the above formulae are remarkably good if r≥ 10t (better than 5%) and they are often used for r > 5t. More accurate equations are available outwith the standards which take account of the thickness. For example for a “long” “thick” cylinder the Lame equations (see any standard text on Mechanics of Materials) may be used to give the following result for internal pressure loading. These are as follows:-
σθ
=
σr
=
σx
=
[ ][ p[1 − (b / r ) ] / [(b / a ) p / [(b / a) − 1]
] − 1]
p 1 + (b / r ) 2 / (b / a) 2 − 1 2
2
2
where a is the internal radius, b the external radius and r any radius within the thickness of the cylinder. A typical distribution of these is shown in the sketch Fig 1.3.
FIG 1.3
Distribution of stresses in a ‘thick’ cylindrical vessel
Again the solution is only valid in the plain part of the cylinder remote from any disturbances. These equations may be useful for high pressure applications although other special considerations may be required. The Lame equations appear in this form in the Australian Pressure Vessel Standard AS1210 Supplement 1. However, for a moderately thick cylinder a reasonable approximation for the circumferential stress, which is usually the largest, is to employ the thin cylinder theory but use the mean radius or mean diameter dm = Di + t. Where Di is the inside diameter i.e., σ θ
If the allowable circumferential stress σ then rearranging gives
pd m 2t
= θ
t
=
p( Di + t ) 2t
is equated to the code allowable design stress f,
=
pDi 2f − p
which is typical of the basic equations, in pressure vessel standards. This is the type of equation, which is employed for the evaluation of the shell thicknesses, and as a starting point for design. Later we will discuss the Tresca criteria of yielding and the associated idea of stress intensities. Simply stated the stress intensity is the maximum difference between the principal stresses. In the case of the cylinder this would be σ θ - σ r where σ r is the radial stress. This latter varies from the pressure on the inside surface to zero on the outside surface. If it is approximated as (p/2) and noted to be negative (compressive) and the stress intensity equated to the allowable (f) then
f
= σθ − σ r
=
pDi 2t
p − − 2
which leads to exactly the same result namely =
t
pDi 2f − p
which is probably fortuitous. The alternative from which also appears =
t
pDo 2f + p
is of course identical. In the case of spherical shells the equations in the standard takes the form t
=
pDi 4 f − 1.2 p
t
=
pDo 4 f + 0.8 p
or
which is again a reasonable approximation for moderately thick spheres. Note the difference in the form for the sphere compared with the cylinder since the stress distribution is more highly non linear through the thickness. However, it must be noted that where the simple idealised cylindrical or spherical geometry is disturbed the membrane stress pattern is totally changed. For example where a cylindrical part is attached to an end closure the stress distribution is more complex. An important case is when a nozzle is located in the main shell. The resulting stress distribution for a given loading is more complex than the simple analysis given above and the results depend on the radius/thickness ratios of both nozzle and shell as well as the size of the nozzle. The objective of this chapter is to introduce the concepts of thin shell analysis for spheres and cylinders and carry it far enough to indicate how it can be used to analyse some classical shell intersection problems and in particular the case of a cylindrical nozzle in a spherical shell under internal pressure. In doing so it attempts to set the scene for the detail discussion which follows in later chapters on various pressure vessel components in order that the reader may appreciate the advantages and the limitations of the theoretical basis of much of the design methodology in the Standards. It is not intended to be a full development of thin shell theory but simply an introduction to facilitate an appreciation of the overall approach in so far as it is relevant to pressure vessel design. Today the approach has been largely (although not entirely) displaced by the finite element method or other advanced techniques and is thus less likely to be used in the future. Nevertheless it remains the essential basis of almost all pressure vessel standards and routine design.
1.1(b) Basic concepts In any general stress analysis there are certain fundamental requirements that have to be satisfied. Briefly these are: Equilibrium When a body is in a state of rest (or uniform motion) the loading both externally and internally must be in equilibrium. Strain Displacement Compatibility When a body deforms it must do so in such a way that movements of adjacent elements of material are compatible with each other and with any external support conditions, i.e. they must fit together in their deformed shape without any gaps or overlapping. This usually reduces essentially to the application of geometrical rules relating displacements to strains throughout the components. Bodies with cracks or defects require special considerations. Constitutive relationships The material behaviour has to be known in so far as there is a relationship between the loads and the resulting deformation which depends on the physical relationships between stress, strain, time and temperature. Where the last two mentioned are unimportant, the term stress/strain relationship is common. The relationship can only initially be found experimentally but may need to be idealised and/or generalised to allow analysis to proceed. When the above three conditions can be satisfied mathematically throughout a body we have what is usually called in Applied Mechanics terms an “exact” solution. However the problem as idealised – the mathematical model – is rarely identical with the real physical problem and one should be clear what is conveyed by the term “exact”. Thus, if the idealised assumptions are restrictive we may have an “exact” solution, which is only an approximate representation of reality. Equally it may be necessary to make mathematical approximations in the solution of the appropriate equations. Most of stress analysis is concerned with approximate solutions of either or both types! Sometimes it is preferred to separate internal and external effects in the above presentation and describe the external conditions (equilibrium and/or compatibility) as “boundary conditions”. There are alternative but similar routes to a solution. For example, energy methods can be used. A number of energy principles have been established mainly for elastic materials which when developed for the appropriate constitutive relationship allow solutions to be found by optimisation of the energy. Usually these are employed to arrive at approximate solutions but they have the advantage over some other approximate methods in that it is sometimes possible to show that the answer obtained (for a load or a displacement) is a “bound”. That is to say it is known to be higher or lower than the exact answer. Two calculations then may allow the exact answer to be bounded, hopefully fairly closely. It should be noted that bounding of local stresses as distinct from generalised quantities such as load or characteristic displacement is not easy. At its best an energy approach may become identical to, or close to, an exact solution if one tries hard enough and obeys all the rules, but in practice energy is usually used as a powerful approximate tool. 1.1(c) Illustration of a beam loaded with direct and bending actions
Before getting involved with pressure vessel geometries, it is worth illustrating the three basic concepts: equilibrium, compatibility and the constitutive relations, as they are used to analyse a simple long straight beam of constant rectangular cross section, breadth b, thickness t and cross sectional area A. The illustration will be doubly informative because as it happens the beam analysis and results provide a simple analogue for the shell analysis. The beam is subject to a direct load P and bending moment M as shown in Fig 1.4a. Compatibility condition Conventionally it is usual to start from geometric considerations and examine possible deformation states for the beam. If each elemental length is to deform in the same way then it is necessary that “plane sections remain plane”, i.e. initially straight sections normal to the axis of the beam remain straight after deformation, for example, the ends of the beam remain flat. This seems plausible from a compatibility point of view since, because of symmetry, any other choice such as curved deformed planes would require gaps in the beam. The strain distribution across the depth of the beam is thus linear (Fig 1.4b) and this can be expressed as ε = C1 + C 2 z
(1.1)
where C1 and C2 are constants and z is measured from the line of the centre of the beam (centroid). Constitutive relation The uniaxial stress/strain relation for the beam material is taken as linear elastic viz. σ =εE
(1.2)
where E is the modulus of elasticity. From equation (1.1) this is expressed as
σ = ( C1 + C 2 z ) E
(1.3)
Equilibrium The internal longitudinal stress σ must be in equilibrium with the externally applied forces P and M:(a)
In the longitudinal direction:P = ∫ σb.dz
(1.4)
Substituting for the longitudinal stress from equation (1.3), equation (1.4) can be written in terms of the constants C1 and C2. Since the case considered is a rectangular cross-section beam the breadth b is independent of z and the intervals are from –t/2 to +t/2.
t/2
∫ (C
P = Eb
1
+ C 2 z ) dz
−t / 2
t/2
z2 P = Eb C1 z + C 2 2 −t / 2
C1 =
(b)
P P = Ebt EA
A = bt )
( where
(1.5)
Moments about the centroidal axis M = ∫ σzbdz
(1.6)
Substituting as before for the longitudinal stress, equation (1.3), equation (1.6) can be written:t/2
∫ (C z + C
M = Eb
1
2
)
z 2 dz
−t / 2
t/2
z2 z3 M = Eb C1 +C2 2 3 −t / 2
C 2 = 12
M M = 3 IE Ebt
where
I=
1 3 bt 12
(1.7)
Equation for stress The final equation for stress (1.3) may thus be written
σ σ
= =
( C1 + C 2 z ) E P M + z A I
From this it is clear that the direct strain ε D is C1 and the bending strain ε may be plotted through the wall thicknesses as shown in Fig 1.4. (b).
(1.8) B
is C2z. These
From equation (1.8), in the absence of axial load, σ = 0 at z = 0. Thus the neutral axis for the bending case is the centroidal axis. The beam in this case, bends to a circular arc, of radius R and the strain can be shown to be, ε = z/R. Thus from equation (1.1) C2 = 1/R, where R is the radius of curvature of the centreline under bending. The term 1/R is known as the curvature. The above equations have been developed for a beam of rectangular cross section in order to highlight the similarity with the thin shell equations given in the next Section. Usually, the beam equations are derived for a singly symmetric cross section under pure bending loading. Whereupon it becomes clear that the neutral axis of bending passes through the centroid of
the cross section. Thereafter, the combined bending and direct loading case is treated by simple superposition. 1.2
SHELLS AND SHELLS OF REVOLUTION
Stress Resultants and their Relation to Stresses A shell may be defined as a curved plate type structure. Usually, but not always, the thickness t is small with respect to the other dimensions. In general the shell is capable of taking bending moments, twisting moments and transverse shear forces through the thickness and forces in its own plane. Consider an infinitesimally small element with some possible force actions, which may arise as shown in Fig 1.5. For clarity the twisting moments, M xy and Myx have been omitted from the sketch at this stage. For convenience, Figs 1.5 and 1.6 are greatly exaggerated in the surface direction compared with the thickness direction to show the stresses. The similarity with the beam will be apparent (compare Figure 1.4 with Figures 1.5 and 1.6). In shell theory as in any other mathematical development the sign convention adopted must be strictly adhered to. The equations here follow Flugge’s convention and development [1.1]1. Other theories and/or conventions will result in slightly different equations. Flugge’s convention is similar to that inherent in BS 5500. As shown the force actions are usually defined in terms of stress resultants which are assumed to act at the mid-surface of the shell. For example, the direct stress resultant N xx is the direct force per unit length acting perpendicular to the face of the element with constant x and in the x direction. Similarly, the shear stress resultant Nyx is on a face of constant y and in the x direction. The symbol Q is used for the through thickness shear resultants. For a detailed explanation of the suffix notation reference should be made to [1.1]. Positive directions are as shown. Now Nxx may be thought of as the resultant of direct stresses, σ xx. Fig 1.6 indicates the stresses at some element of thickness within the shell element. Remembering that dsy, for example, is small but that variations through the thickness are possible, it is found by simple force equilibrium that N xx ds y
=
∫
+t / 2
−t / 2
ry + z ds y dz σ xx r y
(1.9a)
Note that this equation is exactly analogous to the axial equilibrium equation (1.4) for the beam. The other direct stress resultants can be found in a similar way,
1
N xy
=
∫
+t / 2
N yx
=
∫
t/2
−t / 2
−t / 2
ry + z dz ; τ xx r y
Q xz
= −∫
r + z dz; τ yx x rx
Q yz
= −∫
References are given at the end of each chapter.
t/2
−t / 2
t/2
−t / 2
ry + z dz τ xz r y
r + z dz τ yz x rx
N yy
∫
=
t/2
−t / 2
r + z dz ; σ yy x rx
(1.9b,c)
Note the negative sign in Q which is a peculiarity of Flugge’s development because of the positive direction chosen in Fig 1.5. In general the stresses will also give rise to moments about the middle surface or the centre line of the surface. These are represented by moment stress resultants Mxx and Myy and twisting moment resultants Mxy and Myx, again moments per unit length. From equilibrium we find the relationship between them as for example ry + z z dz σ xx r y
+t / 2
M xx
= −∫
M yy
= −∫
t/2
M xy
= −∫
+t / 2
M yx
= −∫
−t / 2
with the others being similar; r + z dz ; σ yy x rx
−t / 2
−t / 2
t/2
−t / 2
ry + z z dz τ xy r y
r + z z dz τ yx x rx
(1.10)
Again this is analogous to equation (1.6) for the beam. The minus signs are again unimportant but are necessary for consistency with the convention used in these notes. The term “stress resultants”, Nxx, Nyy, Nxy, Nyx, Qxz, Qyz, Mxx, Myy, Mxy, Myx, describe the ten force actions on a shell element. In shell theory normally one attempts to solve for these stress resultants and then finally deduces stresses from them. For example, for a linear elastic material a thin shell will have (approximately) a linear distribution of stress across it. The stresses can thus be taken from the simple relations derived for beams of rectangular section, subjected to a normal force P and a bending moment M i.e., equation (1.8):σ
=
P Mz + A I
whereupon:-
σ xx = τ xy
=
N xx t N xy t
−
−
12 M xx z t3 12 M xy z t3
(1.11)
The different sign in equations (1.8) and (1.11), associated with the moment, is due to the convention referred to earlier. In the case of eqn 1.11 positive z results in a compressive bending stress σ xx which is consistent with Fig 1.5.
Shells of Revolution Here we will restrict our attention to shells of revolution by which is meant geometries which can be developed by rotation of some curve (a meridian) about a central axis of revolution (Fig 1.7). A variety of single shapes fall into this category and some are indicated in Fig 1.8 e.g. spherical, conical, toroidal, torispherical, ellipsoidal, cylindrical etc. Notice that the general shell of revolution is denoted by the angle φ and the two radii r1 (the radius of the meridian) and r2 (the conical radius from the centre line). The other familiar shapes can be derived as shown in Fig 1.8 by inserting the specific values of these parameters appropriate to the desired shape. Changing x, y to φ , θ for the shell of revolution would alter equation 1.9(a) for example to
∫
=
Nφ
+t / 2
−t / 2
r + z dz σ φ 2 r2
with a similar expression for Nθ Nθ
=
∫
+t / 2
−t / 2
r + z dz σ θ 1 r1
(1.12a,b)
The equation for moment (1.10) can similarly be altered to:Mφ
= −∫
t/2
−t / 2
r +z z dz σ φ 2 r2
(1.13)
and the others likewise. Now that we have established these preliminaries we will proceed for the shell in exactly the same manner as for the beam, namely, by developing equilibrium, compatibility and constitutive equations and seeking a solution thereafter. 1.3
EQUILIBRIUM EQUATIONS FOR A SHELL ELEMENT
Fig 1.9 shows a finite mid surface shell element with the force actions and their variations across the element. The element of Fig 1.9 is cut by two meridians and two parallel circles. This element is now considered to be of a sufficient size to allow variations of the force actions across the element. For convenience the direct and bending stress resultants have been separated in the two parts of the figure. Actions are considered to increase in the positive co-ordinate direction in proportion to the rate of change of the quantity concerned as shown in Fig 1.9. Notice it is stress resultants which are shown and not forces. Surface loadings, pr, pφ , pz are also shown; these have the units of load/unit area. Fig 1.7 shows a section of the shell midsurface detailing the geometric parameters. Note again that r 1 is the radius of curvature of the meridian, r2 is to the central axis measured normal to the meridian and sweeps out a conical shape. The horizontal radius, when the axis is vertical, is r.
Equilibrium conditions for the element can be established by taking force and moment equilibrium in 3 directions. This results in 6 equations in terms of the 10 stress resultants. These are detailed in what follows. For example taking force equilibrium in the direction of the tangent to the meridian we have the following force contributions ∂N
∂r
φ dφ r + dφ dθ From N φ : N φ + ∂ φ ∂φ
From N θ φ : − N θ φr1 dφ + N θ φ +
N φ rd θ
−
∂N θ φ
dθ r1 dφ ∂θ
=
∂N θ φ
=
∂θ
∂(rN φ )
dφdθ
∂φ
r1 dφdθ
From N θ : − N θ r1 cos φdφdθ From Qφ : − Qφ rd φdθ From pφ :
pφ ( rd θ)( r1 dφ)
to a first approximation
Adding these contributions together and cancelling the terms dθ dφ gives the first equilibrium equation 1.14(a). The others follow by force equilibrium along the circumferential and radial directions together with moments equilibrium in the three mutually perpendicular directions. r1
∂N θ φ
∂θ ∂( rN φθ ) ∂φ
+
+
r1
N θ r1 sin φ ∂M θ φ
∂( rN φ ) ∂N θ ∂θ
+
r1
+
∂φ
N φθ
r1 N θ φ cos φ
+
+ rN φ
∂θ ∂(rM φθ )
− r1 N θ cos φ
∂φ
+
∂Qθ r1 ∂θ
∂(rM φ ) r1
− Nθ
φ
− Qθ r1 sin φ +
− Qφ rr1
∂φ ∂M θ ∂θ
− r1 rQθ
+
Mθ r2
φ
− Qφ r + pφ rr1
∂(rQφ ) ∂φ
+ pθ rr1
− p r rr1
=0
M θ φr1 cos φ
=0
M φθ r1
=0
=0
− M θ r1 cos φ +
−
=0
= 0
(1.14) Before proceeding further it is worth mentioning the special class of membrane shells where bending effects are small, (away from discontinuities) neglected (for convenience) or impossible (e.g. a child’s balloon). 1.3(a)
Membrane shells
Suppose then that Mθ , Mφ , Mθ φ , Mφ θ are zero whereupon Nφ θ (1.14) reduce to
= Nθ φ and equations
r1
∂N θ φ
∂θ ∂( rN θ φ) ∂φ
∂( rN φ )
+
∂φ ∂N θ r1 ∂θ
+
Nθ r2
+
− r1 N θ cos φ
+
pφ rr1 = 0
r1 N θ φ cos φ
+
pθ rr1 = 0
+
Nφ
−
r1
pr = 0
(1.15) There are thus now effectively 3 unknowns, Nθ , Nφ , & Nθ φ and 3 equations of equilibrium. The membrane shell is thus ‘statically determinate’, i.e. a solution can be found from equilibrium alone as we have noted earlier for cylinders and spheres. These equations may be reduced to those for a particular type of shell of revolution by inserting the relevant radii of curvature. For example, for a sphere r1 = r2 = a. However, a useful reduction is for rotationally symmetric loading and we shall proceed with this simplification first of all. It should be noted in passing that for a shell to be truly membrane the boundary support conditions must only sustain membrane actions. 1.3(b)
Rotationally symmetric load on membrane shells
For rotational symmetry all θ sections are considered to be the same and if pθ = 0 then all derivatives with respect to θ vanish. Then, since partial differentials are no longer required, we have d ( rN φ ) dφ
Nφ r1
− r1 Nθ cos φ
+
Nθ r2
−
pr
+
pφ rr1
=0
= 0
(1.16)
Consider the case of internal pressure loading only pr = p Equation (1.16) becomes d (rN φ ) dφ
Nθ
− r1 N θ cos φ
pr2
−
r2 Nφ r1
, Nθ
=
pr2 2
=
= 0
(1.17)
Solving for Nθ and Nφ gives Nφ
=
pr2 2
r 2 − 2 r1
(1.18)
These results are for a general shell of revolution under internal pressure. They can be particularised by inserting the relevant r1 & r2. For example, in a sphere r1 = r2 = a whence Nφ = Nθ = pa/2 or for a cylindrical shell of radius R, r1 = ∞ and r2 = R whence Nφ = pR/2 and Nθ = pR. With N/t = σ these are the same results as obtained earlier from first principles. While membrane solutions are very useful and frequently they are the basis on which design scantlings are settled, it is not long before their limitations are apparent. When one considers the junction of two different shells problems manifest themselves. For example, in the junction region of a pressurised torispherical head and a cylindrical shell, the circumferential stress in the toroidal part turns out to be negative (when r2 > 2r1 for the toroid, see eqn 1.18) while the circumferential stress in the cylindrical part is positive (see Chapter 6). Such a situation cannot exist in a membrane in the absence of shear forces and moments. At many other shell junctions the differences are not so obvious but usually it is found that membrane strains and displacements of the various parts do not match; these may easily be checked by substituting the membrane stress results into the equations given later in Section 1.5. Obviously the membrane condition does not apply near to other disturbances such as local loads, supports, nozzles etc. It will therefore be necessary in general to employ the general equations of equilibrium as appropriate to particular geometric states. 1.4
STRAIN DISPLACEMENT EQUATIONS
Any point on the shell (Fig 1.10) may have a general mid-surface displacement described by the three components u, v, w in the θ ,φ and radial directions respectively. Since the strains represent the same information as the displacements it is important to have the relationships between them. First of all we deal with the mid-surface strains only and use the symbolism ε θ , ε φ and γ θ φ for the normal strains in the circumferential and meridional directions and the shear strain in the θ φ plane, respectively. Then for example using the usual engineering definition of strain, ε
φ
= (change in element length in φ direction)/(original length)
From Fig 1.10 examining the meridional section of the element and noting that the radial movement amplifies the lengths in the ratio (r1 + w)/r1 we have εφ
=
1 r1 dφ
giving εφ
=
∂v ( r1 + w) − r1 dφ r1 dφ + dφ ∂φ r1 1 r1
∂v w + ∂φ
(1.19a)
In a similar way from a plan view sketch the mid-surface circumferential strain can be derived as εθ
=
1 ∂u r ∂θ
+ v cos φ
+ w sin φ
(1.19b)
The shear strain is given by 1 ∂u r1 ∂φ
γθ φ =
u cos φ r
−
+
1 ∂r r ∂θ
(1.19c)
Of course the strains vary not only from point to point on the mid-surface but also through the thickness. The next step is to evaluate the strains at some radial distance z from the midsurface. The procedure is to assign displacements u, v, w to the point at the distance z and to attempt to find u, v, w in terms of the original mid-surface displacements u, v and w. The basic assumption usually made is that during deformation, an original normal to the under-formed middle surface will result in it remaining normal to the deformed surface. This can be understood easily from the so-called “hair brush” analogy; bending a plastic hair brush leaves the bristles straight (and the same length) although rotated in space. The radial distance z is assumed unaltered, although of course the various radii are adjusted accordingly i.e. r1 becomes r1 + z and r2 becomes r2 + z. The secret in the derivation is to draw a clear diagram whereupon equations 1.19 become the strains at any point and u, v, w are used instead of u, v, w. Thus the strains at any point through the thickness can be written in terms of the original mid-surface displacements u, v and w. They are usually further modified by splitting each strain into two parts, a mid-surface direct strain and a bending strain and written in the form ε
= ε mid
+
zk
where k is the change in curvature. It will be noted that this equation has a similar form to that for the simple beam equation (1.1). The detailed derivation will not be given here but the resulting curvatures are given in equation (1.20). kφ
kθ
= −
kθ φ = −
1 r2
2 rr1
1 ∂w v r ∂φ − r 1 1 ∂2w 1 ∂u cot φ ∂w + − − v 2 r1 r2 ∂θ r1 r2 ∂θ ∂φ 2 ∂ w 2 cos φ ∂w 1 ∂v + + + 2 ∂θ∂φ ∂θ rr1 ∂θ r
sin φ rr1
=
−
∂u ∂φ
1 ∂ r1 ∂φ
+
u cot φ r1 r22
( r2 − 2r1 )
(1.20)
These equations give minor inconsistencies in that for (u, v, w) displacements due to rigid body movements all of the strains do not vanish as they ought. Koiter [1.2] discusses the matter in some detail as does Kraus [1.3]. Sanders [1.4] equations are perhaps the most satisfactory since they are a consistent set of equations, however, the differences in the prediction of the stress resultants and displacements are usually unimportant from an engineering point of view. Equations (1.20) are obtained from Kraus [1.3]. They are the same as quoted by Gill [1.5] apart from an overall change in sign. The above expressions at first sight look rather involved but again when dealing with a particular problem they can often reduce considerably. 1.5
CONSTITUTIVE RELATIONSHIPS
Engineers rely largely on mechanical tests to allow them to represent material behaviour. In particular the constitutive relationship is usually determined from uniaxial tensile testing and then generalised for multiaxial behaviour. The main relationship is between stress and strain and it may be elastic, anelastic, linear, non-linear, isotropic, anisotropic, time dependent, etc. In addition temperature may be important. However in many structures elastic conditions dominate and in this section attention will be confined to linear elastic isotropic behaviour. The uniaxial stress strain characteristic for a uniaxial situation is usually written in terms of Young’s modulus E as σ
= Eε
(1.21)
where the stress σ and the strain ε are both in the loading direction. Transverse to the load the strain is negative and equal to vε where v is known as Poisson’s ratio (0.25 to 0.35 for common engineering materials within the elastic range). If an element of material is loaded in three orthogonal directions the relationships are εx
=
εy
=
εz
=
1 [σ x − v (σ y + σ z )] E 1 [σ y − v (σ z + σ x )] E 1 [σ z − v (σ x + σ y )] E
(1.22)
If shear is present γ xy
= τ xy / G
etc, where
G=
E 2(1 + v )
(1.23)
In the case of a thin pressure vessel the radial stress is usually ignored (σ z = 0, plane stress) and the equations are rearranged as E [ε x + vε y ] 1− v2 E = [ε y + vε x ] 1− v2
σx =
σy
(1.24a)
or transformed into the plane of the shell in terms of θ and φ we have
σθ
σφ
E 1− v2 E = 1− v2 =
[ε θ + vε φ ] [ε φ + vε θ ]
(1.24b)
These two dimensional stress/stain relations can be applied to any point on the shell. It will be noted that we have now inherently made two somewhat contradictory assumptions namely zero radial strain (the radial distance z does not change, section 1.4) and zero radial stress; in
fact this has little effect on the outcome of shell analyses since we are generally not too interested in the radial direction. 1.6
METHODS OF SOLUTION
Clearly to solve the general shell problem in the standard fashion would involve unmanageable mathematical complexity and (usually) numerical procedures become necessary. In point of fact the recent advances in Finite Element software mean that the vast majority of the pressurised components discussed in BS 5500, can readily be solved, certainly as thin shells, by the finite element method on a modest desk top computer. However, it is of value to illustrate the classical approach for certain special cases confining ourselves to linear elastic behaviour since many of the rules in the Standards are based on this approach. 1.6(a)
Rotationally Symmetric load on a Spherical Shell
A case, which is of considerable interest in the context of pressure vessel design, is that of the spherical shell. It is simpler and convenient to consider only rotationally symmetric loading. This means all derivatives with respect to θ disappear and also we put pθ = 0. Hence these, together with the sphere geometric relations r1 = r2 = a, reduce the equilibrium equations 1.14 to the following set of ordinary differential equations. d ( N φ sin φ) − N θ cos φ − Qφ sin φ + ap φ sin φ = 0 dφ d (Qφ sin φ) + N θ sin φ + N φ sin φ − ap r sin φ = 0 dφ
(1.25) d ( M φ sin φ) − M θ cos φ − aQ φ sin φ = 0 dφ
Under the same conditions, from equations (1.19 and 1.20), the strain-displacement relations become: dv w + dφ
εφ
=
1 a
εθ
=
1 a sin φ
kφ
= −
1 a2
kθ
= −
cot φ a2
(v cos φ + w sin φ ) dw dφ − v dw − v d φ
d dφ
(1.26)
The stress/strain relations are, as before, for equations 1.24(b)
σθ
σφ
E 1− v2 E = 1− v2 =
[ε θ + vε φ ] [ε φ + vε θ ]
1.24(b)
bearing in mind that in general εφ
= ε φmid
+
zk φ
and
εθ
= εθmid
+
zk θ
Because the shell theory operates with stress resultants we need to recall the relationship between the stress resultants and the stresses. Typically for Nφ we have from equation 1.12(a) for example, with r2 = a
∫
=
Nφ
+t / 2
−t / 2
a+ z σφ dz a
Thus we can write Nφ
=
t/2
∫
−t / 2
E a+ z [(ε φmid + zk φ ) + v(εθmid + zk θ )] dz 2 (1 − v ) a
(1.27)
so that we can relate the strains directly to the stress resultants (rather than the stresses). Similar results are obtained for the other stress resultants. Note in equation (1.27) ε
φ mid
and ε
θ mid
are mid surface strains.
We now have three sets of equations namely equilibrium (1.25) strain-displacement (1.26) and the constitutive relations (1.27 etc). Proceeding in an exactly analogous way to that for the beam and assuming the shell is of constant thickness, substituting equation (1.26) into (1.27) gives Nφ
=
D a
Nθ
=
D a
Mφ
=
K a
Mθ
=
K a
dv dφ + w + v(v cot φ + w) dv + w (v cot φ + w) + v dφ dχ dφ + vχ cot φ dχ χ cot φ + v dφ
(1.28)
where D = Et/(1-v2), is known as the direct stiffness, K = Et3/[12(1 - v2)] is known as the bending stiffness and dw
χ = − v / a dφ
is a subsidiary variable, which is in fact the rotation of the tangent to the meridian. The stress resultants of equation 1.28 may then be substituted into the equilibrium equations 1.25 to seek a solution. Classically the equations have been reduced to a pair of similar equations in Qφ and χ . First of all we will restrict our attention to the case of zero surface loads. Edge Bending
Assuming for the present that pθ = pr = 0 i.e. zero surface loading, the first of the pair of equations is easily derived. Substitution of equation 1.28 (c,d) into equation 1.25 (c) leads to d 2χ dφ
dχ dφ
+
cot φ
−
χ (cot 2 φ + v) =
a2 K
Qφ
with some manipulation detailed in ref [1.1] the second equation can be obtained as d 2 Qφ dφ
+
dQφ dφ
cot φ
− Qφ (cot 2 φ − v)
= − D(1 − v 2 ) χ
This pair of equations describes the state of a spherical shell loaded symmetrically at its edges (Fig 1.11). As it happens the case of a concentrated axial load at the apex is also covered but that is not of interest in the present context. Defining for convenience the linear differential operators L( − −)=
d 2 (− −) d (− −) + cot φ − (− − ) cot 2 φ 2 dφ dφ
we can separate the unknowns Q and χ to obtain an equation in either. For example LL(Qφ )+4k4Qφ = 0
(1.29)
where k 4 = 3(1 − v 2 )
(i)
a2 t2
−
v2 4
Solution for edge bending for large φ
If one assumes that φ is large and cot φ small then a possible solution of equation 1.29, which was attributed to Geckler, is Qφ = [e λ φ( A1 cos λ φ+ A2 sin λ φ) + e −λ φ( A3 cos λ φ+ A4 sin λ φ)]
(1.30)
where λ 4 = 3(1-v2) a2/t2 and the A’s are constants. Usually it is taken to apply if φ > 30°. (ii)
Solution for edge bending for all φ
The Langer asymptotic solution has been used by Leckie & Penny [1.6]. Briefly it is valid for all φ and can be written in terms of Bessel functions as Qφ =
where
Z =
2kφ
φ
sin φ
[ A1ber Z + A2 bei Z + A3 ker Z + A4 kei Z ]
(1.31)
Full Solution Let us return for a moment to the consideration of the loading terms. If p φ and pr are included in equation 1.29 it becomes LL (Qφ ) + 4k 4 Qφ = Φ(φ)
(1.32)
where dp Φ(φ ) = (1 + v)aL ( pφ ) + aL r dφ
dp − v (1 + v) pφ a − va r dφ
Having found the homogeneous solution with the right hand side zero we now require a particular integral. It turns out that the particular integral is well approximated by the membrane solution for many common types of loading. For example if the only loading is pr = p, uniform internal pressure, then it is not difficult to show that Φ (φ ) = 0 i.e. Qφ = 0 or the membrane solution in this case is adequate. For other symmetric loadings the particular integral for Qφ turns out to be approximately proportional to the reciprocal of k 4. Since k is normally fairly large for thin shells it is usual to neglect this solution and approximate the particular integral by the membrane solution. In other words the required solution for spherical shells can be found by superimposing the membrane solution for any surface loads with the edge bending solution neglecting surface loads. Examples of Edge Bending (large φ ) As an example of the edge bending solution, consider the spherical shell in Fig 1.11 subject to an edge force H and moment M. We assume φ to be large whereupon the solution given in equation (1.30) can be used. Also equation (1.29) reduces to d 4 Qφ dφ 4
+ 4λ4 Qφ = 0
(1.33)
with λ4 = 3(1 − v 2 )( a / t ) 2
It turns out that the constants A3 and A4 in equation (1.30) have to be zero since logically one expects the effects to disappear with increasing distance from the edge while expressions like λφ e- increase with decreasing φ (i.e. away from the edge). The full equation 1.30 refers to the situation when there are two edges to the shell and two sets of edge forces. Assuming here that we are concerned only with one edge and changing variables from φ to ψ = (α φ ) – see Fig 1.11, allows equation (1.30) to be written as Qφ = Ae −λ ψ sin( λ ψ+ γ )
where A and γ are new but related constants. The other stress resultants can be found by substituting back.
(1.34)
N φ = − Ae −λ ψ sin( λ ψ+ γ ) cot( α −ψ )
π N θ = − 2λAe −λ ψ sin λ ψ+ γ − 4 a π Mφ = + Ae −λ ψ sin λ ψ + γ + 4 λ 2 M θ = vM φ
(1.35)
The rotation χ is χ=
2λ2 A −λ ψ e cos( λ ψ + γ ) Et
(1.36)
The displacement in the plane normal to the axis of rotational symmetry δ is frequently of interest. Since φ is large, cot φ is small, consequently Nφ can be neglected compared with Nθ so that δ can be found directly from the circumferential strain, (δ = rε θ ). Thus:δ =−
2 Aλa sin φ −λ ψ π e sin λ ψ+ γ − Et 4
(1.37)
We now have a general solution for the edge loaded shell in terms of the two constants. The constants will depend on the actual loading. As examples of edge loading we consider two useful cases of axisymmetric self equilibrating loading, namely edge force, H and edge moment, M. (i)
For the case H applied alone
The constants may be found from the boundary conditions which are,
at
φ = α , (or ψ = 0) ,
Mφ = 0 ,
Qφ = −H sin α
This gives the constants in the above equations as A = 2 H sin α
, γ =−
π 4
(1.38)
The edge rotation and displacement (ψ = 0) are then χψ =0 =
2λH a sin 2 α 2λ2 H sin α , δψ =0 = Et Et
(1.39)
The values of the stress resultants at any location ψ may be obtained from equation (1.35) using the values of A and δ given in equation (1.38):-
π N φ = − 2 H sin αe −λ ψ sin λ ψ− cot( α −ψ ) 4 −λ ψ N θ = 2λH sin αe cos( λ ψ) Mφ =
a
λ
H sin αe −λ ψ sin( λ ψ)
M θ = vM φ
(1.40)
Thus if the loading H is known, the behaviour of the spherical cap under this loading is completely solved; all the stress resultants (and stresses) and the deformations are determined. (ii)
For the case of M applied alone
The boundary conditions are, at
φ = α,
and thus
Mφ = M ,
A=
2Mλ a
Qφ = 0
, γ =0
The edge rotation and displacement therefore become:χψ =0 =
4λ3 M Eat
, δ ψ =0 =
2λ2 M sin α Et
(1.41)
The stress resultants for this case may be obtained from equation (1.35) using the values of A = 2Mλ /a and γ = 0 2 Mλ −λ ψ e sin λ ψcot( α −ψ ) a M −λ ψ π N θ = −2 2λ2 e sin λ ψ − a 4 Nφ = −
π M φ = 2 M e −λ ψ sin λ ψ + 4 M θ = vM φ
(1.42)
Again if the loading M is known all stresses and deformations can be found. It is shown in [1.5] that if the edge force actions H and M are applied to an upper edge of the shell with ψ = (φ - α ) instead of a lower edge the above results are similar except that the expression for Nθ and Mφ above change sign and the values of the constant A changes sign which means that Nθ and Mφ are in fact the same. The sign of the rotations alter but the sense of these should be clear anyway. The end displacement rotations and stress resultants are tabulated below in Tables 1.1 and 1.2. Results such as these may be used to solve problems such as a spherical cap, loaded by pressure or deadload, rigidly attached at its edges or part spheres attached to other shells.
TABLE 1.1 : EDGE SOLUTIONS FOR A THIN SPHERE – LOADS AT LOWER EDGE
δ is horiz displacement
H & M per unit length H ONLY Edge Rotation χ ψ =0
2λ H sin α Et
Edge Deflection δ ψ =0 Qφ
2λH a sin 2 α Et
Nφ Nθ Mφ Mθ
M ONLY 4λ M Eat
2
3
2λ2 M sin α Et M π 2λ e −λ ψ sin λ ψ 2 H e −λ ψ sin α sin λ ψ− a 4 M −λ ψ π e sin λ ψcot( α −ψ ) − 2 H e −λ ψ sin α sin λ ψ − cot( α −ψ ) − 2λ a 4 2λHe −λψ sin α cos λψ
vMφ λ
4
vMφ
a = 3(1 − v ) t 2
2
