1. INTRODUCTION. A shell may be defined as the solid material enclosed. between two closely spaced curved surfaces (1), the distance

Size: px

Start display at page:

Download "1. INTRODUCTION. A shell may be defined as the solid material enclosed. between two closely spaced curved surfaces (1), the distance"

Raymond Doyle
5 years ago
Views:

1 1 1. INTRODUCTION 1.1. GENERAL A shell may be defined as the solid material enclosed between two closely spaced curved surfaces (1), the distance between these two surfaces being the thickness of the shell. If the thickness is small compared to the overall dimensions of the bounding surfaces, then the shell is defined as a "thin" shell; if not, it is termed "thick". A thin shell may be described as a structure in which loads are transferred primarily by direct stresses, with relatively small or localised bending stresses (2). A shell which is formed by translating a curved line along a straight longitudinal axis and which spans longitudinally between supporting diaphragms is termed as a cylindrical shell. The curve in the transverse direction can be circular, parabolic, elliptical or any other shape desired. Reinforced concrete cylindrical shells are widely used as important components of many of the complex structures being designed today. Design of these shells is generally based on what is commonly known as the classical thin shell theory. While it is recognised that the assumptions of elasticity, isotropy and homogeneity, on which the classical thin shell theory is based are not completely fulfilled in a reinforced concrete structure, it is believed that the theory adequately predicts, for design

2 2 purposes, the internal forces in a shell due to a given loading. Unfortunately, because of the great computational effort required in the analysis of shells, solutions in the past have often been restricted to a small class of problems. Published tables greatly facilitate the analysis of uniformly loaded, simply supported single shells of circular cross section and constant thickness. However, even with these tables a large volume of computations is often required (2), if the effect of edge beams is to be taken into account. The importance of shell structures and their generic analysis complexities has naturally led to a reliance on the finite element method for the solution to many types of shell problems. This, of course, required a finite element representation of the shell behaviour and over the last 30 years many elements have been developed and employed is a multitude of programs (3). The main thrust of this research work is to provide design tables for circular cylindrical shells without edge beams and with edge beams using the finite element technique. The quest for a simplified design procedure, in a form which a practical designer would wish it, was a motivating force in the present study BASIC STEPS OF FINITE ELEMENT METHOD(4,5,6) In finite element method the physical body is approximated by replacing it with an assemblage of discrete elements. As these

3 3 elements can be put together in a number of ways, they can be used to represent exceedingly complex geometries. Also nonhomogeneous media, nonlinear material behaviour, dynamic loading, insitu stress conditions, arbitrary geometries, discontinuities and other factors can be handled. 1. Discretizing the continuum: The given body is subdivided into a number of elements which results in an eguivalent body of finite elements. Depending upon (i) the geometry of the body (ID,2D or 3D) or structure and (ii) the number of independent space coordinates (ie. x,y,z) necessary to discribe the problem, various one, two or three dimensional elements can be used. 2. Selecting Element Functions: Simple (trignometric/polynomial) functions are assumed to approximate the displacement of each element. These functions are called displacement models, displacement functions or displacement fields. 3. Calculating Element Properties: The Principle of minimum potential energy is used for the derivation of the following equilibrium equations which describe the characteristics of each element of our system, [K]{ ) = (R) where [K] = J[B]T [D][B]dv (1.1) [K] = Element stiffness matrix [B] = Matrix of (x,y) local coordinates [D] = Elasticity matrix (R) = Load vector {$} = Vector of nodal displacements

4 4 4. Assembling Element Properties: The above equilibrium equations are combined to form a complete set for the complete structure. The system assembly procedure is based on insistency of compatibility at the element nodes, (ie) At nodes where elements are connected the values of the unknown nodal displacement variables are the same for all the elements connecting at that node. 5. Introducing Boundary Conditions: The overall stiffness matrix [K] which we get after assembly is, in fact, singular. Its inverse does not exist and the associated set of simultaneous equations for the unknown displacements cannot be solved. The physical significance of this is that a loaded body or structure is free to experience unlimited rigid body motion unless some support constraints are imposed that will ensure the equilibrium of the loaded system. These constraints are the boundary conditions. 6. Calculating the Primary Unknowns: Important and useful features of the overall system matrix are (i) The system matrix will have nonzero terms clustered about its main diagonal, as a band centered on the diagonal, reflecting the connectivity of the finite element mesh. Appropriate numbering of the nodes causes the system matrices to be closely banded. (ii) System coefficient matrices are usually symmetric. aij aji Hence we need to store only the upper (or lower) triangular matrix.

5 (iii) Diagonal terms are positive definite. 5 One of the key factors in any finite element program is the subroutine for the solution of simultaneous equations [K US) = {R}. The choice of technique depends upon the size of the problem envisaged and upon the type of computer available. Nodal displacements (5), which are the main unknowns involved in the formulation, are calculated in this step. 7. Calculating Secondary Unknowns: The strains, stresses, moments, shear forces, etc. within each element are calculated in this step. First, the nodal displacements for particular elements are extracted from the complete set of nodal displacements and strains are calculated with the help of { } = [B](&} (1.2) After knowing strains, stresses can be calculated by using the relation ( ) - [DJ{6) - [DJ[B]{$). (1.3) 1.3.SHELL ACTIONS & THEORIES (7,8) Membrane Theory of Shells Shells in their ideal form are structures purely in compression. 'Membranes' as we understand them would be able to resist only tension (and no bending). A soap bubble is an ideal

6 6 membrane - thin, tensile and appropriately shaped. The bubble has an inside pressure. If a some what rigid bubble can be made, it will be able to take external pressure and develop pure compression. This is our 'Membrane' shell. Normally we extend this meaning, and include shells developing inplane forces as membrane shells. Referring to Fig.1.1. Nx and Ng can be either tensile or compressive. Nx0 is called the In-plane shear Nx, N0 and Nx0 are forces per unit length and called stress resultants. No real shell behaves like a membrane. But many shells have predominantly membrane action. For such shells, an acceptable solution can be obtained from the equilibrium of the membrane. Usually membrane solutions are very much simpler compared to a 'general solution'. A general solution would take into account all types of stresses, expecially bending and out of plane forces (Fig.1.2). Whereas membrane theory (which ignores the presence of out of plane forces) makes use of equilibrium conditions alone, bending theory makes use of equilibrium of forces and compatibility of displacements. Membrane theory is used as a starting point in many solution routines Bending Theory of Shells Bending theories recognise the presence of bending stress resultants such as Mx, M0, Mx0, Qx and Q0 (Fig. 1.2 )

7 7 DIRECTRIX FIG.1.1 MEMBRANE OR IN PLANE FORCES FIG. 12 BENDING OR OUT-OF-PLANE FORCES

8 besides membrane stress resultants, namely Nx, N0 and NX0. 8 The principles of theory of elasticity are applied in the following sequence to obtain successive relations between 1) Displacements and strains/changes of curvature, ii) Strains & stresses; curvatures and moments, iii) Stresses & stress resultants. Comprehensive equations of equilibrium between all the eight stress resultants are obtained, omitting external forces. There are several methods each differing from the others in the stress resultants ignored and assumptions made. Consequently each theory results in a different differential equation ANALYSIS OF CYLINDRICAL SHELLS BY CLASSICAL METHODS(9,10) The exact relations governing the action of shells must be based upon the equations of the mathematical theory of elasticity. These exact equations of elasticity lead to expressions and equations which are so complicated that it is impossible to obtain solutions for the shell problems of practical interest. In an attempt to obtain solutions to at least some of these problems, the exact equations have been approximated in various ways by different investigators, leading to different final forms of a differential equation, generally in terms of the displacement^. These equations are almost invariably of the eighth order - but there the similarity ends. For the analysis of circular cylindrical shells, of the many theories available, the important and widely used ones are those

9 9 given by Fliigge, Gibson, D-K-J, Dischinger, Aas-Jakobsen, Vlassov, Lundgren, ASCE Manual, Finsterwalder and Schorer. The main object of these theories is to simplify the equations by making assumptions based on the behaviour of the shell. It is known that a long shell resists the external load predominantly through beam action while a short shell resists the load predominantly by arch action. Based on this fact, certain forces and/or displacements can be neglected while developing approximate equations for the analysis of long or short shells. Hence it becomes necessary to carefully study the assumptions and approximations made in a particular theory before applying it to the analysis of cylindrical shells of given dimensions. In general, the approximations made in the various theories can be classified under one or more of the following heads: (a) Neglecting the contribution of radial shear (Q0) in the equilibrium equation. (b) Neglecting some terms (which are small) in force - displacement relations. (c) Neglecting some forces and deformations altogether. (d) Neglecting the Poisson's ratio. Fliigge first presented the rigorous theory for the analysis of cylindrical shells and for assessing the applicability of the other theories, Fliigge's equations have been taken as reference. McNamee, Moe, Holand, Sen, Jenssen and Arya have compared the various theories and the following conclusions are largely based on their results. The theories of Dischinger, Aas-Jakobsen, Lundgren, ASCE Manual, Holand and D-K-J can be applied to

10 10 cylindrical shells of all dimensions. The theory of Vlassov can be applied only to short shells while the theories of Finsterwalder and Schorer can be applied to long shells only (ref.10). The equations of the above theories are occasionally called the exact equations of the shell problem. However, we have no real assurance that any particular solution obtained by an "exact" method is more accurate than one obtained by an approximate method for the shell structure. Moreover, the various "exact" methods do not always give solutions that agree with each other. The apparent agreement between the different methods could be good to poor, according to McNamee (Ref.9) ANALYSIS OF CYLINDRICAL SHELLS USING ASCE TABLES (11,12) Cylindrical shells without edge beams The following procedure is adopted in the analysis. 1. Membrane Analysis: The surface load is assumed to be transmitted to the supports solely by direct stresses, generally called membrane stresses, and the internal stresses and the edge forces created by the surface loading are determined. 2. Errors: "Membrane Analysis" yields displacements and reactions along the longitudinal edges of the shell that do not comply with the boundary requirements. 3. Corrections: To satisfy the boundary requirements, line

11 11 loads must be applied along the longitudinal edges of the shell. 4. Analysis for the effect of line loads: Bending as well as direct stresses created in the shell by the applied line loads are determined. The stresses produced by the line loads must be added to the direct stresses previously evaluated, to obtain the final stress pattern. Because of mathematical difficulties in the bending theory, the uniformly distributed corrective line loads applied along the longitudinal edges are expressed as functions of nlf x sin in which n is any integer. A uniform line load (Fig.1.3 ) may be represented by a Fourier series as (T.) 1 x JL_ T oo 1 _ j n IT x -- Sin ir T* n = lf3p n (1.4) FIG. 1.3

12 12 To provide compatible forces and displacements along the length of free edges under step 4, it is expedient to regard the surface load on the shell also as the sum of partial loads, the intensity of which is defined by a Fourier series of sines as nttx ^>n sin --- The Fourier series for a uniform load Px equals P X 4P x IT OQ n = 1,3,5 1 n sin n TT x 1 (1.5) In tables IB and 1C of the ASCE manual 31, the membrane forces and displacements produced by the sinusoidal loading represented by the first two terms only are given. However when membrane stress resultants obtained by the use of the first two terms of the partial loads are compared with those obtained by the uniform load, it is found that good agreement is obtained for the longitudinal force, Nx. The components of shear stress, NX0 and transverse stress, N0 obtined by the sum of the first two sinusoidal loads are approximately 10% and 15% too small. In designing shells by partial loads, this discrepancy should be borne in mind and adjustments should be made in the final values. For long shells, it is sufficient to use just the first term of the series. For short shells it is sometimes advisable to include the second term as well.

13 13 In tables 2A and 2B of the ASCE Manual 31, the force distribution and displacements at the edge in long barrels for symmetrical line loads are given as represented by the first term of the Fourier series. Extracts of Tables IB, 1C, 2A and 2B of the ASCE Manual are shown in Appendix-A for ready reference regarding the format and use Cylindrical Shell with Edge Beams The analysis of a cylindrical shell fixed to longitudinal edge beams is divided into two parts to facilitate analysis. 1) In the first part, the stresses and displacements along the edge, created by the external loading (dead and live loads) and the corrective line loads are determined, neglecting the presence of the edge beam. 2) The second part consists of applying additional line loads on the shell and equal though opposite loads on the beam so that the longitudinal stresses and displacements at points common to the two members are equal. For the second step, additional formulae for the edge beams are provided in the Manual. The loading on the shell is approximated by partial loadings represented by successive terms of a Fourier series. This same approximation of loading must likewise be applied on

14 14 the beam so that conditions of equal stress and strain are satisfied along the entire length of the beam. To achieve unity between shell and edge beams, pairs of equal but opposite forces like Normal loading (Vb) and shearing load (Sb) are to be applied along the plane of contact. For determination of the unknown values of Vb and Sb the stresses in the shell are equated to those of the beam and the displacement of the shell is equated to that of the beam. Simultaneous equations are formed and solved to get Vb and Sb and these values are incorporated in the further analysis to get the final values of stress resultants in the shell and edge beams FINITE ELEMENT ANALYSIS OF SHELLS <13,14) It is well known from the theory of shells that classical solution involves tedious calculations and is extremely difficult especially for shells of arbitrary shape. The problem becomes much more complicated when the shells have elastic boundaries which is the case in most practical situations. Finite element method is very much suited for the analysis of general shell shapes because of its flexibility in accounting for arbitrary geometry, loading and variation in material properties. The following approaches are possible for the finite element analysis of shells a) Shell as an assembly of flat faceted elements

15 15 b) Shell as an assembly of curved elements c) Shell as an assembly of three dimensional solid elements d) Shell an assembly of degenerated case of three dimensional solid elements. a) Shell as an assembly of flat faceted finite elements: This is based on a physical approximation of the curved shell surface as consisting of a large number of flat pieces of triangular, or quadrilateral shape. In the limit when the number of pieces become infinitely large, the true surface is represented. This is one of the earliest finite elements to oe used for shell analysis and finds a place in the elements libraries of many well known package programs like SAP IV. b) Curved thin shell elements: There are considerable theoretical difficulties in formulating a curved thin shell element. There is a large number of classical theories differing mainly in the approximations introduced in the strain-displacement relations. A thorough examination, as stated in ref.14, by Koiter of available theories disclose that certain formulations do not properly account for the condition of zero strain under rigid body motion in the representation of the twisting curvature term. In the literature there is less emphasis on this approach to finite element analysis of shells when compared to the other approaches.

16 16 c) Three dimensional solid, shell elements: Three dimensional solid elements like 8 noded solid element can be used but in this case more than one layer of elements may be needed across the thickness to simulate the bending behaviour of shell. As an alternative, higher order elements like 20 noded isoparametric solid element can be used. But these approaches to treat the analysis of shell as three dimensional stress analysis are very costly (Ref.13). d) Degenerated shell elements: Shell elements have been developed by degenerating the solid element formulations by introduction of assumptions that normals to the midsurface remain straight and that the normal stress is zero. By this approach the displacements and rotations of the shell midsurface are taken as degrees of freedom. This formulation makes it possible to develop elements for the analysis of moderately thick shells but in the case of thin shells selective and reduced integration technique have to be used due to shear locking effect. (Ref.13) 1.7. ISOPARAMETRIC FINITE ELEMENTS (15) Isoparametric elements make it possible to have non rectangular quadrilateral elements. Their most apparent features are (i) sides that may be curved and(ii) their special coordinate system^, *0 and in Fig.1.4). Isoparametric elements are useful in modelling structures with curved edges and in grading a

17 mesh from coarse to fine. Isoparametric elements are versatile; they have proved effective in two and three dimensional elasticity, shell analysis, and non structural applications. QUADRATIC PLANE ELEMENT (THE SIDES CAN BE STRAIGHT LINES OR QUADRATIC CURVES) A DEGRADED CUBIC QUADRATIC SOLID ELEMENT ELEMENT WITH LINEAR EDGES (THE LEFT AND LOWER SIDES ( A SOLID ELEMENT WITH LINEAR AND CAN MATE WITH LINEAR AND QUADRATIC EDGES ) QUADRATIC ELEMENTS) FIG. 1 4 EXAMPLES OF ISOPARAMETRIC ELEMENTS 1.8. BILINEAR DEGENERATED SHELL ELEMENT (3) FIG. 1*5 FOUR NODED SHELL ELEMENT

18 18 The Bilinear Degenerated Shell (BDS) element (Fig.1.5) evolves from an eight-noded three dimensional brick element. The midsurface, enclosed by four straight sides, forms a hyperbolic paraboloid and, as the name implies, the concept of degeneration is used (Ref.3). Due to the simplicity of the bilinear midsurface geometry of the from Z = xy/c, where c = LB/h (Fig.1.6) and the bilinear displacement field of the form p r * 1 u ui n3i eyi ~ m3i ezi 4, 1 V vi * +-- thi ' x3i ezi " n3i exi / i=l 2 r w m3i exi -^i eyi -* L Cl-6) the strain energy can be integrated analytically over the shell thickness. The integration over the reference midsurface can then be performed numerically. This not only simplifies the derivation but also saves computer effort in forming the element stiffness matrix. The rigid body modes are present since this is xi, yi/ l-3i» m3i/ etc., are defined in Appendix-B.

19 19 an isoparametric element. The transverse shear strain energy is retained. Consequently, the element is applicable to either thick or thin shells. Only one point numerical integration is used for this shear strain energy at the centre of the element to avoid the shear locking effect. The element, when applied to plates is equivalent to the plate element. Several examples are reported to have been tested using the BDS elements (Ref. 3 ). The results show that the element is capable of performing accurately for both thin and thick shells. The details of the formulations of this element are presented in Appendix-B.

Guidelines for proper use of Plate elements

Guidelines for proper use of Plate elements In structural analysis using finite element method, the analysis model is created by dividing the entire structure into finite elements. This procedure is known