),x}2 + [v0,x - (ztp),x]2 + [(itf; + w0 + z),x]2} + i y 2 {[Tt 0 i y - (2<^ ),y]2 + [1 + V0jy - (2^),J\/]2 + [fat + \u2122 0 + 2),y]2} + dz 2{^ 2+V 2 + l} + 2dxdy {[1 + ti0,x - (^ ),x] No,y - (^ ),\u00bb] + [TJ 0, X - (r0),x] [1 + i ; 0 i J \/ - {zrb)M] +(wi + io 0 + z)iX(wi + w 0 + z)>y) Chapter 2. The Formulation of the Finite Strip 37 + 2dydz {(wi + w0 + z),y - (j> [u0jy - (z(j>) x - (z VO,*] 2 + [(Wi + w 0 + z),xf - 1 - \\{wi + z),x]2} (2.45) X f 2 2 + [(Wi+Wo + z ) i y ] 2 - l - [ ( W i + z)iy]2} (2.46) 7*2\/ = { [1 + \u00ab0,x - (Z <\/>),*] k>,y - (Z ** rotation due to the existence of the individual support rotation degrees of freedom that have the same sense. But the xb compatibility cannot be satisfied with the avail-able degrees of freedom. In the case of translatory displacements, the compatibility Chapter 2. The Formulation of the Finite Strip 61 Figure 2.7: Compatibility at a Corner of a Container is easily provided because of the existence of the support degrees of freedom. In order to satisfy the compatibility condition in xp when the fixed-ended strips are to be used, a constrained variational principle was utilised in the problem formu-lation. The Penalty Function method was selected over Lagrange's method as it does not involve additional unknowns. But, as will be shown by numerical examples pre-sented in Chapter 4 this constraint could also be satisfied sufficiently, in the problems considered, by specifying the tp rotation to be zero at the corners. 2.8 .1 T h e C o n s t r a i n t s The Penalty Function formulation introduced by Courant for a conservative problem is based on the minimisation of a new functional instead of the total potential energy. Although the previous derivation was with the use of the Virtual Work principle, here it will be based on the total potential energy approach, and will follow the method Chapter 2. The Formulation of the Finite Strip 62 presented by Zienkiewicz(1977). As mentioned above, the constraints to be satisfied with the Penalty Function method refer to the rotations ib at the corners. Let us consider the joint of the container shown in Figure 2.7. (The 'rotation' R shown in this figure is the \u00a3 derivative of the local v displacement.) If the panel on the left side of the joint, when viewed from outside, is to be denoted by L and that on the right side by R, the constraints to be satisfied along the joint can be written as MIL = \u00b0 <2-i32) *--{%) ^ = \u00b0 \u00ab2-i3) The \u00a3 coordinates in the above equations are local to the respective panels. The subscripts R and L refer to the panel, and \u00a3 = 0 is used to indicate that this constraint is being satisfied only at the reference surface. Now, as all the above quantities are to be evaluated at the corner, and also considering the fact that in the computer code it has been made required that at a given joint the local coordinate ( of the two meeting strips should not be the same, the following expressions can be written with respect to Equations 2.132 and 2.133. fa = Mt = 0) (2-134) fa = fa(t = l) (2-135) SL \u2022 SM. SL \u2022 SM. where the ib and v terms and derivatives are to be obtained from the shape function matrices of the respective finite strips of the respective panels as given below. Chapter 2. The Formulation of the Finite Strip 63 From this point onwards, unless mentioned otherwise, the treatment will be with respect to the two finite strips that correspond to each other in the two panels, and so meet at the common joint. This is on the assumption that all walls of the container are discretised in the same manner. The derived matrices can then be assembled in the finite element manner to get the total matrices. Towards this, let T\/;^ refer to the \u20220 degrees of freedom of the strip R at its support A, which is the support of concern here. Similarly, let ?\/>B be defined for the support B of strip L. Then, the values of ?\/> terms of the above equations can be written as Mt = 0) = W r {V>*} (2-138) Mt = l) = W T K } (2-139) where {N} is the interpolation vector across the strip as defined by Equations 2.6 to 2.10. Also, given that, v = {Nv}T{d} (2.140) where the first vector is obtained from the second row of the shape function matrix [N] of Equation 2.28, the partial derivatives off are given by | = (2-141) In the above, \/ is the length of a finite strip. Now, as the two adjoining sides of the container need not be of the same length, let li and IR refer to the lengths of the left side and right side walls, respectively. Then, assuming that these containers will have only moment resisting corners so that the same shape function vectors and the same number of degrees of freedom are used in all strips, the constraint equations given above (Equations 2.132 and 2.133), can be re-written as {N}T {^i} + U N ^ O V {dL} = 0 (2.142) {N}T {^} + lR{N}(0,fj,0)}T{dR} = 0 (2.143) Chapter 2. The Formulation of the Finite Strip 64 where {dR} and {dL} are the local displacement degrees of freedom vectors for the right hand and left hand side strips, respectively. These can further be simplified as to be expressable purely in terms of the degrees of freedom vectors of the strips by defining a new shape function vector {JV^*} which gives the xb of the supports A and B on substitution of 0 and 1, respectively, for \u00a3. This can be obtained by taking the shape function vector Nv defined above for v, and substituting zero for all the terms that do not correspond to the support xb terms, and substituting unity for the \u2014 \u00a3 terms of the others. This will not be presented here, but will be clear on the examination of Equations 2.28 to 2.36. Then, the above constraint equations become {N*'(i=0)} T {dR} + lL{N}(l,fj,0)}T{dL} = 0 (2.144) {JV '^Ce = 1)}T {d^} + \u00b1 { ^ 0 , ^ 0)} r{*l , i} = 0 (2.145) which, on combination to give a single equation will be {N*-(\u00a3 = O)}T i{w\u00ab-(i,,-,o)}T \" I {<\"} ) > = < I W J 1 t o \/ (2.146) In the above the shape functions are in the local coordinates of the respective strips. For ease in the subsequent developments, Equation 2.146 will be written in a concise form as [C}{d} = {0} (2.147) where, (2.148) Chapter 2. The Formulation of the Finite Strip 65 and [C] = = 0 ) } r i{Jvj } (2.158) 2.9 .1 .2 T h e D i s p l a c e m e n t a n d E l a s t i c F o r c e V e c t o r s Now, if the 5 X 1 vector of reference plane displacements at the support is denoted by {ds(r})}, the elastic forces in the restraining springs can be written as {Fs(v)} = \\kTs(v)} {ds(v)} (2.159) The displacement vector used above can easily be obtained in terms of the strip degrees freedom of vector {d} and the shape function matrix N defined earlier in Chapter 2. The Formulation of the Finite Strip 69 Equation 2.28. It is to be carried out as follows. For the first three members u, v and w, substitute \u00a3 = 0 and \u00a3 = 0 for support A or \u00a3 = 1 for support B. That is, u(rj) v(n) = 0)] {d} (2.160) w(n) where, \u00a3s is the coordinate value for the support of concern. The two rotations (j> and N by if), respectively, can similarly be obtained from the first and second rows of eliminating their -^independent parts and then substituting -1 for \u00a3, in addition to the substitution for \u00a3. Let, the new matrix thus obtained be denoted by [.\/VRS], SO that {ds(v)} = [NRs(v)} {d} (2.161) Then Equation 2.159 can be re-expressed as {Fs(v)} = \\ks(v)\\ NRS(V)\\ {d} (2.162) giving, as a function of 77, the elastic forces in the springs of support S. 2.9.1.3 Virtual Work and Stiffness Matrix Here a virtual work formulation is used to obtain the additional terms contributed by the elastic restraints to the finite strip stiffness matrix. Let there be a virtual displacement {8d} in the finite strip degrees of freedom. Then the virtual work of the restraining springs in support S can be written as rd -d~ giving the additional stiffness from this support as SW = {8d}T fd[NRS}T[krs}[NRS}dr,{d} J \u2014dl (2.163). . [K*] = dJ\\[NBs?[krs][NBs]drj (2.164) where the limits of integration have been non-dimensionalised. Chapter 2. The Formulation of the Finite Strip 70 2.9.1 .4 F o r m o f t h e S t i f fness C o n t r i b u t i o n An examination of Equation 2.161 will reveal that the matrix NRS\\ has non-zero columns only with respect to the degrees of freedom of the particular support. There-fore, the additional contribution as given by Equation 2.164 will have zero terms with respect to all the degrees of freedom that do not correspond to the support of concern. 2.9.2 R e s t r a i n t s A l o n g a N o d a l L i n e o f a F i n i t e S t r i p This section considers the elastically restrained supports that can occur at the plate edges parallel to the finite strip nodal lines. As it is rare for a single finite strip to be used in the analysis of a full plate, it can be asserted that a finite strip will, in general, have at most only one restrained nodal line. 2.9.2.1 M o d e l l i n g o f R e s t r a i n t s Again, here, the restraints will be introduced as independent hnear elastic springs corresponding to each of the degrees of freedom. In this case the spring stiffnesses along the support are to be expressed as functions of the longitudinal coordinate \u00a3. Thus, let Kit) {*r(0> = { kut) \\ (2-165) where each of the members of the vector are expressions in \u00a3 for the respective stiff-nesses along the edge. The subscript T has been introduced to indicate the fact that this is a case of restraints at a nodal hne. The second subscript refers to the degree Chapter 2. The Formulation of the Finite Strip 71 of freedom. As done before, these can be expressed in the form of a diagonal matrix kj denned by KH) o o o o kjw(o o o o 0 0 0 0 0 0 0 0 0 (2.166) 2.9.2.2 T h e D i s p l a c e m e n t V e c t o r Let the 5 X 1 vector of reference plane displacements be denoted by {d{\\. The members of this vector can be obtained by the use of the shape function matrix and the displacement degrees of freedom vector defined in Equation 2.28 in a manner similar to that carried out for the supports. Then, the first three members of the vector will be given by v(0 = [N(t>Vi,0)\\ {d} (2.167) where fji is the coordinate value of the strip nodal line of concern, and will be either -1 or 1. The two rotations (b and tb, respectively, can similarly be obtained from the first and second rows of |iV\"j by eliminating their -^independent parts and then substituting -1 for \u00a3, in addition to the substitution of fji. Let the new matrix thus obtained be denoted by \\NRI , so that {di(t)} = #\u201e(0 {ci} (2.168) Here it may be noted that only the coefficients of {d} that correspond to the par-ticular nodal line will contribute terms towards the forming of {di}, and these will Chapter 2. The Formulation of the Finite Strip 72 form a Fourier series with the bases used in the interpolation. The displacements are expressed as shown above, instead of as an explicit Fourier series, so that the formulation could progress in the matrix form. 2.9.2.3 T h e E l a s t i c F o r c e s i n t h e S p r i n g s Using Equation 2.168, which expresses the displacements in each of the five degrees of freedom at any \u00a3 along the restrained nodal fine, and Equation 2.166 which gives the elastic stiffnesses, the elastic forces can be expressed as F: wl **