THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ELASTIC ANALYSIS OF GRILLAGES INCLUDING TORSIONAL EFFECT AND STABILITY..... -Pin-Yu Chang ~... A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Naval Architecture and Marine Engineering 1967 April, 1967 IP-780

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ACKNOWLEDGEMENTS The author came to this country in 1961 with great interest in ship hydrodynamicso It was Dr. Henry A. Schade, then Chairman and Professor of the Department of Naval Architecture and Marine Engineering, University of California, Berkeley, who aroused the author's interest in ship structures. It was also under Dr. Schade's direction that the author had first contact with the grillage problem. A method had been developed in 1963 to solve this problem before the author came to this University. However, the method then developed is limited to the case with uniform load only. In the seminar for which the author presented the result of his research on the grillage problem, he was informed of the work done by Professor Finn C. Michelsen and Dr. Richard Nielsen, Jr. The application of Laplace Transform removes all the restrictions on the loading and on the boundary conditions. It is under the direction of Professor Michelsen that the author is able to grasp the essential part of the theory of grillages which makes this dissertation possible. In the process of programming for numerical calculation, the guidance of Dro Nielsen is indispensable. The success of any dissertation depends also greatly upon the members of the Doctoral Committee and.other faculty members from whom the author has acquired the knowledge and the ability to undertake independent research work. To all of the Committee members, Professor Finn Co Michelsen, Professor Glen V. Berg, Professor Raymond A. Yagle, Associate Professor Ivor K. McIvor, Associate Professor Roger D. Low ii

and Dr. Richard Nielsen, Jr., the author is deeply grateful for their enthusiasm and valuable advice. To the faculty of the Department of Naval Architecture and Marine Engineering, the author must express his thankfulness for their guidance to his program and their efforts in providing financial support. To Miss Kathryn Wanink, the author must express his appreciation for her accurate and beautiful typing. Finally, the author wants to thank his wife, Frances Chang whose encouragement and understanding have been an important contribution to this thesis. iii

TABLE OF CONTENTS ACKNOWLEDGEMENT.................................................. LIST OF TABLES.................................................... LIST OF FIGURES............................................... LIST OF APPENDICES........................................ NOMENCLATURE................................................... CHAPTER I INTRODUCTION...................................... A. Orthotropic Plate Method..............o............ B. Discrete Field Method............................. C. The Beam-on-elastic-foundation Method.............. D. The Objective of this Dissertation.................o II GENERAL THEORY AND GOVERNING DIFFERENTIAL EQUATIONS..... III SOLUTION OF THE DIFFERENTIAL EQUATIONS FOR DIFFERENT CASES WITH EXAMPLES...... A. The cases with B = 0............................. B. The cases with B i 0, and AB = BA................ C. The cases with B / 0 and AB i BA................. D. On the torsional effect and a comparison with Smith's numerical results....................... IV GRILLAGES WITH ODD STIFFENERS......................... V STABILITY........................................... A. Instability of the first kind...................... B. Instability of the second kind.....................o C. Instability of the third kind....................... VI SUMMARY AND CONCLUSION o............................... REFERENCES.........................................0......o..o.o APPENDICES.o................................................000 ii vi vii viii 1 1 2 3 5 6 12 12 20 23 30 38 46 46 53 59 60 62 63 iv

LIST OF TABLES Table Page III-i Results for Figure 3-3........................... 34 III-2 Results for Figure 3-4........................... 35 III-3 Results for Figure 3-5.......................... 35 II-4 Results for Figure 3-6........................... 35 V

LIST OF FIGURES Figure Page 1-1 ath Stiffener...................................... 3 3-1 Deflection and Bending Moment for Example 3-2............ 19 3-2 Deflection and Bending Moment for Example 3-4...... 30 3-3 Simply Supported (10xlO) Grillage Under Distributed Load o................................................... 33 3-4 Clamped (10xlO) Grillage Under Distributed Load......... 33 3-5 Simply Supported (10xlO) Grillage Under Central Concentrated Load o........................................... 33 3-6 Clamped (10xlO) Grillage Under Central Concentrated Load.................................................. 33 4-1 Reflection and Bending Moment for Girder Number 1 and 2.. 45 vi

LIST OF APPENDICES Appendix Page A Fraction Expansion of [Q+s2I] and the Inverse Transforms o............0........................ 63 B The Alpha Matrix, Unitary Matrix and the Eigenvalues..... 70 C The G Function and Some Formula of Beam on Elastic Foundation................................. 87 vii

NOMENCLATURE U Work done by external loads Vbg)Vtg Potential energy due to the bending, twisting of girders Vbs) Vts Potential energy due to the bending, twisting of stiffeners E Modulus of elasticity G Modulus of rigidity, Ji Torsion constant of the ith girder Ja Torsion constant of the stiffeners Si Spacing between the (i-l)th and the ith girders Sa Spacing of the stiffeners Ii Moment of inertia of the ith girder IS Moment of inertia of the identical stiffeners P. Axial load at the ends of the ith girder 1 qi3 Concentrated normal loads acting on the ith girder at the point x = xg Ria Reactions between the ith girder and the ath stiffeners qci Continuous or concentrated normal load on the ath stiffener dig Deflection of ath stiffener at the intersection with the ith girder due to the lateral load qa alone Pa Axial load at the ends of the ath stiffeners Qd.j Influence coefficients of the stiffeners, it is equal to deflection at i due to unit load at j [ ] Notation for matrices, [a] = [ ij] for example Notation for diagonal matrices [-] Notation for the adjoint matrices { } Notation for the column matrices viii

A, B, C,D, Q, RW Abbreviations of the matrices [Qijl, [Rij] {Wi} etc. [Aij], [Bij, {Ci, {Dij, I The identity matrix, (without subscripts) w(s) The Laplace transform of indicate the elements of of x, s respectively. by (x), (s) follow the d2w = xx- W - XX. dx the matrix W(x), (x), (s) the matrix W are functions Other capital letters followed same convention IW" W it dw dx d3w dx3 d4 d=w dx4 xxxx dx Length of girders Length of stiffeners.L L1(x, ai) L1(x, aibi) Inverse functions for the external load associated with the normal mode yi(x) Gl(x,ai) G,(x,ai bi ) Inverse functions for deflection, slope, moment shear associated with the normal mode Yi(x) o See Appendix Co Subscripts Convention The subscript i, j, and k are used to designate the ordinal number of the girders and a, P are for the ordinal.number of stiffenerso No summation is implied unless specified at each equation. Examples: aij Ii Ij implies aij multiplied by Ii and Ij, no summation. aij Ij (sum on j ) implies ail II +.'. + ain In uij Ij dj (sum on j ) implies Uil Il dl + ui2 I2 d2 + ~o- + uin In dn ix

CHAPTER I. INTRODUCTION Grillages, or gridworks, find wide application in the design of many types of structures. They are of such importance that more than seven hundred works have been contributions to the literature on this subject, (6,7) according to Abrahamsen.' The grillage under consideration in this thesis consists of two sets of mutually orthogonal beams which are interconnected at the intersections. For the convenience of further discussion, one set of the beams, being larger in size and fewer in number, is designated as girders and the other set, consisting of smaller beams, is designated as stiffeners. Since the literature on grillages has been reviewed in great detail elsewhere, (1 267,10) only selected recent works are discussed here for the purpose of comparison. In general, most of the previous analyses can be included in three groups. A. Orthotropic Plate Method The flexural and torsional regidity of the girders and the stiffeners are diffused over their spacings so that the grillage can be regarded as an orthotropic plate. This method is valid only for the uniform grillages with rather close spacing of the beams. Uniformity here implies that the properties and spacing of the members in any one direction are constant, even though they may be different from those of the members in the other direction. (9 -1 -

-2 - B. Discrete Field Method Since the memebers of the grillage are connected only at the intersections, the most realistic approach to the analysis of grillage is to regard it as a discrete field.( This method is valid again only for the uniform grillages, however. C. The Beams-on-Elastic-Foundation Method This method treats both the girders and the stiffeners as simple beams subjected to the external loads and the interactions at the intersections between the stiffeners and the girders. Since the interactions depend upon the deflections, the differential equation, for the deflection of the single girder is similar to the differential equation for a beam on an elastic foundation. With n girders, n coupled equations will result. These differential equations were first applied to the grillage by Vedeler.(ll) Since then many attempts have been made to furnish a solution with various degrees of success. The most important contributions (1,2) are due to Michelsen and Nielsen, because of the following advantages over the other works. In their works, there are no restrictions as to the sizes, spacings and fixities of the girders; and no restrictions on the type of loading. And,most important, the stiffeners may be of different sizes. For the convenience of further discussion, the formulation of the differential equations is introduced briefly below. Under external loads, the girders and stiffeners deform together and offer support to each other. Each member is under the action of the external load and the reactions between the girders and the stiffeners

-3 - at the intersections. If the reactions are known, the deflection of each member can be determined by simple beam theory. However, the reactions can be calculated only after all deflections are known. As a consequence, coupling effect among all the members exist. The reaction can be expressed in terms of the actual deflection and the external loads. Let the member in Figure 1-1 be the ath stiffener. It is under the action of external load, qC(y) and the reactions from the n girders. R1 R2 Ri Rn Figure 1-1. oth Stiffener. Let the deflection at girder i due to qa(y) alone be di. and the deflection due to the reactions be d'., then the actual deflection at is ia= dia - dia ( From simple beam theory did = ij Rjg (sum on j ) (1-2) where cij is the deflection at i due to unit load at j. Thus o.. are the usual beam influence coefficients. From (1-1) and (1-2) we have 0ij Rjp = dia - wia (sum on j ) (1-3) Consider the girders as simple beams acted upon by the interactions and we have

d4w. m EIi 1 = 2 Ria (x a) (1-4) dx where m is the number of the stiffeners. 6(x-xa) is the Dirac delta function. Combining (1-3) and (1-4) we have that d4w. m aij EIj = El (di - wiCi) 8(x-x<X) (sum on j ) (1-4) If the spacing of the stiffeners is small enough, and if it is constant over the whole length of the girders, then the last term on the right of the equal sign can be expressed approximately as m i(x 2 wia 5(x-xa) = (x) (1-5) 0=1 So where SC is the spacing of the stiffeners. Then Equation (1-4) can be written as d4w. wi(x) m. a El. += dia 6(x-xa) (sum on j )(1-6) ia j 4 x dx c c~=l For the cases with axial loads on the girders, or when the torsional regidity of the members is taken into account, Equation (1-6) will become more complicated and will be considered in Chapter II. For uniform load, did is constant, and (1-6) can be approximately written as w.(x) d~.ij EI. w (x ) + - S (sum on ) (1-7) i O j xxxxj SU S7 This set of differential equations can be solved by several methods. ( 3) The Laplace Transform method introduced in (1) and (2) must be regarded as one of the most advantageous, however. By using Laplace Transform, all the restrictions on the end fixities and on the type of loading for the other methods can be completely removed.

-5 - The critical part of this method is the determination of the inverse transform. This has been done and tabulated in (1) and (2) however. D. The Objective of this Dissertation The objective of this study is to extend the beams-on-elasticfoundation method as follows: First, the method is further simplified. It has been discovered that Equations (1-6), and the equivalent equations for more complicated cases can be uncoupled. An easier computation process can therefore be used. Secondly, the effect of the torsional rigidities of members is taken into consideration. Thirdly, a rather complete investigation of the stability of the grillage is introduced. This is one area where no significant contributions have been published in the literature.

CHAPTER II GENERAL THEORY AND THE GOVERNING DIFFERENTIAL EQUATIONS Although the governing differential equations were derived for some fairly general cases by the previous authors, the torsional effect of the girders was not considered. In all of the previous works, the stiffeners are considered to be acted upon by the external lateral loads and the interactions, whereas the girders are subjected to the interactions only. In order to include the torsional effect of the girders and to allow some load acting on the girders the governing differential equations are formulated here by the energy method. The torsional effect of the members is considered approximately without considering the compatibility of the rotation of the members. The change of the moment of inertia of one member due to the deflection of the orthogonal members is neglected. The elastic energy due to bending of the n girders and the m stiffeners is given by L 1 n 2 Vbs = f ~ Z Eli w dx (2-1) o i=l ~ m Vbg = 2 EIa wE 2 dy (2-2) o G~=l The elastic energy due to twisting of the girders and the stiffeners can be written as 1m n+l GJi Vg = 1 ~ S (Yi - yk,-(2-3) 2 Q=l i=l '' -6 -

-7 -m n+l GJ 2 vts 2 z. (Wx i, - Wa i a) (2-4) Vs - ua=l i=l si where Ji and Ja are torsion constants. Regarding the stiffeners as a continuum, it follows that L1 n+l GJ2 Vts = S 2 ~ S S (WX i 'x WXi-l) d (2-5) 2 i=l ScwSi xil) dx If the spacing of stiffeners is small, which is an assumption of this method, Equation (2-5) is a good mathematical model. If furthermore the girders are also closely spaced we can write zia =S (wi a - Wi-l ) (2-6) where small deflections are assumed. From Equation (2-3), we then have '1 ~ 2 m' n+l GJ i i, -. W W. m n+l GJi S 2 0 -= 2 ~. i~ C-. - W i-l7 -a=l i= 1 Xi - 7) Thus we can replace the difference of slopes of adjacent stiffners by the difference of slopes of girders multiplied by a scalar factor in terms of the ratio of their spacings. Equation (2-7) is a good approximation if the girders are spaced closely. With unequal and normally large spacings between girders as found in most ship structures, the accuracy of this approximation can only be determined by experiment. Nevertheless, it should not be worse than the approximation induced by the orthotropic plate method. Since we are essentially replacing the middle term of the orthotropic plate equation as follows

-8 -- " ' i-1 3..2. 2H -2w + Hi+1. W - (2-8) 2 i-1 i ~x2? SSx2 i-i, x2 i 1 x2 i+l Regarding the stiffeners as a continuum, we can now write L n+l GJi SC 2 Vt g z -G- c s [x,i Wxi+l] dx (2-9) i Let q((y) be the lateral load on the stiffeners, qi(t) be the lateral loads on the girders and Pi, Pa be axial loads at the ends of the girders and stiffeners, with RieC being the interaction between the girders and stiffeners at the intersections, then the work done by the loading can be written as follows. I m L n m m1 2 1 2 U f 2 [qa(y)w (y) + 2 PwyY a0(y)1dy + I z [qi(x)wi(x) + 2 PiW (x)]dx o al= o i=l (2-10) In computation all distributed lateral loads are regarded as the loads on the stiffeners. Those concentrated loads which act upon the sections of girders between two stiffeners are considered as lateral loads on the girders. In this manner, any possible loading on a grillage can be considered. Concentrated moments can be regarded as the limit of two large concentrated forces of opposite direction as their separation tends to zero. The sum of Equations (2-1), (2-2), (2-5), (2-9) and (2-10) is then the total potential energy of grillage system. Adding the terms due to reactions L n m L n m 0 = f z Z Ric 5(x-xa)wi(x)dx - I Z Ria 8y-Yi)wa(y)dy i=l 0=1 o i=l 0=1 the total potential energy can be written as

-9 -L n 2 n+l V- U ~2 EI w + 2 + ()2 dx o iv1" '=1 j E o xxxr ~W iidi a =l3 - i- i L n m - f 2 qi(x)Wi + - Pi ix dx - / q~(Y)w + 2Pwi dy o i=l o ci 2 m L n m 1 ( m + / E EIyy dy - E i Z Ri 5(x-xa)wi dx o S-=l o i =ld C=l e n m + f z ' Ria (y-yi) wc, dy (2-11) o i=l a=1 Applying the principle of variational calculus, and letting 6 i and 65w be the variationsof the deflections of the girders and stiffeners respectively, we have n n+l GJ% GSa El. 6w. - z + (w - W il - 1 i=l i Si XX,1 n n m + E (Pi W:x,i - qi) - E 2 R.i 6(x-x6) wi + i=l i=l a==l n m m + E 2 R 5(y-y ) + (EIwyyyy i=l a=l a=1= + PaWyy, - q) 8 = 0 (2-12) Since 6wi and 5w, are independent of each other and are quite arbitrary all terms associated with each 6wi or 65w must vanish. This gives n EI W(yyyyy + P Wyy,a = ca - Ria 5(y-yi) (2-13) i=l and GJLy GJU Ei Wxxx - -Sai(wxxi - wxil) + SUSi+l (WXXi+l -x i) GJiaS GJi+lS. - S3 (W - i-) - + (Wxx, - 1Sxx3 i Wi+ l xi S i + S~ +1 XX~i~l - )i+l

-10 - ki m +P. i - 2 i 5(x-xi) Z Ri a 6(x-x) (2-14) 1 XXP1 =11a=1 A rearrangement of terms in Equation (2-14) gives ija ja JiSa JiSa] EIi xxxx i - Gsi + + i+ + + xxi + P xx i ja JiSu Ja Jis Sa + G SaSi + S W i G Si+ + Sasi Ss xx P Sa1i Sx3l XXP i+l m - ji = Z Ria 8(x-xa) (2-15) a=1 with wxxi = ~0 Wxx i = 0 whenever i < 1 or i > n. We shall at this point assume that the compressive axial force acting on the stiffeners is lower than the critical load of the stiffeners for the case of no support of the girders, then its influence coefficients can be calculated from known relationships and applied to the grillage analysis in the usual manner. From Equation (1-3) we have ij Rja = dia - ia (sum on j ) The cases where the axial loads at stiffeners exceed their critical value is discussed in Chapter V. Let ci = C-+ G and ci+ + i G (2-16) JSi iS c1 ScSi+1 S3 L 1 L 1i+l Equation (2-15) then reduces to ij Ej wjI 4) (cj + cj+ + Pj) WXXi + c 1C wjl+ +l - qi m = Z (di1 - wig) 8(x-x) (sum on j only) (2-17) G=1

-11 - The stiffeners are, by hypothesis, identical and equally spaced, and the last term can therefore be approximated as follows m E WiG b(x-xa) (2-18) a=1 S2( In matrix form, Equation (2-17) can now be written AW(4) + BW" + W D (2-19) where A = [Aij] = [9ij][Ii] ESa B = [Bi] = [aij][[cij] + [Pi]] Su -Cl-C2 C2 0 0 0 0 c2 -c2-c3 c3 0 0 0 Cj = 0 c3 -c3-c4 4 - - 0 0 - m D(x) {Di} = [cij]{qi + Z dia 5(x-x,)}s All continuous loads can in general be regarded as loads on the stiffeners. Only concentrated loads, acting on the portion of girders between two stiffeners are considered as loads acting directly on the girders. The load qi can therefore be expressed as Ni qi = c qip S(x-x) (2-20) 6=1 where Ni = number concentrated load acting on the ith girders

CHAPTER III SOLUTION OF THE DIFFERENTIAL EQUATION FOR DIFFERENT CASES From Chapter I, our problem is reduced to the solution of a set of fourth order linear differential equations. This set of equations can be solved by classical methods. Since the Laplace transformation introduces the boundary conditions as constants, it is more convenient for the problem considered here. Applying the Laplace integral operator to both sides of Equation (2-19), we obtain the following transform, where subscripts have been dropped. 4 2 (As + Bs + I) W(s) = D(s) + C(s) (5-1) where we have defined C(s) = A(s3W(o) + s2W (o) + sW"(o) + W"'(o) + B(sW(o) + W' (o)) = (As5 + Bs)W(o) + (As2 + B)W' (o) + sAW"(o) + AW"' (o) (3-2) Nj m D(s) = [aij]{ e P} SC + { di e- So N = Z dike-sxk (3) k=l Girder deflections are now given by the inverse of Equation (3-1) or 4 Bs2 1 W(x) = 'l{(As4 + Bs + I) (D(s) + C(s))} (3-4) A. The Cases with B = 0 The cases with null B matrix have been solved completely by Michelsen and Nielsen. (1 However, the following method is easier and -12 -

-13 -more interesting. Consider Equation (2-19), which can be written as S, E[a{.wj (x)} + {w(x)} = {d(x)} (3-5) 1 1 Let 7y = I2 w.(x) and multiply Equation (3-5) by [I2] We have Jj J j 1 1 1 S E2 [a] { (} + {yj}:= [] {d} (3-6) 1 1 Since TI2 [a] [I2J is positive definite, there exists a a J unitary matrix [u] such that T 1 1 T [u] I [Ct] I ] = J[], [u] ]= I (3-7) Xi i=l,.o., n are eigenvalues of 1 1 2 2 Let Y i = Zu j then (3-6) reduces to Yi j -J j ( S^Ehi Yi^ + Yi = 1i I2 dj (sum w 0j ) (35-8) The simultaneous equations are uncoupled and can be easily solved. Methods of calculations of Xi and the unitary matrix are given in Appendix B. The Laplace Transform of (3-8) can now be written as (S EX4s 4+ l)y (s) = 2u. d (s) + SEi (syi (o) j iJ j I + s Yi(o) + s y ( o)+ y" (o)) O Yi(X) = u I2~ { dj(s) + ij j UEXis 4+l

-14 - + - s yi() + s yi(o) + syi () + (0) + E xiS (+ 1 ExiSa (3-9) T 1 Since i (x) = u I2 W(x) ij J J Yi(o); yi () Yi (o) ( yoi(o) boundary conditions, wi (o) ditions not given at x = o, From (Sum on l ), it follows that the initial values can be written in terms of the girder wi (o), w (o), w" (o). For boundary conwe have boundary conditions at x = L. (n) n I W (L) ij j j (n = 0, 1, 2, 3) and Equation (3-10) we can therefore determine all required derivatives at x = 0 o By definition, let,. I - d(s) _ I L1 (xai) { ScOEXi 4+1 m —1 Gm(x,a) = {}, S + E\Sa 4a ~ 1 m = 1, 2, 3, 4 Then we can write yL(x) = MTj I L(a) (o) + G3(x,ai) y"(o) X =j I j j 3 i

-15 - + G2(x,ai) Yi(o) + Gl(x,ai) y"o) (sum on j ) (3-10) The G-functions are given in Appendix B and N L(Xai) = E1 dk Gl(x-xk,ai) LE~~Sc k=l Considering the case where all ends are built-in, we have i(o) = yI(o) = Yi(L) = Yi(L) = 0 From Equation (3-10) we have G(Lai) Go(L,ai) Yo(i) ui ijTL(Laj ) Then y!(o) and y"'(o) are readily determined from this equation. We obtain the deflection of girders by I wi = uij Yj (sum on j ) (3-12) Hence I wi (x,a ) + uij[Gl(x,aj)yj(o) + Go(x,a) yj(o) (sum on j ) (3-13) Equation (3-8) is equivalent to the equation of a beam on an elastic foundation, and the solutions are available in the literature for most cases of boundary conditions and loading, Reference (4). The complicated computations as shown in Reference (1) and (2) are therefore reduced to the computations of the eigenvalues, the unitary matrix to obtain Equation (3-8). With Equation (3-8), vi(x), y'(x), y!(x), yA (x)

-16 - can be obtained by simply matching the parameters with the established formula. We then use Equation (3-12) to obtain the deflections, moments, and shears. The following numerical examples show these procedures Example 3-1. A 3-girder and 11-stiffener grillage Length of girders and stiffeners, L = 312", = 288" Spacing of stiffeners, Su = 26" 4 Moment of inertia of stiffeners, Is = 6,100 in Moment of inertias of girders, Ii = 13 19,250 in 12 =38,500 in Spacing of girders, S = S2 = = 72" Uniform load, q = 15 psi. G~IIZ&I E *; ^ F o r:__^jlJ11 _ <er) i 1 I I T..{ 01 (X 6z —A — I ~, 1 t I 1 t 1,-lt 1 2 Stiffener -_ __I I. A I. ----- 312" All ends of stiffeners and girders: —mply-supported By simple beam theory the alpha matrix is 1.52970, aid = 1.86964, i i 1.18977, 1.86964, 2.71948, 1.86964, 1.18977 1.86964 x 10-6.152970 normal load alone are The deflections of the stiffeners due to the d1 1 lc = d3 = 3.5366 -, d 4.936 lSQ"a.6 SUa

-17 - Since the grillage is symmetrical, the computation can be reduced to two girders. The [a] after this contraction is [j] = 11+1' l | 2-.71947, 1.86964. ij] a21 + Ca23 a22 3,73928, 2.71948 This is not symmetrical. In order to make the matrix symmetrical, we can multiply the last column by 2 and divide 12 by 2. This does not change the values of the deflections. Thus ['2][QJ [] - 1 9250 2.71947, 3.73928 -6 [I2][a][I2] = 19250 x x 106 3.73928, 5.43896 The eigenvalues are 1 = 0.00 1932, \2 = 0.1551 The unitary matrix is 0.8191 0.5737 [u] = -0.5737 0.8191 T 1 T u1j I dj(x) = 0.26268, u2j Ij dj(x)= 32.5587 11 di(x) = Z dia 6(x-x- ) Sa = di = 1 The y-equations are then (4) = 2626 Sa El Y1 + Y = 0.2626 So E-2 y4) + Y2 = 32.5587 Comparing with the equation of a beam on elastic foundation (4) EIy + ky = q (b) and using the equation for deflection from (c-4), we have aL aL /L\ - 2cosh -- cos - - y =q( cosha + cosa- ) 'c- 2 coshaL + cosaL

-18 - a. a aL, a2. qsinh -2- sin -2 -y(i) =- aq coshaL + cosaL (d) where 4a4 a = 0.02018, a2 0.006742 a1 SSEi Y1 2 0.2853, y -11758 x 10 -L Y2 L 19.479, L -1.8507 x 10"3 at x = 2 1 1 W1 ulj yj I 2 = 0.082", W2 u y I22 = 0.1148" Ml = EIl W 4.41 x 10 in - lbs M = 12.6 x 106 in - lbs The bending moment of-the stiffeners can be calculated after the deflections at each of the intersections are known. Example 3-2. A grillage of the same properties and the same loading as that in Example 3-1, except the boundary conditions are as follows. s.s fixed.S, I h fixed S.S. fixed.1.1 11.1 For problems like this, the computation is a little more complicated, since the boundary conditions are coupled. The boundary values (o), y (o), y1(o), y'(o), y (o), y1(o) must be determined in one set of equations in order to satisfy the specified boun(Ldar conditions. Except for this step, the computational procedure is the same as shown from Equation (3-5) to (3-13).

-19 - Deflection and moment of the girders at x = L wit =.03623", W2j 2 =.o48l6" 'I2 Mlt) = 2.7827 x 1065 M2 -7 7.4940 x 106 Bending moment of the center stiffene ir MS(in-Lb ) 216" 288" x o 072" 144" MS 0 894, 100 562,940 894,100 0 The distributions of deflections and bending moments are plotted as follows for future reference in Figure 3-1. 1.0 -p 0 *rl ' Bending 2 / (^0.-ft., -,.0 - e (L S_4! ) r-i.06 z, -2. Figure 3-1. Deflection (in inches in broken lines) and Bending Moments (in million peound inches).

-20 - B. The cases with B #. 0, and AB = BA For the cases with axial loads acting on the girders, equation (2-19) takes the following form (4) Sa[cij]{EIj Wj (x) + Pj Wj'(x)} + {Wj(x)} = {dix)} (3-14) If P = qIj then AB = BA Letting yj(x) = I. W(x), no sum on j, then Equation (3-13) reduces to 1- (4) 1 1 SQ, E[ij p[I ]{4yj (x)} + q[Xij ] )} + I]{Y(x)} {d ()} (3-15) If we define Yi(x) = uij Yj(x), (sum on j ) so that i Yi (4) Sa E~i yi (x) + Sa qki yf'(x) + yi(x) = Pij I dj(x) (sum on j ) (3-16) where Xi, i = 1,...,n are eigenvalues of [lij Ii Ij] [u] is the unitary matrix then transformed equation becomes (S Exi s4 + S qi s2+ 1) yi(s) =u dj(s) + S i( 3Y (o) + s y() + sy"(o) + y"'(o) + Sa qxi(syi(o) + y{(o)) from which y(s) = 1 2 S + 1 2I i T uij dj(s) + (s3 +. s). Y(o) 1 SQExi (s + (s -+ E) y.(o) + sy~(o) + y"(o') E i i~~~ (3-17)

-21 - Introducing the parameters a. = / ( SEi) -, bi 2 (SEi) -( 2 -i4E 2 the inverse Laplace transform can be written Yi(x) = ij- L(ajbjx) + [G4(aibi~x) + I G2(ai,bix) y(o) + [G3(ai.bix) + q G(aibix)] y'(o) + G2(aibi,x)y"(o) E i + Gl(ai,bi,x) y"1' (o) (sum on j ) (3-18) The G-functions are given in Appendix C and 1 N L(aibi)x) = 1CE Ii Z dik Gl(aibix-xk) k=l Equation (3-18) is valid for any boundary conditions. For simply supported ends throughout we have Yi(o) = y'(o) = yi(L) = y(L) = 0 From Equation (3-18) we then obtain rG3(aibi)L) + E Gl(aibiL), Gl(aibi'L) Yi(o) - ij L(ajbjL) LG5(aibiL) + E G3(aibi,L), G3(ai,biL)J yi(o) j L"(aj,bj,L) (3-19) When Yi(o) and yi"(o) are known Yi(x) = j L(aj,bj,x) + [G3(ai,bi,x) + E Gl(ai,bix)]Yi(o) + Gl(ai,bi,x)y"'(o) (sum on j ) (3-20)

-22 - Since Equation (3-16) is equivalent to the equation of beam on an elastic foundation with axial load, we can solve Equation (3-16) by using the established formula from the literature by matching the parameters. The following example serves to illustrate this method. Example 3-3. A 3-girder and 11-stiffener grillage with the same properties and the same normal load as in Example 3-1. In addition, the girders are subject to axial loads as follows P2 -- P3 - i i i i I I -1 i -. l. _ m- _ - - O - I P1 = P3 = 1,925,000 pounds P2 = 3,850,000 pounds ii I — I -_ $ 3- 2i.....L — I P. Thus q = 1,000 Since the grillage is the same as that in Example 3-1, the unitary matrix, the eigenvalues are the same. By comparison with Equation (a), we have ScEx1 y 1,000 s +,o s ' + = ScE%2 y) + 1,000 S2 Y + y2 = 0.2626 32.5587 (h) 1 2 4E a1 1- = 0.40726 x 10-3 = 0.8333 x 10-5 1 1 = 0.45460 x 10-4 2 S Ex = (3.9893 x 10-4) = 1.9972 x 10o2 a2 = (0.37127 x 10-4 ) = 0.6093 x 10o b = (4.1559 x 10-4 ) = 2.0386 x 10-2, b2 = (0.53793 x 10-4 = 0.7334 x 10-2

-23 - From Equation (C-5) we have y(x) = q 1- - coshbx cosax' + coshbx'cos ax - coshbL+cosaL b2-a2 1 + ab_ (sinhbx sinhx' + sinhbx' sinax) 2ab r 1 2 2 y"(x)= _ q shbL+-_ 2(b -a )(coshbx cosax' + coshbx' cosax) coshbL+cosaL.(b2a 2)2 + (ab2 (sinhbx sinax' + sinhbx sinax) 2ab where x' = L-x Since the contribution from Yl is very small, it can be neglected. y2(*) = 22 5338, y2(7) - 12.1173 x 1O 4 W (/) = 0.5737 x 22.538/138.9 = 0.09308" W2(2) = 0.8191 x 22.538/138.9 = 0.13290" M1 L) = 3 x 107 x 12.1173 x 10 4 x 138.9 x 0.5737 2.8967 x 106 in - lbs L = 3 x 107 x 121173 x 4 x 277.8 x.819 = 8.2717 x in - bs M2(-) 3 x 10 x 12.1173 x 10-4 x 277.8 x 0.8191 8.2717 x 10 in - lbs C. The cases with B 0 and AB BA For the cases with axial forces and torsional effect accounted for, it is generally true that AB 7 BA and Equation (2-19) cannot be uncoupled as before. In this case, some other method must be applied to find a solution. For the case with B = 0, the method used in References (1) and (2) is excellent. The question is now whether that method can be extended to the case when B 0

-24 - In order to apply the concept in Reference (1) and (2) two essential operations must be accomplished. Firstly, the eigenvalues of [As + Bs2 + I](3-21) have to be determined. Secondly, the partial fraction of [As4 + Bs2 + I]1 will be needed. These two steps are more difficult than the case with B = 0. This will be discussed in Appendix A. The difficulty in the first step can be avoided by changing the form of Equation (2-19). From Equation (2-19) we have (4) 1 1 -1 W(4)x) = A-1 D(x) - A1 W(x) - A BW"(x) Combining this with W"(x) = W"(x) we can write rwcx,~i 1 o,1 i.'rwcx r ) l + 0 (3-22) (X) A-1 -A-1 B W"(X)J A-D(x The Laplace transform becomes "S2I, -I 1 W(S) TSW(O) +W'(o) 1 -1r (2.1 r3-23) lA1 A B+s2I ]W"(s) - LsW(o) + W'l(o)+ A-D(S It follows that |W(x) 1 -1 Pl s2IB, -I 1 LsW(o) +W'(o) (24) = '(3-24),W,(x)J 0 ALAI A B+sI sW"(o) + W"' (o) + A-1D(s)s ~~ j ~ ' - L- -^!~~~~~~~~

-25 - A little algebra can show that W(x) from Equation (3-24) is exactly the same as given by Equation (3-4) and W(s) from Equation (3-23) is exactly the same as that of Equation (3-1). The advantage of Equation (3-24) is however that we have again a standard form for the characteristic root. To show this, let 0, -I Q = i -1 -1 S s = -X,A, A B Then s I, -I det I l = det[Q-XI] = 0 (3-25) A -B+s2I That Equation (3-25) and (3-21) have the same roots follows from the identities ns2 2 o l islI -I 1 IA10: d AS -A det det i det A-B1 A B+s A B+As2 As2 det |As 0 = s (detA-l) det i LI As +Bs2++I 1e 4 2 = det A- det[As + Bs + I] There is another advantage of Equation (3-22) and (3-24), as shown more clearly in the next paragraph, which results from the fact that we can calculate the adjoint of [Q+s I] explicitely. This is necessary to solve Equation (3-24). See Appendix A. In general, the partial fraction expansion is more complicated than that shown in Reference (1) and (2) and complex arithmatic is involved in the computation. From Appendix A, we have

-26 - [Q+s I] n-n = E k=l [F (Xk) 21 + S k k Fl(k) - s +%k m + Z F2(wi) i=l 1 (s2+wi)2 (3-25) m + Z F3(wi) i=l s +Wi where Fl(k) = n-m-1 n j=l j=k [Q-kkI ] ( ij- ~jixk+xk ) i=l (wi-%k)2 (%k- k) F2(wi) = F3(wi) = n-m (pj-wi+wi ) m-1 (wk-wi)2 j=l k=l k/i [Q-wi ]' n ( a jwi+wi ) m (w-wi) j- II (wk-wi) j=i k=l k/i...[Q-wi I] r n -i, I - (1 n-m k=l % k-wi xk-wi n-m nI j=l m-1 + Z k=l kgi (C.w. +w ) 2Wk W.-w n (wk-Wi) k=l kWi In these expressions Xk are the distinct eigenvalues of Equation (3-25) Wi are the double eigenvalues of Equation (3-25) Xk are the conjugates of Xk ak Xk +^ Xk (3-26).k Xk Xk From Equation (3-25)) (3-26) we obtain

-27 -n-m "- {[Q,+s I]- s} = 2 X Re (Fl(\k)cosakx) k=l m m + z F2(w ) sinajx + Z F3(wi)co: i=l i-l ak = /\k for distinct root ak is complex a. = /w. for double root ai is real. 1 1 2 n- m 1 m,7K, {[Q+s I]} = 2 Z Re(Fl(\k) - sinakx) + F3(w) sinaix k=l ak i-l 1 saix m1 + ~ F2(Wi) (sinaix - aixcosaix) (3-28) i=l 2ai3 Let Ro(x), Rl(x) denote the right hand sides of Equations (3-28) and (3-27) respectively. By theorem of differentiation of the Laplace transform we have that Rl(x) = d (R (x)) dx 0 (3-29) With Equations (3-27), (3-28) and (3-29), Equation (3-24) takes the following form rW(x)1 r iW(o)1 r 1]:W'(o)1 | T W"(x) I ~ { w-(o) 1+ w"' o (x where t i Lo (x) = z (x- '~ -) C~=l 1 A- Da (3-30) with [Ro(x-X)] = 0 if x< xc

-28 - The derivative of Equation (3-30) becomes fw'(x~ lW(o) 1 W'(o))i 3= [R Li(x) + [R(x)3 + IL (x)} (3-31) Wm(X) 0 wt(o) 0 LWT(o) 0 where 2(n-m) m [Ro(x)] = 2Re Z Fl(Xk)('ak)sin akx + Z F3(i)(-ai)sin ai x k=l i=l m sinaix + aix co-s ai x + ~ F2(Qi) 2ai (3-32) i=l It is obvious that the boundary conditions are satisfied at x = 0 if [R(o0)] = [I ] (3-33) This serves as a check on the accuracy of the computation program. From Equations (3-30) and (3-31), we obtain the equations for the boundary condition constants "W(L) W(o) R(L) R(L)) Ro(L) (L) W"(L) ~ W"(o)i, -~; = " + ] 'W ) (3-34) W' (L) ' W'(o) 0 R(L)"(L) wRo((L) o (L))' l )) l ( ) / This is the general equation for all types of boundary conditions. For any specified boundary conditions, the associated equations are selected in such a way that all unknown boundary constants can be calculated. For the cases of simply-supported ends throughout we have (0) - [(L)] {0(L)} '( [R fo(L)]_ia (3-35) IW'^o)! ~0

-29 - For other boundary conditions, such simple expressions are not available. The computation is similar however. Whatever values of W(L), W'(L), W"(L), W"' (o), W(o), W'(o), W"(o), W"'(o) are given are substituted into Equation (3-34) and the unknowns determined. There are always a sufficient number of equations. Note that the elements of W(L), W'(L),... and W(o)... etc. can be different from one another, and different from zero. Therefore all possible boundary conditions are admissible in Equation (3-34). Moments loads at the ends are treated as specified boundary conditions. Consider here the case of simply-supported ends throughout with specified end moments {Mi(o)} and {Mi(L)} Since Mi = EIW', it follows from Equation (3-34), that [WI (o ) 0, [R(L)1-{L(L)} + [R(L)] - [R(L)] [Ro(L)] I LW(o)J LW I((L' III(O) c 0WI(L L ]( |M(L)j where W"(L)= IEIi M() ~W't"-~~(Co) j-~ = ____ (3-36) E. Example 3-4. The same grillage as example 3-2 except that the torsion constants of the members are taken as equal to their moment of inertia. The results are plotted in Figure 3-2. The effect of the torsional rigidity of the members is not large on the bending moments of the girders but is significant on the deflection of the grillage and on the bending moment of the stiffeners. The bending moments are listed as follows:

-30 - Bending moments at the center stiffener in pound inches. Location of the intersections 1 2 i = 0, J- = 0, for all ia 894,000 563,000 Ji = Ii,JO = IS for all i,a 834,000 512,000 No conclusion should be drawn from this calculation, since the torsional effects of the grillage depend on many other factors not just the torsional constants of the members. a | Bending Moment 1.0 - 0 ~ 0 Number of tiffener 0~ /~~- 27 — ' \~0 ' //Deflection.04 -2.0 - Figure 3-2. Deflection in Solid Lines Bending Moment in Broken Lines. D. On the Torsional Effect and a Comparison With Smith's Numerical Results. At this point, there are two questions needed to be answered. Firstly, since some kind of approximation has been introduced in setting up the differential equations, we want to know the accuracy of this theory. Secondly, since the computation in cases where torsional effects theory. Secondly, since the computation in cases where torsional effects

-31 - are included is rather complex, it is desirable to know for what kind of grillages this torsional effect can be neglected. The first question can be answered by comparions between the results derived from the method presented here and those from the method of finite elements. In 1964, C.S. Smith has used the finite element (14) method to calculate the responses of grillages under lateral load(4) His results for a 10 x 10 grillage are reproduced here in Figures 3-3, 3-4, 3-5 and 3-6. The parameters used in the figures are as follows. B EI EI = GJS 1.5 L m = Bending moment 5 = Deflection P1= Uniform load taken as a set of equal concentrated forces, P1, applied at the inter sections. P2= Central concentrated load taken as a set of four equal forces, P2, acting at the four center intersections Ii = Ig, i = 1,...,10 Ji = Jg i i =1 10 The curves in solid line are results with the torsional effect taken into consideration, and the curves in broken line are results without the torsional effect. The points shown by the cross marks are the results by the grillage theory developed in this dissertation. The two small circles are from othotropic plate theory.

-32 - The calculation for the cross marks is based on grillage with the following properties Length of girders L = 220" Length of stiffeners = 330" Moment of inertia of girder and stiffeners I = I = 4000 in Girder spacing Si = 30" Stiffener spacing S 20" Uniform load, for Figure 3-3 and 3-4 p = 20 psi Concentration load, Figure 3-5 and 3-6 p = 400,000 pounds For future reference, the results by the grillage theory are given in Table 3-1 to 3-4, All results are referred to the 6th or th 5 stiffener. Girders *__ / / S if ner__ 0 cJ cu 4.'= 330 ~. ~.::. 1

-33 - I 0 Figure 3-3. Simply Supported (10xlO) Grillage Under Distributed Load. Figure 3-5. Simply Supported (10x10) Grillage Under Central Concentrated Load. I0 0. ^I. a. Deflecticn o.oo4 - -- - Lu 2 Figure 3-4. Clamped (1Ox10) Grillage Under Distributed Load. Figure 3-6. Clamped (10xlO) Grillage Under Central Concentrated Load.

-34 - From these figures, the following conclusions can be drawn. (1) If the torsional effect of the members are negligible, the grillage theory developed in this dissertation is as accurate as the method of finite elements as presented by Smith. (2) If the torsional effect of members are not negligible, there may be some discrepancy in the distribution of bending moments. But this discrepancy is significant only on the less stiff members and the discrepancy is usually small. This discrepancy is due to the approximation in setting up the differential equations through Equation (2-7). However, if we note the fact that the difference between including and excluding the torsional effect is small for most grillages, this discrepancy in the distribution of bending moment of the stiffeners is not important except in some few cases as pointed out in the following discussion. TABLE III-1 RESULTS FOR FIGURE 3-3 Location 0 1 2 3 4 5 0.0..04375.08249 31.11311.134034458 (inches) *0.0.02693.05022.06800.07975 0.08554 Bb -0..0-.o41o0.07749.10625.12591.13578 P L3 2 C-O. O2.04 7 1_6:6.:.~_~.~0~~38~ ____~__~~_.~074. O 48a.. _.]-3578 P1L3 _ *0.0252 _.04716.o6386.07489 0.08033 m x 106 0.0.68556 1.0866 1.2843 1.3604 1.3808 (in-lbs) *0.0 __.50210 0.73216.078788 0.76586 0.73723 m 0.0 0.25967 0.41158 0.48646 0.51529 0.52302 P1L *0_o.._ 0.19018 0.27732 0. 29842 0.29008 0.27924 * With torsional effect

-35 -TABLE III-2 RESULTS FOR FIGURE.3-4. Location 0.0.00392 *-0.0.00342 B 0.0.00368 P1L -0.0.00321 m x 106 -1.3596 -0.5061 (in-lbs)*-1.2042 -0.4177 m -0.51497-0.1917 p1L *-0.4561 -0.1582.01183.01975.01016. 016Q 3.01111.01854..00954.01571 0.01218 0.2674 0.00359 0.2359 O.o44619 0.101o 0.01360 0.0893.02564.02870.02148.02390 0.24080.02695.02017.02244 0.3601 0.3812 0.23.29030.2926 0.1364 0.1 43 0.1099 0.1108 * With torsional effect TABLE III-3 RESULTS FOR FIGURE 3-5 Location 6 0.0 0.02495 0.04952 0.07292 0.09346 0.10711 (inches) *0.0 O0.01321 0.02682 o.o4078 0.05410 0.06364 B5 O0.0 0.00281 0.00558 0.00821 0.01053 0.01207 P2L3 *0.0 o.o 8 0.00302 0.0_0459......o6Q9.._ 0.. 00717 m x 105 o'.0 0.42625 1.36809 3.4277 7.8479 20.2735 in-lbs *0.0 -0.63471 -0.66750 0.50564 3.80377...14.5028 0. 0-0 — -0-0..00193 0.00621 0.01558 0.03567 0.09215 P2L *0.0 -0.00288 -0.000303 0.00229 0.01728 o0.659 * With torsional effect TABLE III-4 RESULTS FOR FIGURE 3-6 Location m_ m x (inn P, 6 0.0 __ *0.0 35 ~0.0 DL3 *0.0 106 0.5366 -lbs)*-3.2077 n -0.02439:L *-0.01458 0.00194 0.00123 0.00022 o.ooo14 -o.4848 — -3.4428 -0.02203 -0.01564 0.00750 0.01607 0.02632 0.03468 0.00502 0.01132 o0.01940 0.02634. ooo84. oo18.00296.00391 o.00056 0.00127 0.00218.0096 -0.4138~ -0.2630 0.1149 1.3151 -3.4832 -2.7193 0.1846 11.0657 -0.01879 -0.01195 0.00522 0.05977 -0.01583 -0.01236 0.00083.05030 * With torsional effect

-36 - The influence of the torsional effect depends on the type of loading, the aspect ratio, the ratio of the flexural regidity to the torsional rigidity and the fixities and spacings of the members. (14) Using the method of finite elements, Smith, has calculated more than 180 uniform grillages, from (3 x 3) to (10 x 10) with aspect ratio i/L from 1.0 to 2.0 and with EIs/EIg and 0.01 to 100, T = EIs/GJs = EIg/GJg from 0.2 to 100. The following statement is his conclusion. "It is evident that in the estimation of intersection deflections and of bending moments in flexurally stiffer members, torsional effects may usually be omitted from the analysis of grillages for which T is larger than 20 and may safely be omitted when T is larger than 100. This result will apply to most grillages representing orthogonally stiffened ship's plating, with the probable exception of double bottom structures." This conclusion can will be conceived from the fact that the torsion constant of the open sections are very small in comparison with the moment of inertia. His other conclusion is that torsional influence is consistently greater in simply supported grillages than in clamped edges grillages as shown in Figures 3-3, 3-4, 3-5, 3-6. This conclusion is conceivable from the fact that the torsional effect is greater on the portion of grillages where the difference of slopes of adjacent members are larger. Under the same loading condition, this difference is less in the clamped edge grillages than in the simply supported grillages.

-37 - As for the spacings, it is conceivable from Equation (2-16) and (2-17) that the torsional effect increases as the spacings decreases. This agrees with Smith's results. Regarding the aspect ratio, Smith's calculation does not cover enough values to make a general conclusion. From Schade's curves,(9) it is conceivable that when the aspect ratio is large or small, i.e. e/L > 5 or a/L< 0.2 for example, that the torsional effect is negligible regardless how large the torsion constant of the members may be. From the above discussion the second question can be answered as follows. (1) For grillages with members of open sections, the torsional effect can safely be neglected. (2) For grillages with members of closed sections, the torsional effect may be neglected if the aspect ratio is large under uniform load, or if the edges of the grillages are clamped, or if the spacings of the members are large. The deflection and the bending moments are overestimated

CHAPTER IV GRILLAGES WITH ODD STIFFENERS In the preceding chapter we have assumed that all stiffeners are of the same size and fixity, at each end. Then the reactions Ri. are given by the one Equation (1-3). In cases where some stiffeners are of different size, fixities, or even different length, we must have separate equations for each odd stiffener. Consider the yth stiffener for example, and we have clj Rj7 = d!7 - W. '7 ' (4-1) ' th Where ij = deflection of the y stiffener at unit load at j. latelyth stiffener at die = deflection of the yth stiffener at lateral load only. i due to i due to Obviously a. a.., d' d. 10 1J iy 1Y Thus m m 1A Ra S(X-)= (di-wi ) 6(x-xa) a= 1 - 0c 1 + [ca] [f' ]{d!7}-{ di}+[I- [I]- [' ] ]{Wi7} Let x1 = xy from the end of girders indicate the distance of the first odd stiffener AW )(x) + (x ) BW x) + W(x) = D(x) + D'(xl) -38 - (4-2)

-39 - where Df(xl) = {[l][]cr] di - di + [I-[cla][]- j](xl)}S~ 6(x-xl) (4-3) Regarding Wi(xi) as being constants, Equation (4-2) can be solved by the method of Chapter III. The terms [C][a']-i d' - d. are known ii 1 values can be combined with the A EZD, term. The term [I-O~1 ]W(xl) generate one addition term to put Equation (3-39) in the following form W(x) 7 FW'(o)| W'(o) + ') =, [R oX)l+ = [Ro R(xx) + {L(x)} + [To(x-xl) Iw"(x)j 1- I 0 O0 [T o(-xl)] = [Ro(x-xl)]0 i-oIoX-l s Let x=xl, Equation (4-4) gives additional equations for W(xl) boundary condition equation becomes 0 3< I iW(xl) (4-4) and the W(L) W"(L) W'(L) W"'(L) 0O i, r. I R'(L) gRo(L) IR (xl) Ro(L) To(L-x1) Ro(L) R. (x) Ro(Xl) To (L-xi) -I W(o) W"(o) W'(L) W"' (L) O(x1) W(xl) j T fT \ + Lo(L) (4-5) Lo(xl)j s For more odd stiffeners we have then

W(L) *W(o).R'(L) Ro(L) To(L-x1) - - - To(L-xk) Lo(L) W" (L) W"(o) W'(L) W'(o) R"(L) Ro(L) Too ( T(L-k) L (L) W" I(L) Wi +(o) + 0 0 RI (X1) R,(X1) -I - - 0 - - O L(x1) o W(xl) o R'R(X) Ro(X2) To(x2-xl) -I 0 0 6O Ro0(Xk) RO(xk) o((xk-xl) -I ) W(xk) LO(xk) (4-6) From Equation (4-6) a submatrix is separated in accordance with the type of boundary conditions. We then solve for the unknown boundary constants from the known boundary conditions. For the cases where the torsional rigidity of the members is negligible, Equation (4-2) may be uncoupled. Let the odd stiffeners be of the same length and the axial forces of the girders proportional to their moment of inertia. Then there exists a unitary matrix [u] such that Equation (4-2) can be uncoupled. For this case, Equation (4-2) can be written as m rSE[ ]{l )} + Sr[f]{P J} + {w} = SCY z di b(X-Xa)} c=1 N + SG z I-Ixkl]{wj(xk)} 5(x-xk) (4-7) k=l Where [c]k is the influence matrix for the kt odd stiffener, k and N is the number of odd stiffeners. Since the odd stiffeners are of the same length, it follows that I - [a][a](- k (4-8) k PI

-41 - P. Let 7i(x) = I iwi(x) and q i= Equation (4-9) then reduces to i 7i I~~1 + += 2 i i ll~~~~~~~~il = a= dice (x-xa)} N + sc E k=l (1 - I ) [i )} 5x-xk) (i - 1) 1{yi yi(xk)} 6(x-xk) sS Let furthermore {Yi} = [IiJ{Y} and multiply Equation (4-10) from the left by [Iij. Then we have seEi aj Ii [CIi1jYi + q Ii[q [a ]Ii{Yi } + { Yi} I m {=1 dia 5(x-x-)} + S3 N 2 k=l (4-11) (1 - y ) {yi(xk)} 6(x-xk) Defining [u] to be the unitary matrix:such that [u]T[Ii [a Ii[u] = [Ji] and [uJ [u] = I and letting {Yi = [u]{Yi} Equation (4-11) reduces to I m SaE'%l](y (4)j I S+ q[Xi]yij + yj = [u]l [rI 1 2 c= 1 dia (x-xa) NI + SCa (1 - I ) {Yi(xk)} 6(x-xk) k=l S or SaxEiyi) + SaqXiY' + Yi m 1=l fia (-a N + sa 2 k=l II ( - - ) i( ) (x-x)(41) (4-12) where we have introduced {fic} = S [Uu] I{di

Take Laplace Transform yi (s) 4= s3-+a +is)yi(o) + (s2+ )yI(o) + sy?(o) + y'(o) m N I -xs + i E fia e a + iSa Z (1 - I ) yi(xk)e k (4-13) Ca=l k=l where ca - E6 1 Ei = E 1- S E=Ai Deriving the inverse transform and solving for boundary constants, we obtain y.(x) o For the case of all ends built-in, we have for example Yi(x) = G2(ai,bi,x)y!'(o) + Gl(aibix)y"'i(o) m N I +'i Z fiic Gl(aibix-xa) + PiSc (1- S ) a=?1 k=l Yi(Xk)Gl(aibix-xk) (4-14) Differentiation with respect to x gives Yi(x) = G3(aibix)yi'(o) + G2(aibix)y"(o) + Pi fia G2(ai,bi,x-xa) + PiSa 2 (1 - ) Yi(xk)G2(aibix-xa) (4-15) k=l IS Since Yi(L) = y{(L) = 0, we have

Let yi(x) = Iiwi(x) Pi and q = I-, Equation (4-9) then reduces to Ii E [a]t )} + q[ y + [ Yi c =l EhEM^ } + q1a]{7j'} I [II1]^} = SU a-i:: did 6(x-xa)} N Ik -1 - + a z (1 - - ) [.I ]{7i(xk)} 6(x-xk) k=l IS (4-10o) Let furthermore {Yi} = [ I {i} left by Iij.~ Then we have SoEEIi [a] Ii^i ^} and multiply Equation (4-10) from the i i. I + s q ^i ][a ] [ i7{ yi} + { } I m = gI^{ m di 6(x-x)} + S, a= 1 I1 ( - Is ) {7i(xk)} 5(x-xk) IS N E k=l (4-11) Defining [u] to be the unitary matrix:such that [u]T[IiI[a]CIi][u] = [ki] and [u] [u] = I and letting {yi} = [u]{Yi} Equation (4-11) reduces to SlE[hi (y]{J} + Sq[Xi]{yi} + {y} = [U] [ i] m =1 Xzzl_ di 6(x-xQ N i + Sa ~ (1- Is ) {yi(xk)} 6(x-xk) k=l or SaExiyi + Sq + yi + i m ~ 1=l ria 6(x-x) + Se N k=l Ik (1 - - ) Yi(xk) 5(x-xk) (4-12) where we have introduced {fioJ = Sa [u] TI { dicJ

G2(L), G1(L), G2(L), G3(L) I1 KiSw, (1 - ~iSe, (1- S)G (L-xl), -1 IN - S a (1- IN)G1(L-xN) S - isa (1- iN)G2(L-XN) is / yi(o) y'i(o) G2(xl), G1(x1l ) 0 0 Yi (xl).1~ I1 G2 (x2), G(x ) Pi, (1- -)G(x2-x), IS -1, 0, 0 Yi(x2) G2(xN), G(xN), PiS (1- II)GSXN-xl) iS(l- 1)Gi(xN-X2),... m I Gi ~X fi Gi(L-x ) a=l -1 yi(xN) -. \ m Pi Z a=l fiea G2(L-x ) m Pi Z fio c=l m i z fia 1 a= G1 (x1-xo) G^ If-ol (4-16) (

-44 - Expression for the G functions are given in Appendix C. The deflection of the girders follows from the relationship. Wi(x) = uij(Ii) - yj(x) (sum on j ) (4-17) Example 4-1. A two-girder grillage with the following properties. Length of girders (2) L = 300" Length of stiffeners (9) ~ = 330" girder spacings S = S2 = 110" Stiffeners spacing S = 30" Moment of inertia of stiffeners I = 631 in4 Moment of inertia of girders I1 = I2 = 16,970 in4 Uniform load q = 15 psi All ends of stiffeners and girders are fixed. -The 5th stiffener is much larger than the other identical stiffeners.. The moment of inertia of this stiffener is 107 in4 Figure 4-1 is the plot of the deflection and bending moment of the girder. The solid lines are for the same grillage without the big stiffener and the broken lines are for that with the -big stiffenerf

-45 - a) o d a).rl m) (D x107 -lxlO7.02.04.o6.08 ul U) r! oU aH <^4 Q) Figure 4-1. Deflection and Bending Moment of Girder Number 1 and 2.

CHAPTER V STABILITY Instability of a grillage may occur due to axial loads on the girders, or on the stiffness alone, or due to axial loads in both directions. Let us call these three cases as instability of the first, second and thrid kind respectively. From Equation (3-34), the unknown boundary constants can be solved if the associated submatrix for the specified boundary conditions is non-singular. However, if the values of the axial forces in the girders or in the stiffeners are gradually increased, there exists many possible combinations of the axial forces such that the associated submatrix does become singular. In this case, the grillage system is said to be instable, and the determinant of the associated matrix is then the criterion of instability. Consider the case of all ends of the girders simply supported the criterion becomes det [Ro(L)] = 0 (5-1) To find the critical values of the axial loads, successive values of {Pi} must be put in Equation (3-25) to solve for the eigenvalues and then into Equation (3-28) for Ro(x) e With x = L we then evaluate the determinant (5-1). The values of {Pi} which satisfy Equation (5-1) are the critical values. Even though this is one way to solve the problem, it is complicated and the relation between the rigidity of the members and the

load is obscure. For practical purposes, the torsional rigidity of the members of many grillages can be neglected. In order to find a simple expression for {Pi}, it is necessary to define the relative values of Pi i = 1,.., n. Let us assume that the axial loads on the girders are proportional to their moment of inertias. This assumption is based on the reasoning that since the critical load of a single column is a function of its moment of inertia and that even if the axial loads are not proportional to the moment of inertias at small loads, the stronger girders, with larger I, eventually take up more and more loads as the total load increases, If all the members in one direction are the same, we assume that the axial loads are also the same. A. Instability of the first kind Let R. be the reaction at the intersections of the ith girder and ath stiffeners, since each member is a simple beam, we have m E Ii Wxxxx, i + PiWxx i = E Ria(x-xa) (5-2) a=l for the girders and n E I Wyyyy, a + P Wyy a + ~ Rib(y-yi) = (-35) i=l for the stiffeners. If Pa is lower than the P critical for the stiffener as simple beam, then the influence coefficients aij can be computed. Using the relationship aij Ria = - Wia (5-4)

-48 - Equation (5-2) can be written as Sij(E Ij wxxxx, j + Pjxx, j) + wi = 0 (5-) which is the differential equation for the instabiilty of the first kind. This equation is the same as (4-7) with the terms for lateral loads equal to zero. By the same linear transforms, we have Sm XYi() + Sq iYi +Yi = 0 (5-6) where q = Pi/Ii for all i The Laplace Transform becomes y(s) = 42 [(s3as)yi(O)+(s2+a)yi'(o)-+s yi(o) (5-?) s + 2s + + y't(o)] where = 1 E SaE i For different boundary condtions, the given boundary condition at x = L, give a sufficient number of equations for the unknown boundary constants. As an example consider the case of all ends built in and we have that Yi(s) = - - [s y(0) + y ) (o) (5-8) s +as +pi by which Yi(x) = G2(ai, bi, x) y(O) + Gl(ai, bi. x)yi"(0) (5-9) where ai =1 Pi bi = + a a b a~~~~~~ The G-functions are given in Appendix B.

-49 - The derivative of (5-9) becomes y!(x) = G(ai, bi, x)Y(O) + G2(ai,bix)y(0) (5-10) Introducing the boundary condition yi(L) = 0, y!(L) =, we obtain L2(aibiL), Gl(aibiL)j _y (0) L O0 The condition G2(ai,biL) Gl(ai'biL) det 0 (5-1.2) G (ai,biL) G,(aibiL) I is then the criterion of the critical load. Note the fact that Equation (5-6) is of the same form as the equation of beam on an elastic foundation. Solutions for many cases have been presented in the literature. For such problems, all the steps from (5-7) to (5-13) can be by-passed since the solution can then be obtained by simply matching of parameters. From (5-12), if ai = 0, we have sin biL = + biL (5-13) This condition can be satisfied only if bi = 0. Thus there exist no criterion. For a. f 0, (5-12) reduces to bo sinh aoL + aosin bL = 0 (5-14) where ao, bo are associated with \ Xo = Max {i, i= l...,n}

-50 - This result is the same as that given by Hetenyi(4) From the set of curves in (4) and by matching the corresponding parameters, (5-14) gives the following approximate relation L2 er Pj = 4 + o.o866 L P if < cr,; 3 + 0.202.2 e if > 5 (5-15) EXo SQ P Pr = 5+ 0.-202 Pe if 2>5 (-1) cr^~j ^E0oSa Pe where 2 cEIj Pe ---- L2 For the cases of all ends simply supported, the criterion is!G3(aibiL) Gl(ai,biL) det =0 L- (ai,biL) G3(aibi,L) This can be reduced to cosh aiL + cos biL = 0 (5-16) From Hetenyi's curves again, (5-16) given the approximate relation Pcrj = 1 + 0.0866 L P if Pr < 2 EXoSc Pe 0.202 Tc2E Ij P Pcr,j 2E if cr >2 (5-17) E a0 Sa Pe Each eigenvalue represents a mode of the stiffeners. More about eigenvalues is given in Appendix B. The following example illustrates the application of (5-17). Example 5-1. Find the critical values of the grillage given in Example 3-1.

-51 - From Example 3-1, the properties of this grillage are as follows. Length of stiffeners = 288" Length of girders L = 312" Moment of inertia of girders I2 = 38,500 in4 I = I3 = 19,250 in4 Moment of inertia of stiffeners Is = 6,100 in4 Spacing of stiffeners Sa = 26" From Example 3-1, the maximum eigenvalue o == 0.1551 By Equation (5-17) we have Pcr, = Pcr,3 = [1+0.7643] x 5.8552x107 = 1.03303x108 pounds cr 2 = 2.06606x108 pounds. For the cases with identical girders in equal spacings, the maximum eigenvalues are linear with respect to the number of girders. The following handy formula can therefore be applied. For both ends of the stiffeners simply-supported k\ = [0.020833 + 0.01022(n-l)] i I'g (5-18) For both ends of the stiffeners built in \o = [0.005208 + 0.00182(n-1)] 1 Ig (5-19) EIs From (5-15) and (5-17), we then have Pcrj + ja ll2 / ---- e (5-20) ca + b(n-l) I S ~5

-52 - where t2E I. P - _i re L2 a = 2.0833, b = 1.022 a = 0.5208, b = 0.182 {O.866 6 < 1 p = 2.02 5 > 1 e4 s < 1 3 6 > 1 a = A 1 5 < 1 0 S > 1 for stiffeners with simplysupported ends for stiffeners with built-in ends where 6 is the second term in the bracket of (5-20) for built-in girders for simply-supported girders The following Example 5.2. example illustrates the application of this formula, Find Pcr of the following grillage of 9 girders Is/Ig =0.1, S = t I 10" The stiffeners are built-in both ends. The girders are simply-supported both ends. a = 0.5208, b = 0.182 Try Hence Thus a = 1,p= 0.866. a = 0,p= 2.02. These give These give 6 = 1.2318 6 = 2.8732 Pr = 2.8732 Pe= 2.8732 _2E Ig L2 For grillages with other boundary conditions, the criterion can be obtained analogous.

-53 - In cases where some stiffeners are of different sizes, the procedure used \in this section can be applied in the same manner. Consider, for example, the cases of all girders built-in. The determinant in Equation (4-16) is the criterion. For the case of one add stiffener at x = L/2, we have the stability criterion s G2(L), G1(L), i (i - I1) G1() Si (1- 1) o12(L)_ Is det IGI(L), G2(L), i (1 - Il) G2(L> = (5-21) G2( G1 (), -l B. Instability of the Second Kind The main differences between this problem and that of the first kind are as follows. 1. Since the stiffeners without the supports of the girders are unstable, we can no longer compute their influence coefficients. The influence coefficients of the girders can be used, but this changes our problem from n-dimension to m-dimension, where m is the number of stiffeners. 2. Since the girders are widely spaced, we can no longer treat them the same way as we did the stiffeners, i.e., as a continuous support. Let aa be the influence coefficients of the girder k Since the girders are different from each other only in size, the influence coefficient of the girders are different with only a scalar factor, thus

kat =k = (5-21) aDa = fk a where fk is a scalar factor. From Equation (5-2) and (5-3) and the relationship ao Rip = - wia (sum on 5 ) (5-22) we have that [acp ]{EIp yyyyB + P5 wyy,5} - { i Wa(i) (y-yi)fi} = (5-23) Since Ig = IS and P5 = P for all P, this equation can be uncoupled by letting wa = -up zp (sum on P ) (5-24) where T [u] [aao ][u] = [c] (5-25) Thus Equation (5-23) reduces to (4) n %UEIz 4) +?xPz~ - Z fiZa(yi) 6(y-yi) = 0 (5-26) i=l 2 P Let k EI ' and the Laplace Transform becomes Za(s) 4 [(s3+k2s)(o) + (s2+k2)z'(o) + sz"(o) + zt' (o) s +k s n - fi z a(Yi)e yi (5-27) %aETS i=l Introducing the boundary conditions into the inverse Laplace transform, the determinant of the boundary constants must vanish. This is one of

-55 - the criteria of stability of the grillage. The following examples illustrate this method. Stiffeners are assumed with both ends simply supported. From Equation (5-27) we then have 1 n 1 z(~) = izc(o) + 3 (ke-xinke)z"(o) - i [k(y) - sink(Q-yi)] fi = 0 z ) = 1 sink za'(o) - 1i l z)(y) sink( -yi) fi 0 i=l 1 1111 i=l 1 za(Yi) = Yi %() + k (kYi-sinkyi) z"(o) - I fj 3 z (y) SEI.j=l '[k(yi-yj)-sink(yi-yj)] i = l,...,n (5-28) There are n+2 unknowns in the above homogeneous Equations (5-28). In order to have non-trivial solution for these unknowns the determinant of the coefficients must vanish. For example, the case of one girder at the middle of the stiffeners we have L, (kS-sinke), - C E (- - sin R) Lsink -aEIs in in det 0, s 1 1 sin k 1 sin = k haEIS k 2 ~ 1 kY. ki 2' k3 2 — sin - -, -1 (5-29) which gives. k [(r s kek k kS 1 k sin [(-sin - + - cos -) - 2 cos ]= 0 (5-30) 2 2 2 2 k3XaEIS 2

-56 - This gives us two possible modes. The first mode is a halfwave over the entire length, the second mode is two half waves with the girder as nodal point. Equation (5-29) is satisfied if one of the following two equations is satisfied sin k = 0 (5-31) kY kYk ki 1 k ( cos sin ) 2 cos 2 = 0 2 2 02Sk3~Eis The first condition gives j EIs pcr 2- (5-32) Per ~ 2 ) (2) For any specified \, a second expression for Pr can be obtained. The smaller value of the two is then the critical value. If we want to have the same value for Pcr from the second condition then 1 22EIs ((5-33) Xo (%)3 gives the minimum size of the girder. Where 0 = Max{+, 0=1,...,} From Equation (5-18) and (5-19), Ko is linear to the number of the stiffeners, we then have EIg 1 2EI a+b(n-l) L3 ()3 or Ig = La+b(n-l)J 2j Is (5-34) 2^*

-57 - Where a = 0.020833, b = 0.01021 if the girders are simply supported, a = 0.005208, b = 0.00182 if the girders are built-in at both ends. For other boundary conditions, similar criteria for Pcr and I can be obtained. Note that Pcr can be calculated from the second g cr equation of (5-31) for any specified I less than the value given by Equation (5-34). The value of Pr obtained in this way is less than the value obtained from Equation (5-32).' For more than one girder the determinant equation is further complicated, since more deflection modes are possible. But since we are interested in finding I so as to ensure the highest value of Pcr only. The criterion for one girder, Equation (5-34) is good enough for the cases with girders more than two. More examples are given in Appendix B for cases of one girders with different boundary conditions of the stiffeners. For the case of n girders, n > 2, the maximum possible critical load is n2EI Per= E(5-35) max Smax = MaX{Si,i=,... n} In other words, when the stiffeners are deflected in half waves between girders as shown in the sketch, Smax all girders remain undeflected and act as nodal points. If any one of the girder does deflect, then we may have a lower value of Pr cr

-58 - Sk sk+l k-l k+l Let k be the smallest girder with largest spacings between and k+l, then the portion of stiffeners between k-l and be treated as the case with both ends simple supported at k with one girder k. If k-l, k, k+l can and k+l Sk = k+l = we have 2 it EIS cr S2 2r2L3 'Ikmin = [a+b(n-l)] 2 IS Let us now consider the following mode with the half wave length ex tended to three girder spacings i i+l ~-~_r~-f — ~ i-1 ~ i+2 Si = Si+l = Si+2 = S From Equation (5-28), since the deflection at k and k+l are the same, we have (5-36) 1 3S, 3 (3ks-sin3ks), k S, - (ks-sinks), k3 i\ 1 1 -:I 3 (3ks- sin2ks- sinks) AEI Oj - 1 1 (sin2ks-sinks) 0 XEI k -1 / (5-3'7)

-59 - 2 Let Pcr= -2I Equation (5-37) gives S2 -1 _ EI (5-38) 0o S3 This again gives the same result as Equation (5-33). And; it is concluded that the possible maximum critical load for the grillage with axial forces at the stiffeners which are supported by n girders widely spaced can be made equal to the Euler's load and that the minimum size of the girders to ensure this critical value is given by Equation (5-36). C. Instability of the third kind Consider the girders and stiffeners separately, the axial force on the girders and on the stiffeners can not reach the critical value at the same time. Therefore, it is possible to compute the influence coefficients of either the girders or the stiffeners. If we limited the axial force of stiffeners to below critical, we can solve the problem as the first kind. If we limited the axial force of the girders below critical, then we can solve it as the second kind. It is not difficult to see that the axial forces at either the girders or the stiffeners must be lower than the critical value of the same member as a simple beam. For if the axial forces at the girders are higher than their critical values as simple beams, they can no longer offer constraints to the stiffeners and vice versa.

CHAPTER VI SUMMARY AND CONCLUSION The work in the preceding chapters can be summarized as follows: A. For grillages with members of negligible torsional rigidity, a simple method has been developed to calculate the deflections, bending moments, slopes and shear forces of the members. It is so simple that the computation can be handled with a desk calculator or slide rule. Except for the part of obtaining the eigenvalues and the unitary matrix for the cases with more than three girders, no computer is needed for the computation. The eigenvalues and unitary matrices for the cases of uniform grillages are given in Appendix B. B. oVery simple criteria of stability for various cases have been formulated with the assumption that the axial forces are proportional to the moments of inertia of the members acted upon. C. For the grillages where torsional effect cannot be neglected, the computation is much more difficult because the differential equations cannot be uncoupled. The differential equations are solved by Laplace Transform and a program has been set up for the computation. D. There are no limitations on the types of loading and on the boundary conditions. E. Comparisons between the numerical results obtained by Smith(l4) and those by the present method indicate good agreement between the grillage theory as presented in this dissertation and the method of finite elements as presented by Smith. F. For grillages with members of open section the torsional effect is negligible. For grillages with members of closed section the torsional effect may be important, depending on the values of the -60 -

torsion constants, the spacings of the members, the aspect ratio, ~/L, the types of loading, and the boundary conditions of the grillages.

REFERENCE 1. Michelsen, F.C. and Nielsen, R. Jr., "Grillage Structure Analysis Through Application of the Laplace Integral Transform" SNAME Trans., Nov. 1965. 2. Nielsen, R., Jr., "Analysis of Plane and Space Grillages under Arbitrary Loading by Use of the Laplace Transformation." Doctorate Thesis Danish Technical Press, Copenhages V-Denmark 1965. 3. Wah, Thein, "Analysis of Laterally Loaded Gridwords." J. of the Eng. Mech. Div. Proceeding of the American Society of Civil Engineers, April, 1964. 4. Hetenyi, M., "Beams on Elastic Foundation", U of M Press, Ann Arbor, Michigan, 1946. 5. Tzy Sin-Ben, "The Buckling of Grillages having Transverse Beams of Similar or Dissimilar Section", Chinese Journal of Shipbuilding, Issue No. 2, p. 28, April, 1965. 6. Abrahamsen, E., "Orthogonally Stiffened Plate Fields." First International Ship Structures Congress, Delft, 1961. 7. Abrahamsen, E., "Orthogonally Stiffened Plate Fields." Second International Ship Structures Congress, Delft, 1964. 8. Timonhenko, S., "Theory of Plates and Shells." McGraw-Hill, New York, 1959. 9. Schade, Henry A., "Design Curves for Cross-stiffened Plating Under Uniform Bending Load." Trans. SNAME, Vol. 49, p. 154, 1941. 10. Thein Wah, "A Guide for the Analysis of Ship Structures." U.S. Department of Commerce, Office of Technical Services. 11. Vedeler, G., "Grillage Beams in Ships and Similar Structures", Grondahl & Son, Oslo, 1945. 12. Holman, D.F., "A Finite Series Solution for Grillages," Grondahl & Son, Oslo, 1945. 13. Suhara, J,, "Three-Dimensional Theory of the Strength of Ship Hulls." Memoirs of the Faculty of Engineering, Kyushu University, Vol. 19, No. 4, Fukuoka, Japan. 14. Smith, C.S., "Analysis of Grillage Structures by The Force Method," Trans. RINA, 1964.

APPENDIX A FRACTION EXPANSION OF [Q+s21]-1 AND THE INVERSE TRANSFORMS Suppose we have determined the eigenvalues of the equation det[Q-XI] = 0 Since the coefficients of the characteristic equation are all real, the roots must be in conjugate pairs. If the imaginary components of the conjugate pairs are equal to zero, we have double roots with real value. Then [Q+s2i] — [Q-+sI] (A-l) n-m) (sX)(2+X) n (+ ) k=l i=l where hk, k=l,...,n-m are the complex roots Oi,~ i=l,..., m are the double roots Since all elements of the adjoint matrix are polynomials of s2 one term of the numerator is sufficient to show the nature of the partial fraction expansion. Assuming that the multiplicity of the eigenvalues is not more than two, and letting Xk be the distinct roots and wi be the double roots, then the fraction expansion take the form as follows. n-m h (s2) nm r ak bk 1 n-m m 2 I? (s2+Xk)(s2+Xk) H (s2i )2 k=l i(s k) (s k=l i=l m A B + 2 + - i i=l >s +Wi) (s2 i)2 (A-2) -63 -

-64 - Multiplying sides of Equation (A-2) by (s2+Xk) and letting 2 s =-k we obtain n-m-1 H j=l jik = ak m i=l i=l (wi-2k) (A-3) Multiplying both sides of Equation (A-2) by and letting S =-Nk we similarly obtain (-xk)' n-m-l j=l jik (Xj-7k) (Tj -k) (%k-7k) m i= (l)i -k i=l - bk = ak (A-4) Multiplying both sides of Equation (A-2) by 2 2 s +W i 2 and s = -Wi we obtain (-(______:3i>: Bi n-m m-l 2 1 (j -wi )(Xj i ) I- (k ~-wi) j=l k=l kfi (A- 5) Amd finally multiplying both sides of Equation (A-2) by (s2+ i )2 1 (s2)R - Ai(s2+i ) + Bi n-m k=l (s+k)(s++hk) k=l m-l k=l kWi (s2+wk )2 + ~... (s2+i)2 Derivatives with respect to s on both sides give Derivatives with respect to s on both sides give

-65 - (s2)Q n-m n- (s2+Kk)( 2+Xk) k=l nm-1 n-m II (s2 +)2 n (s2+\k)(s2+Xk) k=l k=l kAi m-1 n (s2 2k) k=l kii n-m r 1 Lk i + S k!.- E 12 + —;k=lL +Xk s 7kj m-1 2 +. Z 2 = k=l S-Wk k7i Ai + ~...(s +i ) With s = -W,. we now have that l A =, ( — - 1 __n n-m m-l n-m n (Xk-i) (k-i ) II (wk- )2 nI k=l k= k= kl kji (-wi ) (\k- wi) (k- 'i) m-1 n (-l"ii)2 k=l k7i n-m k-1 1 k.V + 1 Xk-+'i m-l 2 + - k=l Wkk-Wi kgi (A-6) It follows that [Q+s2I]- n-m k=l 1 S 2 [Q-AkI] n-m-1 _ 2 H (\Xjj-(xj+%j)\k+Xk)(%k-\k) j=l jgk m In i=l (Wi-Xk)2 1 s2 s +%k n-m-1 n j=l j k..... m. [x^j-(%j+Xj)x:k+%k] (k-k:) n (,i-%k)i i=l m + l i=l IlQFOT1 n-m m-1 — (.i) -- ( )2 II (j-j+j )i+i ) (k-i j=1 k=l kfi 1 (s2Hi)2

m i+ i=i n-m n j=l [Q -. ] ' (Xj'j- (\j+\j )coi+ ) rm-1 II k=l k~i (-k li )2 [Q-wiI] n-m n=l j=l 2m-1 (\j\j- (%j+ j)i+wi) I k=l (O_- )2 n-m r(i I k=l xk-Wi +k —i/ kii m-1 2 + ~. ' k=i ~ kC k 1 W~~~~~~; W'i~ — 1 s2+c (A-7) Defining the following parameters %jTj= pi ] j + Xj = J a j,- The above equation can be written as [Q+s2I]-1 n-m = ~ k=l 1 s +Xk s+k [IQ-kI] n-m-1 2 m 2 nI (Pj-ajxk+k) ili (wi-\k) (k-kk) j-=1 i3J__________[& -h X]___ n-m - E k=l n-m-1 _ _2 m nr i (%-oj+Xk k) In j=l i=l j/k ("_i- )2 (k-k ) i=1 i=l [Q-eI] 1 ( s2a+.i)2 n-m 2 ni (j - ji i ) j ~l m-1 k=l k i (Wk-i )2 m 2+ i i=l s +ti i [Q-wiI] I I. n-m 2 n ( j - ji +ci ) j=1 m-1 2 n' (wk-i) k=l k=i

-67 - [Q-'uiI] n-m j=l n-m 1 i k2 E k- + - 'i k=l %k- i % l m-1 ( j-a.j i) n k=]l k=i m-1 + Z k=l k=i 2 lik~w (k k-i 2 k i (A-8) or n-m [Q+s2I]-1 = Z k=l Fl(Xk) s2+k + Fl (%k) S2+'k m F2 (0i) + z (s2e)2 i=l (s2 )2 m F3(iw'i ) i=l S2 1=1 (A-9) Let (a+ib) = a = A+iB 2 2 a2-b2 = A 2ab = B B2+A2 A a = - + 2 2 2 then 2 2 b= B2+A2 A 2 2 4 -1. F(k) F (Xk)! Ls 2+k s2+Ak = Fl(Nk) sin(ak+ibk)x ak+ibk sin(ak-ibk)x F2(X k) - -bk ak-ibk sin akx cosh bkx+i cos akx sinh bkx = 2Re Fl(%k). k.ibk,ak+ibk ci,. Fl(k)xs 2+\k Fl (k) s + = s2+k I +X 2Re Fl(%k) cos(ak+ibk)x = 2Re Fl(\k)[cos akx cosh bkx-i sin akx sinh bkx] -1 <=< Fl (k)s2 Fl (Xk)s2 -. _ F l(k)k sL + S +k j s2+k L i + sF+1 (ik)k S2+~\k J = 2Re Fl(xk)(-ak-ibk)[sinakx cosh bkx + i cos akx sinh bkx] 2 if wi > 0, -b = Woi if 'li < 0 2 Let a = '~i

-68 - -1 i 2a3 1 (sinaix - aix cos aix) 1 72b (-sinh bix + b -1L2 2 (s2. )2 I )ix cosh bix) X 2a sin aix x- sinh b.x 2b 1 2a. (sinaix + aix cos aix) -1 C)<__ s2 L- -i - 1 E -1 ~G, [s2+i 1 2bi 1 a. (sinh bix + bix cosh bix) sin aix sinh bix 1 bi -1; # [s:] = cos aix = cosh bix -1 s2 s2- - 1 -1 [sSXi] = - ai.sin aix =. s n b. x = bi sinh bix (A-10)

-69 - From Equation (A-3) and (A-8), if all eigenvalues are distinct, the numerators of any element of the function expansion can be obtained 2 by simply substituting the associated eigenvalue for s. Therefore it is not necessary to have the expression for the elements of the adjoint matrix. However, if double roots exist, which is possible in general, the first derivative of every element is needed, and therefore it is necessary to have the expression for every element of the adjoint matrix in terms of s. This can be done by using Faddeev's method. [Q+sI] = Is2(n-) + Bls2(n-2) + + B (A-ll) A = Q, P1 = trace A1, B = A1 - P1I A2 = QB1, P2 =! trace A2, B2 = A2 - P2I 2 A QBP 1traQe A B A P E n 0n n n = Bn-ilPn n n n n n

APPENDIX B THE ALPHA MATRIX, UNITARY MATRIX AND THE EIGENVALUES The element, li, of the alpha matrix [a] is the influence coefficient. It is the deflection at point i due to a unit concentrated load at point j. 1 )ij i i From Maxwell's reciprocal law, we see that aij = aji (B-l) Therefore the alpha matrix is real and symmetric. With appropriate simple beam formula the influence coefficients can be easily calculated. For the case of both ends simply supported with three girders equally spaced, we have 0.011719 [a] = L 0.014323 EI S 0.0091146 2 = 0.041089 x Iy EIS 03 %3 = 0.00057769 E 0.014323 0.020833 0.014323. 0091146 ] 0.014323 0.011719 13 I k1 = 0.0026042 -_ y E I Is Is (B-2) where Ai are eigenvalues of [I J][C][I ] -70 -

-71 -The following tables contains the alpha matrices unitary matrices and eigenvalues for the cases up to 17 girders, with all girders identical in size and in spacing. The differences between the sizes and the spacings of the girders cause no difficulty in the theory or in computation. These special cases are chosen as examples only because these are the most common arrangements in structures. The program for this calculation is valid for any combination of the sizes and the spacings of the girders. If the loadings are symmetrical about the center line of the stiffeners, the computation can be simplified by considering only half the structure. The differential equation of a three girders grillage takes the following form ES rl11 + a13 a12 ILw1t 1 Fwl i dl La 21 + 23 ~22 I I2w w )j(4 Lwj 2 or 11 + 13 12 + IlWl( 1w dl (B-3) La21 + 23 6'22 + 22 J 2 Lw2 ld2 Equation (B-3) has the advantage of making the alpha matrix symmetrical 012 = 21 = 23 = 32 Table B-l is for stiffeners which are simply supported at both ends and Table B-2 is for stiffeners which are built-in at both ends. The element of the alpha matrices and the eigenvalues should be multiplied by the factors ~3 ~3 E- and -iIg EIT EIF g

-72 - respectively. Since the arrangement of the girders is symmetric, all the data given in these tables are for the equation after contraction as shown in Equation (B-3). For the case of three girders, we then have for example 13 0.020833 [as] - E EIS 0.020256 a 0 041089 3 y 1 EIS [u] = 1 1 [u 2 1 1:1. 0.020256 0.020833 a2 = 0.00057769 13Iy (B-4) EIS Without this contraction, the alpha matrix and the eigenvalues are as shown in (B-2) and the unitary matrix is v 2 1 [u] = 0 2 1 1 1 12 f2 1 1 (B'5) In Chapter 5, we have used the maximum eigenvalue for the criterion of stability. This value can be obtained from Tables B-l and B-2 directly. We have mentioned before that each eigenvalue represents a mode shape of the deflection of the stiffeners. The mode shape can be easily observed from the unitary matrix. Compare (B-2) with (B-5) we have Eigenvalue xi A2 N3 Eigenvector { 2, 0, 42} { 1, 2, 1 } {1 in 2,1 } Mode =fff

-73 - The contraction eleminates the asymmetric modes but does not affect the maximum, since the maximum eigenvalue is always associated with the symmetric mode \2 Example B-1. One girder grillage. The girder is at of the stiffeners All stiffeners are simply supported at one end and built-in at other end. Find criteria for Ig and Pcr of the stiffeners. k2 P From (5-27) we have EIS z(x) = xz'(o) + - (kx-sinkx)z"' (o) - 1 1 k(x- 2 ) - sink (s- ) z k IsIl k x 2 2 2 z'(x) = z'(o) + 2 (l-coskx)z"' (o) - 1 1 -cosk(x- )) k2 EI k2 2 2 The criterion is 1 1 1 kS kS -, -(k-sinke), - E (- - sin -) kk3 EIS k3 2 2 _ 1 Q det 1,,-_-(l-coskY) 1-:- -(l-cos ) 0 (B-6) k2 SEIS k3 2 detk sin - 1 2 -1 Let r i.e cr we have 2 - s i~e. P cr - (S)2 S 1 XEIS k2 2 ' 2k2 2k

-74 - k2 1 1- 1 XEIs = 0 or 2 2r )EIs Xko (1)3 (~^1 (B-7) The same as (5-33) Example B-2. If the stiffeners are built-in at both ends then the criterion is / det 1 (1 - co 2 k 1 sinkQ, k 1 (1 - co k sk ), 1 (k- sink), 1 1 (k k3 %XEI k2 2 l(1 - cosk), - 1 1 (1 (k2 EI k2 1k _ sin k -1 k 3 T 2 ki sin 2 cos k ) 2 ) / kssT) (B-8) This condition is satisfied if k kQ k_ sin -2 - cos 2 kor 493 or 2- 4.,493 2 (B-9) This gives 4.4932EIs Pcr= - (-) 2 20.19EIs ~2 (2' (B-10) In order to have the same critical value Pcr for the other mode, we can substitute (B-9) into (B-8) and obtain 1 20.19EIs X0 ( )3 2F From Equation (5-18) and (5-19) we have (B-ll) Ig = [a+b(n-l)] 0 where a and b are defined in Equation (5-34). IS (B-12)

-75 - TABLE B-1 SIMPLY SUPPORTED STIFFENERS ALPHA MATRIX.20833E-01 EIGENVALUES OF 1 GIRDERS.20833E-0 1 UNITARY MATRIX OF 1 GIRDERS. 1000E 01 ALPHA MATRIX.30864E-01 EIGENVALUES OF 2 GIRDERS.30864E-01 UNITARY MATRIX OF 2 GIRDERS.10000E 01 ALPHA MATRIX.20833E-01.20256E-01.20256E-01.20833E-01 EIGENVALUES OF 3 GIRDERS.41089E-01 57769E-03 UNITARY MATRIX OF 3 GIRDERS.70711E 00-.70711E 00.70711E 00.70711E 00 ALPHA MATRIX.14667E-01.22667E-01.2266'E-01.373.33E-01 EIGENVALUES OF 4 GIRDERS.65790E-03.51342E-01 UNITARY MATRIX OF 4 GIRDERS.85065E 00.52573E 00 -.52573E 00.85065E 00

-76 - ALPHA MATRIX.10802E-01.17747E-01.14186E-01.11747E-01.30864E-01.25098E-01.14186E-01.258. 25098E-01 0833E-01 EIGENVALUES OF 5 GIRDERS *77162E-03.b1603E-01.12566E-03 UNITARY MATRIX OF 5 GIRDERS.81649E 00.40825E 00.40827E 00,1505b2-04.70711E 00-.70711E 00 -.5773oE 00.57735E 00.57734E 00 ALPHA MATRIX.82604E-02.14091E-01.17007E-01. 14091E-01.25267E-01,31098E-01.1700'7E-01.31098E-01.39359E-01 LtIENVALUES OF 6 GIRDERS.b9329E-0.3.12688E-03.71866E-01 UNITARY MATRIXX OF 6 GIRDERS.73698E 00-.59101E 00.32799E 00.32799E 00.73698E 00.59101E 00 -.59101E 00-.32799E 00.73698E 00 ALPHA MATRIX,65104E-02.11393E-01.14323E-01.10818E-01.11393E-01.20833E-01.26693E-01.20256E-01.14323E-01.26693E-01.35156E- 2631-0126931E-.10818E- 026E-01 2026E-01.26931E-01.20833E-01 EIGENVALUES OF 7 GIRDERS,13770E-03.10177E-02.82131E-01.47331E-04 UNITARY MATRLX OF 7 GIRDERS.65326E 00.65328E 00.27060E 00-.27067E 00 -.49994E 00.50000E 00.50000E 00.50006E 00 -.27068E 00-.27059E 00.65328E 00-.65325E 00.50005E 00-.50001E 00.50000E 00.49994E 00 ALPHA MATRIX.52583E-02.93736E-02.12117E-01.13489E-01.9373bE-02.17375E-01.22862E-01.25606E-01.12117E-01.22862E-01.30864E-01.34979E-01.13489E-01.25606E-01.34979E-01.40238E-01 EIGENVALUES OF 8 GIRDERS.45226E-04.11431E-02.15157E-03.92396E-01

-77 - UNITARY MATRIX OF 8 GIRDERS..42853E 00.57735E 00-.65654E -.656654E 00 57735E 00.22801E _.57735E 00.11779E-06.57735E -.22801E 00-.57735E 00-o42852E 00.22801E 00 00.42853E 00 00.57735E 00 00.65654E 00 ALPHA MATRIX.43333E-02.78333E-02.78333E-02 14667E-0 1.10333E-0l.19667E-01.11833E-01.22667E-01.87210E-02.16735E-01. 103 33E-01 l 19667E-01 27000E-01.31500E-01 23335E-01.11833E-01.87210E-02.22667E-01.16735E-01.31500E-01.23334E-01.37333E-01 27813E-01.27813E-01 -.20833E-01 EIGENVALUES OF 9 GIRDERS.16667E-03.1'2691E-02.46689E-04.10266E 00.22972E-04 UNITARY MATRIX OF 9 GIRDERS 63244E 00 *.51166E 00.51164E 00.24397E-0.60150E 00-.60137E 00 -.63244E 00 *195.44E 0 19521E 00 -.63956E-04-.37174E 00.37197E 00.44727E 00-.44722E 00-.44734E 00.19544E 00 *19559E 00.37175E 00-.37196E 00.51167E 00 *51178E 00.60150E 00-.60137E 00.44721E 00.44702E 00 ALPHA MATRIX I.36314E-02 66366E-02. 88906E-02..10393E-01.11145E-01.66366E-02.88906E-02.12522E-01 -.17030E-01.17030E-01.23666E-01.20035E-01.28174E-01 *21538E-01.30428E-01.10393E-01.11145E-01.20035E-01.21538E-01 o28174E-01.30428E-01 *34060E-01.37065E-01.37065E-01.40696E-01 EIGENVALUES OF 10 GIRDERS *49521E-04.13954t-UZL Zl400E-04.18233E-03..LLZ93t 00 UNITARY MATRIX OF 10 GIRDERS *54853E 00.45573E 00.32602E 00-.59688E 00.16989E 00 -.445573E 00 *59688E 00-.54853E 00-.16989E 00.32602E 00 -.16989E 00.32602E 00 *.59688E 00 *54853E 00 4553E 00.59689E 00-.16989E 00-.45573E 00.32602E 00 *54853E 00 -.32602E 00-.54853E 00.16989E 00-.45573E 00.59688E 00 ALPHA MATRIX *.308 64 E-0 2. 56 906E-0 2 *56906E-02. 10802E-01.77160E-02.14853E-01.91628E-02.17747E-01.10031E-01. 19483E-01.72975E-02.14186E-01.77160E-02.91628E-02 14853E- 01: 17747E-01.20833E-01.25174E-01.25174E-01.30864E-01 2 7778E-01.34336E-01.20256E-01.25-098E-01 10031E-01 * 19483E-01,27778E-01 34336E-01 *38580E-01.28303E-01.72975E-02.14186E-01.20256E-01 * 25098E-01.28303E-01.20833E-01 EIGENVALUES OF 11 GIRDERS *52997E-04.15218E-02.12910E-04.19830E-03.12319E 00.21402E-04

-78 - UNITARY MATRIX OF 11 GIRDERS * 55764E -.28858E -.40829E.49989E * 14963E -.40839E 00.40825E 00.14970E 00.57735E 00-.28911E 00.40825E OC.40865E 00.25592E-05-.50011E 00-.40824E 00.55731E 00-.40826E 00-.40781E 00-.55766E 00 00-.28868E 00 00.40823E 00 00.50000E 00 00-.14938E 00 00-.40830E 00.14943E 00.40823E 00.28868E 00-.57718E 00.40825E 00.40783E 00.50000E 00.55698E-03.55768E 00-.40869E 00.40825E 00.40848E 00 ALPHA MATRIX.26551E-02. 49310E-02.,7516E-02.81171E-02.90275E-02 *94826E-02.49310E-02. 94068E-02 ~ 13048E-01.15779E-01. 1/600E-01. 18510E-01.675 l6E-02 * 13048E-01 18434 —01.22531E-01.25262E-01.2 6627E-01.81 i71E-02.157 9E-01.22531E-01.27917E-01.31558E-01.33379E-01.90275E-02.1 l600E-01.25262E-01.31558E-01.36034E-01 *38310E-01.94826E-02. 18510E-01.26627E-01.33379E-01.38310E-01.40965E-01 EIGENVALUES OF 12 GIRDERS.56777E-04.Z1442E-03.16483E-02.11692 E-04.22089-04.13346E 00 UNITARY MATRIX OF 12 GIRDERS *55066E 00.51l65E JO.36183E -.132751r. 00.36783E 00.55066E -.51865E 00-.25 778E 00.45651E.25778E 0-.55066E 00.13275E.45651E 00-. 13275E 00-.25778E -.36783E 00.45651E 00-.51865E 00-.25778E 00.45651E 00-. 55065E 00.51866E 00-. 3784E 00.13275E 00-.45651t )O.51866E 00-. 13276EF 00-. 3o783E 00.55065E 00-.25778E 00.13275E 00 00.25778E 00 00.36783E 00 00.45651E 00 00.51865E 00 00.55066E 00 ALPHA MATRIX.23081E-02.431 24E-02 59524E-02.722 79E-02.81390E-02. 86856E-02.62705E-02. 43124t-02.82604E-02. 11540E-01 * 1409!E-01.14914E-01.17007E-01. 12283E-01.59524E-02.1154UE-01.16399E-01., O226E-01.22959E-01.24599E-01.17781E-01.722 79E-02. 14091E-01.20226E-01.25267E-01. 28912E-01.31098E-01.22505E-01.81390E-02 159 1 4E-0.22959E-01.28912E-01.33406E-01.36139E-01.26199E-01.86856E-02.17007E-01.24599E-01.31098E-01.36139E-01.39359E-01.28604E-01.62705E-02.12283E-01.1778 E-01.22505E-01.26199E-01.28604E-0.20833E-01 tEGENVALUES OF 13 GIRDERS.60745E-04.23139E-04.17749E-02.79d74E-05.23064E-03.14372E 00.11658E-04 UNITARY MATRIX OF 13 GIRDERS.53449E 00-.48154E 42394kE-04.41'774E -.53449E 00.11914E -. 8520'7E-04-. 52109E. 53449E 00.33292E.1581otE-03.23236E -.37810E 00-. 37824E 00.33327E 00.52112E 00.48159E 00.23192E 00-. 11894E 00-.41790E 00-.37797E 00-. 1934E 00.232olE 00-. 33405E O. 1848 E 00-.48 1oOE 00.52035E 00-. 3 1 14E 00-.48158E 00-.41790E 00.11893E 00.52111E 00.33328E 00-.23189E 00-. 37801 E 00.11894E 00.23192E 00.33327E 00.41791E 00.48159E 00.52112E 00.37796E 00-.33327E 00 00.52095E 00 00-.48104E 00 00.23098E 00 00.11998E 00 00-.41864E 00 00.37833E 00 ALPHA MATRIX.20247E-02.38025E-02.52840E-02.38025E-02.52840E-02. 6491E-02 73086E-02.10272E-01.12642E-01.10272E-01.14667E-01.lt222 -01O.73580E-02.14420E-01.20889E-01.79506kE-02.82469E-02.15605E-01.16198E-01.22667E-01.23556E-01

-79-.64691E-02.12642E-01.18222E-01.22914E-01.26469E-01.28840E-01.73580E-02.14420E-01.20889E-01.26469E-01.30864E-01.33827E-01,79506E-02.15605E-01.22667E-01.28840E-01.33827E-01.37333E-01.82469E-02.16198E-01.23556E-01.30025E-01.35309E-01.39111E-01.30025E-01.35309E-01.39111E-01.41136E-01 EIGENVALUES OF 14 GIRDERS.64803E-04.11795E-04.19016E-02.24366E-04.24691E-03.73265E-05.15399E 00 UNITARY MATRIX OF 14 GIRDERS.51357E * 10737E -.49112E -.21004Et.44721E.30353E -.3837bE 00-.38374t 00.30353E 00.49113E 00-.44721E 00-.21007E 00.51354E 00.49112E 00-.30353E 00-.44721E 00.38380E 00-.30349E 00.49112E 00-.30353E 00.49890E-06-.49115E 00-.10741E 00.30353E 00.49112E 00.44721E 00.51356E 00.44725E 00.81638E-09.41390E-05.44721E 00-.44718E 00-.49115E 00-.30353E 00-.49113E 00-.33917E-06.30349E 00.21005E 00-.49112E 00.30353E 00-.44721E 00-.10734E 00.10737E 00 00.21004E 00 00.30353E 00 00.38376E 00 00.44721E 00 00.49112E 00 00.51357E 00 ALPHA MATRIX.17904E-02.33773E-02.47201E-02.58187E-02.66732E-02. 12835E-02.76497E-02.549 55E-02.33 173E-02.b5104E-02.919O0E-02.11393E-01.13102E-01.14323E-01.15055E-01.10818E-01.47201E-02.91960E-02.13184E-01.16479E-01.19043E-1I.20874E-01.21973E-O1.15796E-01.56187E-02.11393E-01.16479E-01.20833E-01.24251E-01.26693E-01.28158E-01.20256E-01.66732E-02.13102E-01.19043E-01.24251E-01.28483E-01.31535E-01.33366E-01.24025E-01.72835E-02.14323E-01.20874E-01.26693E-01.31535E-01.35156E-01.37354E-01.26931E-01.76497E-02.15055E-01.21973E-01.28158E-01.33366E-01.37354E-01.39876E-01.28801E-01.54955 E-02.10818E-01.15796E-01.20256E-01.24025E-01.26931E-01.28801E-01.20833E-01 EIGENVALUES OF 15 GIRUERS.12150E-04.68937E-04.26324E-03.20282E-02.52906E-05.25716E-04.16426E 00.70930E-05 UNITARY MATRIX OF 15 GIRDKRS.41565t 00.49037E 00.4151736 00.27778E 00.98133E-01-.49035E 00 -.46167L 00.19136E 00.46194E 00.46194t 00-.19239E 00.19124E 00.97117E-01-.41569E 00.97551E-01.49039E 00.27907E 00.41576E 00.35382E 00-.35357E OC-.35354t 00.35355E 00-.35475E 00-.35338E 00 -.49011E 00.27772E 00-.49039E 00.97546E-01.41644E 00-.27793E 00.19056E 00.4619bE 00-.1913oE 00-.19134E 00-.46169E 00.46177E 00.27860E 00-.97423E-01.27776E 00-.41573E 00.48895E 00.97914E-01 -.35404L 00-.35368E 00.35359E 00-.35356E 00-.35211E 00-.35382E 00 ALPHA MATRIX.97545E-01.27784E 00.19134E 00-.46182E 00.27779E 00.48977E 00.35355E 00-.35225E 00.41573E 00.95748E-01.46194E 00.19308E 00.49039E 00-.41685E 00.35355E 00.35407E 00.15944c-02.30192E-02.42405E-02.52582E-02.60723E-02.66830E-02.70900E-02.72936E-02.0 1 9 2-02. 5b349E-02.82774E-Oz,10313E-01.11941E-01.131o26-0 1.13977E-01.14384E-01.42405E-02.82774E-02.11907E-01. 14960E-01.1 74036-01 19235E-01.20456E-01.210b67E-01.52582E-02.10313E-01.14960E-01.18997E-01.22254E-01.24696E-01.26325E-01.27139E-01.60723E-02.11941E-01.17403E-01.22254E-01.26291E-01.29344E-01.31379E-01.32397E-01.66829E-02.13162 -01.19235E-01.24696E-01.29344E-01. 32974E-01.35416E-01.366376-01.70900E-02.13977E-01.20456E-01.26325E-01.31379E-01.35416E-01.38232E-01.39657E-01.72936E-02 *14384E-01.21067E-01.27139E-01.32397E-01.36637E-01.39657E-01.41251E-01 EIGENVALUES OF 16 GIRDERS.70724E-05.73101E-04.27123E-04.21549E-02.48491E-05.2/958E-03.12613b-04.17452E 00

-80 - Ui )ll'A iY iMlA lX. i OF 16 GIRUERS.326dO0 00.466:>'oEt 00-.4300- 00 5.-336- 00.11521E 00-.38709E 00,434226E 00 -.48303[6 00,zLb3oE 00.8913JE-01.43422 00-.326 76c 00-.46655E 00-.38706E 00.3J/14h 00-.32o07E 00.46buoE 00.43J006 00.43418t 00-.175236E 00-.89150E-01 -.i9 17tL-01-.43422E 00-.17522L 00.38710t 00-.482981 00.2553bt 00.4bo57E 00 -.25532L 00.d9J132-01-.4342f ')00.1/523E 00.46o57E 00.48400E 00-.32680E 00.46652E 00.43 0.2553o 00-.ov131E-01-.38714E 00.32619E 00-.17122E 00 -.+3419t: 00.17522t 00.38710E 00-.t2679E O.25541E 00-.89132E-01.48299E 00.17>22E 00-.3t709E 00-.32(I9)L 00-.46655E 00-.8915 7E-01-.43422E 00-.25535E 00 ALPHA MATRIX.89132E-01.17523E 00.25536E 00.32679E 00.38710E 00.43422E 00.46655E 00.48300E 00. 1 42 8' E-02 27149 - 02.3 Z94 E-02.47725h-(0. 554416-02.61443E-02.65729E-02.6t830it-02.48903j-02. -7149E-02. 5258 3E-02.* 746 14E-02.937j6E-02.109177-0O.121176-01 * 1274E-01. 13489E-U. 9t593E-02.3 t2 94c-02.7487 14- 02.10802E-01. 13632E-01.15947L-01.1714 E-01. 19033-0ul.1980) 1:-u01.14186oE-01.4 7725E-02.93736E-02. 13632t-01. 173 15E-01 - 204620 -01.22862t-01.245 776-01.25tj06E-01.L83496-01.55441E-02.109i7L-01.1594 'E-01.204OZE-01.24291iE-01.27292E-01.29435E-01.30 721E-01.22026E-01.61 443 -02.12117E-01.17747E-01.22 62 E-01.27292E-01.30864E-01.33436E-01.34979E-01.25098E-01.65729E-02.12974E-01.19033E-01.24577E-01.29435E-01.33436E-01.3u408E-01.3d209E-01.27442E-01.68301E-02.13489E-01.19805E-01.25606E-01.30721E-01.34979E-01.38209E-01.40238E-01.28937E-01.48903E-02.96593E-02.14186E-01.18349E-01.22026E-01.25098E-01.27442E-01.28937E-01.20833E-01 E ltbiVALUIS F- 17 GIRODRS.13l15I9-04.'7302E-04.46580-7-05.22816E-02.71907E-05.29595E-03.28583E-04.18479E 00.36870E-05 UNITAKY MAfklX OF 17 GIKDL-RS.44289E 00.44295t 00.23580E 00.23570E 00-.36107E 00-.36111E 00-.47136E 00 -.30280E 00.30)3026 00-.40817E (00 40825E 00.46394E 00-.46424E 00-.61237E-04 -.2358t6 00-.235o07E OC.41075E 00.47140E 00-.23505E 00-.23570E 00.47136E 00.46401E 00-.4j424E 00-.40671E 00.40825E 00-.16191E 00.16122E 00.11880E-03 -.81419t-Oi-.81Y694-01.23328t 00.23570E 00.44308E 00.44297E 00-.47135E 00 -.40840e 00.40o2LE 00.O2903E-02.90753E-06-.074740E 00.40825E 00-.18354E-03.360obt 00.3oll5E 00-.23833 E 00-.23570E 00.80408E-01.81871E-01.47137E 00.1619S5 00J-.loll3E 00.40 5E 00-.40825E 00.30428E 00-.30299E 00.30055E-03 -.33382E 00-.33345E 00-.3 404E 00-.33334E 00-.33402E 00-.33337E 00-.33359E 00.81859E-01.82666E-01.16123E 00-. 16271E 00.23570E 00.23760E 00.30301E 00-.30498E 00.36112E 00.36272E 00.40825E 00-.40896E 00.44298E 00.44223E 00.46424E 00-,46180E 00.33333E 00.33102E 00

-81 - TABLE ' -2 BUILT IN STIFFENERS ALPHA MATRIX.52083 E-02 EIGENVALUES UF I GIRDERS.52083E-02 UNITARY MATRIX OF 1 GIRDERS.10000E 01 ALPHA MATRIX.61728E-02 tIGENVALUES UF 2 GIRDERS u 1 728E-02 UNITARY 1MA'IRIX OF 2 GIRDERS.10000E 01 ALPHA MATRIX *,5eL-02. 5,5 6o E-02.3082dt-02. ZOdiE-022 OLLItGE;VALUES (j- 3 GIKDtkRS L 1 S " iq V A L 0 e S l 3 v I K O t R S. -2 iot:-03. bU4i9E-02 UNI'TAAY iMAhI'X UF 3 GIRULRS. 192'b0E 00.60F79E 00 *o)9 IE O0. /92(56 00 ALPHA MlRIX.l 1 67E-02.34t, 7E-02.346'7 tL-02. b53 )3E-02 EIGENVALUES UF 4 GIRULRS,39075E-03. iOUGE-0 1 UiilARY MATRIX uF 4 GIRUERS.92008E 00.39'1?2E OC.39172E 00.92008E 00

-82 - ALPHA MATRIX. 1 74E-02.23148E-02.19096E-02.23148E-02.61728E-02.54561E-02.i9096E-02.54561E-02.52083E-02 EIGENVALUES OF 5 GIRDERS.43082E-03.11997E-01.1 il E-03 UNITAKRY MATRIX OF 5 GIRDERS.79483E 00.26653E 00.54516E 00.21777E 00.71326E 00-.66621E 00 -.56641E 00.64825E 00.50888E 00 ALPHA MATRIX. 16357E-03.15966E-02.20131E-02.15966E-02.44426E-02.61086E-02.20131E-02.61086E-02.93711E-02 EIGENVALUES OF 6 GIRDERS.98217E-04.48904E-03.13990E-01 UNITARY MATRIX OF 6 GIRDERS. 73966E 00.64555E 00 -,62450E O0.55306E 00.25082E 00-.52667E 00.19019E 00.55147E 00.81222E 00 ALPHA MATRIX.52897E-03.11393E-02.11393E-02.32552E-02.15055E-02.47201E-02.11509E-02.3b828E-02 1 5055E-02.47201E-02. 76904E-02.621 48E-02.11509E-02.36828E-02.62148E-02 52083E-02 EIGENVALUES OF 7 GIRDERS.99796E-04.55304E-03.15986E-01.44547E-04 UNITARY MATRIX OF 7 GIRDERS.73818E 00.51977E 00 -.27792E 00.68338E 00 -.37338E 00-.l2970E 00.48830E 00-.49598E 00 14100E 0-4 E 0040625E.43156E 00.51914E 00.69182E 00-.60427E 00.56148E 00.44756E 00 ALPHA MATRIX.38104E-03.83829E-03.83829E-03.24387E-02.11431E-02.36580E-02 L 129 55E-02.42676E-02.11431E-02.12955E-02.36580E-02.42676E-02.61728E-02.75446E-02.75446E-02.97546E-02 EIGENVALUES OF 8 GIRDERS.10668E-03.61925E-03.39097E-04.17982E-01

-83 - UNITARY MATRIX OF 8 GIRDERS.67168E 00.42104E 00.59995E 00.56242E-01.70434E 00-.61892E 00 -o63335E 00.18432E 00.47504E 00.38020E 00-.54097E 00-,l7703E 00 ALPHA MATRIX.10782E 00.34304E 00.58244E 00 72901E 00 28333E-03. 63333E-03 * 88333E-03. 10333E-02. 6603E-03 *63333E-03.88333E-03.i8667E-02 *28667E-02.28667E-02.49500E-02 346o E-02.b3000E-02.25927E-02.47730E-02.10333E-02.76603E-03.34667E-02 *25927E-02.63000E-02.47730E-02 85333E-02 65997E-02.65997E-02 *52083E-02 EIGENVALUES OF 9 GIRDERS.38i57E-04.6866E-0 52E-03 15E-3.19979E-01.22195E-04 UNITARY MATRIX OF 9 GIRDLRS.6499oE 00.34460E 00-.59199E 00 -.45768 E 00.67400E 00-.29411E 00.28111E-01.38804E 00.59670E 00 * 423 9E 00- 28096E 00.95899E-01 -.43344E 00-.44436E 00-.44474E 00. 457iE-01.31811E 00.27100E 00-.41596E 00 49014E 00.50233E 00.65089E 00-.55568E 00.50221E 00.40578E 00 ALPHtA MATRIX *21629E-03.48949E-03.6944J E-03. 8310OE-03.89930 E-03.48949E-03.69440E-03 *14571E-02 *227o7E-02.2 276 7t-022.39956E-02 2bi231E-02 *52251E-02.30963E-02.58398E-02 83i100E-03.28231E-02.52251E-02. 72855E-02.83 783E-02.89930E-03. 30963E-02 ~ 58398E-02.83783E-02.99606E-02 EIGENVALUES OF 10 GIRDERS.39232E-04.75415E-03.19541E-04 12513E-03.21976E-01 UNITARY MATRIX OF 10 GIRDERS.63324E 00.28526E 00 *49657E 00-.51621E 00 -.22308E 00.b2267E 00-.5o2blE 00-.44099E 00 -.34100E 00.50389E 00.8582E 00.42292E 00 *59208E 00-.34883E-Ol-.37343E 00.42317E 00 -.28706L 00-.5256 E 00.13457E 00-.42558E 00.67747E-01.22697E 00.41427E 00 *57419E 00.66526E 00 ALPHA MATRIX ~ 16879E-03.38580E-03.55459E-03 *67515E —03 14749E-03.54561E-03.38580E-03.1 574E-02. 18326E-02.23148E-02.26042E-02 19096E-02. 55459E-03 * 18326E-02.32552E-02. 43403E-02.499 13E-02 ~ 36828E-02.67515E-03.23148E-02 43403E-02.61 728E-02 73302E-02 *545 1E-02.74749E-03.26042E-02.49913E-02.73302E-02 90422E-02.68201E-02.54561E-03.19096E-02.36828E-02.5456 1E-02.68201E-02.52083E-02 EIGENVALUES OF 11 GIRDERS.18691E-04.41270E-04.82211E-03.13624E-03.23974E-01.12635E-04

UNITARY MATRIX OF 11 GIRDERS 56822E -.50665E.25415E * 11521E -.43282E,39400E 00.59063E 00.23875E 00-.56459E-02.56530E 00-.52307E 00.55949E 00.40926E 00.16006E 00.21572E 00-.34657E 00-.40435E 00-.40580E 00-.44915E 00 00-.52093E 00 00.21649E 00 00.55507E 00 00-. 76030E-01 00-.40760E 00.55246E-01-.25786E 00 18845E 00.34178E 00.35237E 00-.42160E 00.50441E 00.48071E 00.61045E 00-.51729E 00.45846E 00.37453E 00 ALPHA MATRIX * 13422E-03.30928E-03 *44933E-03.55437E-03.62439E-03.65941E-03.30928E-03 * 93367E-03.149. 9E-02. 19140E-02.21941E-02.23342E-02.44933E-03.14939E-02.26785E-02.36238E-02.42541E-02.45692E-02.55437E-03 * 19140E-02 36238E-02. 52286E-02.63490E-02.69092E-02.62439E-03.21941E-02.42541E-02.63490E-02 80238E-02.88991E-02.65941 E-03.23342E-02.45692E-02.69092E-02.88991E-02 10084E-01 EIGENVALUES UF 12 GIRDERS.18741t-04.14701E-03.89022E-03.11200E-04.43793E-04.25971E-01 UNITARY MATRIX OF 12 GIRDERS.58020E 00.39149E 00.20190E 00-.41i72E 00.54030E 00 -.35887t 00.55606E 0 50892E 0.592 00 0021E 00.16653E 00 -.69340E-01-. 2 743E-0.5 7662E.45392E 00-.54318E 00 -30040E -.52192E 00-.22649b 00-. 15886E.22656E 00.43641E 00-.50226E 00-.5 1899E 00.44889E 00-. 30353E 00.10718E 00-.54910E 00.10426E 00.50180E 00-.34083E 00 00 00 00.45741E-01.15835E 00.30183E 00.44306E 00.55419E 00.61506E 00 ALPHA MATRIX 1084oE-03.25163E-03.36877E-03.45988E-03. 52495E-03.564 00E-03.40801E-03.5 1 63E-03 76357E-03 1 232 1E-02.15966E-02.18569E-02.20131E-02.14603E-OZ 36877E-03. 12321E-02.22256E-02 3045oE-02.36313E-02.39827E-02.28990E-02.45988E-03. 15966E-02.30456E-02.44426E-02.54838E-02.61086E-02.44o67E-02. 52495E-03. 18569E-02.36313E-02.54838E-02.70500E-02.80262E-02.59054E-02.56400E-03.20131E-02 ~39827E-02.61086E-02.80262E-02.93711E-02.69577E-02.40801E-03.14603E-02.28990E-02.44667E-02.59054E-02.69577E-02.52083E-02 EIGENVALUES OF 13 GIRDERS.19276E-04.46587E-04.95843E-03.78 724E-05.15793E-03.27969E-01.10598E-04 UNITARY MATRIX OF 13 GIRDERS.56301E -. 19028E -.31378E.52564E -.23699E -.27691E.37312E 00.48987E 00.17237E 00-.21452E 00.29193E 00.45655E 00.28681E 00-.47919E O0.57036E 00-.35858E 00-. 17248E 00.39439E 00.41o60E 00.51976E 00.77517E-02-.45955E 00.58c52E-01- 37383E 00.48585E 00-.37663E 00-. 37574E 00-.34980E 00-.34242E 00-. 56268E 00-.12756E 00.44979E 00.41661E 00-. 17680E 00-.37780E 00.38372E-01-.49860E 00 00.13448E 00.50317E 00 00.26038E 00-.36601E 00 00.38997E 00.10230E 00 00.50052E 00.19585E 00 00.57436E 00-.42770E 00 00.42445E 00.36401E 00 ALPHA MATKIX.88889E-04.20741 E- 3 3061 '7E-03.20741E-03.63210E-03.1027'2E-02.30617E-03.38519E-03.10272E-02.13432E-02.18667E-02.25778E-02.44444E-03. 15802E-02.31111E-02.48395E-03.17383E-02.34667E-02.50370E-03.18173E-02.36444E-02

-85-.38519E-03.13432E-02.25778E-02.37926E-02.47407E-02.53728E-02.44444E-03.15802E-02.31111E-02.47407E-02.61728E-02.71605E-02,48395E-03.17383E-02.34667E-02.53728E-02.71605E-02.85333E-02. 50370E-03.l81 73E-02.36444E-02.56889E-02.76543E-02.92444E-02.56889E-02.76543E-02 *92444E-02.10163E-01 EIGtNVALUES OF 14 GIRDERS.20076E-04.49532E-04.10267E-02.70271E-05.16895E-03.10454E-04.29967E-01 UNITARY MATRIX OF 14 GIRDERS.53285E 00.44256E -.34695E-01.37162E -.44970E 00-.36296E.40395E 00-.36206E.12605E 00.36553E -.50513E 00.36366E.28003E 00-.36450E 00.14843E 00.40928E 00.55056E 00.45262E 00.14322E 00-.22944E 00-.47887E 00-.35870E 00-.30080E 00.52723E 00.44303E 00-.55192E 00-.41773E 00-.48727E 00-.24548E 00.11863E 00.46650E 00.32143E 00.24773E 00-.38493E 00.49735E 00-.48054E 00.2?309E 00.80228E-01.45499E 00-. 88201E-01-. 43306E 00-.18479E 00.32560E-01 00.11530E 00 00.22617E 00 00.34433E 00 00.45116E 00 00.53145E 00 00.57441E 00 ALPHA MATRIX.73751E-04.1 7293E-03.*25686E-03.32552E-03.37893E-03.41707E-03.43996E-03.31649E-03.17293E-03.52897E-03 bb466E-03:.11393E-02. 13529E-02. 1 0U55E-02. 1597 1E-U2 1 1509E-02. 25686t-03 * 86466E-03.15793E-02.2 19 73E-02. 267.79E-02.30212E-02.32272E-02.23305E-02.32552E-03.11393E-02.21973E-02.32552E-02.41097E-02.47201E-02. 50863E-02.3o828E-02.37893E-03 13529E-02.26779E-02.41097E-02.54042E-02.63578E-02.69300E-02.50351E-02,41707.E-03.15055E-02.30212E-02.47201E-02.63578E-02.76904E-02.85144E-02.62148E-02.43996E-03.15971E-02.32272E-02.50863E-02.69300E-02.85144E-02.95952E-02.70492E-02.3 1649E-03.11509E-02.23305E-02.36828E-02.50351E-02.62148E-02.70492E-02.52083E-02 EIGENVALULtS OF 15 GIRDERS 10578E-04.T52574E-04.21036E-04.10950E-02.52353E-05.18002E-03.31964E-01.66154E-05 UNITARY MATRIX OF 15 GIRDERS.52710E -.29582E -.11638E.44598kt -.44129E.10o37E.30820E -. 34'809E 00.39946E 00-.49766E 00.12884E 00.18210E 00.43238E 00-.95898E-01.36724E 00-.24492E 00-.23271E 00.49263E 00.52348E 00.30899E 00-.45964L 00-. 19644t 00.48463E 00-.36352t 00.14o21E 00-.3813it 00.24724 724E00.40758E 00.48952E 00.41517E 00-.91454E-01-.43984E 00-.49255E-01.14289E 00-.38288E 00.45943E 00-. 35303t 00-.35168E 00-.35149E 00-.32950E 00-.26545E 00-.53106E 00-.33072E 00.18678E 00.49584E 00.27705E 00-.23587E 00-. 35 354E 00.27908E-01.44038E 00 00.99702E-01-.47909E 00 00.19774E 00.41429E 00 00.30516E 00-.23709E 00 00.40665E 00-.30847E-02 00.48920E 00.24245E 00 00.54289E 00-.41729E 00 00.39704E 00.34026E 00 ALPHA MATkIX.61861E-U4.14567E-03.21751E-03. 27738E-03.32527E-03.361 319E-03.38513E-03.39711E-03.145b7E-03.44699E-03.73435E-03.97361E-03.11654E-02. 13091-02.14048Et-02.14527E-02.2 1751E-03.73435E-03 13470E-02.1 8858E-02.23168E-02.26401E-02.28556E-02.29633E-02.27738E-03.97381E-03 18858E-02.28097E-02.35759E-02.41507E-02.45338E-02.47254E-02.32527E-03.11654E-02.23168E-02.35759E-02.47393E-02.56373E-02.62360E-02.t5353E-02.36119E-03.13091E-02.26401E-02.41507E-02.56373E-02.68965E-02.77585E-02.81896kE-02.3851 3E-03.14048E-02.28556E-02.45338E-02.62360E-02.77585E-02.88980E-02.94846E-02.39711 E-03.14527E-02.29633E-02.47254E-02.65353E-02.81896E-02.94846E-02.10217E-01 EIGENVALUES OF 16 GIRDERS.10856E-04.55t>'3E-04.47049E:-05.11633E-02.22088E-04.19114E-03.64563E-05.33962E-01

-86 - UNiTAKY iMAlAl X lU 16 lG KUR)<S.51192L 00.joOd2 00G.31141c 00.112o3E 00.4618bE 00-.23535E 00-.47860E 00 -. ioood t 00.6t,4UOt OC-.39339F 00.*3013E 00.19933E 00-.O0480E 00.43646E. 00 -214 -.2'1 104b 0-. L 00.,4493,t UU.49303E 00-.46981bE 00-.38922E ()0-.23688E 00.47725E 00-.4b26h3E 00-.457iuL 00.49805E 00-. 11404E-01.61279E-01-.66859E-01 -.2(04tt: 00-.65Y77E-01.41726E 00.32353E 00.47816E 00.44156E 00.34171E 00 -.2420U 0O0.4D97OE 00-.33373E 00.29754E-01-.155991 00.39926E 00-.47029E 00.47961E 00.2L14tk 00.215JO1 00-.2o9ti88 00-.42143h 00-.22744E-01.39753E 00 -.23j001 00-.3/2bbE 00-.74351r-O1-.45b95t JO.30912E 00-.42382E 00-.15457E 00 ALPHA MAIkIX.24134E-01 *86883E-01.17394E 00.27153E 00.36695E 00.44910E 00.50910E 00.54070E 00.S2393c-0q.12384b-03 * l85b1-03.23815-03 *.281021 —Oi. 31431b-03.313811t-03.35246L-03.2 L5260(-03. 1z3641-03.38104L-03.6287t2-03. b3829 -03.100981t-02.11431E-02. 13b4t-02. 2.955Lt-02.92955E-03. 1 i8 76L-03.628 721-03.11574E-02.16289E-02.2014 l1 t02.231431-02.2314dt-U2. 25291E-02.265 73E-02. 19096E-02.23615E-03.b;829E-03.lo289E-02.24387E-02.31245E-02).36580E-02.4J390E-02.426 76E-02. 0716E-02.28102E-03. 10098E-02.20147E-02.31245E-02.416 76E-02.5(001 1-02.55965TE-02.59537E-02.42941E-02.3143oE-03.11431E-02.23148E-02.36580E-02.50011 t-0O.6172 E-02.70302E-02.75446E-02.54561t-02.3381 7E- 0 3.12384E-02.25291E-02.40390E-02.55965E-02.70302E-02.81685E-02.88687E-02.64361E-02.35246E-03.12955E-0?.26578E-02.42676E-02.59537E-02.75446E-02.88687E-02.97546E-02.71131E-02.25260E-03.92955E-03.19096E-02.30716E-02.42941E-02.54561E-02.64361E-02.71131E-02.52083E-02 EIGENVALUES uF 17 bIi)RRS.11234t-04.,)b814t-04.44190E-05.12317E-O0.6455uE-05.20228E-03.23204E-04.35960E-01.36577E-05 UNI TARY MAR7KX CF 17 GIRDERS.46906 (00.32641E 00.39170E 00.99110E-01-.48994E 00-.20962E 00-.42651E 00 -.49781E-01.47900E 00-.44785t 00.29748E 00.35244E 00-.47610E 00-.27797E 00 -.39778t 00.56471E-02.428261 00.46160E 00-.36021E-01-.42682E 00.40680E 00.38941t 00-.45.341E 00-.31418E 00.49661E 00-.30568E 00-.48130E-01.19529E 00.7284oE-o01-23,t51E 00.13369E 00.31694E 00.46228E 00.35837E 00-.44276E 00 -.450281 00.33352E 00.76188E-01.13075E 00-.33952E 00.45420E 00-.11985E 00.30394E 00.40182E 00-.26967E 00-.15619E 00.11562E-01.16075E 00.46323E 00.19600E 00-.12202E 00.40517t 00-.38288E 00.32343E 00-.27053E 00.40415E-01 -.33102E 00-.33311E 00-.32086E 00-.33139E 00-.32754E 00-.33338E 00-.33254E 00.21037E-01.15713E 00.762.42E-01-.21223E 00. 1 5387E 00.26949E 00.24257E 00-.31984E 00.33172E 00.36268E 00.41185E 00-.39693E 00.47509E 00.42180E 00.51550E 00-.43677E 00.37433E 00.31236E 00

APPENDIX C THE G FUNCTION AND SOME FORMULA OF BEAM ON ELASTIC FOUNDATION The G-functions appearing i Gl(ax) = L G2(a,x) = L-1 G3 (a,x) G4(a,x) = I = I G5 (a,x) = G6 (a,x) = G7 (a,x) = L-1 J 4 5 G6 Ls +c - s - [s+c = (a,x) = (a,x) (a,x) = Ln the text are cosh ax sin ax 2a2 sinh ax sin ax 2a2 defined as follows. - sinh ax cos ax 1 2a cosh (sinh ax cos ax + 'cosh ax sin ax) ax cos ax a(sinh ax cos ax - cosh ax sin ax) -2a2 sinh ax sin ax -2a3(sinh ax cos ax + cosh ax sin ax) 4a+ = C = I E2 Since there is no confusion are defined as follows. Gl(a,b,x) = L1 I 4.cmi2. 'i - G2(a,b,x) = L-1 Gj(a,bx) = L1 G4(a,b,x) = L-1 Ls, + S-+CJ s4+ 11s2+c s4+NS2+C = s2+Ns2+c, -87 as in equation (3-8) kS cs involved, the other set of G-functions a cosh ax sin bx - b sinh ax cos bx 2ab(a2+ b2) sinh ax sin bx 2ab ba(b sinh ax cos x + a cosh ax sin bx) a-b2 cosh ax cos bx + sin ax sin bx 2ab

-88 - where 1 - CN b =\/1 c _ N l(b2+ a2)2= V 2 4~ t2(b2-a2) = N b 2 c + 4 as in Equations (3-16) and (3-18). From the above expressions, it is obvious that d d G [Gn(ax G(ax) = dx [Gn-l(ax)] G(ab) [G (abx)] From these relations, we have G x) = 2(a-b2)G3(a,b,x) (a- )G3( - (a2+b2)2Gl(a,bx) G6(a,b,x) = 2(a2-b2)G4(a,b,x) - (a2+b2)2G2(a,b,x) G7(a,bx)= 2(a2-b2)G5(a,b,x) - (a2+b2)2G3(abx) The following formula are taken from Reference (4) with certain changes of the parameters in accordance to the following differential equation. E\Say() + y = q(x) (c-l) a. q(x) = qo, both ends built in. x' = L - x y = q [l- sinh +in L (sinh ax cos ax' + sin ax cosh ax' 0 sinh aL+sin aL + sinh ax' cos ax + sin ax' cosh ax)] aL aL aL aL 2(sinh - cos - + cosh 2 sin -) 2 2 2 2 2 2 sinh aL + sin aLs aL y"(o) = y"(L)= - 2a q sinh aL-sin aL aL aL aL aL L - 2 sin 2 cosh ~ - cos 2' sinh 2 y"(Q) = 4a q (C-2) sinh aL + sin aL

-89 - b. q(x) = P6(x-o), both ends built-in (origin at center) y(o) Pa cosh aL + cos aL - 2 2 sinh aL + sin aL y 2(x) = sin a L + si (sin ax sinh ax' - sinh ax sin ax N 2 sinh aL + sin aL - 2 cos ax cosh ax + cos ax cosh ax' + cos ax' cosh ax) aL aL (L) 4 sinh - sin a) Y (2) = - 4a3P - 2 (C-3) sinh aL + sin aL,,( = 3 cosh aL - cos aL y(o) a sinh aL + sin aL c. q(x) = q - both ends simply supported y(x) = q o (1 cosh as cos ax' + cosh ax' cos ax y(X) ( - cosh aL + cos aL aL aL 2 cosh ~ cos ^) ([) 2 0(C-4) 2 = - cosh aL + cos aL aL aL 2 sinh -=- sin y"() = 4a2qo 2.2 2 cosh aL + cos aL d. q(x) = P6(x-o) both ends simply supported (origin at center of beam) y(x) = Pa (cos ax sinh ax' - cosh ax sin ax 2(cosh aL + cos aL) + sin ax cosh ax' - sinh ax cos ax') Pa sinh aL - sin aL y(o) 2 cosh aL + cos aL y"(o) = a3P sinh aL + sin aL (C-5) cosh aL + cos aL

-90 -e. q(x) = p both ends simply supported, compressive axial load y(x) = p[l - b21 {2ab(cosh bx cosh ax' + cos ax cosh bx') 2ab(cosh bL + cos aL) + (b2-a2)(sinh bx sin ax' + sin ax sinh bx')} y"(x) = - P- [2(b2-a2)(cosh bx cos ax' + cosh bx' cos ax) cosh aL + cos aL b2+a2 + - a (sinh bx sin ax + sinh bx' sin ax)] 2ab r1 1 - a= ~ ES b= 1 1 + V2 DES E 2 E-S E f. q(x) = p, both ends built-in, compressive axial load = P y(x) = p[l - 1i (b cosh bx sin ax' + a sinh bx cos ax' a sinh bL + b sin aL + b sin ax cosh bx' + a cos ax sinh bx')] y (x) = ---- P — [(b3-3a2b)(sin ax cosh bx' a sinh bL + b sin aL + sin ax' cosh ax) + (3b2a-a3)(cos ax sinh bx' + cos ax'sinh ax)] (c-6) The differential equation corresponding to (C-5) and (C-6) is E Su y(4) + \S qy" + y =

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