.1 .. xi ... 1.3.3.1. ‘ . w, w.fi§.».,.... .36.» er... s a :2 L. E ‘ V’V I pf \ . g}: :1? -. 13.3: . . . km? Gk. UK. c. 2.1.. e .94..» v QA‘ A s - 4,0 .1! T V in \I . ‘ 4.». an .1 . a r .42.: ‘ a.“ 15...: 2 .1. . v) man , I (2001/ 5Q ’/ Iii-R 5- 8 This is to certify that the dissertation entitled BUCKLING ARCH UNDER NORMAL PRESSURE presented by JAEGWI GO has been accepted towards fulfillment of the requirements for the Ph.D. degree in Mathematics CWJ—z Major Proféssor’s Signature é/z/ /04/ Date MSU is an Affirmative Action/Equal Opportunity Institution o—h_-O~.-O-.-.-I-.-O-O- ' r AH. r-s‘ - - O C- *- '~- 7‘ *«‘-——* O ‘ “’— ‘O 9 w 9 “r 7"." UBRARY l MiChiQan State University J *— PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/‘DateDuep65-p. 1 5 BUCKLING ARCH UNDER NORMAL PRESSURE By J aegwi Go DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2004 ABSTRACT BUCKLING ARCH UNDER NORMAL PRESSURE By J aegwi Go The behaviors of arches are nonlinear and sensitive to the buckling conditions. This thesis investigates the equilibrium states of a inextensible thin arch with rota- tional resistance at both edges under uniform normal pressure. The results can be applied to the design of pressure-vessels and submarine hatches. The balance of forces of an elemental length is considered to establish the govern- ing nonlinear differential equation. An energy method is taken to derive variational formulation which will be used to prove existence of solution. Bifurcation problems are solved with a perturbation method. Numerical solutions are obtained by applying a shooting method based on the Runge-Kutta. It was found that the behavior of the arch exhibits interesting phenomena such as critical load, non-uniqueness, stability, and snap—through. To my parents and loving wife. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES Introduction 1 Mathematical Formulation 1.1 Introduction . . . .’ ............................. 1.2 Physical assumptions and notations .................... 1.3 Derivation of equation ............................ 1.4 Variational formulations .......................... 2 Existence of Solution 2.1 Introduction .................................. 2.2 Abstract functional preliminaries ...................... 2.3 Existence of solution for nonzero spring constant of buckling arch 3 Analytical Solution 3.1 Introduction .................................. 3.2 Bifurcation theory near trivial solution ................... 3.3 Regular perturbation ............................. 3.4 The stability using perturbation analysis .................. 4 Numerical Results 4.1 Introduction .................................. 4.2 Previous results and extensions to the present problem .......... 4.3 Numerical results ............................... 4.4 Tables of solution ............................... 5 Conclusions and Discussions BIBLIOGRAPHY iv LIST OF TABLES 3.1 Asymptotic solution when T = 0 and T z 00 3.2 Asymptotic solution when 0 < T < 00 4.3.1 Snap-through p. for various T and a 71’ 4.4.1 Solutions when T z co and 01 = 5 7r 4.4.2 Solutions when T 2 co and a = X 4.4.3 Solutions when T = 1000 and a = 7} _7_r_ 4.4.4 Solutions when T = 10 and a = :9- 4.4.5 Solutions when T = 1 and a 2 4.4.6 Solutions when T = 0.5 and o! = 1 4.4.7 Solutions when T = 0 and a 2 4.4.8 Solutions when T = 0 and a 2 4.4.9 Solutions when T = 0 and a 2 4.4.10 Snap-through 1f 4 ml: wlri 3 79 80 81 82 83 84 1.1 1.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 LIST OF FIGURES Normal load uniformly distributed around arch An elemental length The deformation of the arch with small load when T = 0 The deformation of the arch with large load when T = 0 The deformation of the arch with large load when T 2 00 The variation of curvature when T = 0 and a = cal: The variation of curvature when T = 0 and a = wl=t The variation of curvature when T = 0 and a = 23” W The variation of angle when T : co and a 2 Oil The process of deformation in first bifurcation when T 2 00 The process of deformation in second bifurcation when T z oo 4.10 The variations of angle for various T vi 59 59 60 63 65 67 68 70 71 72 Introduction This thesis is concerned with the the nonlinear equations governing the buckled states of elastic, inextensible, and thin circular arches under uniform normal pressure. The study is applicable to the design of many structures in practice, such as dams, under water storages and submarines. First let us define what we meant by the buckling of an arch. An arch is an elastic structural member restrained at the base (Fig 1.1). It could also represent a cross section of a thin shell. A circular arch under normal or hydrostatic pressure may buckle, or deviate from its circular shape, if the pressure is larger than a certain critical pressure. This incipient buckling pressure, since the deformations are infinitesimal, can be obtained analytically, and very difficult to observe experimentally. If the pressure is further increased, large nonlinear deformations of the arch occur. In post buckling, a rich variety of nonlinear phenomena, such as catastrophe, non-uniqueness, and non-existence of solution may occur. We assume the arch is inextensible and the thickness is thin enough such that the Euler-Bernoulli law applies locally, even for large deformations. The buckling of such a ”classical" arch has been considered by Timoshenko, SP. and Gere, J .M. [17], Vlasov, V.Z. [19], and Simitses, G.J. [14]. In general, the elastic equilibrium equations are derived from a small perturbation from the circular arch, and the buck- ling pressure is the first eigenvalue of the linearized system. It was found that the buckling pressure depends on the flexural rigidity of the arch, the opening angle and whether the base is clamped or hinged. The buckling shape tends to be symmetrical for shallow arches and asymmetrical for tall arches. It has been noted by Schreyer, H.L. and Masur, BE [15] that under shallow arch approximation a phenomenon of snap-through happens with sufficient pressure. This phenomenon persists even when pre-buckling displacements, such as axial shortening, are introduced [12]. However, many questions still need to be addressed. In this thesis we shall attempt a compre- hensive study of the buckling and post buckling of the pressurized arch. In early research, Schreyer, H.L. and Masur, E.F. [15] accomplished an exact anal- ysis and obtained analytical solutions for shallow arches with clamped bases. Dickie, J .F. and Broughton, P. [6] performed the buckling of shallow circular pin-ended and fixed arches and used a series method to obtain approximate numerical solutions, while Pi, Y.-L., Bradford, M.A., and Uy, B. [12] investigated the in-plane buckling of circular arches with an arbitrary cross-section under uniformly distributed normal pressure around the arch. They [12] used the energy method to derive analytical solutions for the buckling load and proposed approximations for the symmetric buck- ling of shallow arches. But their study was restricted to hinged and fixed arches. The present research observes the arch under the elastica theory with general bases which are constrained by rotational springs. We investigate, using numerical tools, the nonlinear phenomena and study the susceptibilities of arch for different opening angles and initial base conditions. In chapter 1, section 1.3, the method of equilibrium of elemental segment is used to formulate the governing equation. The balance of normal and shear stresses of the elemental segment leads to a nonlinear ordinary boundary value problem. In section 1.4, the variational formulation is set up using the energy method based on Hamilton’s principal. This principal leads to a minimization of total energy which is used to prove the existence of the solution. In chapter 2, the existence of solution is discussed using the variational formulation found in section 1.4. Previous authors explored the energy principle in Dirichlet and Neumann boundary conditions. In this thesis, we investigate the existence of equilibria using Robin boundary conditions. The work on variational methods for the nonlinear elliptic eigenvalue problem by F.E.Brower [3] serves the proof of the existence of solution. In chapter 3, the regular perturbation method is used to yield exact stability criteria. The critical pressures for various angles are attained with a variety of choices of spring constants. In chapter 4, the shooting method is used to solve boundary value problems based upon the fourth order Runge—Kutta method. To overcome the difficulties in trying to satisfy the multiple two-point boundary conditions, some simplifying techniques are developed. The numerical solutions are calculated for various of values of opening angle and for a variety of spring constants. The sensitivity of curvature in fixed-base buckling and the variations of the angle for different spring constants are found. The results of chapters 1 ~ 4 are discussed and summarized in chapter 5. CHAPTER 1 Mathematical Formulation 1 . 1 Introduction In section 1.2 the physical assumptions to derive mathematical formulation are pro- vided. The non-linear ordinary differential equation is found using the bending theory under the assumptions given in section 1.3. This formulation will be followed in chap- ter 3 to search for bifurcation points and in chapter 4 to determine the exact numerical solutions. Equations based upon Hamilton’s principle for various buckled states are furnished in section 1.4. These forms are derived from an energy minimization prin- ciple and will be used in chapter 2 to prove the existence of equilibrium states for given pressure. 1.2 Physical assumptions and notations Let us consider a circular elastic and inextensible arch, constrained at both ends, under uniform normal pressure. The thickness of the arch is assumed to be thin compared with other length parameters of the problem. By virtue of the assumption of thinness, the arch can be treated as an elastica [9], defined as a material where the local bending moment. being proportional to the local curvature. 4 Let (1:, y) be the coordinates on the cross section, and 6 be the angle between the tangent to the cross section and :r-axis, and s be the normalized arc-length along the cross section of a buckled arch. (see Figure 1.1). The following notations will be used. T = Tension. S = Shear stress. 3’ 2 Arch length. T’ 2 Spring constant. T 2 Normalized spring constant. M 2 Bending moment. q], 2 Normal stress on the surface. (1,, 2 Normalized normal stress on the surface. p 2 Normalized normal pressure. E1 = Flexural rigidity. R = Radius of arch. E = Strain energy. ’ : Potential energy. W 2 Work done. y Figure 1.1: Normal load uniformly distributed around arch T+dT 0+d0 Dd+dNI S+dS Figure 1.2: An elemental length 1.3 Derivation of equation According to the bending theory, the balancing of forces in the normal and tangential directions, and the balance of local moment provides following relations (T + dT) sin(d9) — (S + dS) cos(d0) — qus' + S = 0 (1.1) (T + dT) cos(d6) + (S + (15) sin(d0) — T = 0 (1.2) (M + dM) + (T + dT) sin(d0)ds’ — (S + dS) cos(d6)ds' — q;,ds'c%9 — M = 0, (1.3) where T, S ,and q], are tension, shear, and normal stress on the surface, respectively, 6 is the local angle of inclination (Figure 1.2), and s’ is the arch length. Ignoring of smaller quantities, the above equations are reduced to TdH — dS — qus' = 0 (1.4) dT+SdH =0 (1.5) dill — 5618' = 0. (1.6) Now, assume that the bending moment M is proportional to local curvature (Euler Bernoulli Law of elastica), that is, .6 M =EI£l—, (1.7) ds’ where E] is the flexural rigidity. Then the combination of equations (1.4) ~ (1.7) yields the nonlinear equation 0353309 — 6533633 _ (111635 + 6330;3 : 0: (18) .I ' n3 . . where s = 5—H and qn = 9’;—.,— are the normallzed arc length and stress, respectively. 7 Note that (2:, y), cartesian coordinates normalized by R, are related to 9 by d1? d?! _ ~ E — cos 0(5), 3; —— sm 0(3). (1-9) At the bases with torsional spring constants, the total angle change is resisted by an additional moment, which provides 1 T'(9(—a) + a) — EI[03I(—oz) — R] = 0 (1.10a) 1 T'(0(a) — a) + EI[03r(a) — R] = 0. (1.10b) Thus the normalized boundary conditions are (0(0) + a) — T(03(0) — 20) = 0 (1.11a) (9(1) — a) + T(63(1) — 20) = 0, (1.11b) where T = EI/T’R. Here the T = 0 and T = 00 imply fixed and hinged base buckling, respectively. After dividing the equation (1.8) by 03 and integrating, the above equation is reduced to 1 3 0333 _6168+ 56:9 —p:0, (1.12) where cl is an integral constant. Moreover, multiplying the equation (1.12) by 0,3 and integrating, for some constant 02, yield . 1 933 — 616.? + g6: — [263 'l' Cg :- 0. (1.13) Let w = 0, then the above equation is . l wf—c1w2+Zw4 —pw+(:2:0, (1.14) and so . 1 (.03 :i\/clw2—Zw4+pw—cg. (1.15) 8 But it is impossible to solve the integrand 1 /ds=:l:/ dw \/c1w2 — 1W4 +pw—02 (1.16) analytically, thus numerical tools will be used to obtain the solution of equation (1.12) in chapter 4. 1.4 Variational formulations I. Variation formation in fixed base buckling We consider the equilibria of an arch including rotational resistance at bases under normal hydrostatic pressure p. Let :r(s) and y(s) denote the coordinates of the point 3 on the cross section, and 9(3) the angle between the tangent to the cross section and x-axis, where s is arc-length along the cross section of a buckled arch. Then 113(3) : /8 cos 6(s)ds y(3) = /_8 sin 0(s)ds. and work done W by W’ : —p / (:ryS—yx..)ds—2cr] ..‘—C! = —p /a (1" sin 0(5) — ycosd(s))ds — 2a] . b —a (1.17a) (1.17b) (1.18) (1.19) The work done is considered as the difference in area enclosed by the arch in its deformed and un-deformed states. The dimensionless potential energy V then is V = E — W 2: /-a (1 —- 93(3))2ds + p [[0 (.1: sin 9(3) — ycos9(s))ds — 2a] 0 —O = f [(1— 93(3))2 +p/3 sin(9(s) — 0(0)ng ds+constant. (1.20) C! —a The spring buckled arch gives the boundary conditions 9(—a) = —a (1.21a) 9(0) 2 a, (1.21b) and satisfies the constants base positions /a sin 9(s)ds = 0 (1.22a) a / cos 9(s)ds = 25in a. (1.22b) 0 ~, The transformation 9(9) 2 9(3) - s and dropping the tilde provide the following V = /-0 [93(3) + p/j sin((9(s) — 9(5)) + (8 — f))d£ (is + constant, (1.23) O O and boundary conditions 9(—a) 2 9(0) 2 0 (1.24) with constraints /a sin(9(s) + s)ds = 0 (1.25a) [0 cos(9(s) + s)ds : 2sin a. (1.25b) II. Variational formulation in buckling with non-zero spring constant 10 Let p+ 2 0, p- Z 0, and p E R be constants. For a fixed angle a > 0, let 9: [—a,a] —> R, (1.26) satisfying [a sin 9(s)ds = 0 (1.27a) [0 cos 9(s)ds = 25in a. (1.27b) The potential energy then is defined V(9) = p—(9(—a) + a)? + 1049(0) - 01)2 + [fl[(1-03(8))2+p/_sin(9(s)—9(€))d€ ds. (1.28) a a Note that the first and second terms in equation (1.28) imply the resistance energy to keep un-buckling arch at the both ends. The transform 2(3) = 9(3) — 3 yields 1:31.443) + s)ds = 0 (1.29a) [1. cos(z(s) + 8)ds = 2sin a, (1.2%) and V(z) = p_z(—a)2+p.z(a)2 + f_: [as-)2 + p (.3. sinl(z(s) — z(€) + (s — oldt ds. (1.30) The variational formulations found will be used to show the existence of equilibrium states in chapter 2. 11 CHAPTER 2 Existence of Solution 2. 1 Introduction Chapter 2 is concerned the proofs of the existence of equilibrium states for the variety of spring constants and the upper bound of deflection in hinged case for the equations formulated chapter 1. The variational formulation is used to prove the existence of energy minimization for given pressure and angle. The proofs are based upon a theorem published by F. Browder [3]. In section 2.2 some preliminary functional tools are stated. In section 2.3, the existence of weak solution to the variational formulation is showed using the Browder’s theory outlined in section 2.2. 2.2 Abstract functional preliminaries Definition 2.2.1 A complete normed space is called a Banach space B. Definition 2.2.2 By a Hilbert space H we mean a Banach space B in which there is defined a function (1:, y) on B x B to R with the following properties; Z) (011171+ 02132.31): 01(1171» y) + 02(13‘2. y) 12 ii) ($.31) = (v.13) iii) (13$) = llflvll2 Definition 2.2.3 A sequence {u,,} in a Banach space B is said to converge weakly to u E B if the sequence {f‘(un)} converge to f*(u) for every f‘ E B‘, the dual space of B. We denote this weak convergence by un —A u. Definition 2.2.4 A subset S of a Banach space B is said to be weakly closed if unES andun—\u, thenuES. Definition 2.2.5 Let BI and B2 be Banach spaces. Then B1 is said to be imbedded in B2 if i) every element of B1 is in B2 ii)every convergent sequence in B1 is convergent in B2 The imbedding of B1 in B2 is compact if the imbedding operator I : Bl —> 82 defined by I(:I:) : :1: is compact. Definition 2.2.6 ([3]) For a real Banach space B, a function \11 : B x B —) R is said to be semi-convex if i) For each u E B and each c E R, the subset Sm, defined by SW, = {u E B[\Il(u,v) S c} is convex. ii) For every bounded subset S of B and each weakly convergent sequence {vn} to v in B, \Il(u, v,,) —+ \I/(u,v) uniformly for u in S iii) For fixed v E B, \I/(', v) : B ——> R is continuous. 13 Definition 2.2.7 ([3]) Let H be a Banach space. A function f : H —> R is said to be differentiable at u in H if there exists an element 9 E H“ such that for all h E H f(u+ h) - f(U) - 901): 0(llhlli as llhll —> 0- Definition 2.2.8 Let H be a Banach space. A function \I' : H X H —~> R is difierentiable at (u,w) if there exists a pair 91, 92 in H“ such that for all h1, h2 in H, W“ + how + ’12)‘ @(an) — 91(h1) — 92012) = 0(llhlll + W12”) as llhlll —> 0 and llhgll —> 0. Theorem 2.2.9 ([22]) If In —> a: in a Hilbert space H which is compactly imbedded in a Banach space B, then 512,, —> :1: in B. Theorem 2.2.10 ([3]) Let B be a real reflexive Banach space, \11 : B x B —> R be semi-convex and E(u) = \I/(u, u). Let A be a weakly closed bounded subset of B. Then E is bounded below on A and assumes its minimum on A. Theorem 2.2.11 Let B be a real reflexive Banach space, \IJ : B x B —> R be semi- convea: and E(u) = \Il(u,u). Let f,, i : 1,2,...,N, be a weakly continuous real functions on B. Let C E {lllf1'('ll) : C157; :— 1,2, ..., N} for c = (c1,c2,...,cN) in RN. Suppose further that E(u) —> 00 as llull —+ 00 on C, then E assumes its minimum on C. 14 PI'OOf. LEI. Ci 2 {Ulfi(U) 2 6,}. Then the SOL A = C1 0020 HUN 0 {UlllUll S R}, for all R > 0, is weakly closed and bounded. E] Theorem 2.2.12 ([22/,p279) For u E X, let E : U(u) g X —+ R1 and f,- : U(u) g X —+ R1, i = 1,2, ..., n, be C1 on an open neighborhood U(u) of u, where X is a real Banach space. Supposed that, for each ('w1,w2,...,wn) E R", the system f’(u)h=w,- i=1,2,...,n 1 has a solution h E X and no is a minimizer of E(u) subject to fi(u0) = 0, i = 1, 2, ..., n. Then there exist constants A1, A2, ..., An such that mu.) 2 23;, A,f,-’(u0). 2.3 Existence of solution for nonzero spring con- stant of buckling arch The expression of potential energy derived in 1.4 is 13(9) = [MN-(1)2+10+9(0')2 + [.0 l08(8)2 + p s sin[(9(s) — 9(5) + (s — €)ld£ ds. (2.1) O “'0 We assume here that 0 _<_ p_, 0 S p+, and p_ + p+ > 0. Let us fix pressure p and angle or. We want to minimize the E subject to the constraints / sin(9(s) + s)ds = 0 (2.2a) / cos(9(s) + s)ds = 2sina. (2.2b) 15 We will use the real Sobolev space Wl’2(—a,a) = {u : u, u' E L2(—oz,oz)}, (2.3) with the inner product < u,v >= /a(uv + u'v')ds, (2.4) and the norm lulu; = l f a (u2 + u'2)dsl 5. (2.5) Let H E IV1’2(—oz,a) and define f1(u) = [a cos(u(s) + s)ds (2.6) f2(u) = [0 sin(u(s) + s)ds, (2.7) for u E H. Define j and \II by its) = f: /:sinl(v(s) — 22(5)) + (s — oldtds (2.8) was. s) = p—U(-a)2 + mum)? + /_0 u'ds + mo), (29) for u, v E H, and note E(u) = \Il(u,u). (2.10) Lemma 2.3.1 There exists a constant e such that ll‘ulloo S Clulla for every u E H. 16 Proof. Since u(xo) 2 21(3) — f; u' for any 170 E [—a,a], lu(x0)l S lu(s)l + f: lu’l. The integration and Holder inequality yield 2s)u(s.)) s [0 lu(s)l + 2s [0 )u') -'0 —O < Wallullu + osmium... g V803 +2alulm. (2.11) The last inequality is from the fact [91A + flgBl 3 \/9f + [fit/A2 + 82. D Lemma 2.3.2 The set S = {u E Hlf1(u) = 2sin a,f2(u) = 0} is nonempty and weakly closed in H. Proof. The S is not empty since it contains 11(5) 2 —2s. Let un be a sequence in S such that un ——\ u in H. Then, by the imbedding theorem , un converges to u in L2-norm. On the other hand, |f1('un) - f1(U)| = ff [cos(un(s) + s) — cos(u(s) + s)]ds 0‘ 1 l S 2/ [sin §(un(s) — v(s)) sin —2-(u,,(s) + u(s) + 2s)lds -O a s] [an — ulds —a S cllun — all); —> 0, (2.12) where c is a constant. Thus f1(u) = [:1 cos(u(s) +s)ds = 2 sin (1. Similarly, the proof of f2(u) : f: sin(u(s) + s)ds : 0 can be completed. Hence S is weakly closed. El Lemma 2.3.3 The function \I/(,) is semi-convex function on H X H. Proof. Pick v E H and c E R. Let us take ul and u? in Sm, : {u E Hl‘II(u,v) g c}. For t E [0,1], define g(t) = \Il(tul + (1 —t)u2,v) — t\II(u1,v) —— (1 —t)\IJ(’112,2,?). (2.13) 17 Then g(t) is a parabola with g(0) = 9(1) 2 0, and the second derivative with respect totis 9"(t) = 2p-(‘Ud-Oz) - ltd-0))? + 2p+(ui(a) - U2(a))2 Thus, g(t) g 0 for 0 g t g 1 and \Il(tu1+(1—t)u2,v) g t\11(u1,v)+(1 —t)\Il(u2,v) S c. Therefore tul + (1 — t)u2 E SW, which implies Sm, is convex. Now, assume that vn is a sequence such that converges weakly to v in H. Again, the imbedding implies the convergence of vn to v in L2-norm. Furthermore, for every u E H [‘1’(Usvn) — ‘1’(’usv)| S lpl/_ [— lsin[(v,,(s) — vn(£)) + (s - {ll — sin[(v(s) — v(£)) + (s — {)lldéds < c. /_ l|(vn(s) — s 00 on H. Proof. Note that a E(u) Z p_u2(—a) + p+u2(oz) +/ u'2ds — 2lpla2. —a If f: u’2ds —> 00 then E (u) ——) 00 clearly. If on the other hand ffa u’gds remain bounded then E (u) ——> 00, because lu(ia)l —+ 00 by the following argument, using u(s) = u(-—a) + f; u’, /_:u2 . /_:u<—a Iii/LN: 11:) 2au2(—a )+4\/_3lu(—a )ll/_:lu'%l2l +40. [:lul2. |/\ Lemma 2.3.5 The E : H —> R1 and f, : H —> R1, i = 1,2, are C1 on H. Moreover, E'(u)h = 2p_ u( —)—h( a)+2p+u)(:)h(a)+2/a u'h' '—O + p/i /_ cos[u(s) — u(£ (s — {)l(h(s) — h(§))d€ds (2.14) f[(u )h =/_0 cos(u+ s)h(s)ds (2,15) f2(u )h— - [:1 sin(u + s)h(s)ds, (2.16) a foralluEHandhEH. Proof. Using the Lemma 2.3.1, 0 |p_(u(—a) + h(_..))2 + p+(u(a) + h(a))2 + f (u' + w —O O —lp_u(—a)2 + p+U(a)2 + f u") -0 (I —[2p_u(—a)h(—a) + 2p+u(a)h(oz) + 2/ u'h']l *0 O : p_lh(-a)|2 + p+lh(a)|2 + / h” s lhlis [c‘2(p_ + p+)+1] - 19 Moreover, for some constant c1, | [1 [8 sinl(u(s) + h(s)) — (u(s) + he» + (s — o) _/_:/-:sinlu(S)—U(€) +( 8- ))]—/_:/_:cos[u(s —u(§ )+(3—€)l(h(3l_h(§))l gl:[:).03(h(.)—s( ))—1)+/_:/_:)sin(h(s) )—h(€ )—) (h(s)—he») slow—w S Cllhlia- Thus, using (2.14), [E(u + h) - 1500- 5'10 W < Clhl1, 21 where c is a constant. Hence E (u) is differentiable in H and the derivative is given by (2.14). To Show that E is Cl, pick u, ul, and h in H and note a % a % )/_:(u '—u'))h' s l/ lua—url l/ W] S l'u1—Ul1,2lhl1,2s and [[0 ff {coslu1(s) — u1(£) + (s - {ll — coslu(s) — u({) + (s — €)]} (h(s) — h(€))d§dsl s 2 f .. [1 )sin§[(u1(s) — u(s))— (111(5) ))(slllh) senses S Cll’ui — Ul1,2lhl1,21 where Cl is a constant. Hence, using the Lemma 2.3.1, for some constants k1, kg, and 20 k3. IE'WIW — E’(u)hl S P—lu1(—a) _. u(—O)llh(—a)l + P+|’U1(a) - “(alllhmll + kllul — Ullalhlls S p—kzl’ul — ul'l,2lhll,2 + P+k3lU1— Uli,2lhl1,2 + kilui - Ul1,2lhl1,2 S (k1 + P—k'z + p+k3)lu1 — Ul1,2lhl1,21 and so u —> E’ (u) is continuous. Similar argument can be applied to show the C1 differentiability of the functions f,, i = 1, 2. [3 Theorem 2.3.6 There exists a minimizer uo E H of E(U) = p—U(— a)+p+ 11( + /: u’2ds+p[_:/:Sinl(lt8( ())+ (s—t))d€ds with side conditions a / cos(u(s) + s)ds = 2sina. Moreover, for some (n.1, M) in R2, Elli/0) =M1f[(u(1) + HzfflUOl (2.17) (2.18a) (2.18b) Proof. There exists a closed ball B that contains an element ul in S and, using the Lemma 2.3.4, has a large enough radius R such that E (v.1) < E (u) for any u not in B. Lemma 2.3.2 implies that the set A E B r) S is weakly closed and bounded. It follows from the Theorem 2.2.10 that E is bounded below and assumes its minimum at some no in A. If u E S\B then E(uo) 5 E(ul) < E(u) and therefore the minimum of E on S is attained at no. 21 Now, it is needed to show that, for each (U)1,'w2) E R2, there exist h E H such that f.:’(Uo)h = 10., i=1.2, (2.19) to apply to the surjective implicit function theorem to Lagrange Multiplier Rule 2.2.12. Let h(s) 2 c1 cos(u0(s) + s) + c; sin(uo(s) + s), where c1 and c2 are constants. The (2.19) implies that we need to find c1 and c2 such that —c1/ sin(u0(s) + s) cos(u0(s) + s)ds — c2] sin(u0(s) + s) sin(u0(s) + s)ds = 1.121 c1 / cos(uo(s) + s) cos(u0(s) + s)ds + c2/ sin(u0(s) + s) cos(uo(s) + s)ds = 102. This can be done if the determinant 0 a a 2 det = /_ sin2(uo(s) + s)/ cos2(u0(s) + s) — l]; sin(uo(s) + s) cos(u0(s) + 3)] O —O O z a2 — :1; (l[:cos2(u0(s) + 5)]2 + l/_:si112(u0(s) + s)l2) is nonzero. Holder inequality yields 1 a a 2 det Z (12 — 4 [20/ cos2 2(uo(s) + s) + 2a] sin2 2(u0(s) + s) 2 0, (2.20) L —a —a however, the equality in above inequality holds when sin 2(uo(s) + s) = constant and cos 2(uo(s) + s) = constant. From the side condition f: sin(u.0(s) + s)ds = 0, 110(3) + s 2 mr, n = 0,i1,:l:2, which contradicts the constraint f: cos(u0(s) + s)ds : 2sina. Thus u0(s) + 3 cannot be constant and det > 0. Hence, c,, i = 1,2 will be determined uniquely and Theorem 2.2.12 can be applied. El Theorem 2.3.7 If uo E W’l’2(—a,a) 0 S satisfies E'(u0)h : )i1f[(uo)h + [12f.§(u0)h for all h E I'Vl‘Q(—o',a), (2.21) 22 for some constants H1 and ug, then no E C2l—a, a], and ul)’ — p/S coslu0(s) — u0(£) + (s — {)]df + psinacos(u0(s) + s) a __ ill . , It? — 7 sm(uo(s) + s) — 2 cos(u0(s) + s) (2.22) u[)(—a) — ,o_u0(-a) = 0 (2.23a) u[,(a) + p+u0(a) = . (2.23b) Proof. Let us bring back the equation E'(u0)h = 2p_u0(—a)h(—a) + 2p+u0(a)h(a) + 2/ u.[)h'ds —O A + p/f h(s) f: coslu0(s) - 110(5) + (s - {lldéds — p _a j: 12(5) casinos) — soc) + (s - Oldéds. (2.24) The change of the order of integration of the last term in the equation (2.24) on the right hand side and the constraints furnishes /_a /_s h“) C03(’u0(3) — “(0(5) + (8 — €))d£ds [a _h.(s) l/j COS(UO(8) — no“) + (8 — £))d{ — 2sinacos(u0(s) + 3)l d3. 0 Thus the equation (2.24) is changed to O E’(u0)h 2 2p_u0(-a)h(—a) + 2p+u0(a)h(a) + 2/ ugh' —O O Q +/_" [1.(8l2pl/j Cos[(u0(s) — u0({) + (3 — {)ldé — Sin (1 COS(u0(s) + 8)l ds. Now, define k(s) : 2p /8 coslu0(s) — 210(5) + (3 — {)ldé — 2psinacos(u0(s) + s) O + )1.) sin(u0(s) + s) — [1.2 cos(u0(s) + s). 23 Then the above equation (2.21) is reduced to 2p_uO(—a)h(——a) + 2p+u0(a)h(a) + 2/0 u[,(s)h'(s)ds = — [0 k(s)h(s)ds, -O -O for all h E I'i’01‘2(—oz, 02). Since k(s) is continuous, regularity [10, p65] implies that ill, is absolutely continuous, and thus an E C 2l—a, (1]. Now, integration by parts yields 21pu0(— s)-ui(— s))h(— a)+2(p.uo(a)+ui(a))h(a) (2.25) +/: [— 211"( (3).)i1(s)d 0—. [10, p65] also implies that 2113 = k(s), hence u[)(—a) — p_uo(—Ct) = 0 (2.26a) u[)(cr) + p+u0(a) = 0. (2.26b) 24 CHAPTER 3 Analytical Solution 3. 1 Introduction This chapter considers the approximate solution of the boundary value problem just after buckling. In section 3.2 the necessary conditions to be bifurcation points and some basic bifurcation theory in the neighborhood of the trivial solution are presented. In section 3.3 the regular perturbation method is introduced, and which is used to obtain bifurcation points for a variety of spring constants T and angles 0 presented in section 3.4. The bifurcation points found will be used to complete the solution curves shown in chapter 4. 3.2 Bifurcation theory near trivial solution Assume that the equation F(X, p) = 0. (3.1) has the trivial solution X = 0 for all pressure p in an open neighborhood of po E R. If the Frechet derivative FX (0, p0) is invertible then the implicit function theorem implies uniqueness of the trivial solution [22, p310] and when it is singular biffurcations 25 usually occur. More precisely, let F:U(0,p0)CHxR1—+K be a C2-map an open neighborhood U(O, p0) of the point (0, p0), where H and K are real Banach spaces. The linearized operator F X(0,p0) : H ——) K is assumed to be Fredholm and such that 1. N(FX(0,p0)) = span {v} 2. MHz-(ass) = span (12*) 3. < v", Fxp(0,p0)v ># 0 (bifurcation condition). In [22, p311] it is shown that under the above assumptions (0, p0) a bifurcation point of the equation (3.1). Let us apply this to the boundary value problem derived in section 1.3 6333.965 _ 6553933 + p653 + 65363 Z 0 x3 = cos 9(3) y, = sin 9(3) (9(0) + a) — T(93(0) —— 20) = 0 (0(1) — a) + 1(6.(1) — 2(1) = 0 The transformation 9(3) 2 9(1 — s), and dropping the tilde give 653.9563 _ 0888688 _ P633 + 0930: Z 0 x, = cos 9(s) y, = sin 9(3), 26 (3.4) (3.5a) (3.5b) (3.5c) (3.5a) (3.6a) (3.61)) (3.6c) and boundary conditions (9(0) — a) - 7(0,(()) + 20) = 0 (3.7a) (9(1) +a) +T(9,(1) +20) = 0 (3.7b) 12(0) = y(0) = 21(1) = 0 (3-7C) 33(1) = Sig“. (3.7a) Let x1:9+2as—a 1'2 : 63 + 2C! 373 = 633 4 J74 : 6335 x5 : x(s) -— ilsin a — sin(a — 20.9)] Lx6 : y(s) + ilcosa — cos(a — 2038)] Then the above boundary value problem can be converted into the following form F[X,pl E LX - f(X, p) = 0 (3.8) Ble E BlX(0) + BgX(1) = 0, (3.9) where X : (.131,I2,.’L‘3,I4,175,;176)T (3.10) L[X] = —d—X (3.11) ds 27 ( ) 173 334 f(‘¥7p) = (133.734 +p$3 -' $3011.? — 20):; x2 - 2a cos 9 — cos(a — 2as) \ sin 9 — sin(oz — 2as) ) 00) 00 000000 000010 812 000001 000000 (000000) (000000) 1T0000 000000 32: 000000 [000010 0 Let us define the domain Banach space to be H = {X E (C‘l0,1])6 : Ble = 0}, where B is the boundary operator defined by (3.9). Let the range space be K = {X e (0011)“). 28 Note that the equation (3.8) has the trivial solution X = 0 for all p and 0) 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 FX(07PO) : L _- 0 0 —cy2 0 0 0 —sin(a —- 2023) O 0 0 0 0 (cos(a—2as) 0 0 0 0 0) where '7 = Log—2‘3. Hence FX(0,p0)v = 0 implies [01,3 = 1’2 142,3 : ’U3 U3,s = U4 < ”4.3 ‘~—‘ ‘72'03 v5” 2 — sin(a — 2as)vl [um = cos(a -— 2as)v1, 29 which gives f 1 A2 . v1: —? cos 73 -— T s1n'ys — A38 + .44 I ”U2 = —lSin78 — 2coscys-l-A3 7 7 ”U3 2 A1 C08 ”)8 + A2 sin 73 ’04 = “AWSiIl ’78 + A27 cos 73 sin(a — 2as) sin ’78 20: cos(a - 2023) cos vs 1152/11] l 7(72 - 40?) 72(72 _ 402) + 4 [ sin(a — 2as) cos vs _ 2a cos(a -— 2as) sin 73] 1 2 7(72 — 4012) 72(72 -— 4a?) 3 cos(a — 201.9) sin(oz —- 20.9) cos(a - 205) A — _ _ A A + 3[ 20 402 l 4 20 + 5 A [ cos(a — 20:3) sin 73 2a sin(a — 2as) cos 7.3] v : — _. 6 1 7(72 — 402) 72(72 — 4022) +A [cos(a — 2as) cos 'ys _ 2a sin(a — 2023) sin 73] 2 7(72 - 4042) 72(72 — 402) ' -2.. cos —2,... s' —-2. +AB[_ssm(a as) + (a 2 (29)] _ A4 1n(a as) + AS. ‘ 2a 401 2a 30 The boundary conditions will be satisfied if f 1 ——2A1 + 1.42 — 743 + A4 : 0 7 1’ cos' Tsin sin Tcos A1[— 2” + 7]+A.2[—- ,7 - 7 7]+(1+r).43+.44 =0 "1' “/ 2acosa sina sina cosa A—, 1 +A”‘ +A-. +A— +A=0 1[ iJOW-4aifi 2[ ikfi-4atfi 3[ 402] d 20 l 5 Qasina cosa cosa sina A—., .+As , .+A—+A— +A=0 1[ 73(72—402)] 2[7(72—4022)] 3[ 402] 4[ 2a 1 6 < o a Al sinasiny 2(1cosoicosy1+ [smacosy 2acosasm7] 1 702—4042) 7202-4042) 2 “/(7'2-40/2) 72(72-402) cosa sina cosa A -— -A A :0 + 3[ 20: 402] 4 20 + 5 —cososiny 2asinacos'y cosozcosy 2asinasin7 All" ..2 2 2 2 2l+A2[, 2 2+ 2 2_42l “TU-4a) 7(7 -40) 7(7 --40) 7(7 04) sinoz cosoz sina A . +A A =0. +3[2a+4oi2] 42a+6 Numerical investigations show that this system has one dimensional null set at infinitely many isolated points p0 and it is known [10, p41] that the null space of the adjoint problem has the same dimension. To make sure that bifurcation happens at those points it is enough to verify the bifurcation condition 3. This will be done just for one case bellow, but it appears that the bifurcation condition is always satisfied, except possibly in some special cases. The adjoint F;»(0,p0) of FX(0,p0) is given by [10, p40] Fi'(01P0)Z = —L2 — fi—‘(0.po)z = o. with boundary conditions PZ®)+QZO)=0 where 31 00 000000 001000 19: 000100 000000 [000000) (000000) 7—10000 000000 Q: 000000 001000 (000100) If F;,(0,p0)0* = 0 then r1133 : sin(0‘ — 20.3)0g — cos(a — 2as)vg :- v: < 0313 — —'U§ + 72v: v4.5 : —’U§ v33 2 0 ‘05,, = 0 32 hence 4022 k1 20(72 — 40?) k1 4a2(72 — 4&2) \ cos(a - 208) + k2 2a(y2 — 4a2) k +—:— + k5'y sin 75 — kw cos 73 ”Y sin(a — 2as) — kg 72 k1 162 k2 , k 5: cos(a — 2013) + 2—3— sin(a —— 203) + k3 k. sin(a — 2as) — 402—22- cos(a — 205) — kgs + k4 sin(a — 2023) cos(a — 2as) 4022(72 — 4a?) k —s-l- 4:,- +k5cosys+kfisinys 7 l _ In order that the boundary conditions of the adjoint problem are satisfied k,,i = 1, 2, ..., 6 have to satisfy 2aT cos a + sin a 2aT sin a —- cos a r 191] k1] l+k2[ l 4022 20’)” cos a + sin a + kg] 402 —2aT sin a + cos n- ]+(1+T)k3 4&2 cos a sin a 402 k1[2a('y2 — 402)] sin 02 2[20(72 — 4a?) COS CY 1[402(72 — 402)] + k2[ 402(72 — 4a?) sin a [ cosoz ] +11 1 20472 — 4a?) — sina 2i 20(72 - 402) cos a .k1[402(72 — 402)] + k. A4020)? — 402)] 1 l+Tl€I3+k4=0 —l€4=0 l l + krsl¢l + heal—“(l = 0 1 l + Isl?) + ksl’i sin ‘7’] + kel-icoml == 0 l I . + [ml—'77] + k4l¥l + k5lcos y] + kglsm 7'] = 0. This system has nontrivial solution iff the system for A1, ...,A6 has nontrivial solution [10, p41]. 33 Using Mathematica we will now verify bifurcation condition for angle a = g and spring constant T = 0. FX(0, p0) is non-invertible if "y = 9.42478, in which case we can choose rAl = 1, k1 = 1 A2 = ——3312371257, k2 2 165.9995 A3 = 2.3116318 x10‘”, k3 = —-—23 i 7r A4 = 0.0112579, k4 = —;1; A5 = —4451197.619, k5 = —0.000142586 1A6 2 0.0015886, k6 = 0.0707641. The bifurcation coefficient is then < 11*, FXP(0,p0)v >= —1.29416 x 107. Therefore, (0, p0) with po : 248.05021 is a bifurcation point in this case. 3.3 Regular perturbation We shall briefly make mention of regular perturbation. Assume that the equation ~ F(u) = i (3.12) is unstable or solvable but very tedious. Moreover, the linear problem ~ L(u) = ’9 (3.13) is solvable and the solution is it : T(v). (3.14) If E(u) is sufficiently close to L(u) then 0 can be written by n. v = L(u) + [E(u) — L(u)] = L(u) +1\~l(u), (3.15) 34 where Al (u) is small. The normalization of each term yields the following equation L(u) + 6N(u) = v, (3.16) where e is a small parameter. Now suppose that u can be asymptotically expanded in powers of e, that is, u:u0+cu1+62u2+---, (3.17) then L(u) can be expanded in the power of 6 L(u) = L(uo + eu1+ 6211.2 + - - ) = L(uo) + cL(u1) + €2L(U2) + - - -. (3.18) Again the assumption that N(u) is analytic in u provides N(u) = N(uo + 6u1+ c2u2 + - - ) = N(uo) + 6N1(u0,u1) + €2N2(u0,u1,u2) + (3.19) With the substitution equations (3.17),(3.18) and (3.19) into equation (3.16) and comparison like powers of c, an infinite linear system of equations is obtained f L(UO) : ’U L(Ul) Z —N(ll0) L(Uz) : —]V1(U0, U1) [ll/(uk-l-l) : —N(U(),U1,' ' 'auk)' 35 Since these above equations are linear, we can solve successively f u0 = T(v) u1 = Tl-N(uO)l = Tl-N(T(v))l and u = T(v) + eTl—N(T(v))] + 0(8). (3.20) The symbol 0 is explained well in [40]. 3.4 The stability using perturbation analysis Note that the trivial solution is 90(3) 2 0 — 203, x0(s) = 2—15[sin0 — sin(0 — 203)], y0(s) = 515[COS(0 —— 203) — cos 0]. We perturb the governing equations about the trivial solution. 0333363 — 6333033 — p633 + 03303 Z 0 i = cos 9(3) d3 fil—Z = sin 9(3) The boundary conditions are i) Fixed case(T = 0) 9(0) 2 0 9(1) 2 —0 x(0) = 0 x(1) = we y(0) = 0 y(1) :2 0 (I 36 ii) Hinged case(T = 00) 0,(0) = —20 0,(1) = —20 0(0) 2 0 0(1) = 5—2 y(0) = 0 y(1) = 0 iii)Mixed case (1- ¢ 00,0) (0(0) - 0) — 7(0,(0) + 20) = 0 (0(1) + 0) + T(0,(1) + 20) = 0 01(0) 2 0 11(1) : .-_.._ y(0) = 0 11(1) = 0. The variables are expanded as follows 0(3) 2 60(3) + 561 + 5202 + - -- (3.21) x = 11:0 + ex1+ 521:2 + . -- (3.22) y = 00 + 601+ 5202 + - -- (3.23) 0 =00 +€p1+€202 + - - -. (3.24) The successive differentiation of equation (3.21) gives 0, = —20 + 501,, + 5202,, + - - - (3.25) 9,, 2 591,, + 5262,35 + - - - (3.26) 0,,, = 501,,” + 5202,,” + - - . (3.27) 0,,,, = 501,,” + 5202,”, + - - -, (3.28) and the differentiations of equations (3.22) and (3.23) x, 2 1:0,, + 51:1,, + 520:1, + - - - (3.29) y. = 00,. + 501.. + 62:02,. + - - -- (3-30) 37 However, x, 2 cos9 2 cos(0 — 203) — e91sin(0 — 203) 2 ~52l92 sin(0 —— 203) + Qzl-cosm —— 203)] + - - - (3.31) y, 2 sin 9 2 sin(0 — 203) + 591 cos(0 — 203) 92 +52l92 cos(0 — 203) — —2-1 sin(0 — 203)] + - - -. (3.32) The comparison of like power of above equations (3.25)~ (3.32) yields consecutive equations [)0 + 803 6 3333 — 6 33 Z 3. 1,..+( 20 )1, 0 (33) x1“, 2 ~91 sin(0 — 203) (3.34) 91,5 2 91cos(0 — 203) (3.35) and + 803 1 . 62,3333 + (9270—)92,” : 2—0/[6L301’5838 — 01,3361,sss _ p101,” + 12020136139] (3-36) 62 0:2,, 2 —92 sin(0 — 203) — —21—cos(0 — 203) (3.37) 0‘2 92,5 = 92 cos(0 — 203) - -2-1- sin(0 - 203). (3.38) The boundary conditions also are obtained for i 2 1, 2 i)Fixed case(T 2 0) ii) Hinged case(T 2 oo) 6),,(0) =0 9,,(1) =0 23(0) =0 1:,(1) =0 314(0) =0 iii)Mixed case (7' 2 00,0) 6,-(0) —76,,s(0) 20 0,(1)+70,,S(1) 20 171(0) =0 4734(1) =0 314(0) =0 314(1) =0- Let k 2 (/ 5% then the general solution of 01 is 61 2 01 cos k3 + 02 sin ks + 033 + a4 and the first derivative 9m 2 -—alksin k3 + 02k cos ks + 03. Using the new boundary conditions, we have 61(0) 2 (11+ 04 61(1) 2 alcosk+agsink+ag+a4 91’3“» 2 (12k + (13 61,3(1) 2 —a.1k sin k. + azk cos k + 03, where (1,»,2' 2 1, 2, 3, 4, are integral constants. l\loreover, 1 131(1) 2 / —013in(0—203)ds 0 (3.39) (3.40) (3.41) (3.42) (3.43) (3.44) (11 . . = —— ksmksma + 20coskcos0 — 20cos0 k2 _ 402 l l (12 . . . + ——,— —-kcosksm0 — k51n0 + 20smkcos0 k? — 402 [ 1 cos 0 sin 0 3 i 20 202 i ( ) 39 1 y1(1) = [01COS(a-208)d8 0 2 Z‘E—j—ljgwsinkcosa—20cosksin0—20sin0] + 762:1—24a—J—kcoskcos0+kcos0—205inksin0] + a331n0 a4sm0 = 0 (3.46) 20 61(0) - T9115(0) 3: (11 — Tka'z — 7'03 + (14 (3.47) 61(1) + 701,3(1) 2 al(cosk — TIC sin k‘) + 02(sink + Tk cos k.) + 03(1 + 7') + a4.(3.48) The combination of equations (3.43), (3.44), (3.45), and (3.46) gives the following linear system for Hinged case 0 k 1 0 —k sin k kcosk 1 0 —ksinksin0 kcosksin0 (k2 — 402M? — %¥) 0 —20 coskcos0 +ksin0 2 0 +20 cos0 -20 sin kcos0 ksin kcos0 —k cos k. cos 0 (A?2 — 403%? (k2 — 402)“?m —20 cos k sin 0 +kr cos0 —20 sin 0 —20 sin k sin 0 and the combination of equations (3.45) ~ (3.48) for Mixed case 40 1 cos k — Tk sin k —k sin ksin 0 +20 cos 0 ——20 cos k cos 0 ~20 sin k cos 0 —Tk sin k + Tk cosk kcosksina +k sin 0 k sin k cos 0 —20 cos k sin 0 —20 sin0 —k cos k c080 +k cos 0 —20 sin k sin 0 —T 1 1+ T 1 (Is?- 4424329 —- ———> 0 (k2 — 402)Sl2‘—‘aa (k2 — 402)§i—2—‘l - Solving above linear systems provides critical pressure pcr for various of angle 0, Angle(0) 5|a 05H as]: col: 3 12 tel-‘4 Fixed 42.24 84.208 125.695 166.634 207.212 248.05 Hinge 20.5273 40.1933 58.1368 73.4964 85.4108 93.0188 Table 3.1 (Critical pressures for various open angles) and for the variety of spring constants in mixed case 0/T .0001 .001 .01 .1 1 10 100 T/12 42.2231 42.0717 40.632 31.9219 22.4848 20.7366 20.5484 7T/6 84.1742 83.8714 80.9883 63.4116 44.1913 40.6209 40.2364 7r/4 125.645 125.191 120.859 94.0955 64.3533 58.8019 58.2037 7r/3 166.567 165.963 160.18 123.734 82.2333 74.4316 73.5905 5W/12 207.128 206.377 199.156 152.341 94.1487 86.6682 85.5374 7r/2 247.951 247.061 238.449 180.405 108.523 94.6812 93.1862 Table 3.2 (Critical pressures for various open angles and T) 41 These numbers are the same as obtained in the section 3.2 on Bifurcation theory and the results for the fixed and hinged buckling bases agree with the those of A.N. Dinnik [17], which implies the governing equation is well defined. Now, the 2nd order solution of equation (3.36) is (92 = a,” + 65’, (3.49) where the homogeneous solution is 65’ 2 c1 cos ks + (:2 sin ks + C38 + 6.; (3.50) and for particular solution we assume 0; 2 bls cos ks + bgs sin ks + b3 cos 2ks + b4 sin 2ks, (3.51) Here b,- and Ci, 2' 2 1, 2, 3, 4, are integration constants. Again the continuous differen- tiation of equation (3.51) provides 6.53 2 ()1 (cos ks — s sin ks) + b2(sin k3 + ks cos ks) —b32k sin 2ks + b42k cos 2ks 0.558 2 —b1 (2k sin ks + kzs cos ks) +b2(2k cos k3 — st sin ks) ——b34k2 cos 2ks - b44k2 sin 2ks 6P 2 —b1(2k2 cos ks + k2 cos ks —— kBs sin k3) —()2(2k2 sin ks + k2 sin ks + k33 cos ks) +538k3 sin 2k3 — b48k3 cos 2ks 92:43.93 2 b1(4k3 sin ks + k43 cos ks) ——b2(4k3 cos ks — k43 sin ks) +I)316k4 cos 2ks + 1).,16k4 sin 2ks. 42 Furthermore, the particular solution is needed to satisfy the 2nd order equation 4: ,ssss + 13201) 2 2b1k3 sin ks — 252k3 cos ks + 12b3k4 cos 2ks + 12b4k4 sin 2ks 2.33 E 9(3). (3.52) On the other hand, from the successive differentiation for the first order solution 01(3) 61,, 2 —01k sin ks + Cle cos k3 + 03 91,33 2 —alk2 cos k3 — 02k2 sin ks 91,333 2 01k3 sin ks — 02k3 cos ks 61,3333 2 alk4 cos ks + (lg/$34 sin k3, we obtain 1 62,3333 + k3202,33 : '21—1'[61,361,ssss _ 01,3391,sss - P191,” + 120261,361,88]_ 4 ..2 ’9 (110-3 + P144 (11 — 60/430103) cos ks = ( 20 20 k4 k2 +(__0203 p___1 02 — 60k20203) sin ks 20 20 +(30k3af — 30147303) sin 2k3 — 60k3alag cos 2ks f (s)- (3.53) Hence the equality between 9(3) and f(s) offers [)1 = 4m + (1202 — k2)a3. (3.54) Note that (ll-,2? 2 1, 2, 3, 4, are determined by the solving of lst order equation. More- over, the 2nd solution is 62(3) 2 c1 cos k3 + C2 sin ks + 033 + c4 1 (10.10.42 + —[k2a-zag + kplag — 120Qagag]s cos ks — cos 2ks 40k , 2 2 “(”1 " (12) 1 . . .. + —[1202(L103 — p101 — k72a1(12]3 sm ks + s1n 2ks. (3.50) 4k0 43 The coefficients c,,z’ 2 1, 2, 3, 4, will be determined by boundary conditions for each opening angle. The critical pressures were obtained using MATHEMATICA. 44 CHAPTER 4 Numerical Results 4. 1 Introduction In section 4.2 the forth order Runge-Kutta’ method is described and applied to our boundary value problem. Some techniques are deveIOped in order to reduce the boundary value problem to an initial value problem which is equivalent to finding zeroes problem of one or two parameters. The numerical solutions of the boundary value problem found in section 1.3 are shown for a variety of choices of angle and. spring constants in section 4.3. The perturbation solutions obtained in chapter 3 are compared with the exact numeri- cal solutions, and snap-through pressures defined as the pressure where catastrophic changes in deformations occur, are searched using numerical tools. Finally, tables of solution for many different opening angles and spring constants are established in section 4.4. 45 4.2 Previous results and extensions to the present problem Runge—Kutta’s Method We now recall the differential equation found in section 1.3 033,, —— 063 + $63 — p 2 0 (4.1a) g:— 2 cos6(s) (4.1b) % 2 sin 0(3), (4.1c) where c is an integral constant. With the putting 03 = t 633 2 u 6333 2 ’l), we have first-order equations (93 Z t E F1(S,0,t,U) ts 2 u E 172(3, 0,13, 21.) i 11., 2 '1) E F3(3,6,t, u) T, E F4(s,6,t, u) Lys E F5(8263t3 U), 46 which is integrated ( 191,1 —_— hFi(S,0,t,U) (91,2 = th‘(3 + %h, 9 +%ki,1,t+%ki+1,1,u +%k4+2,1) i 194,3 = hFi(3 + %h, 9 + %ki,2» t + %ki+1,2, U + %ki+2,2) (94,4 = th‘(8 + (1,9 + ki,37t + ki+1,3: U + (91423) k,’ = %(ki,l + 216,32 + 21613 + (9,34). \ When the starting point is (3", 0,,, t,,, 24"), the above system can be read like below F ki,l : hFi(8ny 6n, tn: an) ki,2 = (117431: + at, 9n + #331,731: + %kz‘+1,1, Un + éki+2,1) km 2 th'(Sn + %(l, 9n + ékz'g, tn + %ki+1,2, Un + %ki+2,2) km = th'(Sn + (1,911 + km, tn + ki+1,3: Un + 192423) k and r 9n+1= 9n +%(k1,1+ 2km + 2k1,3 + (171,4) tn-H = tn + g(k'zn + 2km + 2km + (€24) i ”n+1 2 un + éiksn + 2km + 2km + k3,4) f In+1 = In + %(k41 + 2k4,2 + 2,643 + (34.4) Lyn+l : 3111 + 5(k5d + 2k5v2 + 2k5’3 + kS,-l)° Techniques to reduce target boundary conditions Consider the equations (4.1a ~ 4.1c). The transformation 9(1 — s) 2 0(3), and 47 dropping the tilde give 1 3 6335 _Cgs + 50:9 +1): 0 2:, 2 cos 0(3) y, 2 sin 0(3). The boundary conditions are sin0 33(0) 2 0, 23(1) 2 0 , y(0) 2 0, y(1) 2 0 and for i) Fixed case(T 2 0) 0(0) 2 0 0(1) 2 —0 ii) Hinged case(T 2 00) 03(0) 2 —20 05(1) 2 —20 iii)Mixed case (T 2 00,0) (0(0) — 0) — T(63(0) + 20) 2 0 (0(1) + 0) + T(0s(1) + 20) 2 0. (4.2a) (4.2b) (4.2c) It is so difficult to use three parameter shooting to determine the initial values in order to satisfy three target boundary conditions simultaneously. We thus need to reduce the target boundary conditions as follows . Let 32!)? _‘_.’I.' 1’3 3:3. )3 48 (4.4a) (4.4b) (4.4c) Equations (4.2a ~ 4.2c) and boundary conditions for a: and y become a”, — as, + $0? + )3 = 0 (4.5a) if, 2 0030 (4.5b) y, 2 sin 0 (4.5c) 3(0) = 0 3(1) = 5:1; (45(1) y(0) = 0 y(1) = 0, (4.5e) ' (4.51) where E 2 002 and 'p’ 2 p03. Using the fourth—order Runge—Kutta algorithm, we integrate the above equations until 0(l"‘) 2 —0, 01(l") 2 —20, and (0(l“) + 0) + T(0,(l*) + 20) 2 0 in fixed, hinged, and mixed case, respectively. The parameters (011(0), 0, 01(0)) in fixed, (011(0), 0, 0(0)) in hinged, (011(0), 0), and 0(0) or 01(0) in mixed under the condition (0(0) + 0) — T(93(0) + 20) 2 0 will be guessed. Now define the error function Err = (T(l“) — 5(1))2 + (W) — 3(1))2. (4.6) which will be zero when a correct solution is obtained. Then for close enough guesses the above error function will be less than a certain given tolerance and we accept the guessed values as the initial values of the solution. Otherwise, it is necessary to improve the initial values using an algorithm such as Newton’s method. Accuracy is increased by decreasing step size Ah. Another way to increase the accuracy is to define the end point as follows. Observe the error function by adding or subtracting % to l‘. If the error function still unacceptable consider adding or subtracting AT” to new 1*. This process is repeated until the value of error function is stabilizes. Then the step size needs to be decreased again for a further decrease in error. Here we introduce a method to study the change of base angle in the hinged case. 49 Let 19(8) = 0(5) - 9(0) ((3) 2 / cos 1/2(3)ds 0 77(8) = / sin4ds. 0 then in the new variable the governing equations are . 1 3 70.933 _ (32,03 + 55/43 + I) = 0 C3 2 cos 1/1 773 2 sin (D. But 1103(3) 2 03(3) ((3) 2 /08 cos 7,0(3)d3 2 foscos(0(s) — 0(0))d3 2 35(8) cos 0(0) + y(s) sin 0(0) 77(3) 2 [03 sin '¢!(s)ds 2 [Us sin(0(3) — 0(0))d3 2 31(3) cos 0(0) — T(s) sin 0(0). Equations (4.9) ~ (4.11) give two target-boundary conditions sin 0 \/<2(1) + T22(1) = \/$2(1) + 112(1) = satisfying which provides the value 0(0). 50 (4.7a) (4.10) (4.11) (4.12a) (4.12b) (4.13) Now, using the following transform t t 32—, 0:025, 132031.), (=§(—)', 0— 0 t , n = 39, 4(8) = W), 0 we obtain ,—— _— 1—3 _ Wm — “(4 + 5101 + P = 0 (4.14a) Z? = (208$ (4.145) W = sin 0 (4.14c) The initial boundary conditions are ((0) = 0. if) = 0, 44 )= 0. 44(0) = —2. and the target conditions are 2 11240) 2 —2, \/(—(0)2 + 71(0) 2 sin 0. With guessing two parameters (1’)),(0) and (1411(0). we integrate until 114(0) 2 —2. If —2 —2 , E7‘7‘(t*) 2 \/C(t*) + 77(t*) — sin t* is less than given tolerance then we accept the guessed values as the initial values of a solution. Thus the problem is reduced to a simpler one-dimensional shooting problem. The initial value 0(0) is obtained from W“) (0*) For the hinged case we need a different method. The following is useful in the the relation tan 0(0) 2 — determination of the initial conditions for the hinged case. Let us transform 0(3) by 5(5) = 0(3) + 253 (4.15) 51 then the above equations (4.2a ~ 4.2c) is given by . , 1 3 99333 _ C(CDS _ 2(1)47 §(¢s _ 20) +1) 2 0 ms 2 cos(¢(s) — 203) ys : Sin(¢(8) _ 205): and new boundary conditions are 903(0) 2 0 033(1) = 0. Let 32113 7- E". '— l3 :2}: 3 13, then equations (4.16a ~ 4.16c) and boundary conditions become _ l , 3 _ 95111‘ (4951 - 2043) + 5(4): — 203) + p : 0 771 = COS(¢(3) — 203) i], 2 sin((z>(s) — 20s) _ _ _ sin0 T(O) ‘ 0 (1 “ 5,13 3(0) = 0 5(1) = 0 (151(0) = 0 451(1) = 0, (4.16a) (4.16b) (4.16c) (4.18a) (4.18b) (4.18c) (4.134) (4.18e) (4.131) where E 2 c132 and T) 2 11,03. For a given constant k, we start with 13 2 k and integrate Ulltil (Mr) I 0 guessing 0(0), (1511(0), and E. If the error function Err 2 (T(l“) — T(1))2 + (3(0) — 3(1))? is small enough then we regard guessed values as the correct initial values of the solution. Otherwise, iterate 1'3 and integrate up to (251(0) 2 0 to improve the value of error function with guessed values (19(0), 011(0). In this thesis the error bound used is 10‘“. 4.3 Numerical results The equilibrium states of elastic inextensible arches under uniform normal pressure are studied numerically for the various spring constants T. The change of curvature in the fixed case and the change of angle in the mixed case are determined for some choices of angle a with 0 < a < 7r. The boundary values problem is solved mostly numerically by the shooting method (see section 4.2) using iteration, except for the states immediately after bifurcation which is determined from the analytical solution found by the perturbation method. Figures 4.1—(a) and (b) show the stable equilibrium shapes of the arches with very small deflections immediately after bifurcation. For large pressure (Figures 4.2), we can see the mid point is compressed downwards. Figure 4.1-(a) (or 2 385, T 2 0) Figure 4.1-(b)(a 2 g, T 2 0) 53 A A V Figure 4.2-(a) (a 2 g, T 2 0) m N V Figure 4.2-(b) (a 2 T 2 0) 1r 59 Figure 4.2-(c) (2a 2 5.3, T 2 O) In Figures 4.3-(a) ~ (d), hinged base solutions are solved for a 2 g and a 2 g, respectively. Here the solutions are found only in first bifurcation which implies the first deviation of trivial state. We see that theses shapes are antisymmetric. \__/ Figure 4.3-(a) (a 2 A V Figure 4.3-(b) (a 2 g, T 2 00) Figure 4.3-(c) (a 2 g, T 2 oo) 55 Figure 4.3-(d) (a 2 g, T 2 00) Our numerical results show multiple equilibrium solutions which have not been reported previously. The best way to display these solutions is a pressure-base angle plot for the hinged case, or a pressure-base curvature plot for the clamped case. With the clamped base, Figures 4.4, 4.5, and 4.6 shows the variations of a base curvature versus the uniform pressure p. Three types of the behavior of anti-symmetric mode are observed in these figures. Those with a g % belong to first type (closed branch A- D-F-A in Fig 4.4) and bifurcation at a higher pressure, which displays ”snap-through” phenomenon from B to C. If g < a g % we have the second type (closed branch A- D-F-A in Fig 4.5) with bifurcation at a lower pressure , and if a Z 231 we have the third type pitchfork branch in Fig 4.6. The anti-symmetric mode branch proceeds inward in the first type as shown Fig 4.4, while for the second and third types, the anti-symmetric mode moves outward as shown Figs 4.5 and 4.6. Figure 4.6 shows that the variation of curvature of the anti—symmetric mode when a 2 273”- is a pitchfork bifurcation. The deformed shape leans to one side as pressure increases as shown Figure 4.6-(a). However, not all equilibrium solution is stable, or can be realized in practice. For stability we study the work done as follows. As the change of area was used to define work done in section 1.4, we use ”Area” or pressure-volume product surrounded by two 56 arch shapes, the trivial circular arch and the deformed arch for each given pressure, to show the stabilities for each branch. Stable state indicates that an increase in Area is accompanied by an increase of the pressure, while in an unstable state the Area decreases as the pressure increases. The pressure-area plot, Figs 4.4-(c) and 4.5-(c) show only the branch C of symmetric mode (open curve in Figs 4.4and 4.5) is stable. The figures 4.4-(d) ~ 4.4-(h) are representative shapes of each branch. We read A as the bifurcation state in pictures 4.4, 4.5, and 4.7 and B as the snap-through state in figures 4.4 and 4.7. Figure 4.4 (oz 2 g, T 2 O, snap-through p, z 85) 57 Area .07 \ .06» .05» .04» 1),}? .03» .02» .01» OOOOOOO Area .06 .05’ .04 E .03 .02 .01* 0000000 20 40 60 80 100 1&0 E? oq C H m 9‘ *P A a V Q H cal: 21 H O v 58 Figure 4.4-(d) (B, a 2 g, T 2 0) AVA Figure 4.4-(e) (C, a 2 g, T 2 0) Figure 4.4-(f) (D, a 2 ,T 2 0) I 6 Figure 4.4-(g) (E, oz 2 g, T 2 0) Figure 4.4-(h) (F, a 2 %,T 2 0) _5. -7 5 D —10 —12.5> C Figure 4.5 ( a 2 g, T 2 0) Area 0.12 \ 0.1» D, F 0.08 0.06 0.04 0.02> 60 Area 0.1) 1 0.08 ‘ 0.06) . 0.04> ‘ 0.02* 1 0 50 100 150 200 250 Figure 4.5-(b) (a 2 g, T 2 0) Area 0.2 0.19 (3 0.18 ‘ 0.17> 4 0 16L 4 0.15 {// Figure 4.5-(c) (a 2 g, T 2 0) 95(0) 5 {//////////////// 100 200 00 400 ___5_: __________ _lO> -15. Figure 4.6 ( a 2 331, T 2 0) 61 6O 80 100 120 140 16013 Figure 4.6-(a) (a 2 21 T 2 0) In Figure 4.7, we describe the variation of angle at a base for pin-ended arches. For the opening angles that display ”snap.through” phenomenon, similar results to fixed arch (a 2 {35) are obtained. Anti-symmetric mode (closed branch in Fig 4.7) proceeds inward like Fig 4.4. Fig 4.7—(c) shows only the bottom branch C of symmetric mode (open curve in Fig 4.7) is stable. Figures 4.7-(d) ~ 4.7-(g) are representative shapes of each branch. Figure 4.7 ( a 2 g, T 2 oo, snap-through ps 2 40.155) 62 Area 0.15\ 0.125 1),}? 0.1 0.075' 0.05 0.0251 O . . . ~20 -10 0 10 20 3C 40 P Figure 4.7-(a) (a 2 7g, T 2 oo) -20 0 20 40 60 80 Figure 4.7-(b) (a 2 g, T 2 00) Area .168 .167» ////”//flyflf’fl-————1 .166 C3 .165 .164 .163 .162 .161 OOOOOOOO -20 O 20 40 60 P Figure 4.7-(c) (a 2 63 V Figure 4.7-(d) (D, a 2 Figure 4.7-(e) (C, a 2 %,T 2 00) A A V Figure 4.7-(f) (E, a 2 %,T 2 00) Figure 4.7-(g) (F, a 2 %,T 2 oo) 64 For pin-ended buckling arch, the changes in deformation as pressure is decreased when a 2 g are pictured for the first (Fig 4.8) bifurcation modes and second bifur- cation modes (Fig 4.9). Figure 4.8-(a) (a 2 g, T 2 00, p 2 50) Figure 4.8-(b) (a 2 g, T 2 00,1) 2 30) Figure 4.8-(c) (a 2 g T 2 00, p 2 ——20) 65 Figure 4.9-(a) (a 2 {15, T 2 00, p 2 60) Figure 4.9-(b) (a 2 —120) Ala ,T 00,}? Figure 4.10 (a 2 g, curve:T21000, dashed curve:T210, dot-dashed curve:T21, dot curve:T2.5 ) 66 We also study the variations of angle of an edge versus pressure for the variety of choices of spring constants (Figure 4.10). The values 0.5, 1, 10, and 1000 are used for spring constants. We see that snap-through (the largest pressure on the bifurcation branch) increases with decreased spring constant T. It is observed that the changes of angle are almost identical if spring constant T _>_ 10 as shown Fig 4.10. The snap-through pressures are investigated for many different angles and spring constants (Table 4.1). For the clamped base, the phenomenon of snap-through does not occur if the opening angle a 2 7r/5. 67 Table 4.3.1 Snap-through p. for various spring constants and angles T/a 7r/6 7r/5 7r/4 7r/3 00 40.155 47.64 58.2 73.72 10 40.43 48.13 58.867 74.44 5 41.01 48.55 59.53 75.43 1 44.1 52.19 63.94 .5 47.3 56.2 68.765 .4 49.01 58.08 71.9 .3 51.72 62.8 76.86 .2 55.75 75.8 .1 63.64 .01 80.409 0 85 68 4.4 TABLES OF SOLUTION 69 Table 4.4.1 ( T 2 oo, 02 2 %) p 0(0) 98(0) 6 033(0) 40.155 0.497 —2a -37.8282 .98 37 .253 —2a -37.5 8.4358 35 .172 —2a -37254404 10.4 30 .023 —2a -36.88 13.44604 20 -.169 —2a -36.54405 16.03 10 -.304 —2a -36.5 16.60733 0 —.392 —2a -36.77876 15.425 -10 -.435 —2a -37.299 12.32022 -18 -.382 —2a -37.96 6.226 -20 —.253 —2a -37929 .1 90 .564 —2a -86.15653 -4 80 .64 —2a -83.64927 ~15.1 70 .664 —2a -81.74506 -22 60 .66 —2a -80.12241 —26.3 50 .64 —2a -78.70408 -2922 40 .613 —2a -7740019 -31.49 30 .577 —2a -76.23 -32.8647 20 .533 —2a -7514572 -3357 10 .483 —2a -74.10075 -33753 0 .421 —2a —73.07526 -33 -10 .357 —2a -71.95392 -321 -20 .272 —2a -70.6089 298 -30 .162 —2a -68.53317 -25.82 -33 -.169 —2a 51.9520 -7 -30 -.197 —2a -48.42407 -4.84 -20 -.253 —2a —37.929 .1 -10 -.284 -2a -28.3 3.44 0 -.306 —2a -18.7218 6.1 10 -.324 ~26 922023 8.32 20 -.336 —2a .255155 10.22 30 -.349 ——2a 9.634944 11.93 50 -.365 —2a 28.511826 14.856 70 -.374 —2a 47.89747 17.3484 70 Table 4.4.2 ( T 2 00, a 2 %) p 9(0) 03(0) 933(0) C 20 .783 -2a .011442 -11.51 30 .7815 -2a .01268 -17.9 45 .78 -2a .01385 -27.55 58.2(p3) .722232 -2a 2.28 -35.72623 55 .336004 -2a 13.648 -34.65677 50 .1062 -207 18.83 -33.72888 40 -.137 -2a 22.840502 -32.957 30 -.291854 -2a 24.27404 -32.77 20 -.413764 -204 24.72 -32.770713 10 -.513953 -2a 24.38 -32.934968 0 -.590436 -2a 23.09 -33.342665 -10 -.640355 -2a 20.58 -34.02702 -20 -.651 ~20 16.2 -34.98553 71 Table 4.4.3 ( T 2 1000, 01 = '25) :9 9(0) 03(0) 035(0) 6 55 .342774 -1.57123869 13.482601 -34.700021 50 .11425 —1.5714672 18.63315 -33.800015 40 -.132633 -1.571714 22.734 -33.008 30 -.294638 -1.5718761 24.4 -32.713469 20 -.415636 -1.571997 24.82 -32.724487 10 -.509096 -1.572095 24.255801 —33 0 -.58837 —1.5721698 23.081 -33.349965 -10 -.640107 -1.5722215 20.651844 -34 -20 -.649643 -1.572231 16.19999 -34.9879 72 Table 4.4.4 ( T 2 10, a = g) p 9(0) 6’s(0) 033(0) 6 55 .301246 -1.6192112 14.758904 -34.890636 50 .100178 -1.639318 19.162181 -34.099975 40 -.135552 -1.662891 23.041247 -33.31929 30 -.29337 -1.6786728 24.641968 -33.007188 20 -.408572 -1.690193 24.896021 —33.0597821 10 -.502481 -1.6995839 24.345984 -33.28978 0 -.575582 -1.706894 22.9497042 -33.70551 -10 -.624532 -1.711789 20.397615 -34.36031 -20 -.631322 -1.712468 15.864532 -35.31733 73 Table 4.4.5 ( T 2 1, a 2 %) P 6(0) 03(0) 033(0) C 55 .147699 -2.208495 20.187967 -37.18 50 .033185 -2.323009 22.351141 -36.64155 45 -.059608 -2.415802 23.82352 -36.20961 40 -.12917 -2.485364 24.526125 -35.9964 30 -.244923 -2.601117 25.025621 -35.7906 20 -.333529 -2.689723 24.512665 -35.83141 10 -.400954 -2.757148 23.07576 -36.09213 0 -.445862 -2.802056 20.579168 -36.568228 -10 -.457651 -2.813845 16.344204 -37.36509 74 Table 4.4.6 ( r = 0.5, a = %) p 9(0) 93(0) 933(0) 0 55 .105437 -2.930719 22.378046 -39.5348 50 .02532 -3.090952 23.635254 -39.0164 45 -.038909 -3.21941 24.259342 ~38.67689 40 -.091142 -3.323876 24.439219 -38.45663 30 -.172402 -3.486396 23.824888 -38.29383 20 -.238058 -3.617709 22.453325 -38.27972 10 -.278836 -3.699263 19.847078 -38.57587 0 -.295866 -3.733323 15.913716 -39.09426 75 Table 4.4.7 ( T = 0, a = g) P 68 (0) C 933(0) 63 (1) 85 -1.2 -80.6547 .22 -.8943 80 -2.3 -79.3371 -.2 .2066 70 ~3.12 -78.019 -4.2 1.0217 60.1 -3.42 -77. 1256 -9 1.3229 50.1 -3.52 -76.2574 -14 1.4213 39.7 -3.26 -75.815 -19.6 1.1689 30 -2.85 -75.29 -24.9 .749 20 -1.72 -75.187 -31.6 -.345 17.7 -1.3 -74.2005 -33.1 -.9397 136.8 -1.0472 -130.0858 0 130 -.52 129.2127 -3 120 -.22 -125.3618 -6.25 110 .14 -122.1412 -9.8 100 .46 7 -119.166 -13.7 90 .78 -116.5 -17.808 80.2 .85 -112.5192 -20.9 70 .96 -109 ~24.609 60 1.075 -105.8315 -28.4 50 .955 -101.3027 -31.2 40 .7 -95.9898 -33.5 30 .335 -90.007 -35.45 20 -2.387 -62.376 -26.6 30 -3.78 -48.5873 —15.6 40 -4.805 —37.9618 -6.53 50 -5.6 -28.9556 1.8 60 -6.283 —20.4263 9.91 70 -6.913 -I2.2265 17.758 80 -7.493 -4.19563 25.5 100 -8.558 11.000402 40.32 110 -9.048 18.279398 47.504 130 -9.938 34.5697461 63.122 76 Table 4.4.8 ( 7‘ = 0, oz = 132) p 03(0) 0 055(0) 05(1) 45.9 -1.9 -58.2979 66.22 -1.9228 50 -.58 -57.8025 -66.08 -3.5396 60 .6 -58.1096 -64.236 -4.7668 80 1.88 -59.177 -58.6 -6.0658 100 2.59 -60.868 -51.1 -6.7715 120 2.92 -63.1 -41.793 -7.1132 140 2.7 -66.873 -29.5 -6.8854 160 1.12 -73.4 -11.473 -5.3096 166 -.885 -77 -2.4851 -3.3006 250 -0.69 -124.295 -9 230 0.14 -119.453 -16.55 210 1 -114.498 -25.5 200 1.29 -111.8051 -29.6 180 1.75 -106.3007 -37.83 160 2.04 -100.6955 -45.7 140 2.12 -95.053 -52.75 120 2 -89.3322 -59 100 1.7 -83.4009 -64.3 80 1.07 -76.903 -68.02 60 0 -68.8727 -69.65 50 -1.11 -62.7008 -68.25 45.3 -2.94 -53.6 -62.5023 50 -4.42 46.3 -55.7076 60 -5.85 ~39.3 ~48.0631 80 -7.75 -29.34865 -36.63 100 -9.2 -21 -26.9159 120 -10.452 -13.15 -18.184 140 -11.57 -5.60287 -9.9 160 42.591 1.813 -1.88 77 Table 4.4.9 ( T 2 0, oz 2 5) p 03(0) c 6...(0) 05(1) 261 -444 16 -1215575 -3.8028 263 -5155 16 .123.202 -3.128 265 -5.56 16 -12411 -27545 270 -6.26 16 -125.7568 -2.0569 280 .7.25 16.3492 -128 -1.1818 300 -8.58 16.2078 -130.85 0.2174 320 -9.624 16.2 -132.69865 1.2637 340 -1053 16.1916 -133.85 2.1423 360 -1133 16 -134.3776 2.9488 380 -12.08 15.7927 -1343 3.6738 400 -12.76 15.1 -1334017 4.3971 420 -1343 14.4 -1317057 5.0505 440 -1407 13 -1285528 5.703 460 -1471 10.6 -1229318 6.3349 480 —15.39 1.59535 .1042 6.9782 78 Table 4.4.10 Snap-through for various spring constants and angles 07 T 123 6(0) 65(0) c 033(0) 77 /6 00 40.155 .4964 -2a -37.8282 .98 77/6 10 40.43 .471575 -1.0524 -38.227782 1.9 77/6 5 41.01 .469087 -1.0581 -38.61398 2.1 77/6 1 44.1 .464296 -1.1065 -41.63451 2.5 77/6 .5 47.3 .463198 -1.168 -44.8358 2.8 77/6 .4 49.01 .478478 -1.16 -46.38068 2.21 77/6 .3 51.72 .502658 -1.117 -48.87495 1.104 77/6 .2 55.75 .503038 -1.15 -52.71645 1.22 77/6 .1 63.64 .504819 -1.235 -60.277 1.39844 77/6 .01 80.409 .516971 -1.71 -76.9004 1.14 77/6 0 85 -1.2 -80.6547 .22 -.8943 77/5 00 47.64 .627 —2a -37.11 .01 77/5 10 48.13 .578689 -1.2616 ~37.49456 1.88 77/5 5 48.55 .577004 -1.2669 -37.9 1.9483 77/5 1 52.19 .529856 -1.3551 -40.89262 4 77/5 .5 56.2 .520137 -1.473 -44.1724 4.84 77/5 .4 58.08 .531773 -1.498 -45.7 4.50622 77/5 .3 62.8 .62431 -1.27 -49.18353 .2 77/5 .2 75.8 .627646 -1.26 -59.4 .09438 77/4 00 58.2 .722232 —2oz -35.72623 2.28 77/4 10 58.867 .781358 -1.5712 -36.210096 .169 77/4 5 59.53 .733378 -1.5812 -36.58276 1.98 77/4 1 63.94 .640194 -1.716 -39.534 5.7101 77/4 .5 68.765 .582796 -1.976 -42.64989 8.61 77/4 .4 71.9 .701717 -1.78 -44.52 3.99565 77/4 .3 76.86 .776637 -1.6 -47.69544 .44 77/3 00 73.72 .483 —2a -30.91 17.0013 77/3 10 74.44 1.022147 -2.0969 -33.33359 .9 77/3 5 75.43 .969672 -2.1099 -33.71 2.7946 79 CHAPTER 5 Conclusions and Discussions In this chapter previous works are discussed and summarized. In chapter 1 the ordinary differential equation was formulated using the balance of normal and tangential stresses. This provided the nonlinear boundary value problem for the buckling arch under uniform normal load. The Hamilton’s principal yielded a variational formulation which was used to prove the existence of equilibrium states in chapter 2. In chapter 2 the existence of equilibria were proved using the variational for- mulation obtained in chapter 1. We studied the existence of equilibria using Robin boundary conditions, while previous authors explored the energy principle in Dirichlet and Neuman boundary conditions. In chapter 3 we investigated criteria for various opening angle and buckling spring constants. In the special case of fixed and hinged buckling bases our results agree with the works of A.N. Dinnik [17]. Furthermore, we investigated criteria for different choices of buckling spring constants. In chapter 4 the boundary value problem formulated in chapter 1 was solved using numerical tools. The fourth order Runge-Kutta Method and the shooting method are used to find exact numerical solutions away from the bifurcation points. For the 80 f... _. variety of choices of angle a and spring constant T, stable and unstable equilibrium states were calculated. Perturbation in chapter 3 and numerical results in chapter 4 showed the theoretical development for existence of equilibria in chapter 2 was appropriate. The solutions found in fixed case were agreed with the results of R.W. Dickey and .1 . J. Roseman [6] We studied the changes of curvature and angle at a base for clamped and pin-ended arches respectively. The shape of bifurcation curves was an ellipse, which is different from pitchfork bifurcation of previous work. Y.-L. Pi, M.A. Bradford, and B. Uy [12] studied typical variations of the dimensionless central vertical displacement with the dimensionless axial load for pin-ended arches. Our investigations were matched partially with the Y.L. Pi, M.A. Bradford, and B. Uy’s [12]. But, in practice, the variation of angle in a hinged base arch is discontinuous and our study explained the discontinuity through the snap—through phenomena. Moreover, we investigated the snap-through using numerical tools for various opening angle and base conditions. The asymptotic solutions were combined in a small regions near the singular points with exact numerical solutions. The tables of solutions for various opening angles and springs constants were established. In this thesis, we developed the energy principle with Robin boundary conditions in chapter 2 and explored the bifurcation criteria for numerous opening angles and spring constants in chapter 3. In chapter 4 an elliptic bifurcation curve was obtained using numerical tools. The buckling of a thin circular arch is complex. The nonlinear governing equa— tions offer many conceptual, analytical, numerical challenges. In this thesis we have advanced in all of these areas and examined in depth many interesting phenomena such as bifurcation, non-uniqueness, catastrophe, and instability. In spite of these results, some extensions of the thesis is still possible. These include studies in 81 {T 1) Uniqueness of the solution 2) Extending the perturbation analysis to higher order 3) Relaxing the assumption of inextensibility . 82 BIBLIOGRAPHY 83 [2] [3] [4] [5] [6] [7] [8] [9] [10] BIBLIOGRAPHY Antrnan, SS. and Dunn, J.E.(1980, July). Qualitative behavior of buckled non- linearly elastic arches, J. of Elasticity, Vol.10, No. 3, pp. 225-239. Atanackovie, TM. and Milisavljevie, B.M.(1981). Some Estimates for a Buckling problem, Acta Mechania 41, pp. 63-71. Browder, F.E.(1965). Variational Methods for Nonlinear Elliptic Eigenvalue Problems, Bull. Amer. Math. Soc., Vol. 71, pp. 176-183. Conte, SD. and Boor, C.D.(1980). Elementary Numerical Analysis, McGraw- Hill, Third Edition. Dadeppo, D.A.( 1969). Nonlinear Analysis Of Buckling And Postbuckling Behav- ior 0f Circular Arches, J. Appl. Math. and Phys, Vol. 20, pp. 847-857. Dickey, R.W. and Broughton, P.(1971). Stability Criteria for Shallow Arches, Journal of the Engineering Mechanices Division ASCE 97. (EM3), pp.951-965. Elgindi, M.B.M. and Yen, D.H.Y.(1992). Existence of Equilibrium States of Hol- low Elastic Cylinders Submerged In a Fluid, Internat. J. Math. Math. Sci. Vol. 15 No. 2, pp. 385-396. Granas, A., Guenther, R.B., and Lee, J.W.(1979). The Shooting Method For The Numerical Solution Of A Class Of Nonlinear Boundary Value Problems, SIAM. J. NUMER. ANAL., Vol. 16 No. 5, pp. 828-836. Love, A.E.H.(1944). A Treatise On The Mathematical Theory Of Elasticity , New York Dover Publications. Miklavcic, Milan.(1998). Applied Functional Analysis And Partial Difierential Equations, World Scientific. 84 [11] Mirmiran, A. and Amde, A.M.(1993). Inelastic Buckling 0f Prestressed Sandi- wich 0r Homogemeous Arches, J. Struct. Engineering, Vol. 119, No. 9, pp. 2733- 2743. [12] Pi, Y.L., Bradford, M.A.(2002). and Uy, B., In-plane Stability 0f Arches, Inter. J. Solids and Structures, Vol. 39, 2002, pp. 105—125. [13] Pi, Y.L. and Trahair, N.S.(1996). In-plane Inelastic Buckling And Strengths Of Steel Arches, J. of Structural Engineering, pp. 734-747. [14] Simitses, G.J.(1976). Elastic Stability of Structures, Prentice-Hall, New Jersey. [15] Schreyer, H.L. and Masur, E.F.(1966). Buckling of Shallow Arches, J. Eng. Mech., Vol. 92(EM4), pp. 1-19. [16] Tadjbakhsh, I. and Odeh, F.(1967). Equilibrium States Of Elastic Rings, J. Math. Anal. Appl., Vol. 18, pp. 59-74. [17] Timoshenko, SP. and Gere, J.M.(1961). Theory Of Elastic Stability, McGraw- Hill Second Edition. [18] Ugural, AC. and Fenster, S.K.(1987). Advanced Strength And Applied Elasticity, P T R Prentice Hall, Englewood Cliffs, New Jersey, Second SI Edition. [19] Vlasov, V.Z.(1963). Thin- Walled Elastic Beams, 2nd Edition, Israel Programme for Scientific Translation, Jerusalem, Israel. [20] Wang, C.Y., Watson, L.T., and Kamat, M.P.(1983). Buckling, Postbuckling, and The Flow Through a Tethered Elastic Cylinder Under External Pressure, J. Appl. Mech., Vol. 50, pp. 13-18. [21] Wang, C.Y.(1997). Post-Buckling Of a Clamped-Simply Supported Elastica, Int. J. Non-Linear Mechanics, Vol. 32, No. 6, pp. 1115-1122. [22] Zeidler, Eberhard.(1995). Applied Functional Analysis : Main Principal and Their Applications, Springer-Verlag. General References [23] Agarwal, RP. and Chow, Y.M.(1984). Iterative Methods For A Forth Order Boundary Value Problem, J. Compu. Appl. Math, Vol. 10, pp. 203-217. 85 [24] Antman, S.S.(1970). Existence Of Solutions Of The Equilibrium Equations For Nonlinearly Elastic Rings And Arches, Indiana Univ. Math. J., Vol. 20, No. 3, pp. 281-302. [25] Atanackovic, T.M.(1986). Buckling Of A Heavy Compressed Column With Im- perfections, Q. JI Mech. Appl. Math, Vol. 39, Pt. 3, pp.361-379. [26] Atanackovic, T.M.(1984). Estimates Of Maximum Deflection For A Rotating Rod, Q. JI Mech. Appl. Math., Vol. 37, Pt. 3, pp.515-523. [27] Carrier, G.F.(1947). On The Buckling Of Elastic Rings, J. Math. and Phys, Vol. 26, pp. 94-103. [28] Crandall, MC. and Rabinowitz, P.H.(1971). Bifurcation From Simple Eigenval- ues, J. Functional Analysis 8, pp. 321-340. [29] Decker, D.W. and Keller, H.B.(1981). Path Following Near Bifurcation, Commu. Pure and Appl. Math., Vol. XXXIV, pp. 149-175. [30] Dickey, R.W. and Roseman, J .J .(1993). Equilibria Of The Circular Elastica Un- der A Uniform Central Force Field, Q. Appl. Math., Vol. LI, pp. 201-216. [31] Elgindi, M.B.M. and Yen, D.H.Y.(1993). On The Existence Of Equilibrium States Of An Elastic Beam On A Nonlinear Foundation, Internat. J. Math. and Math. Sci, Vol. 16, No. 1, pp. 193-198. [32] Flaherty, J.E., Keller, J.B., and Rubinow, S.I.(1972). Post Buckling Behavior Of Elastic Tubes And Rings With Opposite Sides In Contact, Siam J. Math, Vol. 23, No. 4, pp.446-455. [33] Gilding, B.H.(1987). The First Boundary Value Problem For —u” = Au” , J. Math. Analy. and Appl. 128, pp. 419-442. [34] Grans, A., Geunther, RB, and Lee, J.\lV.(1978). On A Theorem Of S. Bernstein, Pacific J. Math, Vol. 74, No. 1, pp. 67-82. [35] Gupta, C.P.(1988) Solvability OfA Fourth Order Boundary Value Problem With Periodic Boundary Conditions, Internat. J. Math. and Math. Sci. Vol. 11, No. 2, pp. 275-284. [36] Isaacson, E. and Keller, H. B.(1966). Analysis Of Numerical Methods, John Wiley and Sons, Inc. 86 [37] Kevorkian, J. and Cole, J.D.(1981). Perturbation Methods In Applied Mathemat- ics, Springer-Verlag. [38] Langford, W.F.(1977). Shooting Algorithm For The Best Least Squares Solution Of Two-Point Boundary Value Problems, Siam. J. Numer. Anal, Vol. 14, No. 3, pp. 527-542. [39] Matkowsky, B.J. and Reiss, E. L.(1997). Singular Perturbations Of Bifurcations, Siam. J. Appl. Math., Vol. 33, No. 2, pp. 230-255. [40] Nayfeh, A.(1973). Perturbation Methods, New York, WILEY. [41] Reddy B.D.(1998) Introductory Functional Analysis:with applications to bound- ary value problems and finite elements , Springer. [42] Pi, Y.L. and Trahair, N.S.(1999, November). In-Plane Buckling and Design of Steel Arches, J. Structural Engineering, pp. 1291-1298. [43] Wang, C.Y.(1982). On The Bifurcation Solutions Of An Axially Rotating Rod, Q. JI Mech. Appl. Math., Vol. XXXV, Pt. 3, pp.391-402. 87 t 'l‘ l. ‘1' 1H]]]i]]]]|]i][]]i[)])]