Exact solution for an approximate differential equation of a straight bar under conditions of a non-linear equilibrium

Inc. J. Non-Linew Mechanics. Vol. 21, No. 5, pp. 421-429. Printed in Great Britain

1986

002&7462:86/f3.00+.00 Q 1986 Pergamon Journals Ltd.

EXACT SOLUTION FOR AN APPROXIMATE DIFFERENTIAL EQUATION OF A STRAIGHT BAR UNDER CONDITIONS OF A NON-LINEAR EQUILIBRIUM D. E. PANAYOTOUNAK~S National

Technical

and P. S. THEOCARIS

University

of Athens,

Athens,

Greece

(Received 4 February 1986; received for publication 14 April 1986) Abstract-In

this paper

a closed-form

solution

of the strongly

non-linear

differential

equation

of

the form: y”’ + y’y”tany

- Ay’lcosy = 2Ccos2y

(A; C are positive

constants)

is presented under special physical conditions of the problem. This type of differential equation governs the equilibrium of a straight and prismatic bar subjected to a conservative system of arbitrary discrete and distributed co-planar loads in the most general case of response. Whereas all previous solutions are based on several convenient functional transformations, the procedure developed in the paper presents the advantage to yield a straightforward closed-form solution of this general and complicated problem. Special cases of two particular problems, concerning the mode of loading, have been solved. The one of them was selected conveniently from the already solved otherwise problems for reasons of comparison. The results are in complete agreement with the already existing solutions.

1. INTRODUCTION

The analysis of large elastic deformations of beams and beam-columns has been the subject of several investigations commencing with Euler’s studies of the elastica [1,2]. Analytic solutions of the strongly non-linear differential equation of the elastic curve have been given for the case of straight or circular prismatic rods, due only to concentrated loads, [l-7]. These solutions are presented under the form of elliptic integrals of the first and second kind. Love [l] was the first to establish the methodology to work out this problem and calculate the constants of integration, by solving a system of simultaneous non-linear equations. Thus, solutions to specific problems of bars loaded, either by a single axial concentrated force, or by an axial concentrated load and a bending moment have been given in [1,2,6]. Reference [S] yields also the exact form of the elastic line of a straight bar submitted to a general concentrated co-planar loading consisting of an axial compressive force, a transverse force and a bending moment, by taking into account also the influence of the shear deformation. Lee, Manuel and Rossow [4] have developed a systematic and general method for analysing large deflections of rigid-elastic frame-type structures, subjected to a conservative system of arbitrary discrete loads, by neglecting the effects of axial and shear deformations. An analytic also solution was given in [7] for the problem of large elastic deformations of a planar curved bar, with an arbitrary curvature function, submitted to a concentrated loading. On the other hand, the numerical integration of the problem of in-plane deflections of plane elastic frame-type structures was analyzed in [8,9]. In both previous papers, distributed loads on segments of the structure were considered. The objective of the present paper is to give a closed-form solution for the problem of a non-linear elastic and buckling analysis for a straight prismatic bar, due to arbitrary coplanar generalized concentrated forces, as well as to distributed loads, in the most general case of response. The analysis was based on the closed-form integration of the exact differential equation, governing the equilibrium of a free body, cut out of the bar. This differential equation, neglecting the effects of axial and shear deformations, is of the form: y”’ + y’y”tany

- Ay’/cosy

= 2Ccos’y

(A; Care positive constants)

and it was firstly presented by Heinzerling [lo] on 1938. We notice here that, since then, this differential equation was solved in a closed-form only for the special case when the 421

422

D. E. PANAYOTOUNAKOS and P. S. THEOCARIS

bars are submitted to terminal generalized concentrated forces, because, in this case, it may be readily reduced to another second-order non-linear differential equation of simpler form. In the present investigation, a closed-form solution of the differential equation was achieved by applying convenient functional transformations. It is worthwhile mentioning that the general solutions derived in this paper can be further used for liberating the analysis of the relative problems appearing in [8,9] from the need of long computer cods. Finally, two special cases for different loading modes of bars were examined, the one of which yielded an analytic solution, which was in complete agreement with the solutions given in [4,6]. 2. THE GOVERNING

NON-LINEAR

EQUILIBRIUM

EQUATION

Figure 1 shows a part of a plane elastic rigid frame-type of structure in its undeformed and deformed states respectively. Let x,, y,; 8, be the generalized global co-ordinates of a generic joint II, where 8, is the rotation of the joint; let P,,; P,,. and M, be co-planar internal forces and couple applied on the n-joint, and q, a distributed load applied on the part n(n + 1). Also, P,.,+ G P,,.+ 1, M,+ 1 denote the corresponding quantities applied on the (n + l)-joint. The distance of the n-joint from the origin is given by s, where, in general, s is measured along the centroid axis of the structure. Figure 2 shows a free body cut-out of the deformed segment n(n + l), in which all the generalized forces are positive as indicated in the figure. The arc length, s, is always positive and 0 is positive counterclockwise with respect to the co-ordinate Ox-axis. The equilibrium of the deformed free body, shown in Fig. 2, requires that:

M = Py,(x - x,) - Px,cV- Y,) - q(x - x,,)‘P + M,,

(2.1)

and the exact strongly non-linear differential equation of the elastic curve is given by:

Wds = - CP& - x,1 - PAY - Y.) - 4x - x,)*/2+ MnlIWn

(2.2)

with: dx = coseds,

dy = sineds.

(2.2a)

In equation (2.2) E,,J, represents the constant flexural rigidity of the segment n(n + 1). This equation can be further written under the form: P = A(y - y.)- B(x - x,) + C(x - x,)2 - D,

Fig. 1. Geometry and sign convention of a segment n(n + 1) of a plane rigid framed structure.

(2.3)

Exact solution

for an approximate

423

differential equation

0

Fig. 2. Geometry

and sign convention

of a free body cut out of the segment n(n + 1).

in which the constant coefficients A; B; C; D are real positive numbers, much smaller than the unit. Furthermore, 0’(s) represents the curvature function of the deformed elastic part and primes always denote differentiations with respect to the arc-length s. We notice here, that all functions are continuous and continuously differentiable with respect to the variable S.

Now, differentiating equation (2.3) once with respect to s and using relations (2.2a), the following differential equation may be obtained: 8” = Asin - BcosfI + 2C(x - x,)cos0.

(2.4)

From the literature it may be derived that a closed-form solution of equation (2.4) has been given only in the case when the coefficient C is equal to zero. This happens for the case when the bar-segment is loaded only by terminal co-planar concentrated generalized forces. This is equivalent to the problem of the non-linear and buckling analysis of a thin, straight elastic rod by using the theory of elastica [1,2], where the prebuckling state, is a membrane, or a bending state, but only due to concentrated loading. As it has been already shown (see [l-6]) the analytic solution of the strongly non-linear differential equation (2.4), for C = 0, is given by elliptic integrals of the first and second kind. On the other hand, numerical solutions by using convenient computer cods, have also been recently given for the case when C # 0 in [8-g]. Taking into account that, in almost all the structures of engineering practice, a distributed load, q, appears, the aim of this paper is to establish a closed-form solution of equation (2.4). Then, by using this solution, the problem of non-linear and buckling analysis of plane rigid frame-type structures can be completely solved. However, it may be observed that equation (2.4) includes the term (x(s) - x,), which must be replaced by a function of 0. So, solving (2.4) with respect to C(x - x,) and differentiating the resulting new equation with respect to s, yields: tY + 6YYtantJ - AB’/cose = 2cc03e.

(2.5)

This equation was firstly presented in [lo], but since then it has been never analytically solved. In the following we shall try to give a closed-form solution of equation (2.5). 3. CLOSED-FORM

SOLUTION

OF

EQUATION

(2.5)

We introduce first the transformation: 8’ = ,(,)1’2

- (z = I92 # O),

(3.1)

which yields:

28’8”=z’=!?g = i(z)‘“,

(3.2)

424

D. E. PANAYOTOUNAKOSand P. S. THE~CARIS

where dots represent differentiation with respect to 8. Solving equation (3.2) with respect to 8” and using relation (3.1) we have: 8” = i/2,

(3.3)

and furthermore:

e,,,=

$(i/2)

= &i/2)1

= i(z)“2/2.

(3.4)

Introducing the expressions for B; 66”; 8” and t9”’from equations (3;1)-(3.4) into equation (2.5) we derive the following differential equation: i + tan& - ~CCOS~~/(~)‘~~ = ZAlcos&

(3.5)

Thus, by using the functional transformation (3.1), the third-order strongly non-linear differential equation (2.5) was transformed to an equivalent non-linear second-order differential equation with variable coefficients. Further, in order to simplify the last equation, we use a new functional transformation of the form: z(6) = q(r),

r = sine.

(3.6)

Then, based on relations:

d*z d = ,,(Q cost7) = y cos*l3 - ij sin& de* where stars indicate differentiations equivalent form:

(3.7)

with respect to {, equation (3.5) takes the following

4 - 4C(rl)_ iI2 = 2Afcos38.

(3.8)

The last equation can be written under a dimensionless form as follows: (ij/2A)” - 7

(1,2q)-i’* = l/(1 - t2)3’2,

(3.9)

in which I, represents the total length of the undeformed n(n + l)-segment. Based on relations (3.6) and (3.1) the non-linear term (1,2$-i’*, appearing in the last equation, can be replaced by the equal term (ljf~*)-~‘*, in which K denotes the curvature-function in the deformed state of the bar. By now, we can readily prove the inequality: 0 < 1*ic* n < 2.

(3.10)

Since the curvature K is always not equal to zero, the validity of the left member of this relation is obvious. Because the bar is submitted to large elastic deformations, it is well known that, in the classical problem of elastica in cantilevers, the curvature-function K* can be replaced by the equivalent expression: zp,,(cose - cosa)/EJn

(see Ref. [2], p. 78, first equation).

So, the right member of inequality (3.10) can be equivalently written in the form: py,1,2(c0se- cosa)/E,J,, < 1,

(3.11)

425

Exact solution for an approximate differential equation

which a represents the angle of rotation of the free end of the deformed cantilever. If we substitute the term P,,,lf/E,J, by a complete elliptic integral of the first kind X(p), given in [2], which is greater than unity, we obtain: in

x*@xc0se - cosa) < 1;

w-*(P)> 1).

It has been shown in [2] by a series of applications that the validity of this inequality in the case of thin and long bars is obvious. So, the non-linear term (1.2q)-i’* of equation (3.9) can be written in an absolutely converging Maclaurin series, fact that allows the following simplification on the form of this equation: (3.12) Taking into account that: c2 = sin*8

1 =- (1 - c2)-3/2 = 1 + it2 + o(t4);

<

5*+ 1,

relation (3.12) can be written in the following final form: (3.13) The integration of the last ordinary second-order algebra, gives:

linear differential equation, after some

~(5) = C,coskS + C,sinkS + F<* + G,

(3.14)

in which C,; C2 are arbitrary constants of integration, whereas F; G are constant coefficients given by: k* = 2Cl;.

F = 3A/k*;

By now, based on relation (3.6), equation (3.14) becomes: z(e) = C,cos(ksin@ + C2sin(ksin@ + Fsin*8 + G,

(3.16)

which, by using equation (3.1), yields: 6’(s) = +[C,cos(ksinB) + C,sin(ksine) + Fsin*O + Cl’/*. Introducing

(3.17)

the constraint forces at the boundary conditions, being expressed by: fore

= 8,;

and:

e:: 1 =

M,Z+,IE:J:

fore = em+l,

we may now calculate the integration constants C,; C2 as follows: C r = [Ksin(ksinB,+ i) - Lsin(ksine,)]/A

C2 = Kcos(ksin0,) - Lcos(ksine,+ JA

(3.18)

in which: K = M~/E~J~ - Fsin*O, - G; L = Mi+ ,jE,2J,2 - Fsin*O,+ 1 - G,

(3.19)

D. E. PANAYOTOUNAKOS and P. S. THE~CARIS

426

and: A = cos(ksin&)sin(ksin& + 1) - cos(ksin0, + ,)sin(ksine,). Neglecting the positive sign in relation (3.17), because d0/ds is in fact negative (see Fig. l), we derive the following elliptic equation: [C,cos(ksin@ + &sin(ksinB) + Fsin28 + G] - l12d0 = - ds.

(3.20)

The last integral can be evaluated by a convenient numerical integration. If we want to express the function e(s) under an analytic approximate form, the following consideration can be done. Using the transformation: sin0 = t s de = dt/(l - t2)‘j2, and developing the functions cos(kt), sin(kt) in a Maclaurin series, we find that the lefthand side member of equation (3.20) takes the following general hyperelliptic form:

s

Cw 2p+1 +

alt2” + a2t2P-2 + ... + a2p+l]-“2dt

= -ds,

(3.21)

in which p is a natural number (including zero). This integral can be evaluated in a closed form, based on [14]. Now, it is clear that since the function 0(s) is known, the co-ordinates x(0’); y(P) may be determined by using equations (2.2a). In conclusion, we must underline now, that one, based on equations (3.20), (3.21), may obtain the direct closed-form solution of the problem of large elastic deformations for thin bars submitted to concentrated terminal loads and to a uniformly distributed loading. This closed-form solution is more accurate than the previously developed, that is the solution using computer cods [S, 91, as well as the solution using the substitution of the distributed loading by a finite number of discrete loads [l 11. 4. SPECIAL

CASES

In this section we shall examine some special cases concerning the loading of a segment of an elastic plane frame-type structure; for these cases the analytic solution of the governing non-linear equilibrium differential equation (2.3) may be derived from the general solution, which was given in the previous paragraph. 4.1. A structure loaded by a distributed loading4 Firstly, we consider the case where a part of the structure is loaded only by a distributed load q. A simple example for this case is the non-linear and buckling analysis of a nonsymmetric portal rigid frame, loaded by two equal co-planar compressive forces (acting centrically along its two vertical members) together with a distributed vertical load on the horizontal member. Then, the coefficient A of the governing equilibrium equation (2.3) for the horizontal member becomes equal to zero. So, the equivalent to equation (2.3) differential equation (3.8) takes the form: y - 4C(fJ)--“2 = 0.

(4.1)

Using the substitution: (4.2) we conclude that: Y = dpldt

= (dp/WWdt)

= PP‘,

(4.3)

Exact solution for an approximate differential equation

where primes indicate differentiation to:

427

with respect to rl. Consequently, equation (4.1) leads dp* = 8Q-“*dq.

(4.4)

A first integration of the last equation gives: (cl + 16C$‘*)- “* drl = d&

(4.5)

which, based on relations (3.6) and (3.1), can be written under the following equivalent form: (ci + 16C8’)-“* .28’d@ = cos6d6.

(4.6)

Integrating equation (4.6), one may obtain (see Ref. [15]) PI* - 3c,h”* = B(sin0 + sin&),

(4.7)

where: /I”* = (ci + 16CB’)“*;

0 < B = 3(16C)*/4 << 1,

and sine,, is a constant of integration. If we consider equation (4.7) as a cubic equation with respect to h”* (of Cardan form), one may assume, without loss of generality, that the discriminant satisfies the inequality: Q = -CT + [B(sinfI + sinf&J/2]* < 0. So, the three roots of equation trigonometric form [16]:

(4.7) are real and the first one has the following

hi’* = 2c;‘* cos(a/3)

(4.8)

cosa = B(sin6 + sin6,)/2c:12.

(4.8a)

where:

By now, it may be noticed that the quantity B(sin6 + sin&,)/2c:‘* is much smaller than unity (B* < c:) and so the angle CLapproaches the value n/2. Consequently, developing the functions cosa;cos(a/3) in Maclaurin series, we have: cos(a/3) = 819 + cosaf9 - 8a4/(9* x 4!) + *e*.

(4.9)

Because a -, x/2 one may write, by introducing a maximum error of about 2.5%, the following relation: cos(a/3) = 819 + cosa/9.

(4.10)

Introducing the expression (4.8a) into the last relation we have: rdB/[(a + Bsine)(b - &in@] = -ds,

(4.11)

in which: I- = 1296Cc:;

a = 251~:‘~+ Bsine O**

b = -77~:~~ - Bsineo.

(4.1 la)

a + BsintJ -- r (b-a2Bsin6)de + c a + b s (a + Bsin@o( - Bsine) 3’ - Bsine

(4.12)

The integration of equation (4.11) gives (see Ref. [15]), Is = B(alnb

where c3 is a new constant of integration.

428

D. E. PANAYOTOUNAKOS and P. S. THE~CARIS

The elementary integrals in the right-hand side member of equation (4.12) can be calculated, based on the procedure included in [lS]. In conclusion, one may underiine that in the case of a segment of the structure loaded only by a distributed load (without axial forces), the closed form solution of the governing equilibrium non-linear differential equation is given by the relation (4.12). An important fact here is that the form of the solution includes a maximum error of about 2.5%. 4.2. A structure loaded by terminal co-planar concentrated loads As a second case we consider now the problem of a segment of the structure loaded only by terminal co-planar concentrated forces. In this case the coefficient C is equal to zero and equation (3.8) becomes: Tj = 2A/cos%

(4.13)

Substituting the function y, given in equations (3.7), into the last equation we derive: (4.14)

i’ + tan& = tA/cost?

This equation is a linear first-order differential equation with respect to i, the general integral of which can be easily derived as follows: z = 2c,sinO - 2AcosB + Zc,

(4.15)

where c,; c2 are arbitrary constants of integration. By now, based on relation (3.1), equation (4.15) can be transformed into the equivalent relation: dO/ds = -[2(c,sinO

- Acose +

cJJ~/~.

(4.16)

The final solution of equation (4.16) can be achieved by a direct integration being included in [4-61, through elliptic integrals of the first and second kind. 5. CONCLUDING

REMARKS

It is worthwhile pointing out that the methodology which is developed in this paper gives a closed-form solution of the strongly non-linear differential equation governing the equilibrium of a straight and prismatic bar, subjected to a conservative system of arbitrary and distributed loading, contrariwise to the existing literature on this subject. The closed-form solution was derived by using successive functional transformations, which changed the governing strongly non-linear differential equation to a directly integrated one. Since the bars may be considered as segments of an elastic rigid frame like structure, the general closed-form solution, which is presented here, can be applied to analyse nonlinear and buckling problems in rigid frames without the need of computer codes or numerical integrations. 1. 2. 3. 4. 5. 6. 7. 8. 9.

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Exact solution for an approximate differential equation

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