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A Closed Form Solution for Reissner's Planar Finite-Strain Beam Using Jacobi Elliptical Functions

In the article we introduce an analytical solution for Reissner's large-deflection finite-strain planar beam subject to an end force and a bending moment. The solution is given in terms of Jacobi elliptical functions. The obtained analytical solution is enhanced with numerical examples. A buckling and post buckling behavior of a beam under axial compressive load applied at the end and subject to various boundary conditions is also discussed in some details. In particular, the buckling factor is derived for each case of the boundary conditions.


Introduction
In this paper we are concerned with analytical solution of the equations for large deflection of initially straight, weightless, uniformly isotropic, linear elastic beam subject to end load which was in 1972 purposed by Reissner (Reissner, 1972). The theory behind these equations enhanced the well-known Euler-Bernoulli large-deflections beam theory with starching and shearing strain. For convenience we will call a beam which fit this extend theory the Reissner's beam, although the equations are formally identical to the equations, which can be obtained from the planar Cosserat beams theory by assuming linear constitutive relations (Antman, 2005).
Available literature on this very specific subject is relatively rare. The elliptic integrals solution for simple supported extensional beam under axial compressive force has been given by Pflüger (Pflüger, 1950), Stoker (Stoker, 1968) and Magnusson with coauthors (Magnusson et al., 2001). The elliptic integral solution for the Reissner's beam the solution was provided by Humer (Humer, 2013) . It is of order to mention also two other solutions. Goto and coworkers (Goto et al., 1990) published a closedform solutions for elastic beam with axial and shear deformations, using elliptic integrals. However underlying theory the authors adopt was the Timoshenko beam theory of finite displacements with finite strains and that with small strains. Unlike Hummer solution which involve only elliptic integrals of first and second kind their solution also include elliptic integral of third kind. The elliptic integral solution for extensional beam was also given by Stemple (Stemple, 1990), however, the equations that he integrates were derived from his own beam theory.
This brief review shows that available analytical solutions for extensional and shear-deformable beams are given only in terms of elliptic integrals. A shortcoming of an elliptic integral solution is that it is implicit, meaning that in the formulas for the beam coordinates the independent variable is beam cross-section inclination and not the beam arc length.
The aim of this paper is to provide a solution for the Reissner's beam in terms of Jacobi elliptical functions. For the Euler-Bernoulli beam such solution was proved more suitable for both numerical computation as well as for the analytical treatment than the solution using elliptic integrals (Batista, 2014(Batista, , 2015aGoss, 2003;Levyakov, 2001;Love, 1944). We will also give an applications of the solution primarily as an indication of its ability.
Before proceeded we note that since we consider only the integration of Reissner's equation we omit reviewing some important topics. Thus, for the history of the large deflections of beams beside mentioned works, we refer to Antman's article (Antman, 1972) , Gorski survey paper (Gorski, 1976) and Goss dissertation (Goss, 2003). For qualitative treatment of the solutions for nonlinear elastic beams the primary reference is Antman's book (Antman, 2005), and for numerical treatment we refer to Saje (Saje, 1991) and Batista and Kosel (Batista and Kosel, 2005) .

Basic equations 2.1 Problem statement
We consider an initially straight Reissner's beam of length  , which will be used as unit of length, with one end fixed and a force and a bending moment acting at the other end. In the Cartesian coordinate system OXY the shape of the deformed base curve of the beam is described by the following differential equations (Reissner, 1972), (Eq 14a,14b,10) ( ) In these equations X, Y are coordinates of deformed beam base curve, φ is angle between X axis and the outward normal to sheared cross section of the beam, ε , γ, and κ are successively axial, transverse and bending strains and parameter 0 1 s ≤ ≤ is length parameter of undeformed beam, measured from the beam immovable end to the beam movable end (Figure 1). Thru this paper we will assume that at initial state beam is on X axis and that the immovable beam end is at the coordinate's origin Since the beam under load cannot became a point the physical limitation for ε is (Antman, 2005) 1 0 This condition also prevent that the normal to the sheared cross-section become orthogonal to the base curve.
Beam equilibrium equations are (Reissner, 1972) (Eqa 2*a,b and 3*) where N and Q are respectively normal and shear forces with respect to deformed cross section and M is a bending moment.

Figure 1. Geometry and load of the beam
We assume that the forces and moment are related to deformations by the following linear constitutive equations (Reissner, 1972) (Eqs 32a,32b, 32c) where EA , s GA and EI are positive constants which represent respectively axial, shear and bending stiffness of the beam. For the interpretation of these equations and future references we refer to Irschik and Gerstmayr (Irschik and Gerstmayr, 2009) and Humer (Humer, 2013) If the force F is acting on the movable end of beam under clockwise angle α measured from the negative X axis, then the solution of the force equilibrium equations (5) are (Reissner, 1972) Substituting these expressions for N and Q in constitutive equations (7)1,2 we get Substituting for M from constitutive relation (7)3 into moment equilibrium equation (6) we obtain, using expression for forces (8), In this way the problem is reduced to the integration of the system of two nonlinear differential equations (2) and (10) for unknowns φ and κ . Once these are known X and Y can be obtained by integration of Eqs (1).

Equations transformation
We introduce the new variable ψ defined by which is a counter clockwise angle between the direction of force and inward normal to the beam deformed cross section. Then we rewrite Eqs (9), (2) and (10) in the form The non-dimensional parameters in these equation are the load parameter ω, the generalized slenderness ratio λ and the stiffness ratio ν which are defined by (Batista and Kosel, 2005) The parameter 2 ω represent dimensional force while ω has no physical meaning. In limit when 0 s EA GA = we have 1 ν = and EA EI λ =  . Such beam is shear-less ( ) 0 γ ≡ and allow only stretching. We note that the ordinary slenderness ratio λ′ (Humer, 2013;Timoshenko, 1961) and the stiffness ratio ν ′ (Humer, 2013) are define by The connections of these parameters to the present parameters λ and ν are the following (The later by (7)2 imply 0 Q → which is incompatible with beam equilibrium equations.) For ν ′ = ∞ we have λ λ′ = and 1 ν = . Thus with present parameters we obtain solution for the shear-less beam directly and not thru a limit process.
Using ε from Eq (12) in condition (4) and taking 0 ψ = (maximal ε ) we obtain upper limit for the load In particular for shear-less beam load parameter is limited by the slenderness to 2 2 ω λ < , while for stretch-less beam the load parameter is unlimited 2 ω < ∞ .
We next transform Eqs (1) for X and Y in the following way. By rotating the coordinate system about the origin by α in the clockwise direction we obtain new coordinate system Oxy. The transformations between coordinates of the systems are Note that conditions (3) imply Substituting above expressions for X and Y in to the Eq (1) and using Eq (11) for ψ and Eq (12) for ε and γ , we obtain ( ) Now, once we solve differential equations (13) and (14) for unknowns ψ and κ we obtain φ from Eq Further, from Eqs (21) , by integration, using conditions (20), we obtain coordinates . Knowing x and y we obtain X and Y from Eqs (19).

General solution
We eliminate s from Eq (14) The result of integration of this equation with respect to ψ and subject to condition Future, using Eq (29)1, we can express κ as Further simplification of this equation is obtained by introducing yet another variable t defined by Gauss type transformation (Groebner and Hofreiter, 1961) Substituting this in Eq (36) , we get As is well-known the general solution of Eq (38) is were sn is Jacobi elliptic function and C is constant of integration. From this we obtain by consecutive substitutions into Eq (37) , then obtained expression into Eq (35) , and finally into Eq (29), the final solution of Eqs (13) and (14) ( where cn is Jacobi elliptic function. Using these expressions for ψ and κ we from Eqs (23) and (24) obtain coordinates. Both integrals in Eq (23) was found by the Maple program. We note that complete elliptic integral of third kind vanish from expression for x once obtained result is simplified. The final result can be written in the form Here K and E are complete elliptic integrals of first and second kind, respectively, and dn and Z are Jacobi's elliptic functions.
The remaining unknown of some interest is the length of deformed beam L which is given by In general this integral has no closed form solution. However, there is a special case 0 Integration yield the following expression, valid for 1 (Lawden, 1989) and am is Jacobi's amplitude function (Reinhardt and Walker, 2010).

Properties of the solution 4.1 Solution domain
The solution given by Eqs (41), (42), (43) contains three parameters λ , ν , ω and two constants C and k by means of which we can calculate parameters 2 m , ω  , k  by using Eqs (34) and (39) respectively. Since we interested only in real solution we assume that k is real and that ( ) This inequality ensures that ω  is real and also that 2 1 0 m + > , both of which are easily verified by Eq (34). From inequality (18) The inequalities (18), (47)  we can like in the case of elastica (Batista, 2014;Goss, 2003;Love, 1944) classify three different types of underlying curves: for 1 k <  the underlying curve is inflectional (Fig. 4a), for 1 k =  the curve is homoclinic (Fig. 4b) and for 1 k >  the curve is non-inflectional (Fig. 4c)

Formulas for k
The solution do not depend on 1 κ , which appears in Eq (31). To see that we substitute A from Eq (29) 2 into Eq (31). In this way we obtain This relation is automatically satisfied by ( ) from Eq (41) When 2 0 ν λ = the above expression reduce to which is relation for elastica (Batista, 2014). Alternatively, we can write Eq (49) in the factorized form When 1 0 κ = then above relation is satisfied if This relation is valid for any inflection point. When 2 1 k = then 1 κ and ω are not independent. As follows from Eq (52), they are related by

Symmetry
By replacing k in Eqs (41) This is solution which describe the beam which is symmetric with respect to x axis. However in order to obtain a symmetric shape in the system OXY we must also replace α in Eq (19)

Some special cases
When 2 1 0 λ = then we have, from Eqs (34) and (39) , In this case Eqs (41), (42), (43) reduce to known solution for elastica (Batista, 2014) In the case when 0 k = we obtain by means of Eqs (34) and (39) Hence the solution given by Eq (41) When 0 ω = then we obtain the solution for the pure bending of the beam. From Eqs (34) and (39) It follows from Eq (50) that k → ∞ as 0 ω → , however, in this limit we have This results shows that the beam subject to pure bending deforms into a circular arc. In this case α is indeterminate and represent rigid body rotation.

Numerical examples
In this section we will use the present solution for calculation of a shape of deformed cantilever under end load. The boundary conditions of the problem are where 1 κ is expressed in terms of moment M by 1 M EI κ =  . We assume that parameters λ , ν , ω , 1 κ are given and that the unknowns of the problem are k, C and possible α . We will consider only the nontrivial cases when 0 k ≠ .

Cantilever under follower force
As the first example we consider a cantilever subject only to a follower force. The direction of the force with respect to beam is in this case ( ) , therefore, k is given by Eq (53). By means of Eqs (11) and (41)  1 cn , Substituting κ given by Eq (42)  , and therefore in particular for ( ) In this way we obtain an explicit expressions for all three unknowns' k, α and C. The results obtained by the present analytical methods presented in Table 1 (Batista, 2013) .  Table 1

Follower force and moment
As second example we consider a cantilever subject to the end follower force and the end moment. We therefore have 1 0 κ ≠ and thus k is given by Eq (50) Again, we obtain the explicit expressions for all three unknowns' k, α and C. We note that when 1 k = then 1 κ is given by Eq (54).
For comparison with present analytical solution we conduct numerical integration of Eqs. (13), (14) and (21) for three cases by using the Shvartsman method (Batista, 2013;Shvartsman, 2007). Again, the agreement between results of analytical and numerical solution are excellent (Table 2). . The analytic solution were obtained by the Maple program with 14 digits = . Numerical solution were obtained by routine dopri5 (Hairer et al., 1987) with all tolerances set to   Table 2

Cantilever under constant force
In this case α is given. From the boundary conditions ( ) 0 0 φ = , on using Eqs (11) and (41) By means of Eq (42) the boundary condition ( ) This is transcendental equation for unknown k which can be solved by numerical methods.
Alternatively, we can in this case take 1 sin 2 k ψ = so the equation is to be solved for 1 ψ .
For numerical calculation we consider the example due to Saje (Saje, 1991) and was also treated by present author (Batista and Kosel, 2005). Agreement between results is perfect (Table 3 and Fig 5).  (Saje, 1991). Numerical values are from (Batista and Kosel, 2005) . Diff is difference between solutions

Beam under compressive axial force
In this section we will demonstrate an analytical capabilities of the present solution by treating the classical problem of the beam under axial compressive force subject to various boundary conditions ( Figure 6). Our main objective is derivation of critical (buckling) factor β defined by 2 2 ω β π ≡ (72) With β the critical force is expressed in the following way (Ziegler, 1977) We will also consider post buckling behavior of doubly spurted beams in some details. As is wellknown, a doubly supported elastica undergoes the secondary loss of stability with increasing force (Levyakov, 2001). We make conjecture that for Reissner's beam secondary loss of stability occurs under the same condition as for elastica. We will call corresponded β the secondary critical factor. We will consider only first buckling mode since higher buckling modes are most probably unstable and thus has no practical values.

Simply supported beam
The boundary conditions at beam's simpley supported ends are ( ) ( ) From ( ) 0 0 κ = we obtain, on using of Eq (42), Similarly, from ( ) This is characteristic equation which relate k and ω when the beam is in equilibrium. Figure 7 shows an examples of pitchfork bifurcation diagrams of Eq (76) . For 0 ν > the bifurcation point become unstable by lowering the slenderness (Fig 7b). For extensible beam this was observed by Magnuson and coauthors (Magnusson et al., 2001). The transition point can be calculated analytically in the following way. If we denote ( ) For 2 0 ν λ = we obtain well-known Euler buckling factor 1 β = (Timoshenko, 1961;Ziegler, 1977). The formula also imply that when 0 ν > then the beam with 2 λ π ν < will not buckle. In particular for shear-lees beam with 1 ν = we obtain 2 λ π < . This agree with result of other authors (Britvec, 1973;Magnusson et al., 2001;Stemple, 1990). Some numerical values for β for data provide by Humer (Humer, 2013) are given in Table 4. In Table 5 a comparisons of results for Reissne's beam and Huddelson's model of beam (Huddleston, 1972) is shown. We note that unlike the Huddelson's beam the Reisner's beam has the critical force for any 0 ν < .
The expression for ( ) 1 0 Y = is obtained by means of Eqs (19) and (43). To fulfil this boundary condition we have two possibilities. The first one is when sin 0 α = , which is satisfied by The second possibility is In this case the beam forms a loop, that is, we have ( )

Clamped beam
The boundary conditions for this case are ( ) ( ) Using Eqs (22) and (41) we find from the first two boundary conditions that ( ) ( ) One way to satisfy this equation is to choose ω  to be the multiple of period of sn and cn functions ( ) For 2 0 ν λ = this reduce to well-known Euler critical factor 4 β = (Timoshenko, 1961;Ziegler, 1977).
As in the case of simply supported beam, the boundary condition ( ) This, together with ( ) The second case yield to Eq (79) which is the case when ( ) 1 0 X = , that is, when the beam forms a ribbon. We again use solution ( ) * * ,k ω of Eqs (76) and (79) and (82)   We obtain asymmetric solution of Eq (81) by observing that The equation connecting α and k is therefore It can be easily shown that the symmetric and asymmetric solution intersect at the point where the beam form a ribbon.

Case when one end is clamped and hinged
The boundary conditions for this case are From the boundary condition ( ) 1 0 κ = , on using Eq (42), we obtain equation ( ) By using well known trigonometric identities we can express sinα and cosα through sin 2 α . On using Eq (90), we then obtained from the boundary condition ( )

Cantilever
The boundary conditions at clamped and free end are These conditions yields two equations ( ) Solution of the first is and of the second ( ) ( ) For 2 0 ν λ = we have 1 4 β = which is well-known Euler critical factor (Timoshenko, 1961;Ziegler, 1977) . When 0 ν > then beams with λ π ν < cannot buckle.

Cantilever with guided end
We assume, that the beam guide is frictionless so 0 α = . The boundary conditions at clamped end and guided ends are The condition ( ) For 0 ν = this reduce for well-known Euler critical factor 1 β = (Timoshenko, 1961;Ziegler, 1977).

Conclusions
We give a new closed form solution for weightless Raissner's beam subject to end load in terms of Jacobi elliptical functions. We demonstrate that solution is efficient for numerical calculation and also for analytical investigations. In particular we demonstrate, that with present solution the derivation of the formulas for force critical factor for beam under compression force is for all discussed boundary condition almost trivial. We also derive the critical and lower limit slenderness under which the rod can't buckle for all the cases of doubly supported beam. These slenderness occurs only when 0 ν > .