A multi-step linearization technique for a class of ... - Springer Link

Comput. Mech. (2006) 39: 73–81 DOI 10.1007/s00466-005-0009-6

O R I G I N A L PA P E R

Rajesh Kumar · L. S. Ramachandra · D. Roy

A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

Received: 15 March 2005 / Accepted: 22 September 2005 / Published online: 22 November 2005 © Springer-Verlag 2005

Abstract Non-linear finite element analyses of structures (such as beams) involve construction of weak solutions for the governing equations. While a weak approach weakens the differentiability requirements of the so-called shape functions, the governing equations are only satisfied in an integral sense and not point-wise, or, even path-wise. Moreover, use of a finite mesh leads to a stiffening of the numerical model. While strong solutions obtained through some of the existing mesh-free collocation methods overcomes some of these lacunae to an extent, the quality of the numerical solutions would be considerably improved if the computational algorithm were able to faithfully reproduce (or approximate or preserve) certain geometrical features of the response surfaces or manifolds. This paper takes the first step towards realizing this objective and proposes a multistep transversal linearization (MTL) technique for a class of non-linear boundary value problems, which are treated as conditionally dynamical systems. Numerical explorations are performed, to a limited extent, through applications to large deflection analyses of planar beams with or without plastic deformations. 1 Introduction There is no dearth of numerical techniques for the solution of nonlinear boundary value problems (BVPs) of interest in engineering mechanics, viz. large deflection problems of beams with geometric and/or material non-linearity. Of particular prominence among these techniques is the finite R. Kumar Department of Civil Engineering, Banaras Hindu University, Varnasi, India L. S. Ramachandra Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, India D. Roy (B) Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India E-mail: [email protected]

element method [8,11], which obtains a weak approximation to the true solution using a Galerkin-like projection principle. While a weak approximation weakens the differentiability requirements of the basis functions used to span the function space, the closeness to the true solution is only in an integral sense and not point-wise. Moreover, usage of a finite mesh size as well as an absence of sufficient continuity conditions across inter-element boundaries may lead to an artificial stiffening of the numerical model and accordingly an underestimation of the computed response levels. Many of these problems can however be remedied if, for instance, moving (windowed) collocation algorithms are used to obtain strong solutions within a mesh-free framework [6,20]. Yet another advantage of strong solutions, obtained through a point-wise satisfaction of the governing differential equations, is that no requirement to numerically evaluate integrals, which is often done using a quadrature procedure. This in turn requires a set of quadrature points to be properly chosen and a wrong choice for these points may lead to a sharp loss of accuracy. For instance, in problems having localized forms of nonlinearity (as in acute plastic deformations near a so-called plastic hinge), errors in solutions owing to a wrong choice of the quadrature rule may quickly propagate and may even result in the numerical model being ill-conditioned. Apart from the collocation procedure, several other techniques for computing strong solutions are also available. For instance, a perturbation method [21,4] has been used to solve the large elastic-plastic BVPs of a strut. Vaz and Silva [17] have recently employed a Runge-Kutta numerical integration technique to solve a two-point BVP. Wavelet-based collocation procedures have also been attempted [16]. Governing equations for nonlinear BVPs in structural mechanics are almost always in the form of differential equations (ordinary or partial), whose solution surfaces (manifolds) have certain geometrical characteristics. While a strong approach based on a collocation-type algorithm seems to be a better choice than a weak approach (especially for strongly non-linear structural systems), it would have been far more desirable if it were to automatically preserve (some of) the geometrical features of the solution manifolds. It is thus

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natural to look for a linearization of the governing nonlinear differential equations so that one can explicitly construct and modify (if necessary) the linearized manifold to preserve any invariants, isospectrality or other features of relevance. Moreover, since the point-wise solution of the linearized differential equation is either known or easier to obtain, a collocation-type algorithm should automatically result from such a procedure. It is worth noting that geometry-preserving integrators have recently aroused quite a bit of interest for integrating nonlinear dynamical systems [5]. However, these procedures are yet to be explored and applied in the context of systems of engineering interest. As a first step in realizing the above objective, an implicit family of semi-analytical integration techniques, referred to as multi-step transversal linearization (MTL), was developed for solving non-linear ordinary differential equations (ODEs) governing initial value problems (IVPs), especially those of interest in structural dynamics [13,18]. In this paper, an attempt is made to modify and adapt the MTL family of procedures for solving non-linear ODEs governing nonlinear static deflections of beams in a plane. The overall procedure of applying the MTL techniques basically remains the same in that the non-linear parts of the governing equations are converted into a set of equivalent and conditionally known forcing terms. The equivalent forcing terms are non-uniquely constructed in such a way that the linearized vector field remains identical with the original one only at a chosen set of discretization (grid) points distributed spatially across the domain of the problem. These terms are derivable via a suitably chosen discretized projection of the non-linear parts of the vector field on a finite-dimensional space spanned by any available set of basis functions (interpolating or otherwise) valid within the domain of interest. Presently a set of polynomial basis functions is used to discretize the non-linear part of the operator. The coefficients associated with these basis functions are so chosen that the linearized vector field exactly reproduces its non-linear counterpart at the discretization points and also remains transversal to the original vector field at all the points of discretization. The unknown discretized solution vectors may then be viewed and determined as the transversal points of intersections of the linearized and non-linear solution manifolds in the associated phase space. The numerical enforcement of such transversal intersections is then ensured by constructing a set of coupled, algebraic constraint equations, derivable from the conditional solutions of the linearized ODEs. Unlike the well-known technique of tangential linearization, the MTL algorithm does not require any differentiation of the nonlinear vector functions and is thus applicable even when such functions are not differentiable. The family of MTL procedures is numerically illustrated with a couple of BVPs on the static, geometric and material non-linear deflection analyses of beams. Comparisons with other standard solutions, available in the literature and those obtained via ANSYS (version 5.4), a commercially available finite element software, are also provided to numerically verify the accuracy and demonstrate the relative advantages of the presently developed algorithm.

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2 The methodology The basis of the MTL approach, considered in some detail in the following, is the replacement of the non-linear part of the governing ODEs through a set of conditionally linearized ones so that the non-linear and linearized forms of the governing equations remain identical at a set of (arbitrarily chosen) points of discretization along the axis of the independent variable (which is presently the spatial variable). However, unlike the locally transversal linearization (LTL) approach proposed earlier for a class of non-linear BVPs [9, 10], the MTL-based linearization also strives to achieves a uniform ‘closeness’ of the non-linear and linearized solutions for all points (not necessary the discretization points) of the domain of interest. Whereas the LTL approach, being a single-step technique, has to be so applied as to convert the given nonlinear BVP to a conditional IVP, it should be possible to apply the MTL approach, which is global or semi-global in nature, in a more direct manner. In order to achieve these objectives, it is important to arrive at a suitable discretized and (globally) linearized form of the non-linear parts of the vector fields (see [1] for a definition of vector fields). Further discussion would be facilitated and considerably simplified by taking up an example problem and building the technique around it. Thus consider the planar, static, large-deflection (Elastica) problem of a cantilever beam subjected to a transverse, concentrated load, P , at its free end [15]. The governing second order non-linear ODE is given by: P d2 θ cos θ = 0, (1) + 2 ds EI where θ, P and EI are, respectively, the slope, lateral load and flexural rigidity of the beam and s denotes the independent spatial coordinate along the deformed longitudinal (centroidal) axis. It is now intended to find a suitably linearized replacement for cos θ, valid uniformly over the entire domain  = [0, L], where L denotes the undeformed length of the centroidal axis. Towards this, consider cos θ = cos θ (s) as a functional of the independent variable, s, and express it through a Taylor expansion as: d cos θ (s)|s=s0 h ds h2 d2 + ··· , (2) + 2 cos θ (s)|s=s0 ds 2 where h is a spatial increment given by h = (s − s0 ). The above expansion may be viewed as taking place in a function (vector) space, which is spanned by the basis functions 1, h, h2 , h3 , . . . and so on. Accordingly, the following should be a valid expansion for cos θ (s) : cos θ (s0 + h) = cos θ (s0 ) +

cos θ (s) ≈ cos θM (s) = b0 + b1 (s − s0 ) + b2 (s − s0 )2 +b3 (s − s0 )3 + · · · ,

(3)

where s0 is the left boundary of the domain , i.e., for the present problem, one has s0 = 0. At this stage, the domain 

A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

is discretized using p equal intervals, each of size h, or p + 1 equi-spaced discretization points, sj , j = 0, 1, . . . , p with h = sj +1 − sj . The assumption on a uniform interval-size is made only for convenience and may readily be relaxed. Now, the following approximation for the replacement of cos θ (s) based on the pth level MTL procedure is introduced: (p)

cos θM = b0 + b1 (s − s0 ) + · · · + bp (s − s0 )p +bp+1 (s − s0 )p+1 .

(4)

The first (p + 1) coefficients viz. b0 , b1 , . . . , bp , are presently determined in such a way as to interpolate the functional (p) approximant, cos θM , over the discretized function values, cos θ(sj ) cos θj , j = 0, 1, . . . , p. Thus one readily has: b0 = cos θ0 = 1

(for a cantilever fixed at s=0)

(5)

The last co-efficient bp+1 is found from the known boundary condition at s = L, i.e., dθd(L) = θ  (L) = 0. Now, with a s constant mesh size h = sJ +1 − sj , j = 0, 1, · · · , p − 1 (with s0 = 0 and sp = L), one has the following system of linear equations to determine b1 , b2 , . . . , bp+1 in terms of cos θk , k = 0, 1, . . . , p :    h h2 · · · hp+1 b1      2h (2h)2 · · · (2h)p+1   b2    .     .   ..  .  . .  . .    .   ..      ph (ph)2 · · · (ph)p+1    b   p   2 p+1 b h 2ph · · · (p + 1)(ph) p+1   cos θ1 − 1     cos θ2 − 1            ..  . = . . (6)   ..             cos θp − 1   0 It may be noted that in deriving the last equation in the above system of equations, the condition θ  (L) = 0 has been multiplied by the mesh size, h, on both sides. This has been done in order to avoid a possible ill-conditioning in the above system of equations for small values of h. It is worth mentioning and important to note that one is not restricted in choosing only Taylor-like polynomials (p) {Pk } to arrive at the replacement function cos θM in Eq. 4. For instance, one may use Lagrangian polynomials, any windowed reproducing kernel approximation as used in a meshfree approach [4], distributed approximating functionals [19] or wavelet-based (such as the Deslaurier–Dubuc) interpolating functions, sometimes referred to as interpolets [3], to (p) obtain cos θM . Thus a possible form of the linearized ODEs corresponding to the original ODE (1) via the pth level MTL procedure becomes: d2 θ¯ (s) (p) = −λ2 cos θM (s) (7) ds 2

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where, λ2 = P /EI and EI denotes the (uniform) flexural rigidity of the cantilever beam. In the above equation, θ¯ (s) stands for the MTL-based approximation for θ (s). It is noted that an explicit computation of the right hand side of Eq. 7 (p) is not possible as cos θM (s) is a function of discretized state variables, θk , k = 0, 1, . . . , p, which are not known. Nevertheless, Eq. 7 may be conditionally integrated twice (with respect to the independent variable, s), thereby leading to:  s ξ

(p)

cos θM (ξ1 )dξ1 dξ .

θ¯ (s) = −λ2 0

(8)

0

The above integration may be conditionally performed (i.e., given some guess values for θk , k ∈ [0, p]) using Simpson’s, Gaussian quadrature or any other quadrature rules. At this stage, the unknown state variables, θk , may be determined as transversal points of intersections [12] of the paths traced by θ (s) and θ¯ (s) at s = sk , k = 1, . . . , p. Towards this, the following constraint equations may be readily introduced: θk = θ¯k sk ξ = −λ

(p)

cos θM (ξ1 )dξ1 dξ,

2 0

k = 1, . . . , p.

(9)

0

In other words, the transversality of the solution essentially implies [12] that the discretized solutions θk = θ¯k belong to ¯ where M and M ¯ are manifolds containing the true soluM∩ M tion θ (s) and the approximated solution θ¯ (s) respectively. Equations 9 constitute a set of p non-linear algebraic equations for {θk |k = 1, . . . , p } and the associated roots may be found with a standard solver, e.g. the Newton-Raphson method. 2.1 Numerical examples As considered in Sect. 2, the cantilever beam subjected to a transversal tip load is a classical instance of a nonlinear BVP with a purely geometric form of non-linearity. A numerical exploration of this simple example, as undertaken in this sub-section, makes way for a more versatile application of MTL method to different type of problems. The cantilever beam represented by Eq. 1 is considered here. The MTLbased results for a cantilever beam are compared with those via an exact method (i.e., closed form solutions obtainable through elliptic integrals) and ANSYS (a commercially available finite element code, version 5.4) in Table 1. The element used in ANSYS is the two-noded beam element (2D, Elastic 3). From the table it is clear that results obtained by the proposed method are closer to the exact results as compared withANSYS results. It has also been seen that the MTL-based displacements (or slopes) are consistently on the higher side as compared to ANSYS results. This is probably due to the MTL-based numerical model being less stiff (and thus closer to the nonlinear model) than the one based on a classical finite element method. Unlike a weak, finite element-based

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Table 1 Deflections and slopes of a tip loaded Cantilever beam PL2 /EI 1.0 2.0 3.0 0 5.0 6.0 7.0 8.0 9.0 10.0

w/L

Exact [6] u/L

α

w/L

ANSYS u/L

α

w/L

MTL u/L

α

0.30172 0.49346 0.60325 0.66996 0.71379 0.74457 0.76737 0.78498 0.79906 0.81061

0.05643 0.16064 0.25442 0.32894 0.38763 0.43459 0.47293 0.50483 0.53182 0.55500

0.46135 0.78175 0.98602 1.12124 1.21537 1.28370 1.33496 1.37443 1.40547 1.43029

0.30000 0.49197 0.60132 0.66605 0.70951 0.73994 0.76235 0.77922 0.78743 0.79454

0.05510 0.15823 0.25070 0.32178 0.37814 0.43433 0.47274 0.50136 0.53229 0.5515

0.45985 0.77892 0.97534 1.11675 1.20234 1.27234 1.31967 1.35764 1.38762 1.41865

0.30183 0.49378 0.60364 0.67139 0.71533 0.74637 0.76985 0.78521 0.80432 0.82263

0.05651 0.16058 0.25432 0.32946 0.38915 0.44338 0.48246 0.51107 0.54371 0.56323

0.47132 0.79024 0.98923 1.14278 1.23645 1.29830 1.34626 1.38428 1.41678 1.44827

where, w/L is non-dimensional lateral deflection, u/L is non-dimensional axial deflection and α is slope

Fig. 1 Deflected shape of cantilever beam (subjected to tip load) at different values of P/Pcr

method, the MTL method preserves the forms of the vector fields in the linearized and non-linear equations at the discretization points. Moreover, it preserves the virtue of a meshless collocation method by completely doing away with the requirements of preserving continuity at the inter-element boundaries. This could possibly be one reason to explain why MTL-based results are closer to the true solution, which is presently obtainable in the form of elliptic integrals (Fig. 4). However, the MTL-based approach presently employed is in a strong form vis-`a-vis the weak approach employed in ANSYS. This precludes a direct comparison of accuracy of the two approaches and further investigations are certainly necessary before providing a more scientific basis for the validity of the above observations. The deflected shapes of cantilever beam at different values of P L2 /EI are shown in Fig. 1. In Fig. 2, comparisons of slope (at the free end) with those obtained through ANSYS have been shown, in graphical form, for different values of

the load, P L2 /EI . Comparisons of convergence results with an increasing number of discretization points (nodes) using both the MTL and ANSYS have been shown in Fig. 3. The rate of convergence appears to be nearly the same in both the cases. The percentage errors in lateral tip deflection (w) with respect to the exact values, for both ANSYS and MTL using the same number of nodes, are shown in Fig. 4 at different values of the loading parameter. It is clear from Fig. 4 that errors in MTL-based results are conspicuously less as compared with ANSYS results. 3 An MTL approach for elasto-plastic analysis of beams In this section, the realm of application of the MTL technique is extended to a problem with material nonlinearity, i.e., the elasto-plastic analysis of beams. Governing equations of elasto-plastic analysis of a cantilever beam (strut) may be

A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

Fig. 2 Variation of slope at free end with the load for cantilever beam

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Fig. 4 Percentage error of lateral deflection (w/L) from exact values at different PL2 /EI

The Plastica equation [21] for a cantilever beam (Figs. 5, 6) may be written as: β dθ =  , 0 ≤ s ≤ l, (10a) ds (1 − 2f δp + 2fy) dθ = β [1 − f (y − δp )], l ≤ s ≤ 1, (10b) ds dx = cos θ, (10c) ds dy = sin θ, (10d) ds where θ is the local angle of inclination, s is the non-dimensional arc length =s • /Lp , s • is the arc length, Lp is length of

Fig. 3 Convergence of slope at free end with number of points for cantilever beam at load PL2 /EI = 9.0

Fig. 5 The elastic-plastic stress distribution across a section

obtained by extending of the theory of the Elastica. The equations governing large plastic deformations of beams, known as Plastica equations, have been reported by Yu and Johnson [21]. These equations, wherein an elastic-plastic stress-strain effect is incorporated, provide a more realistic description of large deformation behaviour of beams than the theory of Elastica. Presently, an effort is made to numerically solve the Plastica equations via the MTL method. In the process, the versatility of the MTL method is checked, once again, through a problem whose vector fields are only continuous but not differentiable.

Fig. 6 A vertical strut loaded by a vertical force at its free end

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R. Kumar et al.

plastic region, β is the non-dimensional parameter = Me Lp / EI , δp = δp• /Lp = y(1) is the non-dimensional deflection of  point C, δp• = y • s • =Lp is the deflection of point C, l = Lp /L, x = x • /Lp and y = y • /Lp are non-dimensional Cartesian co-ordinates and f ≡ F Lp /Me is a non-dimensional load parameter. The boundary conditions for the cantilever beam are: θ (0) = x(0) = y(0) = 0 and y(1) = δp or θ  (1) = β, 0 ≤ s ≤ l

(11)



where θ = dθ/ds It is now intended to solve Eqs. 10 along with boundary conditions Eq. 11. Equations 10, consist of two parts, viz. Eqs. 10a and 10b, and may thus be solved sequentially in a piecewise manner. First, Eq. 10a may be solved along with the associated boundary conditions in Eq. 11. Equation 10a, with the help of Eq. 10d, may now be written as:   f dθ 3 d2 θ =− 2 sin θ = γ (θ, θ  ) 0 ≤ s ≤ l. (12) ds 2 β ds Equation 12 with boundary conditions Eq. 11 leads to the boundary value problem in the form of a non-linear second - order ordinary differential equation. These equations need to be solved using an MTL approach on the lines similar to those adopted for the Elastica problem (in Sect. 2). For a better illustration of implementation of the MTL procedure for this problem, assume that the cantilever beam is descretized in p equal parts (Fig. 7). Following the MTL approach, the interpolation schemes for θ(s) and θ  (s) may be written as: θ (s) = θ  (s) =

p  i=1 p



θi qi (s),

(13a)

originally non-linear ODE (Eq. 12) may now be transversally linearized as:  p  p     θi qi (s), θi qi (s) . (14) θˆ = γ i=1

i=1

The right hand side of the linearized form of Eq. 14 may be interpreted as being a conditional function of s alone, provided that the knowledge of the discretized state variables  θi , θi | i = 1, 2, . . . , p is available. Subject to such availability, θˆ and θˆ  can be found by directly integrating the right hand side modulo the evaluation the additional constants of integration. Let the MTL-based linearized form and its corresponding (conditional) solutions be written as: θˆ  (s) = (s), s  θˆ (s) = (ξ )dξ + c1 ,

(15a) (15b)

0

θˆ (s) =

 s ξ

(ξ1 )dξ1 dξ + c1 s + c2 , 0

0

where the notation (s) = γ



p 

i=1

θi qi (s),

(15c) p  i=1

 θi qi (s) is

used for convenience. The arbitrary constants of integration, c1 and c2 , may be obtained (conditionally) using the known   boundary conditions θ (0) = θ0 = 0 and θ  (1) = θp = β (as was done while solving the Elastica problem) or even using any other pair of discretized state variables. Thus,   treating, for instance, the first unknown state variable pair, θ1 , θ1 , as the conditions to determine c1 and c2 , one has:   (16) c1 = (s)ds s=s1 − θ1 . s

θi qi (s),

(13b)

i=1

where θi and θi are the state variables, i = 1, 2. . ., p and {qi (s)} is a set of polynomial interpolation functions. It is noted that the integer, p, which denotes the number of interpolation points, determines the order of accuracy of the MTL method. Analogous to the case of initial value problems, the

This enables expressing c1 in terms of the state variables. Similarly,  

(ξ )dξ ds|s=s1 + c1 s1 − θ1 . (17) c2 = s

ξ

Thus, c2 is also expressed in terms of the state variables. Now the conditions of transversal linearization may be invoked at all the discretization points excepting the first one (i.e., the fixed boundary point, j =0). These conditions may be expressed in terms of the following non-linear algebraic equations.    (18) θj = (θk , θk , s)ds s=sj + c1 (θk , θk ), s 

(θk , θk , s)ds s=sj + c1 (θk , θk ) + c2 (19) θj = s

Fig. 7 A schematic representation of discretized cantilever beam

with k = 0, 1, . . ., p and j = 1, 2, . . ., p. Equations 18 and 19 yield 2p equations in 2p + 2 variables with 2 of them [θ (0)and θ  (1)] known. Thus the roots of these equations may be determined using any established search procedure,

A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

such as the Newton-Raphson procedure. Finally coordinates of deflected shape (including Lp , i.e., the length of the plastic zone) are determined with the help of Eqs. 10c and d. Numerical integrations of these last two equations are performed using a Simpson’s rule. The boundary conditions for the entire elastic-plastic problem (i.e., over the whole length of the beam) are: θ (0) = x(0) = y(0) = 0

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Table 2 Slope (α) and deflection (δ) at the free end of the cantilever beam: Exact and MTL results (β = 0.1) f = 1.0 2.0 3.0 0 4

FL Me

α = θ (1) Exact [20] 0.10371 0.10850 0.11531 0.12775 0.14124

MTL 0.10368 0.10844 0.11524 0.12666 0.13995

δ = y(1) Exact [20] 0.05231 0.05537 0.05982 0.06823 0.07779

MTL 0.05205 0.05534 0.05978 0.06765 0.07708

and θ  (1) = 0.

(20)

For the elastic plastic deflection problem for whole beam, Eqs. 10 along with boundary conditions (Eq. 20) are required to be solved. Equation 10 has been solved by using results obtained from the solution of Eq. 12 and then applying Simpson’s rule for numerical integration.

will appear and spread from the root. For this elastic-plastic problem, the governing Eq. 10 is solved with boundary conditions (Eq. 20). The load deflection relationship has been shown for different value of β in Fig. 9. Figure 10 shows the deformed shape for β = 0.3. In Fig. 10 the locus of the plastic hinge is also shown.

3.1 Numerical results

4 Conclusions

Figure 7 (schematically) shows the large deflection of a cantilever beam (vertical strut or column) under the action of a vertical force after initial elastic buckling. With reference to this figure, the Plastica problem may be viewed as a combination of following two problems. If the bending moment at any cross-section of the strut is less than the maximum elastic bending moment, Me , deflection follows the Elastica equations. Otherwise, there must exist a point C with bending moment Mc = Me , so that the lower region CA is in the elastic-plastic bending state. The lower region CA may then be regarded as a vertical column loaded by Me and a vertical force F at its ‘free’ end, as shown in Fig. 8. Presently, Eq. 12 which corresponds to Fig. 8, subjected to boundary conditions (Eq. 11), is solved by the MTL method for the large plastic deformations. The results have been compared with the results of so-called ‘exact’ numerical integration [21] for β = 0.1 in Table 2. Results obtained by the proposed method are very close to the ‘exact’ results. Now, for the original problem, as depicted in Fig. 6 and covering the entire beam length, it is assumed that the beam is initially straight and subjected to a vertical force at its free end. If the condition F · δ • ≥ Me holds true, a plastic region

The MTL technique, recently proposed by Roy and Kumar [13] and Viswanath and Roy [18] for non-linear oscillators, has presently been extended and explored, to a very limited extent, for solving a class of non-linear BVPs, governed by ODEs. In particular, the present effort has been directed towards solving the Elastica and Plastica equations for a cantilever beam and comparing the results through MTL with those via several other alternative tools. The presently developed MTL method may be viewed as an improvement over and further generalization of the LTL technique developed earlier by Ramachandra and Roy [9,10] for non-linear IVPs and Ramachandra and Roy [9,10] for non-linear boundary value problems. More specifically, the LTL technique may be looked upon as a special case of its MTL counterpart

Fig. 8 A vertical strut loaded by the maximum elastic bending moment and a vertical force at its free end

Fig. 9 Load-deflection relationship for different value of β

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Fig. 10 The deformed shapes of an elastic-plastic strut with β = 0.3, where C is the boundary point between the elastic and plastic regions

wherein the linearized solution manifold transversally intersects the non-linear solution manifold at a single point in the discrete solution space (in which the desired solution vector exists). The development of the MTL technique is premised on the following argument. Since transversal intersections of the linearized and non-linear solution curves (or, equivalently manifolds, which are homeomorphic to these curves through appropriately constructed atlases) at multiple grid points are ensured, there is supposed to be a ‘closeness’ of the linearized and non-linear vector fields. Consequently, the linearized and non-linear solution curves, al least within a subset of the domain containing these grid points, should be sufficiently ‘close’. A more mathematically rigorous development of this argument is skipped in this study as the present focus in on a numerical verification. An implementation of the MTL strategy finally results in a reduction of the non-linear part of the operator to a set of conditional forcing terms expressible in terms of the discretized, unknown state variables. The discretization needed to derive such forcing terms is non-unique and may be performed via any available set of basis functions (valid within the domain of interest), which constitute a partition of unity. While an MTL method has herein been developed for a rather restricted class of BVPs governed by nonlinear ODEs, the concept may as well be extended to a far more general class of such problems governed by nonlinear partial differential equations. Here again, the nonlinear terms in the governing equations may be discretized through a set of interpolating basis functions and then considered as conditionally known forcing functions. Even parts of the linear operators may be discretized in this way and the transversality of the original and approximated solution manifolds would be ensured for almost all choices of discretization points. Unlike the usual form of an FE procedure, there should be

R. Kumar et al.

no requirements to differentiate the vector fields through the MTL approach and the choice of the transversally linearized equations would also be non-unique, thereby adding considerable flexibility to this class of methods. Indeed the authors have already conducted studies on these lines and the findings would soon be reported in another article. There are actually quite a few other possible avenues of exploiting the MTL principle. For instance, no specific effort has been made in this study to convert the MTL-based procedure into a geometric integration technique. Nevertheless the non-uniqueness and flexibility in deriving the linearized equations enable one to obtain the linearized solution as an exponential map, which evolves on a Lie manifold and thus has the power to automatically preserve certain aspects of the solution. For instance, a typical situation arises while dealing with the finite-rotation (Simo–Reissner) beam theory [13], wherein the spatial as well as temporal updates of rotation require the orthogonality of the associated transformations to be preserved. Here the true solution evolves on a specially orthogonal [SO(3)] manifold. It has recently been shown by Viswanath and Roy [18] in the context of low-dimensional non-linear oscillators that the MTL-based linearized system may be so derived that the elements of the linearized system matrix are conditionally known functions of time. In such a case, the linearized solution is obtainable as a composition of exponential maps through the usage of Magnus or Fer expansions [2]. Precisely a similar idea may be extended in both time and space such that the associated linearization procedure becomes truly geometry-preserving. The importance of developing such a procedure in the context of non-linear finite element analyses can hardly be over-emphasized and is presently under progress. References 1. Arnold VI (2000) Ordinary differential equations. Springer, Berlin Heidelberg New York 2. Blanes S, Casas F, Oteo JA, ROS J (1998) Magnus and Fer expansions for marix differential equations: the convergence problem. J Phys A 31:259–268 3. Deslauris G, Dubuc S (1989) Symmetric iterative interpolation processes. Constr Approx 5:49–68 4. He J-H (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Non-Linear Mech 35:37–43 5. Iserless A, Munthe-Kaas HZ, Norsett SP, Zanna A (2000) Lie group methods. Acta Numer 9:215–365 6. Liu WK, Chen Y, Jun S, Chen JS, Belytschko T (1996) Overview and applications of the reproducing kernel particle method. Arch Comput Mech Eng State Art Rev 3:3–80 7. Mattiasson K (1981) Numerical results from large deflection beam and frame problems analyzed by means of elliptic integrals. Int J Numer Meth Eng 17:145–153 8. Noor AK, Peters JM (1980) Non-linear analysis via global-local mixed finite element approach. Int J Numer Meth Eng 15:1363– 1380 9. Ramachandra LS, Roy D (2001a) A new method for non-linear two-point boundary value problems in solid mechanics. ASME J Appl Mech 68:776–786 10. Ramachandra LS, Roy D (2001b) A novel technique in the solution of axisymmetric large deflection analysis of circular plates. ASME J Appl Mech 68:814–816

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11. Reddy JN, Singh IR (1981) Large deflections and large-amplitude free vibrations of straight and curved beams. Int J Numer Meth Eng 17:829–852 12. ROY D (2001) A new numeric-analytical principle for nonlinear deterministic and stochastic dymnamical systems. Proc R Soc Lond A 457:539–566 13. Roy D, Kumar R (2005) A multi-step transversal linearization (MTL) method in non-linear structural dynamics. J Sound Vib (in press, Available online 2 February 2005) 14. Simo JC, Vu-Quoc L (1988) On the dynamics in space of rods undergoing large motion: a geometrically exact approach. Comput Meth Appl Mech Eng 66:125–161 15. Timoshenko SP, Gere JM (1961) Theory of elastic stability. McGraw-Hill, New York 16. Vasilyev OV, Bowman C (2000) Second generation wavelet collocation method for the solution of partial differential equations. J Comput Phys 165:660–693

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17. Vaz MA, Silva DFC (2003) Post-buckling analysis of slender elastic rods subjected to terminal forces. Int J Non-linear Mech 38:483– 492 18. Viswanath A, Roy D (2005) A multi-step transversal linearization (MTL) method in nonlinear dynamics through a Magnus characterization. Int J Nonlinear Mech (in review) 19. Wei GW, Zhang DS, Kouri DJ, Hoffman DK (1998) A robust and reliable approach to nonlinear dynamical problems. Comput Phys Commun 111:87–92 20. Wang QX, Li H, Lam KY (2005) Development of a new meshless – point-weighted least squares (PWLS) method for computational mechanics. Comput Mech 35:170–181 21. Yu TX, Johnson W (1982) The plastica: the large elastic-plastic deflection of a strut. Int J Non-Linear Mech 17:195–209