Computational Mechanics (1993) 11,207-227
Computational
Mechanics
@ Springer-Verlag 1993
Non-linear analysis of wave propagation using transform methods
D. S. Pipkins ~ and S. N. Atluri 2
1 FAA Technical Center, ACD-220 Bldg. 201, Atlantic City, NJ 08405, USA
Georgia Institute of Technology, French Bldg., Atlanta, GA 30332, USA
Abstract. Non-linear wave propagation/transient dynamics in latnce structures is modeled using a technique which combines
the Laplace transform and the Finite element method. The first step in the technique is to apply the Laplace transform to the
governing differential equations and boundary conditions of the structural model. The non-linear terms present in these
equations are represented in the transform domain by making use of the complex convolution theorem. Then, a weak
formulation of the transformed equations yields a set of element level matrix equations. The trial and test functions used in
the weak formulation correspond to the exact solutions of the linear parts of the transformed governing differential equations.
Numerical results are presented for a viscoelastic rod and von Karman type beam.
1 Introduction
The combination of the Laplace transform and the Finite element method (FEM) has been shown
to have the ability to produce numerical results which are in many cases superior to those of the
traditional semi-discrete FEM when performing wave propagation and vibration analysis of lattice
structures whose governing differential equations (GDE's) are linear. For example, see ArM and
Gfilqat (1977), Narayanan and Beskos (1982), Beskos and Narayanan (1983) and von Flowtow
(1984). Not only are the results of the transform domain FEM superior, but the solutions may in
general be accomplished using considerably fewer degrees of freedom (DOF) than would be
necessary in an equivalent semi-discrete FEM model. The success of the method stems principally
from the fact that the exact solution of the transformed linear GDE is usually known and hence
can be incorporated into the finite element formulation. Motivated by this success of the transform
domain FEM for linear wave propagation problems, the intent of the present paper is to extend
the technique to non-linear problems.
In extending the transform domain FEM to non-linear wave propagation problems, the
obvious difficulty lies in the application of the Laplace transform to a non-linear partial differential equation (PDE). While at first, this extension to non-linear PDE's may seem impractical, it is
noted that the application of the Laplace transform to non-linear ordinary differential equations
(ODE's) has been investigated by Weber (1956), Bellman et al. (1966), Sato and Asada (1988) and
Karmakar (1980). In addition, Hilmer and Schmid (1988) combined the Laplace transform with
FEM in order to solve non-linear problems involving soil-structure interaction while Wolf (1988)
used the Fourier transform to obtain non-linear solutions. Following the idea of Weber, the
complex convolution integral will be used in this work to represent non-linear terms in the
transform domain and thus facilitate the transformation of non-linear PDE's. Then, using a weak
formulation of the transformed PDE and associated boundary conditions, a completely general
methodology for transform domain FEM formulations is presented. In particular, transform
domain FEM formulations for a viscoelastic rod and von Karman type beam are developed.
2 Solution of non-linear O D E ' s in the transform domain
Before considering non-linear PDE's it is instructive to study some of the numerical aspects of the
proposed method for the simpler case of non-linear ODE's. By working with ODE's, numerical
procedures for evaluating the complex convolution integral may be developed without having to
208
Computational Mechanics 11 (1993)
introduce numerical issues pertaining to spatial discretization which will arise when PDE's are
considered. As an example of a non-linear ODE consider:
y,, + y = y2
(1)
with initial conditions:
y(0) = 0, y'(0) = 1
(2)
where the dependent variable, y, is a function of time, t, and primes denote differentiation. In order
to solve this ODE with the Laplace transform, the non-linear term, y2, must be correctly
represented in the transform domain. The application of the Laplace transform to this ODE yields:
-y'(O) - sy(O) + s2~ " + ~" = a(y2).
(3)
Here, the term, A(y2), represents the Laplace transform of the non-linear term, y2, and s is the
Laplace parameter. To correctly represent a non-linearity of this type in the transform domain,
use is made of the complex convolution theorem which states that the Laplace transform of the
product of two functions, f(t) and g(t), is (Smith 1966):
1 a+ioo
A[f(t)g(t)]_ = ~ i ~ !
F(fl)G(s - fl)dfl
(4)
--'00
where the path of integration in the complex plane must be to the right of all singularities of the
integrand and is usually taken parallel to the imaginary axis (i.e. ~ constant). Using the complex
convolution integral to represent the Laplace transform of y2, and applying the initial conditions
to Eq. (3), an expression in the transform domain for the solution, Y(s), is:
=Y(s) =
1 + 2ni,
_.00
Y(fl)=Y(s - fl)dfi
[s 2 + 1].
(5)
Observing the form of Eq. (5), it is evident that the non-linear ODE in the time domain has been
transformed into a non-linear integral equation in the transform domain. In order to solve this
equation, the following iterative scheme is proposed:
[ 1,?
1+
-- p)d
Is 2 + 1]
k -- 1, 2, 3 , . . . .
(6)
1
$2-~- 1'
which is the solution of the linearized problem. The implementation of this iterative scheme gives
rise to the principal numerical issue when attempting to represent non-linear terms in the transform
domain. This issue is the numerical evaluation of the complex convolution integral. A major
concern in choosing a method with which to evaluate the convolution integral is computational
efficiency. Computational efficiency is deemed necessary if the present method is to be competitive
with explicit and implicit time integrators. Although it is improbable that the present transform
domain method would ever replace time integration methods for the general purpose solution of
ODE's, it is a frequency domain technique and hence does not suffer any degradation when applied
to numerically stiff ODE's, unlike many time integration routines (Kirksey 1988). Therefore, for
this special class of ODE's, the transform domain method may have some practical utility.
The method chosen for the evaluation of the convolution integral comes from the common use
of the Fourier transform to solve convolution type integral equations (Sneddon 1972). Consider
the following integral equation:
The initial solution (guess) used to start the iterative process represented by Eq. (6) is ~'0 -
h(x) = S k ( x -
t)f(t)dt.
(7)
--00
Applying the Fourier transform to this integral equation transforms it into:
q(2) =
(8)
D. S. Pipkins and S. N, Atluri: Non-linear analysis of wave propagation using transform methods
209
where 2 is the Fourier parameter. Using this result, the evaluation of the convolution integral can
be achieved by applying the inverse Fourier transform to/t(2), which is the product of the Fourier
transforms of the components, f(t) and k(t), of the integrand of the convolution integral appearing
in Eq. (7). This method is readily extended to the convolution integral in Eq. (5), which may be
written in the form:
•
co
S Y(~ + i{t Y(o~ + i(co -- {))d{
(9)
27t -
by taking s = o- + ico and fl = ~ + i~ and setting o"= 2~ (for a reason to be made clear shortly).
Equation (9) is the starting point for the numerical evaluation of the complex convolution integral
in Eq. (5) using Fourier transforms. The first step being to take the Fourier transform of the
complex function Y, by applying a complex fast Fourier transform (FFT). The product of the
Fourier transform of ~" with itself is then numerically inverted, again using the FFT, to yield the
result of the complex convolution integral.
As mentioned in the above discussion, the Re(s) = o- is chosen such that o- = 2~. The reason is
the iteration proposed by Eq. (6) is such that each updated solution, Yk(s),is based upon Yk- l(fl)
and ~'k- 1(s -- fl)" If the Re(s - fl) were not equal to the Re(fl) then ~'k-1 would have to be known
along two contours parallel to the imaginary axis rather than one contour as is the case if (r = 2a.
With this restriction on ~r, it is evident that with each iteration, ~'k is determined along a contour
parallel to the imaginary axis and whose real part is twice that of the contour on which ~'k-1 is
known. This iterative progression in the complex plane is demonstrated in Fig. 1. Carrying out
the iteration proposed by Eq. (6) and using the Fourier transform procedure to evaluate the
convolution integral, the transform domain method converges to the solution, y(t), of Eq. (1) as
calculated using a Runge-Kutta solver. The nature of the iterative convergence is shown in Fig.
2 where yt(t) (k = 1, 2, 3), is compared to the Runge-Kutta solution.
The solution of Eq. (1) demonstrates that the use of the complex convolution integral to
represent quadratic non-linearities in the transform domain is a viable technique. However, there
is no need to limit the method to quadratic non-linearities. In fact, the method is completely general
and can be extended to product non-linearities of higher order. For example, consider the following
ODE with cubic non-linearity:
y" + y = - - l O y 3
(10)
and initial conditions:
y(O) = O, y'(O)-- 1
(11)
where the dependent variable, y, is again considered to be a function of time, t. Applying the
Laplace transform to the ODE given by Eq. (10) and taking initial conditions into account yields:
Y(s) = [1 - lOA(y3)]/[s: + 1].
(12)
Here, as previously, the term A(y 3) represents the Laplace transform of the non-linear term, y3.
However, the non-linearity now involves the product of three functions which causes the complex
convolution representation of the non-linearity to become:
1 ~+i|
I-
1 ~+i~
]
The Fourier transform method may be adapted to evaluate this double complex convolution
integral. Using a successive application of the Fourier transform on Eq. (13), it is observed
that A(y 3) may_be evaluated by an inverse Fourier transform of the triple product of the Fourier
transform of Y. Again, a FFT routine is used to carry out the Fourier and inverse Fourier
transforms. Introducing the iterative scheme:
(14)
210
Computational Mechanics 11 (1993)
and again using the solution of the linearized problem as an initial guess, the solution of Eq. (10)
may be computed. Analogous to the solution process for the quadratically non-linear problem,
it is necessary to judiciously choose the Re(s) and Re(fl) in order to efficiently carry out the iteration
proposed by Eq. (14). The best choice of these variables in the case of cubic non-linearity is
Re(fl) = 2 Re(y) and Re(s) = 3 Re(y). This choice allows for Ykto be determined from known values
of Irk- * on a single contour whose real part is the Re(7). Figure 3 shows this iterative progression
in the complex plane in which Yk is determined on a contour whose real part is three times that
of the contour on which Yk- ~ is known. Figure 4 shows the iterative convergence of successive
solutions, yk(t) (k = 1, 2, 3), of Eq. (14) to the non-linear solution, y(t), which was calculated using
a Runge-Kutta routine.
The methodology of representing product non-linearities in the transform domain by convolution integrand should now be clear and its application to higher order product non-linearities a
straight forward procedure. The two structural model considered in the next section have quadratic
and cubic non-linearities for which the treatment has been given explicitly in this section. Although
not pursued in the current paper, the possible application of the transform domain techniques to
systems in which the non-linearity is manifested in the form of a trigonometric function of the
dependent variable is an interesting topic and is currently under investigation. This type of
non-linearity is important as it quite often is present in systems where arbitrarily large motions are
modeled such as that discussed in Kondoh and Atluri (1985). The methodology being investigated
for representing trigonometric non-linearities in the transform domain is that given by Novikov
(1984). Novikov's work presents a method which accounts for the effect of a change of variable
by an arbitrary non-linear function in the Laplace transform and its inverse. Like the technique
discussed here, Novikov's method also involves the use of the complex convolution integral.
3 Solution of non-linear PDE's in the transform domain
The previous sections use of Laplace transforms to solve non-linear ODE's has laid the groundwork for the extension of the transform domain FEM to non-linear PDE's. It is hoped that this
extension will provide a FEM procedure capable of modeling a wave front as it propagates through
a lattice structure which exhibits non-linear behavior. At present, very few FEM formulations
capable of modeling non-linear wave propagation exist (Celia et al. 1980; Oden and Wellford 1975).
In this section, the two non-linear structural models for which the transform domain FEM
formulation is carried out are a viscoelastic rod and von Karman type beam. In these formulations,
the methods of the previous section are used to represent the non-linear GDE's and boundary
conditions of these models in the transform domain. Then a weak formulation in the transform
domain of the GDE's and associated boundary conditions will produce a system of element level
equations for each structural model. The weak formulation provides a completely general setting
for the development of FEM representations of systems. It is hoped that this use of a general
methodology will make obvious the application of the transform domain FEM to other non-linear
systems.
3.1 Viscoelasticrod model
The viscoelastic rod modeled in this section was presented by Engelbrecht (1979). This one
dimensional model considers only axial motion, u(x, t), while ignoring the effect of lateral contractions. The linear momentum balance of the rod is expressed by:
(2 + 2#) { u,x~ -t. g /[-~Ju,x(Z)L~z
o exp (
t-Zldzl t+~(2+2#)(l+mo)[(u,x)2], x pu,,=O
TO
/
(15)
J,xJ
and the traction boundary condition by:
~'=(2 +2/,)
[
j~3u'x(r) ( t-Z)dzl+~(2+2p)(l+mo)(U,:,)2
-exp
(16)
D. S. Pxpkinsand S. N. Atluri:Non-linearanalysisof wave propagation usingtransformmethods
211
where mo = 2(Vl + v2 + v3) 2 and # are the Lame constants, Y1, v2, and v3 are 2nd order elastic
(2 + 2#)
moduli, p is the material density, g is the damping coefficient, "Cois the relaxation constant, and
is the prescribed boundary traction. Note that the integral term in Eqs. (15) and (16) accounts for
the viscoelastic effect with the exponential function being the relaxation function of the model.
The benefit which are derived from using a transform domain solution procedure for viscoelastic
models is another attractive feature of the present method.
Applying the Laplace transform to the linear momentum balance and traction boundary
condition gives (assuming homogeneous initial conditions):
ECgt,xx- sZpgt+ 3(2 +
z
2#)(1 +
mo)(A[(u,~)2]),x
= 0;
(17)
~F= ECft,x + ~(2 + 2#)(1 + mo)nE(u,x) 2]
(18)
where E c is the complex modulus which represents the viscoelastic effect in the transform domain
and is given by:
EC=(2+2#)
and
I
"CoS 1
(19)
l+gzo~ ~j
A[(u,x)2], the
Laplace transform of the non-linear term, is given explicitly by:
1 o~+ioo
A[ (u'x)2] = ~igi =~oo (kx(fl)U'x(S -- fl)dfl
(20)
which indicates that, as was done for the ODE's, the complex convolution integral is used to
provide a legitimate representation of the non-linear terms in the transform domain. Having a
description of the linear momentum balance and traction boundary condition in the transform
domain, the FEM formulation is carried out by writing the weak form of the linear momentum
balance and traction boundary conditions:
i{ECu,~,,--s2pu+~(2+2~)(l+mo)(A[(u.x)2]),x} ~Sgtdx
+ { T -- ECff.x - ~(,L + 2#)( l + mo)A [ (u.x)2] }~Sffl:- Lo= O.
(21)
Integrating the weak formulation by parts gives:
L
o
{ECfl,,~aft~+ s2pf~aft}dx =
-
~
L
(2 + 2#)(1 + mot 5 AE(u,x)2](~a,xdx
o
+ {T}
-
-
(~x=L
Ix=o.
(22)
At this point, specific choices must be made for the trial, zi, and test, &i, functions to be used in the
symmetric weak form given by Eq. (22). The function to be used for both the trial and test functions
is the solution of the linearized momentum balance equation. The linear portion of Eq. (17) is just
the simple wave equation which is often used to model a longitudinal rod undergoing small
deformations. With this choice, the trial and test functions are:
sinh(S(LcX) )
~i(x, s) =
sinh ( ~ )
~i, (s) +
sinh ( ~ )
(23)
sinh ( ~ )
ti2(s)'
212
Computational Mechanics 11 (1993)
sinh(S(LcX))
sinh ( ~ )
sinh(~)
6~,(s)-~ sinh(~)a~2(s)
&7(x,s)=
(24)
where c2 - (2 + 2#), L is the element length and the coefficients of the shape functions correspond
P
to nodal displacements of the rod element. The hyperbolic shape functions in Eqs. (23) and (24)
are actually compositions of the permissible wave modes which may exist in the linear rod model
(Pipkins 1992). Inserting the chosen trial and test functions into Eq. (22) and carrying out the
integrations, the following element level matrix equations for the non-linear viscoelastic rod model
are obtained:
]c
~Sc~
uscS/sinh ( ~ ) 1 ( ~ , ) =
EC~_~cS/sinh(S_~Lc
) S__cCOth(S_~Lc
) ]\u2j ( ~ i ) -
G(~I)
(25)
where G = 3(2 + 2#)(1 + mo). The matrix on the left hand side of these equations is the same as
would appear in the linearized problem. The right-hand side vector having components
and
QD 2 arises as a result of the non-linearity. These components are given explicitly by:
QD~
L
C
\
QDI=-5o~27zi=-i~o(t'x(fl)ft'x(s-fl)dfl
s
_
QD2 = o~ (2zti~-i~o
-
sinh
C
(SLc) dx
(26)
. /sx\
c o s h / - - /
sinh
()
c
The numerical evaluation of the terms given by Eqs. (26) and (27) is critical to the success of the
proposed method. An important difference between these non-linear terms and those which arose
in the solution of ODE's is that Eqs. (26) and (27) contain spatial integrations over the domain of
the element in addition to the complex convolution integrals which were present in the ODE case.
In the previous section dealing with ODE's, the Fourier transform method was identified as an
accurate and computationally efficient method of evaluating the complex convolution integral.
However, the spatial integration which has arisen in the PDE formulation is a new situation which
must now be dealt with if the transform domain FEM is to be successfully extended to non-linear
PDE's. The primary effect of this spatial integration is to make necessary the introduction of a
spatial discretization. In order to identify the best procedure with which to introduce the spatial
discretization, the following methods were tested: linear interpolation, quadratic interpolation and
substitution. For clarity in the discussion to follow, let:
exact
1
5 ~7,x(fl)tTx(s-fl)dfl
P(s'x)=A[(u'x)2]= -2rci=-,~
which allows QD1,and QD2to be written in the form:
(28)
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
_
L
QD1 = - ~P(s,x)
C
dx;
o
Scosh
L
QD2 = ~fi(s,x) c
sinh ( ~ )
213
o
dx.
(29, 30)
sink ( ~ - )
The method of linear interpolation introduces an approximation of P(s, x) over the domain of
each finite element. The approximation_, ~L(S,X), linearly interpolates the nodal values of/5(s, x)
over each element. The evaluation of P(s, x) at the element nodes is carried out by utilizing the
Fourier transform method of evaluating convolution integrals. The explicit form of the approximation is:
Using/~c(s, x) in place of/5(s, x) in Eqs. (29) and (30), approximate expressions for QD 1 and QD2
are obtained. These expressions are given by Eqs. (A.1) and (A.2) in Appendix A.
Next, consider the method of quadratic interpolation. As is inferred from the name, quadratic
interpolation introduces a quadratic approximation of/5(s,x) over the element domain. The
quadratic interpolation is defined by using evaluations of P(s,x) at the nodes as well as the
midpoint of the element. The quadratic approximation, ,be(s, x), is of the form:
fio(s,x)=I(L-2x)(L-x)Jfi(s,O)+ [_
L~X)_jfi,,s'~./l( L I
+[X(2L~L)Ip(s,L).
(32)
The approximate expressions for QD 1 and QD 2 which result from using iSQ(s,x) in place of P(s, x)
in Eqs. (29) and (30) are given by Eqs. (A.3) and (A.4) in Appendix A.
The approximate evaluation of QD1 and Q_D2 is quite different in the exact substitution
approach in that no attempt is made to calculate P(s, x) at discrete values ofx and then interpolate
these over the element domain. Instead, the orders of integration in Eqs. (26) and (27) are
interchanged. The proof of the validity of this interchange appears in Pipkins (1992). After
interchanging the orders of integration, Eqs. (26) and (27) become:
QD1-
1
t
2-ni~i~ !O.x(fl)~,~,(s-fi)
Scosh
c
( s(L c x) )
sinh(@)
}
dx dfi
(33)
Insertion of the partial derivatives with respect to x of the trial function (Eq. 23) into Eqs. (33) and
(34) and subsequent evaluation of the spatial integration results in approximate expressions for
QDt and QD2. These expressions are given by Eqs. (A.5) and (A.6) in Appendix A. The convolution
integrals which appear in these expressions are evaluated using the Fourier transform method. It
is noted that due to the increase in the number of corivolution integrals which must be evaluated,
the exact substitution method requires a much greater computational effort per element than either
the linear or quadratic interpolation methods. For this reason, unless exact substitution is able to
provide accurate results using a coarse mesh, then it is not likely to be practical when compared
to the linear and quadratic interpolation methods.
A few final comments should be made about the element equations (Eq. 25) derived for the
viscoelastic rod model. The interesting feature of these equations as compared to their linear
214
Computational Mechanics 11 0993)
counterpart is the presence of the QD vector which contains the non-linear terms. It is the numerical
evaluation of the QD vector by either linear interpolation, quadratic interpolation or exact
substitution which is responsible for the introduction of a spatial discretization. The form of the
QD vector also gives rise to a provocative thought about the way in which mathematics sometimes
mimics physical phenomenon. Consider the physical implication of the convolution integrals
which appear in the QD vector. The linear rod problem, with no QD vector, essentially consists of
equations which are uncoupled with respect to frequency. The presence of the QD vector in the
non-linear problem causes the equations to be coupled with respect to frequency. This coupling
of frequencies which arises mathematically through the convolution integral is an experimentally
observed phenomenon (Nayfeh and Mook 1979). For instance, beams excited at one frequency
often have responses which consist of several frequencies. This behavior is unexplained by linear
models, but the presence of the QD vector in the non-linear model presented in this section would
permit such a response.
3.2 yon Karman beam equations
The von Karman beam equations constitute a non-linear model which couples the bending and
axial deformations of a beam. Consider a beam element oriented along the x-axis and of length
L, undergoing transverse deflections, w(x, t), and axial deflections, u(x, t) in the x-y plane. The von
Karman equations which govern this motion are:
EA[u,~ + 89 ,x)23 , pAu,u = O; E L.w ~. x -. EA[u
x w x + ~(w,x)
1
3],x + pAw,. = 0
.
with force boundary conditions given by:
]V = EA[u x +~(wx)
1
2.
1
3
],
)(/I -~- Elzw,x~;
?/= -Elzw,~xx+EA[u.xW~+~(W,x)
]
x
- -
(35, 36)
(37-39)
where E is Young's Modulus, I s is the moment of inertia about the z-axis, A is the cross-sectional
area, p is the mass density, ]V is the prescribed axial force, 2~t is the prescribed bending moment
about the z-axis, and V is the prescribed shear force. Applying the Laplace transform to equations
(35) and (36) which are the beam GDE's yields:
1
EA(f~,~ + gA[(w,x
)2 ]),x - PAszf~ = 0;
(40)
EIz~,xx~ - EA(a[uxw,x] + 89
(41)
x)3]),x + pAs2# = 0
where homogeneous initial conditions have been assumed. The Laplace transforms of the force
boundary conditions are:
N=EA(ft,~+ 89
M=EIz#,xx;
~'=-EI=#~+EA(a[u,~wx]+ 89
(42-44)
The now familiar convolution integral representations of the Laplace transforms of the non-linear
terms are:
A[ (w'x)z] = ~ i ~ - ioo #,x(fl)#,~,(s - fl)dfl;
1 ~+i~
A[(w'x)3] =
~i~ffioo ~'x(s -
flV 1 ~+i~
A[u,xw,~] = 2ni~-i~o a,:,(fl)#,~(s - fi)dfi
(45, 46)
]
)L 2~i~-~ioo ~'x(fl - 7)w'~(7)dT_ldfl"
(47)
Using the description of the GDE's and force boundary conditions in the transform domain
afforded by Eqs. (40)-(44), the FEM formulation is carried out by writing the weak form of these
equations:
L
{EA(f~.~ + 89
,x - sZpAu} 6ftdx + {N - EA(ft,~ + -~a(w~,)}U
f oll 2
0
L
+ ~ {EI#,x~x~ -- EA(A(u,xW~,) + 89
0
+ s2pA~}g)~dx
-L
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
- { M - EI~,x,,}b~,,,lz6 - { ~" + EI@,x,,x - EA(A(u,xw,~,) + 89
215
= 0.
(48)
Integrating the weak form by parts in order to obtain the symmetric form gives:
L
Nc~ftlLo-- ~ {EA(ft,~ + 89
- sZpAft6~}dx - M6wx]~- [-'6~]~
0
L
+ ~ {EI~.~x6~,xx + EA(A(u,xW,x)+ 89
o
+ s2pA~b~}dx=O.
(49)
Specific choices can now be made for the trial, ~7and ~, and test, 6~ and 6~, functions to be used
in Eq. (49). As was done in the viscoelastic rod formulation, the functions chosen for the trial and
test functions will correspond to the exact solution of the linear portions of the transformed GDE's
(Eqs. (40) and (41)). The linear portions of the two GDE's are uncoupled and correspond to the
longitudinal rod and Bernoulli-Euler beam. The trial and test functions corresponding to the
solution of the longitudinal rod were given in Eqs. (23) and (24) while those for the Bernoulli-Euler
beam are:
ff~(x)= Nlff~t + NzOI + N3w2-+-N402;
bw(x)= Nl~SWl + N260a q- N36ff~2+ N4~O2
(50,51)
where 0 = w x and the shape functions, N j, are given by Eqs. (B.1)-(B.4) in Appendix B. Again, it
is pointed out that the chosen trial and test functions are actually compositions of the permissible
wave modes which can exist in the longitudinal rod and Bernoulli-Euler beam (Pipkins 1992).
Inserting these choices for the trial and test functions in Eq. (49) and carrying out the integrations,
the following element level matrix equations for the von Karman beam are obtained:
-Bll
~'QD1
B12
0
0
0
0
{/~1'
/1Nil ~
B21
B22
0
0
0
0
i/~2
N2
QD2
0
0
Dll
D12
D13
D14
wl
171
0
0
D21
022
023
D24
191
M1
QD3I
I
0
0
031032033034
W2
~'2
_ 0
0
D41
\02,
\]~2'
D42
Da4-
D43
QD4[
(52)
[005 I]
l\QD6 I
where the matrix on the left-hand side of these equations is made up of components arising from
linear terms. Specifically, the Bij represent the components of the stiffness matrix of the longitudinal
rod which were given in Eq. (25) and Dij the components of the Bernoulli-Euler beam stiffness
matrix that are given by Eqs. (B.5)-(B.10) in Appendix B. The non-linear terms are once again
accounted for in the QD vector. The components of the QD vector are given by:
QD1-
EA}o
~
c
\
c
sinh ( ~ )
dx;
EA~A[(w,x)2]c
Q D 2 = 2 -b
sinh ( ~ )
L
QDj = EA [. (A [u,~w,x] + 1A [(wx) 3])Mjdx, j = 3, 4, 5, 6.
o
dx
(53,54)
(55)
The functions M j, which are derivatives of the shape functions used in the expression for 6~
(Eq. (51)), are given by Eqs. (B.11)-(B.14) in Appendix B. The spatial integrations present in the
components of the QD vector once again make it necessary to introduce a spatial discretization.
The only discretization method implemented for the von Karman beam equations was linear
interpolation. The linear interpolation procedure is analogous to that used for the viscoelastic rod.
The procedure being to interpolate nodal evaluations of the convolution integrals appearing in
Eqs. (53)-(55) linearly over the domain of the beam element. The resulting approximations of the
216
Computational Mechanics 11 (1993)
components of the QD vector generated by the linear interpolation method are given by Eqs.
(B.15)-(B.20) in Appendix B.
The physical implication of the convolution integrals which appear in the QD vector is worth
mentioning again. The convolution integrals cause the non-linear problem to be coupled with
respect to frequency. As discussed in the previous section detailing the formulation of the viscoelastic rod model, this frequency coupling is an experimentally observed phenomenon.
4 Numerical implementation
The elements derived for the viscoelastic rod and von Karman beam may be used to model
complicated structures. In order to do this, the element equations for either the viscoelastic rod
or yon Karman beam must be assembled into a global system of equations and solved in the
transform domain. Then the solution must be obtained in the time domain by use of the inverse
Laplace transform. The method of global assembly of the element equations is similar to that
discussed by Weaver and Gere (1980). The numerical inverse Laplace transform utilized is detailed
in Crump (1976). Thus, the principal numerical issue which must be addressed here is the solution
in the transform domain of the non-linear global equations.
A generic representation in the transform domain of the global equations of either a viscoelastic
rod or a plane frame modeled by the von Karman beam equations is:
[KSr]uSr = f s r _ Q_
(56)
where [K st] is the global stiffness matrix, u sr the vector of unknown nodal displacements, f s r
the vector of nodal loads, and (~ is the vector consisting of the element level QD vectors. In order
to solve these non-linear equations the (~ vector which contains the non-linear terms is treated as
Irn(s)
S PLANE
5
k=l k=2
k=3
k=4
RUNGE-KU'FrA
H ~/
3 ITERATIONS,y3(l}
4
/ /
/F
--~//
2 ITERATIONS,y2(1)
1 ITERATION;yl(t)
--~'// / " ~
>'2
a
2a
4a
8a
Re(s)
0.5
1
1.5
2
2.5
3
2
Ira(s)
S PLANE
k=l k=2
a
3
3a
k=3
~, Re(s)
Figs. 1-3. 1 Iterative stepping in the s-plane for quadratic non-lineanty.
2 Convergence of Laplace transform method to Runge-Kutta solution for
quadratic non-linearity. 3 Iterative stepping in the s-plane for cubic nonlinearity
D. S. Pipkins and S. N. Atluri: Non-hnear analysis of wave propagation using transform methods
217
an artificial loading which is updated during the simple iterative process given by:
[KST ]u.ST
k =
fST
-Qk-1,
-
-
k=1,2,3,...
(57)
where (~o is computed using the solution of the linearized problem. The iterations are carried out
until the solution has converged. As was the case for the iterative solution of non-linear ODE's
by Eqs. (6) and (14), each iteration of Eq. (57) will in general determine the updated solution on a
different contour in the complex plane than that on which the old solution was known. The
relationship between the updated solution's contour and that of the old solution depends on the
order of the non-linearities in the problem. The viscoelastic rod, which has a quadratic nonlinearity, has a contour progression like that shown in Fig. 1. For the von Karman beam, which
has both cubic and quadratic non-linearity, an artificial construct is necessary to make the iterative
contour progression of the different order non-linearities match. This construct involves the
introduction of a function, r t), which allows the quadratically non-linear terms in Eqs. (35) and
(36) to be treated as cubic non-linearities but does not alter these equations. The simple choice for
t) is:
~(x,t)= 1, x~E0, L], tel0, oo).
(58)
With this transformation of the quadratically non-linear terms into pseudo cubic non-tinearities,
the contour progression associated with the von Karman equations is like that shown in Fig. 3.
Because of this progression in the complex plane associated with the iteration of Eq. (57), it is
difficult to determine when satisfactory convergence has been obtained by simply comparing the
updated transform domain solution to the old solution. Therefore, convergence must be verified
by using the inverse Laplace transform to compare the time domain solution of successive
iterations.
5 Numerical results
This section will demonstrate the ability of the transform domain FEM to perform wave propagation and transient dynamic analysis of the viscoelastic rod and von Karman type beam. The
formulations presented have revealed that the primary difference between the application of the
transform domain FEM to linear and non-linear problems is the need to introduce a spatial
discretization in the non-linear case. In order to observe the effect of this spatial discretization,
most of the examples presented will involve mesh refinement. Each of the examples presented
needed three iterations (k = 3 in Eq. (57)) to converge for the time windows for which the solutions
are plotted. The parameters used in the Fourier transform evaluation of the complex convolution
integrals and the inverse Laplace transform are example dependent. These parameters are the
number of FFT sample points, the sample increment, A~o, along the complex plane contours and
the constant, a (see Figs. 1 and 3) which defines the location of the iteration contours in the complex
plane.
5.1 Example 1
A viscoelastic rod is modeled using the three spatial discretization schemes (linear interpolation,
quadratic interpolation and exact substitution) in order to observe their performances relative to
one another. The geometry, load history and material parameters of the problem are shown in
Fig. 5. The Jux [ at x = 0.0 feet was computed using 5, 10 and 20 elements of equal length for each
type of spatial discretization. In all cases, the number of FFT sample points used was 2048,
Am = 0.5 and a = 0.5. The linear interpolation results are shown in Fig. 6, quadratic interpolation
in Fig. 7 and those of exact substitution in Fig. 8. The methods of linear and quadratic interpolation
exhibit convergence to the expected solution (boundary condition) as the number of elements is
increased. However, the method of exact substitution is subject to severe oscillatory behavior,
particularly for small t, as the number of elements is increased. In order to investigate the cause
218
Computational Mechanics 11 (1993)
I
I
0
0.5 84
I 0.25
>'-0.25
5 ~
\ t---,- \~
RUN~E-~UTTA
-0.5-
- -- - 3 ,TERA'r,ONS; Y:I
--
-1
\
~t
~
- - 1 ITERATION; Yl(t)
^
~
%-j
:
I
0.3
I
%
I
I I
I I I
I i I
II
5 Elements
I
l
\
-- -- -- 2 ITERATIONS; Y2(t)
-0.75
4
~
O.
o . 1~
I
[
0.5
0.4
u,
5ft
J/
P
2+2~=1001blft2
A=I ft2
p=l slug/ff3
I o.3
\
'J
I
"-a o . 2
i
I
10 Elements
o.
mo=-1.2
0.5
g=0.2
P (Ibs)
ro=O01
t o.,a
~0.2
20 Elements
O.l'
I
i
2
[
3
I
4
I
5
t (sec)
I 9
0.5
6
02
5
1
1.5
2
3
3.5
,,.-
t (s)
6
2.5
Figs. 4-6. 4 Convergence of Laplace transform method to Runge-Kutta solution for cubic non-linearity. 5 Fixed-free viscoelastic rod subjected to tip loading. 6 l u,xl at x = 0.0 ft using linear interpolation and 5, 10 and 20 elements
0..5
I
i
!i i
i\:,.-.../'-,,:-v',:',
i i
"d"
0.4
,-.---...
0.3
'
0.3"
=x O . 2 .
5 Elements
I
I
0.1
i
I O.4"
I
t
t
I
!
:~0.2
,,
0 . 5 84
O.l
r
:
5 Elements
J
V,
i
0 . 5 84
l
0.4'
0.3
i
/"
/
iv!:J
o.~
V
o.3
-~o.2
V
"~
i 0.2
10 Elements
0.1
I 0 Elements
0.1
0.~.
I
0.5-
0.4
0.3
0.3
~0.2
2
=~-"0 .
o 1
20 Elements
0
1
o.s
7
z
z.s
t (s) ~
2
2.s
3
3.5
20 Elements
t (sec)
o.s
8
Figs. 7 and 8. lu.xl at x = 0.0ft using quadratic interpolation 7 and using
exact
z
1.5
2
t (s)
=
2.5
3
3.5
substitution 8 and 5, 10 and 20 elements
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
0.5
0.4
EXACTSUBSTITUTION
_
~
~
0.5
........ . . ~ / ~
0.4
0.3
.3
0.2
.2
10 ELEMENTS(non-uniform)
20 Elements
0.i
0.5
219
.i
'
'
'
~
~
'
0[5
'
i
1:5
t (s)-
10
LINEAR INTERPOLATION
2'.5
:~
~-
~
3'.s
0.4
T 0.3
x
0 . 2 84
20 Elements
0.1 ~
0:5
1:5
i
9
~
2:s
~
Figs. 9 and 10. [u.~l at x =0.0ft 9 with A~o=0.1, using linear
interpolation and e x a c t substitution; 10 using non-uniform 10
element mesh
3:s
t (s)------..-
5 84
. . . .
Linear Response
Non-Linear Response
4
/#/
= 21
./
\ ',
//
1
- 2
/
0.5
11
!
1.5
t (s)
2
=
2.5
"',
3
12
:
i
.....
Linear
~
Non-Linear
i -
i
0.25 0.5 0.75
1
t
(s} ~
Response
~''~
Response
i
"~
i
i
1,25 1.5 1,75
2
,=
Figs. 11 and 12. u at x = 0.0 ft 11 as a result of a compressive load; 12 as a result of a tensile load
of this oscillatory behavior, the 20 element linear interpolation and e x a c t substitution solutions
were recomputed. All parameters remained the same with the exception of Ac9. The new value of
Ae) was 0.1. The results shown in Fig. 9 indicate that decreasing Aco has removed the oscillations
from the e x a c t substitution solution. As these results seem to indicate that the linear interpolation
scheme is more robust with respect to sampling parameters than the e x a c t substitution scheme
and nearly equivalent to the quadratic interpolation scheme, the remaining calculations in this
section are done using the linear interpolation scheme which is the computationally most efficient
method of spatial discretization. One final perturbation of this problem is to use a non-uniform
mesh. This is done for a 10 element model using 4 equal length elements from x = 0.0 ft to x = 1.0 ft
and 6 equal length elements from x = 1.0ft to x = 5.0 ft. The results, using Aco = 0.5 are shown in
Fig. 10. These results indicate that the transform domain F E M procedure performs well using a
non-uniform mesh. The non-uniform mesh has significantly improved the 10 element uniform
mesh solution shown in Fig. 6. This property could be advantageous if a fine resolution of a wave
220
Computational Mechanics 11 (1993)
I=0.1473s
501
1 O0
30
20
-
I6o
Linear Response
~.
Non-LinearResponse
~
40
D
.
i0
20
5O
3O
I=0.3007 s
Linear Response
ear Response
10
~2o
10
.
.
.
.
.
.
.
.
Linear Response
Non-LinearResponse
t=0.7609s
..........
LinearResponse
-
Non-Linear Response
1
14
1=0.4541 ~
.
l 80
~'~=' 60
.~ 40
D
20
g2o
~30-
.
- -
100
-'"-~'.~,
..........
"'\'",~-~,~on-LJn
50 84
I=0.6075s
2
3
4
5
x (II)
Linear
Respo~
Non-Lmear Response
".,% ~
Figs. 13 and 14. Stress wave profiles resulting from a compressive
2
13
x (.)
3
4
5 load; 13 t=0.1473, 0.3007 and 0.4541 seconds; 14 t=0.6075 and
0.7609 seconds
is necessary only at specific points in a lattice structure such as the actuator locations in an active
control model.
5.2 Example 2
Again, consider the viscoelastic rod whose geometry, material parameters and load history are
shown in Fig. 5. Here, the response computed is that of the displacement at x = 0.0 feet resulting
from a compressive and tensile load. The number of F F T sample points used in the calculations
is 2048, Ao~ = 0.5 and a = 0.5. The rod is subdivided into 20 equal length elements and the linear
interpolation scheme is used. The displacement, as a result of a compressive load, is shown in
Fig. 11 where it is compared to the linear response. The magnitude of the displacement resulting
from a tensile load is shown in Fig. 12 where it is also compared to the linear response. Note in these
results the effect that the non-linearity has on the maximum displacements and the times at which
they occur.
5.3 Example 3
One final calculation using the viscoelastic rod whose geometry, material parameters and load
history are shown in Fig. 5 is carried out. Here, stress wave profiles are computed until shortly
after reflection from the fixed boundary at x = 5.0 feet has occurred. The number of FFT sample
points used in the calculations is 2048, Aco = 0.5 and a = 0.5. The rod is subdivided into 20 equal
length elements and the linear interpolation scheme is used. The linear and non-linear stress wave
profiles resulting from a compressive load are shown in Figs. 13 and 14. The linear and non-linear
stress wave profiles resulting from a tensile load are shown in Figs. 15 and 16. These plots are very
interesting from a wave propagation point of view. Notice in Fig. 13 how the non-linear compressive stress wave is steepening in relation to the linear wave. If this compressive wave wag allowed
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
50-
50
1=0,1473 s
I
4O
30
..........
- -
~2o
30"
Linear Response
Non-LinearResponse
20-
..........
Linear Response
Non-Linear Response
10
3_0O
50-
!
4o I
I=0.3007 s
~
30-
,
.
t=0.7609 s
J' 8 0
.
. . . . . . . . . . Linear Response
~~Non-Linear
Response
~20-
i
60
~
40
~-~
...........
20
t0-
1
50
I
t=0.6075 s
4o]
10
i
221
t=o.4s41 s
16
Linear Response
Non-Linear Response
2
3
x (ft) ~
,-
4
5
40.
30-
~,,,
Linear
--~__.20t~
..........
Non.linear Respolse ~ ~
10
i
2
3
4
Figs. 15 anti 16. Stress wave profiles resulting from a tensile load;
15 t = 0.1473, 0.3007 and 0.4541 seconds; 16 t = 0.6075 and 0.7609
seconds
x (ft)
15
s
P(t)
//
\
\
3T
Ib/ft2
2.
NICKELL
-
....
I
TRANSFORMDOMAIN ,,', j
A= 1.0 ft2
\
\
/
4ft
~
p=0.1 slug/f{
I=1.0 f{
"
~0.5.
P 0bs)
I
0.5
4o
-0.
1
.
.
.
.
1.5
2
/.
2.5
t (s)
2.0
I
I
1
I
',
',
1
2
3
4
5
6
~t(sec)
18
1'7
Figs. 17 anti 18. 17 Fixed-fixed beam subjected to mid-span loading. 18 Mid-span deflection of fixed-fixed beam; 2 element
transform domain and Nickell solution
to continue down the rod, without impacting the rigid wall, a shock wave may eventually form.
On the other hand, the steepness of the non-linear tensile wave shown in Fig. 15 is diminishing
with respect to the linear wave. This differing behavior of the compressive and tensile waves is
controlled by the material parameter, too, which for this problem is - 1.2. As is discussed in Jeffry
and Engelbrecht (1980), when m o < - 1 only compressive shock waves are possible and when
mo > - 1 only tensile shock waves may form.
5.4 Example 4
A model of a fixed-fixed beam is analyzed in order to compare the transform domain FEM solution
of the von Karman beam equations against an existing approximate solution. The geometry, load
222
Computational Mechanics 11 (1993)
history and material parameters of the problem are shown in Fig. 17. Due to symmetry, only one
half of the beam is modeled. The number of FFT sample points used is 2048, Am = 0.25 and
a = 0.33. The time history of the mid-span deflection using 2 elements is shown in Fig. 18 where
it is compared to the approximate solution given by Nickell (1973). The transform domain FEM
solution is in excellent agreement with the solution of Nickell until around t --- 1.8 seconds.
6 Concludingremarks
The application of transform domain FEM to linear problems is a well established topic. Because
the transform domain FEM is exact with respect to linear problems, the method does not require
a spatial discretization which results in a substantial savings in the number of D O F necessary to
model a structure as compared to other FEM techniques. Despite the success of the transform
domain FEM in solving linear problems, its possible use for the solution of non-linear problems
has received little attention. In an attempt to full this void, the present paper has taken these
transform domain FEM techniques and applied them successfully to non-linear problems. The
principal issue in doing this was to find a legitimate representation of non-linear terms in the
transform domain. This was done by making use of the complex convolution integral, which allows
product non-linearities to be written in the transform domain. Once the governing differential
equations and boundary conditions of the problem were written in the transform domain, a weak
formulation yielded the element level equations. The non-linear element equations were found to
require a spatial discretization. It was found that a simple linear interpolation scheme could be
used to carry out this spatial discretization. Numerical results obtained for a viscoelastic rod and
von Karman beam indicate that the transform domain FEM is a viable solution technique in the
area of non-linear wave propagation where short time responses are of interest.
Acknowledgements
The authors wish to express their gratitude to the U.S. Air Force Office of Scientific Research which administered the NDSEG
fellowship which helped to financially support this research.
Appendices
A Expressions related to the viscoelastic rod model
A.1 Linear interpolation approximation of the QD vector
-
-
.
.
.
.
oo -Li j
A.2 Quadratic interpolation approximation of the QD vector
(A.1)
(A.2)
+
O
I
+
+
§
a
9
~
-~
L.~-'~
t~
~
O
o
L
+
v
A
I
-t-
I
II
9
i
+
+
,,I
~
I I
I
I
I
+
+
I
v
I
O
i
@
.z
224
Computational Mechanics 11 (1993)
sinh(~)cosh(~)
sin~(~)~os~(~))
+
+
,a
s
r
C
+ ~(t~)~,(s-/~)
sinh((s7)L)cosh(~L-) sinh
~-
+
s-fl
C
sinh(~)
-o~(~)o~(s-~)
+
+
sinh((Scfi)L)cosh(S~Lc)
sinhI
Ic~
!
d~
(A.5)
s B s-/~
I~
~)
QD 2 "~ -
2;i"~'~4sinh(~) sinh(@) sinh
x [Lcosh(~)+sinh(~)sinh((scfl)L)c~ (~)
(Sc) ~
I Lcosh
--~1(~)U2($--~)
+ sinh((s -~)L)
( ~ ) c~
s-fl
+
-t
~-sinh(~) c~} ((ss 7)L) -
C
•
~:,
sinh
C
Lcosh(~s~'L)+sinh('sT'L) + sinh(~)cosh(~)
(~)
-~c
s
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
{
sinh(~)cosh(~)
-~bi2(fl)/~2(S--fl) L +
snh/:
c
/
s-~
\
c
/
sinh ( @ ) cosh ( @ )
+
(-Sc)
225
-~c
~t,aR-
(A.6)
e
B Expressions related to the yon Karman beam model
B.1 Shape functions used in von Karman beam weak formulation
NI=
[cos (Hx) + cosh (Hx)]
2
[cos (Hx) - cosh(Hx)] [sin(HL) sinh (HL)]
211 - cos (HL) cosh (/-/L)]
[sin (Hx) - sinh (Hx)] [sinh (EEL) cos (HL) + cosh (HL) sin (HL)]
211 - cos (HL) cosh (F/L) ]
N 2 ~--
[sin (Hx) + sinh (Hx)]
2I/
[sin (Hx) - sinh (Hx)] [sin (HL) sinh (HL)]
2HI1 - cos (HL) cosh (HL)]
[cos (Hx) - cosh (Hx)] [sinh (HL) cos (HL) - cosh (HL) sin (F/L)]
2 H [ 1 - cos (HL) cosh (HL)]
N3=
(B.2)
[cos (Hx) - cosh (Hx)] [cos (HL) - cosh (HL)]
211 - cos (EEL) cosh (EEL)]
+ [sin (Hx) - sinh (Hx)] [sin (HL) + sinh (HL)]
211 - cos (HL) cosh (HL)]
N 4 =
(B.1)
(s.3)
[cos (Hx) - cosh (Hx)] [sin (HL) - sinh (HL)]
2H[1 - cos (HL) cosh (HL)]
+
[sin (Hx) - sinh (Hx)] [cos (HL) - cosh (HL)]
2HI1 - cos (HL) cosh (EEL)]
(B.4)
pAs2) 1/4.
where H = ( - -
EIz
B.2 Components of the Bernoulli-Euler beam stiffness matrix
1511 =/533 = H3[sinh (EEL) cosh (HL) + cosh (HL) sin (HL)]
1 - cos (HL) cosh (HL)
(B.5)
/512 =/)zl = - i53,, = - i543 = H2 sinh (HL) sin (HL)
1 - cos (HL) cosh (F/L)
(B.6)
/)13 = O31 = - -
Hz [sinh (HL) + sin (HL)]
1 - cos (HL) cosh (HL)
(EEL) - cos (HL)]
1 - cos (HL) cosh (FIL)
/514 ~" /)41 ~- - - / ) Z 3 -~- - - / ) 3 2 ~ - - - / " / 2 [ c ~
(B.7)
(B.8)
226
Computational Mechanics 11 (1993)
/)22 = D44 =
/)24 =
H[sin (F/L) cosh (F/L) - cos (F/L) sinh (F/L)]
1 - cos (ILL) cosh (F/L)
(B.9)
D42 = F/[sinh (F/L) - sin (F/L)]
1 - cos (F/L) cosh (F/L)
(B.10)
pAs2)l/4
EIz J
where/7 =
B.3 Shape function derivatives which arise in the QD vector expression
H [ - s i n ( H x ) - sinh(Flx)][sin(HL)sinh(HL)]
2
2 [ 1 - cos (ELL)cosh (F/L)]
//[.cos (Hx) - cosh (Hx)] [sinh (HL) cos (F/L) + cosh (HL) sin (F/L)]
211 - cos (ILL) cosh (F/L)-]
[cos (Fix) + cosh (F/x)] [-cos(F/x) - cosh (F/x)] [-sin (F/L) sinh (HL)-I
m4
2
211 - cos (F/L) cosh (HL)]
[ - sin (Fix) - sinh (F/x) ] [.sinh (F/L) cos (HL) - cosh (HL) sin (HL)]
211 - cos (HL) cosh (/TL)]
H [ - sin (Fix) - sinh (Hx)] [cos (ILL) - cosh (F/L) ]
Ms=
211 - cos (ILL) cosh (ELL)]
//[cos (Fix) - cosh (F/x)] [sin (F/L) + sinh (F/L)]
+
211 - cos (ILL) cosh (F/L)3
[ - sin (F/x) - sinh (F/x)] [sin (ELL) - sinh (//L)]
M6=
211 - cos (ILL) cosh (F/L)]
M3=
F/[.-sin(Hx) + sinh(Hx)]
q
[cos (Elx) - cosh (Elx)] [cos ( I L L ) - cosh (F/L)3
211 - cos (ILL) cosh (F/L)]
p As 2 ~ 1/4
where /-f=
k ~ / 9
B.4 Linear interpolation approximation of the QD vector
+
(B.11)
(B.12)
(a.13)
(B.14)
(
QDa ,,~ EA I1 + 1_ - c_~ _
--2-
~ L s i n h ( ~ ) ] ff(s'O)+-2 k ~ L s i n h ( ~ ) J
EA1l~c~
QD2,~--~QD~
EA I 1 - c o s h (s-~Lc)l
s~hnh(~Lc)jfi(s,o)+EA
P(s'L)
[ l-c~ 1
l
(B.16)
1+ . . . . .
EA[~ - 1 q sin (HL)(i -( cHoLs )h)~f /~os
~ (/TL-)
+ sinh
(HL)(lc~sh
(~))-c~
/~(s, 0)
EA ~ sin (HL)(1 - cosh (HL)) + sinh (HL)(I - cos (ILL)) ~ R(s, L)
k
(B.15)
_1
(B.17)
D. S. Pipkins and S. N. Atluri: Non-linear analysis of wave propagation using transform methods
227
~sin (ILL) sinh (IIL) + cos (HL) - cosh (HL)]~( s O)
+ EA [ sin (HL)f_I~L(lsin(hH L )+~ cos
c o ~(HL)
( i 7 i-~ cosh (HL) ]~R'st,L)
(B. 18)
QD5 ~ EA [ sin (HL)(1 - cosh (rlL)) + sinh (HL)(1 - cos (HL))]/](s, 0)
+ EA[1
sin(HL)(1-cosh(HL))+sinh(HL)(1-cos(FIL))lKs
H - ~ --- c o ~ - H L ~ ~ ) )
J (,L)
EA [ sin (HL) sinh (/-/L) +
_
cos ( n r ) - cosh ( n c ) ]
EA~sin (ILL) sinh (HL) + cos (HL) - cosh (FIL)~K(s
L
where H =
(
(B.19)
n--FL]--foos(h--i)c-0 sh
J.,
L)
_
(B.20)
, x)3]).
pAs 2) 1/4 fi(s, x)= A[(w,x) 2] and/#(s, x)= (A[u,xWx] + ~A[(w
EIz
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Communicated by D. E. Beskos, August 13, 1992