Download NONLINEAR DYNAMIC ANALYSIS OF BEAMS WITH VARIABLE

NONLINEAR DYNAMIC ANALYSIS OF BEAMS WITH VARIABLE STIFFNESS.
AN ANALOG-EQUATION SOLUTION
J.T. Katsikadelis and G.C. Tsiatas
Department of Civil Engineering
National Technical University of Athens, GR-15773 Athens, Greece
1. SUMMARY
In this paper the Analog Equation method (AEM), a BEM-based method, is employed to the
nonlinear dynamic analysis of an initially straight Bernoulli-Euler beam with variable
stiffness undergoing large deflections. In this case the cross-sectional properties of the beam
vary along its axis and consequently the coefficients of the differential equations governing
the dynamic equilibrium of the beam are variable. The formulation is in terms of the
displacement components. Using the concept of the analog equation, the two coupled
nonlinear hyperbolic differential equations are replaced by two uncoupled linear ones
pertaining to the axial and transverse deformation of a substitute beam with unit axial and
bending stiffness, respectively, under fictitious time dependent load distributions. A
significant advantage of this method is that the time history of the displacements as well as
the stress resultants are computed at any cross-section of the beam using the respective
integral representations as mathematical formulae. Beams with constant and varying
stiffness are analyzed under various boundary conditions and loadings to illustrate the merits
of the method as well as its applicability, efficiency and accuracy.
2. INTRODUCTION
In recent years a need has been raised in engineering practice to predict accurately the
nonlinear response of beams subjected to dynamic loads, especially when the properties of
their cross section are variable. The nonlinearity results from retaining the square of the
slope in the strain-displacement relations (intermediate nonlinear theory). In this case the
transverse deflection influence the axial force and the resulting equations, governing the
dynamic response of the beam, are coupled nonlinear with variable coefficients. Moreover,
the pertinent boundary conditions of the problem are also nonlinear.
Many researchers have been involved in the solution of the problem using various
techniques. In all cases only linear boundary conditions were considered (immovable edges).
Approximate analytical solutions are restricted only to large amplitude free vibration of
uniform beams [1,2,3,4] using elliptic integral, perturbation or Ritz-Galerkin methods. The
Galerkin method has also been used by Raju et al. [5] to analyze the large amplitude free
vibrations of tapered beams while Sato [6] investigated the nonlinear free vibrations of
stepped height beams using the transfer matrix method. Numerical methods have been
developed for the solution of the problem like the Finite Difference Method [7]. The Finite
Element Method was used for free nonlinear vibrations of uniform beams [8,9] and of
tapered beams [10] as well as for forced nonlinear vibrations of uniform beams
[11,12,13,14].
The complexity of the problem, compelled many of the above researchers to make gross
simplifications. In the case of nonlinear free vibrations [1,2,3,4,5,6,8,9,10] either the time
function or the space function is assumed and a characteristic nonlinear frequency is
computed. Although this is correct in the linear vibration theory, reducing the problem to a
conventional eigenvalue problem by using the fact that the solution variables separable in
space and time, the nonlinear free vibration problem do not admit this separation since the
mode shape varies with time. The FEM has been successfully employed to the solution of
forced nonlinear vibrations of beams with constant cross-section [11,12,13,14]. However,
beams with non-uniform cross-section are often approximated by a large number of small
uniform elements replacing the continuous variation with a step law. In this way it is always
possible to obtain acceptable results and the error can be reduced as much as desired by
refining the mesh.
In this paper, an accurate direct solution to the governing coupled nonlinear differential
equations of hyperbolic type is presented, which permits the treatment also of nonlinear
boundary conditions. The solution is achieved using the analog equation method [15] as it
was developed for the nonlinear static analysis of beams with variable stiffness [16].
According to this method the two coupled nonlinear differential equations are replaced by
two equivalent uncoupled linear ones pertaining to the axial and transverse deformation of a
substitute beam with unit axial and bending stiffness under fictitious time dependent load
distributions. Several beams are analyzed under various boundary conditions and load
distributions, which illustrate the method and demonstrate its efficiency and accuracy.
3. GOVERNING EQUATIONS
Consider an initially straight beam of length l having variable axial EA and bending
stiffness EI , which may result from variable cross-section, A A( x ) , and/or from non
homogeneous linearly elastic material, E E ( x) ; I I x is the moment of inertia of the
cross-section. The x axis coincides with the neutral axis of the beam, which is bent in its
plane of symmetry xz by the combined action of the distributed load px px ( x) and
pz pz ( x, t ) in the axial and transverse directions. Moderate large deflections result from
the nonlinear kinematic relation, which retains the square of the slope of the deflection,
while the strain component remains still small compared with the unity. Namely,
x
u, x
1
2
w,2x z
(1)
where u u ( x, t ) and w w( x, t ) are the axial and transverse displacement, respectively,
w, xx is the curvature of the deflected axis. Neglecting the axial inertia force and
and
considering the equilibrium of the beam element in the deformed configuration [16] we
obtain the following equations of motion
Nx ,x
px
w N z ,x
M ,x Q
pz
(2a)
(2b)
(2c)
subjected to boundary conditions
a1u 0
1
1
a2 N x 0
w 0
a3 ,
Nz 0
2
w, x 0
2
M 0
a1u l
3
,
3
,
1
1
w l
a2 N x l
2
w, x l
a3
(3a)
Nz l
3
(3b)
M l
3
(3c)
2
and the initial conditions
w x, 0
w( x) ,
w x, 0
w( x)
(4a,b)
x mass density per unit length and ak , ak ,
where
k
,
k
,
k
,
k
(k 1, 2,3) are given
constants; w0 ( x), w0 ( x) are prescribed functions. Eqns (3a,b,c) describe the most general
boundary conditions associated with the problem and can include elastic support or restrain.
The quantities N x , N z are [16]
Nx
N Qw, x ,
Nz
Nw, x Q
(5a,b)
where N and Q are the axial and shear force tangential and normal to the deformed axis of
the beam, respectively, and M the bending moment; N and M are expressed in terms of
the displacements as
N
1
2
EA u , x
M
w,2x
(6)
EIw, xx
(7)
Note that on the base of eqns (2c), (5) and (6), the boundary conditions (3) are in general
nonlinear.
Substituting eqns (5) into eqns (2a,b) and using eqn (2c) to eliminate Q yield
N ,x
M , x w, x , x
w M , xx
px
Nw, x , x
(8a)
pz
(8b)
which by virtue of eqns (5) and (7) become
EA u , x
w
1
2
w,2x
EIw, xx , xx
,x
EIw, xxx w, x , x
EA u , x
1
2
(9a)
px
w,2x w, x , x
pz
(9b)
4. THE AEM SOLUTION
Eqns (9) are solved using the AEM, which for the problem at hand is applied as follows. Let
u u ( x, t ) and w w( x, t ) be the sought solution of eqns (9), which are two and four times
differentiable, respectively, in 0,l . Noting that eqn (9a) is of the second order with respect
to u and of fourth order with respect to w we obtain by differentiating
u , xx b1 x, t
(10a)
w, xxxx b2 x, t
(10b)
where b1 , b2 are fictitious loads depending also on time. Equations (10) indicate that the
solution of eqns (9) can be established by solving eqns (10) under the boundary conditions
(2)-(3), provided that the fictitious load distributions b1 , b2 are first determined. Note that
eqns (9) are quasi-static, that is the time considered as a parameter. The fictitious loads are
established by developing a procedure based on the boundary integral equation method for
one-dimensional problems. Thus, the integral representations of the solutions of eqns (10)
are written as
b
u ( x, t ) c1 x c2
a
G1 x,
,t d
b1
b
w( x, t ) c3 x 3 c4 x 2 c5 x c6
a
(11a)
G2 x,
b2
,t d
(11b)
where ci (i 1, 2,...6) are arbitrary constants to be determined from the boundary conditions
and Gi ( x, ) (i 1, 2) are the fundamental solutions of eqns (4a,b), that is a particular
singular solution of each of the following equations
G1 , xx
x
(12a)
G2 , xxxx
x
with
) being the Dirac function. After integration of eqns (12) we obtain [17]
(x
G1
1
2
G2
1
12
(12b)
x
(13a)
x
2
x
(13b)
The derivatives of u and w are obtained by direct differentiation of eqns (11). Namely
l
u , x x, t
c1
u, xx x, t
b1 x, t
w, x x, t
3c3 x 2 2c4 x c5
0
G1 , x x,
b1
(14a)
,t d
(14b)
w, xx x, t
6c3 x 2c4
w, xxx x, t
6c3
w, xxxx x, t
b2 x, t
l
0
l
0
l
0
G2 , x x,
G2 , xx x,
G2 , xxx x,
b2
b2
,t d
b2
,t d
,t d
(15a)
(15b)
(15c)
(15d)
Subsequently, the interval 0, l is divided into N equal constant elements, having length
l / N , on which the N nodal points are placed at the middle of the elements. Equations (14)
and (15) are descritized and applied to the N nodal points. This yields
u c1x1 c2 x 0 G 1b1
u,x c1x 0 G 1 ,x b1
u,xx b1
w c3 x3 c4 x 2 c5 x1 c6 x0 G 2b 2
w,x 3c3 x2 2c4 x1 c5 x0 G 2 ,x b 2
w,xx 6c3 x1 2c4 x0 G 2 ,xx b 2
(16a)
(16b)
(16c)
(17a)
(17b)
(17c)
w,xxx 6c3x0 G 2 ,xxx b 2
w,xxxx b 2
(17d)
(17e)
where G1 , G1 ,x , G 2 ,xxx are N N known matrices, originating from the integration of
the kernels G1 ( x, ) , G2 ( x, ) and their derivatives on the elements; u , u,x ,… w,xxxx are
vectors including the values of u , w and their derivatives at the nodal points; b1 , b 2 are
also vectors containing the values of the fictitious loads at the nodal points and
x k {x1k , x2k , xNk }T are vectors containing the k th power of the abscissas of the nodal
points.
Finally, collocating eqns (9) at the N nodal points and substituting the derivatives from
eqns (16) and (17) yields the following system of equations for b1 , b 2
K1 (b1 , b 2 , c) p x
(18a)
Mb 2 K 2 (b1 , b 2 , c) p z (t )
(18b)
where M is the known N N generalized mass matrix; K i (b1 , b 2 , c) generalized stiffness
vectors and c {c1 , c2 , c6 }T . After substituting the relevant derivatives in the boundary
conditions, we obtain six additional equations
f1 (b1 , b 2 , c)
0
(19)
i 1, 2,...6
Eqn (18b) is the semi-discretized equation of motion of the beam and its initial conditions
result from eqns (17a) when combined with eqns (4). Thus, we have
b2 0
G 2 1 w 0 c3 x3 c4 x 2 c5 x1 c6 x0
(20a)
b2 0
G 2 1w 0
(20b)
The time step integration method for nonlinear equations of motion can be employed to
solve eqn (18b). In each iteration for b 2 within a time step, the current value of b 2 is
utilized to update the vectors b1 and c on the basis of eqns (18a) and (19) solving a
nonlinear system of algebraic equations. In this paper, the average acceleration time step
integration method was employed to solve eqn (18c). Once the vectors b1 , b 2 , c have been
computed the displacements vectors u , w and their derivatives at any instant t are
computed from eqns (16) and (17).
5. NUMERICAL EXAMPLES
On the base of the procedure described in previous section a FORTRAN program has been
written for the nonlinear dynamic analysis of beams with arbitrarily varying stiffness. Three
types of boundary conditions are considered (i) hinged-hinged, (ii) fixed-hinged and (iii)
fixed-fixed. The employed initial conditions for the free vibrations study are
w( x) 16 w0
respectively;
4
2
3
/ 5 , w( x)
4 w0 2
4
5
3
3
2
, w( x) 16 w0
4
2
3
2
,
x / l and w0 is the central deflection of the beam. In all three cases it is
assumed w( x) 0 .
5.1 Uniform cross-section. Free Vibrations
For the comparison of the results with ones available from the literature, the free vibrations
of a uniform beam with length l 1.0 m ( N 21 ) have been studied. The employed data are
E 2.1 108 kN/m 2 , b 0.01m , h 0.03m and
2355 kg/m . In Table 1 results for the
frequency ratios 0 / at various amplitude ratios w0 / are presented compared with
existing solutions; 0 is the respective frequency of the linear vibration and
I / A is
the radius of gyration
TABLE 1
Frequency ratios 0 / at various amplitude ratios w0 / in example 5.1.
w0 /
0.2
0.6
1.0
3.0
5.0
AEM
1.003
1.032
1.089
1.623
2.347
hinged-hinged
Ref. [9] Ref. [14]
1.004
1.004
1.033
1.036
1.089
1.070
1.626
1.673
2.350
2.350
AEM
1.003
1.019
1.052
1.364
1.815
fixed-hinged
Ref.[9] Ref. [14]
1.002
1.002
1.017
1.021
1.047
1.057
1.362
1.416
1.829
1.916
AEM
1.002
1.008
1.021
1.179
1.429
fixed-fixed
Ref. [9] Ref. [14]
1.001
1.001
1.008
1.011
1.022
1.030
1.183
1.322
1.447
1.556
5.2 Uniform cross-section. Forced Vibrations
A fixed-fixed uniform beam with length l 0.508 m has been analyzed ( N 21 ) under a
concentrated force of 2.844 kN ( t 0 ) acting at the midspan of the beam with zero initial
conditions. The employed data are: E 2.07 108 kN/m 2 , b 0.0254 m , h 0.003175 m
and
0.2186 kg/m . Many investigators have studied this problem. McNamara [11] used
five beam bending elements based on a central–difference operator, Mondkar and Powell
[12] used five eight-node plane stress elements to model one-half of the beam, Yang and
Saigal [13] used six beam elements and Leung and Mao [14] used six beam elements for the
one-half of the beam. In Table 2 the maximum deflection and the period of the first cycle are
presented as compared with the above existing solutions.
TABLE 2
Maximum deflection wmax (m) and period T ( sec) of the first cycle in example 5.2.
AEM Ref. [11] Ref. [12] Ref. [13] Ref. [14]
wmax ( m) 0.01960 0.02286 0.01956 0.01956 0.01946
T ( sec)
2280
2884
2300
2300
2151
5.3 Variable cross-section. Free and forced Vibrations
The free and forced vibrations of a steel I-section beam with length l 10.0 m ( N 21 )
have been studied. The cross-section is constructed from a pair of identical flange plates
b 300 mm wide by t f 30 mm thick and a web plate tw 12 mm thick with variable
height hw ( x) . The other data are: E 2.1 108 kN/m2 , hw (0) 500 mm , g 0 3000 kN/m ,
7850 A( x) kg/m . Two cases are considered (a) Constant web height, hw hw (0) and (b)
Linearly varying web height, hw hw (0)(0.5 x / l ) . In both cases the volume of the
material, i.e. V
t w hw (0) 2t f b l , was kept unchanged. In Figures 1 and 2 results for the
natural vibrations ( w0 0.5 m ) are presented for case (a) and (b), respectively. Finally, the
forced vibrations have been studied under the so-called “static” load g g 0t / t1 if 0 t t1
and g g 0 if t1 t ( t1 0.02sec ) with zero initial conditions. The time histories of the
central deflection for case (a) and (b) are shown in Figure 3 and 4, respectively.
B.C. type (i)
B.C. type (ii)
B.C. type (iii)
1
1
0.5
0.5
0
-0.5
0
-0.5
-1
-1
-1.5
-1.5
0
0.01
0.02
t (sec)
0.03
B.C. type (i)
B.C. type (ii)
B.C. type (iii)
1.5
W(l/2,t) (m)
W(l/2,t) (m)
1.5
0.04
0
0.01
0.02
t (sec)
0.03
0.04
Figure 1: Free vibrations. Time history of the Figure 2: Free vibrations. Time history of the
central deflection: Case (a).
central deflection: Case (b).
B.C. type (i)
B.C. type (ii)
B.C. type (iii)
1
0.8
W(l/2,t) (m)
W(l/2,t) (m)
0.8
B.C. type (i)
B.C. type (ii)
B.C. type (iii)
1
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0
0.02
0.04
t (sec)
0.06
0.08
Figure 3: Forced vibrations. Time history of
the central deflection: Case (a).
0
0.02
0.04
t (sec)
0.06
0.08
Figure 4: Forced vibrations. Time history of
the central deflection: Case (b).
6. CONCLUSIONS
In this paper a direct solution to dynamic problem of beams with variable stiffness
undergoing large deflections has been presented. The governing equations have been
derived considering the dynamic equilibrium in the deformed configuration. The presented
solution is based on the concept of the analog equation, which converts the two coupled
nonlinear equations of motion into two quasi-static uncoupled linear equations with
fictitious loads. These equations are subsequently solved using the one-dimensional integral
equation method. From the presented analysis and the numerical examples the following
main conclusions can be drawn.
(a) Only static fundamental solutions are employed to derive the integral representation of
the solution.
(c) The displacements and the stress resultants are computed at any point using the respective
integral representation as mathematical formulas.
(d)Accurate numerical results for the displacements and the stress resultants are obtained.
(e) The solution of the static problem can be obtained using the same computer program if
the inertia forces are neglected.
7. REFERENCES
[1] Woinowsky-Krieger S., The effects of an axial force on the vibration of hinged
bars, Transactions of the ASME, (1950), 35-36.
[2] Evensen, D.A., Nonlinear vibrations of beams with various boundary conditions,
American Institute of Aeronautics and Astronautics Journal, (1968), 370-372.
[3] Srinivasan, A.V., Large amplitude free oscillations of beams and plates, American
Institute of Aeronautics and Astronautics Journal, (1965), 1951-1953.
[4] Ray, J.D. and Bert, C.W., Nonlinear vibration of a beam with pinned ends,
Transactions of the ASME, Journal of Engineering for Industry, (1969), 997-1004.
[5] Raju, L.S., Rao, G.V. and Raju K.K., Large amplitude free vibrations of tapered
beams, American Institute of Aeronautics and Astronautics Journal, (1976), 280282.
[6] Sato, H., Non-linear free vibrations of stepped thickness beams, Journal of Sound
and Vibration, (1980), 415-422.
[7] Abhyankar, N.S., Hall, E.K. and Hanagud, S.V., Chaotic vibrations of beams:
numerical solutions of partial differential equations, Journal of Applied Mechanics,
(1993), 167-174.
[8] Bhashyam, G.R. and Prathap, G., Galerkin finite element method for non-linear
beam vibrations, Journal of Sound and Vibration, (1980), 191-203.
[9] Singh, G., Rao, G.V. and Iyengar, N.G., Re-investigation of large amplitude free
vibrations of beams using finite elements, Journal of Sound and Vibration, (1990),
351-355.
[10] Raju, K.K., Shastry, B.P. and Rao, G.V., A finite element formulation for the large
amplitude vibrations of tapered beams, Journal of Sound and Vibration, (1976),
595-598.
[11] McNamara, J.E., Solution schemes for problems of nonlinear structural dynamics,
Journal of Pressure Vessel Technology, ASME, (1974), 96-102.
[12] Mondkar, D.P. and Powell G.H., Finite element analysis of nonlinear static and
dynamic response, International Journal for Numerical Methods in Engineering,
(1977), 499-520.
[13] Yang, T.Y. and Saigal, S., A simple element for static and dynamic response of
beams with material and geometric nonlinearities, International Journal for
Numerical Methods in Engineering, (1984), 851-867.
[14] Leung, A.Y.T. and Mao, S.G., Symplectic integration of an accurate beam finite
element in non-linear vibration, Computers & Structures, (1995), 1135-1147.
[15] Katsikadelis, J.T., The analog equation method-A powerful BEM-based solution
technique for solving linear and nonlinear engineering problems. in Boundary
Elements XVI, CLM Publications, Southampton (1994), 167-182.
[16] Katsikadelis, J.T. and Tsiatas, G.C., Large Deflection Analysis of Beams with
Variable Stiffness. An Analog Equation Solution, Proc. of the 6th National
Congress of Mechanics, E.C Aifantis and E.N. Kounadis (Eds.), Thessaloniki,
Greece, (2001), 172-177.
[17] Banerjee, P.K. and Butterfield, R., Boundary Element Methods in Engineering
Science, Mc Graw-Hill, London (1981).