Non-linear damping of slender wood bridges

Compurers& Snucrures Vol. 61, No. 4. pp. 651-664. 1996 Copyright 0 1996 Bleviu Science Ltd Printedin Great Britain. All rights rewed 0045-7949/96 515.00 + 0.00

Pergamon PIIt !%04!5-7949(96)00049-l

NON-LINEAR

DAMPING

OF SLENDER

WOOD BRIDGES

A. Jr&Bat and A. Tesart: tHelsinki University of Technology, Department of Structural Engineering, Laboratory of Bridge Engineering, Otaniemi, Finland SWasser- und Schiffahrtsamt Kiel-Holtenau, Schleuseninscl 2, 24 159. Kiel, Germany (Received

8 March

1995)

Abstract-The paper is concerned with the solution of non-linear damping behaviour of wood bridges subjected to traffic and wind forcing. The development of reliable and efficient techniques for the numerical handling of non-linear vibrations of wood bridges is emphasized. Viscoelastic material behaviour is considered theoretically, numerically and experimentally. Illustrative numerical and experimental verifications are presented. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION

Non-linear and damping properties of slender wood bridge structures can initiate significant dynamic displacements even in cases of common traffic, wind or technology forcing of such constructions. Vibrations of slender non-linear structural systems can generally result in unpredictable and often catastrophic instabilities, especially during the action of smooth or turbulent wind flows. The dynamic behaviour of non-linear structural systems can result both in a loss of stability and predictability in the limit state response. There can arise a loss of global parametric stability as critical parameters are varied. From a theoretical point of view there can almost be revealed an immense complexity in the intricate patterns of bifurcating limit states of even relatively simple non-linear systems. The above effects include sudden jumps to resonance, super or subharmonic oscillations, period doubling bifurcations, etc. The need for treatment of the above problems appeared during theoretical, numerical and experimental investigations made in ramifications of the research project ‘Development of Wood Bridges’, undertaken in the Laboratory of Bridge Engineering of Helsinki University of Technology in 1992. Some results obtained were published in Refs [l-3]. The results from the above problems are presented below. 2. GOVERNING

INCREMENTAL

EQUATIONS

OF MOTION

The governing incremental equation of motion for the non-linear dynamic analysis of slender wood bridges is given by M, da, + C, dv, + P, dr, = dR,, CA6 61/6-D

where da,, dv, and dr, denote the increments of nodal accelerations, velocities and deformations, respectively. M, and C, are structural mass and damping matrices, dR, is the vector of external loads and P, dr, is the vector of non-linear external forces defined by P, dr, = K, + N, dr, - dV, +d,,

(2)

where N, dr, is the vector of non-linear pseudoforces [4] and dV, +dris the local approximation error. At each frequency or time step an estimate of N, dr, is computed and iterations are performed until dV, +dr becomes sulhciently small when compared to the adopted tolerance norm. As an estimate of N, dr, for the first iteration at time step t an extrapolated value from previous solutions can be used, i.e. N, dr, = (1 + B)N,-dr drr-dr - 3N,-2 dt dr,-2.d,, (3) where /I is extrapolation parameter ranging from oto 1. Applying the deflections of non-linearly behaving wood bridges studied, the internal nodal forces are not in equilibrium with the acting external forces. Only the application of extra forces to support the incorrect version of the deformations while seeking to improve them allows the pseudo-forces to be reduced to zero. Principally, the problem of solving non-linear incremental equations is simply that of finding a technique for reducing pseudo-forces to zero [S]. 3. DAMPING

INFZUENCRS

In the field of slender wood bridges the limit state dynamic behaviour is generally influenced by the

(1) 657

A. Jutila and A. Tesar

658

damping properties of the material, structure, joints and environment. Generally, the reduction of the mechanical energy of a vibrating wood bridge can be simulated by hysteretic and viscous damping models. In the hysteretic model the damping forces generated by internal material friction are proportional to the deformations occurring. Damping thus depends on the stress through the displacement dependence of the incremental stiffness. The specific work D of total damping can be expressed as the function of stress [5]: (4)

D = J . a”,

where J and II are experimentally found structural parameters. The total work of damping is then given

CONFIGURATION

1

CONFIGURATION

2

by integration of specific works of individual elements of the discretization mesh used. The variability of stress causes each particle of the structure to have its own hysteresis curve contributing to the total damping of the wood bridge studied. In the element volume VBthe work of damping is given by

D8=I”DdY=[mXDdV,dcd..

The maximal stress Q- corresponds to maximal damping D,, and DE can be expressed as

D, = DmxVg81,

!

Plaatlo

Tube

:

spruce

wood

&

Fin8

@

Top Disc

0

Bottom

Part

0 110 mm

Plate

(5)

8 99 mm

Fig. 1. Cross-section and elevation of the model studied (all dimensions in milliieters).

(6)

Non-linear damping of slender wood bridges

0

Wind Tunnel

@

Model of Transporter

@

Artificial

Terrain

@

Dynamio

Moment

@

Gravity

659

Shell Bridge

Transducer

Block

Fig. 2. Model set-up in wind tunnel.

with parameter /I, :

IL= The energy

with nondimensional

I’(z+#co)d(~). C7) cumulated

expressed then as

in volume analysed can be

82=

parameter j$ :

6’(+J($)/d(~)d(~). t9)

The loss factor of damping for an analysed element can he then expressed by the following

660

A. Jutila and A. Tesar

which is to be incorporated complex elasticity moduli:

into

the following

E = Eo (1 + in).

(11)

G = Go (1 + jr,+,

(12)

and

oscillating at the frequency 00 with the amplitude 10. Every complex model can be transformed into a convenient equivalent viscous formulation [7J The damping capacity !P of a material of wood bridges can be defined as the ratio of dissipated energy 6 W per cycle of vibration and the maximum stored energy W per cycle: Y=6w/w=2rr~=4rr[.

or into the corresponding

k = k0 (1 + iv,“),

(13)

in nodes of joints or structural resistances [6]. The loss factor is equivalent to the dissipated energy and the equation of motion, eqn (l), can be therefore modified as M,dn,+P,(l+i&dr,=dR,.

(14)

Problems such as noncausalities can occur and therefore hysteretic damping should be employed especially for the steady-state harmonic forcing of slender wood bridges. It is common practice in wood structures to substitute the complex stiffness by an equivalent viscous damping. Such a shift from one model of damping to another one has to ensure the same amount of energy dissipation 6W in the hysteretic model:

cycle of vibration.

The

system

The damping capacity can be related to the hysteretic loss factor q, based on the complex elasticity moduli or complex spring characteristics as mentioned above, and to the viscous damping ratio 4.

4. VISCOELASTK

MATERIAL

MODEL

The dissipative properties of wood bridges can be approached in a more realistic way when a viscoelastic material model is used. Principally, such a model is described in a time domain and no limitations concerning material non-linearities have to be made. In this paper a numerical procedure proposed by Simo [S] is applied to study the viscoelastic damping properties of slender wood bridges. The free energy function of the wood material

F0 = Y$ + Y&,

studied

(19)

is

Table 1. Physical parameters of the model Physical parameter Weight of 692 mm plastic tube Weight of 735 mm spruce wood part weight of iins Total weight of model Length of tlow-exposed part Total length of model Modulus of elasticity of spruce wood Modulus of elasticity of plastic Area of spruce wood part of cross-section Area of plastic part of cross-section Areaoffins Total area of cross-section Distance of centre of gravity T from reference point A (Fig. 1) Moment of inertia in direction y Moment of inertia in direction x

(18)

is split up into a volumetric part !P$ and a deviatoric part Yk. The volumetric energy part is considered as being simply elastic. The stress d is split up into elastic and dissipative components expressed as

and in the viscous damping model,

per

(17)

spring characteristics

Coniiguration

1

Configuration 2 (with fins)

97 g 265 g 415 g 696 mm 739 mm 118,500 kg cm-’ 97,820 kg cm-* 360.85 mm* 156.29 mm* 537.14 mm* 11.951 mm

97 g 265 g 7g 422 g 696 mm 139 mm 118,500 kg cm-* 97,820 kg cm-* 360.85 mm* 156.29 mm* 2x30mm2 597.14 mm* 11.705mm

57,258.650 mm* 97,420.094 mm*

60,027.334 mm* 153,420.090 mm*

a)

tlEWLETT.PACKARD

‘4

Fig. 3. Plucking of the model (configuration without tins): (a) torsional vibration; (b) croISS,-wind vibration; (c) along-wind vibration.

where 1 is the viscosity constant related to the elasticity modulus: A = zE through

of the material

(20)

where c and 1 are the elasticity and viscosity tensors, respectively. Through the principle of virtual displacements one obtains a system of differential equations:

the relaxation time r and discrete strain BTcB d Vr, + L = Br,

(21)

and with the strain rate C = Bv,

WpN dva,

BTABd Vv, + 0 J1

s0

= K, r, + C, v, + M, a1= R,, (22)

arc the functions of the matrix B consisting of the derivatives of corresponding shape functions N. Now, eqn (19) can be modified as u = cBr, + ~Bv,,

sD

(23)

(24)

where p is the material density. Equation (24) relates the structural damping models to the material level and vice versa. The two time-dependent effects, the material law and the inertial forces, are both solved within the algorithm loop employing the Newmark method [6] for direct time integration.

A. Jutila and A. Tesar

662

A

!..,

I

.,,:,

A.3

I

._

+._A-

I

Fig. 4. Plucking of the model (configuration with fins): (a) along-wind vibration; (b) cross-wind vibration.

In a viscoelastic formulation a different solution strategy of the constitutive eqn (19) is performed. The material law formulated as a convolution integral results in

E=

h(t - s)

aa/aa ds,

(25)

using the kernel with parameter q = -t/r:

substructures in space and time. Mixed multigrid schemes of discretization in space and time consist of a problem-oriented variability of substructure sizes and time steps in various regions of the structure as well as in various time steps. The mixed NewmarkWilson technique for direct time integration of the corresponding non-linear equations of motion can be applied in combination with the FETM-method of mixed space discretization when adopting the substructuring of the above problems in space and time [5].

h(t) = f (1 - eq). 5. NUMERICAL AND EXPERIMENTAL VERIFICATIONS

The volumetric and deviatoric parts of the deformations have to be handled separately. The model can be extended to more sophisticated formulations by employed extended kernels h(t). With some modifications the above considerations can be applied also for the solution of more complicated wave propagation problems of structural engineering [9] or ultimate states of slender wood bridge structures [lo]. The solution of such problems is concerned with the generalized dynamic simulation of complex structures consisting of an assemblage of

For verification of the present theoretical considerations the numerical and experimental studies of special model as shown in Fig. 1 were made. The model was originally designed and used for the experimental investigation of aeroelastic response of special types of transporter shell bridges in the wind tunnel of the Norwegian Institute of Technology in Trondheim [l 11. The model was built using a combination of wood and plastic components glued together. Such a combination increases the viscoelastic properties of the simulated wood bridges. The

Table 2. Calculated and measured natural frequencies of the model studied Experiment

Theory Without Model Cross-wind Along-wind Torsion

fills

16.721 Hz 21.932 Hz 88.145 Hz

With tills

16.973 Hz 23.642 Hz 88.146 Hz

Without 6llS

16.21 Hz 20.83 Hz 89.29 Hz

With fitlS

15.62 Hz 20.84 Hz -

663

Non-linear damping of slender wood bridges LOG. DECR. 0.1000 0.0953

0.0750

0.0599

-

0.0500

-

0.0250

-

0.0200

DYNAMIC

INCLINATION

OF MODEL

AXIS IN MM OF RECORDER

SCALA

Fig. 5. Numerical and experimental evaluation of the damping level-amplitude vs logarithmic decrement of damping for: (a) cross-wind flexural vibration; (b) torsional vibration; (c) along-wind flexural vibration (numerical values, experimental values).

bottom part of the model cross-section was made of spruce wood. The top shell part of the model was made of a half-circular plastic tube as shown in Fig. 1. The model was provided with exchangeable spruce wood tins and a plywood top disc for influencing the aeroelastic parameters during smooth or turbulent air flows. The massive model base plate made of spruce wood was connected with a dynamic flexure transducer, with fins oriented in the along-wind direction as shown in Fig. 2. The physical parameters of the model set-up studied are summarized in Table 1. The details of the aeroelastic analysis are not treated here; they are given in Ref. [l I]. In this paper the attention is focused on the solution of the actual damping behaviour of the model studied, taking into

account the viscoelastic material effects theoretically treated above. The top plucking of the model and evaluation of Figs 3 and 4 submit the values of natural frequencies and of the damping level in along-wind and cross-wind directions of flexural vibrations as well as in the torsional oscillations of both configurations of the model set-up studied. The experimental and numerical results obtained are summarized and compared in Table 2 as well as in Fig. 5. The numerical evaluations were performed using the above theoretical considerations incorporated into the algorithms of the FETM-method in accordance with the flow-charts in Ref. [S]. The results obtained present good coincidence of the theoretical, numerical and experimental approaches.

A. Jutila and A. Tesar

664 6. CONCLUSIONS

Some new concepts were developed recently in the form of new ideas in discretization and vectorizing procedures and in the automation of numerical procedures when treating the non-linear damping and viscoelastic behaviour of slender wood bridges. The prediction is simply that these are going to be developed further, thoroughly evaluated and included into commonly available software. Because the efficiency of the available solution procedures is highly case-dependent, an optimally efficient computer program for the solution of the viscoelastic damping of slender wood bridges must contain a number of options and consequently requires an increased degree of user sophistication

together with a way to automatically select from the available options the procedures that are best suited for each particular case.

3.

4.

5. 6. I.

8.

9. REFERENCES

1. A. Jutila and A. Tesar, Vibration of slender wood bridges. In: Proc. Structural Dynamics-EURODYN ‘93, pp. 10591066, Trondheim (1993). 2. A. Jutila, P. Haakana, L. Salokangas and A. Tesar, Numerical and experimental analysis of static and dynamic behaviour of wood bridges. Technical report of the research project Development of Wood Bridges,

10. 11.

Laboratory of Bridge Engineering, p. 80, Helsinki University- of Technology, Helsink~( 1992). H. Rautakorni. A. Tesar. A. Jutila, R. Miikinuro. P. Haakana and ‘L. Salokangas, Prospects of wood in various types of bridges. Technical report of the research project Development of Wood Bridges, Laboratory of Bridge Engineering, Publication No. 5, p. 54, Helsinki University of Technology, Helsinki (1993). A. Tesar, Energy deletion silent boundary in non-linear wave propagation. Int. J. Struct. Engng Mech. (in press). A. Tesar, L. Fillo, Transfer Matrix Method, p. 240. Kluwer Academic, Dordrecht (1988). A. Tesar, Non-linear dynamic response of thin shells. Acta Technica CSAV 6, 754115 (1983). M. Kaliske, J. Jagusch, N. Gebbeken and H. Rothert, Damping models in finite element computations. In: Proc. Structural Dynamics-EVRODYN ‘93, pp. 585-591, Trondheim (i993). J. C. Simo. On a fullv three-dimensional finite strain viscoelastic’ damage model-formulation and computational aspects. Comput. Meth. Engng 29, 1595-1638 (1990). A. Tesar and J. Svolik, Wave distribution in fibre members subjected to kinematic forcing. Int. J. Commun. Numer. Meth. Engng 9, 189-196 (1993). E. Isoksela, A. Jutila, L. Salokangas and A. Tesar, Numerical and experimental analysis of wood bridges. Comput. Struct. (in press). A. Tesar, Aeroelastic response of transporter shell bridges in smooth air flow. Technical report, p. 75, Institute of Steel Structures, The Norwegian Institute of Technology, Trondheim (1978).