Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 4, pp. 931-942, Warsaw 2007 THEORETICAL AND EXPERIMENTAL ASSESSMENT OF PARAMETERS FOR THE NON-LINEAR VISCOELASTIC MODEL OF STRUCTURAL POUNDING Robert Jankowski Gdańsk University of Technology, Faculty of Civil and Environmental Engineering, Poland e-mail: jankowr@pg.gda.pl Impacts between adjacent structures during earthquakes have been recen- tly intensively studiedwith thehelp ofdifferentmodels of the impact force. It has been verified through comparisons that the non-linear viscoelastic model is the most accurate one among them. One of the aims of the pre- sent paper is to derive a formula relating the impact damping ratio, as a parameter of the non-linear viscoelastic model, with a more widely used coefficient of restitution. Another aim is to determine the range of the coefficients of restitution and the impact stiffness parameters for different buildingmaterials, such as: steel, concrete, timber and ceramics, based on the results of an impact experiment. Both aims are new and original ele- ments of the study in the field of earthquake-induced structural pounding. The results of the analysis show a wide range of the model parameters obtained for various prior-impact velocity and mass values. The use of these parametric values in numerical simulations allows us to study the behaviour of colliding structures with the increased accuracy. Keywords: structuralpounding, earthquakes,non-linearviscoelasticmodel 1. Introduction Interactions between insufficiently separated structures with different dyna- mic characteristics have been repeatedly observed during earthquakes. This phenomenon, often referred as the earthquake-induced structural pounding, may lead to someminor damage at contact locations in the case of moderate ground motions (see, for example, Zembaty et al., 2005) and may result in substantial destruction or even collapse of interacting structures during severe earthquakes (see Rosenblueth and Meli, 1986; Kasai and Maison, 1997). The 932 R. Jankowski problem of earthquake-induced structural pounding has been recently intensi- vely studiedwith the use of differentmodels of the impact force. The simplest model applies a linear elastic spring (see, for example, Maison and Kasai, 1992) and does not take into consideration energy dissipation during impact due to plastic deformations, local cracking or crushing, fracturing, friction, etc. A more precise linear viscoelastic model (see Anagnostopoulos and Spi- liopoulos, 1992; Jankowski et al., 1998) accounts for some energy loss, but the force-deformation relation is still simplified. In order to simulate this relation more realistically, a non-linear elastic model, which follows the Hertz law of contact, has been adopted by a number of researchers (see Jing and Young, 1991; Chau and Wei, 2001). This model, however, does not account for the energy dissipation during contact. In order to overcome the disadvantages of the models mentioned, a non- linear viscoelastic model of earthquake-induced structural pounding has been proposed by Jankowski (2005c). It has been verified through experiments that the model is the most precise one in simulating the impact force time history during an impact as well as in simulating the pounding-involved structural response during earthquakes. Themodel has been successfully used for study- ing earthquake-induced poundingbetween two adjacentmulti-storey buildings (Jankowski, 2005b) as well as for the analysis of the pounding force response spectrum under earthquake excitation (Jankowski, 2005a). According to the non-linear viscoelastic model, the impact force, F, be- tween two structural members with masses m1 and m2 is expressed by the following formula (Jankowski, 2005c) F =        0 for δ¬ 0 (no contact) β √ δ3+ cδ̇ for δ > 0 and δ̇ > 0 (contact – approach period) β √ δ3 for δ > 0 and δ̇¬ 0 (contact – restitution period) (1.1) δ=x1−x2−d c=2ξ √ β √ δ m1m2 m1+m2 where β is the impact stiffness parameter, ξ denotes the impact damping ratio, which accounts for the energy dissipation during the impact, x1, x2 are displacements of the structural members and d is the initial in-between separation gap. Although the abovemodel has been proposed for earthquake- induced structural pounding, due to its general form it can be also successfully used to study impacts between other types of colliding bodies. The precise determination of the parameters of the non-linear viscoelastic model: β and ξ is essential in order to enhance the accuracy of the numerical Theoretical and experimental assessment of parameters... 933 analysis. Therefore, one of the aims of this paper is to derive a formula relating the impactdamping ratiowith a coefficient of restitution,which is aparameter widely used and studied in the literature (Goldsmith, 1960). The analogous formula defined for the linear viscoelasticmodel has confirmed its applicability (see Anagnostopoulos and Spiliopoulos, 1992). Another aim of the paper is to assess the range of the coefficients of restitution and the impact stiffness parameters for different building materials, such as: steel, concrete, timber and ceramics, based on the results of an impact experiment. Both aims are new and original elements of the study conducted by the author in the field of earthquake-induced structural pounding. 2. The formula between the impact damping ratio and coefficient of restitution The coefficient of restitution is a well-known parameter used in the classical theory of impact. It defines the relation between the post-impact relative ve- locity, δ̇f (δ̇f ¬ 0), and the prior-impact relative velocity, δ̇0 (δ̇0 > 0), of two colliding bodies (Goldsmith, 1960) e= |δ̇f| δ̇0 (2.1) The formula for the relation between the impact damping ratio, ξ, and the coefficient of restitution, e, for the non-linear viscoelastic model can be obta- ined by equating the loss in the kinetic energy (seeGoldsmith, 1960) with the energy loss through the work done by the damping force during the impact m1m2 2(m1+m2) (1−e2)(δ̇0)2 = δmax ∫ 0 cδ̇ dδ=2ξ √ β m1m2 m1+m2 δmax ∫ 0 4 √ δδ̇ dδ (2.2) where δ̇ is the relative velocity between colliding structures during the appro- ach period (δ̇ > 0) and δmax denotes the maximum deformation. In order to determine the formula for δ̇ during the approach period (required to evaluate the integral of equation (2.2)) let us first look at the energy balance during the restitution period of collision, which is considered to be elastic (see equ- ation (1.1)). Due to the energy transfer from the accumulated elastic strain energy at the beginning of the period to the kinetic energy at the end of it, 934 R. Jankowski the following condition holds for each value of deformation δ ∈ 〈0,δmax〉 in the restitution period δ ∫ 0 β √ δ3 dδ+ m1m2 2(m1+m2) δ̇2 = m1m2 2(m1+m2) (δ̇f) 2 (2.3) Solving the above equation allows us to determine the formula for the relative velocity, δ̇, during the restitution period (δ̇¬ 0) as equal to δ̇=− √ (δ̇f) 2− 4β(m1+m2) 5m1m2 √ δ5 (2.4) Moreover, for the point of maximum deformation, when δ= δmax and δ̇=0, from equation (2.3) we obtain δmax = 5 √ √ √ √ ( 5m1m2(δ̇f) 2 4(m1+m2)β )2 (2.5) Assuming that a similar expression as equation (2.4) concerns also the appro- ach period of collision and ensuring that the relation between the post-impact and prior-impact relative velocities, defined by equation (2.1), is satisfied, we can express the formula for the relative velocity, δ̇, during the approach period (δ̇ > 0) as δ̇= 1 e √ (δ̇f) 2− 4β(m1+m2) 5m1m2 √ δ5 (2.6) Substituting equation (2.6) as well as the formula for (δ̇f) 2 obtained from equation (2.5) into equation (2.2), yields m1m2 2(m1+m2) (1−e2)(δ̇0)2 = 4 √ 5 5 ξβ e δmax ∫ 0 4 √ δ √ √ δ5max− √ δ5 dδ (2.7) The detailed evaluation of the integral of equation (2.7) has been presented in Appendix. Substituting equation (A.10), for b = δmax and c = 4 √ δ5max, into equation (2.7) leads to m1m2 2(m1+m2) (1−e2)(δ̇0)2 = 4 √ 5 25 π ξβ e √ δ5max (2.8) Substituting equations (2.5) and (2.1) into equation (2.8) and solving for ξ gives ξ= √ 5 2π 1−e2 e (2.9) Theoretical and experimental assessment of parameters... 935 3. Experimental determination of the coefficients of restitution and impact stiffness parameters for different building materials The experimental study has been carried out in order to determine the range of the coefficients of restitution and the impact stiffness parameters for the most commonly used building materials, such as: steel, concrete, timber and ceramics. The experiment has been conducted by dropping balls of different masses, m1, on a rigid surface (m2 → ∞) of the same material and obse- rving the impact force time histories as well as recording the prior-impact and the post-impact velocities. The properties of balls used in the experiment are specified in Table 1. The experimental setup is shown in Fig.1. Table 1.Properties of balls used in the experiment Material Type/grade/ Ball diameter Ball mass, No. of balls class [mm] m1 [kg] tested 21 0.053-0.054 2 Steel 18G2A 50 0.538-0.541 2 83 2.013 2 103 1.329-1.350 5 Concrete C30/37 114 1.763-1.835 5 128 2.531-2.636 5 55 0.065-0.066 2 Timber pinewood 71 0.109-0.112 2 118 0.493-0.497 2 58 0.243-0.247 2 Ceramics 25 69 0.372-0.377 2 80 0.538-0.572 2 From the experimental results concerning the prior-impact and the post- impact velocities, the coefficients of restitution, e, have been first calculated with the help of equation (2.1). The values of e for different prior-impact velocities are summarised in Table 2. It should be mentioned that due to exceeding the allowable acceleration limit, the steel balls could not be tested for the impact velocities higher than 2m/s, and therefore these results are not given in the table. A graphical presentation of the relation between the mean value of the coefficient of restitution and the prior-impact velocity for different materials is also shown in Fig.2. The results obtained indicate that 936 R. Jankowski Fig. 1. Experimental setup the value of e does not depend on the mass of the balls tested but it is much sensitive to the prior-impact velocity. The highest values of the coefficient of restitution have been obtained for ceramic balls, whereas the lowest for timber ones.Moreover, the general trend for all materials shows a decrease in the coefficient of restitution with an increase in the prior-impact velocity. Fig. 2. Coefficient of restitution vs. prior-impact velocity After determination of the coefficients of restitution, the impact stiffness parameters, β, have been determined by fitting the experimentally obtained impact force time histories using the method of the least squares. The values of β for different masses of the balls tested are presented in Table 3. The results obtained indicate that the impact stiffness parameter does not depend Theoretical and experimental assessment of parameters... 937 Table 2.Values of the coefficient of restitution, e, obtained from the expe- riment Material Prior-impact Range of coefficient velocity, δ̇0 [m/s] of restitution, e [–] 0.2 0.6415-0.7438 0.5 0.5264-0.7142 Steel 1.0 0.5140-0.6239 1.5 0.4967-0.5652 2.0 0.4863-0.5057 0.2 0.7230-0.7840 0.5 0.6580-0.6859 1.0 0.5858-0.5981 Concrete 1.5 0.5378-0.5658 2.0 0.4419-0.5659 3.0 0.4311-0.5001 4.0 0.4278-0.4826 0.2 0.6567-0.6934 0.5 0.5852-0.6356 1.0 0.5334-0.5797 Timber 1.5 0.4851-0.5354 2.0 0.4281-0.5347 3.0 0.4076-0.4708 4.0 0.4002-0.4428 0.2 0.7575-0.7996 0.5 0.6870-0.7681 1.0 0.6270-0.6697 Ceramics 1.5 0.5770-0.6456 2.0 0.5115-0.5642 3.0 0.4737-0.5208 4.0 0.4330-0.4921 on the prior-impact velocity but it shows a dependence onmass of the tested balls. The general trend for all tested materials shows a small increase in β with an increase in the ball mass.Moreover, in the case of the impact stiffness parameter, the highest values of β have been obtained for steel balls, whereas the lowest for timber ones. 938 R. Jankowski Table 3. Values of the impact stiffness parameter, β, obtained from the experiment Material Ball mass, Impact stiffness parameter, β [N/m3/2] m1 [kg] Range Mean 0.053-0.054 1.1-1.5 ·1010 1.30 ·1010 Steel 0.538-0.541 2.4-4.4 ·1010 3.55 ·1010 2.013 3.8-6.6 ·1010 5.44 ·1010 1.329-1.350 3.9-10.0 ·109 7.90 ·109 Concrete 1.763-1.835 4.7-11.2 ·109 8.13 ·109 2.531-2.636 6.4-13.0 ·109 10.45 ·109 0.065-0.066 0.7-1.8 ·108 1.38 ·108 Timber 0.109-0.112 0.9-2.8 ·108 2.16 ·108 0.493-0.497 1.0-5.2 ·108 2.97 ·108 0.243-0.247 1.1-2.3 ·109 1.80 ·109 Ceramics 0.372-0.377 2.2-4.0 ·109 3.13 ·109 0.538-0.572 2.8-5.8 ·109 4.57 ·109 4. Concluding remarks In this paper, the determination of parameters for the non-linear viscoelastic model of structural pounding has been carried out. The formula relating the impact damping ratio with a coefficient of restitution has been first derived. Then, values of the coefficients of restitution and impact stiffness parameters have been determined for different building materials based on the results of an impact experiment. The paper deals with new and original elements of the study conducted in the field of earthquake-induced structural pounding. The results of the study show a wide range of parameters of the non- linear viscoelastic model determined for steel, concrete, timber and ceramics for various prior-impact velocities andmasses of balls tested. The application of the obtained parametric values to numerical simulations allows us to study the behaviour of colliding structures with increased accuracy. In this paper, the results of the experiment conducted by dropping re- latively small balls on a rigid surface have been used. Further experimental studies involving larger elements with different contact surface geometries are therefore required to verify the results obtained.The confirmation of the range of parameters of the non-linear viscoelastic model should also be done thro- Theoretical and experimental assessment of parameters... 939 ugh experiments of pounding between models of real structures conduced on a shaking table under real earthquake excitations. The use of the results of the study presented in this paper does not have to be limited to simulation of pounding-involved behaviour of structures during earthquakes. They can be also applied to study impacts between different types of colliding bodies in other conditions. A. Appendix The present appendix shows the evaluation of the definite integral of the form b ∫ 0 4 √ δ √ c2− √ δ5 dδ (A.1) where b and c are positive constants and c2 ­ √ δ5. Let us start the evaluation by making a substitution, 4 √ δ = y. Then, we can write b ∫ 0 4 √ δ √ c2− √ δ5 dδ=4 4 √ b ∫ 0 y4 √ c2−y10 dy (A.2) After making the second substitution, y5 = t, we receive (for c> 0) b ∫ 0 4 √ δ √ c2− √ δ5 dδ= 4 5 4 √ b5 ∫ 0 √ c2− t2 dt= 4 5 c 4 √ b5 ∫ 0 √ 1− (t c )2 dt (A.3) The next substitution, t/c= z, gives b ∫ 0 4 √ δ √ c2− √ δ5 dδ= 4 5 c2 4 √ b5 c ∫ 0 √ 1−z2 dz (A.4) Let usnow try to evaluate the integral of the right-hand side of equation (A.4). Note that 4 √ b5 c ∫ 0 √ 1−z2 dz= 4 √ b5 c ∫ 0 1−z2√ 1−z2 dz= 4 √ b5 c ∫ 0 1√ 1−z2 dz− 4 √ b5 c ∫ 0 z z√ 1−z2 dz (A.5) 940 R. Jankowski Indeed 4 √ b5 c ∫ 0 √ 1−z2 dz=arcsinz ∣ ∣ ∣ 4 √ b5 c 0 − 4 √ b5 c ∫ 0 z z√ 1−z2 dz= (A.6) = arcsin 4 √ b5 c − 4 √ b5 c ∫ 0 z z√ 1−z2 dz In order to evaluate the integral of the right-hand side of equation (A.6), let us apply the method of integrating by parts. It yields 4 √ b5 c ∫ 0 √ 1−z2 dz=arcsin 4 √ b5 c − ( −z √ 1−z2 ∣ ∣ ∣ 4 √ b5 c 0 + 4 √ b5 c ∫ 0 √ 1−z2 dz ) (A.7) So 4 √ b5 c ∫ 0 √ 1−z2 dz=arcsin 4 √ b5 c + 4 √ b5 c √ 1− √ b5 c2 − 4 √ b5 c ∫ 0 √ 1−z2 dz (A.8) Hence 4√ b5 c ∫ 0 √ 1−z2 dz= 1 2 ( arcsin 4 √ b5 c + 4 √ b5 c2 √ c2− √ b5 ) (A.9) Substituting the above into equation (A.4), finally gives b ∫ 0 4 √ δ √ c2− √ δ5 dδ= 2 5 ( c2arcsin 4 √ b5 c + 4 √ b5 √ c2− √ b5 ) (A.10) References 1. Anagnostopoulos S.A., SpiliopoulosK.V., 1992,An investigation of ear- thquake inducedpoundingbetweenadjacentbuildings,Earthquake Engineering and Structural Dynamics, 21, 289-302 2. Chau K.T., Wei X.X., 2001, Pounding of structures modelled as non-linear impacts of two oscillators, Earthquake Engineering and Structural Dynamics, 30, 633-651 Theoretical and experimental assessment of parameters... 941 3. Goldsmith W., 1960, Impact: the theory and physical behaviour of colliding solids, Edward Arnold, London, UK 4. Jankowski R., 2005a, Impact force spectrum for damage assessment of earthquake-inducedstructuralpounding,KeyEngineeringMaterials,293/294, 711-718 5. Jankowski R., 2005b, Non-linear modelling of earthquake induced pounding of buildings,Mechanics of the 21st Century – Proc. of 21st ICTAM, Springer, Dordrecht, Netherlands, CD-ROM, Paper ID 12659 6. JankowskiR., 2005c,Non-linear viscoelasticmodelling of earthquake-induced structural pounding, Earthquake Engineering and Structural Dynamics, 34, 595-611 7. Jankowski R.,WildeK., FujinoY., 1998,Pounding of superstructure seg- ments in isolated elevated bridge during earthquakes,Earthquake Engineering and Structural Dynamics, 27, 487-502 8. Jing H.-S., Young M., 1991, Impact interaction between two vibration sys- tems under random excitation, Earthquake Engineering and Structural Dyna- mics, 20, 667-681 9. Kasai K., Maison B.F., 1997, Building pounding damage during the 1989 Loma Prieta earthquake,Engineering Structures, 19, 195-207 10. Maison B.F., Kasai K., 1992, Dynamics of pounding when two buildings collide,Earthquake Engineering and Structural Dynamics, 21, 771-786 11. Rosenblueth E., Meli R., 1986, The 1985 earthquake: causes and effects in Mexico City,Concrete International, 8, 23-34 12. Zembaty Z., Cholewicki A., Jankowski R., Szulc J., 2005, Earthquakes of September 21, 2004 in north-eastern Poland and their effects on structures, Inżynieria i Budownictwo, 1, 3-9 [in Polish] Analityczne i eksperymentalne szacowanie wartości parametrów nieliniowego lepkosprężystego modelu zderzeń pomiędzy konstrukcjami budowlanymi Streszczenie Zjawisko zderzeń pomiędzy sąsiednimi konstrukcjami budowlanymi podczas trzę- sień ziemi jestwostatnimczasie intensywniebadane zwykorzystaniemróżnychmode- li numerycznych siły zderzenia w czasie kontaktu.Wyniki badań eksperymentalnych pokazują, iż nieliniowymodel lepkosprężysty jest najdokładniejszywśródmodeli sto- sowanych do analizy. Model ten zdefiniowany jest poprzez dwa parametry: liczbę tłumienia zderzenia oraz parametr sztywności zderzenia. Jednym z celów niniejszego 942 R. Jankowski artykułu jest wyprowadzeniewzoru na zależność pomiędzy liczbą tłumienia zderzenia a współczynnikiem odbicia, który jest parametrem często stosowanym i opisywanym w literaturze. Kolejnym celem jest wyznaczenie zakresuwartości współczynnikówod- bicia i parametrów sztywności zderzenia dla różnychmateriałów budowlanych (stali, betonu, drewna i ceramiki) na podstawie wyników badań eksperymentalnych. Wy- niki analizy pokazują szeroki zakres wartości parametrów nieliniowego modelu lep- kosprężystego zderzeń otrzymanych dla różnych wartości prędkości zderzenia oraz masy testowanych elementów. Zastosowanie tych wartości w symulacjach numerycz- nych prowadzi do zwiększenia dokładności uzyskiwanychwynikówwanalizie zjawiska zderzeń pomiędzy konstrukcjami budowlanymi podczas trzęsień ziemi. Manuscript received February 21, 2007; accepted for print May 23, 2007