Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 3, pp. 705-725, Warsaw 2011 EXACT SOLUTIONS FOR FREE VIBRATIONS OF A STEEL-CONCRETE BRIDGE Luca Della Longa Antonino Morassi Anna Rotaris University of Udine, Department of Civil Engineering and Architecture, Udine, Italy e-mail: antonino.morassi@uniud.it This paper deals with a class of free vibrations for a two-span, two-lane steel-concretebridge.Thedeck structure ismodeledasa thin, homogene- ous, orthotropic plate stiffened by beams running along the longitudinal direction of the bridge. The method of separation of variables is used to find exact solutions for a class of free vibrations of the structure. A comparisonbetweenanalytical and experimental natural frequencies and vibrationmodes of the bridge is presented and discussed. Key words: vibrations, bridges, orthotropic plate, eigenvalue problem, exact solutions 1. Introduction The derivation of accurate mechanical models is one of the main challenges in modern analysis of structural dynamics. The development of such models is of great interest in the study of bridge structures, especially for structural control, damage detection and health monitoring purposes, see (Aktan et al., 1997; Gentile and Cabrera, 1997; Gentile, 2006; Morassi and Vestroni, 2008; Dilena andMorassi, 2011). In the context of structural identification based on dynamic testing, for example, analytical modelling plays a crucial role both in the interpretation of measurements and in the application ofModel Updating procedures. Moreover, it has been recognized in the Structural Identification applications that one can hardly obtain anything good without a relatively simple class of physicalmodelswhich, however,mustbeable todescribewithin a certain degree of accuracy the behavior of the real system, see, for example, the papers (Jaishi andRen, 2005; Gorman andGaribaldi, 2006; Catbas et al., 2007; Morassi and Tonon, 2008a,b; Marcuzzi andMorassi, 2010). 706 L. Della Longa et al. This paper deals with the dynamic behavior of the two-span, two-lane steel-concrete bridge shown in Fig.1. The bridge belongs to the new high- way connecting the cities of Pordenone and Conegliano, in the Friuli Venezia Giulia, a region located in the North East of Italy. Steel-concrete composi- te structures are largely employed in bridge engineering, especially for bridge decks where a severe control of deformability is needed under important we- ights during operation. The bridge under study belongs to the rather common class of steel-concrete bridges in which the reinforced concrete slab deck is stiffened transversally by means of steel beams, and these beams, in their turn, are connected at the ends to longitudinal steel plate girders of large cross-section. Fig. 1. Zigana bridge: (a) plan, (b) side view and (c) transversal cross-section (lengths in meters) Exact solutions for free vibrations... 707 The object of the present study is twofold. First, to develop a simplified analyticalmodel able to accurately describe the dynamic behavior of this class of steel-concrete bridges. Secondly, to derive an exact solution for a class of free vibrations of the bridge, namely for vibrating modes which are antisym- metric about the transversal symmetry axis of the bridge (e.g., the x2-axis in Fig.3). In themodelingprocess, thebridgedeck is describedas two identical homo- geneous, orthotropic rectangular plates. Eachplate is assumed tobe connected by means of cylindrical hinges along the external edge to a reinforcing beam and is simply supported along the two orthogonal edges. Moreover, the two deck plates are connected together by means of cylindrical hinges to the cen- tral reinforcing beam running along the common edge, which coincides with the longitudinal bridge axis. Concerning the plate eigenvalue problem, there is a vast literature on exact solutions for rectangular plates, see, for example, the paper Leissa (1973) for a comprehensive treatment of the isotropic case and Li (2000) for exact so- lutions for multi-step orthotropic shear plates. The determination of exact solutions for the free vibration of rectangular plates reinforced with beams running along their boundary has undergone growing interest in the recent years. The paper by Cox and Benfield (1959) seems to be the first contribu- tion on this subject. In (Cox andBenfield, 1959), the finite differencemethod was used to estimate the first frequencies of a uniform isotropic square pla- te having pinpoint supports at the four corners and flexible beams along the edges. Elishakoff and Sternberg (1980) investigated the vibration of a thin, homogeneous, isotropic, rectangular plate stiffened along two parallel edges and simply supported along the other two. In their study, the torsional vi- bration of the beam stiffeners was also considered. Gorman (2003) presented a comprehensive study of analytical type solutions for the free vibration of a thin, homogeneous, isotropic corner-supported rectangular plate with symme- trically distributed reinforcing beams attached to the plate edges. The beam reinforcement was treated in the most general case in (Gorman, 2003) by ta- king into account both bending and rotational stiffness of the beams as well their lateral and rotational inertia. All these results are valid for a single rectangular plate. In the present case, the study of the free vibration of the bridge leads to an eigenvalue problem for a system formed by two uniform, orthotropic rectangular plates. Themethod of separation of variables is used to find the exact eigenpairs of the system. A comparison between the analytical and experimental natural frequencies of the bridge is also presented and discussed. 708 L. Della Longa et al. 2. Description of the bridge Ziganabridge is a two-span, two-lane steel-concrete composite structure shown inFig.1.Thebridgedeck is formedbyareinforced concrete (r.c.) slab, of thick- ness 0.25m, supported by transverse double-T steel beams of height 0.63m. Thedeck has a composite structure and the r.c. slab is connected to transverse steel beams by means of steel connectors welded on the upper flange of the beams.The end section of each transverse beam is connected by a bolted joint to the longitudinal main beams, see Fig.2b. In fact, the whole bridge deck is supported by three continuous, two-span double-T girders of height 2.85m. The cross-section of each girder was obtained by welding the web panel with the upper and lower flange, and additional flange splices have been inserted to stiffer the beams on the region around the inner support, see Fig.2a. Finally, five beam elements, each 12.64m long, were assembled bymeans of bolted jo- ints to obtain longitudinal continuous beams resting on three supports. Each Fig. 2. Constructional details of Zigana bridge: (a) bolted joint of the longitudinal plate girders and (b) bolted connection between longitudinal plate girders and transverse beams. In figure (b), the measures in parentheses are referred to the central plate girder (lengths in millimeters) longitudinal beam is supported on steel Polytetrafluoroethylene bearing de- vices at the ends and at the middle point. The inner support of the central girder does not allow any longitudinal or transversal displacement, while at the end supports the transversal (horizontal) movement is constrained. The longitudinal displacement of the two lateral longitudinal beams is restrained at the inner support, while the in-planemotions are free at the end supports. The inner support is on a pile foundation with two or four cast-in-place r.c. piles of 1.2mdiameter and 31.6m length, for the lateral and the central girder respectively. The abutments consist of an horizontal r.c. beam of solid square Exact solutions for free vibrations... 709 cross-section of side 1.3m, mounted on a single pile at both ends and on two piles on themiddle. Construction of the bridgewas completed in the Fall 2004. It was designed under the ItalianStandardSpecifications forHighwayBridges inSeismicAreas (D.M. 4/5/1990, Circ. Min. LL.PP. 34233, 25/2/1991, and others). Dynamic and static tests on the bridge were carried out in December 2004 and June 2006, respectively. With reference to Fig.3, our mechanical modelling of Zigana bridge is based on the following assumptions: i) The bridge deck is described as a plate reinforced by a set of equidistant ribs along the x2-direction (orthotropic plate). ii) The connections between the deck and themain longitudinal beams are modeled as ideal continuous cylindrical hinges allowing rotation around the x1-direction. The hinges are assumed to be located on the vertical plane containing the shear center line of the main plate girders. iii) Thebridge deck is assumed to be simply supported at the external edges (parallel to the x2-direction) and at the central line support (x2-axis). Under the above hypotheses and assuming uniformmechanical properties, the analytical model of the bridge is symmetric about the x2-axis and, therefore, the vibrating modes of the vertical free vibrations split into two sets of sym- metric and antisymmetricmodes about the x2-axis. The analysis developed in this paperwill concernwith an exact solution for the class of x2-antisymmetric free vibrations of the bridge. Fig. 3. Schematic view of the bridge deck. Thick line: longitudinal beams; dashed line: supports 710 L. Della Longa et al. 3. Formulation of the free vibration problem Consider the bridge deck structure of Fig.3 formed by two rectangular plates Ω+ = {(x1,x2) | 0 0, D22 > 0, D12 > 0 are the two flexural and the torsional rigidity of the orthotropic plate, respectively, and D1 > 0 is the termwhich takes into account the transverse contraction of the plate, see (Timoshenko, 1959). InEq. (3.10), J and J0 are the uniformbending stiffness of the two lateral longitudinal beams and of the central beam, respectively. Further, ρp, ρb, ρb0 are the surface mass density of the plate, the linear mass density of the two lateral beams and of the central beam, respectively. The differential formulation of the free vibration problem for the system is obtained by imposing the stationarity of Rayleigh quotient (3.9) on the set D defined by Eqs. (3.7) and (3.8), see (Weinberger, 1965). The eigenvalue pro- blem consists in determining the eigenpair {ω2 > 0; w∈H4(Ω+)∪H4(Ω−)}, with w(x1,±b)∈H4(0,a) and w(x1,0)∈H4(0,a), such that D11w,1111+2Hw,1122+D22w,2222=ω 2ρpw in Ω +∪Ω− w,22=0 on Γ + b ∪Γ− −b ∪Γ0 [[w]] = 0 on Γ0 H̃w,112+D22w,222= { Jw,1111−ω2ρbw on Γ+b −(Jw,1111−ω2ρbw) on Γ−−b (3.12) H̃[[w,112 ]]+D22[[w,222 ]] =−(J0w,1111−ω2ρb0w) on Γ0 w=0 on Σ + 0 ∪Σ − 0 ∪Σ + a ∪Σ − a w,11=0 on Σ + 0 ∪Σ − 0 ∪Σ + a ∪Σ − a where H = 2D12 + D1, H̃ = 4D12 + D1, [[w]] ≡ w(x1,0+) − w(x1,0−), [[w,112 ]] ≡ w,112 (x1,0+)−w,112 (x1,0−). Here, Σ + 0 = [0,b]. From the me- chanical point of view, boundary condition (3.12)2 expresses the vanishing of bendingmoment and conditions (3.12)4−6 express the equilibriumof the forces acting in the direction transversal to the plane (x1,x2) along the lines Γ + b , Γ− −b , Γ0, respectively. The differential operator governing eigenvalue problem (3.12) is a self- adjoint compact operator in D and, therefore, by general results (see Brezis, Exact solutions for free vibrations... 713 1986) (i) there exists a countable number of real eigenvalues ω2n, n=1,2, . . ., with ω2n > 0, and such that limn→∞ω 2 n = ∞; (ii) eigenfunctions associated to distinct eigenvalues are orthogonal with respect to the mass distribution; (iii) there exists a countable base of the space D formed by the eigenfunctions of the problem. It is worth noticing that if w(x1,x2) is an eigenfunction of boundary value problem (3.12), then it can be extended as w(−x1,x2)=−w(x1,x2) (x1,x2)∈Ω +∪Ω− (3.13) to obtain the corresponding x2-antisymmetricmode shape of thewhole bridge structure in the rectangle [−a,a]× [−b,b], see Fig.3. 4. Exact free vibration solutions The exact solutions to eigenvalue problem (3.12) will be determined bymeans of a variant of themethod of separation of variables, see, e.g., (Voigt, 1893). It is assumed that a non-trivial solution to boundary-value problem (3.12) can be found in the form w(x1,x2)=X(x1)Y (x2) (4.1) By substituting Eq. (4.1) into Eq. (3.12)1 and dividing by w, it follows that D11 XIV X +2H XII X Y II Y +D22 Y IV Y =ω2ρp (4.2) where ϕI denotes the first derivative of the function ϕ with respect to its argument. Taking the partial derivative of Eq. (4.2) with respect to x2, it follows that 2H XII X (Y II Y )I +D22 (Y IV Y )I =0 (4.3) which can be separated, yielding XII +βX =0 in (0,a) (4.4) where β is the separation parameter. The non trivial solutions to differential equation (4.4) subject to boundary conditions (3.12)7, e.g. X(0)= 0=X(a) (4.5) are given by Xn(x1)= sin nπx1 a βn = (nπ a )2 n=1,2, . . . (4.6) 714 L. Della Longa et al. It is worth noticing that the eigensolutions Xn(x1) satisfy also boundary con- ditions (3.12)8, since X II n (0) = X II n (a) = 0. Moreover, from Eq. (4.4) one has XIVn =−βnXIIn =β2nXn in (0,a) (4.7) Therefore, by Eq. (4.2), the function Y ∈ H4(−b,0)∪H4(0,b) must satisfy the ordinary differential equation D22Y IV −2HβnY II +(D11β2n−ω2ρp)Y =0 in (−b,0)∪ (0,b) (4.8) the boundary conditions at x2 =±b D22Y III(b)− H̃βnY I(b)= (Jβ2n−ω2ρb)Y (b) D22Y III(−b)− H̃βnY I(−b)=−(Jβ2n−ω2ρb)Y (−b) (4.9) Y II(b)= 0 Y II(−b)= 0 and the boundary and jump conditions at x2 =0 Y II(0+)= 0 Y II(0−)= 0 [[Y ]] = 0 (4.10) D22[[Y III]]− H̃βn[[Y I]] =−(J0β2n−ω2ρb0)Y (0) where [[Y ]]≡Y (0+)−Y (0−). Again, by general results, see Brezis (1986), eigenvalue problem (4.8)- (4.10) has a countable number of real, positive eigenvalues for every given n, n=1,2, . . ., with the accumulation point at infinity. Therefore, the eigenfunc- tions of problem (3.12) can be written as wn,m(x1,x2)=Xn(x1)Ym(x2) n=1,2, . . . m=1,2, . . . (4.11) Looking for solutions of the form Y (x2) = exp(qx2), the characteristic equ- ation for differential equation (4.8) becomes f(z)= z2−4Az+B=0 (4.12) where z= q2 A≡ Hβn 2D22 > 0 B≡ D11β 2 n−ω2ρp D22 (4.13) To find the roots of f(z), one has to distinguish the following cases i) B> 0 “low” frequency case ii) B< 0 “high” frequency case iii) B=0 “limit” case (4.14) Exact solutions for free vibrations... 715 Case i). When B> 0, the following three subcases have to be considered i1) 4A2−B> 0 i2) 4A2−B< 0 i3) 4A2−B=0 (4.15) i1) If 4A2 −B > 0, then there exist two real positive roots of f(z) such that 0 2A. Therefore, putting q1 = −i √ −z1, q2 = i √ −z1, q3 =− √ z2, q4 = √ z2, the general solution to Eq. (4.8) is Y (x2)=C1cos( √ −z1x2)+C2 sin( √ −z1x2) (4.23) +C3cosh( √ z2x2)+C4 sinh( √ z2x2) with four constants of integration Ci, i=1, . . . ,4. Case iii). Finally, when B = 0 there are two real solutions z1 = 0 and z2 =4A, and the general solution to Eq. (4.8) is of the form Y (x2)=C1+C2x2+C3cosh(2 √ Ax2)+C4 sinh(2 √ Ax2) (4.24) with Ci, i=1, . . . ,4, constants of integration. By substituting the general solution to equation (4.8) (e.g., the expression (4.18), (4.21), (4.22), (4.23), (4.24) for cases i1), i2), i3), ii), iii), respectively) into the eight boundary and jump conditions (4.9) and (4.10), a homogene- ous linear system of eight equations for Ci, C ′ i, i = 1, . . . ,4, is obtained, e.g. M(ω2)c = 0. To find non-trivial solutions for Y (x2), the determinant detM(ω2) of the correspondingmatrix should be set equal to zero. The above analysis can be simplified by adopting suitable symmetry con- siderations. Because of the symmetry of the eigenvalue problem with respect to the x1-axis, the eigenfunctions of the system (4.8)-(4.10) split into the two sets of x1-symmetric, S, and x1-antisymmetric, A, eigenfunctions, namely S = {Y (x2) eigenfunction | Y (−x2)=Y (x2), x2 ∈ (−b,b)} (4.25) A= {Y (x2) eigenfunction | Y (−x2)=−Y (x2), x2 ∈ (−b,b)} Byway of an example, it will be shown how the analysis simplifies in case i1) (B> 0,4A2−B> 0). Symmetric eigenfunctions have the expression Y (x2)=    C1 sinh( √ z1x2)+C2cosh( √ z1x2) +C3 sinh( √ z2x2)+C4cosh( √ z2x2) in (0,b) −C1 sinh( √ z1x2)+C2cosh( √ z1x2) −C3 sinh( √ z2x2)+C4cosh( √ z2x2) in (−b,0) (4.26) Exact solutions for free vibrations... 717 where the four constants C1, C2, C3, C4 can be determined by imposing the four linearly independent boundary conditions (4.9)1,3 and (4.10)1,3. The corresponding 4×4 coefficient matrix MS(ω2) is   q21 sinh(q1b) q 2 1 cosh(q1b) q 2 2 sinh(q2b) q 2 2 cosh(q2b) m21 m22 m23 m24 0 q21 0 q 2 2 2q31 − H̃βnq1 J0β2n−ω2ρb0 2q32 − H̃βnq2 J0β2n−ω2ρb0   (4.27) where q1 = √ z1, q2 = √ z2 and m21 =(D22q 3 1 − H̃βnq1)cosh(q1b)− (Jβ2n−ω2ρb)sinh(q1b) m22 =(D22q 3 1 − H̃βnq1)sinh(q1b)− (Jβ2n−ω2ρb)cosh(q1b) (4.28) m23 =(D22q 3 2 − H̃βnq2)cosh(q2b)− (Jβ2n−ω2ρb)sinh(q2b) m24 =(D22q 3 2 − H̃βnq2)sinh(q2b)− (Jβ2n−ω2ρb)cosh(q2b) Similarly, and always in case i1), the antisymmetric eigenfunctions can be written in the form Y (x2)=    C1 sinh( √ z1x2)+C2cosh( √ z1x2) +C3 sinh( √ z2x2)+C4cosh( √ z2x2) in (0,b) C1 sinh( √ z1x2)−C2cosh( √ z1x2) +C3 sinh( √ z2x2)−C4cosh( √ z2x2) in (−b,0) (4.29) and four constants of integration Ci, i = 1, . . . ,4, will be determined from boundary conditions (4.9)1,3, (4.10)1,3. The corresponding 4× 4 coefficient matrix MA(ω 2) is   q21 sinh(q1b) q 2 1 cosh(q1b) q 2 2 sinh(q2b) q 2 2 cosh(q2b) m21 m22 m23 m24 0 1 0 1 0 q21 0 q 2 2   (4.30) Remaining cases i2), i3), ii), iii) have been studied analogously. Finally, in order to determine the natural pulsations as roots of the cha- racteristic equations detMS(ω 2)= 0, detMA(ω 2)= 0, a numerical procedure was used. The essential steps of the algorithm can be summarized as follows. Tofix the ideas, the case of x1-symmetric eigenfunctionswill be considered. Let the x1-wave number βn be given, e.g. βn = (nπ/a) 2, n= 1,2, . . .. Once a value for ω was set, say ω̃, fourth order polynomial equation (4.12) in the 718 L. Della Longa et al. variable q = √ z, was solved and the expression of the general solution to ordinary differential equation (4.8) was determined. Next, by imposing the boundary conditions, the value of detMS(ω̃ 2) was calculated. By repeating this procedure for ω̃ +∆ω, where ∆ω is a proper increment, a graph of detMS(ω 2) was reconstructed in the given frequency interval. Eigenfrequency values were evaluated by a bisection method applied between two consecutive values of the ω variable corresponding to the change of sign of detMS(ω 2). For each eigenfrequency value, after solving MScS = 0, the vector cS of the constants of integrationwas calculated and, therefore, the correspondingmode of vibration was determined. 5. Application to Zigana bridge The analysis presented in the preceding Section was applied to determine a class of eigenpairs of Zigana Bridge, namely the vibrating modes antisymme- trical about the x2-axis in Fig.3. Fig. 4. Dimensional details of Zigana bridge: (a) longitudinal lateral double-T steel beams, (b) longitudinal central double-T steel beam, (c) transverse double-T steel beam (lengths in millimeters) In Table 1, the geometrical, inertial and mechanical parameters of the bridge are presented. In particular, by neglecting the effect of transverse con- traction, see (Timoshenko, 1959), the rigidities of the deck are calculated by means of the expressions Exact solutions for free vibrations... 719 D11 = Ech 3 12(1−ν2c) D22 = EsI (hom) s a1 (5.1) D12 = Gch 3 12 + C 2a1 Table 1.Geometrical, inertial andmechanical parameters of Zigana bridge Parameter Symbol Value Span length a 31m Deck width b 13m Plate thickness h 0.25m Spacing between stiffeners a1 2.528m Young’s modulus of steel Es 2.1 ·1011Nm−2 Young’s modulus of concrete Ec 3.0605 ·1010Nm−2 Homogenization factor steel-concrete n=Es/Ec 5.833 Poisson’s ratio for steel νs 0.3 Poisson’s ratio for concrete νc 0.2 Mass density of steel ρs 7850kgm −3 Mass density of concrete ρc 2450kgm −3 Plate bending stiffness (x2-axis) D11 4.89 ·107Nm Plate bending stiffness (x1-axis) D22 7.41 ·108Nm Plate torsional stiffness D12 1.97 ·107 Nm In the foregoing formulas, Ec,Es areYoung’smoduli of concrete and steel, respectively; Gc is the shear modulus of concrete; νc is Poisson’s coefficient of concrete; h is the plate thickness; a1 is the spacing of the steel transversal beams in the x1 direction. InEq. (5.1), I (hom) s is themomentof inertia, forunit length, of the section formedby the r.c. slab andone stiffener evaluated for the whole section homogenized to steel (with homogenization factor n=Es/Ec), see Fig.4. The quantity C is the torsional rigidity of a single transversal steel beamand it has been evaluated accordingly to classical theories for thinwalled beams with open cross sections, see, for example, (Timoshenko, 1959). In Table 2, for every value of n, n = 1, . . . ,5, the natural frequencies of the first four free x1-symmetric and x1-antisymmetric vibration modes of the bridge are collected. As an example, the shape of vibrating modes correspondingto the lower sixnatural frequenciesarepresented inFig.5 for the whole bridge deck. Generally speaking, lowest vibration modes are influenced by deformation of the longitudinal plate girders. In the first mode wS1,1, at f =3.23Hz, the deformed shape of all the three longitudinal beams is similar 720 L. Della Longa et al. Table 2. Theoretical values of natural frequencies of x1-symmetric modes wSn,m and of x1-antisymmetric modes w A n,m of Zigana bridge, for n,m = =1, . . . ,4 n m Frequency Frequency wSn,m [Hz] w A n,m [Hz] 1 1 3.23 4.13 2 5.40 14.89 3 19.86 46.96 4 54.11 99.07 2 1 8.77 9.70 2 19.79 24.78 3 30.35 49.54 4 57.59 100.04 3 1 11.26 11.53 2 34.25 37.81 3 55.09 65.03 4 74.62 104.55 4 1 13.56 13.65 2 40.36 41.71 3 78.52 85.50 4 109.76 125.28 to the first bendingmode of the continuous two-span beam.The secondmode wA1,1, at f = 4.13Hz, has torsional character since the vertical deflection of the central girder is negligible and the two lateral beams vibrate out-of-phase following the fundamentalmode of the simply supported two-span continuous beam. In the thirdmode wS1,2, at f =5.40Hz, the two lateral girders vibrate in-phase according to the fundamental mode of the simply supported two- span beam,whereas the central girder has a similar shape but opposite phase. Higher modes have more pronounced wavy character. Table 3.Comparison between experimental and theoretical values of the first natural frequencies. Error=100 · (ftheor −fexp)/fexp Experimental Experimental Theoretical Theoretical Error mode order value [Hz] mode order value [Hz] 1 3.13 wS1,1 3.23 3.19 2 4.00 wA1,1 4.13 3.25 5 6.04 wS1,2 5.40 −10.60 Exact solutions for free vibrations... 721 Fig. 5. 3D view of the first six theoretical x2-antisymmetrical mode shapes of Zigana bridge Finally, Table 3 compares the computed and experimental frequencies of the lower modes of the bridge. The latter were extracted from frequency re- sponsemeasurements carried out on Zigana bridge bymeans of a stepped-sine technique, see (Morassi and Tonon, 2008b) for more details on dynamic te- sting. In brief, the bridge was excited with an electro-mechanical actuator mounted in the vertical direction on the bridge deck, at one fourth of themid span of the lateral side, see Fig.6. The transversal motions of the deck we- re measured by using twelve accelerometers and four seismometers. Based on this experimental set-up, the frequency response functions (FRFs) of the brid- ge deck were simultaneously determined within the frequency range 0-15Hz with resolution 0.1Hz in 1.0-2.5Hz, 0.02Hz in 2.5-9.0Hz and 0.04Hz for fre- quencies greater than 9.0Hz. Each FRF termwas computed according to the stepped-sine technique, and formally it was obtained as the ratio between the discrete Fourier transform (DFT) of the accelerometer output signal and the DFT of the force input signal. Figure 7 shows an example of measurements. Modal components andnatural frequencieswere extracted fromthemeasu- red FRFs evaluated between the excitation point and the response points by means of a multiple curve-fitting modal analysis technique, see Ewins (2000). The curve fitting procedure assumes a FRF term given as the ratio between the numerator polynomial and the denominator polynomial of a suitable or- der. A numerical algorithm based on an iterative least-squared method was used to obtain the optimal values of themodal parameters, see, as an example, 722 L. Della Longa et al. Fig. 6. Instrumental layout and bridge supports Fig. 7. Zigana bridge: example of comparison betweenmeasured (continuous line) and synthesized (dashed line) point inertance Exact solutions for free vibrations... 723 Fig.7. More precisely, five, three and five vibrating modes were identified in the frequency intervals 2.5-7Hz, 7-9Hz and 9-15Hz, respectively. The experimental mode shapes in Fig.8 correspond to linear interpolation of themeasuredmodal components. Visual comparison andModal Assurance Criterion allow one to confirm the correspondence between the first (3.13Hz), second (4.00Hz) and fifth (6.04Hz) experimental modes and the theoretical modes wS1,1, w A 1,1, w S 1,2, respectively, as it is shown in Table 3. The results of Table 3 show that the percentage differences between the experimental and theoretical natural frequency values are rather small, even when compared with the refined finite element model of the whole bridge developed by Mo- rassi andTonon (2008b). Therefore, the analytical model of the bridge can be considered satisfactory for practical engineering applications. Fig. 8. 3D view of the first six experimental mode shapes of Zigana bridge 6. Concluding remarks In this paper, a class of free vibrations of a two-span, two-lane steel-concrete bridge has been studied. The eigenvalue problem concerns with vibration of two thin, homogeneous, orthotropic, rectangular plates connected by cylindri- cal hinges to a reinforcing beam running along the common edge. Each plate is stiffened along the external edge parallel to the common edge and simply supported along the other two orthogonal edges. This combination of boun- dary and jump conditions seems not to be considered in the literature before. The classical method of separation of variables has been used to find a class of exact eigenpairs of the system. Analytical values of the first lower natural frequencies agree well with those obtained in dynamic tests carried out on the 724 L. Della Longa et al. real bridge. The analytical model of the bridge will be of valuable importance in the design of new bridges of the same class and on its use as the reference model for future structural identification analyzes. 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(Eds.), 2008,DynamicMethods for Damage Iden- tification in Structures, CISMCourses and Lectures No. 499, Springer,Wien 19. Timoshenko S.P., Woinowsky-Krieger S., 1959, Theory of Plates and Shells, McGraw-Hill, NewYork 20. VoigtW., 1893,Bemerkung zu demProblemder transversalenSchwingungen rechteckiger Platten,Nach. Ges. Wiss. (Göttingen), 6, 225-230 21. Weinberger H.F., 1965, A First Course in Partial Differential Equations, Dover Publications Inc., NewYork Dokładne rozwiązania zagadnienia drgań swobodnych konstrukcji mostu stalowo-betonowego Streszczenie Praca dotyczy analizy pewnej klasy drgań swobodnych dwuprzęsłowego, dwujez- dniowego mostu stalowo-betonowego. Przęsło zamodelowano jako cienką, jednorod- ną, ortotropową płytę usztywnioną belkami zamocowanymi w kierunku wzdłużnym mostu. Do znalezienia dokładnych rozwiązań problemu drgań swobodnych modelu zastosowanometodę rozdzielenia zmiennych. Na koniec zaprezentowano i przedysku- towano porównanie częstości własnych i postaci drgań własnychmodelu z wynikami zarejestrowanymi na rzeczywistej konstrukcji. Manuscript received March 14, 2011; accepted for print May 13, 2011