A Damage Analysis of Steel-Concrete Composite Beams Via Dynamic Methods: Part I. Experimental Results
Descrição do Produto
1 234356 17389 668 766 4 6 634
3 29734 6 3 17389 38 68 37 234356 266 7 MICHELE DILENA ANTONINO MORASSI 1234567286 9 8 8225 8 8 25 6 9 82 4 22 282 82 64 (Received 4 November 2000; accepted 30 January 2002)
!6546" This paper is the second part of an experimental–analytical investigation on the dynamic behavior of damaged steel–concrete composite beams. In the first part of the research, we presented and discussed the experimental results of a comprehensive series of dynamic tests performed on composite beams with damage in the connection. Experimental observations suggested the formulation of a composite beam analytical model, where the strain energy density of the connection also includes an energy term associated to the occurrence of relative transversal displacements between the reinforced concrete slab and the steel beam. A comparison with experimental results shows that the model enhances accuracy in describing the undamaged state of composite beams and that it is also appropriate to accurately predict the dynamic behavior under damaged conditions. A damage detection technique based on the measurement of variation in the first flexural frequencies was then applied to the suggested model and gave positive results.
#2 $95" Steel–concrete composite beams, structural diagnostics, small vibrations, analytical models
!"2# " This paper is the second part of an experimental-analytical investigation on the dynamic behavior of damaged steel–concrete composite beams and on its links with non-destructive dynamic methods of damage detection. In the first part of the research, we presented and discussed the experimental results coming from a comprehensive series of dynamic tests performed on composite beams with damage in the connection; see Morassi and Rocchetto (2003) for an introduction to the subject and for detailed references. Here, an improved analytical model of a composite beam is developed, so as to accurately describe the dynamic behavior of the beam under damaged conditions. Moreover, a diagnostic technique based on frequency measurements is offered to locate damage. The present research originates from a recent paper by Biscontin et al. (2000). In that work, a one-dimensional model of a composite beam was offered, where the elements connecting the steel beam and the reinforced concrete (r.c.) slab were described by means of a strain energy density function defined throughout the longitudinal axis, and where the two beams were forced to maintain equal transversal displacements. The analytical model was
%9&584 9 !546 98 48 98659 9: 529–565, 2003 Sage Publications
1 2003 1
DOI: 10.1177/107754603030695
530 M. DILENA and A. MORASSI
used to interpret a series of dynamic tests performed on (undamaged) composite beams whose connections have different linear densities. Analytical results proved consistent with experimental tests, and this suggested that the theoretical model (and its possible improvements) could be useful in dealing with damage detection from dynamic experimental data. The work, nevertheless, left some important questions unsolved, and namely the capability to accurately describe small vibrations of composite beams with damaged connection. It is known that this aspect is crucial in structural diagnostics. In fact, the findings of most diagnostic techniques based on dynamic measurements are strongly dependent upon the accuracy of the analytical model used to interpret measurements; see, for example, Davini et al. (1995), Vestroni and Capecchi (1996), Ruotolo (1997), Ruotolo and Surace (1997), and Morassi and Rovere (1997). Considering the one-dimensional vibrating model as a limit of the three-dimensional model when the beam is ‘‘infinitely long’’, the correct way to estimate the accuracy of a one-dimensional analytical model should be to evaluate the error made upon dimensional reduction. Results of this kind would be desirable, but analytical estimates of the approximation resulting from the dimension reduction process seem to be available for the elastostatic case of a single beam element only; see, for example, Anzellotti et al. (1994). Taking these aspects into account, dynamic experimentation has been carried out in order to establish the degree of accuracy of theoretical models of composite beams with damage in the connection. In Morassi and Rocchetto (2003) a wide experimental investigation was performed on four composite beams, two having a partial connection (T1PR and T2PR) and two having a total connection (T1CR and T2CR). Both the undamaged configuration and three degrees of damaged configurations for each beam were analyzed; damage was induced by removing concrete around some elements connecting the steel beam and the r.c. slab and consequently causing a lack of structural solidarity between the two beams. The most significant modal parameters were assessed for each configuration. In addition to other aspects, carefully described in Morassi and Rocchetto (2000), tests revealed that the connection, besides hindering sliding on the steel–concrete interface, also plays a crucial role in reducing transversal displacements between the r.c. slab and the metallic beam. In undamaged areas of the beam, the difference between transversal displacement components of flexural modes of vibration in the two beams proved to be negligible (at least in the first four flexural modes being tested). Conversely, in damaged areas, the components of transversal displacement in the beam and in the r.c. slab are significantly different. Therefore, the hypothesis of cinematic coincidence of transversal displacements in the two beams formulated in Biscontin et al. (2000), and commonly accepted in static investigation (see, for example, Kristek and Studnicka, 1982; Giuriani, 1983; Gattesco, 1999), proved inappropriate to describe the behavior of a composite beam in a damaged area. These experimental observations suggested the formulation of a composite beam analytical model, where the strain energy density of the elements connecting the steel beam and the r. c. slab also includes an energy term associated with the occurrence of relative transversal displacements between the two beams. A comparison with experimental results shows that the model enhances accuracy in describing the undamaged state of a composite beam and it also accurately predicts dynamic behavior under damaged conditions, for a large range of frequencies. As a matter of fact, frequency modelization errors for damaged configurations are comparable to those for the undamaged system, and in all cases discrepancies between theoretical and experimental values follow a pattern similar to that of a single homogeneous beam having equal slenderness. The suggested model also allows us to reproduce carefully
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 531
some features arising from dynamic tests, such as, for example, the shift of the nodal points of flexural modes produced by damage, or the local behavior of flexural modes near the damaged region. A damage detection technique based on the measurement of variation in the first flexural frequencies was then applied to the suggested model and gave positive results. The remainder of the paper runs as follows. First, we present an improved one-dimensional model of a composite beam (Section 2) and we study the related problem of small free vibrations in the case of damage in the connection (Section 3). Then, in Section 4 we present an interpretation of dynamic tests aimed at the identification of an accurate analytical model of damaged composite beams. Finally, the results of a damage detection technique are described and discussed in Section 5.
$ 1 !"%2 " %&2 % " 1' "2%' "( 1 "" % %1 Let us consider a composite beam of length L as shown in Figure 1 in Morassi and Rocchetto (2003). The notation used hereafter to formulate the dynamic problem is essentially the same as in Biscontin et al. (2000, Section 2). However, because the formulation of the mechanical model has been modified, and also to help comprehension, a brief presentation of the problem is given hereafter. The two beams forming the system, the upper beam made of concrete and the lower of steel, are connected by means of metallic connectors or 6&, whose bottom ends are welded on the beam flange and whose top ends are embedded in the r.c. slab. Hereafter, the quantities relative to the concrete beam and to the steel beam are defined by indices 1 1 22 3, respectively. 431 2 41 2 51 5 corresponds to a system of Cartesian coordinates whose origin 31 is the barycenter of the 1th beam transversal section; we assume that the two axes 51 coincide with the same vertical direction 5 and are oriented downwards and that the two axes 41 are parallel to one another. The direction of the 6 longitudinal axis of each beam, which coincides with the barycenter line, is defined by the versor 11 1 12 1 13 , where 12 and 13 are the versors of the Cartesian axes included in the section plane and ‘‘1’’ is the usual vector product in 21 . The longitudinal axes of the two beams run parallel at a distance 7. Moreover, we assume that the axes 41 and 51 are parallel to the inertia principal directions of the transversal section as calculated in the barycenter. Let us suppose that the system vibrates freely in the plane 413 2 11 5, with small oscillations around a straight and not stressed configuration of equilibrium. We accept that the two beams forming the system will slide on the steel–concrete interface and, in contrast to the hypothesis made in Biscontin et al. (2000), we admit the occurrence of relative transversal displacements between the two beams in direction 5. Consequently, in accordance with the traditional cinematic hypotheses of the beam theory under small axial and flexural deformations, the actual configuration is thoroughly defined by assigning axial displacements 81 492 5, 1 1 22 3, and transversal displacements 1 492 5, 1 1 22 3, of the 9 abscissa’s transversal section at a moment of time . In the event of a linear elastic constitutive relation and of isotropic materials, the strain energy in each beam is equal to 33 33 2 3 2 1 2 1 2 2 2 1 492 5 81 492 5
1 495 1 495 1 4 5 1 9 6 92 (1) 9 3 9 4 3 4 3
532 M. DILENA and A. MORASSI
Figure 1(a). Analytical modes 111 1223 12 1223 41 1223 41 1222 relative to free vibrations of the free–free composite beam, with elastic and inertial properties of Table 1, in undamaged configuration.
where 1 3 1 1 and 1 3 1 1 are the section’s flexural stiffness and axial stiffness respectively. 1 and 1 are the area and the moment of inertia as to axis 41 of the 1th beam’s transversal section and 1 is the relative Young’s modulus of the material. Connectors hinder the relative displacements occurring between the two beams. If the distance between two consecutive studs is almost constant and short enough compared with the beam length, then the strain energy in the connection can be described as approximately
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 533
Figure 1(b). Analytical modes 111 1223 12 1223 41 1223 41 1222 relative to free vibrations of the free–free composite beam, with elastic and inertial properties of Table 1, in undamaged configuration.
equal to an interface energy density 3 4 5 along the beam axis. Under this simplification and in the event of a linear constitutive equation of the connection, we have 2 33 1 24 2 3 492 5 495 83 492 5 4 82 492 5 6 74 3 4 5 1 3 9 4 52 33 2 33 2 32 36 2 492 5 3 492 5 2 495 733 3 492 5 2 492 5 6 6 6 3 7 9 9 9 9
534 M. DILENA and A. MORASSI
7 2 3 6 495 4 2 492 5 4 3 492 55 92 3
(2)
where 495 and 495 are, respectively, the shearing and the axial stiffness for unit length of the connection. As remarked in Biscontin et al. (2000), the first term 2 of expression (2) considers the strain energy caused by the occurrence of relative sliding u3 492 5 4 82 492 5 3 3 492 5 74 between the base and the head of the connecting elements, where the stud 6 9 base displacement was calculated considering that it is 74 far from the steel beam barycentric axis. The second term of expression (2) considers the strain energy caused by the studs ends rotation; the energy value was calculated by referring to an equivalent beam, being 73 1 7474 long and having a 495 shearing stiffness. The top end of the beam is fixed to the slab axis 2 492 5 3 492 5 and its bottom end to the beam flange, and it is subjected to rotations 2 9 9 applied on both ends, respectively. Finally, the third term represents the strain energy caused by the occurrence of relative transversal displacement 4 2 492 5 4 3 492 55 between the r.c. slab and the steel beam. If we neglect the kinetic energy associated with the rotation of the beam transversal sections, the overall kinetic energy of the system is equal to 82 33 2 33 9 1 24 2 82 492 5 2 492 5 2 495 4 5 1 6 3
4 82 33 2 33 9
83 492 5 2 3 492 5 6 3 495 6 (3) 92 3
where 1 is the linear mass density of the 1th beam. To fix ideas and considering that the experimental part has been developed for this configuration, we will hereafter closely examine the case of a free beam. In accordance with Hamilton’s principle, the equations of free and undamped motion and their corresponding natural boundary conditions are obtained by imposing the stationarity of the action functional 1 51 1 4 4 5 4 4 2 4 5 6 3 4 5 6 3 4 555
(4) 52
on the following set of admissible configurations of the system
1 5482 2 83 2 2 2 3 5 682 472 5 8 482 5 2 1 472 5 8 3 482 5 2 1 1 22 32 81 492 75 8 3 4 2 2 3 5 2 1 492 75 8 3 4 2 2 3 5 2 1 1 22 32 81 492 5 1 1 492 5 3 8 in 482 5 for 1 2 and for 1 3 2 1 1 22 39
(5)
In expression (5), 6 495, where is a non-negative integer number, is the space of the functions defined on 9, being continuous and having partial continuous derivatives up to the th order. We assume hereafter that the coefficients in the action functional expression (4) are as regular as is needed so that all the operations in which they are involved are meaningful.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 535
It is worth noticing that this assumption of regularity does not allow for the study of cases in which rather abrupt variations of the stiffness coefficients 495 and 495 occur, such as for damage in the connecting elements. This situation will be considered in detail in the last part of this section. The stationarity condition on the action functional implies the vanishing of its first variation. By applying a standard procedure of calculus of variations, as illustrated in Biscontin et al. (2000), the free vibrations in an &84742 composite beam are regulated by the following system of partial differential equations 2 3 2 3 82 3 8 2 3 2 6 83 4 82 6 74 1 2 3 2 (6) 9 9 9
2 3 2 3 83 3 8 3 3 3 4 83 4 82 6 74 1 3 3 2 9 9
2 2 33 2 3 3 2 3 2 2 3 3 733 3 6 4 4 2 4 3 5 1 2 3 2 4 3 2 3 6 9 9 9
9 9
9
2 2 2 3 3 3 3 3 3 3 83 4 82 6 7 4 74 4 3 3 3 6 9 9 9 9 2 2 33 3 2 3 3 733 3 6 6 4 2 4 3 5 1 3 3 2 6 9
9 9
(7)
(8)
(9)
for 9 8 482 5 and 8. In the case of a free beam, the natural conditions on the boundary impose the annulment of the axial force and shear force , and of the bending moment applied on each end of the beam
82 1 82 9 2 3 3 2 3 2 3 2 733
2 3 6 3 6 1 82 3 4 9 9
9 9
2 3 2
(10)
2
(11)
2
3 3 3
3 2 1 82 (12) 93 83 1 82 3 3 (13) 9 2 3 3 2 3 2 3 3 2 3 3 733
3 3 6 83 4 82 6 74 7 4 6 3 6 1 82(14) 3 4 9 9 9
9 9 3 4 2
3 4 3
3 3 1 82 93
(15)
in 9 1 8 and 9 1 , and for 8. To simplify notation, in Equations (6)–(15) we omitted to explicitly indicate that functions depend on 9 and variables. Unless otherwise stated, this notation will also be maintained in the following section.
536 M. DILENA and A. MORASSI
Equations (6)–(15) show that small free vibrations in a composite beam are regulated by a system of differential equations, where a coupling takes place between axial (82 and 83 ) and transversal ( 2 and 3 ) motions. The coupling in system (6)–(15) is both of a mechanical and of a geometrical nature. In fact, axial and transversal motions are unrelated if and only if the connection stiffness coefficient is null or if the characteristic distance 74 is null. Note that the model proposed by Biscontin et al. (2000) is found by formally considering 2 3 3 (or, equivalently, 1 ) in Equations (6)–(15). The same procedure will be now adapted to formulate the problem of small vibrations for a composite beam with damage in the connection. To do this, a damage model is needed. It is known that the description of the interaction effects arising between the concrete and the connection in the neighborhood of a damaged connecting element is extremely complex and difficult to include in a one-dimensional model. The present research is limited to a simplified and macroscopic description, where the effects of damage on the metallic beam and the r.c. slab would be neglected and where damage would only be considered as a sudden reduction of the stiffness coefficient of the connection around the damaged area. This point of view is not uncommon in structural diagnostics and is often adopted also in simpler conditions, such as, for instance, in carved rods (Adams et al., 1978; Vestroni and Capecchi, 1996; Ruotolo, 1997; Ruotolo and Surace, 1997; Cerri and Vestroni, 2000) or in r.c. beams with artificially induced cracks (Casas and Aparicio, 1994). In these cases too, a careful description of damage is neglected – and it would be hardly worth doing, since it would require a detailed knowledge of degradation, which is not always available in advance – and a simpler model is often preferred, provided that it includes the essential aspect of the phenomenon. Considering the type of damage studied in the experiments led by Morassi and Rocchetto (2003), 7 495 and 7 495 coefficients of the damaged connection are assumed to be as follows
7 495 1 495 ! 5482176 495 2
(16)
7 495 1 495 ! 5482176 495 2
(17)
where ! 5482176 495 is the characteristic function of a 482 4 5 interval determined in 482 5, e.g., ! 5482176 495 1 2 if 9 8 482 4 5 and ! 5482176 495 1 8 if 9 8 4 4 2 5, being the length of the damaged area. This kind of damage can be compared to the extreme case where concrete surrounding adjacent studs on one end of the beam is thoroughly degraded and the plasticized connectors cannot hinder relative displacements between the slab and the metallic beam. In Dilena (2001, Section 4) more generic types of damage were considered, such as, for instance, the occurrence of one or more damaged areas, not necessarily located to the end of the beam, see also Section 5. For the sake of simplicity and considering that the experimental part has been developed for these configurations, the damage scenarios described by expressions (16) and (17) will be closely examined hereafter. Finally, it is worth noting that other mechanical models were introduced to interpret measurements on damaged composite beams; see Dilena (2001, Sections 2 and 3) for a comprehensive discussion. The problem of small undamped free vibrations in a damaged composite beam can now be formulated as above. Indefinite equations (6)–(9) are still valid in 482 4 5 4 4 2 5 where 495, 495 coefficients are replaced by 7 495, 7 495. The same is true for the boundary conditions (10)–(15) in the end cross-sections of the composite beam. On the other hand, geometrical and mechanical jump conditions appear in the 9 1 4 abscissa section separating the undamaged and damaged areas. Using the same notation as before, we have
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 537
1 7 81 4 4 5 2
1 81 4 4 5 2 2
(18)
1 7 1 4 4 5 2
1 1 4 4 5 2 2
(19)
1 1 1 7 4 4 5 2
4 4 5 2 2 1 9 9 1 7 1 4 4 5 2
1 1 4 4 5 2 2 1 7 1 4 4 5 2
1 1 4 4 5 2 2 1
1 4 4 5 2
1
1
7 4 4 5 2 2
(20) (21) (22) (23)
1 1 22 3, for 8. In the following section, the case of a damaged composite beam having initial constant elastic and inertial properties will be closely examined. This case is simple but rather meaningful, and will be used in Section 4 to interpret the dynamic tests carried out in Morassi and Rocchetto (2003).
) (!%% !1 " "( 1 "" % %1 1232 4567879 6 7
The infinitesimal small undamped free vibrations affecting a uniform composite beam are regulated by the coupled system of partial differential equations (6)–(9) with constant coefficients. To these equations, the boundary conditions (10)–(15) are added in the event of free ends. For vibrations that are harmonic in time and whose frequency is ", longitudinal displacements and transversal displacements can be considered as follows
81 492 5 1 81 495 " 2 1 492 5 1 1 495 " 2
(24)
1 1 22 3, where the vibration spatial amplitude values 482 495 2 83 495 2 2 495 2 3 4955 3 482 2 83 2 2 2 3 5 fulfill the differential system 2 8222 6 483 4 82 6 23 74 5 6 2 "3 82 1 82
(25)
3 8223 4 483 4 82 6 23 74 5 6 3 "3 83 1 82
(26)
4 2 9
2 6
733
43 222 6 223 5 4 4 2 4 3 5 6 2 "3 2 1 82
2 2 22 4 3 9
3 6 74 483 4 82 6 3 74 5 6
733 43 223 6 222 5 6 4 2 4 3 5 6 3 "3 3 1 82
(27) (28)
and where the free–free boundary conditions are
821 485 1 8 1 821 45 2
(29)
538 M. DILENA and A. MORASSI
221 485 1 8 1 221 45 2
(30)
4 2 999 485 6 2
733 43 22 485 6 23 4855 1 82
(31)
4 2 999 2 45 6
733 43 22 45 6 23 455 1 82
(32)
2 4 3 999 3 485 6 483 485 4 82 485 6 3 485 74 5 74
6
733 43 23 485 6 22 4855 1 82
(33)
2 4 3 999 3 45 6 483 45 4 82 45 6 3 45 74 5 74
6
733 43 23 45 6 22 455 1 8
(34) 2
To simplify notation, in previous expressions and hereafter, the symbol 4# 4955 is used to # 495 of the function #495. indicate the ordinary derivative 9 The complete primitive 482 495 2 83 495 2 2 495 2 3 4955 of the system (25)–(28) takes the form of
482 495 2 83 495 2 2 495 2 3 4955 1
23
5 6
$ 4%2 2 %3 2 %1 2 %9 5
7 1 2
(35)
82
23 5 6 where 5$ 9 82 is a vector of constants. In expression (35), & 2 25 6 3 4%2 2 %3 2 %1 2 %9 5 is the ' th eigenpair of the eigenvalue problem in the 9 space variable. It is obtained by seeking solutions for system (25)–(28) having the form of 7 2. The twelve complex numbers 23 5& 9 82 are the roots of the characteristic polynomial ( 4& 5, where ( 4& 5 1 ) 4& 5 * 4& 5 6 + 4& 5
(36)
and
3 3 2 & 6 2 "3 4 3 & 6 3 "3 4 4 3 2 (37) 2 2 3 2 3 3 73 73 * 4& 5 1
2 & 9 4 & 3 3 6 4 2 "3
3 & 9 4 & 3 3 6 734 6 4 3 "3 7 7 2 33 73 4 & 3 3 6 2 (38)
3 2 3 3 3 3 3 3 9 3 73 3 + 4& 5 1 4& 74 " 4 2 6 3 5 6 & 42 6 3 5 2 & 4 & 6 4 2 " (39) 7
) 4& 5 1
In expression (36), ) 4& 5 and * 4& 5 are the characteristic polynomials that are obtained when separate (approximate) solutions are sought for system (25)–(28) with 2 495 1 3 495 1 8 and with 82 495 1 83 495 1 8. These two cases can be matched with the composite beam
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 539 ‘‘uncoupled’’ 4)4& 55 axial vibrations and 4 *4& 55 flexural variations. The +4& 5 term can be therefore considered as the ‘‘coupling term’’ of these two kinds of variation. A study of the roots of polynomial (4& 5 similar to that presented in the appendix of Biscontin et al. (2000) shows that for any elastic and inertial coefficients, if "3 is bigger than "3 , " a suitable threshold frequency value, there will be three negative roots & 3 and one positive, and two complex conjugates; conversely, there will be two positive roots and two negative, and two complex conjugates. This property allows the representation of the exponential factors present in expression (35) through harmonic functions if & 3 root of (4& 5 is negative, through hyperbolic functions if & 3 root is positive, and through suitable combination of products of harmonic and hyperbolic functions if & 3 root of (4& 5 is complex. Note that the latter situation did not occur under the assumption of equal transversal displacements. Considering that the 25 6 eigenvector, ' 1 22 2 23, is proportional to the vector of components
%2 1 22 %3 1 * 4& 5 2 %1 1 , 4& 5 2 %9 1 - 4& 5 2
(40)
where
2 . 3 6 2 "3 2 ' 3 . 3 6 3 "3 2 3 - 4. 5 1 4 2 . 6 2 "3 6 4 * 4. 5 4 25 2 . 74 * 4. 5 1 4
733 3 . 6
, 4. 5 1 4 - 4. 5 2 733 3 9 3 . 6 2" 4 4 2 . 6 7
(41) (42)
(43)
the characteristic polynomial can be then formed for the eigenvalue problem (25)–(34), by imposing that the general solution (35) must fulfill boundary conditions (29)–(34). In so doing, 23 we obtain a homogeneous linear system in real constants 5$ 9 82 , for example, 34"54 1 5, where 34"5 is a 23 1 23 matrix depending on " (see Dilena, 2001, Equation 2.150). Natural cyclic pulsations correspond to those special " values that cancel out the determinant of 34"5. In order to determine the composite beam natural pulsations as roots of the characteristic polynomial 34"5, a numerical procedure was used, the essential steps of which can be summarized as follows. Once a value for " was set, say "3 , the sixth degree polynomial equation in & 3 (4& 5 1 8 was solved, where (4& 5 was assumed as in expression (36). Once the twelve & 5 6 roots of (4& 5 were found, the expressions of the eigensolutions 7 2 were determined by imposing the natural boundary conditions, as well as the value of 34"3 5. By repeating this procedure for "3 6 ", where " is a proper increment, the graph of 34"5 was reconstructed in a given frequency interval. Eigenfrequency values were calculated by a bisection method applied between the two values of the " variable, corresponding to a change of sign of 34"5. For each eigenfrequency value, after solving 34 1 5, the vector 4 was calculated and therefore the corresponding mode of vibration was determined.
540 M. DILENA and A. MORASSI 12 2 7879 6 7
The study of small vibrations of a uniform composite beam can be easily extended to include the case of damage in the connection. Therefore, in this section we will briefly point out what essential changes to the procedure above are needed; see Dilena (2001, Section 2.4.2) for a comprehensive discussion. Let us assume that damage in the connection is determined by expressions (16) and (17), i.e. that damage only affects the beam sector defined by the 4 4 2 5 interval. The interpretation of the findings offered in Section 4 will show how , i.e. the length of the damaged area, can be dependent upon the number of damaged studs. For free harmonic vibrations in time, with " cyclic pulsation, longitudinal and transversal displacements of each beam are also found in the form (24), separately in the two 482 4 5 and 4 4 2 5 intervals. In the first interval, the problem of eigenvalues in the 9 spatial variable is solved by the system of expressions (25)–(28). Then, the complete primitive is described by expression (35), where & are the roots of (4& 5 polynomial (Equation (36)) and eigenfunctions 25 6 495 are described by expressions (40)–(43), totaling twelve real constants 23 5$ 9 82 . In the damaged area, we have 495 1 495 1 8, and axial and flexural vibrations are separately uncoupled in the r.c. slab and in the metallic beam. For each single beam, the general mode shape of axial and transversal displacements is the classical one, totaling 23 twelve more real coefficients 5$ 1 9 1 82 . Now, imposing the twelve boundary conditions (29)–(34) and as many jump conditions (18)–(23) in 9 1 4 , the linear homogeneous 6 6 system 34"5 542 42 9 1 5 is deduced, having a 34"5 3 1 3 matrix. As before, natural 6 cyclic pulsations correspond to those " values for which the determinant of 34"5 vanishes. Finally, natural frequencies and their corresponding vibration modes were assessed with the same procedure as described at the end of Section 3.1. 1212 5 78
We will conclude this section with an application of the results described above on the study of free vibrations of a composite beam whose geometrical and inertial properties are illustrated in Table 1. We consider the undamaged beam and two damage states corresponding to 1 8 32 m (configuration D1) and 1 8 7 m (configuration D2). This example essentially corresponds to the T1PR beam tested during the experiments. Although the choice of the system coefficients will be discussed in detail in the following section, some preliminary remarks seem to be necessary here, before offering the results. As a matter of fact, the main difficulty concerns the correct selection of the connection parameters and , and of Young’s modulus of the concrete 2 . In Biscontin et al. (2000) it was shown how dynamic measurements can be used to estimate and 2 . Moreover, some standard experimental tests prove useful here, and namely the single-axle compressive stress test for 2 and the push-out test for , to estimate the value for these two parameters. Finding an estimate for is more complicated, especially because of the difficulties in experimentally reproducing the structural behavior of a connector under axial extensions. On the other hand, the effect of the connection axial stiffness on the global behavior of a composite structure has been poorly investigated so far, also regarding the static case; see Aribert and Abdel Aziz (1985). Considering these aspects, was assessed on the basis of the nominal axial stiffness of a single stud, distributed for each length unit of the beam in a width interval
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 541
Table 1. Physical parameters used in the analysis of free vibrations of the composite beam: (a) steel beam; (b) r.c. slab; (c) shearing and axial stiffness of a single stud; (d) length of the damaged region for configurations D1 and D2.
Parameter
Value (a)
3 3 3 3
7 8 m 2 1 2811 m3 2 1 281 m9 23 8 kg m12 3 2 1 2822 N m13 (b)
2 2 2 2
7 8 m 7 88 1 2813 m3 88 1 281 m9 7 2 kg m12 333 1 2824 N m13 (c)
/ 3 3 073
3 7 1 28 N m12 8 1 28 N m12
(d)
8 32 m 8 7 m
d (D1) d (D2)
equal to the distance separating two consecutive studs. The single stud was described as an equivalent beam, being 73 long and having an 3 3 073 axial stiffness, where 3 is the Young’s modulus and 3 is the transversal section area of the connector. The top and bottom ends of the equivalent beam are considered as being respectively fixed on the slab axis and on the beam flange. This simplified description is consistent with the (sometimes rigid) hypotheses that were formulated in Section 2 of this work and previously in Biscontin et al. (2000) to deduce a one-dimensional model of a composite beam. Therefore, will be hereinafter assumed as equal to
1
3 3 2 73 40) 5
(44)
in the event of equal and equidistant studs. In the appendix, a justification is offered for the choice of the nominal value (44) of , i.e. the connection axial stiffness. The 0–2000 Hz frequency range was carefully analyzed. A comparison between eigenfrequencies and their corresponding mode shapes is offered in Table 2 and in Figures 2 and 3, for the undamaged configuration and for D1, D2 damaged configurations. Mode shapes are 2 normalized so that 4 4 2 4832 6 32 5 6 3 4833 6 33 559 1 2 and, for the sake of simplicity, the comparison is restricted to the lower modes. Generally speaking, no displacement components were equal to zero, although we can observe a prevalence of longitudinal or transversal components, following the mode order. (Wherever ambiguities are not likely to arise, we will deal hereafter with flexural or longitudinal modes.) For example, for the undamaged case il-
542 M. DILENA and A. MORASSI
Figure 2.
Analytical modes 111 1223 12 1223 41 1223 41 1222 relative to free vibrations of the free–free
composite beam, with elastic and inertial properties of Table 1, in damaged configuration D1.
lustrated in Figure 1, in 1–5, 7–9, 11, 12 and 14 modes, the transversal displacement prevails, while 6, 10 and 13 modes correspond to mainly longitudinal vibrations. Among the latter, a distinction can be made between vibrations where motion components of the two beams are phase coincident (6 and 10 modes) and vibrations where motion components are phase different (13 mode). When the mode order rises, the coupling of longitudinal and transversal vibrations also increases.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 543
Table 2. Theoretical natural frequencies of flexural (f) and longitudinal (l) vibrations of the composite beam, with elastic and inertial properties of Table 1, for undamaged and damaged configurations D1, D2 (rigid vibration modes are omitted): 1 2884 * 76 4 * 76 50* 76 and 526 1 2884 * 76 4 1 5 4 * 76 4 1 550* 76 4 1 5.
Mode Number
1f 2f 3f 4f 5f 6l 7f 8f
Undamaged Model Model 1 1 * (Hz) * (Hz) 526 60.56 60.53 0.05 137.98 137.54 0.3 246.98 244.40 1.0 387.01 377.52 2.4 559.30 532.53 4.8 617.06 617.04 0.0 764.59 702.04 8.2 1003.27 877.93 12.5
Damage D1
* (Hz) 59.41 134.49 235.72 355.41 486.04 616.31 629.59 788.56
1.85 2.2 3.6 5.9 8.7 0.1 10.3 10.2
Damage D2
* (Hz) 57.04 122.99 200.03 294.73 417.02 616.23 553.44 676.65
5.77 10.6 18.2 21.9 21.7 0.1 21.2 22.9
According to general theorems of monotony, damage induces a reduction in natural frequencies. It can be proven that longitudinal frequencies are very insensitive to the presence of damage in the connection, with average percentage changes of about 0.1% for both D1 and D2 damaged configurations. In particular, this is true for longitudinal frequencies corresponding to the axial motions where the r.c. slab and the metallic beam are phase coincident, e.g., modes 6 and 10 in Figure 1. Conversely, flexural frequency variations are fairly marked as early as in the first damaged configuration and, owing to the specific location of the damaged area, they rise when the mode order increases. Considering that the attention in the experimental part was focused on lower flexural modes, we will hereafter closely examine the effects of damage on transversal motions. For undamaged configurations, differences between transversal components of flexural modes evaluated for the r.c slab and the steel beam are almost negligible in lower modes, e.g. up to modes 4–5. Relative transversal components of flexural modes become important as the mode order increases, especially in the areas close to the ends and to the maximum/minimum peaks of mode shapes. A similar situation occurs in damaged configurations, except for the damaged area where significant differences between transversal components are already apparent as early as in the second flexural mode. As shown in Figures 2 and 3, relative components of flexural modes in the damaged area rapidly grow as the mode order rises and also as damage increases. Moreover, a comparison between flexural modes corresponding to the undamaged configuration and to one of the damaged ones has revealed that the nodes tend to displace towards the damaged area. For example, Figure 4 illustrates the behavior of the transversal component in the slab when damage incidence grows. For the sake of completeness, Table 2 illustrates flexural frequencies in the undamaged beam with the model 1 . The differences as compared to the suggested model rise as the mode order increases and become negligible for low (flexural) frequencies. It is also to be noted that frequencies of the first longitudinal modes shown in the two models are virtually coincident.
544 M. DILENA and A. MORASSI
Figure 3.
Analytical modes 111 1223 12 1223 41 1223 41 1222 relative to free vibrations of the free–free
composite beam, with elastic and inertial properties of Table 1, in damaged configuration D2.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 545
Figure 4. First four analytical flexural modes 41 122 (relative to r.c. slab axis) for undamaged and damaged configurations D1, D2 of the free–free composite beam, with elastic and inertial properties of Table 1.
546 M. DILENA and A. MORASSI
* !# #!1' 2 1+ " 2 !% !"'% %!!% 1 " "( 2, 1 % 1 2 2% ( 1 " "( 1 1#!1 % 1 1', 1' "2%' In this section, the experimental results obtained from tests on the T2CR and T1PR beams, which correspond to a total and a partial connection respectively (see Morassi and Rocchetto, 2003) will be thoroughly described and interpreted. T1CR and T2PR samples revealed no substantial new elements and are closely examined in Dilena (2001). To interpret dynamic tests performed on undamaged composite beams, the vibrating model described by Equations (6)–(9) was used, with boundary conditions (10)–(15). The steel beam mass linear density 3 and the concrete mass linear density 2 were evaluated from the measurement of each beam total mass, under the condition of assuming a homogeneous material (see Table 3). Mechanical and geometrical sizes concerning the steel beam, that is its Young’s modulus 3 , its moment of inertia 3 , and the transversal section area 3 , were assumed as being equal to nominal values, that is 3 1 3 2 1 2822 N m13 , 3 1 2 1 281
m9 , and 3 1 2 1 2811 m3 . As for the r.c. slab, the transversal section area 2 and its moment of inertia 2 were calculated for the case of a uniform beam having an uncracked cross-section. Considering the slab nominal size and neglecting the contribution of the weak metallic reinforcing, 2 1 1 281 m9 and 2 1 7 1 2813 m3 were obtained. The connection axial stiffness is evaluated via Equation (44), with 3 1 3 2 1 2822 N m13 and 3 1 2 333 1 2819 m3 (the stud diameter is equal to 12.5 mm). As for the Young’s modulus 2 of the concrete and for the connection stiffness constant, the values deduced from experimental tests were used, for a first approximation. 2 was assumed equal to the origin tangent value deduced in a series of single-axle compressive stress tests performed on some material samples drawn from slab casting. The shearing stress stiffness / of each stud was calculated as being 30% of the 24 secant value, where 24 is the ultimate shearing resistance of a connector, resulting from some 3&'(9&6 tests performed on samples reproducing the mechanical interaction between the slab and the steel beam in the stud neighborhood. Starting from these estimated values, as in Biscontin et al. (2000, Section 4.4), the two parameters 2 and / were identified by means of vibration measurements, by imposing that analytical and experimental frequency values of both flexural and axial fundamental modes in the undamaged configuration coincide. In doing so, the limit model 1 was used. This choice is motivated by the observation that the two analytical models 1 and finite supply virtually coincident fundamental (flexural and longitudinal) modes, as was remarked at the end of Section 3.3. Solutions for values of / ranging from 2 1 28 to 1 28 N m12 , and for values of 2 ranging from 7 1 2824 to 1 2824 N m13 were looked for. The evaluation of this parameter variation range was considered as being sufficient, because the values that had been experimentally deduced were equal to 3 8 1 28 N m12 and to 8 1 2824 (T2CR beam) to 3 1 2824 (T1PR beam) N m13 , respectively. In this range, a unique solution was found for both models: for both 2 and / this solution shows small deviations from the values obtained during the first trial, see Table 3. As reasonably expected, the identified Young’s modulus values of concrete nearly coincide in both cases, while some differences arise in the evaluation of the connection stiffness constant. Tables 4 and 5 contain a comparison between experimental frequencies and analytical frequencies of undamaged beams for the identified models with 1 and finite . With
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 547 Table 3. Measured linear mass densities of the composite beams T1PR () 1 2 ) and T2CR () 1 37), and comparison between experimental and analytical identified values of constants 2 and / (/ 1 0) ) used in the interpretation of dynamic tests.
Parameter
2 (kg m12 ) 3 (kg m12 ) 2 (N m13 ) / (N m12 )
526
Experimental Value T1PR T2CR 7 2 23 8 23 8 3 1 2824 526 8 1 2824 526 3 8 1 28 536
Analytical Identified Value T1PR T2CR – – – – 33 1 2824 8 1 2824 3 7 1 28 3 8 1 28
mean origin tangent value determined by single-axle compressive stress tests on cylinders. 0.3 Fsu mean secant value of the ultimate shearing resistance of the connectors, as determined by pushout tests. 536
Table 4. Comparison between experimental and analytical natural frequencies of flexural vibration modes (rigid vibration modes are omitted) for the T2CR beam in undamaged configuration: 1 2884 * 4 * 50* .
Mode Number
2 3 7
Experimental Value * (Hz)
3 87 2 7 3 72 3 8 3 7
8
Analytical Value for 1 * (Hz)
3 87 8 8 27 8 4 3 3 8 7 78 28 7 7 32 2 3 77
Analytical Value for 1 * (Hz)
3 88 48 8 27 2
4 8 3 7 48 2 78 2 3 7 2 2 22 3 3 2 73
Table 5. Comparison between experimental and analytical natural frequencies of flexural vibration modes (rigid vibration modes are omitted) for the T1PR beam in undamaged configuration: 1 2884 * 4 * 50* .
Mode Number
2 3 7
Experimental Value * (Hz)
8
27 32 8 7
Analytical Value for 1 * (Hz)
8
8 8 27 4 8 3 3 2 7 82 23 8 78 33 7 2887 3
Analytical Value for 1 * (Hz)
8 7 48 8 27 4 7 3 8 2 8 7 3 3 73 7 2 83 8 37 7 3
548 M. DILENA and A. MORASSI
the exception of the second mode, both mathematical models overestimate frequencies associated to flexural vibration modes and disparities are even higher when the mode order rises, following a pattern similar to that of single homogeneous beams having equal slenderness. It is worth observing that the analytical model with finite leads to higher accuracy in reproducing flexural frequencies. Figures 5 and 8 suggest a comparison of the first four experimental and theoretical (normalized) flexural modes. As the two analytical models 1 and finite supply virtually coincident vibration modes for the undamaged configuration, at least within the lower frequency range, Figures 5 and 8 contain vibration modes referred to the model with finite . For the sake of clarity, comparison in the figures is limited to the modal components assessed along the r.c. slab. This is not a restrictive choice since, as shown in Section 3.3, differences between transversal components of flexural modes evaluated for the r.c. slab and the steel beam are almost negligible in lower modes throughout undamaged portions of the composite beam. It can be observed that measurements are highly consistent with theoretical estimates. In Tables 4 and 5 and further on, the greatest care will be devoted to the frequencies of flexural modes. In fact, experimental tests have revealed that the effect of damage on frequencies of axial modes is marginal. Anyway, for the sake of completeness, it should be recalled that frequencies of longitudinal motions are described with great accuracy through the analytical model, with percentage errors not exceeding 1% for frequencies lower than 2000 Hz. As already observed in Biscontin et al. (2000), the analytical model includes some axial frequencies that remain undetected in inertance measurements. These frequencies correspond to the longitudinal motions where the vibrations of r.c. slab and the steel beam are phase different (this is, for instance, the case of mode 13 in Figure 1). It can be shown that the difficulty in detecting these frequencies is owed to the presence of important energy dissipation, which is caused by relative displacements at the connection interface and which considerably reduces their corresponding vibration modes. This is why they are so difficult to detect in inertance measurements. To interpret dynamic tests performed on damaged composite beams, the vibrating model described in Section 3.2 was used. Damage is assumed as in expressions (16) and (17), and the extension of the damaged area was assessed by attributing to each stud a width interval corresponding to the distance separating two consecutive studs. Therefore, the extension of the damaged area in D1 (D2) configuration proved to be equal to 0.21 m (0.43 m) and to 0.175 m (0.325 m) for the beams with a partial connection and a total connection respectively, see Figure 1 in Morassi and Rocchetto (2003). Tables 6 and 7 contain a comparison between experimental frequencies and analytical frequencies of the beams for the identified models of damaged configurations D1 and D2. Generally speaking, the analytical model includes absolute variations in the flexural frequencies, which are fairly similar to those measured during tests. Relative variations in the examined modes, too, show a consistent behavior if compared to experimental findings. As a matter of fact, frequency modelization errors for damaged configurations are comparable to those for the undamaged system, and discrepancies between theoretical and experimental values follow a pattern similar to that of a single homogeneous beam having equal slenderness, see Figure 11. The suggested model can furthermore thoroughly reproduce some other features arising from dynamic tests, such as, for example, the shift of nodal points of the flexural modes produced by damage, or the separation of the transversal components in the neighborhood of the damaged region, see Figures 6–7 and 9, 10 for T2CR and T1PR beams, respectively.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 549
Figure 5. Comparison among the first four analytical (solid line) and experimental flexural modes of the T2CR beam in undamaged configuration.
To conclude this section, justification for an (apparent) contradiction in the experimental part seems worth offering. This contradiction arises between the significant sensitivity to damage shown by flexural frequencies and the weak differences in these frequencies resulting from a comparison between the partial connection beam and the total connection beam. To simplify the discussion, let us focus on the undamaged configuration of the beams. Besides small differences between the mass density of the slab and the Young’s modulus of
550 M. DILENA and A. MORASSI
Figure 6. Comparison among the first four analytical (r.c. slab, solid line; steel beam, dashed line) and experimental (r.c. slab, 1) flexural modes of the T2CR beam in damaged configuration D1.
the concrete beam, these two types of composite beam are essentially different in the linear density of their connectors. In fact, considering that the studs forming the connection are all equal, the coefficients of shearing stiffness and axial stiffness of the connection in total connection beams are about 1.5 times greater than stiffness coefficients in partial connection beams. On one hand, the experiments have shown that flexural frequencies in total connector beams are slightly higher than flexural frequencies in partial connector beams; in fact, as
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 551
Figure 7. Comparison among the first four analytical (r.c. slab, solid line; steel beam, dashed line) and experimental (r.c. slab, 1; steel beam, 1) flexural modes of the T2CR beam in damaged configuration D2.
shown in Tables 4 and 5, differences in the first five frequencies are equal to about 1.5, 2, 5.5, 6.5 and 4 Hz, respectively. On the other hand, the variations induced by damage prove to be meaningful as early as in the first damaged configuration. In fact, as shown in Tables 6 and 7, in D1 differences in the first five frequencies of T1PR and T2CR beams are equal to about 0.4, 3.5, 15, 32 and 46 Hz and to 0.14, 2.1, 7, 19 and 30, respectively. Regarding
552 M. DILENA and A. MORASSI
Figure 8. Comparison among the first four analytical (solid line) and experimental flexural modes of the T1PR beam in undamaged configuration.
configuration D2, these variations grow to 2.7, 23.3, 52, 72 and 98 Hz and 1.15, 10.3, 38, 59 and 71 Hz, respectively. These two experimental findings may seem to be in contrast. In fact, even if the connection stiffness is totally annulled, damage only affects a limited sector of the beam, extending to about 1/23 (total connection) or to 1/16 (partial connection) of the total length of the beam in D1 configuration and to twice as much in D2 configuration.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 553
Table 6. Comparison between experimental and analytical natural frequencies of flexural vibration modes (rigid vibration modes are omitted) for the T2CR beam in damaged configurations D1, D2: 1 2884 * 4 * 50* .
Mode Number 1 2 3 4 5 6 7
Damage D1 Exp. Value Anal. Value * (Hz) * (Hz)
2
2 82 2 7 27 37 78 38 773 7 7 2 78 27 8 73 73 88
2
7 2
42 4 3 8 28 2 2 37 3 2
Damage D2 Exp. Value Anal. Value * (Hz) * (Hz)
8
27 37 272 2 38 3 332 33 8 737 7 7 8 7 7 7 3 77
43 8 47
3 28 2 33 72
Table 7. Comparison between experimental and analytical natural frequencies of flexural vibration modes (rigid vibration modes are omitted) for the T1PR beam in damaged configurations D1, D2: 1 2884 * 4 * 50* .
Mode Number 1 2 3 4 5 6 7
Exp. Value * (Hz)
8 27 28 2 33 727 28 8
27 7
22
Damage D1 Anal. Value * (Hz) 2 27 37 3 7 2 8
3
42 38 4 2 7 27 3 2 33
3
Exp. Value * (Hz) 238 7 2
37 72 7 8 2 7 7 7
Damage D2 Anal. Value * (Hz) 8 233 388 87 3 7 2 83 7
42 43 3 7 2 3 7 3
The use of the analytical model of a composite beam, as illustrated in Section 3, allows an interpretation for this difference in flexural frequency sensitivity to a variation in the connection stiffness. A parametric analysis, in fact, shows that the first flexural frequencies of a composite beam in the undamaged configuration are hardly sensitive to uniform variations in stiffness, also in a quite wide neighborhood of nominal values, see Figures 12 and 13. For example, other parameters and structural coefficients being equal, 8 variations of the nominal value produce only a few Hz variations in the first two frequencies. The same is true for similar variations of . Generally, these variations rise as vibrational modes increase. The situation is completely different, though, when the coefficients of stiffness in the connection are annuled to simulate damage, even in a small portion of the beam. In fact, if 8, the cinematic constraint by which the transversal displacements of the r.c. slab and the concrete beam are coincident is lost in the damaged area. This aspect proved to be crucial to accurately describe the dynamic behavior of the damaged composite beam. Regarding this, if the damaged configuration was simply described by assuming 1 8 in the damaged area and by maintaining the coefficient equal to the value in the undamaged configuration, then the analytical model would give very low frequency variations, which
554 M. DILENA and A. MORASSI
Figure 9. Comparison among the first four analytical (r.c. slab, solid line; steel beam, dashed line) and experimental (r.c. slab, 1; steel beam, 1) flexural modes of the T1PR beam in damaged configuration D1.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 555
Figure 10. Comparison among the first four analytical (r.c. slab, solid line; steel beam, dashed line) and experimental (r.c. slab, 1; steel beam, 1) flexural modes of the T1PR beam in damaged configuration D2.
556 M. DILENA and A. MORASSI
Figure 11. Frequency modelization errors for the T2CR beam (a) and the T1PR beam (b) in undamaged and damaged D1, D2 configurations.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 557
Figure 12. Variations of the first four flexural frequencies of the T1PR undamaged beam versus shearing stiffness connection (51 is the analytical identified value of the shearing stiffness coefficient for unit length of the connection, see Table 3): modes 1 and 2 (a); modes 3 and 4 (b).
558 M. DILENA and A. MORASSI
Figure 13. Variations of the first four flexural frequencies of the T1PR undamaged beam versus axial stiffness connection (61 is the analytical identified value of the axial stiffness coefficient for unit length of the connection, see Table 1 and expression (44)): modes 1 and 2 (a); modes 3 and 4 (b).
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 559
would be inconsistent with experimental findings. Similarly, the mode shapes would be described with a high degree of inaccuracy and would not consider the particular features (such as, for instance, the displacement of nodes, or the considerable difference between transversal modal components of the r.c. slab and of the metallic beam in the damaged area) that were revealed during measurements (Dilena, 2001, Section 3.3).
- !# #!1' 2 1+ " %!% !"'% 211+% 2% % " The vibrational characteristics of an elastic system are altered by a decrease in strength induced by damage in a sector of the structure. The ultimate scope of structural diagnostics via modal analysis is to interpret the changes induced in the modal parameters and to use them to predict the location and degree of degradation. This section deals with the identification of damage located in the connecting elements of a composite beam. Two main techniques will be offered: the first is based on the measurement of the changes induced by damage in lower flexural frequencies, and the second makes use of a property concerning the shift of nodal points of the lower flexural modes. 232 7879 77 5 8 9 5 5
It was shown in Morassi (1993) that, for small damage, variation in frequency in a onedimensional element induced by localized damage, such as for instance a carving or a crack, is proportional to the elastic stress strain energy density of the corresponding mode shape measured in the undamaged configuration in the same location as damage. In particular, frequency variations in the different orders are only dependent upon damage location. In the event of single and limited damage, this result suggests a simple diagnostic strategy. First of all, some ‘‘reasonable’’ damage conditions can be foreseen and a number of maps describing frequency sensitivity to damage can be guessed by relating them to frequency variations to limited damage. Secondly, frequency variations are measured and compared to the sensitivity maps, ultimately choosing the map that can ‘‘better’’ reproduce experimental results. Initially, the selection pattern can be a qualitative one, so as to eliminate the simulated damage configurations where frequency variations are significantly different from the measured ones. Later on, another selection pattern can be chosen and namely based on the (somehow) minimum distance between experimental variations and analytical variations. The method is based on a simple idea, which is quite widespread in the literature; see for example, Hearn and Testa (1991) and Liang et al. (1992), and for a slightly different version Armon et al. (1994). The method proved to be useful in a number of studies on beam elements and in simulations on simple structures, while a thorough investigation on real and quite complex systems and an estimation of the degree of accuracy of the original model to build maps of sensitivity to damage are lacking. Some findings from this technique applied to damage detection in the connection of a composite beam are described hereafter. Initially, the derivative of proper frequencies was used with respect to damage parameters and ( 52)&28 28 6 6 ) to describe the first-order variations induced by damage type (16) and (17) in the composite beam. This technique has the advantage of using frequency sensitivity to damage as the explicit expression of modal properties of an undamaged system. On the other hand, though, owing to the linearization procedure on which it is based,
560 M. DILENA and A. MORASSI
the technique does not allow the analysis of such serious damage conditions as those that were induced onto the composite beams during the experimental work. In fact, in all damage configurations studied in the present work, the frequency variations calculated from the sensitivity expression proved to be much lower than the experimental ones. As a consequence, the application of this technique did not give any useful indications for diagnostic purposes. It was therefore necessary to calculate frequencies directly on all damage configurations and this was done by adapting the procedure as described in Section 3.3 to the case of the damaged interior areas of the beam. Let us now consider the T1PR beam and solve the problem of damage detection in D1 configuration, i.e. damage to stud 16 to the right end of the beam. Since the undamaged beam is symmetric, damage at any one of a set of symmetrically placed areas will produce identical changes in natural frequencies. Indeterminacy can be removed by using information on mode shapes, as we will see in Section 5.2. For the time being, let us assume that damage is known beforehand to be located in the right-hand section of the beam. Let us identify eight damage configurations that are likely to occur on each single stud, from 9 to 16. It is to be noted that, in doing so, we are also assuming that damage in the beam affects only one connector (whose location is unknown). In real applications, this may entail that we know beforehand that damage affects a group of neighboring studs. For each damaged configuration, the absolute variations associated to the first five flexural vibration modes were calculated. These were later compared to the measured ones, see Table 8(a). Variations depend on the location of damage and on the considered vibration mode. This proves that a relation exists between damage location and frequency variations. It is worth noting that, from the fourth mode onwards, frequency variations induced by damage in an ‘‘interior’’ stud, say from stud 9 to 15, are fairly limited if compared to those induced by damage in the ending stud 16. This aspect had already been observed in a series of preliminary dynamic tests carried out by Biscontin and Wendel (1998, Sections 1.4 and 1.5). A selection pattern consists of comparing experimental variations to analytical variations of frequency in the different damage configurations, via their ‘‘Euclidean distance’’. This distance was assumed equal to the square root of the sum of squared differences between the experimental and simulated variations in flexural frequencies. Damage in stud 16 corresponds to the minimum distance and this pattern allows the univocal localization of damage. This technique was also applied to detect D2 configuration, which corresponds to damage in the two ending studs 15 and 16. Under the hypothesis that damage affects two consecutive studs, the two ending studs correspond to the minimum distance of experimental variations, see Table 8(b). Similar results were obtained for the T2CR beam in configurations D1 and D2. 2 2 7879 77 5 8 759 5 56 5
The effect of damage on the nodes of free vibration modes of a vibrating system was first considered by Gladwell and Morassi (1999). They showed that in a thin rod with a notch, the nodes of the longitudinal vibration modes move toward the notch. This means that every node located to the left of the notch in the undamaged configuration moves to the right, and every node on the right moves to the left. This property is clearly very useful in practical terms for diagnostic purposes. In fact, for every vibration mode, which has at least two nodes, there is exactly one neighboring pair that moves toward each other, and the notch is located in between.
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 561 Table 8. Damage detection in the T1PR beam. Comparison between experimental (* ) and analytical (* ) frequency variations for damaged configurations D1 (a) and D2
(b): 1
82 4*
4 * 53 .
Damaged Connector(s)
* 2 (Hz)
* 3 (Hz)
16 15 14 13 12 11 10 9
2 23 2 23 2 2 2 8
8 8 8 37 8 8
7 8 3 82 2 8 8 33 8 2 3
* 1 (Hz)
* 9 (Hz)
* (Hz)
(Hz)
3 7 2 72 8 7 3 3 7
3
2 3
33 22 3 3 3 82 3 7 7 3 7 77
2 8 2 7 73
22 8 2 3 2 3 3 73
7
2 7
8
3 23 22 2 8 22 27 23 23 2 8 27
22 2 2 2 33 7 2 37 32 33 2 3
37 2
8 28 222 28 72 228 8 –
(a)
*3+1
+,
+-
-+
+
,-+
(b) 15–16 14–15 13–14 12–13 11–12 10–11 9–10 *3+ 1
7 3 3 3 2 3 8 2 7 8 8 77 +./
2 22 3 7 2 7 2 3 3
+/
- +/,
0+
/ +
Numerical simulations suggest that this property can be also extended to flexural vibrations of the beam or to more complex vibrating systems, where damage is localized. Up to now, anyway, the finding has been thoroughly proved only for the case of axial vibrations. Experimental results in Morassi and Rocchetto (2000) suggested that an equivalent property can also be valid for steel–concrete composite beams. In particular, a comparison between experimental flexural modes in the undamaged configuration and in one of the damaged ones revealed that the nodes tend to displace towards the damaged area. Let us consider, for example, the T1PR beam. A comparison of the first four modes shows that they displace towards the damaged area and, as already observed in Gladwell and Morassi (1999), regarding uniform axially vibrating rods, for a given mode changes in node position increase as the node approaches damage location. An analysis of Figure 8 in Morassi and Rocchetto (2003) shows that both D1 and D2 damage are localized in the interval (3.00 m, 3.50 m), which corresponds with good approximation to the exact location of the damaged area. Similar situation occurs for the T2CR beam, see Figure 7 in Morassi and Rocchetto (2003). An extensive parametric analysis based on the analytical model as described in Section 2 confirmed that the nodes of flexural modes tend to displace towards the damaged area even when damage is localized in the central area of the beam. In these cases, with equal damage conditions, it was observed that the nodes displace less than when damage is located to the end of the beam. This observation is consistent with experimental results in Biscontin and Wendel (1998, Sections 1.4 and 1.5). The reason for this is to be found in the fact that the ending
562 M. DILENA and A. MORASSI
sectors of the beam are highly sensitive to stiffness variations in the connection; conversely, the central area of the beam shows a lower and oscillating sensitivity to damage, as shown in frequency variations illustrated in Table 8. In Dilena (2001, Section 4.4) a comprehensive description of findings is given. Generally speaking, the differences between transversal components in the slab and in the beam are significant only for severe levels of damage – at least two damaged connectors – and only from the fourth mode onwards. Node displacement towards the damaged area usually becomes significant only from the third vibrational mode and they are less important than those produced by damage in one or two ending studs, i.e. the cases studied during experiments. This peculiar dynamic behavior of a damaged composite beam suggests that variations of the nodes of flexural modes should be considered and used with care when considering the problem of damage localization.
. " '# " This paper is the second part of an experimental–analytical investigation on the dynamic behavior of damaged steel–concrete composite beams. In the first part of the research, we presented and discussed the experimental results of a comprehensive series of dynamic tests performed on composite beams with damage in the connection. Damage was induced by removing concrete around some elements connecting the steel beam and the r.c. slab and consequently causing a lack of structural solidarity between the two beams. Experimental observations suggested the formulation of an analytical model of a composite beam, where the strain energy density of the connection includes also an energy term associated with the occurrence of relative transversal displacements between the r.c. slab and the steel beam. A comparison with experimental results has shown that the model significantly enhances accuracy when describing the undamaged state of composite beams and that it is also appropriate to accurately predict their dynamic behavior under damage conditions. A damage detection technique based on the measurement of variation in the first flexural frequencies was then applied to the suggested model and gave positive results. In conclusion, analytical results prove consistent with experimental tests, and this suggests that the theoretical model can be useful in dealing with structural diagnostics concerning composite systems with more realistic typologies of damages, such as those due to fatigue phenomena induced by heavy repeated loads.
1% 2 / 1 1 1', 1' % 1 % "( 0% " % " 1/ 1' (( % "%(( % In describing the analytical model of a composite beam offered in Section 3, the axial stiffness of a single connector was assumed as being equal to 3 3 073 , where 3 and 3 are the Young’s modulus of the material and the area of the stud transversal section, respectively, and 73 is the distance between the r.c. slab axis and the metallic beam extrados. This appendix contains an explanation for this choice. Since our aim is to assess the connection axial stiffness, we shall skip general considerations and focus our attention on transversal motions of the slab and of the metallic beam only. Therefore, the following assumptions are made: 82 1 8 e 83 1 8. In all the considerations
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 563
hereafter, we suppose that the materials of the composite beam have a linear elastic behavior, in accordance with the hypotheses made at the beginning of Section 2. The relative motion % 3 3 4 2 occurring between the slab and the metallic beam is hindered by the connection. The single connector can be approximately described as a cylindrical element of the beam embedded in the r.c. slab, with its bottom end welded to the extrados of the metallic rod. The latter shows considerable stiffness to hinder deformations induced by the forces that act in the plane where the middle flange of the steel beam is located; consequently, the base of the connector can be considered with good approximation as a stiff element. When the system is free to vibrate, transversal inertia forces in a portion of the slab are immediately balanced by elastic forces produced by the connectors embedded in the same portion of the slab. Superficial bond forces transmitted between the stud lateral area and the surrounding concrete produce these forces. A comprehensive description of the force transmission phenomenon is extremely complicated and obviously has a significant three-dimensional nature. In general terms, however, the reaction of each single stud can be compared to that of an elastic spring (having an unknown stiffness), which hinders the relative transversal displacement w between the r.c. slab and the metallic beam, see Figure A1. Under this hypothesis, the axial force from a stud, say 31 , is proportional via the constant to the relative displacement % in the 91 abscissa section where the stud is located, 1 5. i.e. 31 1 %49 It is then possible to calculate , by imposing a congruence condition, i.e. by making %491 5 equal to a ‘‘characteristic’’ axial displacement of the stud. Let us now determine the field of axial displacements %3 in the stud. Similarly to the bond stress characterizing steel rods embedded in reinforced concrete, the 31 force can be considered with good approximation as being uniformly distributed along the lateral area of the stud, so as to distribute evenly the axial force to each length unit and with an intensity of 4 1 31 05, where 5 is the stud length. In doing so, it was implicitly assumed that a perfect bond exists between the lateral area of the stud and the surrounding reinforced concrete. This condition is absolutely met when the system is subject to low stress, as usually happens in dynamic characterization tests. In our case, the worst stress condition was reached during static tests performed on experimental samples; see Morassi and Rocchetto (2003, Section 3). In these cases, the maximum bond stress was equal to 28 N m13 . This value is much less than the limit stress that is likely to produce a separation between steel and reinforced concrete. Solving the problem of extensional deformation in the stud as follows
3 3 %223 4 65 6 4 1 82 for 8 1 6 1 52 %3 485 1 82 %23 455 1 82
(A1)
where the 6 abscissa has its origin in the fixed base of the stud, the following is found: 4 6 %3 4 65 1 6 54 (A2) 3 3 3 Now, the congruence condition can be imposed; for instance, by making the displacement of the stud at the slab axis ( 6 1 705 in our case, since 5 1 8 8 m and 73 1 8 87 m) equal to the %491 5 transversal displacement of the slab axis. Consequently, we find that
564 M. DILENA and A. MORASSI
Figure A1. Simplified model of the interaction between a stud and the r.c. slab.
1
78 3 3 2 32 73 40) 5
(A3)
i.e. a 50% higher value than the nominal value (44). Anyway, it is worth noting that the bond stress can be significantly reduced near the stud base (where the reinforced concrete has a weak compactness) and can concentrate towards the free end of the stud (also because the stud head is found here, an aspect that was not considered in this study for the sake of simplicity). Both aspects cause a significant reduction of stiffness as compared to (A3) assessment and make the analytical evaluation closer to the nominal value (44).
A DAMAGE ANALYSIS OF STEEL–CONCRETE COMPOSITE BEAMS 565
1892227286+ 3'2 4&6'95 2 ' 69 6'481 15+ 446629 95 ' 9865 !&6 98 8 2 8 8 6'2 98826 98 4* 4 6
82 92
286+ 32 4 6'481 452 &2 69 9&5 5 28 48 924&2 15+ 5& 69'2669 48 15+ 7 94 69 252 95 5& 6 & & 98 98 6'2 &!826 9 6' 34325+
!%(%!% % Adams, R.D., Cawley, P., Pye, C.J., and Stone, B.J., 1978, ‘‘A vibration technique for non-destructively assessing the integrity of structures,’’ %9&584 9 92'48 4 8 8225 8 282 20(2), 93–100. Anzellotti, G., Baldo, S., and Percivale, D., 1994, ‘‘Dimension reduction in variational problems, asymptotic development in -convergence and thin structures in elasticity,’’ 73696 84 9, 61–100. Aribert, J.M. and Abdel Aziz, K., 1985, ‘‘Calcul des poutres mixtes jusqu’1 l’2tat ultime avec un effect de soul3vement 1 l’interface acier-b2ton,’’ 9865&6 98 9164 )&2 4, 3–36. Armon, D., Ben-Haim, Y., and Braun, S., 1994, ‘‘Crack detection in beams by rank-ordering of eigenfrequency shifts,’’ 92'48 4 627 48 84 :592 8 8, 81–91. Biscontin, G. and Wendel, P., 1998, ‘‘Structural diagnostics of steel-concrete composite beams via dynamic methods,’’ 3'2 (in Italian), University of Udine, Udine, Italy. Biscontin, G., Morassi, A., and Wendel, P., 2000, ‘‘Vibrations of steel–concrete composite beams,’’ %9&584 9 !546 98 48 98659 6, 691–714. Casas, J.R. and Aparicio, A.C., 1994, ‘‘Structural damage identification from dynamic-test data,’’ ASCE %9&584 9
65&6&54 8 8225 8 120(8), 2437–2450. Cerri, M.N. and Vestroni, F., 2000, ‘‘Detection of damage in beams subjected to diffused cracking,’’ %9&584 9 9&8 48 !546 98 234(2), 259–276. Davini, C., Morassi, A., and Rovere, N., 1995, ‘‘Modal analysis of notched bars: tests and comments on the sensitivity of an identification technique,’’ %9&584 9 9&8 48 !546 98 179(3), 513–527. Dilena, M., 2001, ‘‘Vibrations of steel–concrete beams with damage in the connection: Part II. analytic models and damage detection,’’ 3'2 (in Italian), University of Udine, Udine, Italy. Gattesco, N., 1999, ‘‘Analytical modeling of nonlinear behavior of composite beams with deformable connection,’’ %9&584 9 9865&6 984 622 62245' 52(2), 195–218. Giuriani, E., 1983, ‘‘Deformable connectors in steel-concrete composite beam,’’ 6& 2 6 25'2 5, 259–295. Gladwell, G.M.L. and Morassi, A., 1999, ‘‘Estimating damage in a rod from changes in node positions,’’ 8 252 :59!( 27 8 8 8225 8 7, 215–233. Hearn, G. and Testa, R.B., 1991, ‘‘Modal analysis for damage detection in structures,’’ %9&584 9 65&6&54 8 8225 8 117(10), 3042–3063. Kristek, V. and Studnicka, J., 1982, ‘‘Analysis of composite girders with deformable connectors,’’ :5922 8 9 6'2 86 6&6 98 9 8 8225 73(2), 699–712. Liang, R.Y., Hu, J., and Choy, F., 1992, ‘‘Quantitative NDE technique for assessing damages in beam structures,’’ %9&584 9 8 8225 8 92'48 118(7), 1468–1487. Morassi, A., 1993, ‘‘Crack-induced changes in eigeparameters of beam structures,’’ %9&584 9 8 8225 8 92( '48 119(9), 1798–1803. Morassi, A. and Rovere, N., 1997, ‘‘Localizing a notch in a steel frame from frequency measurements,’’ %9&584 9 8 8225 8 92'48 123(5), 422–432. Morassi, A. and Rocchetto, L., 2003, ‘‘A damage analysis of steel-concrete composite beams via dynamic methods: Part I. Experimental results,’’ %9&584 9 !546 98 48 98659 9, 507–527. Ruotolo, R., 1997, ‘‘Tecniche basate sulle vibrazioni per l’identificazione del danno strutturale,’’ :'1 3'2 (in Italian), Polytechnic of Turin, Turin, Italy. Ruotolo, R. and Surace, C., 1997, ‘‘Damage assessment of multiple cracked beams: numerical results and experimental validation,’’ %9&584 9 9&8 48 !546 98 206(4), 567–588. Vestroni, F. and Capecchi, D., 1996, ‘‘Damage evaluation in cracked vibrating beams using experimental frequencies and finite element models,’’ %9&584 9 !546 98 48 98659 2, 69–86.
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