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- Title: 475
- Type: Notes
- School: Washington
- Course: SPAN 475
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ASCE 16th Engineering Mechanics Conference July 16-18, 2003, University of Washington, Seattle DAMAGE AND RESIDUAL LIFE ASSESSMENT OF STRUCTURES USING FRACTURE MECHANICS Trisha Sain 1 , J. M. Chandra Kishen2 , Associate Member, ASCE ABSTRACT A damage and residual life assessment method has been developed in this paper using the concepts of inverse method and fracture mechanics. The objectives are to detect, locate and quantify any damage that has occurred within a beam during its service life using displacement measurements taken at some arbitrary points along it. Grobner bases is used to check the compatibility among the measured displacement data. Any incompatibility among those measured values indicate some damage. The curvatures are computed using nite difference technique. These curvatures show abrupt change at damaged locations. The damage is quanti ed in terms of crack length and an analytical model is developed using a cracked beam nite element. Finally, the residual life of damaged beam has been assessed, using fatigue fracture concepts. The fatigue life has been determined in terms of number of applied load cycles, the beam can withstand, before reaching the critical crack length. Keywords: Inverse method, Damage detection, Grobner bases, Residual life, Fatigue crack propagation. INTRODUCTION All civil engineering structures are initially designed depending on certain design criteria, such as design loads, allowable stresses etc. But, damage due to an extreme event is always possible in a structure during its design life. Sometimes, undetected and un-repaired damage may lead to structural failure demanding costly repair and a huge loss of lives. Therefore, the problem of maintenance and repair of existing engineering structures involves damage detection at an early stage. For massive structures like bridges and dams that were constructed some 50 - 60 years ago, it is necessary to test its functionality under the present load situation and quantify damage if any, since it involves huge expenditure to demolish and reconstruct them. It is also important to evaluate the residual life of these structures. Therefore, a standard damage detection algorithm includes four different stages of analysis, as follows: 1. 2. 3. 4. 1 2 Detection of damage if present in the structure. Determination of the geometric location of the damage. Quanti cation of the severity of the damage. Prediction of the remaining service life of the structure. Graduate Student, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India. Corresponding author. Assistant Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India. E-mail:chandrak@civil.iisc.ernet.in. Tel:+91-80-394-3117. Fax:+91-80-360-0404. Damage can be de ned as the change in structural performance, which can be identi ed in terms of discrete cracks or a weak zone formation and a consequent stiffness reduction. In this work, an attempt is made to detect damage and estimate the residual life of structures using concepts of fracture mechanics and inverse method. For detecting the existence of damage and to identify its location, displacements under service loads are measured at certain points. Grobner basis is used to check the compatibility among the measured displacement data. Any incompatibility among those measured values indicate some damage. The curvatures are computed using nite difference technique. These curvatures show abrupt change at damaged locations. The damage is quanti ed in terms of crack length and an analytical model is developed using a cracked beam nite element. The nal step of a damage detection algorithm, i.e. residual life assessment of a damaged structure has been performed on the basis of fatigue fracture phenomena, as most of the real structures are frequently subjected to fatigue loading. In the foregoing study, residual life of the damaged structure is quanti ed in terms of the number of load cycles required to grow any detected crack up to certain critical length. The fatigue behavior of concrete depends on a number of parameters, such as, tensile strength, fracture energy of concrete, relative size of the structure and on loading history the structure is subjected to. The numerical model for fatigue crack propagation in concrete structures, proposed in this study, as a modi cation of Slowik et. al. model, incorporates all the above described parameters. Further the model has been validated against some already available experimental results. ANALYTICAL MODEL FOR DAMAGE DETECTION: AN INVERSE APPROACH Inverse problems are concerned with the determination of inputs or causes (loads, stresses) from the observed output or responses (i.e. strains, displacements, natural frequencies, mode shapes, modal curvatures etc.) in contrast with the direct problems, in which outputs or responses are determined using knowledge of the inputs or sources. The following assumptions are made in the development of an analytical model: 1. Field measurement of displacements are perfect and contain no error. 2. Damage occurs in the form of a discrete crack. 3. Structures remain linearly elastic even after experiencing the damaging event. The analytical displacement curve for any structural element is a function of its geometry, its material properties and the loading. In an inverse problem, the loading and the material properties are unknown quantities. The displacements measured at site assumed as error free has to satisfy the analytical displacement curve. If they do not, we may assume that some damage has occurred. The damage may be in the form of changed material property or a change in the geometry. In this work, we assume that damage has occurred in the form of a discrete crack, which alters the geometry or more speci cally the moment of inertia. Since, displacements are the only known data from the site, we make use of Grobner bases (Ioakimidis 1998; Ioakimidis 1994) to arrive at a so called compatibility equation which is a function of the known quantities. As an example, we consider a simply supported beam subjected to a uniformly distributed load of magnitude w per unit length. The de ection curve for the beam is given by, g y = (x4 2Lx3 + L3 x) (1) 24 In Equation 1, a new variable g = w/EI is introduced to reduce the two unknown quantities w and EI into one. The problem now reduces for the determination of the unknown 2 FIG. 1. Crack beam element quantity g. Assuming that the displacements y1 and y2 are measured at two points x1 and x2 . From the mathematical package Maple, using Grobner basis, we obtain, 24y1 + gx4 2gLx3 + gL3 x1 = 0 1 1 24y2 + gx4 2gLx3 + gL3 x2 = 0 2 2 L3 x1 y2 2Lx3 y2 L3 x2 y1 + 2Lx3 y1 x4 y1 + x4 y2 = 0 1 2 2 1 (2) (3) (4) Equation 4 contains only the known quantities and is known as the compatibility equation. If there is no crack and the measured displacements are free of error Equation 4 will be satis ed. Any error in satisfying the equation will indicate presence of some aw or damage within the structure. Once damage is detected within the structure, we have to identify its location. We make use of the curvatures to nd the location of damage. Assuming small de ections, the curvatures are computed from the known eld measurement of displacements using the central difference approximation: yi+1 2yi + yi 1 d2 y = (5) 2 dx h2 where h is the distance between two adjacent measurement points. The computed curvatures are plotted, and the points of abrupt changes in the plot indicate a likely site of damage. Degree Of Damage Determination by Finite Element Method: Cracked beam element To determine the degree of damage at the identi ed damaged zone, nite element analysis is performed using cracked beam element (Tharp 1987) Cracked beam element is a two noded, zero length element, having three degrees of freedom per node as in a regular two dimensional beam element. The stiffness matrix for the crack element is obtained in terms of shear and bending compliances using the energy approach. Neglecting axial deformations, the stiffness matrix for the crack element for the degrees of freedom as shown in Figure 1, is given by (Tharp 1987) 1 vv 0 1 mm 1 vv 0 1 mm 1 mm 2 [k] = where vv = and mm = 0 1 vv 0 1 vv 0 1 mm A 0 0 (6) 0 0 2(1 2 ) E 2(1 2 ) E KII V KI M dA 2 (7) A 0 dA (8) 3 FIG. 2. Geometry and loading pattern or the beam In the above equations, A is the area of cross-section, vv and mm are the compliances with respect to the shear and bending moment respectively, KI and KII are stress intensity factors corresponding to mode I and mode II respectively. Brown and Srawley (1966) have presented an equation for KI under pure bending of an edge cracked strip, valid for a/b 0.6. KI = 6M (( a)1/2 /b2 )(1.12 1.39(a/b) + 7.32(a/b)2 13.1(a/b)3 + 14(a/b)4 ) Numerical expression for KII as given by Tharp (1987) and valid for 0 a/b 1.0 is KII = V (b a) 1/2 (1.993a/b + 4.513(a/b)2 9.516(a/b)3 + 4.482(a/b)4 ) where, a is the crack length and b the depth of section as shown in Figure 1. VALIDATION OF THE PROPOSED METHOD To validate the proposed method of obtaining the location of damage and quantifying the same, two case studies are considered. In these studies, simply supported beams with known location of damage are analyzed. The damage is introduced in the form of discrete cracks. Case 1 In this example a simply supported beam with a crack of size 50 mm at midspan is modeled using the fracture mechanics code FRANC (FRANC-2D 1998). The geometry and loading condition of the beam is given as gure 2. The nite element mesh for the enlarged crack position is shown in gure 3. The displacements are obtained at some equidistant points along the length of the beam. The curvatures are computed and plotted as shown in Figure 4. A peak is observed in the curvature plot at the mid-span region indicating the location of damage. Case 2 This is a case similar to Case 1 except that cracks of different sizes are introduced at three different loactions. At left quarter and mid-span locations, crack of size 50mm and at right quarter span a crack of size 100mm were introduced, all in the bottom region. as in the earlier examples, the displacements are obtained from nite element analysis and the curvatures are 4 (10) (9) FIG. 3. Finite element mesh indicating enlarged crack location FIG. 4. Curvature along the length of beam (Case 1) 5 FIG. 5. (Curvature along the length of beam (Case 2) TABLE 1. Predicted crack lengths Case 1 2 Position mid-span left mid-span right mid-span Crack length (mm) Exact Computed 50 50.75 50 51.10 50 49.20 100 103.00 4 3.90 Error % 1.5 2.2 1.6 3.0 1.2 Experiment computed and plotted as shown in Figure 5. This plot again shows the presence of peaks at the cracked locations. In Cases 1 and 2 above, the displacements of damaged beam to be obtained from the eld are simulated through a nite element analysis wherein the effect of cracks are taken into account. The curvature plots could identify the locations of damage. To quantify the damage, that is to obtain the crack sizes introduced, the proposed nite element formulation using the cracked beam element is used. Analyses are carried out by introducing crack element at the already identi ed locations. The crack size is varied until the displacements obtained agree with the simulated displacements. Table 1 shows the crack sizes obtained from the analysis. It is seen that the results are within 3% error in all the cases, which is encouraging. EXPERIMENTAL VALIDATION An experimental study is done on a steel beam having a square cross-section under threepoint bending (Figure 6). 6 FIG. 6. Experimental beam model: loading con guration and geometry FIG. 7. Displacements the along length of beam (experimental results) A constant load of 200 N is applied and de ections at seven equidistant points are measured. The beam was damaged by introducing a crack of size 4 mm at the mid-span of the beam and the vertical de ections were measured upon loading. Figure 7 shows the plot of displacements measured on the undamaged and the damaged beam. The curvatures are computed and plotted as shown in Figure 8. It is seen that the curvatures show abrupt changes near the damaged zone. Further, a nite element analysis is performed on this beam by introducing a crack element at the mid-span. It is seen from Table 1 that the displacements obtained matched very closely (within 3% error) with those observed in the experiment, indicating the effectiveness of the proposed method and the robustness of the crack element. 7 FIG. 8. Curvature along the length of beam (experimental) RESIDUAL LIFE ASSESSMENT: FATIGUE FRACTURE PHENOMENON The fourth and nal stage of damage detection algorithm is to assess the residual life of the structure depending upon the present damaged state of the members. Residual life prediction for concrete structures broadly de nes the available lifetime; the structure can still withstand an appreciable load under the presence of cracks. The assessment also requires that either the damage be detected before it has developed to a dangerous size or load be restricted in such a way, that the crack will never reach a critical size causing failure. Very seldom does a fracture occur due to an unforeseen overload on the undamaged structure. Usually, it is caused by a structural aw or a crack, which develops and grows to a critical size, due to repeated or sustained normal service loads. For most of the civil engineering structures, the external load is of varying nature. Under the action of varying load, failure is governed by fatigue fracture phenomena for majority of structures. In case of metals, fatigue is a well-understood phenomenon, causing an irreversible, internal material damage (Broek 1978), whereas for concrete structures, fatigue is a complex phenomenon, due to its heterogeneous property. Fatigue behaviour of concrete depends on a number of parameters, such as grade of concrete (i.e. tensile/compressive strength), fracture properties of concrete (fracture toughness, characteristic length), relative size of the structures(Bazant and Kangming 1991),(Bazant and Schell 1993), frequency of the externally applied load cycle and the loading history (Broek 1978). Based on linear elastic fracture mechanics concepts, the fatigue crack propagation law proposed by Slowik et. al. (Slowik et al. 1996) includes all of the above-mentioned parameters, except the frequency of externally applied load and is described by, da KImax m KI n =C + F (a, ) dN (KIC KIsup )p (11) where KIsup is the maximum stress intensity factor ever reached by the structure in its past 8 loading history, KIC the fracture toughness, m, n, p, are constants. These constants m, n, and p are determined by Slowik et. al. through an optimization process using the experimental data to be 2.0, 1.1, and 0.7 respectively. Discussion on Parameter C The parameter C in empirical equation 11 basically gives a measure of crack growth per load cycle(Kumar 1999). In concrete members, this parameter indicates the crack growth rate for a particular grade of concrete and is also size dependent. Slowik et. al. (Slowik et al. 1996) have determined the value of C to be equal to 9.5 10 3 and 3.2 10 2 mm/cycle for small and large size specimen respectively. It should be noted here that the stress intensity factor is expressed in M N m 3/2 . These values were determined for a particular loading frequency. Since the parameter C gives an estimation of crack propagation rate in fatigue analysis, it should also depend upon the frequency of loading. Further the fatigue crack propagation takes place primarily within the fracture process zone and hence C should be related to the relative size of the fracture process zone, which itself is related to characteristic length. Therefore, C should depend on the characteristic length lch and ligament length L, where lch = EGf ft 2 (12) Here E is the elastic modulus of concrete, ft is the tensile strength of the concrete and Gf is the speci c fracture energy. According to Slowik (Slowik et al. 1996) a linear relationship exists between parameter C and the ratio of ligament length to characteristic length, and is expressed as, L C = 2 + 25 10 3 mm/cycle (13) lch This equation does not account for the frequency of fatigue loading. Hence, we have modi ed this equation to account for the loading frequency. This is done through a regression analysis, using experimental results of different investigators and hence accounting for the loading frequency. Figure 9 shows a plot of C times f (Cf ) versus the ratio of ligament to characteristic length ( lL ), where f is the frequency of loading. In this plot experimental results of Slowik et. ch al. (Slowik et al. 1996) and Bazant and Kangming (1991) have been used for normal strength concrete with load frequencies 3 and 0.033 Hz. respectively, for small, medium, and large size specimens. The resulting best t curve represents a quadratic polynomial given by, Cf = 0.0193 L lch 2 + 0.0809 L lch + 0.0209mm/sec (14) From this equation one can obtain the value of parameter C for any load frequency, grade of concrete and size. Effect of overload Unlike in metals wherein an overload causes an increase in the plastic zone size thus retarding the rate of subsequent crack growth, in concrete the size of process zone is increased due to overload and the rate of crack propagation also increases suddenly. In Equation 11 the function F (a, ) describes the sudden increase in equivalent crack length due to an overload. It should be noted that the function F (a, ) is not directly related to fatigue, it only takes care of the structural response due to overloads. Based on a nonlinear interpretation, Slowik et. al. (1996) concluded that overloads cause a sudden propagation of the ctitious crack tip. 9 FIG. 9. Relation between Cf and the ratio of ligament length to characteristic length They obtained the values of function F (a, ), for certain specimen geometries, by unloading and reloading at several load levels in the prepeak region and calculating the equivalent crack lengths from the corresponding compliances. No closed-form equation for computing F (a, ) was developed by them. In this work, we have developed an analytical expression to compute the sudden increase in crack length due to overloads. Since the rate of crack propagation due to overload depends on inherent property of concrete and stress amplitude, function F accounting for these is proposed as KI a (15) F= KIC where, KI , indicates the instantaneous change in stress intensity factor, from the normal load cycle, to certain overload cycle, i. e. KI = KI overload KI normal load (16) Here, KI overload represents the maximum stress intensity factor due to overload, and KI normal load is the maximum stress intensity factor due to normal load, just before the overload. a is the crack length, the structure has reached and KIC is the fracture toughness of the concrete. Validation of the proposed fatigue model The proposed fatigue law given by Equation 11 together with Equations 14 and 15 are validated using the experimental results of Slowik et. al. for small and large specimens and Bazant and Kangming s results for small, medium and large specimens. Compact tension specimens were used by Slowik, whereas three point bend specimens were used by Bazant and Kangming. Figure 10 shows the fatigue crack propagation curve of relative crack depth (a/d) versus number of load cycles (N ) for three point bend specimens of Bazant and Kangming. 10 FIG. 10. Fatigue crack propagation curves for three point bend specimens Analytical results using the proposed model are also plotted on the same graph, for small, medium, and large specimens. The results seem to match well with the experimental ones. Figures 11 and 12 show the fatigue crack propagation curves of crack length (a) versus the number of load cycles (N ) for the experimental results reported by Slowik et. al. for their compact tension small and large sized specimens, along with the proposed fatigue law. Again, a good agreement is seen with the experimental results, thus validating the proposed law. Residual life assessment The objective of this study is to predict the remaining life of a structural member at any instant during its service. This is ful lled through a fatigue life assessment by determining the number of loading cycles required for a dominant crack to become unstable and causing fracture. As the crack propagates, the stress intensity factor gradually approaches the fracture toughness of the material. In this study, a linear elastic fracture mechanics criteria is used to determine unstable crack propagation. Accordingly, a crack becomes unstable, when the mode I stress intensity factor reaches the fracture toughness of the material. At this point, the crack propagation curve becomes asymptotic as shown in Figures 10, 11 and 12. Thus, as mentioned before one can predict the residual fatigue life of the structure, in terms of the number of load cycles, the structure can still withstand, before fracture failure occurs. CONCLUSIONS In this study an analytical model is proposed for identifying, locating and quantifying damage using an inverse approach. It is seen that the curvatures would give a fair idea about the location of damage in beams. The compatibility equation obtained using Grobner bases serves as a means to identify damage from eld data of displacements. The crack beam element used in the nite element analysis and formulated using the concepts of fracture mechanics gives good results for quantifying damage in the form of discrete cracks. Further the residual life of 11 FIG. 11. Fatigue crack propagation curve for compact tension specimens (G05, G16, G18) FIG. 12. Fatigue crack propagation curve for compact tension specimens (G06, G07, G13, G15, G16) 12 a crack member is estimated using the concepts of linear elastic fracture mechanics through a fatigue crack propagation law. This law describes the effect of overloads on the rate of crack propagation. It is shown that the proposed law agrees well with the experimental results of different investigators for varying specimen sizes and loading frequencies. REFERENCES Bazant, Z. and Kangming, X. (1991). Size effect in fatigue fracture of concrete. ACI Materials Journal, 88(4), 427 437. Bazant, Z. and Schell, W. (1993). Fatigue fracture of high strength concrete and size effect. ACI Materials Journal, 90(5), 472 478. Broek, D. (1978). Elementary Engineering Fracture Mechanics. Kluwer Academic Publishers. Brown, W. and Srawley, J. (1966). Plane strain crack toughness testing of high strength metallic materials,. Report No. ASTM 513, American Society for Testing and Materials, Philadelphia. FRANC-2D (1998). A Two-Dimensional Fracture Mechanics Analysis Code. Cornell Fracture Group, Cornell University. Ioakimidis, N. (1994). Symbolic computations for the solution of inverse/design problems with maple. Computers and Struc., 53, 63 68. Ioakimidis, N. (1998). Application of computer-generated nite-difference equations to decision and inverse problems in elasticity. Computers and Struc., 68, 529 541. Kumar, P. (1999). Elements of Fracture Mechanics. Wheeler Publishing. Slowik, V., Plizzari, G., and Saouma, V. (1996). Fracture of concrete under variable amplitude loading. ACI Materials Journal, 93(3), 272 283. Tharp, T. (1987). A nite element for edge-cracked beam columns. Int Jl. for Numerical Methods in Engg., 24, 1941 1950. 13
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