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Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams
Mechanical Systems and Signal Processing 25 (2011) 630–643
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Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams Maosen Cao a,c, Lin Ye a,b,n, Limin Zhou a, Zhongqing Su a, Runbo Bai c a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China Laboratory of Smart Materials and Structures (LSMS), Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW2006, Australia c Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, People’s Republic of China b
a r t i c l e i n f o
abstract
Article history: Received 12 November 2009 Received in revised form 17 June 2010 Accepted 28 June 2010 Available online 3 July 2010
Fundamental mode shape and static deflection are typical features frequently used for identification of damage in beams. Regarding these features, an interesting question, still pending, is which one is most sensitive for use in damage identification. The present study addresses the key sensitivity of these features for damage identification in cantilever beams, wherein these features are extremely similar in configurations. The intrinsic relation between the fundamental mode shape and static deflection is discussed, and in particular, an explicit generic sensitivity rule describing the sensitivity of these features to damage in cantilever beams is proposed. The efficiency of this rule in identifying damage is investigated using Euler–Bernoulli cantilever beams with a crack. The validity of the approach is supported by three-dimensional elastic finite element simulation, incorporating the potential scatter in actual measurements. The results show that the generic sensitivity rule essentially provides a theoretical basis for optimal use of these features for damage identification in cantilever beams. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Fundamental mode shape Static deflection Crack modeling Damage identification Cantilever beams
1. Introduction Damage identification in beam-like structures has been a basic research topic in structural health monitoring for the late decades [1]. Various algorithms have been derived for damage diagnosis in a cantilever beam [2,3]. In studies of damage identification in cantilever beams, fundamental mode shape is the most typical dynamic property [3,4–9] employed for damage localization and quantification. The extensive use of fundamental mode shape in damage identification is, to a large extent, attributed to its good sensitivity, reliability and relative convenience in experimental acquisition using a standard modal testing method. Although the higher mode shapes may theoretically be more sensitive to small damage, difficulties in acquisition considerably decrease their practicability in damage diagnosis. Parallel to fundamental mode shape, static deflections including deflection under tip-concentrated loading and deflection under uniformly distributed loading are alternative simple properties [10–14] for damage identification in cantilever beams. Hereafter, for conciseness in statement, fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading are symbolized as FMS, dtcl and dudl, respectively.
n Corresponding author at: Laboratory of Smart Materials and Structures (LSMS), Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW2006, Australia. Tel.: +61 02 93514798; fax: + 61 02 93517060. E-mail address: [email protected] (L. Ye).
0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.06.011
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For damage identification in cantilever beams based on FMS, dtcl or dudl, a key question behind the popular algorithms is which of these characteristic properties is most sensitive for use in damage identification in a cantilever beam. This issue presents a challenge for the optimal uses of these properties for damage identification. In this study, sensitivity analyses of FMS, dtcl and dudl for crack identification in cantilever beams are investigated, using analytical models in conjunction with a three-dimensional finite element method. A generic sensitivity rule characterizing these features in cantilever beams is developed. The outcomes are beneficial for optimal selection of these features for crack identification in cantilever beams. 2. Formulation The free vibration of a cantilever beam with given initial displacement and velocity can be found by superposing contributions from each mode as follows [15]: yðx,tÞ ¼
1 X
Yn ðxÞðAn cos on t þ Bn sin on tÞ
ð1Þ
n¼1
where on and Yn (in Appendix A) are the nth order natural frequency and mode shape, respectively, and coefficients An and Bn are in the following form: RL RL y0 ðxÞYn ðxÞdx v0 ðxÞYn ðxÞdx , Bn ¼ 0 R L ð2Þ An ¼ 0 R L 2 2 0 Yn ðxÞdx 0 Yn ðxÞdx with y0 and v0 being initial displacement and velocity, respectively. As the discussion should be independent of the initial velocity in this study, v0 = 0 with Bn = 0 for all n [15] are adopted in the discussion. Essentially, 9An9 or 9Bn9 quantitatively signifies the contribution of the nth mode shape Yn to displacement y. An index, RAn, can be established as follows: 9An 9 1 P 9Am 9
RAn ¼
ð3Þ
m¼1
which reflects the contribution ratio of the nth mode shape Yn to the displacement y. The specific dtcl and dudl in Fig. 1, Yc and Yd, with the subscripts ‘c’ and ‘d’ identifying ‘concentrated’ and ‘distributed’ loadings, respectively, expressed in Appendix A, are taken as initial displacements, respectively. According to Eq. (2), substitutions of y0 =Yc and y0 = Yd into An individually, one obtains coefficients Acn and Adn , respectively, as follows: Acn ¼
4PL3 Hc1 ðgn Þ 3CEI Hc2 ðgn Þ
ð4-aÞ
Adn ¼
qL4 Hd1 ðgn Þ 2CEI Hd2 ðgn Þ
ð4-bÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where E is the elastic modulus, I the second moment of inertia, C the arbitrary constant in Yn, and gn ¼ L 4 o2n rA=EI with r, A and L being the density of material, the cross-sectional area and the length of the beam, respectively. Hc1 ðgn Þ, Hc2 ðgn Þ, Hd1 ðgn Þ and Hd2 ðgn Þ are, respectively, detailed as Hc1 ðgn Þ ¼ ðsin gn þ sinh gn Þðg3n þcosh gn ðg3n cos gn 3sin gn Þ þ3 cos gn sinh gn Þ, Hc2 ðgn Þ ¼ g3n ½3 sinh2gn cos2 gn þ 6 sinh gn cos gn gn cos 2gn 3 cosh gn ð2 sin gn þcosh gn sin 2gn Þ þ gn ðcosh 2gn þ 4 sin gn sinh gn Þ,
Hd1 ðgn Þ ¼ fg4n 8cos gn þ ½ðg4n þ 8Þcos gn 8cosh gn þ 8gðsin gn þsinh gn Þ, Hd2 ðgn Þ ¼ g4n ½3 sinh2gn cos2 gn þ 6 sinh gn cos gn gn cos 2gn 3cosh gn ð2 sin gn þ cosh gn sin 2gn Þ þ gn ðcosh 2gn þ 4sin gn sinh gn Þ:
Fig. 1. Static deflections of a cantilever beam. (a) Tip-concentrated loading and (b) uniformly distributed loading.
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Table 1 Contributions of first five mode shapes to vibration displacement. Mode order
Type of initial displacement Tip-concentrated loading
First Second Third Fourth Fifth
Uniformly distributed loading
Acn
RAcn ð%Þ
Adn
RAdn ð%Þ
0.9707 0.0247 0.0032 0.0008 0.0003
97.08 2.472 0.315 0.082 0.030
1.0134 0.0143 0.0011 0.0002 0.0001
98.42 1.389 0.104 0.019 0.005
Note: Acn andAdn are calculated using the normalized static deflections in Appendix A.
Fig. 2. Comparison between static deflection and fundamental mode shape. (a) FMS versus dtcl and (b) FMS versus dudl.
Thus the relative contribution ratios, RAcn and RAdn , can be defined as RAcn ¼
9Acn 9 9fc ðgn Þ9 ¼ 1 1 P P c 9Am 9 9fc ðgm Þ9
m¼1
RAdn ¼
m¼1
9Adn 9 1 P m¼1
ð5-aÞ
9Acm 9
¼
9fd ðgn Þ9 1 P 9fd ðgm Þ9
ð5-bÞ
m¼1
where fc ¼ Hc1 =Hc2 and fd ¼ Hd1 =Hd2 . RAcn or RAdn is a constant as a result of the constant gn in Hc1 , Hc2 , Hd1 and Hd2 , and these constants are irrelevant to material properties, geometrical dimensions and magnitude of external loads. Herein, the first 20 mode shapes are calculated, from which the RAcn and RAdn for the first five mode shapes, with respect to the two loading cases in Fig. 1, are defined according to Eq. (5), listed in Table 1. Clearly, the relative contribution ratios, RAc1 and RAd , of FMS is nearly two orders of magnitude 1 higher than those of the second mode and the others, being dominant in the dynamic displacement. As a consequence, the static deflections, dtcl and dudl, in Fig. 1 closely resemble the FMS, as depicted in Fig. 2. This implies that for damage identification in cantilever beams, the FMS acquired by dynamic testing extremely approximates dtcl and dudl, and vice versa.
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3. Generic sensitivity Curvature has often been used as a feature for effectively quantifying damage in beam-type structures [16]. The curvatures derived from dtcl, Yc, dudl, Yd and FMS, Y1 (in Appendix A), are
jYc ¼
d2 Yc P ¼ ðxLÞ EI dx2
ð6-aÞ
jYd ¼
d2 Yd q ðLxÞ2 ¼ 2EI dx2
ð6-bÞ
jY1 ¼
d2 Y1 2 ¼ C l1 ðcos l1 x þ cosh l1 xÞ þ0:7341ðsin l1 x þsinh l1 xÞ dx2
ð6-cÞ
In Eq. (6), P, q and C are the parameters that are responsible for a global magnitude of individual curvatures. Normalization of curvature magnitude is necessary for reflecting the sensitivity of the three deformation features to damage in a quantitatively comparable manner. The following normalized forms, YcN , YdN and Y1N , with beam tip deformation being unity, Fig. 3(a), are used in the subsequent discussion: YcN ¼
x2 ð3LxÞ 2L3
ð7-aÞ
YdN ¼
x2 ð6L2 4Lx þ x2 Þ 3L4
ð7-bÞ
Y1N ¼
1 ðcos l1 xcosh l1 xÞ0:7341ðsin l1 xsinh l1 xÞ 2
ð7-cÞ
The curvatures resulting from Eq. (7) for dtcl, dudl and FMS are expressed as follows and as shown in Fig. 3(b)
jc ¼
3ðLxÞ L3
ð8-aÞ
jd ¼
4ðLxÞ2 L4
ð8-bÞ
1 2
j1 ¼ l21 ðcos l1 x þ cosh l1 xÞ þ0:7341ðsin l1 x þ sinh l1 xÞ
ð8-cÞ
where the asterisk marks the derivation based on the normalized deformation. The points of intersection marked by K1, K2 and K3 in the curvature curves in Fig. 3(b) are obtained through solving different combination of two simultaneous equations in Eq. (8): xK1 ¼ 0:2058L,
xK2 ¼ 0:2500L,
xK3 ¼ 0:2866L
Fig. 3. Normalized FMS, dtcl, and dudl (a) and their curvatures (b).
ð9Þ
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Thus, the curvature of the beam is divided into four segments: 0 o x o xK1 , xK1 o x oxK2 , xK2 o x o xK3 and xK3 ox o L. The sensitivity of the three deformation features with respect to damage is dependent on the segment. For example, for an arbitrary value of x within xK3 ox o L, we have
jd ðxÞ o j1 ðxÞ o jc ðxÞ
ð10Þ
Owning to the constitutive equation of the Euler–Bernoulli beam, M =EIj, the following relation is obtained via multiplying each term in Eq. (10) by EI Md ðxÞ o M1 ðxÞ o Mc ðxÞ
ð11Þ
Assuming that a damage event occurs at location x=xt and it alters the local flexural stiffness from EI(xt) to EIðxt Þ ¼ ð1KÞEIðxt Þ with 0 o K o 1, variations in curvature at the site of damage from the initial intact state are formulated as 1 1 K ¼ Md ðxt Þ ð12-aÞ Djd ðxt Þ ¼ Md ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
1 1 K ¼ M1 ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
ð12-bÞ
1 1 K ¼ Mc ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
ð12-cÞ
Dj1 ðxt Þ ¼ M1 ðxt Þ
Djc ðxt Þ ¼ Mc ðxt Þ
where L/(1 L)EI(xt) is a constant. Based on Eqs. (11) and (12), variations in curvature at the site of damage from the initial intact state are obtained as the following first inequality:
Djd ðxt Þ o Dj1 ðxt Þ o Djc ðxt Þ, Djc ðxt Þ o Dj1 ðxt Þ o Djd ðxt Þ, Djc ðxt Þ o Djd ðxt Þ o Dj1 ðxt Þ, Djd ðxt Þ o Djc ðxt Þ o Dj1 ðxt Þ,
xK3 o xt o L 0 o xt oxK1 xK1 o xt o xK2
ð13Þ
xK2 o xt o xK3
Then, the relations of the damage-induced variation in curvature to the other three segments can be elaborated using the rest of Eq. (13). In Eq. (13), the magnitude of Dj quantitatively characterizes the sensitivity of the curvature to damage in a cantilever beam with respect to each deformation feature. Thus, a generic sensitivity rule can be established as follows: for damage location xt 2 ð0 xK1 Þ, the degree of sensitivity to the crack for three deformation features is ranked with dudl first, FMS second and dtcl third; for xt 2 ðxK1 xK2 Þ, FMS first, dudl second and dtcl third; for xt 2 ðxK2 xK3 Þ, the ranking is in the order FMS, dtcl and dudl; and for xt 2 ðxK3 LÞ, dtcl has the greatest sensitivity, followed by FMS and dudl. Moreover, in the narrow interval ðxK1 ¼ 0:2058L, xK3 ¼ 0:2866LÞ, the sensitivity of the different features becomes less distinguishable.
Fig. 4. Cracked cantilever beam models with a single-edge crack. (a) A cantilever beam with a single-edge crack, (b) a two-segment model with a massless rotational spring, (c) tip-concentrated loading and (d) uniformly distributed loading.
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4. A cracked beam model A single-edge Mode I crack in a cantilever beam, as depicted in Fig. 4, is commonly evaluated as a massless rotational spring [17–19] in analytical analysis of the beam. The equivalent stiffness of the rotational spring, KT, obtained from the crack strain energy function and Castigliano’s theorem, can be expressed in the following form, as deduced in Appendix B: KT ¼
6p
Ra 0
EIh af 2 ða=hÞda
ð14Þ
where b and h are the width and height of the beam, a is the depth of the crack, and f(a/h) is the geometric function [20] given as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2h pa 0:923 þ 0:199ð1sin ðpa=2hÞÞ4 ¼ tan f h 2h pa cosðpa=2hÞ The presence of the crack is represented by an additional rotation, yn(Lc)= M(Lc)/KT with M(Lc)= EId2YI/(Lc)/dx2 at the crack location (x =Lc), resulting in a discontinuity in the slope of the beam. The transition of slope is consequently expressed as [21] dYI ðLc Þ EI d2 YI ðLc Þ dYII ðLc Þ þ ¼ dx KT dx2 dx
ð15Þ
where YI and YII are the amplitudes of the flexural deformation of the beam segments I and II, respectively. 4.1. Dynamic response Based on the Euler–Bernoulli beam theory, the governing equations of flexural vibration of beam segments I and II, Fig. 4(b), are expressed as d4 YI 4 ln YI ¼ 0; dx4
0 r x rLc
ð16-aÞ
d4 YII 4 ln YII ¼ 0; Lc r x rL dx4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ln ¼ 4 on 2 rA=EI The general solutions to Eq. (16) can be written as Yin ðxÞ ¼ Bi,1 cos ln x þBi,2 sin ln x þ Bi,3 cosh ln x þ Bi,4 sinh ln x
ð16-bÞ
ði ¼ I,IIÞ
ð17Þ
where BI,j and BII,j with j =1,2,3,4 are constants to be determined from the boundary and continuity conditions. The boundary conditions at the clamped and free ends are specified as YI ð0Þ ¼ 0,
dYI ð0Þ ¼ 0, dx
d2 YII ðLÞ ¼ 0, dx2
d3 YII ðLÞ ¼0 dx3
ð18Þ
Eq. (15) gives the compatibility in slope at the crack location; moreover, the continuity conditions of the displacement, bending moment and shear force are stated as YI ðLc Þ ¼ YII ðLc Þ,
d2 YI ðLc Þ d2 YII ðLc Þ ¼ , dx2 dx2
d3 YI ðLc Þ d3 YII ðLc Þ ¼ dx3 dx3
ð19Þ
An 8 8 matrix of coefficients for the eight constants (Bi,j, i= I, II and j = 1,2,3,4) can be constructed from the general solution in conjunction with the boundary and continuity conditions (Eqs. (15), (18) and (19)). The associated set of simultaneous equations with respect to the eight constants will have nontrivial solutions only if the determinant of the coefficients becomes zero. Expansion of the determinant leads to the frequency equation, from which the natural frequencies can be numerically determined. The natural frequencies prompt the solution of the constants in Eq. (17). 4.2. Static models dtcl in Fig. 4(c) for the cracked cantilever beam is obtained from a particular bending equation EId4Yc/dx4 =0 considering the boundary conditions Yc;I ð0Þ ¼ 0,
Yc;I u ð0Þ ¼ 0,
00 Yc;II ðLÞ ¼ 0,
EIYc;II u00 ðLÞ ¼ P
ð20Þ
with the prime denoting differentiation, and compatibility conditions including Eq. (15) with the shear effect neglected, and Yc;I ðLc Þ ¼ Yc;II ðLc Þ
ð21Þ
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Fig. 5. Numerical model of a cantilever beam with a single-edge crack at Lc = 0.1L. (a) Finite element mesh, (b) zoomed-in crack.
The resultant solution is in the form: Yc;I ðxÞ ¼ ðx3LÞ=6EI; Yc;II ðxÞ ¼ P½6EIðLLc ÞðLc xÞ þ K T x 2 ðx3LÞ=6EIKT ;
0 r x r Lc Lc r x r L
ð22Þ
where KT is given in Eq. (15). dudl in Fig. 4(d) for the cracked cantilever beam is obtained from EId4Yd/dx4 = q considering zero shearing force at x =L and other boundary and compatibility conditions similar to those for the tip-concentrated loading case. The resultant solution has the form: Yd;I ðxÞ ¼
Yd;II ðxÞ ¼
qx2 ð6L2 4Lx þx2 Þ , 24EI
0 r x r Lc
q½12EIðLLc Þ2 ðLc xÞKT x2 ð6L2 4Lx þx2 Þ , 24EIKT
ð23-aÞ
Lc rx rL
ð23-bÞ
4.3. Numerical simulation A three-dimensional elastic finite element simulation is performed to further validate the generic sensitivity rule. The beams are modeled using 20-node 3D structural solid elements (SOLID 95) in the commercial software ANSYSs. In particular, the single-sided crack is modeled via pairs of coincident nodes with coordinates identical to those of adjoining elements. An illustration of the finite element mesh of a cracked beam is shown in Fig. 5. dtcl with P= 100 N, dudl with q= 100 kN/m2 and FMS are evaluated, followed by a crack identification procedure implemented on the acquired dynamic and static deformation responses. 5. Results and discussion A cracked cantilever steel beam is considered with dimensions: length (L)=0.30 m, width (b)=0.02 m, and depth (h)=0.02 m, and Young’s modulus (E), Poisson’s ratio (v), and mass density (r) are taken as 206 GPa, 0.3, and 7850 kg m 3, respectively. Four crack scenarios of Lc =0.1, 0.25, 0.5 and 0.75L from the clamped end with a/h=0.15 for analytical models and a/h=0.3 for numerical simulation are considered. The adoption of larger a/h for numerical simulation means to offer a clearer graphic presentation of the sensitivity of FMS, dtcl and dudl to a crack under considerations of modeling and simulation error. 5.1. Sensitivity For the scenario with the crack located at 0.5L, Fig. 6(a) presents the resultant three deformation features, with the slopes in Fig. 6(b) and the curvatures in Fig. 6(c), arising from the first-order difference and the second-order central difference, respectively. The effect of the crack on the slope is barely visible, with only a tiny increase in the local slope at the crack location, which cannot be used as an effective damage signature, especially taking into account the presence of noise and systematic errors. However, a sharp singular peak in the curvature makes the crack clearly identifiable. The height of bars in Fig. 6(c) denotes the magnitude of the singular peak for curvature. Clearly, the magnitudes of the singular peak for curvature are ranked with dtcl first, FMS second and dudl third. Since the magnitude of the singular peak indicates the degree of sensitivity to the crack, this ranking implies that when the crack is at x= 0.5L, dtcl is most sensitive to the
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Fig. 6. Curvatures of FMS, dtcl and dudl with a crack at Lc =0.5L. (a) Deformations for Lc = 0.5L, (b) slopes for Lc = 0.5L and (c) curvatures for Lc = 0.5L.
presence of the crack, followed by FMS and dudl. Fig. 7 depicts the curvatures for the scenarios when the crack is located at 0.1, 0.25 and 0.75L, respectively. For the crack located at 0.1L (Fig. 7(a)), the magnitude of the singular peak for dudl is greater than that of FMS and dtcl, respectively. This implies that dudl is most sensitive to the presence of the crack, followed by FMS and dtcl. Similarly, the ranking of sensitivity for the crack located at 0.25L can be observed in Fig. 7(b), in the order of FMS, dudl and dtcl. In addition, the ranking of sensitivity for the crack located at 0.75L is the same as that for the crack located at 0.5L, because the two crack scenarios occur in the same segment of the beam. For the numerical simulation, one-dimensional LoG (Laplacian of Gaussian) [22], rather than the second-order central difference, is employed to acquire the curvature from deformation responses with greater accuracy. This operator is expressed as ðx2 s2 Þ
cðxÞ ¼ pffiffiffiffiffiffi
2 2ps5
2
eðx
=2s2 Þ
ð24Þ
where s is the standard variance of normal distribution. In essence, the one-dimensional LoG, as shown in Fig. 8, is really a Gaussian wavelet of two-order vanishing moments. The features of convolution manipulation, smoothing, and flexibility in choosing s to match the signal under investigated entails it a more versatile differential operator than the second-order central difference. In the present analysis, a particular LoG with s = 1.8 (Fig. 8) is optimally chosen as the operator. The distortion near the end of beam induced by the convolution is treated by a cubic spline extrapolation, as stated in [3]. For a crack located at 0.1L from the clamped end, the curvatures from the normalized deformation responses with normalization manipulation similar to that adopted in Eq. (7), are shown in Fig. 9(a). It can be observed that the magnitudes of the singular peaks in curvature are ranked in the order of dudl, FMS, and dtcl. This is the same result as that obtained from the theoretical analysis, as shown in Fig. 7(a). The curvatures for the other scenarios with the crack located at Lc = 0.25, 0.5 and
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Fig. 7. Comparison of curvatures of FMS, dtcl and dudl for a crack at Lc = 0.1 (a), 0.25 (b), 0.75L (c), respectively.
Fig. 8. One-dimensional Laplacian of Gaussian (LoG).
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Fig. 9. Comparison of curvatures of FMS, dtcl and dudl for a crack at Lc = 0.1 (a), 0.25 (b), 0.5 and 0.75L (c), respectively.
0.75L, respectively, are shown in Fig. 9(b)–(d). The rankings of the magnitudes of the singular peaks in curvature for the crack located at 0.5 and 0.75L, are firstly dtcl, secondly FMS and thirdly dudl, which is consistent with the rankings obtained from the theoretical analysis (Figs. 6(c) and 7(c), respectively). For the crack located at 0.25L, the discrepancy in magnitude of the singular peak of curvature between the three deformation features is relatively small and the sensitivity of three deformation features to the crack is almost identical. Thus the results yielded by both theoretical analyses and numerical simulation fall into the scope of the generic sensitivity rule, validating the generic sensitivity rule well.
5.2. Effects of scatter in measurement In the actual measurement of FMS, dtcl, and dudl, there is always some scatter in the data because of random noise and systematic error. To assess the influence of potential noise, the actual measurements are mimicked by adding Gaussian white noise to these deformation responses obtained from the finite element simulation. The noise level is defined by the signal-to-noise ratio (SNR), SNR =20 log10(As/An), with As and An denoting the root-mean-square (RMS) magnitude of the simulated deformation response and added noise, respectively. The Monte Carlo method [23] is employed to simulate the deformation responses taking into account noise. In the analysis, the LoG operator (Eq. (24)) is applied to acquire curvatures from the mimicked responses. A correlation coefficient between the local portions of curvatures with and
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Fig. 10. Curvatures from dtcl for signifying effective (a) and ineffective (b) identification of the crack located at Lc = 0.1L.
Fig. 11. Curvatures from FMS for signifying effective (a) and ineffective (b) identification of the crack located at Lc = 0.1L.
Fig. 12. Curvatures from dudl for signifying effective (a) and ineffective (b) identification of the crack located at Lc =0.1L.
without noise is adopted as a measure for assessing the impact of the noise. The correlation coefficient adopted is expressed as Pn i ¼ 1 ðUi UÞðVi VÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð25Þ Pn Pn 2 2 i ¼ 1 ðUi UÞ i ¼ 1 ðVi VÞ where U and V are the local curvatures with a window size of 0.1L (covering 30 nodal points of the FE model) around the crack location for the intact and noisy responses, respectively. A value of correlation coefficient below 0.5 is used to
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Table 2 Effects of random noise on three deformation features for crack identification. SNR
100 90 85 80 75 70
[R0.1L(%), RL(n)]
[2.385, 0.351] [8.858, 1.302] [13.355, 1.960] [30.867, 4.531] [35.637, 5.234] [101.596,14.910]
S
dtcl
FMS
dudl
Effective (%)
Ineffective (%)
Effective (%)
Ineffective (%)
Effective (%)
Ineffective (%)
97.96 93.10 88.25 79.20 64.25 41.09 77.31
2.04 6.90 11.75 20.80 35.75 58.91 22.69
98.28 93.61 88.58 80.79 66.39 45.91 78.93
1.72 6.39 11.42 19.21 33.61 54.09 21.07
98.03 94.23 88.72 80.99 66.3 45.74 79.00
1.97 5.77 11.28 19.01 33.7 54.26 21.00
identify the threshold when the noise masks the presence of the crack. For the crack at 0.1L and the SNR =90, the influence of uncertainty on the identification of the crack is illustrated in Figs. 10–12, in which two sets of 200 profiles of curvature along the beam axis are presented for each deformation feature. In the figures, similar profiles of curvature with respect to three deformation features exist for either the effective or ineffective crack identification situation, and the discrepancy is almost indiscernible. To comprehensively investigate the impact of noise level, 10,000 noisy deformation responses for each deformation feature at one noise level are further constructed using the Monte Carlo method in order to approach a stable statistic character for the impact of noise. The noise levels of SNR =100, 90, 85, 80, 75 and 70 are considered. In Table 2 the percentages of effective and ineffective crack identification for each deformation feature at each noise level are evaluated and listed, where R0.1L and RL denote, at a noise level, the ratio of the mean magnitude of noise to the magnitude of deformation responses at 0.1L and L of the beam, respectively. For SNR =80, approximately 80% successful identification of the crack was achieved for each deformation feature, which may be regarded as an acceptable boundary, below which the efficiency of crack identification deteriorates markedly and has little reliability. At such a level of signal-to-noise ratio, the mean noise magnitude approximately equals 3.1% of the deformation response magnitude at the 0.1L of the beam, which provides a prior precision criterion for the optimal selection of deformation features from a perspective of experimental measurement. For conventional linear variable differential transformer (LVDT)-based measurement methods, dtcl is potentially the most feasible to meet this precision requirement, due to convenience in measurement, whereas use of FMS is probably more difficult because of the relatively complex procedure involved in the modal analysis. Advanced optical or electromagnetic spectrum-based measurement apparatus, such as a laser vibrometer, may provide the opportunity to meet this precision requirement in actual measurement of all deformation features. Clearly, all the deformation features are able to identify the crack for the SNRZ80, but their capabilities decrease significantly when SNRr75. At the level of SNR =70, these features are almost inadequate to act as signatures for crack identification. In Table 2, the average effective and ineffective ratios for each deformation feature imply that the impact of scatter is most significant for dtcl, followed by FMS and ducl. In other words, in the case of a cantilever beam with a crack, ducl has greatest immunity to noise, followed by FMS and dtcl. This result coincides with that for the sensitivity of the three deformation features to the presence of the crack. The finding that higher sensitivity corresponds to greater immunity to noise also applies to the other crack cases with the crack at 0.25, 0.5, and 0.75L, respectively. 6. Conclusions The most typical dynamic and static deformation features used in various algorithms for the detection of damage in cantilever beams are fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading. A fundamental question behind these popular algorithms is which is most sensitive for use in damage identification. With the Euler–Bernoulli beam theory and finite element simulation, the sensitivity of these features to damage in cantilever beams is comprehensively investigated in this study, and a generic sensitivity rule characterizing the sensitivity characteristics of the three deformation features to an edge crack in a cantilever beam is developed, addressing the potential scatter of actual measurement of these features. This generic sensitivity rule holds promise for the rational and optimal use of fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading for damage detection in cantilever beams.
Acknowledgements L. Ye, L. Zhou and Z. Su thank the Hong Kong Polytechnic University for a special research grant on ‘‘Advanced Composite and Fundamental Structure’’. L. Zhou also thanks the Research Grant Council of the Hong Kong SAR for the research Grant of PolyU5333/07E. M.S. Cao is grateful for National Natural Science Foundation of China (Grant no. 50978084).
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Appendix A Yc ð x Þ ¼
Px2 ðx3LÞ 6EI
Yd ðxÞ ¼
qx2 ð6L2 4Lx þ x2 Þ 24EI
Yn ðxÞ ¼ C½ðcos ln xcosh ln xÞwn ðsin ln xsinh ln xÞ
ðA1Þ
ðA2Þ ðA3Þ
where C is an arbitrary constant and, wn ¼ ðcos ln L þ cosh ln LÞ=ðsin ln Lþ sinh ln LÞ with lnL= gn being the solutions of cosðgn Þ coshðgn Þ ¼ 1 for an Euler–Bernoulli cantilever beam. Eq. (A3) gives the fundamental model shape when n = 1: Y1 ðxÞ ¼ C½ðcos l1 xcosh l1 xÞw1 ðsin l1 xsinh l1 xÞ
ðA4Þ
with w1 = 0.7341 arising from g1 = 1.87510406871. Appendix B The stress intensity factor of mode I for a single-edge crack due to bending moment is expressed as pffiffiffiffiffiffi a K ¼ KI ¼ s paf h
ðA5Þ
2
where s =6M/bh with M being the bending moment at the cross-section of crack, a is the crack depth, and f(a/h) is the geometric function. If the crack can be represented by a spring, the additional work due to the bending moment equals to the crack strain energy: Z a M2 DU ¼ ¼ Js bda ðA6Þ 2KT 0 where KT is the equivalent stiffness of the rotational spring, and Js ¼ KI2 =E is the strain energy density function in the planestress condition [20]. Substituting Eqs. (A5) into Eq. (A6), KT can be formulated as KT ¼
¼
M2 M2 M2 M2 ¼ Ra 2 ¼ R a s2 paf 2 ða=hÞ ¼ R Ra 2 Þ2 paf 2 ða=hÞ a ð6M=bh 2 0 Js bda 2 0 ðKI =EÞbda 2 0ð Þbda bda 2 0 E E Ebh4 EIh ¼ Ra 6p 0 ðaf 2 ða=hÞÞda 72p 0 ðaf 2 ða=hÞÞda Ra
ðA7Þ
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams Maosen Cao a,c, Lin Ye a,b,n, Limin Zhou a, Zhongqing Su a, Runbo Bai c a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China Laboratory of Smart Materials and Structures (LSMS), Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW2006, Australia c Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, People’s Republic of China b
a r t i c l e i n f o
abstract
Article history: Received 12 November 2009 Received in revised form 17 June 2010 Accepted 28 June 2010 Available online 3 July 2010
Fundamental mode shape and static deflection are typical features frequently used for identification of damage in beams. Regarding these features, an interesting question, still pending, is which one is most sensitive for use in damage identification. The present study addresses the key sensitivity of these features for damage identification in cantilever beams, wherein these features are extremely similar in configurations. The intrinsic relation between the fundamental mode shape and static deflection is discussed, and in particular, an explicit generic sensitivity rule describing the sensitivity of these features to damage in cantilever beams is proposed. The efficiency of this rule in identifying damage is investigated using Euler–Bernoulli cantilever beams with a crack. The validity of the approach is supported by three-dimensional elastic finite element simulation, incorporating the potential scatter in actual measurements. The results show that the generic sensitivity rule essentially provides a theoretical basis for optimal use of these features for damage identification in cantilever beams. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Fundamental mode shape Static deflection Crack modeling Damage identification Cantilever beams
1. Introduction Damage identification in beam-like structures has been a basic research topic in structural health monitoring for the late decades [1]. Various algorithms have been derived for damage diagnosis in a cantilever beam [2,3]. In studies of damage identification in cantilever beams, fundamental mode shape is the most typical dynamic property [3,4–9] employed for damage localization and quantification. The extensive use of fundamental mode shape in damage identification is, to a large extent, attributed to its good sensitivity, reliability and relative convenience in experimental acquisition using a standard modal testing method. Although the higher mode shapes may theoretically be more sensitive to small damage, difficulties in acquisition considerably decrease their practicability in damage diagnosis. Parallel to fundamental mode shape, static deflections including deflection under tip-concentrated loading and deflection under uniformly distributed loading are alternative simple properties [10–14] for damage identification in cantilever beams. Hereafter, for conciseness in statement, fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading are symbolized as FMS, dtcl and dudl, respectively.
n Corresponding author at: Laboratory of Smart Materials and Structures (LSMS), Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW2006, Australia. Tel.: +61 02 93514798; fax: + 61 02 93517060. E-mail address: [email protected] (L. Ye).
0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.06.011
M. Cao et al. / Mechanical Systems and Signal Processing 25 (2011) 630–643
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For damage identification in cantilever beams based on FMS, dtcl or dudl, a key question behind the popular algorithms is which of these characteristic properties is most sensitive for use in damage identification in a cantilever beam. This issue presents a challenge for the optimal uses of these properties for damage identification. In this study, sensitivity analyses of FMS, dtcl and dudl for crack identification in cantilever beams are investigated, using analytical models in conjunction with a three-dimensional finite element method. A generic sensitivity rule characterizing these features in cantilever beams is developed. The outcomes are beneficial for optimal selection of these features for crack identification in cantilever beams. 2. Formulation The free vibration of a cantilever beam with given initial displacement and velocity can be found by superposing contributions from each mode as follows [15]: yðx,tÞ ¼
1 X
Yn ðxÞðAn cos on t þ Bn sin on tÞ
ð1Þ
n¼1
where on and Yn (in Appendix A) are the nth order natural frequency and mode shape, respectively, and coefficients An and Bn are in the following form: RL RL y0 ðxÞYn ðxÞdx v0 ðxÞYn ðxÞdx , Bn ¼ 0 R L ð2Þ An ¼ 0 R L 2 2 0 Yn ðxÞdx 0 Yn ðxÞdx with y0 and v0 being initial displacement and velocity, respectively. As the discussion should be independent of the initial velocity in this study, v0 = 0 with Bn = 0 for all n [15] are adopted in the discussion. Essentially, 9An9 or 9Bn9 quantitatively signifies the contribution of the nth mode shape Yn to displacement y. An index, RAn, can be established as follows: 9An 9 1 P 9Am 9
RAn ¼
ð3Þ
m¼1
which reflects the contribution ratio of the nth mode shape Yn to the displacement y. The specific dtcl and dudl in Fig. 1, Yc and Yd, with the subscripts ‘c’ and ‘d’ identifying ‘concentrated’ and ‘distributed’ loadings, respectively, expressed in Appendix A, are taken as initial displacements, respectively. According to Eq. (2), substitutions of y0 =Yc and y0 = Yd into An individually, one obtains coefficients Acn and Adn , respectively, as follows: Acn ¼
4PL3 Hc1 ðgn Þ 3CEI Hc2 ðgn Þ
ð4-aÞ
Adn ¼
qL4 Hd1 ðgn Þ 2CEI Hd2 ðgn Þ
ð4-bÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where E is the elastic modulus, I the second moment of inertia, C the arbitrary constant in Yn, and gn ¼ L 4 o2n rA=EI with r, A and L being the density of material, the cross-sectional area and the length of the beam, respectively. Hc1 ðgn Þ, Hc2 ðgn Þ, Hd1 ðgn Þ and Hd2 ðgn Þ are, respectively, detailed as Hc1 ðgn Þ ¼ ðsin gn þ sinh gn Þðg3n þcosh gn ðg3n cos gn 3sin gn Þ þ3 cos gn sinh gn Þ, Hc2 ðgn Þ ¼ g3n ½3 sinh2gn cos2 gn þ 6 sinh gn cos gn gn cos 2gn 3 cosh gn ð2 sin gn þcosh gn sin 2gn Þ þ gn ðcosh 2gn þ 4 sin gn sinh gn Þ,
Hd1 ðgn Þ ¼ fg4n 8cos gn þ ½ðg4n þ 8Þcos gn 8cosh gn þ 8gðsin gn þsinh gn Þ, Hd2 ðgn Þ ¼ g4n ½3 sinh2gn cos2 gn þ 6 sinh gn cos gn gn cos 2gn 3cosh gn ð2 sin gn þ cosh gn sin 2gn Þ þ gn ðcosh 2gn þ 4sin gn sinh gn Þ:
Fig. 1. Static deflections of a cantilever beam. (a) Tip-concentrated loading and (b) uniformly distributed loading.
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Table 1 Contributions of first five mode shapes to vibration displacement. Mode order
Type of initial displacement Tip-concentrated loading
First Second Third Fourth Fifth
Uniformly distributed loading
Acn
RAcn ð%Þ
Adn
RAdn ð%Þ
0.9707 0.0247 0.0032 0.0008 0.0003
97.08 2.472 0.315 0.082 0.030
1.0134 0.0143 0.0011 0.0002 0.0001
98.42 1.389 0.104 0.019 0.005
Note: Acn andAdn are calculated using the normalized static deflections in Appendix A.
Fig. 2. Comparison between static deflection and fundamental mode shape. (a) FMS versus dtcl and (b) FMS versus dudl.
Thus the relative contribution ratios, RAcn and RAdn , can be defined as RAcn ¼
9Acn 9 9fc ðgn Þ9 ¼ 1 1 P P c 9Am 9 9fc ðgm Þ9
m¼1
RAdn ¼
m¼1
9Adn 9 1 P m¼1
ð5-aÞ
9Acm 9
¼
9fd ðgn Þ9 1 P 9fd ðgm Þ9
ð5-bÞ
m¼1
where fc ¼ Hc1 =Hc2 and fd ¼ Hd1 =Hd2 . RAcn or RAdn is a constant as a result of the constant gn in Hc1 , Hc2 , Hd1 and Hd2 , and these constants are irrelevant to material properties, geometrical dimensions and magnitude of external loads. Herein, the first 20 mode shapes are calculated, from which the RAcn and RAdn for the first five mode shapes, with respect to the two loading cases in Fig. 1, are defined according to Eq. (5), listed in Table 1. Clearly, the relative contribution ratios, RAc1 and RAd , of FMS is nearly two orders of magnitude 1 higher than those of the second mode and the others, being dominant in the dynamic displacement. As a consequence, the static deflections, dtcl and dudl, in Fig. 1 closely resemble the FMS, as depicted in Fig. 2. This implies that for damage identification in cantilever beams, the FMS acquired by dynamic testing extremely approximates dtcl and dudl, and vice versa.
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3. Generic sensitivity Curvature has often been used as a feature for effectively quantifying damage in beam-type structures [16]. The curvatures derived from dtcl, Yc, dudl, Yd and FMS, Y1 (in Appendix A), are
jYc ¼
d2 Yc P ¼ ðxLÞ EI dx2
ð6-aÞ
jYd ¼
d2 Yd q ðLxÞ2 ¼ 2EI dx2
ð6-bÞ
jY1 ¼
d2 Y1 2 ¼ C l1 ðcos l1 x þ cosh l1 xÞ þ0:7341ðsin l1 x þsinh l1 xÞ dx2
ð6-cÞ
In Eq. (6), P, q and C are the parameters that are responsible for a global magnitude of individual curvatures. Normalization of curvature magnitude is necessary for reflecting the sensitivity of the three deformation features to damage in a quantitatively comparable manner. The following normalized forms, YcN , YdN and Y1N , with beam tip deformation being unity, Fig. 3(a), are used in the subsequent discussion: YcN ¼
x2 ð3LxÞ 2L3
ð7-aÞ
YdN ¼
x2 ð6L2 4Lx þ x2 Þ 3L4
ð7-bÞ
Y1N ¼
1 ðcos l1 xcosh l1 xÞ0:7341ðsin l1 xsinh l1 xÞ 2
ð7-cÞ
The curvatures resulting from Eq. (7) for dtcl, dudl and FMS are expressed as follows and as shown in Fig. 3(b)
jc ¼
3ðLxÞ L3
ð8-aÞ
jd ¼
4ðLxÞ2 L4
ð8-bÞ
1 2
j1 ¼ l21 ðcos l1 x þ cosh l1 xÞ þ0:7341ðsin l1 x þ sinh l1 xÞ
ð8-cÞ
where the asterisk marks the derivation based on the normalized deformation. The points of intersection marked by K1, K2 and K3 in the curvature curves in Fig. 3(b) are obtained through solving different combination of two simultaneous equations in Eq. (8): xK1 ¼ 0:2058L,
xK2 ¼ 0:2500L,
xK3 ¼ 0:2866L
Fig. 3. Normalized FMS, dtcl, and dudl (a) and their curvatures (b).
ð9Þ
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Thus, the curvature of the beam is divided into four segments: 0 o x o xK1 , xK1 o x oxK2 , xK2 o x o xK3 and xK3 ox o L. The sensitivity of the three deformation features with respect to damage is dependent on the segment. For example, for an arbitrary value of x within xK3 ox o L, we have
jd ðxÞ o j1 ðxÞ o jc ðxÞ
ð10Þ
Owning to the constitutive equation of the Euler–Bernoulli beam, M =EIj, the following relation is obtained via multiplying each term in Eq. (10) by EI Md ðxÞ o M1 ðxÞ o Mc ðxÞ
ð11Þ
Assuming that a damage event occurs at location x=xt and it alters the local flexural stiffness from EI(xt) to EIðxt Þ ¼ ð1KÞEIðxt Þ with 0 o K o 1, variations in curvature at the site of damage from the initial intact state are formulated as 1 1 K ¼ Md ðxt Þ ð12-aÞ Djd ðxt Þ ¼ Md ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
1 1 K ¼ M1 ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
ð12-bÞ
1 1 K ¼ Mc ðxt Þ ð1KÞEIðxt Þ EIðxt Þ EIðxt Þ
ð12-cÞ
Dj1 ðxt Þ ¼ M1 ðxt Þ
Djc ðxt Þ ¼ Mc ðxt Þ
where L/(1 L)EI(xt) is a constant. Based on Eqs. (11) and (12), variations in curvature at the site of damage from the initial intact state are obtained as the following first inequality:
Djd ðxt Þ o Dj1 ðxt Þ o Djc ðxt Þ, Djc ðxt Þ o Dj1 ðxt Þ o Djd ðxt Þ, Djc ðxt Þ o Djd ðxt Þ o Dj1 ðxt Þ, Djd ðxt Þ o Djc ðxt Þ o Dj1 ðxt Þ,
xK3 o xt o L 0 o xt oxK1 xK1 o xt o xK2
ð13Þ
xK2 o xt o xK3
Then, the relations of the damage-induced variation in curvature to the other three segments can be elaborated using the rest of Eq. (13). In Eq. (13), the magnitude of Dj quantitatively characterizes the sensitivity of the curvature to damage in a cantilever beam with respect to each deformation feature. Thus, a generic sensitivity rule can be established as follows: for damage location xt 2 ð0 xK1 Þ, the degree of sensitivity to the crack for three deformation features is ranked with dudl first, FMS second and dtcl third; for xt 2 ðxK1 xK2 Þ, FMS first, dudl second and dtcl third; for xt 2 ðxK2 xK3 Þ, the ranking is in the order FMS, dtcl and dudl; and for xt 2 ðxK3 LÞ, dtcl has the greatest sensitivity, followed by FMS and dudl. Moreover, in the narrow interval ðxK1 ¼ 0:2058L, xK3 ¼ 0:2866LÞ, the sensitivity of the different features becomes less distinguishable.
Fig. 4. Cracked cantilever beam models with a single-edge crack. (a) A cantilever beam with a single-edge crack, (b) a two-segment model with a massless rotational spring, (c) tip-concentrated loading and (d) uniformly distributed loading.
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4. A cracked beam model A single-edge Mode I crack in a cantilever beam, as depicted in Fig. 4, is commonly evaluated as a massless rotational spring [17–19] in analytical analysis of the beam. The equivalent stiffness of the rotational spring, KT, obtained from the crack strain energy function and Castigliano’s theorem, can be expressed in the following form, as deduced in Appendix B: KT ¼
6p
Ra 0
EIh af 2 ða=hÞda
ð14Þ
where b and h are the width and height of the beam, a is the depth of the crack, and f(a/h) is the geometric function [20] given as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2h pa 0:923 þ 0:199ð1sin ðpa=2hÞÞ4 ¼ tan f h 2h pa cosðpa=2hÞ The presence of the crack is represented by an additional rotation, yn(Lc)= M(Lc)/KT with M(Lc)= EId2YI/(Lc)/dx2 at the crack location (x =Lc), resulting in a discontinuity in the slope of the beam. The transition of slope is consequently expressed as [21] dYI ðLc Þ EI d2 YI ðLc Þ dYII ðLc Þ þ ¼ dx KT dx2 dx
ð15Þ
where YI and YII are the amplitudes of the flexural deformation of the beam segments I and II, respectively. 4.1. Dynamic response Based on the Euler–Bernoulli beam theory, the governing equations of flexural vibration of beam segments I and II, Fig. 4(b), are expressed as d4 YI 4 ln YI ¼ 0; dx4
0 r x rLc
ð16-aÞ
d4 YII 4 ln YII ¼ 0; Lc r x rL dx4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ln ¼ 4 on 2 rA=EI The general solutions to Eq. (16) can be written as Yin ðxÞ ¼ Bi,1 cos ln x þBi,2 sin ln x þ Bi,3 cosh ln x þ Bi,4 sinh ln x
ð16-bÞ
ði ¼ I,IIÞ
ð17Þ
where BI,j and BII,j with j =1,2,3,4 are constants to be determined from the boundary and continuity conditions. The boundary conditions at the clamped and free ends are specified as YI ð0Þ ¼ 0,
dYI ð0Þ ¼ 0, dx
d2 YII ðLÞ ¼ 0, dx2
d3 YII ðLÞ ¼0 dx3
ð18Þ
Eq. (15) gives the compatibility in slope at the crack location; moreover, the continuity conditions of the displacement, bending moment and shear force are stated as YI ðLc Þ ¼ YII ðLc Þ,
d2 YI ðLc Þ d2 YII ðLc Þ ¼ , dx2 dx2
d3 YI ðLc Þ d3 YII ðLc Þ ¼ dx3 dx3
ð19Þ
An 8 8 matrix of coefficients for the eight constants (Bi,j, i= I, II and j = 1,2,3,4) can be constructed from the general solution in conjunction with the boundary and continuity conditions (Eqs. (15), (18) and (19)). The associated set of simultaneous equations with respect to the eight constants will have nontrivial solutions only if the determinant of the coefficients becomes zero. Expansion of the determinant leads to the frequency equation, from which the natural frequencies can be numerically determined. The natural frequencies prompt the solution of the constants in Eq. (17). 4.2. Static models dtcl in Fig. 4(c) for the cracked cantilever beam is obtained from a particular bending equation EId4Yc/dx4 =0 considering the boundary conditions Yc;I ð0Þ ¼ 0,
Yc;I u ð0Þ ¼ 0,
00 Yc;II ðLÞ ¼ 0,
EIYc;II u00 ðLÞ ¼ P
ð20Þ
with the prime denoting differentiation, and compatibility conditions including Eq. (15) with the shear effect neglected, and Yc;I ðLc Þ ¼ Yc;II ðLc Þ
ð21Þ
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Fig. 5. Numerical model of a cantilever beam with a single-edge crack at Lc = 0.1L. (a) Finite element mesh, (b) zoomed-in crack.
The resultant solution is in the form: Yc;I ðxÞ ¼ ðx3LÞ=6EI; Yc;II ðxÞ ¼ P½6EIðLLc ÞðLc xÞ þ K T x 2 ðx3LÞ=6EIKT ;
0 r x r Lc Lc r x r L
ð22Þ
where KT is given in Eq. (15). dudl in Fig. 4(d) for the cracked cantilever beam is obtained from EId4Yd/dx4 = q considering zero shearing force at x =L and other boundary and compatibility conditions similar to those for the tip-concentrated loading case. The resultant solution has the form: Yd;I ðxÞ ¼
Yd;II ðxÞ ¼
qx2 ð6L2 4Lx þx2 Þ , 24EI
0 r x r Lc
q½12EIðLLc Þ2 ðLc xÞKT x2 ð6L2 4Lx þx2 Þ , 24EIKT
ð23-aÞ
Lc rx rL
ð23-bÞ
4.3. Numerical simulation A three-dimensional elastic finite element simulation is performed to further validate the generic sensitivity rule. The beams are modeled using 20-node 3D structural solid elements (SOLID 95) in the commercial software ANSYSs. In particular, the single-sided crack is modeled via pairs of coincident nodes with coordinates identical to those of adjoining elements. An illustration of the finite element mesh of a cracked beam is shown in Fig. 5. dtcl with P= 100 N, dudl with q= 100 kN/m2 and FMS are evaluated, followed by a crack identification procedure implemented on the acquired dynamic and static deformation responses. 5. Results and discussion A cracked cantilever steel beam is considered with dimensions: length (L)=0.30 m, width (b)=0.02 m, and depth (h)=0.02 m, and Young’s modulus (E), Poisson’s ratio (v), and mass density (r) are taken as 206 GPa, 0.3, and 7850 kg m 3, respectively. Four crack scenarios of Lc =0.1, 0.25, 0.5 and 0.75L from the clamped end with a/h=0.15 for analytical models and a/h=0.3 for numerical simulation are considered. The adoption of larger a/h for numerical simulation means to offer a clearer graphic presentation of the sensitivity of FMS, dtcl and dudl to a crack under considerations of modeling and simulation error. 5.1. Sensitivity For the scenario with the crack located at 0.5L, Fig. 6(a) presents the resultant three deformation features, with the slopes in Fig. 6(b) and the curvatures in Fig. 6(c), arising from the first-order difference and the second-order central difference, respectively. The effect of the crack on the slope is barely visible, with only a tiny increase in the local slope at the crack location, which cannot be used as an effective damage signature, especially taking into account the presence of noise and systematic errors. However, a sharp singular peak in the curvature makes the crack clearly identifiable. The height of bars in Fig. 6(c) denotes the magnitude of the singular peak for curvature. Clearly, the magnitudes of the singular peak for curvature are ranked with dtcl first, FMS second and dudl third. Since the magnitude of the singular peak indicates the degree of sensitivity to the crack, this ranking implies that when the crack is at x= 0.5L, dtcl is most sensitive to the
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Fig. 6. Curvatures of FMS, dtcl and dudl with a crack at Lc =0.5L. (a) Deformations for Lc = 0.5L, (b) slopes for Lc = 0.5L and (c) curvatures for Lc = 0.5L.
presence of the crack, followed by FMS and dudl. Fig. 7 depicts the curvatures for the scenarios when the crack is located at 0.1, 0.25 and 0.75L, respectively. For the crack located at 0.1L (Fig. 7(a)), the magnitude of the singular peak for dudl is greater than that of FMS and dtcl, respectively. This implies that dudl is most sensitive to the presence of the crack, followed by FMS and dtcl. Similarly, the ranking of sensitivity for the crack located at 0.25L can be observed in Fig. 7(b), in the order of FMS, dudl and dtcl. In addition, the ranking of sensitivity for the crack located at 0.75L is the same as that for the crack located at 0.5L, because the two crack scenarios occur in the same segment of the beam. For the numerical simulation, one-dimensional LoG (Laplacian of Gaussian) [22], rather than the second-order central difference, is employed to acquire the curvature from deformation responses with greater accuracy. This operator is expressed as ðx2 s2 Þ
cðxÞ ¼ pffiffiffiffiffiffi
2 2ps5
2
eðx
=2s2 Þ
ð24Þ
where s is the standard variance of normal distribution. In essence, the one-dimensional LoG, as shown in Fig. 8, is really a Gaussian wavelet of two-order vanishing moments. The features of convolution manipulation, smoothing, and flexibility in choosing s to match the signal under investigated entails it a more versatile differential operator than the second-order central difference. In the present analysis, a particular LoG with s = 1.8 (Fig. 8) is optimally chosen as the operator. The distortion near the end of beam induced by the convolution is treated by a cubic spline extrapolation, as stated in [3]. For a crack located at 0.1L from the clamped end, the curvatures from the normalized deformation responses with normalization manipulation similar to that adopted in Eq. (7), are shown in Fig. 9(a). It can be observed that the magnitudes of the singular peaks in curvature are ranked in the order of dudl, FMS, and dtcl. This is the same result as that obtained from the theoretical analysis, as shown in Fig. 7(a). The curvatures for the other scenarios with the crack located at Lc = 0.25, 0.5 and
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Fig. 7. Comparison of curvatures of FMS, dtcl and dudl for a crack at Lc = 0.1 (a), 0.25 (b), 0.75L (c), respectively.
Fig. 8. One-dimensional Laplacian of Gaussian (LoG).
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Fig. 9. Comparison of curvatures of FMS, dtcl and dudl for a crack at Lc = 0.1 (a), 0.25 (b), 0.5 and 0.75L (c), respectively.
0.75L, respectively, are shown in Fig. 9(b)–(d). The rankings of the magnitudes of the singular peaks in curvature for the crack located at 0.5 and 0.75L, are firstly dtcl, secondly FMS and thirdly dudl, which is consistent with the rankings obtained from the theoretical analysis (Figs. 6(c) and 7(c), respectively). For the crack located at 0.25L, the discrepancy in magnitude of the singular peak of curvature between the three deformation features is relatively small and the sensitivity of three deformation features to the crack is almost identical. Thus the results yielded by both theoretical analyses and numerical simulation fall into the scope of the generic sensitivity rule, validating the generic sensitivity rule well.
5.2. Effects of scatter in measurement In the actual measurement of FMS, dtcl, and dudl, there is always some scatter in the data because of random noise and systematic error. To assess the influence of potential noise, the actual measurements are mimicked by adding Gaussian white noise to these deformation responses obtained from the finite element simulation. The noise level is defined by the signal-to-noise ratio (SNR), SNR =20 log10(As/An), with As and An denoting the root-mean-square (RMS) magnitude of the simulated deformation response and added noise, respectively. The Monte Carlo method [23] is employed to simulate the deformation responses taking into account noise. In the analysis, the LoG operator (Eq. (24)) is applied to acquire curvatures from the mimicked responses. A correlation coefficient between the local portions of curvatures with and
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Fig. 10. Curvatures from dtcl for signifying effective (a) and ineffective (b) identification of the crack located at Lc = 0.1L.
Fig. 11. Curvatures from FMS for signifying effective (a) and ineffective (b) identification of the crack located at Lc = 0.1L.
Fig. 12. Curvatures from dudl for signifying effective (a) and ineffective (b) identification of the crack located at Lc =0.1L.
without noise is adopted as a measure for assessing the impact of the noise. The correlation coefficient adopted is expressed as Pn i ¼ 1 ðUi UÞðVi VÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð25Þ Pn Pn 2 2 i ¼ 1 ðUi UÞ i ¼ 1 ðVi VÞ where U and V are the local curvatures with a window size of 0.1L (covering 30 nodal points of the FE model) around the crack location for the intact and noisy responses, respectively. A value of correlation coefficient below 0.5 is used to
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Table 2 Effects of random noise on three deformation features for crack identification. SNR
100 90 85 80 75 70
[R0.1L(%), RL(n)]
[2.385, 0.351] [8.858, 1.302] [13.355, 1.960] [30.867, 4.531] [35.637, 5.234] [101.596,14.910]
S
dtcl
FMS
dudl
Effective (%)
Ineffective (%)
Effective (%)
Ineffective (%)
Effective (%)
Ineffective (%)
97.96 93.10 88.25 79.20 64.25 41.09 77.31
2.04 6.90 11.75 20.80 35.75 58.91 22.69
98.28 93.61 88.58 80.79 66.39 45.91 78.93
1.72 6.39 11.42 19.21 33.61 54.09 21.07
98.03 94.23 88.72 80.99 66.3 45.74 79.00
1.97 5.77 11.28 19.01 33.7 54.26 21.00
identify the threshold when the noise masks the presence of the crack. For the crack at 0.1L and the SNR =90, the influence of uncertainty on the identification of the crack is illustrated in Figs. 10–12, in which two sets of 200 profiles of curvature along the beam axis are presented for each deformation feature. In the figures, similar profiles of curvature with respect to three deformation features exist for either the effective or ineffective crack identification situation, and the discrepancy is almost indiscernible. To comprehensively investigate the impact of noise level, 10,000 noisy deformation responses for each deformation feature at one noise level are further constructed using the Monte Carlo method in order to approach a stable statistic character for the impact of noise. The noise levels of SNR =100, 90, 85, 80, 75 and 70 are considered. In Table 2 the percentages of effective and ineffective crack identification for each deformation feature at each noise level are evaluated and listed, where R0.1L and RL denote, at a noise level, the ratio of the mean magnitude of noise to the magnitude of deformation responses at 0.1L and L of the beam, respectively. For SNR =80, approximately 80% successful identification of the crack was achieved for each deformation feature, which may be regarded as an acceptable boundary, below which the efficiency of crack identification deteriorates markedly and has little reliability. At such a level of signal-to-noise ratio, the mean noise magnitude approximately equals 3.1% of the deformation response magnitude at the 0.1L of the beam, which provides a prior precision criterion for the optimal selection of deformation features from a perspective of experimental measurement. For conventional linear variable differential transformer (LVDT)-based measurement methods, dtcl is potentially the most feasible to meet this precision requirement, due to convenience in measurement, whereas use of FMS is probably more difficult because of the relatively complex procedure involved in the modal analysis. Advanced optical or electromagnetic spectrum-based measurement apparatus, such as a laser vibrometer, may provide the opportunity to meet this precision requirement in actual measurement of all deformation features. Clearly, all the deformation features are able to identify the crack for the SNRZ80, but their capabilities decrease significantly when SNRr75. At the level of SNR =70, these features are almost inadequate to act as signatures for crack identification. In Table 2, the average effective and ineffective ratios for each deformation feature imply that the impact of scatter is most significant for dtcl, followed by FMS and ducl. In other words, in the case of a cantilever beam with a crack, ducl has greatest immunity to noise, followed by FMS and dtcl. This result coincides with that for the sensitivity of the three deformation features to the presence of the crack. The finding that higher sensitivity corresponds to greater immunity to noise also applies to the other crack cases with the crack at 0.25, 0.5, and 0.75L, respectively. 6. Conclusions The most typical dynamic and static deformation features used in various algorithms for the detection of damage in cantilever beams are fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading. A fundamental question behind these popular algorithms is which is most sensitive for use in damage identification. With the Euler–Bernoulli beam theory and finite element simulation, the sensitivity of these features to damage in cantilever beams is comprehensively investigated in this study, and a generic sensitivity rule characterizing the sensitivity characteristics of the three deformation features to an edge crack in a cantilever beam is developed, addressing the potential scatter of actual measurement of these features. This generic sensitivity rule holds promise for the rational and optimal use of fundamental mode shape, deflection under tip-concentrated loading and deflection under uniformly distributed loading for damage detection in cantilever beams.
Acknowledgements L. Ye, L. Zhou and Z. Su thank the Hong Kong Polytechnic University for a special research grant on ‘‘Advanced Composite and Fundamental Structure’’. L. Zhou also thanks the Research Grant Council of the Hong Kong SAR for the research Grant of PolyU5333/07E. M.S. Cao is grateful for National Natural Science Foundation of China (Grant no. 50978084).
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Appendix A Yc ð x Þ ¼
Px2 ðx3LÞ 6EI
Yd ðxÞ ¼
qx2 ð6L2 4Lx þ x2 Þ 24EI
Yn ðxÞ ¼ C½ðcos ln xcosh ln xÞwn ðsin ln xsinh ln xÞ
ðA1Þ
ðA2Þ ðA3Þ
where C is an arbitrary constant and, wn ¼ ðcos ln L þ cosh ln LÞ=ðsin ln Lþ sinh ln LÞ with lnL= gn being the solutions of cosðgn Þ coshðgn Þ ¼ 1 for an Euler–Bernoulli cantilever beam. Eq. (A3) gives the fundamental model shape when n = 1: Y1 ðxÞ ¼ C½ðcos l1 xcosh l1 xÞw1 ðsin l1 xsinh l1 xÞ
ðA4Þ
with w1 = 0.7341 arising from g1 = 1.87510406871. Appendix B The stress intensity factor of mode I for a single-edge crack due to bending moment is expressed as pffiffiffiffiffiffi a K ¼ KI ¼ s paf h
ðA5Þ
2
where s =6M/bh with M being the bending moment at the cross-section of crack, a is the crack depth, and f(a/h) is the geometric function. If the crack can be represented by a spring, the additional work due to the bending moment equals to the crack strain energy: Z a M2 DU ¼ ¼ Js bda ðA6Þ 2KT 0 where KT is the equivalent stiffness of the rotational spring, and Js ¼ KI2 =E is the strain energy density function in the planestress condition [20]. Substituting Eqs. (A5) into Eq. (A6), KT can be formulated as KT ¼
¼
M2 M2 M2 M2 ¼ Ra 2 ¼ R a s2 paf 2 ða=hÞ ¼ R Ra 2 Þ2 paf 2 ða=hÞ a ð6M=bh 2 0 Js bda 2 0 ðKI =EÞbda 2 0ð Þbda bda 2 0 E E Ebh4 EIh ¼ Ra 6p 0 ðaf 2 ða=hÞÞda 72p 0 ðaf 2 ða=hÞÞda Ra
ðA7Þ
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