- Email: [email protected]
Modeling damaged structural members for vibration analysis
Journal of Sound and Vibration (1987) 112(3), 541-543
LETTERS T O T H E
EDITOR
MODELING DAMAGED STRUCTURAL MEMBERS FOR VIBRATION ANALYSIS In a recent paper [1], Yuen has modelled structural damage on a cantilever by way of a beam element with reduced modulus o f elasticity. A similar study was reported also by Harrison [2]. Yuen computed the eigenvalues and eigenvectors for the modified structure and compared them with the original, undamaged structure. The results of this work could be o f better utilization if the change in the elasticity modulus were directly related with some physical parameter describing the natural damage, such as crack depth, reduction in beam section (due to corrosion for example), etc. Such work was reported earlier by Chondros and Dimarogonas [3]. To relate Yuen's results [1] with a cracked cantilever, the method of the direct stiffness formulation will be used, as described by the author and co-workers [4-8]. To obtain numerical results, the section used in Yuen's paper [1] (see Figure 1) will be utilized here. It is a tubular section 1 in 2 with 1-2 mm wall thickness. A surface crack will be considered first extending just over the thickness of the section wall 1.2 mm (see Figure 1). The crack depth ratio a/w is then 1.2/25.4=0.047. Now, with the side walls
~//////////////////~
I,
25.4
'-I
Figure 1. Beam section. Dimensions in ram.
considered approximately, as cracked strips o f rectangular cross-section Wb, from Figure 2 of reference [4], the flexibility parameter w/EIc=30, where c = d~o/M is the local flexibility o f the cracked section, the ratio of angular deformation to the applied bending moment. For the data given by Yuen [1], c = 1.16x 10 -5 r a d / N m and the corresponding bending stiffness krc=l/c=5.18x106Nm/rad. The same stiffness of the uncracked element in the Yuen analysis is kre = 2EI/L = 1-75 x 105 N m / r a d for one element oflength L = 50 mm. The combined stiffness of the cracked element is
l/kT = (1/kre) + (1/krc),
(1)
or kr = 169281 Nm/rad. Since the stiffness is proportional to the modulus of elasticity, the modified modulus has to be in proportion to the stiffness modification. Therefore, the crack in Figure 1(a) corresponds to a modified modulus ofelasticity, for kr/kr~ = 0"966, E -- 0"966 Eo.
(2)
Such a small modification could not have been detected in Yuen's model, although the structural damage is already substantial. 541 0022=460x/87/030541+03 $03.00/0 9 1987 Academic Press Inc. (London) Limited
542
LETTERS TO THE EDITOR
Further, a 50% deep crack was considered. A similar procedure leads to an equivalent modulus of elasticity of E = 0.33 Eo, quite within Yuen's results. To generalize these results, the bending stiffness o f the cracked member is
1/kT = ( 1 / k r , ) + (l/krc) = ( L / 2 E I ) + (h/AEI) = (L/EI)(0"5+ h/AL),
(3)
where A = w / c E I = wkrc/EI is the dimensionless crack stiffness [4], L the structural element length, h the section height, kre the bending stiffness of the uncracked element, = L/2EI, and krc the crack stiffness. Since the stiffness is proportional, in elastic structures, to the elasticity modulus,
E / Eo = kr/kre = 0.5/(0.5 + h/AL) -
1 {l+2(h/L)/A}
(4)
The dimensionless stiffness A for a rectangular cross-section is given in references [4, 5] and for circular cross-section in references [6, 7] as function of the relative crack depth a/h. By using equation (4) and the values of dimensionless spring constant for the particular cross-section, the modulus of elasticity ratio can be computed for the given structural element length. Any structural analysis method can then yield the dynamic response of the modified (damaged) structure. 10 "--~
O8 ~uo6
01
02
03
04
05
06
07
08
o/h
Figure 2. Modulus of elasticity ratio E/Eo vs. crack depth ratio a/h for beams of tubular (A), solid square (B) and solid circular (C) cross-section. L/h =2. In Figure 2, the modulus of elasticity ratio is plotted against crack depth for tubular (A), solid square (B) and solid circular cross-section (C) for the element used by Yuen, that is with L / h ~ 2 . A. DIMAROGONASt
School of Engineering, University of Patras, Patras, Greece (Received 7 July 1986) REFERENCES
1. M. M. F. YUEN 1985 Journal of Sound and Vibration 103, 301-310. A numerical study of the eigenparameters of a damaged cantilever. 1 Now at the Schoolof Engineeringand AppliedScience,Departmentof Mechanical Engineer ng, Washington University, Campus Box 1185, St. Louis, Missouri 63130, U.S.A.
LE'VI'ERS TO TIlE EDITOR
543
2. H. B. HARRISON 1976 Journal of the Structural Division, American Society of Civil Engineers 102. Interactive non-linear structural analysis. 3. T. CIIONDROS and A. D. DIMAROGONAS 1980 Journal of Sound and Vibration 69, 331-538. Identification of cracks in complex structures by vibration analysis. 4. N. ANIFANTIS and A. D. DIMAROGONAS 1984 Computers and Structures 18, 351-356. Post buckling behavior of transverse cracked columns. 5. N. ANIFANTIS and A. D. DIMAROGONAS 1983 International Journal of solids and Structures 19, 281-291. Stability of columns with a single edge crack to follower and vertical loads. 6. A. D. DIMAROGONAS and S. A. PAIPETIS 1983 Rotor Dynamics. London: Elsevier-Applied Science Publishers. 7. A. O. DIMAROGONAS and C. A. PAPADOPOULOS 1983 Journal of Sound and Vibration 91, 583-593. Vibration of cracked shafts in bending. 8. A. DENTSORAS and A. D. DIMAROGONAS 1983 Engineering Fracture Mechanics 17, 381-386. Resonance controlled fatigue crack propagation.
LETTERS T O T H E
EDITOR
MODELING DAMAGED STRUCTURAL MEMBERS FOR VIBRATION ANALYSIS In a recent paper [1], Yuen has modelled structural damage on a cantilever by way of a beam element with reduced modulus o f elasticity. A similar study was reported also by Harrison [2]. Yuen computed the eigenvalues and eigenvectors for the modified structure and compared them with the original, undamaged structure. The results of this work could be o f better utilization if the change in the elasticity modulus were directly related with some physical parameter describing the natural damage, such as crack depth, reduction in beam section (due to corrosion for example), etc. Such work was reported earlier by Chondros and Dimarogonas [3]. To relate Yuen's results [1] with a cracked cantilever, the method of the direct stiffness formulation will be used, as described by the author and co-workers [4-8]. To obtain numerical results, the section used in Yuen's paper [1] (see Figure 1) will be utilized here. It is a tubular section 1 in 2 with 1-2 mm wall thickness. A surface crack will be considered first extending just over the thickness of the section wall 1.2 mm (see Figure 1). The crack depth ratio a/w is then 1.2/25.4=0.047. Now, with the side walls
~//////////////////~
I,
25.4
'-I
Figure 1. Beam section. Dimensions in ram.
considered approximately, as cracked strips o f rectangular cross-section Wb, from Figure 2 of reference [4], the flexibility parameter w/EIc=30, where c = d~o/M is the local flexibility o f the cracked section, the ratio of angular deformation to the applied bending moment. For the data given by Yuen [1], c = 1.16x 10 -5 r a d / N m and the corresponding bending stiffness krc=l/c=5.18x106Nm/rad. The same stiffness of the uncracked element in the Yuen analysis is kre = 2EI/L = 1-75 x 105 N m / r a d for one element oflength L = 50 mm. The combined stiffness of the cracked element is
l/kT = (1/kre) + (1/krc),
(1)
or kr = 169281 Nm/rad. Since the stiffness is proportional to the modulus of elasticity, the modified modulus has to be in proportion to the stiffness modification. Therefore, the crack in Figure 1(a) corresponds to a modified modulus ofelasticity, for kr/kr~ = 0"966, E -- 0"966 Eo.
(2)
Such a small modification could not have been detected in Yuen's model, although the structural damage is already substantial. 541 0022=460x/87/030541+03 $03.00/0 9 1987 Academic Press Inc. (London) Limited
542
LETTERS TO THE EDITOR
Further, a 50% deep crack was considered. A similar procedure leads to an equivalent modulus of elasticity of E = 0.33 Eo, quite within Yuen's results. To generalize these results, the bending stiffness o f the cracked member is
1/kT = ( 1 / k r , ) + (l/krc) = ( L / 2 E I ) + (h/AEI) = (L/EI)(0"5+ h/AL),
(3)
where A = w / c E I = wkrc/EI is the dimensionless crack stiffness [4], L the structural element length, h the section height, kre the bending stiffness of the uncracked element, = L/2EI, and krc the crack stiffness. Since the stiffness is proportional, in elastic structures, to the elasticity modulus,
E / Eo = kr/kre = 0.5/(0.5 + h/AL) -
1 {l+2(h/L)/A}
(4)
The dimensionless stiffness A for a rectangular cross-section is given in references [4, 5] and for circular cross-section in references [6, 7] as function of the relative crack depth a/h. By using equation (4) and the values of dimensionless spring constant for the particular cross-section, the modulus of elasticity ratio can be computed for the given structural element length. Any structural analysis method can then yield the dynamic response of the modified (damaged) structure. 10 "--~
O8 ~uo6
01
02
03
04
05
06
07
08
o/h
Figure 2. Modulus of elasticity ratio E/Eo vs. crack depth ratio a/h for beams of tubular (A), solid square (B) and solid circular (C) cross-section. L/h =2. In Figure 2, the modulus of elasticity ratio is plotted against crack depth for tubular (A), solid square (B) and solid circular cross-section (C) for the element used by Yuen, that is with L / h ~ 2 . A. DIMAROGONASt
School of Engineering, University of Patras, Patras, Greece (Received 7 July 1986) REFERENCES
1. M. M. F. YUEN 1985 Journal of Sound and Vibration 103, 301-310. A numerical study of the eigenparameters of a damaged cantilever. 1 Now at the Schoolof Engineeringand AppliedScience,Departmentof Mechanical Engineer ng, Washington University, Campus Box 1185, St. Louis, Missouri 63130, U.S.A.
LE'VI'ERS TO TIlE EDITOR
543
2. H. B. HARRISON 1976 Journal of the Structural Division, American Society of Civil Engineers 102. Interactive non-linear structural analysis. 3. T. CIIONDROS and A. D. DIMAROGONAS 1980 Journal of Sound and Vibration 69, 331-538. Identification of cracks in complex structures by vibration analysis. 4. N. ANIFANTIS and A. D. DIMAROGONAS 1984 Computers and Structures 18, 351-356. Post buckling behavior of transverse cracked columns. 5. N. ANIFANTIS and A. D. DIMAROGONAS 1983 International Journal of solids and Structures 19, 281-291. Stability of columns with a single edge crack to follower and vertical loads. 6. A. D. DIMAROGONAS and S. A. PAIPETIS 1983 Rotor Dynamics. London: Elsevier-Applied Science Publishers. 7. A. O. DIMAROGONAS and C. A. PAPADOPOULOS 1983 Journal of Sound and Vibration 91, 583-593. Vibration of cracked shafts in bending. 8. A. DENTSORAS and A. D. DIMAROGONAS 1983 Engineering Fracture Mechanics 17, 381-386. Resonance controlled fatigue crack propagation.