A Continuous Vibration Theory for Beams with a Vertical Edge Crack

Transcription

1 Transaction B: Mechanical Engineering Vol. 17, No. 3, pp. 194{4 c Sharif University of Technology, June 1 A Continuous Viration Theory for Beams with a Vertical Edge Crac Astract. M. Behzad 1;, A. Erahimi 1 and A. Meghdari 1 In this paper, a continuous model for exural viration of eams with an edge crac perpendicular to the neutral plane has een developed. The model assumes that the displacement eld is a superposition of the classical Euler-Bernoulli eam's displacement and of a displacement due to the crac. The additional displacement is assumed to e a product etween a function of time and an exponential function of space. The unnown functions and parameters are determined ased on the zero stress conditions at the crac faces and the concept of J-integral from fracture mechanics. The governing equation of motion for the eam has een otained using the Hamilton principle and solved using a modied Galerin method. The results have een compared with nite element results and an excellent agreement is oserved. Keywords: Viration; Craced eam; Vertical crac; J-integral. INTRODUCTION Fatigue and crac initiation and propagation in structures and machinery sujected to dynamic loading are one of the main concerns of designers and users. An uncontrolled crac can lead to a catastrophic failure under certain conditions. The importance of early detection of cracs maes researchers study various aspects of the ehavior of structures defected y cracs. One of these aspects is the viration of craced structures. Crac creation and development in a system changes the dynamic and viration ehavior of that system. With measurement and analysis of these virations, the cracs can e identied well in advance and appropriate actions can e taen to prevent more damage to the system. The viration ehavior of craced structures has een investigated y many researchers. Dimaragonas presented a review on the topic of the viration of craced structures [1]. His review contains viration of craced rotors, ars, eams, plates, pipes, lades and shells. Two more literature reviews are also availale on the dynamic ehavior of craced rotors y Wauer and Gasch [,3]. Beams are important elements in structures and 1. School of Mechanical Engineering, Sharif University of Technology, Tehran, P.O. Box , Iran. *. Corresponding author. m ehzad@sharif.edu. Received 1 May 9; received in revised form Feruary 1; accepted 6 April 1 machinery, so the viration ehavior of craced eams has een studied y researchers widely. One may dene a crac with its edge parallel or perpendicular to the neutral axis as horizontal or vertical cracs, respectively, as shown in Figure 1. There exist three methods for the viration modeling of eams with horizontal transverse cracs: 1. Discrete models with a local exiility model for cracs.. Continuous models with a local exiility model for cracs. 3. Continuous models with a continuous model for the crac. The local exiility model for the crac has een suggested y Dimaragonas for the rst time [4]. He replaced the craced eam with two undamaged half eams connected y a rotational spring. The stiness of this spring is otained from the concept of the J-integral in fracture mechanics. Papadopoulos presented a complete literature review on the method of using the J-integral for nding the local exiility of Figure 1. A eam with (a) horizontal edge crac and () vertical edge crac under ending.

2 A Viration Theory for Beams with Vertical Edge Crac 195 cracs [5]. The local exiility idea has een followed y several researchers till now. Some researchers modeled two undamaged half eams discretely and added the exiility of the rotational spring to the exiility matrix of the system [6,7]. Others modeled two undamaged half eams continuously and used appropriate oundary conditions for each part to lin them through the rotational spring [,9]. Some other researchers have tried to modify and improve the local exiility model of the crac y adding one or two linear springs esides the rotational one [1]. These methods have also een extended for eams with more than one crac [11-13]. The local exiility model for the crac is a simple approach and has a relatively good result in nding the fundamental natural frequency of a craced eam. However, this method cannot e implemented for nding stress at the crac area under dynamic loads, mode shapes in free virations and operational deformed shapes in forced virations. Another approach to the viration analysis of craced eams is continuous modeling of the crac. Christides and Barr developed a continuous theory for the viration of a uniform Euler-Bernoulli eam containing one or more pairs of symmetric cracs [14]. They suggested some modications on the familiar stress eld of an undamaged Euler-Bernoulli eam in order to consider the crac eect. The dierential equation of motion and corresponding oundary conditions are given as the results. However, in their model, two dierent and incompatile assumptions have een made for displacement and strain elds. Although the accuracy of the results in nding the natural frequencies is acceptale for some applications, their model is still not reliale for more accurate analyses such as stress analysis near the crac tip under dynamic loading and mode shape analysis. In addition, the otained partial dierential equation is complicated and dependent on some constants that are unnown and must e calculated y correlating the analytically otained results with those calculated y nite element in each case. Several researchers followed the Christides and Barr approach y modifying their method and gained some improvements [15-19]. However, there still exists the inconsistency etween strain and displacement elds, which causes inaccuracy in the results, especially in mode shapes and stress analysis. Behzad et al. presented a new continuous theory for the ending analysis of a craced eam []. A ilinear displacement eld has een suggested for the eam strain and stress calculations and the ending dierential equation has een otained using equilirium equations. The model can predict the loaddeection relation of the eam near or far from the crac tip accurately and can e also used for stressstrain analysis in a craced eam. This model is also used for the viration analysis of a craced eam and showed an excellent performance in dynamic loading too [1,]. They used this method also for the force viration analysis of eams with a horizontal edge crac [3]. In all the aove approaches, the crac is assumed to e horizontal. This type of crac is more proale to e created and other forms of crac tend to grow horizontally in ending. However, y using a pre-craced element with specic orientation in a structure, the crac may ecome horizontal. The other application of this research is in the area of rotor dynamics, where a craced eam rotates. Figure 1 shows a eam with horizontal and vertical cracs. In this paper, a continuous approach for the exural viration analysis of a eam with a vertical edge crac has een presented for the rst time. The crac is assumed to e an open edge notch and the crac edge is perpendicular to the neutral plane. A quasilinear displacement eld has een suggested for the craced eam and the strain and stress elds have een calculated. The dierential equation of motion of the craced eam has een otained using the Hamilton principle. This partial dierential equation has een solved with a special numerical algorithm ased on the Galerin projection method. The constants needed in this model can e otained using fracture mechanics. The results of this study are compared with the nite element results for verication. CRACK BEHAVIOR ANALYSIS IN BENDING AND HYPOTHESES The asic assumption in the Euler-Bernoulli viration theory for eams is that the plane sections of a eam which are perpendicular to the neutral axis remain plane and perpendicular to the neutral axis after deformation. In the presence of an edge crac, this assumption, especially near the vicinity of the crac, is no longer correct. Behzad et al. suggested that for a horizontal crac, the crac faces have an additional displacement due to the asence of the normal stress [,1]. They discussed that this additional displacement is inherited y the adjacent area with lesser magnitude, and dissipates away with an exponential regime along the eam length. Consequently, for planes far from the crac tip, this warping will e negligile and the displacement eld can e assumed linear. This idea can e followed here for a vertical crac with some modications. In order to have a etter sense of the exural virations in a craced eam, a nite element model has een produced in this research, and the mid span vertical crac exural ehavior can e seen in Figure. This nite element model is made using ANSYS software [4]. The crac is modeled as a vertical U-shape notch at the mid-span

3 196 M. Behzad, A. Erahimi and A. Meghdari as a crac, and a ne singular mesh is used. Note that the grid lines of Figure are only some hypothetical lines that show the deformation eld of the eam, and these lines are not referring to the nite element mesh. Near the crac area, the plane sections will no longer remain plane. In fact, the crac faces have an additional rotation, in comparison with the remaining part of the section, due to the asence of normal stress. This additional rotation dissipates gradually, while the distance from the crac tip increases. With a good approximation, it can e supposed that each plane section turns into two straight planes after deformation. Each straight plane section turns into two planes with dierent slopes, one on the right side and the other on the left side of the crac edge. The slope dierence etween these two planes decreases with distance from the crac tip. These two straight planes connect to each other through a nonlinear part near the xz- Figure. Displacement eld illustration in a eam with a vertical edge crac suject to ending. plane. Figure 3 shows the coordinate system and the parameter denitions, graphically. In order to nd the stress, strain and deformation functions for a eam with a vertical crac in exural viration, a displacement eld for the eam has een suggested in this research. In fact, it is assumed that each plane section turns into two straight planes and a nonlinear connector after deformation. In this research, the eam is assumed to e a slender prismatic eam and the crac is considered as an open edge U-shape notch. The cross section of the eam is assumed to e symmetric aout the y-axis, so the y-axis can e assumed to e the neutral axis in pure ending. The displacements and stresses are supposed to e small and the crac does not grow. Finally, the material is assumed to e linear elastic. DISPLACEMENT FIELD DEFINITION With reference to Figures and 3 and the aove assumptions, the displacement eld for a eam with a vertical edge crac can e dened. It is well nown that the displacement and stress elds near the crac tip are 3D functions. In this paper, at rst, a 3-dimensional displacement eld has een introduced for the eam ut, afterwards, the equations are integrated over the cross-section area of the eam and a 1-dimensional relation is otained for the eam virations. This Figure 3. Coordinate system and parameters denition.

4 A Viration Theory for Beams with Vertical Edge Crac 197 equation is not an exact equation, ut the results of this research show the good engineering approximation of this relation. The crac section consists of two parts: The crac face, which is denoted in Figure y A c, and the remaining part of the section, which is denoted y A h in this research. Under pure ending, the healthy part of the cross section (A h ) rotates aout its neutral axis, which is coincident with the y-axis in this research. This planar part remains plane after rotation and perpendicular to the neutral axis. The crac face rotates aout the y-axis too, ut more than the remaining part of the section and, consequently, does not remain perpendicular to the neutral axis due to the shear stress near the crac tip. The crac face can also e assumed to remain plane after deformation, except at a small area near the crac tip. The rotation dierence etween the crac face and the remaining part of the section inherits to the adjacent cross sections ut, gradually, the magnitude of this dierence decreases. As a side eect, deformation of the eam along the z-axis is a function of y. In fact, the parts of the eam sections which have more rotation cause more vertical displacement, too. The numerical simulations conrm this phenomenon. Based on the aove explanations, the following displacement eld is introduced for a eam with a vertical edge crac in exural viration: u = w = ( v = ( z(x; t) y < z((x; t) + (x; t)) y > w (x; t) y < w (x; t) + (x; t) y > (1) In which u; v and w are the displacement components along x; y and z axes. (x; t) is the rotation of that part of the section with y <, as shown in Figure 3. (x; t) is the additional rotation of that part of the section with y > and (x; t) is the additional vertical displacement of the eam for y >. By assuming that the plane sections in y < remain perpendicular to the neutral axis, one has: (x; t) (x; t) : The additional rotation (x; t) of that part of the plane sections with y > has its maximum value at the crac face and decreases gradually with distance from the crac tip. This additional rotation is a nonlinear and complicated variale with respect to x. Here, in this research, an exponential regime has een assumed for function (x) along the x-axis as follows: (x; t) = m(t)e jx xcj sgn(x x c ): (3) In Equation 3, m(t) is the magnitude of additional rotation of the crac faces, is a dimensionless exponential decay rate, which will e otained later in this paper, x c is the crac position, is the depth of the eam and sgn(x x c ) is the sign function, which is 1 for x < x c and +1 for x > x c. The application of a sign function is due to the fact that the additional rotation function has a discontinuity at the crac position and the sign of its value changes when passing through the crac tip. In order to nd the value of m(t), zero normal stress conditions at the crac faces can e used. The normal strain function can e found using Equation 1: " x = u ;x = zw ;xx y < z w ;xx m(t) jx xcj e +m(t)e jx xcj (x x c )! y > (4) in which (x x c ) is the Dirac delta function and the suscript ; x denotes the partial derivative with respect to x. The normal stress at the crac faces where y > and x = x + c or x c should e zero, so one has: m(t) = w ;xx(x c ; t): (5) The additional displacement, (x; t), which also decreases gradually with distance from the crac tip can e assumed to e a function similar to (x; t) as follows: (x; t) = n(t)e jx x cj ; (6) where n(t) is the magnitude of additional displacement at the crac face, which can e found using a zero shear stress condition at the crac faces. The shear strain function, xz, can e found using Equation 1: xz = 1 (u ;z + w ;x ) ( y < = m(t)+n(t) e jx x cj sgn(x x c ) y > (7) The shear stress at the crac faces, where y > and x = x + c or x c, should e zero so one has: n(t) = m(t) = w ;xx(x c ; t): () To avoid discontinuity and considering the nonlinearity at the crac tip, it is assumed that the displacement eld at y > transforms into the dened functions in Equation 1 with an exponential regime from the

5 19 M. Behzad, A. Erahimi and A. Meghdari displacement eld at y <. So, the displacement eld is modied in this paper as follows: u = zw ;x y < z (w ;x + 1 e y d w ;xx (x c )e jx xcj sgn(x x c ) y > w (x; t) y < w = w (x; t) 1 e y d w ;xx (x c )e jx xcj y > ( y < v = d ze y d e jx xcj y > (9) In Equation 9, is a dimensionless parameter and will e discussed later in this paper. The term (1 e y d ) prevents the discontinuity at the crac tip. The displacement component, v, is modied in order that yz ecomes zero at the crac faces. EQUATION OF MOTION Now, the strain eld can e extracted from the displacement eld. The normal strain component of the stress eld can e written using Equation 9 as follows: " x = zw ;xx y < z w ;xx 1 e y d 1 (x x c) w ;xx (x c )e jx xcj y > (1) The normal stress energy of the eam can e otained using the following relation: Z Z V = 1 x " x dv = 1 E " xdv: (11) V V In which V is the normal strain energy function, V is the volume of the eam and E is the modulus of elasticity. In this research, the craced eam is assumed to e slender. So, the Euler-Bernoulli assumption can e used and one can neglect the shear strain energy in comparison with the normal strain energy. The displacement eld is dened in order to force average shearing strain components to e zero; similar to an undamaged Euler-Bernoulli eam. The inetic energy of the craced eam can e also calculated as follows: T = 1 Z V w ;tdv: (1) In Equation 1, the rotational inertia has een neglected, similar to the undamaged Euler-Bernoulli eam theory. Using the Hamilton principle, one has: Z t1 t (T V )dt = : (13) Now, using Equations 1 to 13 and performing appropriate calculations, the following equations can e otained: jx Iw ;xx w ;xx (x c ; t)e +Aw ;tt =; Z Z = z 1 e y d da = I c z e y d da: (14) A c A c Equation 14 is the governing equation of motion of a eam with a vertical edge crac. In this equation, is a geometrical factor which can e found for every given cross-section and crac. I and I c are the moment inertia of the cross-section and the crac face, respectively. In an undamaged eam, the geometrical parameter,, is zero and, hence, Equation 14 turns into a familiar form of the Euler-Bernoulli viration equation for slender eams. The dimensionless exponential decay rates (; ) are the only factors that have not een discussed yet. In the next section, the parameters, and, are calculated and then, the solution for the partial dierential Equation 14 is presented. EXPONENTIAL DECAY RATES AND CALCULATION The exponential decay rates presented in this research can e otained using the concepts of additional remote point rotation and the J-integral, which are two familiar concepts in fracture mechanics. Several researchers used a similar method to evaluate the crac properties [5,]. When a pair of static ending moments, M, are applied to the craced eam, an additional relative rotation,, will exist etween two ends of the eam, due to the crac, as shown in Figure 4. For an Euler-Bernoulli simply supported eam, the slope function of the neutral axis is as follows: h = dw dx = M EI x l : (15) Figure 4. Additional rotation of a eam with a vertical crac under ending.

6 A Viration Theory for Beams with Vertical Edge Crac 199 For a eam with a vertical crac under pure static ending, the time derivatives vanish from Equation 14. Integrating two sides of the otained equation and using appropriate oundary conditions for pure ending, the following equation will e otained: d w dx = M EI 1 + I e jx xcj : (16) Solving Equation 16 will result in the load-deection relation of a eam with vertical crac under static pure ending. The results are as follows []: w = M EI M EI x + c 1x + c + x + c 3x + c 4 + I e x x c I e x x c xx c x>x c (17) in which constants c 1, c, c 3 and c 4 will e as follows []: c 1 = c 3 I c = I e xc c 4l 1 l c 3 = l c 4 = c x c I I e l xc (1) Using Equations 15 and 17, one can otain the additional remote point rotation,, as follows: = ( c () h ()) ( c (l) h (l)) = M e x c EI I e l x c ; (19) where c and h are the rotation of a craced eam and an undamaged or healthy eam under static ending, respectively. In Equation 19, parameter is a function of. However, the nite element results, in comparison with those otained y this model, show that parameter is a large enough parameter and, accordingly, it can e assumed that parameter tends to innity. It must e noticed that, despite the fact that the exponential decay rate,, is otained here y nite element analysis and correlating analytical and nite element results, this value for is a general value and, in other cases, can e used without separate calculations. On the other hand, additional rotation means that the craced eam accumulates more strain energy compared with an undamaged eam. This extra strain energy which is called U T here is stored at the vicinity of the crac. The additional rotation of a eam sujected to a pair of ending moments at two ends, as shown in Figure 4, can e otained using Castigliano's theorem as follows: : () This additional strain energy is due to the crac and can also e written in the following form [1,5]: U T = Z A c J s (a)da: (1) In Equation 1, A c is the crac face area. Equation 1 is called the Paris equation, and J s in this equation is the strain energy release rate. There are several experimental, analytical and numerical formulas to calculate the value of the J-integral ased on the geometry, loading and type of crac [5,6]. In this case, the value of the J-integral could e otained from the following equation [1,5]: J s = 1 E +m 4 6 X 6X K Ii! + 6X! K IIi K IIIi! 3 5 ; () where K Ii, K IIi and K IIIi are the Stress Intensity Factors (SIF), corresponding to three modes of fracture, which result for every individual loading mode, i. In pure ending, SIF is nonzero only for mode I. In Equation, if the plane stress assumption is used, then, E = E and, if the plane strain assumption is used, then E = E=(1 ). In this article the plane strain assumption is used. The eam with a vertical edge crac can e assumed to consist of a set of thin plates along a z- axis, and each plane to contain an edge crac and e sujected to axial tension or compression. This Stress Intensity Factor (SIF) for such plates is [6]: K I = p af a = Mz I ; F a = ; 1 + :1 cos 4 a r a tan a : (3) Equation 3 has an accuracy of.5% for any a=. From Equations and 3, the energy release rate is: Mz J s = K I E = 1 E I af a : (4)

7 M. Behzad, A. Erahimi and A. Meghdari Now, sustituting Equation 4 into 1 and, then, using Equation, the additional rotation of the craced eam,, can e otained in terms of ending moment and geometrical parameters. For a rectangular cross section, this additional rotation is: = 1 ' = E Z d d Z a M I '; z sf s dsdz: (5) The additional rotation can e evaluated using the otained relation in Equation 19 too. Comparing the two sides of Equations 5 and 19, one has: e x c e l x c = (1 ) I I ': (6) The numerical solution of Equation 6 will lead to nding the value of exponential decay rate for any values of geometrical parameters and simply supported ends. From Equation 6, it can e shown that exponential decay rate is a function of a=d, l=d and x c =l. However, Behzad et al. discussed that, for slender eams (l=d 1), slenderness factor (l=d) and crac position ratio x c =l have a minor eect on for horizontal cracs []. It can e shown that exponential decay rate is only a function of crac depth ratio (a=) for slender eams with vertical cracs too. Figure 5 shows versus a= for slender eams (l=d 1). EIGEN SOLUTION FOR SIMPLY SUPPORTED BEAM WITH VERTICAL CRACK In order to nd the natural frequencies and mode shapes of a eam with vertical crac, the equation of motion presented in Equation 14 must e solved. However, this equation cannot e solved analytically, and a Figure 5. Exponential decay rate () versus crac depth ratio (a=) for a slender simply supported eam with a vertical edge crac. numerical method must e used. The especial form of Equation 14 in which the solution at the crac position is appeared in the governing equation prevents one from using the ordinary Galerin projection method. Behzad et al. presented a modied Galerin projection algorithm for solving this type of equation [1]. In this paper, a similar approach has een used. The eam is assumed to e simply supported in this section. However, for every desired oundary condition, the presented solution can e used. It can e assumed that the solution is a harmonic function, so one has: w(x; t) = X(x)e i!t ; (7) in which! is the natural frequency of the eam. Sustituting Equation 7 into Equation 14 and assuming EI to e constant along the eam, the following eigenvalue prolem will e otained: d dx X I X (x c )e jx x cj X() = X(l) = X () = X (l) = A EI! X = () In Equation, simply supported oundary conditions have een used. In a normal Strum-Louiville prolem, one P can easily consider function X to e in the form of c i S i (x) in which S i (x) are shape functions that satisfy the physical oundary conditions. However, in this research, the results show that such an approach will lead to a divergence of the results. Since the function e jx x cj d in Equation is not a smooth function, it seems that the solution, especially for larger crac depth ratios, tends to have large derivatives near the crac tip. Accordingly, extracting the value of X (x c ) from X y derivation can lead to large uctuations in the results and divergence. In order to avoid the divergence prolem, function X and the value of X (x c ) are not extracted from X y direct derivation. Instead, X is discretized independently from X and, then, a constraint equation is provided to lin X to X. Considering the aove discussion, the following relations can e written: X I X (x c )e P X = N c i S i(x) jx xcj d P = N c i S i (x) (9) in which c i and c i are two independent sets of constants, functions S i (x) are shape functions, which must satisfy physical oundary conditions, and N is the numer of shape functions. Sustituting Equation 9 into Equation, multiplying two sides of the equation y

8 A Viration Theory for Beams with Vertical Edge Crac 1 S j (x), then, integrating along the length of the eam, one has: NX c i S i (x)s j (x)dx A NX EI! c i S i (x)s j (x)dx = ; j = 1; ; ; N: (3) Or in the matrix form: A Kc! Pc = ; EI K ij = S i (x)s j (x)dx; P ij = S i (x)s j (x)dx: (31) On the other hand, if one sustitutes the second equation of Equation 9 into the rst one, the following relation will e otained: NX c i S i (x) I S i (x c )e jx x cj = NX c i S i (x): (3) Multiplying two sides of Equation 3 y S j (x) and, then, integrating along the length of the eam, one has: NX c i = I S i (x c ) NX c i S i (x)s j (x)dx S j (x)e jx! x cj dx S i (x)s j (x)dx; j =1; ; N: (33) Rearranging Equation 33 into matrix form, the following equation can e written: Qc = Rc; Q ij = R ij = S i (x)s j (x)dx I S i (x c ) S j (x)e jx x cj dx; S i (x)s j (x)dx: (34) Now, from Equation 34, coecients c i can e related to c i, as follows: c = Q 1 Rc: (35) Sustituting Equation 35 into 31, the following equation will e otained: K A EI! M c = ; M = PQ 1 R: (36) The natural frequencies and corresponding mode shapes for the craced eam can e calculated solving the matrix eigenvalue prolem of Equation 36. In the next section, the results are presented for a simply supported eam with a rectangular crosssection. RESULTS FOR A SIMPLY SUPPORTED BEAM WITH RECTANGULAR CROSS-SECTION In this section, the eigenvalue prolem of Equation 36 has een solved for free viration analysis of a simply supported slender prismatic eam with a vertical edge crac and rectangular cross-section. In such a eam, the exponential decay rate,, can e assumed to e innite and the exponential decay rate,, can e calculated from Equation 6. The values of versus crac depth ratio (a=) have een shown in Figure 5. In a simply supported craced eam, shape functions S i (x) can e assumed to e in the form of sin(ix=l), which satisfy physical oundary conditions. The natural frequency and mode shapes can e calculated using the eigenvalue prolem of Equation 36. In this research, the numer of shape functions, N, is set to e 1. In order to generalize the results, the natural frequencies of the craced eam have een divided into the corresponding values for an undamaged eam (! ). Figures 6, 7 and show the fundamental, second and third natural frequency ratios of the craced eam, respectively. In Figures 6 to, the natural frequency ratios have een plotted versus the crac depth ratio (a=) for several crac positions. In Figures 6 to, the results of Finite Element (FE) analysis are also presented for verication. The nite element results have een otained using ANSYS software. In order to have an accurate and reliale model, the PLANE13 singular element has een used in the craced area []. This element is an -node quadratic solid singular element, especially designed for crac analysis. In this research, a ne mesh has een used at the vicinity of the crac, and the dependency of the results on mesh size has een checed. In all results,

9 Figure 6. Fundamental natural frequency ratio for a eam with a vertical crac versus crac depth ratio. M. Behzad, A. Erahimi and A. Meghdari there is good agreement etween analytical results and those otained y FE analysis. As can e seen in Figure 6, the reduction rate of the fundamental natural frequency has a direct relation with the position of the crac. This rate reduces for cracs that have more distance from the mid span of the eam. For the cracs at xc =l = :1, the fundamental natural frequency drops less than 1 percent when the crac reaches half of the eam depth, while for the cracs at the mid span, this value is aout 4 percent. The dependency of the reduction of the natural frequency on the crac position is also seen in the rst few natural frequencies. For cracs at the mid span, the second natural frequency remains nearly constant with the crac depth, ecause this point coincides with the node of the second viration mode of the eam. Figures 9 and 1 compare the natural frequency Figure 9. Fundamental natural frequency ratio versus Figure 7. Second natural frequency ratio for a eam with a vertical crac versus crac depth ratio. Figure. Third natural frequency ratio for a eam with a vertical crac versus crac depth ratio. crac depth ratio for vertical crac and horizontal crac [1] at xc =l = :5. Figure 1. Second natural frequency ratio versus crac depth ratio for vertical crac and horizontal crac [1] at xc =l = :5.

10 A Viration Theory for Beams with Vertical Edge Crac 3 drop for horizontal and vertical cracs. In Figure 9, the fundamental frequency ratio for the mid span vertical crac has een compared with a horizontal one in the same position. It can e seen that a horizontal crac has much more eect than a vertical one on the natural frequency. This result was predictale, ecause a horizontal crac reduces ending stiness more than a vertical crac. Figure 1 shows a similar comparison for the second natural frequency at x c =l = :5. It can e seen that a horizontal crac has more eect on the second natural frequency, too. Figures 11 to 13 show the rst three normalized mode shapes for a craced eam with a=d = :5 and x c =l = :1,.3 and.5. Comparison of the analytic and nite element results in this set of gures shows the eciency of the model presented in this research. CONCLUSIONS A continuous model for exural viration analysis of a eam with a vertical edge crac has een developed in this paper. It is assumed that the crac face rotates Figure 11. First three normalized mode shapes of a craced eam with x c =l = :1 and a= = :5. ( ): Analytical results; ( ): Finite element results. Figure 13. First three normalized mode shapes of a craced eam with x c=l = :5 and a= = :5. ( ): Analytical results; ( ): Finite element results. more than other parts of the section, as well as its adjacent area. The additional rotation decays with an exponential regime along the eam length. On the ase of this assumption, a displacement eld for the eam has een suggested and modied for compatiility, continuity and consistency. The strain and stress elds are calculated y direct derivation of the displacement eld and y using the linear elastic material model. Then, the partial dierential equation of motion has een otained using the Hamilton principle. This equation has een evaluated for static conditions and the exponential decay rate has een otained with the aid of the J-integral concept in fracture mechanics. The otained governing equation of motion for a simply supported eam with a rectangular crosssection and vertical edge crac has een solved with a modied Galerin projection method. The otained results have een compared with nite element results for a few rst natural frequencies and mode shapes, and an excellent agreement has een oserved. The otained results have also een used for studying the eect of crac parameters on natural frequencies and mode shapes. The calculated natural frequencies for a eam with vertical crac have een compared with those otained for horizontal crac, and it is oserved that the natural frequencies are more sensitive to horizontal cracs. Finally, it must e noticed that the developed theory in this research is only correct for open edge cracs without extension. ACKNOWLEDGMENT The authors would lie to acnowledge the nancial assistance of the \Iranian Gas Transmission Co." throughout this research. Figure 1. First three normalized mode shapes of a craced eam with x c=l = :3 and a= = :5. ( ): Analytical results; ( ): Finite element results. REFERENCES 1. Dimarogonas, A.D. \Viration of craced structures- A state of the art review", Eng. Fract. Mech., 5, pp.

11 4 M. Behzad, A. Erahimi and A. Meghdari (1996).. Wauer, J. \On the dynamics of craced rotors: A literature survey", Appl. Mech. Rev., 43(1), pp (199). 3. Gasch, R. \A survey of the dynamic ehavior of a simple rotating shaft with a transverse crac", J. Sound. Vi., 16(), pp (1993). 4. Dimarogonas, A.D. and Paipetis, S.A., Analytical Methods in Rotor Dynamics, London, Applied science pulisher (193). 5. Papadopoulos, C.A. \The strain energy release approach for modeling cracs in rotors: A state of the art review", Mech. Syst. Signal Pr.,, pp (). 6. Zheng, D.Y. and Fan, S.C. \Viration and staility of craced hollow-sectional eams", J. Sound. Vi., 67, pp (3). 7. Yang, J., Chen, Y., Xiang, Y. and Jia, X.L. \Free and forced viration of craced inhomogeneous eams under an axial force and a moving load", J. Sound. Vi., 31, pp ().. Lin, H.P. \Direct and inverse methods on free viration analysis of simply supported eams with a crac", Eng. Struct., 6(4), pp (4). 9. Zheng, D.Y. and Fan, S.C. \Viration and staility of craced hollow-sectional eams", J. Sound. Vi., 67, pp (3). 1. Loya, J.A., Ruio, L. and Fernandez-Saez, J. \Natural frequencies for ending virations of Timosheno craced eams", J. Sound. Vi., 9, pp (6). 11. Orhan, S. \Analysis of free and forced viration of a craced cantilever eam", NDT&E Int., 4, pp (7). 1. Yang, X.F., Swamidas, A.S.J. and Seshadri, R. \Crac identication in virating eams using the energy method", J. Sound. Vi., 44(), pp (1). 13. Wang, J. and Qiao, P. \Viration of eams with aritrary discontinuities and oundary conditions", J. Sound. Vi., 3, pp. 1-7 (7). 14. Christides, S. and Barr, A.D.S. \One-dimensional theory of craced Bernoulli-Euler eams", J. of Mech. Sci., 6(11/1), pp (194). 15. Shen, M.H.H. and Pierre, C. \Natural modes of Bernoulli-Euler eams with symmetric cracs", J. Sound. Vi., 13(1), pp (199). 16. Shen, M.H.H. and Pierre, C. \Free virations of eams with a single-edge crac", J. Sound. Vi., 17(), pp (1994). 17. Carneiro, S.H.S. and Inman, D.J. \Comments on the free viration of eams with a single-edge crac", J. Sound. Vi., 44(4), pp (1). 1. Chondros, T.G., Dimarogonas, A.D. and Yao, J. \A continuous craced eam viration theory", J. Sound. Vi., 15(1), pp (199). 19. Chondros, T.G., Dimarogonas, A.D. and Yao, J. \Viration of a eam with reathing crac", J. Sound. Vi., 39(1), pp (1).. Behzad, M., Meghdari, A. and Erahimi, A. \A linear theory for ending stress-strain analysis of a eam with an edge crac", Eng. Fract. Mech., 75(16), pp (). 1. Behzad, M., Meghdari, A. and Erahimi, A. \A new continuous model for exural viration analysis of a craced eam", Pol. Mar. Res., 15(), pp ().. Behzad, M., Meghdari, A. and Erahimi, A. \A new approach for viration analysis of a craced eam", Int. J. of Eng., 1(4), pp (5). 3. Behzad, M., Meghdari, A. and Erahimi, A. \A continuous model for forced viration analysis of a craced eam", ASME Int. Mech. Eng. Cong. and Exp., Orlando, Florida USA (5). 4. ANSYS User's Manual for Rev.,, ANSYS Inc. (4). 5. Barani, A. and Rahimi, G.H. \Approximate method for evaluation of the J-integral for circumferentially semi-elliptical-craced pipes sujected to comined ending and tension", Scientia Iranica, 14(5), pp (7). 6. Tada, H., Paris, P.C. and Irvin, G.R., The Stress Analysis of Cracs Handoo, Hellertown, Pennsylvania, Del Research Corp. (1973). BIOGRAPHIES Mehdi Behzad has a PhD in Mechanical Engineering from the University of New South Wales, Sydney, Australia and is a faculty memer of the mechanical engineering department of Sharif University of Technology in Tehran, Iran. Professor Behzad is also chairman of the Iran Maintenance Association. Alireza Erahimi has a PhD in Mechanical Engineering from Sharif University of Technology in Tehran, Iran and is also a researcher of the condition monitoring center at that university. Ali Meghdari has a PhD in Mechanical Engineering from the University of New Mexico, Aluquerque, U.S.A. and is a faculty memer of the Mechanical Engineering Department of Sharif University of Technology in Tehran, Iran. He is also a Distinguished Professor of Mechanical Engineering (MSRT), Vice-President of Academic Aairs and a Fellow of the American Society of Mechanical Engineers (ASME).