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A bending theory for beams with vertical edge crack
ARTICLE IN PRESS International Journal of Mechanical Sciences 52 (2010) 904–913
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A bending theory for beams with vertical edge crack A. Ebrahimi , M. Behzad, A. Meghdari Sharif University of Technology, Mechanical Engineering Department, P.O. Box 11365-9567, Azadi Ave., Tehran, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 December 2008 Received in revised form 6 March 2010 Accepted 12 March 2010 Available online 18 March 2010
In this paper a linear continuous theory for bending analysis of beams with an edge crack perpendicular to the neutral plane subject to bending has been developed. The model assumes that the displacement field is a superposition of the classical Euler–Bernoulli beam’s displacement and of a displacement due to the crack. It is assumed that in bending the additional displacement due to crack decreases exponentially with distance from the crack tip. The strain and stress fields have been calculated using this displacement field and the bending equation has been obtained using equilibrium equations. Using a fracture mechanics approach the exponential decay rate has been calculated. There is a good agreement between the analytical results from solving the differential equation of cracked beam and those obtained by finite element method. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Cracked beam Vertical crack Bending Stress–strain Load–deflection
1. Introduction Since late 1950s the crack issue and the behavior of structural elements contain one or more cracks have been considered by researchers seriously. The first attempts for finding an analytical solution for stress and displacement fields near the crack tip were performed by Irwin [1,2] and Williams [3]. In these works an Airy stress function for an infinite thin plate with a through crack has been suggested and a general governing equation for stress state near the crack tip has been derived. Later, some researchers tried to find a solution for this equation. Sedov presented the general solution for an internal crack in an infinite plate using plane state stress assumption for symmetric (mode I) and antisymmetric (mode 2) cases [4]. In spite of many efforts for finding an exact solution for stress state in cracked bodies, exact closed form analytical solutions is only found for cracks in infinity large bodies under pure tension. Furthermore this exact solution is only accurate near the crack tip and far from the crack tip this solution cannot be implemented. The necessity of finding stress and displacement fields in cracked bodies especially near the crack tip in real cases persuade the researchers to develop numerical and empirical methods. A large number of empirical and numerical formulations have been reported for several continua with different forms of crack under various forms of loading. The most important and useful forms of these formulae has been collected by Tada et al. [5]. Further
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E-mail address: [email protected] (A. Ebrahimi). 0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.03.004
developments in elastic and elasto-plastic fracture mechanics can be found in the literature [6–8]. The load–deflection behavior of a cracked beam in bending and the stress–strain analysis are engineering problems considered by researchers. Typically, in order to have load deflection relation for such a beam one should transform the load to stress field, then relate the stress field to strain field and finally transform back the strain field into the deflection or start this cycle from the deflection towards the load. However, it is hard to relate the load to stress and deflection to strain in the whole of a cracked body because of the complex nature of the crack and nonlinear spatial variation in stresses. Accordingly, developing a comprehensive and exact model which can describe all phenomena occurring at the vicinity and far from the crack simultaneously is not a simple task. Yet researchers tried to develop some models to evaluate the behavior of a cracked beam in bending. These models are mostly not exact but have a reasonable error for engineering applications. One of the simplest models for beams with edge crack in bending has been presented by Dimarogonas [9]. He assumed the cracked beam consists of two normal parts connected by a rotational spring at the position of the crack. This local flexibility idea has been followed by several researchers till now [10–14]. This model is only applicable for load-deflection approximation of a cracked beam far from the crack tip and cannot be considered for stress analysis. Some researchers preferred to use a continuous displacement and stress field rather than the local flexibility model. Such an approach can lead to a better result at the vicinity of the crack [15–19]. In their approach the crack effect can be taken into account by applying some modifications on the stress and displacement fields of an ordinary Euler–Bernoulli beam. The
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Nomenclature a crack length A cross-sectional area area of the crack face Ac b width of the beam d height of the beam ci(i¼1,2,3) integration constants E modulus of elasticity F(a/d) crack shape factor I second moment of the section area about y-axis second moment of the crack face area about y-axis Ic strain energy density function (J-integral) Js k a geometrical factor for the cross section of a cracked beam Ki(i¼I,II,III) stress intensity factors L length of the beam M bending moment m, n real constants u displacement component along x-axis
main defect in this approach is that two independent functions for stress and displacement fields are considered by guesswork. In fact the suggested functions are incompatible and the model can only be used for a specific application such as estimation of the first natural frequency of the cracked beams. Behzad et al. [20] presented a new continuous theory for bending analysis of a beam with an edge crack. A bilinear displacement field suggested for the beam strain and stress calculations and the bending differential equation has been obtained using equilibrium equations. The model can predict the load–deflection relation of the beam near or far from the crack tip accurately and can be also used for stress–strain analysis in a cracked beam. This model is also used for vibration analysis of a
Neutral Plane
Crack front Bending moment Crack front
UT v V w w0 xc a,b
D
ex gxz, gyz j n y y* yc yh
s0 sx c
905
additional strain energy due to the crack displacement component along y-axis dimensionless deformation of the cracked beam displacement component along z-axis displacement along z-axis for yo0 crack position exponential decay rates additional deformation along z-axis normal strain shear strain shape coefficient for additional remote point rotation Poisson ratio slope function of a cracked beam subject to bending additional remote rotation of a cracked beam rotation of a cracked beam under bending rotation of an undamaged beam under bending maximum normal stress of an undamaged beam subject to bending normal stress additional rotation of the beam cross-sections
cracked beam and showed an excellent performance in dynamic loading too [21–23]. In all previous continuous approaches it is assumed that the crack front is parallel with the neutral plane of the uncracked beam and with the bending moment vector too (Fig. 1(a) and (b)). This type of crack is more probable to be created and other forms of cracks tend to grow in this mode in bending. However it is possible that the crack front make an arbitrary angle with the bending moment vector. For example suppose a rotating cracked rotor under a constant unidirectional bending moment. When the rotor rotates, the crack front makes different angles with the moment vector in each position. In bending of a cracked beam one may define a crack with its front line perpendicular to the neutral plane of the uncracked beam, a ‘‘vertical crack’’ as shown in Fig. 1(c). In this case, the crack front is perpendicular to the bending moment vector too and the crack surfaces pass vertically through the uncracked beam’s neutral plane, with the upper half of the crack surface subjected to opening due to beam tension and the lower half closing due to compressive beam stresses. In practice, when a crack exists in an element, one may end up with a vertical crack in structures depending on the orientation of the element. The other application of the study of this type of cracks is in the area of rotor dynamics where a cracked beam rotates. In this paper a continuous model for bending analysis of a beam with an edge vertical crack has been presented for the first time. From experimental and numerical observations, a quasilinear displacement field has been introduced for a beam with an edge vertical crack. The strain and stress fields have been calculated from the displacement. The bending differential equation of the cracked beam has been extracted from equilibrium equations. The required constants needed in this model can be obtained using fracture mechanics. The results of this study are compared with the finite element results for verification.
2. Definitions and assumptions
Bending moment Fig. 1. (a) A normal beam and the neutral axis in bending, (b) a cracked beam with parallel crack front and (c) a cracked beam with perpendicular crack front.
The basic assumption in the Euler–Bernoulli bending theory for beams is that the plane sections of beam which are perpendicular to the neutral axis remain plane and perpendicular to the neutral axis after deformation. In the presence of an edge
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crack, the planes will not remain plane after deformation particularly near the vicinity of the crack due to a shear stress near the crack tip which leads to warping in plane sections. Thus at the vicinity of the crack the displacement field is completely nonlinear. For the planes far from the crack tip, the warping will be smaller and the displacement filed can be assumed linear. In order to have a better sense of the bending in a cracked beam, a finite element model has been produced in this research. The details of this model are presented in Section 7 of this paper. The mid-span vertical crack behavior under a pure bending moment can be seen in Fig. 2. In this model the beam is assumed to be a slender prismatic beam with rectangular cross-section made of a linear elastic material with small deformations. A vertical narrow notch with parallel faces at the mid-span is used for modeling the crack in this research. Note that the grid lines of Fig. 2 are only some hypothetical lines which show the deformation field of the beam and these lines are not referring to the finite element mesh discussed in Section 7. Near the crack area the plane sections will no longer remain plane. In fact in the crack faces due to lack of normal stress there exists an additional rotation in comparison with the remaining part of the section. This additional rotation dissipates gradually, while the distance from the crack tip increases. With a good
approximation it can be supposed that each plane section turns into two straight planes after deformation. Each straight plane section turns into two planes with different slopes one in the right side and the other in the left side of the crack edge. The slope difference between these two planes decreases with distance from the crack tip. These two straight planes connect to each other through a nonlinear part near the horizontal line passing through the crack tip. For a crack in a location other than the midspan of the beam, similar discussion can be made. Fig. 3 shows the coordinate system and the parameter definitions graphically for a typical cracked beam with arbitrary position of the crack. In order to find the stress, strain and deformation functions for a beam with a vertical crack in pure bending a displacement field for the beam has been suggested in this research. In fact it is assumed that each plane section turns into two straight planes and a nonlinear connector after deformation. The essential assumptions used in this research can be listed as follows:
The beam is slender and prismatic. The beam is symmetric with respect to the x–y plane so the y-axis can be assumed to be the neutral axis in pure bending.
The crack is considered to be an open edge narrow notch with parallel faces and wedge-shaped front.
Fig. 2. Displacement field illustration in a beam with a vertical edge crack subject to bending.
z
z b
Ah
d Ac
y
x
a
d
L
b a
xc y u(x,y,z)
x
b a
zψ(x)
z
y ) ψ(x
x
θ(x)
y
z
d
Fig. 3. Coordinate system and parameters definition.
x
w(x,y)
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The deformations are supposed to be small. The plane strain assumption has been used in this research.
Consequently the strain component along z-axis has been neglected. The applied load is small enough and the crack does not grow. The material is assumed to be linear elastic. It is assumed that the crack faces do not contact in bending. This assumption is true when the crack faces have enough distance from each other and the bending moment is small enough, or if the bending moment is superimposed with axial tension, preventing crack surface contact.
3. Displacement field With refer to Figs. 2 and 3 and the above assumptions the displacement field for a beam with vertical edge crack can be defined. It is well known that the displacement and stress fields near the crack tip are 3D functions. In this paper at first a 3-dimensional displacement field has been introduced for the beam but afterwards the equations are integrated over the crosssection area of the beam and a 1-dimensional load–deflection relation has been obtained for the beam. This equation is not an exact equation but the results of this research show the good engineering approximation of this relation. The crack section consists of two parts: the crack face which is shown in Fig. 2 by Ac and the remaining part of the section which is shown by Ah in this research. Under pure bending the healthy part of the cross section (Ah) rotates about its neutral axis which is coincident with y-axis in this research. This planar part remains plane after rotation and perpendicular to the neutral axis. The crack face rotates about y-axis too but it rotates more than the remaining part of the section. Consequently, the crack face does not remain perpendicular to the neutral axis due to the shear stress near the crack tip. The crack face can also be assumed to remain plane after deformation except at the small area near the crack tip. The rotation difference between the crack face and the remaining part of the section inherits to the adjacent cross sections but gradually the magnitude of this difference decreases. As a side effect, the deformation of the beam along z-axis is a function of y. In fact the parts of the beam sections which have more rotation cause more vertical displacement too. The numerical simulations with ANSYS software confirm this phenomenon. On the base of the above explanations, the following displacement field is introduced for a beam with a vertical edge crack subject to pure bending: ( 8 yo 0 zyðxÞ > > >u¼ > > y ðxÞ þ c ðxÞÞ y4 0 zð > > < ( ð1Þ w0 ðxÞ yo0 > > w¼ > > ðxÞ þ D ðxÞ y 4 0 w 0 > > > : v¼0 In which u, v, w are the displacement components along x, y and z axes. y(x) is the rotation of the part of the section with yo0 as shown in Fig. 3. c(x) is the additional rotation for that part of the section with y40 and D(x) is the additional vertical displacement of the beam for y40. By assuming that the plane sections in yo0 remain perpendicular to the neutral axis one has dw0 ðxÞ yðxÞ ¼ dx
ð2Þ
The additional rotation c(x) of that part of plane sections with y40 has its maximum value at the crack faces and decreases gradually with distance from the crack tip. This additional rotation is a nonlinear and complicated variable with respect to
907
x. Here in this research an exponential regime has been assumed for c(x) along the x-axis as follows:
cðxÞ ¼ meaððjxxc jÞ=bÞ sgnðxxc Þ
ð3Þ
In Eq. (3) m is a real constant, a is a dimensionless exponential decay rate which will be obtained later in this paper, xc is the crack position, d is the depth of the beam and sgn(xxc) is the sign function which is 1 for xoxc and +1 for x4xc. The application of sign function is due to the fact that the additional rotation function has a discontinuity at the position of the crack and the sign of its value changes when passing through the crack tip. In order to find the real constant m, zero normal stress condition at the crack faces can be used. The normal strain function can be found using Eq. (1) 8 d2 w0 > > > z yo0 dx2 @u < 2 ¼ ex ¼ d w a @x > 0 > > z m eaððjx-xc jÞ=bÞ þ 2me-aððjx-xc jÞ=bÞ dðx-xc Þ y40 : b dx2 ð4Þ In which d(x xc) is the Dirac delta function. The normal stress at the crack faces where y40 and x¼xc + or xc should be zero so one has b d2 w0 ð5Þ m¼ a dx2 xc The additional displacement D(x) which also decreases gradually with distance from the crack tip can be assumed to be a function similar to c(x) as follows:
DðxÞ ¼ ne-aððjx-xc jÞ=bÞ
ð6Þ
Where n is a real constant which can be found using zero shear stress condition at the crack faces. The shear strain function gxz can be found using Eq. (1) 8 <0 y o0 1 @u @w 1 a aððjxxc jÞ=bÞ gxz ¼ þ ¼ : sgnðxxc Þ y 40 mþ n e 2 @z @x b 2 ð7Þ The shear stress at the crack faces where y40 and x ¼xc + or xc should be zero so one has b b2 d2 w0 ð8Þ n¼ m¼ 2 a a dx2 xc To avoid discontinuity at the crack tip and considering the nonlinear spatial variation of displacement at the crack tip, it is assumed that the displacement field at y40 transforms into the defined functions in Eq. (1) with an exponential regime from the displacement field at yo0. So the displacement field is modified in this paper as follows: 8 dw0 > > y o0 > z < dx u¼ 2 dw0 b > bðy=dÞ d w0 > > Þ 2 eaððjxxc jÞ=bÞ sgnðxxc Þ y 40 : z dx þ a ð1e dx xc
w¼
8 w ðxÞ > < 0
b2
bðy=dÞ
d2 w0 Þ 2 eaððjxxc jÞ=bÞ dx xc
> : w0 ðxÞ a2 ð1e 8 <0 v ¼ b2 b bðy=dÞ aððjxxc jÞ=bÞ : 2 ze e a d
yo 0 y4 0
y o0 y 40
ð9Þ
In Eq. (9) b is a dimensionless parameter and will be discussed later in this paper. The term (1 e b(y/d)) prevents the discontinuity at the crack tip. The displacement component v is modified in order to gyz become zero at the crack faces.
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4. Equilibrium equations Now the strain field can be extracted from the displacement field. The normal strain component of the stress field can be written using Eq. (9) as follows: 8 > d2 w0 > > yo0 < z dx2 2 2 ex ¼ > d w b d w 0 0 > aððjxxc jÞ=bÞ > y40 ð1ebðy=dÞ Þ 12 dðxxc Þ : z x e a dx2 dx2 c
ð10Þ In each cross section one has the static equilibrium if the following conditions are met: 8R > < A ex dA ¼ 0 R ð11Þ M > : A zex dA ¼ E where A is the cross section area of the beam, E is the modulus of elasticity and M is the bending moment. The first equation of (11) is automatically satisfied if the y-axis is chosen to the centroidal axis of the cross-section. Substituting Eq. (10) into the second equation of (11) and performing appropriate calculations one has Z M d2 w0 d2 w0 k 2 xc eaððjxxc jÞ=bÞ ¼ zex dA ¼ I 2 E dx dx A Z 2 bðy=dÞ k ¼ Ic z e dA ð12Þ Ac
In Eq. (12) I and Ic are the second moment of the cross-section area and the crack face area about y-axis, respectively, and Ac is the area of the crack face as shown in Fig. 3. k is a geometrical constant which can be found for every cross section and crack size easily. Evaluating Eq. (12) at x¼ xc one has d2 w0 M ¼ ð13Þ EðIkÞ dx2 xc Now the bending differential equation of a beam with a vertical edge crack can be obtained in an explicit form using Eqs. (12) and (13) as follows: d2 w0 M k aððjxxc jÞ=bÞ ð14Þ ¼ 1þ e 2 EI Ik dx This equation is the main result of this investigation. In an undamaged beam the geometrical parameter k is zero and hence Eq. (14) turns into the familiar form of Euler–Bernoulli bending equation for slender beams. The dimensionless exponential decay rates (a,b) are the only factors which has not been discussed yet. In the next section the solution for the differential Eq. (14) is presented and then parameters a and b are calculated.
5. Solution The exact solution of differential Eq. (14) can be obtained by integration and using appropriate boundary conditions. In this paper the beam is assumed to be simply supported. However any desired boundary conditions might be used with the obtained bending equation. The right-hand side of Eq. (14) is not a smooth function therefore definite integration of this function is not possible unless one divides this function into two smooth parts as follows: 8 M k aððxxc Þ=bÞ > > x r xc 1 þ e > Ik d2 w0 < EI ¼ ð15Þ M k aððxxc Þ=bÞ > dx2 > > x 4 xc : EI 1 þ Ik e
Integrating both parts of Eq. (11) will lead to the following results: 8 2 M x b2 k aððxxc Þ=bÞ > > x r xc þc1 x þ c2 þ 2 e > < EI 2 a Ik 2 w0 ¼ ð16Þ >M x b2 k aðxxc Þ=b > > x 4 xc þc3 x þ c4 þ 2 e : EI 2 a Ik In Eqs. (16) c1, c2, c3 and c4 are constants which can be calculated using both boundary conditions and continuity conditions at the crack location as follows: 8 0 w0 ð0Þ ¼ 0 and w0 ðlÞ ¼ 0 < B:C: s : dw0 dw0 þ þ : Continuity : w0 ðxc Þ ¼ w0 ðxc Þ and ðx Þ ¼ ðx Þ dx c dx c ð17Þ Consequently, the constants c1, c2, c3 and c4 will be as follows: 8 b2 k aðxc =bÞ > > e > c2 ¼ 2 > > a Ik > > > > b k > > < c4 ¼ c2 2xc a Ik ð18Þ 2 > > c ¼ l c4 1 b k eaðlxc =bÞ > > 3 > 2 l l a2 Ik > > > > b k > > : c1 ¼ c3 2 a Ik Eqs. (16) and (18) give the load–deflection relation of the cracked beam in pure bending.
6. Exponential decay rates a and b calculation When a pair of bending moments M are applied to the cracked beam an additional relative rotation y* will exist between two ends of the beam due to the crack as shown in Fig. 4. For an Euler–Bernoulli simply supported undamaged beam the slope of the neutral axis is as follows: dw M l yh ¼ ¼ x ð19Þ dx EI 2 Using Eqs. (16) and (19) one can obtain the additional remote point rotation y* as follows:
y ¼ ðyc ð0Þyh ð0ÞÞðyc ðlÞyh ðlÞÞ ¼
lxc Mb k 2eaðxc =bÞ ea b EI a Ik ð20Þ
where yc and yh are the rotation of a cracked beam and an undamaged beam under bending, respectively. On the other hand, the additional rotation y* means that the cracked beam accumulates more strain energy compared with an undamaged beam. The extra strain energy which is called UT here, is stored at the vicinity of the crack. The additional rotation of a beam subject to a pair of bending moments at two ends as shown in Fig. 4 can be obtained from Castigliano’s theorem
y ¼
@UT @M
ð21Þ
M
M θ θ
Fig. 4. Additional rotation of a beam with a vertical crack under bending.
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Eq. (24) has an accuracy of 0.5% for any a/b [5]. From Eqs. (23) and (24) the energy release rate is a K2 1n2 Mz 2 Js ¼ I0 ¼ paF 2 ð25Þ E E I b
This additional strain energy due to the crack has the following form [9]: Z ð22Þ UT ¼ Js ðaÞdA Ac
In which Ac is the crack face area. Eq. (22) is called Paris equation and Js in this equation is the strain energy release rate which could be obtained from the following Eq. [9]: 2 !2 !2 !2 3 6 6 6 X X 14 X K Ii þ KIIi þm KIIIi 5 Js ¼ 0 ð23Þ E i¼1 i¼1 i¼1
Now substituting Eq. (25) into (22) and then using Eq. (21), the additional rotation of the cracked beam y* can be obtained in terms of bending moment and geometrical parameters. For a rectangular cross section this additional rotation is 1n2 2M pj E I2 R d=2 R a 2 2 s j ¼ ðd=2Þ 0 z sF dsdz b
y ¼
In Eq. (23), if the plane stress assumption is used then E0 ¼E and if the plane strain assumption is used then E0 ¼E/(1 u2). In this article the plane strain assumption is used. In addition, in Eq. (23) KIi, KIIi and KIIIi are the stress intensity factors (SIF) corresponding to three modes of fracture, which result for every individual loading mode i. In general a cracked structure can be loaded with three force vectors and three moment vectors. In pure bending there exist only one moment vector so the only nonzero SIF is for mode I. The beam with a vertical edge crack can be assumed to be consisting of a set of thin plates along z-axis and each plane contains an edge crack subject to axial tension or compression. This stress intensity factor for such plates is [5] pffiffiffiffiffiffi a Mz s0 ¼ KI ¼ s0 paF b I rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2b pa 4 pa tan ¼ 1þ 0:122cos F pa b 2b 2b
909
ð26Þ
The additional rotation can be evaluated using results of Eq. (20) too. Comparing two sides of Eqs. (26) and (20) one has b k
a Ik
2 ð2eaðxc =bÞ eaððlxc Þ=bÞ Þ ¼ ð1n2 Þ pj I
ð27Þ
Numerical solution of Eq. (27) will lead to finding the value of exponential decay rate a for any values of geometrical parameters and simply supported ends. However in Eq. (27) the parameter k is a function of the unknown parameter b. By comparing the finite element results and those obtained by this analytical model, one can conclude that b is a comparatively large parameter. Table 1 shows the percentage error in stress and deformation evaluation for several values of b. It can be seen in Table 1 that the deformation is not too sensitive to the value of b but the stress predicted by the developed model of this paper is very sensitive to the value of b especially near the crack tip. When the parameter b
ð24Þ
Table 1 The effect of b value on the deformation and stress evaluation errors. Percentage error in deformation
b ¼1
b ¼10
b ¼N
a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
(x xc)/l ¼0.01
(x xc)/l ¼0.1
(x xc)/l¼ 0.2
(x xc)/l ¼ 0.4
(x xc)/l ¼0.01
(x xc)/l ¼0.1
(x xc)/l¼ 0.2
(x xc)/l ¼ 0.4
5.4 6.4 7.1 5.7 4.5 5.8 3.0 3.2 4.1 1.0 2.1 3.0
4.5 5.4 6.6 4.7 4.4 4.9 2.1 2.7 2.6 0.3 0.8 1.1
4.0 4.7 5.0 4.1 4.2 4.2 1.4 1.9 2.1 0.1 0.1 0.2
2.8 3.4 4.1 2.9 4.3 5.9 1.0 2.1 1.9 0.0 1.0 0.3
18.7 44.4 80.1 16.9 38.4 65.3 8.3 14.1 18.8 2.8 4.0 5.6
11.5 15.1 22.0 6.6 4.4 11.7 0.1 2.3 6.3 0.0 0.8 1.3
11.0 14.3 18.2 4.0 2.7 9.8 0.0 0.1 2.2 0.0 0.0 0.1
9.9 13.0 17.1 1.4 1.0 5.3 0.0 0.0 1.0 0.0 0.0 0.0
6 5.5
Exponential decay rate (α)
b ¼0.1
Maximum percentage error in stress
5 4.5 4 3.5 3 2.5 2 1.5 1 0.1
0.2
0.3
0.4 0.5 0.6 Crack-Depth ratio (a/b)
0.7
0.8
0.9
Fig. 5. Exponential decay rate (a) versus crack depth ratio (a/b) for a slender simply supported beam with a vertical edge crack.
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increases, the percentage error decreases in both stress and deformation. Accordingly it can be assumed that the parameter b tends to infinity. In fact, the exponential decay rate b is obtained here by finite element analysis and correlating the analytical and finite element results. In spite of this fact, it must be noticed that this value for b is a general value and in other cases the same value can be used without separate calculations. From Eq. (27) it can be shown that the exponential a decay rate is a function of a/b, l/b and xc/l or
aa
a l xc , , b b l
ð28Þ
From Eq. (27) it can also be shown that for slender beams (l/bZ10) the slenderness factor (l/b) and the crack position ratio xc/l have a minor effect on a. In this case a is a function of crack depth ratio a/d. Fig. 5 shows a versus a/b for slender beams (l/bZ10).
7. Finite element model In order to verify the developed theory of this research, a finite element model was created and the results obtained from the developed model is compared with those obtained by finite element. The ANSYS software was utilized for this purpose. First, a 2-D model of the cracked beam face was created and meshed. The crack was modeled as a narrow notch with parallel faces and wedge-shaped front as shown in Fig. 6. In order to have an accurate and reliable model the PLANE183 singular element has been used in the cracked area [24]. This element is an 8-node quadratic solid singular element which specially designed for crack analysis. In this research a fine mesh has been used at the vicinity of the crack and dependency of the results to the mesh size has been checked. The meshed model can also be seen in Fig. 6. Then the model was extruded into a 3-D model as shown in Fig. 7. After extrusion along normal to the beams face, the 2-D PLANE183 singular element was replaced by 3-D respective
Fig. 6. Cracked beam face mesh.
Fig. 7. Cracked beam 3-D mesh.
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singular element SOLID186 [24]. This 20 node brick element is recommended for 3-D fracture model analysis [24]. Finally, the boundary conditions and bending moment were applied to the model. The displacement and stress fields are obtained solving the model and the results are presented in the next section.
8. Results for a simply supported beam with rectangular cross-section Eq. (16) and the calculated value of a while assuming b to be infinite, illustrate the solution for deflection of a beam with an edge crack. In this section a slender simply supported beam with a rectangular cross section as shown in Fig. 3 has been used for illustrative example. The load–deflection relation for the cracked beam is the direct result of this research. In order to perform a comprehensive analysis a dimensionless deflection function is defined as follows: V
x a x EI 1 c w0 ðxÞ , , ¼ l b l M l2
ð29Þ
Fig. 8 shows the function V for xc/l ¼0.5 and a/b¼0.2, 0.5 versus x/l calculated from (29) and finite element method. Fig. 9 shows the results for xc/l ¼0.8. In all of the results there is a good
911
agreement between analytical results and those obtained by FE analysis. Stress analysis of the cracked beam has also been investigated in this research. The normal stress component of a beam with a vertical crack can be calculated using Eqs. (10), (13) and (14) and assuming the decay rate constant b to be infinity as follows: 8 Mz k aððjxxc jÞ=bÞ > > yo 0 > < I 1 þ Ik e sx ¼ Mz b 2Idðxxc Þ aððjxxc jÞ=bÞ > aððjxxc jÞ=bÞ > > y4 0 þ e : I 1e a ðIkÞ ð30Þ The term d(x xc) in Eq. (30) shows the singularity at the crack pffiffiffi tip. It is not the usual 1= r type of stress singularity usually associated with stresses near a crack. However, the stress is infinite at the crack tip according to Eq. (30) similar to usual linear elastic fracture mechanics stress models. The stress is transformed into a dimensionless form by dividing to the maximum normal stress of an undamaged beam (s0 ¼Md/2I). The results for the stress ratio function are shown in Fig. 10 for xc/l ¼0.5 and a/b¼0.5, 0.2 at horizontal line at the upper cord of the cracked beam passing near the crack tip. These results are obtained by finite element method and the method presented in this article. Fig. 11 illustrates the stress regime near the crack tip at the upper cord of the beam for xc/l¼0.8. A good agreement between the
Dimensionless deformation (V)
0.16 0.14
a/b=0.5
0.12 0.1 a/b=0.2
0.08 0.06 0.04 0.02 0 -0.02 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position ratio (x/l) Fig. 8. Dimensionless deformation function V versus x/l for xc/l ¼0.5. (——): Analytical results; (): FE results.
Dimensionless deformation (V)
0.14 a/b=0.5
0.12 0.1 a/b=0.2
0.08 0.06 0.04 0.02 0 -0.02 0
0.1
0.2
0.3
0.4 0.5 0.6 Position ratio (x/l)
0.7
0.8
0.9
1
Fig. 9. Dimensionless deformation function V versus x/l for xc/l ¼0.8. (——): Analytical results; (): FE results.
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2.2
Stress ratio (σ/σ0)
2 1.8 1.6
a/b=0.5
1.4
a/b=0.2
1.2 1 0.8 0
0.1
0.2
0.3
0.4 0.5 0.6 Position ratio (x/l)
0.7
0.8
0.9
1
Fig. 10. Stress ratio (s/s0) versus position ratio x/l for xc/l ¼0.5. (——): Analytical results; (): FE results.
2.2
Stress ratio (σ/σ0)
2 1.8 1.6 a/b=0.5 1.4
a/b=0.2
1.2 1 0.8 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position ratio (x/l) Fig. 11. Stress ratio (s/s0) versus position ratio x/l for xc/l ¼0.8. (——): Analytical results; (): FE results.
analytical and finite element results can be realized from Figs. 10 and 11. One can see that even near the crack tip the results are almost the same. It must be noticed that the Figs. 10 and 11 do not contain the stress value exactly at the crack tip because of the singularity.
Acknowledgment The authors would like to acknowledge the ‘‘Iranian Gas Transmission Co.’’ for their financial assistance throughout this research. References
9. Conclusions A new continuous model for bending analysis of a beam with a vertical edge crack has been developed in this paper. This model can be used for load–deflection and stress–strain assessment of a beam with an edge vertical crack subject to pure bending. It is assumed that the crack faces and their adjacent area rotate more than an undamaged beam about the neutral axis and this additional rotation decreases exponentially with distance from the crack tip. The displacement field of the beam has been obtained using this assumption and has been modified for compatibility and continuity. The bending differential equation of the cracked beam has been calculated using static equilibrium equations. The final equation is a simple and accurate mathematical model for load– deflection analysis of beams with vertical crack under bending. An explicit expression for the normal stress function of the beam has also been presented. The calculated results for deflection and stress of the cracked beam have been compared with finite element results for verification and an excellent agreement has been observed.
[1] Irwin GR. The crack extension force for a part through crack in a plate. J Appl Mech 1962;29:651–4. [2] Irwin GR. Encyclopedia of physics, vol. 6. Berlin: Springer; 1958. [3] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech 1957;24:109–14. [4] Sedov LI. A course in continuum mechanics, vol. 4. Groningen: VoltersNoordhoff; 1972. [5] Tada H, Paris PC, Irvin GR. The stress analysis of cracks handbook, 3rd ed.. Professional engineering publishing; 2000. [6] Meguid SA. Engineering fracture mechanics. Elsevier Applied Science; 1989. [7] Hellan K. Introduction to fracture mechanics. McGraw-Hill; 1984. [8] Sanford RJ. Principles of fracture mechanics. Prentice Hall; 2003. [9] Dimarogonas AD. Vibration of cracked structures—a state of the art review. Eng Fract Mech 1996;5:831–57. [10] Zheng DY, Fan SC. Vibration and stability of cracked hollow-sectional beams. J Sound Vib 2003;267:933–54. [11] Lin Hai-Ping. Direct and inverse methods on free vibration analysis of simply supported beams with a crack. Eng Struct 2004;26(4):427–36. [12] Zheng Y, Kessissoglou NJ. Free vibration analysis of a cracked beam by finite element method. J Sound Vib 2004;273(3):457–75. [13] Dado MHF, Abuzeid O. Coupled transverse and axial vibratory behaviour of cracked beam with end mass and rotary inertia. J Sound Vib 2003;261: 675–96. [14] Loyaa A, Rubiob L, Ferna´ndez-Sa´ez J. Natural frequencies for bending next term vibrations of Timoshenko previous term cracked beams. J Sound Vib 2006;290:640–53.
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[15] Christides S, Barr ADS. One-dimensional theory of cracked Bernoulli–Euler beams. J Mech Eng Sci 1984;26(11/12):639–48. [16] Shen MHH, Pierre C. Free vibrations of beams with a single-edge crack. J Sound Vib 1994;170(2):237–59. [17] Carneiro SHS, Inman DJ. Comments on the free vibration of beams with a single-edge crack. J Sound Vib 2001;244(4):729–37. [18] Chondros TG, Dimarogonas AD, Yao J. A continuous cracked beam vibration theory. J Sound Vib 1998;215(1):17–34. [19] Yang XF, Swamidas ASJ, Seshadri R. Crack identification in vibrating beams using the energy method. J Sound Vib 2001;244(2):339–57.
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[20] Behzad M, Meghdari A, Ebrahimi A. A linear theory for bending stress–strain analysis of a beam with an edge crack. Eng Fract Mech 2008;75(16):4695–705. [21] Behzad M, Ebrahimi A, Meghdari A. A new continuous model for flexural vibration analysis of a cracked beam. Pol Marit Res 2008;15(2):32–9. [22] Behzad M Meghdari A, Ebrahimi A. A new approach for vibration analysis of a cracked beam. Int J Eng 2005;18(4):319–30. [23] Behzad M, Meghdari A, Ebrahimi A. A continuous model for forced vibration analysis of a cracked beam. In: Proceeding of 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida USA. [24] ANSYS User’s Manual for rev. 8, ANSYS Inc., 2004.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A bending theory for beams with vertical edge crack A. Ebrahimi , M. Behzad, A. Meghdari Sharif University of Technology, Mechanical Engineering Department, P.O. Box 11365-9567, Azadi Ave., Tehran, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 December 2008 Received in revised form 6 March 2010 Accepted 12 March 2010 Available online 18 March 2010
In this paper a linear continuous theory for bending analysis of beams with an edge crack perpendicular to the neutral plane subject to bending has been developed. The model assumes that the displacement field is a superposition of the classical Euler–Bernoulli beam’s displacement and of a displacement due to the crack. It is assumed that in bending the additional displacement due to crack decreases exponentially with distance from the crack tip. The strain and stress fields have been calculated using this displacement field and the bending equation has been obtained using equilibrium equations. Using a fracture mechanics approach the exponential decay rate has been calculated. There is a good agreement between the analytical results from solving the differential equation of cracked beam and those obtained by finite element method. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Cracked beam Vertical crack Bending Stress–strain Load–deflection
1. Introduction Since late 1950s the crack issue and the behavior of structural elements contain one or more cracks have been considered by researchers seriously. The first attempts for finding an analytical solution for stress and displacement fields near the crack tip were performed by Irwin [1,2] and Williams [3]. In these works an Airy stress function for an infinite thin plate with a through crack has been suggested and a general governing equation for stress state near the crack tip has been derived. Later, some researchers tried to find a solution for this equation. Sedov presented the general solution for an internal crack in an infinite plate using plane state stress assumption for symmetric (mode I) and antisymmetric (mode 2) cases [4]. In spite of many efforts for finding an exact solution for stress state in cracked bodies, exact closed form analytical solutions is only found for cracks in infinity large bodies under pure tension. Furthermore this exact solution is only accurate near the crack tip and far from the crack tip this solution cannot be implemented. The necessity of finding stress and displacement fields in cracked bodies especially near the crack tip in real cases persuade the researchers to develop numerical and empirical methods. A large number of empirical and numerical formulations have been reported for several continua with different forms of crack under various forms of loading. The most important and useful forms of these formulae has been collected by Tada et al. [5]. Further
Corresponding author. Tel.: + 98 21 66165556; fax: + 98 21 66165509.
E-mail address: [email protected] (A. Ebrahimi). 0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.03.004
developments in elastic and elasto-plastic fracture mechanics can be found in the literature [6–8]. The load–deflection behavior of a cracked beam in bending and the stress–strain analysis are engineering problems considered by researchers. Typically, in order to have load deflection relation for such a beam one should transform the load to stress field, then relate the stress field to strain field and finally transform back the strain field into the deflection or start this cycle from the deflection towards the load. However, it is hard to relate the load to stress and deflection to strain in the whole of a cracked body because of the complex nature of the crack and nonlinear spatial variation in stresses. Accordingly, developing a comprehensive and exact model which can describe all phenomena occurring at the vicinity and far from the crack simultaneously is not a simple task. Yet researchers tried to develop some models to evaluate the behavior of a cracked beam in bending. These models are mostly not exact but have a reasonable error for engineering applications. One of the simplest models for beams with edge crack in bending has been presented by Dimarogonas [9]. He assumed the cracked beam consists of two normal parts connected by a rotational spring at the position of the crack. This local flexibility idea has been followed by several researchers till now [10–14]. This model is only applicable for load-deflection approximation of a cracked beam far from the crack tip and cannot be considered for stress analysis. Some researchers preferred to use a continuous displacement and stress field rather than the local flexibility model. Such an approach can lead to a better result at the vicinity of the crack [15–19]. In their approach the crack effect can be taken into account by applying some modifications on the stress and displacement fields of an ordinary Euler–Bernoulli beam. The
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Nomenclature a crack length A cross-sectional area area of the crack face Ac b width of the beam d height of the beam ci(i¼1,2,3) integration constants E modulus of elasticity F(a/d) crack shape factor I second moment of the section area about y-axis second moment of the crack face area about y-axis Ic strain energy density function (J-integral) Js k a geometrical factor for the cross section of a cracked beam Ki(i¼I,II,III) stress intensity factors L length of the beam M bending moment m, n real constants u displacement component along x-axis
main defect in this approach is that two independent functions for stress and displacement fields are considered by guesswork. In fact the suggested functions are incompatible and the model can only be used for a specific application such as estimation of the first natural frequency of the cracked beams. Behzad et al. [20] presented a new continuous theory for bending analysis of a beam with an edge crack. A bilinear displacement field suggested for the beam strain and stress calculations and the bending differential equation has been obtained using equilibrium equations. The model can predict the load–deflection relation of the beam near or far from the crack tip accurately and can be also used for stress–strain analysis in a cracked beam. This model is also used for vibration analysis of a
Neutral Plane
Crack front Bending moment Crack front
UT v V w w0 xc a,b
D
ex gxz, gyz j n y y* yc yh
s0 sx c
905
additional strain energy due to the crack displacement component along y-axis dimensionless deformation of the cracked beam displacement component along z-axis displacement along z-axis for yo0 crack position exponential decay rates additional deformation along z-axis normal strain shear strain shape coefficient for additional remote point rotation Poisson ratio slope function of a cracked beam subject to bending additional remote rotation of a cracked beam rotation of a cracked beam under bending rotation of an undamaged beam under bending maximum normal stress of an undamaged beam subject to bending normal stress additional rotation of the beam cross-sections
cracked beam and showed an excellent performance in dynamic loading too [21–23]. In all previous continuous approaches it is assumed that the crack front is parallel with the neutral plane of the uncracked beam and with the bending moment vector too (Fig. 1(a) and (b)). This type of crack is more probable to be created and other forms of cracks tend to grow in this mode in bending. However it is possible that the crack front make an arbitrary angle with the bending moment vector. For example suppose a rotating cracked rotor under a constant unidirectional bending moment. When the rotor rotates, the crack front makes different angles with the moment vector in each position. In bending of a cracked beam one may define a crack with its front line perpendicular to the neutral plane of the uncracked beam, a ‘‘vertical crack’’ as shown in Fig. 1(c). In this case, the crack front is perpendicular to the bending moment vector too and the crack surfaces pass vertically through the uncracked beam’s neutral plane, with the upper half of the crack surface subjected to opening due to beam tension and the lower half closing due to compressive beam stresses. In practice, when a crack exists in an element, one may end up with a vertical crack in structures depending on the orientation of the element. The other application of the study of this type of cracks is in the area of rotor dynamics where a cracked beam rotates. In this paper a continuous model for bending analysis of a beam with an edge vertical crack has been presented for the first time. From experimental and numerical observations, a quasilinear displacement field has been introduced for a beam with an edge vertical crack. The strain and stress fields have been calculated from the displacement. The bending differential equation of the cracked beam has been extracted from equilibrium equations. The required constants needed in this model can be obtained using fracture mechanics. The results of this study are compared with the finite element results for verification.
2. Definitions and assumptions
Bending moment Fig. 1. (a) A normal beam and the neutral axis in bending, (b) a cracked beam with parallel crack front and (c) a cracked beam with perpendicular crack front.
The basic assumption in the Euler–Bernoulli bending theory for beams is that the plane sections of beam which are perpendicular to the neutral axis remain plane and perpendicular to the neutral axis after deformation. In the presence of an edge
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crack, the planes will not remain plane after deformation particularly near the vicinity of the crack due to a shear stress near the crack tip which leads to warping in plane sections. Thus at the vicinity of the crack the displacement field is completely nonlinear. For the planes far from the crack tip, the warping will be smaller and the displacement filed can be assumed linear. In order to have a better sense of the bending in a cracked beam, a finite element model has been produced in this research. The details of this model are presented in Section 7 of this paper. The mid-span vertical crack behavior under a pure bending moment can be seen in Fig. 2. In this model the beam is assumed to be a slender prismatic beam with rectangular cross-section made of a linear elastic material with small deformations. A vertical narrow notch with parallel faces at the mid-span is used for modeling the crack in this research. Note that the grid lines of Fig. 2 are only some hypothetical lines which show the deformation field of the beam and these lines are not referring to the finite element mesh discussed in Section 7. Near the crack area the plane sections will no longer remain plane. In fact in the crack faces due to lack of normal stress there exists an additional rotation in comparison with the remaining part of the section. This additional rotation dissipates gradually, while the distance from the crack tip increases. With a good
approximation it can be supposed that each plane section turns into two straight planes after deformation. Each straight plane section turns into two planes with different slopes one in the right side and the other in the left side of the crack edge. The slope difference between these two planes decreases with distance from the crack tip. These two straight planes connect to each other through a nonlinear part near the horizontal line passing through the crack tip. For a crack in a location other than the midspan of the beam, similar discussion can be made. Fig. 3 shows the coordinate system and the parameter definitions graphically for a typical cracked beam with arbitrary position of the crack. In order to find the stress, strain and deformation functions for a beam with a vertical crack in pure bending a displacement field for the beam has been suggested in this research. In fact it is assumed that each plane section turns into two straight planes and a nonlinear connector after deformation. The essential assumptions used in this research can be listed as follows:
The beam is slender and prismatic. The beam is symmetric with respect to the x–y plane so the y-axis can be assumed to be the neutral axis in pure bending.
The crack is considered to be an open edge narrow notch with parallel faces and wedge-shaped front.
Fig. 2. Displacement field illustration in a beam with a vertical edge crack subject to bending.
z
z b
Ah
d Ac
y
x
a
d
L
b a
xc y u(x,y,z)
x
b a
zψ(x)
z
y ) ψ(x
x
θ(x)
y
z
d
Fig. 3. Coordinate system and parameters definition.
x
w(x,y)
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The deformations are supposed to be small. The plane strain assumption has been used in this research.
Consequently the strain component along z-axis has been neglected. The applied load is small enough and the crack does not grow. The material is assumed to be linear elastic. It is assumed that the crack faces do not contact in bending. This assumption is true when the crack faces have enough distance from each other and the bending moment is small enough, or if the bending moment is superimposed with axial tension, preventing crack surface contact.
3. Displacement field With refer to Figs. 2 and 3 and the above assumptions the displacement field for a beam with vertical edge crack can be defined. It is well known that the displacement and stress fields near the crack tip are 3D functions. In this paper at first a 3-dimensional displacement field has been introduced for the beam but afterwards the equations are integrated over the crosssection area of the beam and a 1-dimensional load–deflection relation has been obtained for the beam. This equation is not an exact equation but the results of this research show the good engineering approximation of this relation. The crack section consists of two parts: the crack face which is shown in Fig. 2 by Ac and the remaining part of the section which is shown by Ah in this research. Under pure bending the healthy part of the cross section (Ah) rotates about its neutral axis which is coincident with y-axis in this research. This planar part remains plane after rotation and perpendicular to the neutral axis. The crack face rotates about y-axis too but it rotates more than the remaining part of the section. Consequently, the crack face does not remain perpendicular to the neutral axis due to the shear stress near the crack tip. The crack face can also be assumed to remain plane after deformation except at the small area near the crack tip. The rotation difference between the crack face and the remaining part of the section inherits to the adjacent cross sections but gradually the magnitude of this difference decreases. As a side effect, the deformation of the beam along z-axis is a function of y. In fact the parts of the beam sections which have more rotation cause more vertical displacement too. The numerical simulations with ANSYS software confirm this phenomenon. On the base of the above explanations, the following displacement field is introduced for a beam with a vertical edge crack subject to pure bending: ( 8 yo 0 zyðxÞ > > >u¼ > > y ðxÞ þ c ðxÞÞ y4 0 zð > > < ( ð1Þ w0 ðxÞ yo0 > > w¼ > > ðxÞ þ D ðxÞ y 4 0 w 0 > > > : v¼0 In which u, v, w are the displacement components along x, y and z axes. y(x) is the rotation of the part of the section with yo0 as shown in Fig. 3. c(x) is the additional rotation for that part of the section with y40 and D(x) is the additional vertical displacement of the beam for y40. By assuming that the plane sections in yo0 remain perpendicular to the neutral axis one has dw0 ðxÞ yðxÞ ¼ dx
ð2Þ
The additional rotation c(x) of that part of plane sections with y40 has its maximum value at the crack faces and decreases gradually with distance from the crack tip. This additional rotation is a nonlinear and complicated variable with respect to
907
x. Here in this research an exponential regime has been assumed for c(x) along the x-axis as follows:
cðxÞ ¼ meaððjxxc jÞ=bÞ sgnðxxc Þ
ð3Þ
In Eq. (3) m is a real constant, a is a dimensionless exponential decay rate which will be obtained later in this paper, xc is the crack position, d is the depth of the beam and sgn(xxc) is the sign function which is 1 for xoxc and +1 for x4xc. The application of sign function is due to the fact that the additional rotation function has a discontinuity at the position of the crack and the sign of its value changes when passing through the crack tip. In order to find the real constant m, zero normal stress condition at the crack faces can be used. The normal strain function can be found using Eq. (1) 8 d2 w0 > > > z yo0 dx2 @u < 2 ¼ ex ¼ d w a @x > 0 > > z m eaððjx-xc jÞ=bÞ þ 2me-aððjx-xc jÞ=bÞ dðx-xc Þ y40 : b dx2 ð4Þ In which d(x xc) is the Dirac delta function. The normal stress at the crack faces where y40 and x¼xc + or xc should be zero so one has b d2 w0 ð5Þ m¼ a dx2 xc The additional displacement D(x) which also decreases gradually with distance from the crack tip can be assumed to be a function similar to c(x) as follows:
DðxÞ ¼ ne-aððjx-xc jÞ=bÞ
ð6Þ
Where n is a real constant which can be found using zero shear stress condition at the crack faces. The shear strain function gxz can be found using Eq. (1) 8 <0 y o0 1 @u @w 1 a aððjxxc jÞ=bÞ gxz ¼ þ ¼ : sgnðxxc Þ y 40 mþ n e 2 @z @x b 2 ð7Þ The shear stress at the crack faces where y40 and x ¼xc + or xc should be zero so one has b b2 d2 w0 ð8Þ n¼ m¼ 2 a a dx2 xc To avoid discontinuity at the crack tip and considering the nonlinear spatial variation of displacement at the crack tip, it is assumed that the displacement field at y40 transforms into the defined functions in Eq. (1) with an exponential regime from the displacement field at yo0. So the displacement field is modified in this paper as follows: 8 dw0 > > y o0 > z < dx u¼ 2 dw0 b > bðy=dÞ d w0 > > Þ 2 eaððjxxc jÞ=bÞ sgnðxxc Þ y 40 : z dx þ a ð1e dx xc
w¼
8 w ðxÞ > < 0
b2
bðy=dÞ
d2 w0 Þ 2 eaððjxxc jÞ=bÞ dx xc
> : w0 ðxÞ a2 ð1e 8 <0 v ¼ b2 b bðy=dÞ aððjxxc jÞ=bÞ : 2 ze e a d
yo 0 y4 0
y o0 y 40
ð9Þ
In Eq. (9) b is a dimensionless parameter and will be discussed later in this paper. The term (1 e b(y/d)) prevents the discontinuity at the crack tip. The displacement component v is modified in order to gyz become zero at the crack faces.
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4. Equilibrium equations Now the strain field can be extracted from the displacement field. The normal strain component of the stress field can be written using Eq. (9) as follows: 8 > d2 w0 > > yo0 < z dx2 2 2 ex ¼ > d w b d w 0 0 > aððjxxc jÞ=bÞ > y40 ð1ebðy=dÞ Þ 12 dðxxc Þ : z x e a dx2 dx2 c
ð10Þ In each cross section one has the static equilibrium if the following conditions are met: 8R > < A ex dA ¼ 0 R ð11Þ M > : A zex dA ¼ E where A is the cross section area of the beam, E is the modulus of elasticity and M is the bending moment. The first equation of (11) is automatically satisfied if the y-axis is chosen to the centroidal axis of the cross-section. Substituting Eq. (10) into the second equation of (11) and performing appropriate calculations one has Z M d2 w0 d2 w0 k 2 xc eaððjxxc jÞ=bÞ ¼ zex dA ¼ I 2 E dx dx A Z 2 bðy=dÞ k ¼ Ic z e dA ð12Þ Ac
In Eq. (12) I and Ic are the second moment of the cross-section area and the crack face area about y-axis, respectively, and Ac is the area of the crack face as shown in Fig. 3. k is a geometrical constant which can be found for every cross section and crack size easily. Evaluating Eq. (12) at x¼ xc one has d2 w0 M ¼ ð13Þ EðIkÞ dx2 xc Now the bending differential equation of a beam with a vertical edge crack can be obtained in an explicit form using Eqs. (12) and (13) as follows: d2 w0 M k aððjxxc jÞ=bÞ ð14Þ ¼ 1þ e 2 EI Ik dx This equation is the main result of this investigation. In an undamaged beam the geometrical parameter k is zero and hence Eq. (14) turns into the familiar form of Euler–Bernoulli bending equation for slender beams. The dimensionless exponential decay rates (a,b) are the only factors which has not been discussed yet. In the next section the solution for the differential Eq. (14) is presented and then parameters a and b are calculated.
5. Solution The exact solution of differential Eq. (14) can be obtained by integration and using appropriate boundary conditions. In this paper the beam is assumed to be simply supported. However any desired boundary conditions might be used with the obtained bending equation. The right-hand side of Eq. (14) is not a smooth function therefore definite integration of this function is not possible unless one divides this function into two smooth parts as follows: 8 M k aððxxc Þ=bÞ > > x r xc 1 þ e > Ik d2 w0 < EI ¼ ð15Þ M k aððxxc Þ=bÞ > dx2 > > x 4 xc : EI 1 þ Ik e
Integrating both parts of Eq. (11) will lead to the following results: 8 2 M x b2 k aððxxc Þ=bÞ > > x r xc þc1 x þ c2 þ 2 e > < EI 2 a Ik 2 w0 ¼ ð16Þ >M x b2 k aðxxc Þ=b > > x 4 xc þc3 x þ c4 þ 2 e : EI 2 a Ik In Eqs. (16) c1, c2, c3 and c4 are constants which can be calculated using both boundary conditions and continuity conditions at the crack location as follows: 8 0 w0 ð0Þ ¼ 0 and w0 ðlÞ ¼ 0 < B:C: s : dw0 dw0 þ þ : Continuity : w0 ðxc Þ ¼ w0 ðxc Þ and ðx Þ ¼ ðx Þ dx c dx c ð17Þ Consequently, the constants c1, c2, c3 and c4 will be as follows: 8 b2 k aðxc =bÞ > > e > c2 ¼ 2 > > a Ik > > > > b k > > < c4 ¼ c2 2xc a Ik ð18Þ 2 > > c ¼ l c4 1 b k eaðlxc =bÞ > > 3 > 2 l l a2 Ik > > > > b k > > : c1 ¼ c3 2 a Ik Eqs. (16) and (18) give the load–deflection relation of the cracked beam in pure bending.
6. Exponential decay rates a and b calculation When a pair of bending moments M are applied to the cracked beam an additional relative rotation y* will exist between two ends of the beam due to the crack as shown in Fig. 4. For an Euler–Bernoulli simply supported undamaged beam the slope of the neutral axis is as follows: dw M l yh ¼ ¼ x ð19Þ dx EI 2 Using Eqs. (16) and (19) one can obtain the additional remote point rotation y* as follows:
y ¼ ðyc ð0Þyh ð0ÞÞðyc ðlÞyh ðlÞÞ ¼
lxc Mb k 2eaðxc =bÞ ea b EI a Ik ð20Þ
where yc and yh are the rotation of a cracked beam and an undamaged beam under bending, respectively. On the other hand, the additional rotation y* means that the cracked beam accumulates more strain energy compared with an undamaged beam. The extra strain energy which is called UT here, is stored at the vicinity of the crack. The additional rotation of a beam subject to a pair of bending moments at two ends as shown in Fig. 4 can be obtained from Castigliano’s theorem
y ¼
@UT @M
ð21Þ
M
M θ θ
Fig. 4. Additional rotation of a beam with a vertical crack under bending.
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Eq. (24) has an accuracy of 0.5% for any a/b [5]. From Eqs. (23) and (24) the energy release rate is a K2 1n2 Mz 2 Js ¼ I0 ¼ paF 2 ð25Þ E E I b
This additional strain energy due to the crack has the following form [9]: Z ð22Þ UT ¼ Js ðaÞdA Ac
In which Ac is the crack face area. Eq. (22) is called Paris equation and Js in this equation is the strain energy release rate which could be obtained from the following Eq. [9]: 2 !2 !2 !2 3 6 6 6 X X 14 X K Ii þ KIIi þm KIIIi 5 Js ¼ 0 ð23Þ E i¼1 i¼1 i¼1
Now substituting Eq. (25) into (22) and then using Eq. (21), the additional rotation of the cracked beam y* can be obtained in terms of bending moment and geometrical parameters. For a rectangular cross section this additional rotation is 1n2 2M pj E I2 R d=2 R a 2 2 s j ¼ ðd=2Þ 0 z sF dsdz b
y ¼
In Eq. (23), if the plane stress assumption is used then E0 ¼E and if the plane strain assumption is used then E0 ¼E/(1 u2). In this article the plane strain assumption is used. In addition, in Eq. (23) KIi, KIIi and KIIIi are the stress intensity factors (SIF) corresponding to three modes of fracture, which result for every individual loading mode i. In general a cracked structure can be loaded with three force vectors and three moment vectors. In pure bending there exist only one moment vector so the only nonzero SIF is for mode I. The beam with a vertical edge crack can be assumed to be consisting of a set of thin plates along z-axis and each plane contains an edge crack subject to axial tension or compression. This stress intensity factor for such plates is [5] pffiffiffiffiffiffi a Mz s0 ¼ KI ¼ s0 paF b I rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2b pa 4 pa tan ¼ 1þ 0:122cos F pa b 2b 2b
909
ð26Þ
The additional rotation can be evaluated using results of Eq. (20) too. Comparing two sides of Eqs. (26) and (20) one has b k
a Ik
2 ð2eaðxc =bÞ eaððlxc Þ=bÞ Þ ¼ ð1n2 Þ pj I
ð27Þ
Numerical solution of Eq. (27) will lead to finding the value of exponential decay rate a for any values of geometrical parameters and simply supported ends. However in Eq. (27) the parameter k is a function of the unknown parameter b. By comparing the finite element results and those obtained by this analytical model, one can conclude that b is a comparatively large parameter. Table 1 shows the percentage error in stress and deformation evaluation for several values of b. It can be seen in Table 1 that the deformation is not too sensitive to the value of b but the stress predicted by the developed model of this paper is very sensitive to the value of b especially near the crack tip. When the parameter b
ð24Þ
Table 1 The effect of b value on the deformation and stress evaluation errors. Percentage error in deformation
b ¼1
b ¼10
b ¼N
a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼ a/b¼
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
(x xc)/l ¼0.01
(x xc)/l ¼0.1
(x xc)/l¼ 0.2
(x xc)/l ¼ 0.4
(x xc)/l ¼0.01
(x xc)/l ¼0.1
(x xc)/l¼ 0.2
(x xc)/l ¼ 0.4
5.4 6.4 7.1 5.7 4.5 5.8 3.0 3.2 4.1 1.0 2.1 3.0
4.5 5.4 6.6 4.7 4.4 4.9 2.1 2.7 2.6 0.3 0.8 1.1
4.0 4.7 5.0 4.1 4.2 4.2 1.4 1.9 2.1 0.1 0.1 0.2
2.8 3.4 4.1 2.9 4.3 5.9 1.0 2.1 1.9 0.0 1.0 0.3
18.7 44.4 80.1 16.9 38.4 65.3 8.3 14.1 18.8 2.8 4.0 5.6
11.5 15.1 22.0 6.6 4.4 11.7 0.1 2.3 6.3 0.0 0.8 1.3
11.0 14.3 18.2 4.0 2.7 9.8 0.0 0.1 2.2 0.0 0.0 0.1
9.9 13.0 17.1 1.4 1.0 5.3 0.0 0.0 1.0 0.0 0.0 0.0
6 5.5
Exponential decay rate (α)
b ¼0.1
Maximum percentage error in stress
5 4.5 4 3.5 3 2.5 2 1.5 1 0.1
0.2
0.3
0.4 0.5 0.6 Crack-Depth ratio (a/b)
0.7
0.8
0.9
Fig. 5. Exponential decay rate (a) versus crack depth ratio (a/b) for a slender simply supported beam with a vertical edge crack.
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increases, the percentage error decreases in both stress and deformation. Accordingly it can be assumed that the parameter b tends to infinity. In fact, the exponential decay rate b is obtained here by finite element analysis and correlating the analytical and finite element results. In spite of this fact, it must be noticed that this value for b is a general value and in other cases the same value can be used without separate calculations. From Eq. (27) it can be shown that the exponential a decay rate is a function of a/b, l/b and xc/l or
aa
a l xc , , b b l
ð28Þ
From Eq. (27) it can also be shown that for slender beams (l/bZ10) the slenderness factor (l/b) and the crack position ratio xc/l have a minor effect on a. In this case a is a function of crack depth ratio a/d. Fig. 5 shows a versus a/b for slender beams (l/bZ10).
7. Finite element model In order to verify the developed theory of this research, a finite element model was created and the results obtained from the developed model is compared with those obtained by finite element. The ANSYS software was utilized for this purpose. First, a 2-D model of the cracked beam face was created and meshed. The crack was modeled as a narrow notch with parallel faces and wedge-shaped front as shown in Fig. 6. In order to have an accurate and reliable model the PLANE183 singular element has been used in the cracked area [24]. This element is an 8-node quadratic solid singular element which specially designed for crack analysis. In this research a fine mesh has been used at the vicinity of the crack and dependency of the results to the mesh size has been checked. The meshed model can also be seen in Fig. 6. Then the model was extruded into a 3-D model as shown in Fig. 7. After extrusion along normal to the beams face, the 2-D PLANE183 singular element was replaced by 3-D respective
Fig. 6. Cracked beam face mesh.
Fig. 7. Cracked beam 3-D mesh.
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singular element SOLID186 [24]. This 20 node brick element is recommended for 3-D fracture model analysis [24]. Finally, the boundary conditions and bending moment were applied to the model. The displacement and stress fields are obtained solving the model and the results are presented in the next section.
8. Results for a simply supported beam with rectangular cross-section Eq. (16) and the calculated value of a while assuming b to be infinite, illustrate the solution for deflection of a beam with an edge crack. In this section a slender simply supported beam with a rectangular cross section as shown in Fig. 3 has been used for illustrative example. The load–deflection relation for the cracked beam is the direct result of this research. In order to perform a comprehensive analysis a dimensionless deflection function is defined as follows: V
x a x EI 1 c w0 ðxÞ , , ¼ l b l M l2
ð29Þ
Fig. 8 shows the function V for xc/l ¼0.5 and a/b¼0.2, 0.5 versus x/l calculated from (29) and finite element method. Fig. 9 shows the results for xc/l ¼0.8. In all of the results there is a good
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agreement between analytical results and those obtained by FE analysis. Stress analysis of the cracked beam has also been investigated in this research. The normal stress component of a beam with a vertical crack can be calculated using Eqs. (10), (13) and (14) and assuming the decay rate constant b to be infinity as follows: 8 Mz k aððjxxc jÞ=bÞ > > yo 0 > < I 1 þ Ik e sx ¼ Mz b 2Idðxxc Þ aððjxxc jÞ=bÞ > aððjxxc jÞ=bÞ > > y4 0 þ e : I 1e a ðIkÞ ð30Þ The term d(x xc) in Eq. (30) shows the singularity at the crack pffiffiffi tip. It is not the usual 1= r type of stress singularity usually associated with stresses near a crack. However, the stress is infinite at the crack tip according to Eq. (30) similar to usual linear elastic fracture mechanics stress models. The stress is transformed into a dimensionless form by dividing to the maximum normal stress of an undamaged beam (s0 ¼Md/2I). The results for the stress ratio function are shown in Fig. 10 for xc/l ¼0.5 and a/b¼0.5, 0.2 at horizontal line at the upper cord of the cracked beam passing near the crack tip. These results are obtained by finite element method and the method presented in this article. Fig. 11 illustrates the stress regime near the crack tip at the upper cord of the beam for xc/l¼0.8. A good agreement between the
Dimensionless deformation (V)
0.16 0.14
a/b=0.5
0.12 0.1 a/b=0.2
0.08 0.06 0.04 0.02 0 -0.02 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position ratio (x/l) Fig. 8. Dimensionless deformation function V versus x/l for xc/l ¼0.5. (——): Analytical results; (): FE results.
Dimensionless deformation (V)
0.14 a/b=0.5
0.12 0.1 a/b=0.2
0.08 0.06 0.04 0.02 0 -0.02 0
0.1
0.2
0.3
0.4 0.5 0.6 Position ratio (x/l)
0.7
0.8
0.9
1
Fig. 9. Dimensionless deformation function V versus x/l for xc/l ¼0.8. (——): Analytical results; (): FE results.
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2.2
Stress ratio (σ/σ0)
2 1.8 1.6
a/b=0.5
1.4
a/b=0.2
1.2 1 0.8 0
0.1
0.2
0.3
0.4 0.5 0.6 Position ratio (x/l)
0.7
0.8
0.9
1
Fig. 10. Stress ratio (s/s0) versus position ratio x/l for xc/l ¼0.5. (——): Analytical results; (): FE results.
2.2
Stress ratio (σ/σ0)
2 1.8 1.6 a/b=0.5 1.4
a/b=0.2
1.2 1 0.8 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position ratio (x/l) Fig. 11. Stress ratio (s/s0) versus position ratio x/l for xc/l ¼0.8. (——): Analytical results; (): FE results.
analytical and finite element results can be realized from Figs. 10 and 11. One can see that even near the crack tip the results are almost the same. It must be noticed that the Figs. 10 and 11 do not contain the stress value exactly at the crack tip because of the singularity.
Acknowledgment The authors would like to acknowledge the ‘‘Iranian Gas Transmission Co.’’ for their financial assistance throughout this research. References
9. Conclusions A new continuous model for bending analysis of a beam with a vertical edge crack has been developed in this paper. This model can be used for load–deflection and stress–strain assessment of a beam with an edge vertical crack subject to pure bending. It is assumed that the crack faces and their adjacent area rotate more than an undamaged beam about the neutral axis and this additional rotation decreases exponentially with distance from the crack tip. The displacement field of the beam has been obtained using this assumption and has been modified for compatibility and continuity. The bending differential equation of the cracked beam has been calculated using static equilibrium equations. The final equation is a simple and accurate mathematical model for load– deflection analysis of beams with vertical crack under bending. An explicit expression for the normal stress function of the beam has also been presented. The calculated results for deflection and stress of the cracked beam have been compared with finite element results for verification and an excellent agreement has been observed.
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