Effects of an edge crack on the free vibration and lateral buckling of a cantilever laminated composite slender beam

Abstract

The objective of this paper is to investigate the effects of laminate scheme, crack ratio (a/h) and crack position (L₁/L) on the free vibration and lateral buckling of a cantilever slender rectangular beam such as an airplane wing or a wind turbine blade with a single edge crack. The local flexibility approach is adopted to model the crack, based on linear fracture mechanics and the Castigliano theorem. The governing matrix equations for free vibration and buckling load are derived from the standard beam element and cracked beam element combined with the local flexibility concept. The results are obtained numerically using the finite element method, based on the energy approach and are given for crack ratios up to 0.5 at crack positions up to 0.9 ratio. Polynomial approximation for displacements gives results with reasonable accuracy.

Keywords

Buckling load cantilever cracked beam flexibility concept laminated composite natural frequency

1. Introduction

The advantages of composites such as high strength-to-weight ratio, excellent fatigue and corrosion resistance, good impact resistance, design flexibility and lower part count enable them to be used in new arenas. Commercial composites find applications in industries such as automotive components, boats, consumer goods and corrosion-resistant industrial parts, whereas advanced composites, initially developed for the military aerospace industry, offer performance superior to that of conventional structural metals and now find applications in communications satellites, aircraft, sporting goods, transportation, heavy industry and also in the energy sector, in both wind turbine construction and oil and gas exploration (Sourcebook 2007, 2006).

Composites have enabled the makers to build very light and very strong, long-endurance aircrafts. For example, unmanned aerial vehicles (UAVs) are currently the fastest-growing segment of the aerospace sector and composites are the materials of choice for these vehicles, which can range from a few inches in length to the size of a commercial airliner (Black, 2006). UAVs may have wings with high aspect ratio and large wingspans compared to their fuselage. Similarly, the wind energy industry uses composite blades with high aspect ratio on large, utility-scale wind turbines.

Consequently, beams and beamlike elements are principal constituents of many mechanical structures and are used widely in the aforementioned areas.

The lateral-torsional buckling performance of deep slender beams like turbine blades, helicopter rotor blades and aircraft wings is a limiting state that may often be a control factor in beam designs. However, during operation, all structural elements are subjected to degenerative effects that may initiate defects such as cracks which, as time progresses, lead to the catastrophic failure or breakdown of the structure (Kisa, 2004).

So, the knowledge of the behavior of a defective element is of great importance to be able to maintain the structural integrity and present safety parameters. Also, the weakened buckling capacity due to the existence of cracks is an important issue that deserves special attention (Yang and Chen, 2008). As has been stated in almost all papers related to the cracked structures, the existence of a crack reduces the local stiffness and consequently the static, dynamic and stability behaviors of a structural element are altered such that these alterations may be used to detect the crack location and its size (Qian et al., 1990; Shen and Pierre, 1994; Bamnios and Trochides, 1995; Saavedra and Cuitino, 2001).

The crack effects on dynamic behaviors such as changes in natural frequencies and modes of vibrations have been the subject of many investigations. Since it is impossible to mention them all, the reader is referred to the survey paper by Dimarogonas (1996), where abundant information and the literature can be found until the year the paper is released. In addition, several prominent works (Gudmundson, 1983; Gounaris and Dimarogonas, 1988; Ghoneam, 1995; Krawczuk and Ostachowicz, 1995; Chondros et al., 1998; Yokoyama and Chen, 1998; Chaudhari and Maiti, 1999; Krawczuk et al., 2000; Song et al., 2003; Orhan, 2007) can be mentioned. As can be seen from the existing literature, unfortunately, much less investigation on lateral-torsional buckling capacity and dynamics of cracked slender beams has been reported.

To cite a few; Wang et al. (2005) have investigated the coupled bending and torsional vibration of a fiber-reinforced composite cantilever slender beam with an edge crack. They have modeled the crack with a local flexibility matrix. Karaagac et al. (2009) have investigated, both experimentally and numerically, the effects of crack ratios and positions on the fundamental frequencies and buckling loads of cantilever slender beams, made of isotropic material, with a single-edge crack. Prasad et al. (2010) have investigated the vibration frequency affecting the crack position and crack grow rate of a vibrating cantilever beam. Recently, Bouboulas and Anifantis (2011) have developed a finite element model in order to study the vibrational behavior of a beam with a non-propagating edge crack and have applied to a cantilever beam subjected to dynamic loadings. They have conducted parametric studies to investigate the sensitivity of vibrational behavior to the crack angle, depth, and position.

To the authors’ knowledge, both the lateral-torsional buckling capacity and free vibration of a cracked cantilever slender beam made of composite material together has not been studied prior to the work presented in this paper. Therefore, the objective of this paper is to study the effects of laminate scheme, crack ratio (a/h) and crack location (L₁/L) on the free vibration and lateral buckling capacity of a cantilever slender rectangular beam with an open single-edge crack.

In the present study, a finite element algorithm based on energy method is developed. Matlab software is used for numerical calculations. The local flexibility approach is adopted to model the crack, based on linear fracture mechanics and the Castigliano theorem. The Euler-Bernoulli assumptions are used in modeling the beam.

The crack is assumed to be always open and non-propagating by time. It is also assumed that the crack affects only the stiffness of the beam, whereas the mass (Gounaris and Dimarogonas, 1988; Krawczuk and Ostachowicz, 1995) and damping (Gounaris and Dimarogonas, 1988) of the beam remain unchanged.

2. Stress intensity factors

The cracked laminated beam configuration considered is shown in Figure 1. A cracked beam element of rectangular cross-section has an edge crack with a tip line parallel to the z-axis, i.e. with a uniform depth. A generalized loading is indicated by six general forces: an axial force P₁, shear forces P₂ and P₃, bending moments P₄ and P₅, a torsional moment P₆ (Gounaris and Dimarogonas, 1988; Wang et al., 2005) as seen in Figure 2.

Figure 1.

Lateral-torsional buckling of cantilever laminated composite beam loaded with a vertical tip force.

Figure 2.

Schematic view of cracked beam under generalized loading.

The stress intensity factors (SIFs) of the three modes of fracture (opening, sliding and tearing types) correspond to the generalized loading $P_{n}$ . Since the SIFs for composites can not be taken in the same way as in the case of isotropic material having the same geometry and loading, it was suggested by Bao et al. (1992) that SIFs can be expressed as

K_{jn} = σ_{n} \sqrt{π a} F_{jn} (a / h, λ^{1 / 4} L / h, δ), j = I, II, III n = 1, 2, \dots, 6

(1)

where

σ_{n}

is the uniform remote stress at the crack-section away from the crack due to the n-th independent loading; a, h and L are the crack depth, beam width and beam length, respectively,

F_{jn} (a / h)

represents the correction function, which takes into account finite dimensions of the beam (Krawczuk and Ostachowicz, 1995) and takes the form for different geometry and loading modes but is the same as the corresponding function for an isotropic material (Bao et al., 1992). λ and δ are dimensionless parameters characterizing the in-plane orthotropy and are defined in terms of the engineering elastic constants (Bao et al., 1992; Bazant et al., 1996):

λ = \frac{E_{2}}{E_{1}}, δ = \frac{\sqrt{E_{1} E_{2}}}{2 G_{12}} - \sqrt{ν_{12} ν_{21}}

(2)

where

E_{1}, E_{2}, G_{12}, ν_{12}

and

ν_{21}

are the elastic constants of the orthotropic material and are given in Reddy (1997) and Wang et al. (2005).

Bao et al. (1992) have shown that the effect of combined parameter $λ^{1 / 4} L / h$ is negligible when $λ^{1 / 4} L / h \geq 2$ . This condition is fulfilled for the composite beam whose aspect ratio is greater than four. In this case, considering the effect of δ on the stress intensity factors, the following form of equation (1) is suggested

K_{jn} = σ_{n} \sqrt{π a} Y_{j} (δ) F_{jn} (a / h), j = I, II, III n = 1, 2, \dots, 6

(3)

where

Y_{j} (δ)

is the correction factor due to the material orthotropy.

As explained in detail in Chajes (1974), the beam can be in equilibrium in a slightly buckled form when the critical load is acting. As a result of buckling, three moments, namely $M_{x'}, M_{y'}$ and $M_{z'}$ are induced at any section, about a second set of local coordinate axes $x', y', z'$ that move relatively with the section as it deforms (see Figure 1). $M_{x'}$ , $M_{y'}$ and $M_{z'}$ describe twisting, lateral bending and vertical bending of the beam, respectively. Correspondingly, the loads that the beam could take are the vertical bending (P₄), the lateral bending (P₅) and the torsional moment (P₆) as shown in Figure 3. So, only the components in the flexibility matrix related to $i, j = 4, 5, 6$ are needed. The stress intensity factors given with equation (3) are determined (Krawczuk and Ostachowicz, 1995; Wang et al., 2005) as

K_{I 4} = σ_{4} \sqrt{π a} Y_{I} (δ) F_{I 4} (a / h), σ_{4} = \frac{6 P_{4}}{{bh}^{2}}, K_{I 5} = σ_{5} \sqrt{π a} Y_{I} (δ) F_{I 5} (a / h), σ_{5} = \frac{12 P_{5}}{{hb}^{3}} z, K_{I 1} = K_{I 2} = K_{I 3} = K_{I 6} = 0 K_{II 1} = K_{II 2} = K_{II 3} = K_{II 4} = K_{II 5} = K_{II 6} = 0 K_{III 6} = σ_{6} \sqrt{π a} Y_{III} (δ) F_{III} (a / h), σ_{6} = \frac{24 P_{6} π^{3}}{π^{5} {hb}^{2} - 192 b^{3}} \cos (\frac{π}{b} z) K_{III 1} = K_{III 2} = K_{III 3} = K_{III 4} = K_{III 5} = 0

(4)

where

F_{I 4} (a / h) = \sqrt{\frac{\tan η}{η}} [0.923 + 0.199 (1 - \sin η)^{4}] / \cos η, η = \frac{π a}{2 h}

(4a)

F_{I 5} (a / h) = \sqrt{\frac{\tan η}{η}} [0.752 + 2.02 (a / h) + 0.37 (1 - \sin η)^{3}] / \cos η

(4b)

F_{III} (a / h) = \sqrt{\frac{\tan η}{η}}

(4c)

Y_{I} (δ) = 1 + 0.1 (δ - 1) - 0.016 (δ - 1)^{2} + 0.002 (δ - 1)^{3} Y_{II} (δ) = Y_{III} (δ) = 1

(4d)

and

σ_{4}, σ_{5}

denote bending stresses induced by bending,

σ_{6}

denotes the shear stress induced by torsion along the short edge of the cross-section, determined using the classical theory of elasticity (Wang et al., 2005).

Figure 3.

Three moments present at any section during buckling.

3. The local flexibility matrix

A crack introduces considerable local flexibility due to the strain energy concentration in the vicinity of the crack tip under load. The idea of an equivalent spring, i.e., a local compliance is used to quantify, in a macroscopic way, the relation between the applied load and the strain concentration around the tip of the crack (Gounaris and Dimarogonas, 1988).

According to the Castigliano’s theorem and the Paris equation, the relation between the additional displacement along the direction of loading P_i and the strain energy U_s is given by

u_{i} = \frac{\partial U_{s}}{\partial P_{i}} = \frac{\partial}{\partial P_{i}} \int_{0}^{a} J (α) d α

(5)

where

J (α) = \partial U_{s} / \partial α

is the strain energy release rate and a is the crack depth. Thus, the local compliance, by definition, is

c_{ij} = \frac{\partial u_{i}}{\partial P_{j}} = \frac{\partial^{2}}{\partial P_{i} \partial P_{j}} \int_{0}^{a} J (α) d α

(6)

Considering the generalized loading, for composite materials the strain energy release rate $J (α)$ is given as (Gounaris and Dimarogonas, 1988; Nikpur and Dimarogonas, 1988; Dimarogonas, 1996; Chondros et al., 1998);

J (α) = D_{1} (\sum_{n = 1}^{6} K_{I n}) 2 + D_{2} (\sum_{n = 1}^{6} K_{II n}) 2 + D_{12} (\sum_{n = 1}^{6} K_{I n}) (\sum_{n = 1}^{6} K_{II n}) + D_{3} (\sum_{n = 1}^{6} K_{III n}) 2

(7)

where

K_{I n}, K_{II n}, K_{III n}

given with equation (4), are the stress intensity factors defined in the preceding chapter.

D_{1}, D_{2}, D_{12}

and D₃ are constants defined by

D_{1} = - \frac{{\bar{S}}_{22}}{2} Im (\frac{μ_{1} + μ_{2}}{μ_{1} μ_{2}}), D_{2} = \frac{{\bar{S}}_{11}}{2} Im (μ_{1} + μ_{2}), D_{12} = {\bar{S}}_{11} Im (μ_{1} μ_{2}), D_{3} = 0.5 \sqrt{S_{44} S_{55}}

(7a)

with

μ_{1}

and

μ_{2}

are the roots of the characteristic equation, which is given by equation

{\bar{S}}_{11} μ 4 - 2 {\bar{S}}_{16} μ 3 + (2 {\bar{S}}_{12} + {\bar{S}}_{66}) μ 2 - 2 {\bar{S}}_{26} μ + {\bar{S}}_{22} = 0

(7b)

for compliances

{\bar{S}}_{11}, {\bar{S}}_{22}, S_{44}

and

S_{55}

, see Reddy (1997) and Wang et al. (2005).

To obtain the elements of the flexibility matrix, the beam is considered as an assembly of strips ordered along the z-axis as shown in Figure 4. Hereby, the strain energy release rate $J (α)$ has to be integrated along the crack width (or thickness of the beam b) to give (Dimarogonas, 1996)

c_{ij} = \frac{\partial u_{i}}{\partial P_{j}} = \frac{\partial 2}{\partial P_{i} \partial P_{j}} \int_{- b / 2}^{b / 2} \int_{0}^{a} J (α) d α d z

(8)

Figure 4.

Geometry of cracked section showing integral limits.

Substituting equation (7) into equation (8) yields the general equation for the local compliances as follows:

c_{ij} = \frac{\partial 2}{\partial P_{i} \partial P_{j}} {\int_{- b / 2}^{b / 2} \int_{0}^{a} [D_{1} (\sum_{n = 1}^{6} K_{I n}) 2 + D_{2} (\sum_{n = 1}^{6} K_{II n}) 2 + D_{12} (\sum_{n = 1}^{6} K_{I n}) (\sum_{n = 1}^{6} K_{II n}) + D_{3} (\sum_{n = 1}^{6} K_{III n}) 2] d α d z}

(9)

equation (9) gives the elements of the complete flexibility matrix of the cracked section.

Combining the SIFs in equation (4) with equation (9) yields

c_{ij} = \frac{\partial 2}{\partial P_{i} \partial P_{j}} {\int_{- b / 2}^{b / 2} \int_{0}^{a} [D_{1} (K_{I 4} + K_{I 5}) 2 + D_{3} (K_{III}) 2] d α d z}

(10)

Based on equation (10), the final local flexibility matrix with the elements of interest can be formed as

c_{ij} = [c_{44} 000 c_{55} 000 c_{66}]

(11)

By inversing equation (11) the local stiffness matrix is obtained, i.e. $k_{ij} = c_{ij}^{- 1}$ .

4. The finite element model

A finite element model is developed to represent the cracked beam element of length d and the crack is located at a distance d₁ from the left end of the element as shown in Figure 5. The element is then considered to be split into two segments by the crack. The left and right segments are represented by non-cracked subelements. The crack represents net ligament effect created by loadings. This effect can be related to the deformation of the net ligament through compliance expressions $c_{ij}$ , by replacing the net ligament with a fictitious spring connecting both faces of the crack (Yokoyama and Chen, 1998). So, the spring effects are introduced to the system by using the local flexibility matrix given by equation (11). The cracked or non-cracked element has two nodes with five degrees-of-freedom in each node. They are denoted as vertical bending displacements $(v_{1}, v_{2})$ , lateral bending displacements $(w_{1}, w_{2})$ , slopes $(v'_{1}, v'_{2}, w'_{1}, w'_{2})$ and rotations about x- axis $(γ_{1}, γ_{2}$ , where prime $(')$ denotes differentiation with respect to the longitudinal coordinate x.

Figure 5.

The cracked beam finite element.

The displacements $v (x)$ and $w (x)$ due to the vertical and lateral bending are approximated by using cubic polynomials (Krawczuk et al., 2000), while the rotation $γ (x)$ can be expressed by linear functions, for the left and right segments of the beam.

for 0 \leq x \leq d_{1}, v_{1} (x) = a_{1} + a_{2} x + a_{3} x 2 + a_{4} x 3 w_{1} (x) = b_{1} + b_{2} x + b_{3} x 2 + b_{4} x 3 γ_{1} (x) = c_{1} + c_{2} x

(12a)

for d_{1} \leq x \leq d, v_{2} (x) = a_{5} + a_{6} x + a_{7} x 2 + a_{8} x 3 w_{2} (x) = b_{5} + b_{6} x + b_{7} x 2 + b_{8} x 3 γ_{2} (x) = c_{3} + c_{4} x

(12b)

The coefficients $a_{1 - 8}, b_{1 - 8}, c_{1 - 4}$ of the polynomials can be expressed uniquely in terms of the boundary conditions shown in Figure 5 and the local flexibility concept at the crack location. Eventually, the following expressions are obtained for a cracked element:

4.1. Vertical bending

v_{1} (0) = q_{1}, v'_{1} (0) = q_{2} v_{2} (d) = q_{3}, v'_{2} (d) = q_{4}

(13a)

4.2. Lateral bending

w_{1} (0) = q_{5}, w'_{1} (0) = q_{6} w_{2} (d) = q_{7}, w'_{2} (d) = q_{8}

(13b)

4.3. Twisting about x-axis

γ_{1} (0) = q_{9}, γ_{2} (d) = q_{10}

(13c)

At the crack location d₁, the flexibility concept requires:

4.4. Vertical bending

Continuity of the vertical displacement

v_{1} (d_{1}) = v_{2} (d_{1})

(14a)

Discontinuity of the cross-sectional rotation

v'_{2} (d_{1}) = v'_{1} (d_{1}) + c_{44} M_{z 1} (d_{1})

(14b)

where

M_{z 1} (d_{1}) = E_{ef} I_{z} v ″_{1} |_{x = d_{1}}

and

I_{z}

is the second moment of area of cross-section about z- axis.

Continuity of the vertical bending moment

M_{z 1} (d_{1}) = M_{z 2} (d_{1})

(14c)

Continuity of the shear force

S_{y 1} (d_{1}) = S_{y 2} (d_{1})

(14d)

4.5. Lateral bending

Continuity of the lateral displacement

w_{1} (d_{1}) = w_{2} (d_{1})

(15a)

Discontinuity of the cross-sectional rotation

w'_{2} (d_{1}) = w'_{1} (d_{1}) + c_{55} M_{y 1} (d_{1})

(15b)

where

M_{y 1} (d_{1}) = E_{ef} I_{y} w ″_{1} |_{x = d_{1}}

and

I_{y}

is the second moment of area of cross-section about y- axis.

Continuity of the lateral bending moment

M_{y 1} (d_{1}) = M_{y 2} (d_{1})

(15c)

Continuity of the shear force

S_{z 1} (d_{1}) = S_{z 2} (d_{1})

(15d)

4.6. Twisting about x-axis

Discontinuity of the torsional angle

γ_{2} (d_{1}) = γ_{1} (d_{1}) + c_{66} T_{1} (d_{1})

(16a)

where

T_{1} (d_{1}) = G_{ef} J γ'_{1} |_{x = d_{1}}

and J is the torsional constant.

Continuity of the torsional moment

T_{1} (d_{1}) = T_{2} (d_{1})

(16b)

By considering equation (12), describing the displacements for the left and right part of the element and rearranging equations (12) to (16), the nodal displacements can be expressed in matrix forms as

q_{v} = D_{v} a, q_{w} = D_{w} b, q_{γ} = D_{γ} c

(17)

where

q_{v}, q_{w}, q_{γ}

D_{v}, D_{w}, D_{γ}

a, b, c

matrices are given as

q_{v} = [q_{1} q_{2} 0000 q_{3} q_{4}] T q_{w} = [q_{5} q_{6} 0000 q_{7} q_{8}] T q_{γ} = [q_{9} 00 q_{10}] T

(17a)

D_{v} = [1000000001000000001000 - 1 00000000 - 1 - 1 0 S 1 S 2 10000 - 1 S 3 S 4 010000001 d d 2 d 3 000001 2 d 3 d 2], D_{w} = [1000000001000000001000 - 1 00001000 - 1 - 1 0 S 5 S 6 10000 - 1 S 7 S 8 010000001 d d 2 d 3 000001 2 d 3 d 2] D_{γ} = [1000010 - 1 - 1 S 9 10001 d],

where

S 1 = 2 c_{44} E_{x} I_{z} d_{1} S 2 = 6 c_{44} E_{x} I_{z} d_{1}^{2} S 3 = - 2 c_{44} E_{x} I_{z} S 4 = - 6 c_{44} E_{x} I_{z} d_{1} S 5 = 2 c_{55} E_{x} I_{y} d_{1} S 6 = 6 c_{55} E_{x} I_{y} d_{1}^{2} S 7 = - 2 c_{55} E_{x} I_{y} S 8 = - 6 c_{55} E_{x} I_{y} d_{1} S 9 = - c_{66} GJ

(17b)

a = [a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} a_{7} a_{8}] T b = [b_{1} b_{2} b_{3} b_{4} b_{5} b_{6} b_{7} b_{8}] T c = [c_{1} c_{2} c_{3} c_{4}] T

(17c)

Matrices in equation (17) can be written in compact form as

{q v q w q γ} 20 \times 1 = D 20 \times 20 {a b c} 20 \times 1

(18)

Taking inverse of equation (18), the matrix giving the constants $a 1 - 8, b 1 - 8, c 1 - 4$ is obtained as

{a b c} 20 \times 1 = C 1 (20 \times 20) {q v q w q γ} 20 \times 1

(19)

where

C 1 = D - 1

. Using equation (19), the expressions relating the constants to the nodal displacements can be presented as follows:

a 1 = C 11 q 1 + C 12 q 2 + C 17 q 3 + C 18 q 4 + C 19 q 5 + C 110 q 6 + C 115 q 7 + C 116 q 8 + C 117 q 9 + C 120 q 10 a 2 = C 21 q 1 + C 22 q 2 + C 27 q 3 + C 28 q 4 + C 29 q 5 + C 210 q 6 + C 215 q 7 + C 216 q 8 + C 217 q 9 + C 220 q 10 • • b 1 = C 91 q 1 + C 92 q 2 + C 97 q 3 + C 98 q 4 + C 99 q 5 + C 910 q 6 + C 915 q 7 + C 916 q 8 + C 917 q 9 + C 920 q 10 • • c 1 = C 171 q 1 + C 172 q 2 + C 177 q 3 + C 178 q 4 + C 179 q 5 + C 1710 q 6 + C 1715 q 7 + C 1716 q 8 + C 1717 q 9 + C 1720 q 10 • •

(20)

or in matrix form

{a b c} 20 \times 1 = C 20 \times 10 {q v q w q γ} 10 \times 1

(21)

Then, substituting equation (21) into equations (12a and b), the displacements at any point of the element are obtained in matrix form:

for 0 \leq x \leq d 1 v 1 = P a 1, w 1 = Pb 1, γ 1 = Pc 1

(22a)

for d 1 \leq x \leq d v 2 = P a 2, w 2 = Pb 2, γ 2 = Pc 2

(22b)

in which the matrices

P, R, a 1, b 1, c 1, a 2, b 2, c 2

are given as

P = [1 x x 2 x 3], R = [1 x] a 1 = [a 1 a 2 a 3 a 4] T, b 1 = [b 1 b 2 b 3 b 4] T, c 1 = [c 1 c 2] T

(22c)

So, the generalized displacement vector can be expressed as

q e = [q 1 q 2 q 5 q 6 q 9 q 3 q 4 q 7 q 8 q 10] T

(23)

5. Energy equations

Energy equations should be expressed separately for the cracked element and intact elements on the left and right side of cracked element. The elastic potential energy U, with the warping and shear effects neglected, due to vertical and lateral bendings and twisting, of an Euler beam with an elemental length d is given (Chondros et al., 1998)for the intact elements on the left side of the cracked element as:

U l = \frac{1}{2} (\int_{0}^{d} E ef I z (v ″ 1) 2 d x) + \frac{1}{2} (\int_{0}^{d} E ef I y (w ″ 1) 2 d x) + \frac{1}{2} (\int_{0}^{d} G ef J (γ' 1) 2 d x)

(24a)

for the cracked element:

U c = \frac{1}{2} (\int_{0}^{d 1} E ef I z (v ″ 1) 2 d x) + \frac{1}{2} (\int_{0}^{d 1} E ef I y (w ″ 1) 2 d x) + \frac{1}{2} (\int_{0}^{d 1} G ef J (γ' 1) 2 d x) + \frac{1}{2} (\int_{d 1}^{d} E ef I z (v ″ 2) 2 d x) + \frac{1}{2} (\int_{d 1}^{d} E ef I y (w ″ 2) 2 d x) + \frac{1}{2} (\int_{d 1}^{d} G ef J (γ' 2) 2 d x)

(24b)

for the intact elements on the right side of the cracked element:

U r = \frac{1}{2} (\int_{0}^{d} E ef I z (v ″ 2) 2 d x) + \frac{1}{2} (\int_{0}^{d} E ef I y (w ″ 2) 2 d x) + \frac{1}{2} (\int_{0}^{d} G ef J (γ' 2) 2 d x)

(24c)

Similarly, the kinetic energy T of an element in length d of an Euler beam is given as:for the intact elements on the left side of the cracked element:

T l = \frac{1}{2} ρ [\int_{0}^{d} A ({\overset{\cdot}{v}}_{1}^{2} + {\overset{\cdot}{w}}_{1}^{2} + I yz {\overset{\cdot}{γ}}_{1}^{2}) d x]

(25a)

for the cracked element:

T c = \frac{1}{2} ρ [\int_{0}^{d 1} A ({\overset{\cdot}{v}}_{1}^{2} + {\overset{\cdot}{w}}_{1}^{2} + I yz {\overset{\cdot}{γ}}_{1}^{2}) d x] + \frac{1}{2} ρ [\int_{d 1}^{d} A ({\overset{\cdot}{v}}_{2}^{2} + {\overset{\cdot}{w}}_{2}^{2} + I yz {\overset{\cdot}{γ}}_{2}^{2}) d x]

(25b)

for the intact elements on the right side of the cracked element:

T r = \frac{1}{2} ρ [\int_{0}^{d} A ({\overset{\cdot}{v}}_{2}^{2} + {\overset{\cdot}{w}}_{2}^{2} + I yz {\overset{\cdot}{γ}}_{2}^{2}) d x]

(25c)

in which the first, second and third terms represent the energies due to vertical, lateral and twisting motions, respectively.

I yz

is the polar moment of inertia.

The work done, V, by the vertical displacement of a tip load P as the beam buckles is given (Karaagac et al., 2009):for the intact elements on the left side of the cracked element:

V l = \int_{0}^{d} M (x) γ 1 (w ″ 1) d x

(26a)

for the cracked element:

V c = \int_{0}^{d 1} M (x) γ 1 (w ″ 1) d x + \int_{d 1}^{d} M (x) γ 2 (w ″ 2) d x

(26b)

for the intact elements on the right side of the cracked element:

V r = \int_{0}^{d} M (x) γ 2 (w ″ 2) d x

(26c)

where,

M (x) = P (L - x) .

6. Equation of motion

By substituting the expressions of equation (22) into the energy equations (24) to (26), the elastic stiffness matrix $k e$ , geometrical stiffness matrix k_g and mass matrix m_e are obtained for both cracked finite element and intact finite elements. Mass and stiffness matrices of each beam element are used to form global mass and stiffness matrices. By performing the required operations for the entire system, one obtains the following governing matrix equations giving:the natural frequencies

(K - p_{n}^{2} M) q = 0

(27a)

and the critical (lowest) buckling loads,

(K - P cr K g) q = 0

(27b)

where

K, K g

and M represent global elastic stiffness, geometrical stiffness and mass matrices, respectively.

p n

stands for the natural frequencies and

P cr

for the critical (lowest) buckling loads.

7. The effective flexural moduli of the laminated composite beam

According to classical lamination theory (CLT), it is assumed that the interlaminar displacements and forces are zero throughout the laminated beam and no slip occurs between the lamina interfaces. Considering the energy equations (24), it is needed to find equivalent elasticity and shear moduli due to bending and twisting, respectively.

For symmetric laminates, the equations for bending deflection are uncoupled from those of the longitudinal displacements. The in-plane forces are zero only for small deformations, and not necessary for stability analysis. So, the constitutive equation, in the absence of in-plane forces, is reduced to

{M x M y M xy} = [D 11 D 12 D 16 D 21 D 22 D 26 D 61 D 62 D 66] {κ x κ y κ xy}

(28)

for symmetric laminates [26–28] where

M x, M y

are bending moments per unit length and

M xy

is the twisting moment per unit length,

κ x, κ y, κ xy

are the midplane curvatures.

[D]

is bending stiffness matrix and its elements

D ij

are given by

D ij = \frac{1}{3} \sum_{k = 1}^{N} {\bar{Q}}_{ij}^{k} (z_{k + 1}^{3} - z_{k}^{3}) i, j = 1, 2, 6

(29)

where N is the number of laminates, z is the coordinate variable of the kth laminate in the cross-section and

{\bar{Q}}_{ij}^{k}

are stiffnesses (see Kaw, 1997; Reddy, 1997; Kollar and Springer, 2003; Wang et al., 2005). It should be noted that, in case of antisymmetric lay-ups,

D 16 = D 26 = 0

By taking inverse of equation (28)

{κ x κ y κ xy} = [D_{11}^{*} D_{12}^{*} D_{16}^{*} D_{21}^{*} D_{22}^{*} D_{26}^{*} D_{61}^{*} D_{62}^{*} D_{66}^{*}] {M x M y M xy}

(30)

is obtained, where

D_{ij}^{*} = D_{ij}^{- 1}

Equation (30) is used to define effective flexural moduli in terms of the bending compliance matrix as follows: First applying $M x \neq 0, M y = 0, M xy = 0$ and then substituting in equation (30) gives

κ x = D_{11}^{*} M x

(31)

by using equation (31) along with known moment M(x) expressions

M y (x) = h M x

(32)

M y (x) = E x I y κ x

(33)

and taking

I y = hb 3 / 12

for rectangular cross-section, the effective flexural longitudinal modulus is

E ef = \frac{12}{b 3 D_{11}^{*}}

(34)

Similarly, taking $M x = 0, M y = 0, M xy \neq 0$ , it can be shown that the effective flexural shear modulus is,

G ef = \frac{12}{b 3 D_{66}^{*}}

(35)

and torsional rigidity (Kollar and Springer, 2003) is

G ef J = \frac{4 h}{D_{66}^{*}}

(35a)

where the torsional constant

J = 4 I y = hb 3 / 3

8. Results and discussions

Wang et al. (2005), performed the free vibration analysis considering only the unidirectional composite beams consisting of several plies having the same fiber orientation changing from 0° to 90° and discussed the vibrational behaviour due to unidirectional fiber orientation. In this study, both buckling and free vibration analyses are performed by considering six different laminate schemes that are [0₄], [90₄] unidirectional, [0 90]_s, [90 0] _s cross-ply symmetric and [0 45–45 0], [0 60–60 0] angle-ply antisymmetric. Material properties of the laminates and the geometry of beam are taken same as those given in the paper by Wang et al. (2005). The aim of doing so is to compare the results of free vibration obtained for [0₄], [90₄] unidirectional laminates with those given in the paper by Wang at al. (2005) and further to investigate the effect of mixed fiber orientation on the static and vibrational behavior of cracked beams. The composite material properties taken are given in Table 1. The geometry of the beam is taken to be: length

L = 0.5 m

; width

h = 0.1 m

and thickness

b = 0.005 m

. As seen from laminate schemes, four laminates of equal thickness of 1.25 mm constitute the beam with a constant thickness (b) of 5 mm.

Table 1.

Composite material properties

Property	Symbol*	Matrix	Fiber
Moduli of elasticity (GPa)	E_m, E_f	2.76	275.6
Moduli of rigidity (GPa)	G_m, G_f	1.036	114.8
Poisson’s ratio	v_m, v_f	0.33	0.2
Mass density (kg/m³)	ρ_m, ρ_f	1600	1900
Fiber volume fraction	V	0.5

The subscript m stands for matrix and f for fiber.

In many works related to the cracked beam, the crack ratio is generally considered up to 0.6. This consideration may be due to the fact that the correction functions in equations (7a and b) are valid for crack ratios up to 0.6 (Yokoyama and Chen, 1998). Since they are functions of crack ratio only and go to infinity with crack ratio (a/h) approaching unity, they may not be able to describe the vibration characteristics for crack ratios close to unity (Wang et al., 2005). So, to extend the investigation for crack ratios larger than 0.6 is found impractical (Saavedra and Cuitino, 2001). Thus, in the present study, numerical analyses are performed considering crack ratios up to 0.5 and the beam is discretized with 11 finite beam elements.

The laminated composite beams, which have six different laminate schemes, are for simplicity denoted by C1, C2, C3, C4, C5 and C6 as shown below.

C 1 = [04], C 2 = [904], C 3 = [0 90] s, C 4 = [90 0] s, C 5 = [045 - 450], C 6 = [0.60 - 60.0]

In order to validate and confirm the accuracy of the present method, numerical comparisons for both natural frequencies and critical buckling loads of uncracked beams are performed among the results of the present model and those of available in literature.

8.1. Natural frequency changes

Numerical comparison for the first three natural frequencies of uncracked beam is performed with the results obtained by Wang et al. (2005). As seen from Table 2, comparison shows that the agreements between the related results are very good.

Table 2.

Comparison of the first three natural frequencies of the cantilever laminated composite uncracked beam for θ = 0° and 90°

		Laminate
		[0₄]			[90₄]
	rad/s	1st	2nd	3rd	1st	2nd	3rd
Present (11 elem.)	Bending	181.034	1134.545	3177.242	43.580	273.114	764.844
	Torsion	413.571	1249.158	2110.216	413.571	1249.158	2110.216
Wang et al.	Bending	181	1134.5	3176.7	43.6	273.1	764.7
(2005) (Analytic)	Torsion	413.5	1240.6	2067.7	413.5	1240.6	2067.7

Since the stiffness parameters EI and GJ are determined by the laminate schemes, and no crack involved, natural frequencies of the beam depend not only on the crack ratio and crack location, but also on the material properties (Wang et al., 2005).

Three situations are selected in terms of crack ratio. Assume that the crack ratio is fixed at a/h = 0.1, 0.3 and 0.5. Figures 6 to 11 show the effects of the crack location, crack ratio and laminate schemes on the natural frequencies. The first three natural frequencies versus crack locations are plotted in Figures 6 to 11 and the first three natural frequency values of the uncracked beams are also included in the figures.

Figure 6.

Variation of first three natural frequencies of laminated cracked beam C1 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

Figure 7.

Variation of first three natural frequencies of laminated cracked beam C2 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

Figure 8.

Variation of first three natural frequencies of laminated cracked beam C3 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

Figure 9.

Variation of first three natural frequencies of laminated cracked beam C4 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

Figure 10.

Variation of first three natural frequencies of laminated cracked beam C5 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

Figure 11.

Variation of first three natural frequencies of laminated cracked beam C6 having crack ratios a/h = 0.1, 0.3, 0.5 versus the crack location. ♦, ▪, ▴: 1st, 2nd and 3rd natural frequencies of uncracked beam, ⋄, □, Δ: 1st, 2nd and 3rd natural frequencies of cracked beam, ——: bending, - - - -: torsion.

It is obvious from the figures that, among the laminated beams, C1 has the largest frequency value (181.034 Hz) followed by C5, C3, C6 (approx. 170 Hz) and then C4 (75.981 Hz) and C2 (43.580 Hz). At this point, it is worth to mention that C3, having plies with fiber angle of 0° outsides has a natural frequency 2.24 times higher than that of C4 having plies with fiber angle of 90° outsides. This shows that arranging the plies with fiber angle of 0° outsides in all laminates considered, give rise to an increase in the natural frequencies.

In general, the natural frequencies controlled by bending experience further reduction with the crack ratio increased at a certain crack location. But the same trend is not observed in the natural frequencies controlled by torsion, even in the case of large crack ratio. They almost remain unchanged for different crack locations along the beam length compared to those of the uncracked beams. The drop in natural frequencies is greater for cracks close to the clamped end. In other words, the frequency drop is greatest for a crack located where the bending moment is the largest (Shen and Pierre, 1994).

When the crack ratio a/h = 0.1 (Figures 6(a) to 11(a)), the natural frequencies are very close to those of uncracked beams. A slight reduction (by ∼ %2) occurs in the first and third natural frequencies of C1, C3, C4, C5, C6 and first and second frequencies of C2, which are controlled by bending, compared to the corresponding natural frequencies of uncracked beams. Second natural frequencies are controlled by torsion except for C2, where the third natural frequency is torsional. A slight increase is observed in the bending frequencies as the crack location moves to the free end of the beams.

When the crack ratio a/h = 0.3 (Figures 6(b) to 11(b)), reduction in bending controlled frequencies becomes apparent. The first and third natural frequencies of C1, C3, C5, C6 and the first and second natural frequencies of C2, C4 are controlled by bending. Again, a slight increase is observed in bending controlled frequencies. But in C4 the second natural frequency approaches conspicuously to the third natural frequency as the crack location moves along the length of the beam.

When the crack ratio a/h = 0.5 (Figures 6(c) to 11(c)), reduction in bending controlled frequencies are more apparent and changes in the second and third frequencies become interesting. At this crack ratio C1 and C3 laminates have very similar plots as seen from Figures 6(c) and 8(c). A transient state (Wang et al., 2005) is observed in the second and third frequency variations of the laminates C1 and C3. At this transient state (crack locations up to $L_{1} / L = 0.1$ in C1 and 0.2 in C3), as can be shown in enlarged views in Figures 6(d) and 8(d), the frequency variation previously controlled mainly by the bending mode (or the torsional mode) becomes controlled by the torsional mode (or the bending mode). This may be due to the shape and increasing distortion at the crack location, and the higher modes seem more sensitive to the crack depth, even though the crack is not located at the large curvature position (Wang et al., 2005). Also, it is noticed that the second and third frequencies occur very close each other. Plots of C5 and C6 laminates also resemble each other as seen in Figures 10(c) and 11(c) due to the similar laminate schemes they have. But, different from C1, C3, while, especially at the lower crack ratios, the third frequencies are controlled by bending mode, they are controlled by torsion mode at the higher crack ratios. At this crack ratio, C4 behaves in a similar way as C5 and C6 do, but C2 does not.

Interestingly, all three natural frequencies of C2 are controlled by corresponding bending modes. Also note that the bending controlled frequencies, increasing for crack locations along the half length of the beam, reach a maxima and then decrease to the values between the initial and maxima values along the way to the beam end. This is a common feature for all laminates.

8.2. Buckling load

For each laminated beam, the buckling load was predicted by numerical analysis.

Again, three situations are selected in terms of the crack ratio. Assume that the crack ratio is fixed at a/h = 0.1, 0.3 and 0.5. Figure 12(a) to (f) plotted with respect to crack location show the effects of the crack location, crack ratio and laminate schemes on the buckling loads. The critical buckling load values of uncracked beams are also included in the figures. Curves in the figures reflect the combined effects of both lateral bending and torsion (lateral-torsional buckling) exerted on the beams by the critical buckling loads.

Figure 12.

Variation of buckling loads versus crack position for the cracked beams with different crack ratio, ---⋄---: a/h = 0.1, ---□---: a/h = 0.3, ---Δ---: a/h = 0.5, ×: uncracked.

The plots are similar to those plotted for natural frequencies, meaning that similar observations can be made for buckling loads. In general, the buckling loads experience further reduction with the crack ratio increased at a certain crack location, comparing to the uncracked P_cr values. It is clear from the figures that reduction in the buckling capacity is higher when the crack is near the clamped end of the beam.

However, this reduction becomes less after $L_{1} / L = 0.4$ as the crack position is moved towards the free end. This means that the maximum reduction in buckling load occurs at sections subjected to large bending moments.

Considering the laminate schemes, C5 has the largest buckling capacity (998.015 N) followed by C6 (918.051 N), C1 (688.699 N), C3 (647.701 N) and then C4 (289.05 N) and C2 (165.786 N). Among all the laminates, the ones with angle-plies in between have the highest buckling capacity. This arises from the fact that the stresses created by bending and torsion are well met by the longitudinal fibers of outside plies of 0° and angled fibers of plies in between, since the directions of stresses coincide with the directions of fibers. Considering the laminate schemes investigated, in general, configuring laminates with plies of 0° outsides increase the rigidity and therefore the buckling capacity of beams. This is verified easily by the example; although C3 and C4 have the same plies, C3 with plies of 0° outsides has a buckling capacity 2.24 times higher than that of C4.

9. Conclusions

In this study, the effects of laminate schemes, crack ratio and crack positions on the natural frequency and the critical buckling loads of cracked cantilever beams were investigated numerically. The numerical method based on the energy method presented herein, is a straightforward one involving no tedious mathematics and polynomial approximation for the displacements have given results with reasonable accuracy. Results show that the decrease of both the natural frequency and buckling load depends not only on the crack ratio and crack position, but also on the laminate schemes. In the case of the natural frequency, the first natural frequencies of all laminates are controlled by the bending mode, independent from the crack ratio, while the higher ones are controlled either by torsion or by bending, depending on the crack ratio. Arranging laminates with plies of 0° fiber angle outsides considerably increases the bending controlled natural frequencies, regardless of whatever plies are in-between. The laminates with angle plies in-between (C5, C6) have higher torsion controlled frequencies than those of the other laminates (C1–C4).

In the case of the buckling load, the laminates with angle plies in-between (C5, C6) have higher buckling load capacity than those of the other laminates (C1–C4), due to their ability to meet both longitudinal and torsional stresses created during lateral-torsional buckling.

For small crack ratios, the reductions in natural frequency and buckling load are small, becoming progressively greater at larger crack ratios. The higher drops in natural frequency and buckling load are observed when the crack is located near the clamped end. The numerical method offered, can be applied easily to other structural elements with single or multiple cracks, having various geometry and boundary conditions, provided that the necessary stress intensity factors are known.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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