Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 543828, 16 pages
http://dx.doi.org/10.1155/2012/543828
Research Article

Modified and Simplified Sectional Flexibility of a Cracked Beam

Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan

Received 25 April 2011; Revised 20 July 2011; Accepted 27 September 2011

Academic Editor: George Jaiani

Copyright © 2012 Chih-Shiung Wang and Lin-Tsang Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a new sectional flexibility factor to simulate the reduction of the stiffness of a single-edge open cracked beam. The structural model for crack of the beam is considered as a rotational spring which is related to the ratio of crack depth to the beam height, a/h. The mathematical model of this single-edge open crack beam is considered as an Euler-Bernoulli beam. The modified factor, f(a/h), derived in this paper is in good agreement with previous researchers' results for crack depth ratio a/h less than 0.5. The natural frequencies and corresponding mode shapes for lateral vibration with different types of single-edge open crack beams can then be evaluated by applying this modified factor f(a/h). Using the compatibility conditions on the crack and the analytical transfer matrix method, the numerical solutions for natural frequencies of the cracked beam are obtained. The natural frequencies and the mode shapes with crack at different locations are obtained and compared with the latest research literature. The numerical results of the proposed cracked beam model obtained by this method can be extended to construct frequency contour. The natural frequencies measured from field can be used in solving the inverse problem to identify cracks in structures.

1. Introduction

A crack in the structure will reduce the structural strength and result in severe damage under critical loading conditions. The major issue of the structural health monitoring is to detect crack depth and location in the present study. The model with linear elastic fracture mechanics and Euler-Bernoulli beam theory are being widely used in the recent research literatures. The cracked beam is modeled as two-segment beam with the crack simulated as a rotational spring. The crack of the beam is considered as a local flexibility which is a function of the crack depth.

Sih [1] proposed strain energy density factor theory to discuss the all mixed-mode crack extension problems. The strain energy density factor is a linear elasticity function of the mixed-mode stress intensity factors. Tada et al. [2] presented the stress intensity factors of different modes due to general loading. The presented model introduced the local flexibility matrix from the stress intensity factors. Nobile [3] proposed a simple method for obtaining approximate stress intensity of straight cracked beams. The system takes account of the elastic crack tip stress singularity while using the elementary beam theory. Dimarogonas and Paipetis [4] introduced a general stiffness matrix for cracked structural members to model the respective dynamic system. The local flexibility can be derived further from the general stiffness matrix. Chondros et al. [5] developed a continuous cracked beam vibration theory for the lateral vibration of cracked Euler-Bernoulli beams with single-edge or double-edge open cracks. The crack was modeled as a continuous flexibility using the displacement field. Anifantis and Dimarogonas [6] studied the system stability of the cracked column with vertical load. The method developed a general flexibility matrix to express the local flexibility of a beam with a single-edge crack. Ostachowicz and Krawczuk [7] developed a new local flexibility which was derived from the stress intensity factor by Anifantis and Dimarogonas [6].

In order to obtain the natural frequencies of the crack beam, finite element method was used to compute the eigensolutions in the recent literatures. The order of the determinants increases as the degree of freedom increases in finite element method. In order to reduce the order of the determinants, Lin et al. [8] proposed using transfer matrix for beams with arbitrary number of cracks. The method uses only four unknown constants which can be solved through satisfying four boundary conditions. Lin and Chang [9] used the analytic transfer matrix method to solve eigensolutions of a cracked cantilever beam. The eigenfunctions obtained in this method are analytical solutions. The dynamic responses can be obtained by this method, and the solutions converge quite fast. Alsabbagh et al. [10] presented a new simplified formula for the stress correction factor by using strain energy density approach. A modified factor for local flexibility was used in solving the characteristic equation of the cracked beam. Lin [11] used the Timoshenko beam theory and transfer matrix method to solve the direct and inverse problems of simply supported beam with a single-edge open crack. The location and crack size of the beam can be determined by the method presented. The theoretical results are also validated by a comparison with experimental measurements.

2. Derivation of Stress Intensity Factor

A prismatic beam is considered with an open and nonpropagating crack of depth 𝑎, length 𝐿, height , and width 𝑡. The singular stress distribution at the crack tip takes the form [3, 4, 10]𝜎𝑠𝑥=𝐾𝐼2𝜋𝑟,(2.1) with the conditions that 𝜎𝑠𝑥 acts at a distance 𝑟=𝑏 from the tip and 𝐾𝐼 is the stress intensity factor. The normal stress acting on the reduced cross-section passing through the crack tip is given as 𝜎𝑥=𝑀𝐼𝑦,(2.2) where 𝑀 is the bending moment and 𝐼=𝑡(𝑎)3/12 is the moment of inertia of the remaining part of the cracked beam. The distance 𝑦 in (2.2) is found from Figure 1 as𝑦=𝑦𝑏=2𝑎2𝑏,(2.3) where 𝑦 is the distance from the neutral axis of the reduced cross-section to the tip as shown in Figure 1.

543828.fig.001
Figure 1: Geometry of a beam with a single-edge crack.

The stress condition is considered as 𝜎𝑠𝑥=𝜎𝑥 at the crack tip 𝑟=𝑏. Substituting (2.3) into (2.2), 𝜎𝑥 and 𝐾𝐼 can be expressed as𝜎𝑥=12𝑀𝑡(𝑎)32𝑎2𝐾𝑏,(2.4)𝐼=2𝜋𝑏𝜎𝑥=12𝑀2𝜋𝑏𝑡(𝑎)32𝑎2𝑏.(2.5) The distance 𝑏 can be determined from the equilibrium condition of forces along the 𝑥-axis:𝑏0𝐾𝐼2𝜋𝑟𝑑𝑟=𝑦𝑦𝑏𝜎𝑥𝑑𝑦.(2.6) The left-hand side of (2.6) is evaluated, using (2.5), to be𝑏0𝐾𝐼2𝜋𝑟𝑑𝑟=24𝑀𝑏𝑡(𝑎)32𝑎2𝑏.(2.7) The right-hand side of (2.6) is evaluated, using (2.4), to be𝑦𝑦𝑏𝜎𝑥𝑑𝑦=6𝑀𝑏𝑡(𝑎)3(𝑎𝑏).(2.8) Substitution of (2.7) and (2.8) into (2.6) leads to24𝑀𝑏𝑡(𝑎)32𝑎2=𝑏6𝑀𝑏𝑡(𝑎)3(𝑎𝑏).(2.9) So1𝑏=3(𝑎).(2.10) Substituting (2.10) into (2.5), the stress intensity factor can be expressed as𝐾𝐼=6𝑀𝑡3/2𝐹𝑎,(2.11) where𝐹𝑎=2𝜋127(1𝑎/)3=0.482(1𝑎/)3.(2.12)

3. Calculation of the Equivalent Flexibility in the Crack Beam

Let 𝑈𝑇 be the strain energy due to the crack. According to Castigliano’s theorem, the additional displacement is 𝑢𝑖=𝜕𝑈𝑇/𝜕𝑃𝑖 under general loading 𝑃𝑖. In this work, the displacement will reduce to𝜃=𝜕𝑈𝑇𝜕𝑀,(3.1) where the displacement 𝑢𝑖 is taken as the rotation 𝜃 since the bending moment 𝑀 is the only load of the structure.

The strain energy has the form [46, 10]𝑈𝑇=𝑎0𝜕𝑈𝑇𝜕𝑎𝑑𝑎=𝑡𝑎0𝐽𝑑𝑎,(3.2) where 𝐽 is the strain energy density function. Therefore,𝑢𝑖=𝜕𝑈𝑇𝜕𝑃𝑖=𝜕𝜕𝑃𝑖𝑡𝑎0𝐽(𝑎)𝑑𝑎(Paris'sequation).(3.3) The moment 𝑀 substitutes for the generalized load 𝑃𝑖𝜕𝜃=𝑡𝜕𝑀𝑎0𝐽(𝑎)𝑑𝑎.(3.4) The flexibility influence coefficient will be written as𝑐=𝜕𝜃=𝜕𝜕𝑀2𝜕𝑀2𝑡𝑎0𝐽(𝑎)𝑑𝑎.(3.5) The strain energy density function 𝐽 has the form𝐾𝐽=2𝐼𝐸,(3.6) where 𝐸=𝐸/(1𝜐2) for plane strain and 𝐸 and 𝜐 are Young’s modulus and Poisson’s ratio, respectively. The flexibility scalar is𝜕𝑐=2𝜕𝑀2𝑡𝑎0𝐾2𝐼𝐸=𝑑𝑎2𝜋1𝜐21(1𝑎/)29𝐸𝐼(1𝑎/)2,(3.7) where Poisson’s ratio is taken as 𝜐=0.3 and the area moment of inertia is taken for the whole cross-section as 𝐼=𝑡3/12. The nondimensional cracked section flexibility can be found from (3.7) as𝑐=𝐸𝐼𝑐𝐿=2𝜋1𝜐21(1𝑎/)29(1𝑎/)2𝐿.(3.8)

4. Free Vibration of a Cracked Beam

A simple beam with length 𝐿 and an open-edge crack at position 𝑋1 is considered as shown in Figure 2. Euler-Bernoulli beam bending theory was used in solving the free vibration problem. According to [8, 9], the differential equation of motion for each segment is𝜕𝐸𝐼4𝑌𝑖(𝑋,𝑇)𝜕𝑋4𝜕+𝜌𝐴2𝑌𝑖(𝑋,𝑇)𝜕𝑇2=0𝑋𝑖1<𝑋<𝑋𝑖,𝑖=1,2,(4.1) where 𝜌 is the density of the material and 𝐴 is the cross-section area of the rectangular beam. The boundary conditions of the simply supported beam are𝑌𝑌(0,𝑇)=𝑌(𝐿,𝑇)=0,(0,𝑇)=𝑌(𝐿,𝑇)=0.(4.2) The beam at the crack position is simulated as a rational spring with sectional flexibility as shown in Figure 3. The continuous conditions at the crack position are𝑌1𝑋1,𝑇=𝑌2𝑋+1𝑌,𝑇,(4.3)1𝑋1,𝑇=𝑌2𝑋+1𝑌,𝑇,(4.4)1𝑋1,𝑇=𝑌2𝑋+1,𝑇,(4.5) and the compatibility condition due to the rational flexibility is𝑌2𝑋+1,𝑇𝑌1𝑋1,𝑇=𝐸𝐼𝑐𝑌2𝑋+1,𝑇.(4.6) From the above equations, the following quantities are introduced for nondimensional analysis:𝑌𝑦=𝐿𝑋,𝑥=𝐿,𝑡=𝑇,𝑥𝑖=𝑋𝑖𝐿,𝑙1=𝐿1𝐿,𝑙2=𝐿2𝐿.(4.7) Equation (4.1) can then be expressed in a nondimensional form as𝐸𝐼𝐿4𝜕4𝑦𝑖(𝑥,𝑡)𝜕𝑥4𝜕+𝜌𝐴2𝑦𝑖(𝑥,𝑡)𝜕𝑡2=0𝑥𝑖1<𝑥<𝑥𝑖,𝑖=1,2.(4.8) The continuous conditions at the crack position as non-dimensional form are𝑦1𝑥1,𝑡=𝑦2𝑥+1,𝑦,𝑡1𝑥1,𝑡=𝑦2𝑥+1,𝑦,𝑡1𝑥1,𝑡=𝑦2𝑥+1,𝑦,𝑡2𝑥+1,𝑡𝑦1𝑥1,𝑡=𝑐𝑦2𝑥+1,,𝑡(4.9) where 𝑐 is the non-dimensional cracked section flexibility as (3.8).

543828.fig.002
Figure 2: Schematic diagram of a simply supported cracked beam.
543828.fig.003
Figure 3: Model for the cracked beam with sectional flexibility.

5. Calculation of Natural Frequencies

The equation of motion for the cracked simple beam system can be expressed as (4.8). Using the method of separation of variables, 𝑦𝑖(𝑥,𝑡)=𝑤𝑖(𝑥)𝑒𝑗𝜔𝑡, in (4.8), the differential equation for free vibration can be written as𝑤𝑖(𝑥)𝜆4𝑤𝑖(𝑥)=0,𝑥𝑖1<𝑥<𝑥𝑖,𝑖=1,2,(5.1) where𝜆4=𝜌𝐴𝜔2𝐿4𝐸𝐼.(5.2) From (4.9), the continuous conditions at the crack position are𝑤1𝑥1=𝑤2𝑥+1,𝑤1𝑥1=𝑤2𝑥+1,𝑤1𝑥1=𝑤2𝑥+1,𝑤2𝑥+1𝑤1𝑥1=𝑐𝑤2𝑥+1.(5.3) A closed-form solution to this eigenvalue problem can be obtained by employing transfer matrix methods [8, 9]. The general solution of (5.1), for each segment, is𝑤𝑖(𝑥)=𝐴𝑖sin𝜆𝑥𝑥𝑖1+𝐵𝑖cos𝜆𝑥𝑥𝑖1+𝐶𝑖sinh𝜆𝑥𝑥𝑖1+𝐷𝑖cosh𝜆𝑥𝑥𝑖1,𝑥𝑖1<𝑥<𝑥𝑖,𝑖=1,2,(5.4) where 𝐴𝑖, 𝐵𝑖, 𝐶𝑖, and 𝐷𝑖 are constants associated with the 𝑖th segment (𝑖=1,2). These constants of the second segment (𝐴2, 𝐵2, 𝐶2, and 𝐷2) are related to those of the first segment (𝐴1, 𝐵1, 𝐶1, and 𝐷1) through the continuous conditions in (5.3) and can be expressed as𝐴2𝐵2𝐶2𝐷2=𝑡11𝑡12𝑡13𝑡14𝑡21𝑡22𝑡23𝑡24𝑡31𝑡32𝑡33𝑡34𝑡41𝑡42𝑡43𝑡44𝐴1𝐵1𝐶1𝐷1=𝑇4×4𝐴1𝐵1𝐶1𝐷1,(5.5) where 𝑇4×4 is a 4×4 transfer matrix which depends on eigenvalue 𝜆 and the elements are derived from [8].

Using (5.5), the four constants of the first segment (𝐴1, 𝐵1, 𝐶1, and 𝐷1) can be mapped into those of the second segment (𝐴2, 𝐵2, 𝐶2, and 𝐷2); thereby, the number of independent constants can be reduced to four. For the case of a simply supported beam, the corresponding boundary conditions of (4.2) and (4.3) can be written as𝑌𝑌(0,𝑇)=0𝑤(0)=0,(5.6)𝑌(𝐿,𝑇)=0𝑤(1)=0,(5.7)(0,𝑇)=0𝑤𝑌(0)=0,(5.8)(𝐿,𝑇)=0𝑤(1)=0.(5.9) Due to (5.6) and (5.8), (5.4) yields𝐵1=0,𝐷1=0.(5.10) Satisfying the boundary conditions (5.7) and (5.9), (5.4) leads to the following equations:𝐴2sin𝜆𝑙2+𝐵2cos𝜆𝑙2+𝐶2sinh𝜆𝑙2+𝐷2cosh𝜆𝑙2=0,𝐴2sin𝜆𝑙2𝐵2cos𝜆𝑙2+𝐶2sinh𝜆𝑙2+𝐷2cosh𝜆𝑙2=0,(5.11) which can be expressed in matrix form as00=sin𝜆𝑙2cos𝜆𝑙2sinh𝜆𝑙2cosh𝜆𝑙2sin𝜆𝑙2cos𝜆𝑙2sinh𝜆𝑙2cosh𝜆𝑙2𝐴2𝐵2𝐶2𝐷2=𝐵2×4𝐴2𝐵2𝐶2𝐷2,(5.12) where𝐵2×4=sin𝜆𝑙2cos𝜆𝑙2sinh𝜆𝑙2cosh𝜆𝑙2sin𝜆𝑙2cos𝜆𝑙2sinh𝜆𝑙2cosh𝜆𝑙2.(5.13) Substituting (5.5) into (5.12) and applying (5.10), one obtains00=𝐵2×4𝐴2𝐵2𝐶2𝐷2=𝐵2×4𝑇4×4𝐴1𝐵1𝐶1𝐷1=𝑅2×4𝐴1𝐵1𝐶1𝐷1,(5.14) where𝑅2×4=𝐵2×4𝑇4×4=𝑟11𝑟12𝑟13𝑟14𝑟21𝑟22𝑟23𝑟24.(5.15) A nontrivial solution of the simply supported beam requires𝑟det11(𝜆)𝑟13𝑟(𝜆)21(𝜆)𝑟23(𝜆)=0.(5.16) The characteristic equation of the cracked simply supported beam can be obtained as𝑐𝜆𝑛𝜆sinh𝑛𝑙1𝜆sinh𝑛𝑙2𝜆sin𝑛𝑙1+𝑙2𝑐𝜆𝑛𝜆sin𝑛𝑙1𝜆sin𝑛𝑙2𝜆sinh𝑛𝑙1+𝑙2𝜆+2sinh𝑛𝑙1+𝑙2𝜆sin𝑛𝑙1+𝑙2=0,(5.17) where 𝜆𝑛 is the eigenvalues of the system. This characteristic equation can be solved by using the Newton-Raphson method to obtain the eigenvalues and corresponding eigenfunctions.

6. Numerical Results

In order to verify the procedure presented in this paper, results obtained by applying this method are compared with the available data for single-edge open cracked beam. A 300 mm simple supported beam of cross-section 20×20mm2, with modulus of elasticity 𝐸=2.06×1011N/m2, the density 𝜌=7800kg/m3, the crack is located at the position 240mm and crack depth 𝑎=8 mm.

A simplified stress intensity factor 𝐾𝐼 in (2.11) is expressed in this paper. The function 𝐹(𝑎/) in (2.12) can be compared with the expression given by [2]𝐾𝐼=6𝑀𝑡3/2𝐹2𝑎,𝐹2𝑎=𝜋𝑎2𝜋𝑎tan𝜋𝑎20.923+0.199(1sin(𝜋𝑎/2))4.cos(𝜋𝑎/2)(6.1) Another stress intensity factor is [12]𝐾𝐼=6𝑀𝑡3/2𝐹3𝑎,𝐹3𝑎=𝜋𝑎𝑎1.1221.44𝑎+7.332𝑎13.083𝑎+144.(6.2)

In this research, small crack depth ratio which implies early stage of structure damage is considered. The comparison of data obtained by the proposed method and previous research is shown in Figure 4. The stress intensity factor of the present model and those in [2, 12] are rather close to each other for small crack depth ratio 𝑎/.

543828.fig.004
Figure 4: Stress intensity factor variations.

The nondimensional cracked section flexibility 𝑐 in (3.8) can be compared with the expression given by [5]𝑐2=6𝜋1𝜐2𝐿𝑎Φ,Φ𝑎𝑎=0.62722𝑎1.045333𝑎+4.59484𝑎9.97365𝑎+20.29486𝑎33.03517𝑎+47.10638𝑎40.75569𝑎+19.610.(6.3) Another non-dimensional cracked section flexibility is [7]𝑐3𝑎=6𝜋2𝐿𝑓𝐽𝑎,𝑓𝐽𝑎𝑎=0.63841.035𝑎+3.72012𝑎5.17733𝑎+7.5534𝑎7.3325𝑎+2.49096.(6.4)

Three generalized loading conditions, bending, tension, and torsion, on a cracked beam were considered to evaluate sectional flexibility in the past literature. Beams are mainly affected by bending moment in most loading cases; therefore only bending effects are considered in evaluating the simplified cracked section flexibility. The results of the proposed method of non-dimensional cracked section flexibility are compared with those of previous research in Figure 5. The results of this simplified method and those of [5, 7] are in good agreement for small crack depth ratio 𝑎/. A cracked beam with 𝑎/ greater than 0.5 is already severely damaged and is not suitable for applying this sectional flexibility model.

543828.fig.005
Figure 5: Nondimensional cracked section flexibility variations.

The first five natural frequencies of the uncracked beam are calculated as 𝜔01=517.85, 𝜔02=2071.4, 𝜔03=4660.64, 𝜔04=8285.58, and 𝜔05=12946.22Hz. The ratios of natural frequencies between cracked beam and uncracked beam are listed in Table 1. The variation of natural frequency ratio with the crack depth of this simply supported beam with a crack located at the section (𝐿1/𝐿=0.8) for the first five modes is plotted as in Figure 6.

tab1
Table 1: Natural frequency ratio with the crack depth at location 0.8.
fig6
Figure 6: Variation of natural frequency ratio with the crack depth of a simply supported beam (𝐿1/𝐿=0.8): (a) first frequency, (b) second frequency, (c) third frequency, (d) fourth frequency, and (e) fifth frequency.

With a crack located at the center of the beam, the ratios of natural frequencies between cracked beam and uncracked beam are listed in Table 2. The variation of natural frequency ratio with the crack depth of a simply supported beam with a crack located at (𝐿1/𝐿=0.5) is shown in Figure 7 for the first five modes.

tab2
Table 2: Natural frequency ratio with the crack depth at location 0.5.
fig7
Figure 7: Variation of natural frequency ratio with the crack depth of a simply supported beam (𝐿1/𝐿=0.5): (a) first frequency, (b) second frequency, (c) third frequency, (d) fourth frequency, and (e) fifth frequency.

It is quite obvious that the natural frequencies decrease due to the existence of cracks. That is due to the cracked beam becoming more flexible due to the reduction of moment of inertia of the section property.

Figures 8(a)8(e) show the first five mode shapes of a cracked simply supported beam with single open crack at 𝐿1/𝐿=0.8, crack ratio 𝑎/=0.4, and normalized amplitude 𝑌/𝑌max. It is obvious that the mode shapes all show turning point at crack location 𝐿1/𝐿=0.8.

fig8
Figure 8: (a) Normalized mode shape of the first mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/=0.4. (b) Normalized mode shape of the second mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/=0.4. (c) Normalized mode shape of the third mode with a crack location 𝐿1/𝐿=0.8and crack depth ratio a/h = 0.4. (d) Normalized mode shape of the fourth mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/=0.4. (e) Normalized mode shape of the fifth mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/=0.4.

7. Conclusion

The simplified stress intensity factor and flexibility were derived utilizing the crack beam theorem of Nobile [3] and Dimarogonas [4, 5]. The order of polynomial functions for crack depth ratio 𝑎/ is reduced, because, with a crack depth ratio 𝑎/<0.5, the high-order terms will approach to 0. Therefore, these higher-order arithmetic terms can be neglected and, in order to predict the early stage of structural damage, crack depth ratio 𝑎/ should be less than 0.5. The simplified stress intensity factor and flexibility are compared with the recent literature, and the numerical results are found to be in good agreement.

If the crack is right on the position of nodal point of certain modes, frequencies ratio shows no difference with 𝜔𝑖/𝜔0𝑖=1. For example, the crack at 0.8 L is also a nodal point of the fifth mode which gives the numerical result as 𝜔5/𝜔05=1. The numerical results obtained by this method are in good agreement with the actual vibration response of a simply supported beam. The turning points of certain mode shape function reveal the information about crack location. For the case with a crack at 0.8 L, it can be seen obviously from shape function of mode two, mode three, and mode four with a turning point at 0.8 L.

The simplified stress intensity factor and flexibility of this method can be further extended to construct frequencies ratio contours for beams with cracks. The natural frequencies obtained by applying this model can be used to verify the experimental measurements in a similar way to that in [11]. The location and crack depth of a beam can then be identified as an inverse problem by matching up field measurement of frequencies of a cracked beam.

References

  1. G. C. Sih, Method of Fracture Analysis and Solutions of Crack Problems, Noordhoff International, Leiden, The Netherlands, 1973. View at Zentralblatt MATH
  2. H. Tada, P. C. Paris, and G. R. Irwin, The Stress Analysis of Cracks Handbook, Paris Productions, St. Louis, Mo, USA, 2nd edition, 1985. View at Zentralblatt MATH
  3. L. Nobile, “Mixed mode crack initiation and direction in beams with edge crack,” Theoretical and Applied Fracture Mechanics, vol. 33, no. 2, pp. 107–116, 2000. View at Publisher · View at Google Scholar
  4. A. D. Dimarogonas and S. A. Paipetis, Analytical Methods in Rotor Dynamics, Applied Science, New York, NY, USA, 1983.
  5. T. G. Chondros, A. D. Dimarogonas, and J. Yao, “A continuous cracked beam vibration theory,” Journal of Sound and Vibration, vol. 215, no. 1, pp. 17–34, 1998. View at Publisher · View at Google Scholar
  6. N. Anifantis and A. Dimarogonas, “Stability of columns with a single crack subjected to follower and vertical loads,” International Journal of Solids and Structures, vol. 19, no. 4, pp. 281–291, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. W. M. Ostachowicz and M. Krawczuk, “Analysis of the effect of cracks on the natural frequencies of a cantilever beam,” Journal of Sound and Vibration, vol. 150, no. 2, pp. 191–201, 1991. View at Publisher · View at Google Scholar
  8. H. P. Lin, S. C. Chang, and J. D. Wu, “Beam vibrations with an arbitrary number of cracks,” Journal of Sound and Vibration, vol. 258, no. 5, pp. 987–999, 2002. View at Publisher · View at Google Scholar
  9. H. P. Lin and S. C. Chang, “Forced responses of cracked cantilever beams subjected to a concentrated moving load,” International Journal of Mechanical Sciences, vol. 48, no. 12, pp. 1456–1463, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. S. Y. Alsabbagh, O. M. Abuzeid, and M. H. Dado, “Simplified stress correction factor to study the dynamic behavior of a cracked beam,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 127–139, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. H. P. Lin, “Direct and inverse methods on free vibration analysis of simply supported beams with a crack,” Engineering Structures, vol. 26, no. 4, pp. 427–436, 2004. View at Publisher · View at Google Scholar
  12. W. F. Brown and J. E. Srawley, Plane Strain Crack Toughness Testing of High Strength Metallic Material, American Society Testing and Materials, 1966.