Modified and Simplified Sectional Flexibility of a Cracked Beam

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 ℎ, and width 𝑡. The singular stress distribution at the crack tip takes the form [3, 4, 10]𝜎^𝑠𝑥=𝐾^𝐼√2𝜋𝑟,(2.1) with the conditions that 𝜎^𝑠𝑥 acts at a distance 𝑟=𝑏 from the tip and 𝐾_𝐼 is the stress intensity factor. The normal stress acting on the reduced cross-section passing through the crack tip is given as 𝜎_𝑥=𝑀𝐼𝑦,(2.2) where 𝑀 is the bending moment and 𝐼=𝑡(ℎ−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=𝑦−𝑏=ℎ2−𝑎2−𝑏,(2.3) where 𝑦 is the distance from the neutral axis of the reduced cross-section to the tip as shown in Figure 1.

Figure 1 

Geometry of a beam with a single-edge crack.

The stress condition is considered as 𝜎^𝑠𝑥=𝜎_𝑥 at the crack tip 𝑟=𝑏. Substituting (2.3) into (2.2), 𝜎_𝑥 and 𝐾_𝐼 can be expressed as𝜎_𝑥=12𝑀𝐾𝑡(ℎ−𝑎)³ℎ2−𝑎2−𝑏,(2.4)_𝐼=√2𝜋𝑏𝜎_𝑥=12𝑀√2𝜋𝑏𝑡(ℎ−𝑎)³ℎ2−𝑎2−𝑏.(2.5) The distance 𝑏 can be determined from the equilibrium condition of forces along the 𝑥-axis:^𝑏0𝐾_𝐼√2𝜋𝑟𝑑𝑟=^{𝑦𝑦−𝑏}𝜎_𝑥𝑑𝑦.(2.6) The left-hand side of (2.6) is evaluated, using (2.5), to be^𝑏0𝐾_𝐼√2𝜋𝑟𝑑𝑟=𝑡(2ℎ4−𝑀𝑎𝑏)³ℎ2−𝑎2−𝑏.(2.7) The right-hand side of (2.6) is evaluated, using (2.4), to be^𝑦_𝑦−𝑏𝜎_𝑥𝑑𝑦=6𝑀𝑏𝑡(ℎ−𝑎)³(ℎ−𝑎−𝑏).(2.8) Substitution of (2.7) and (2.8) into (2.6) leads to24𝑀𝑏𝑡(ℎ−𝑎)³ℎ2−𝑎2−𝑏=𝑡(ℎ6𝑀−𝑏𝑎)³(ℎ−𝑎−𝑏).(2.9) So𝑏=13(ℎ−𝑎).(2.10) Substituting (2.10) into (2.5), the stress intensity factor can be expressed as𝐾_𝐼=6𝑀𝑡ℎ^3/2𝐹𝑎ℎ,(2.11) where𝐹𝑎ℎ=2𝜋27⋅√(1−1𝑎/ℎ)³=0.482√(1−𝑎/ℎ)³.(2.12)

3. Calculation of the Equivalent Flexibility in the Crack Beam

Let 𝑈_𝑇 be the strain energy due to the crack. According to Castigliano’s theorem, the additional displacement is 𝑢_𝑖=𝜕𝑈_𝑇/𝜕𝑃_𝑖 under general loading 𝑃_𝑖. In this work, the displacement will reduce to𝜃=𝜕𝑈^𝑇𝜕𝑀,(3.1) where the displacement 𝑢_𝑖 is taken as the rotation 𝜃 since the bending moment 𝑀 is the only load of the structure.

The strain energy has the form [4–6, 10]𝑈_𝑇=^𝑎0𝜕𝑈_𝑇𝜕𝑎𝑑𝑎=𝑡^𝑎0𝐽𝑑𝑎,(3.2) where 𝐽 is the strain energy density function. Therefore,𝑢_𝑖=𝜕𝑈^𝑇𝜕𝑃_𝑖=𝜕𝜕𝑃_𝑖𝑡^𝑎0𝐽(𝑎)𝑑𝑎(Paris'sequation).(3.3) The moment 𝑀 substitutes for the generalized load 𝑃_𝑖𝜃=𝜕𝜕𝑀𝑡^𝑎0𝐽(𝑎)𝑑𝑎.(3.4) The flexibility influence coefficient will be written as𝑐=𝜕𝜃𝜕𝑀=𝜕𝑀𝜕²²𝑡^𝑎0𝐽(𝑎)𝑑𝑎.(3.5) The strain energy density function 𝐽 has the form𝐽=𝐾^2𝐼𝐸^,(3.6) where 𝐸^=𝐸/(1−𝜐²) for plane strain and 𝐸 and 𝜐 are Young’s modulus and Poisson’s ratio, respectively. The flexibility scalar is𝑐=𝜕²𝜕𝑀²𝑡^𝑎0𝐾^2𝐼𝐸^𝑑𝑎=2𝜋1−𝜐²ℎ1−(1−𝑎/ℎ)²9𝐸𝐼(1−𝑎/ℎ)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=𝑡ℎ³/12. The nondimensional cracked section flexibility can be found from (3.7) as𝑐^∗=𝐸𝐼𝑐𝐿=2𝜋1−𝜐²1−(1−𝑎/ℎ)²9(1−𝑎/ℎ)²⋅ℎ𝐿.(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 is𝐸𝐼𝜕⁴𝑌_𝑖(𝑋,𝑇)𝜕𝑋⁴+𝜌𝐴𝜕²𝑌_𝑖(𝑋,𝑇)𝜕𝑇²=0𝑋_𝑖−1<𝑋<𝑋_𝑖,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 are𝑌𝑌₁𝑋⁻¹,𝑇=𝑌₂𝑋⁺¹,𝑇,(4.3)_″1𝑌𝑋⁻¹,𝑇=𝑌^″₂𝑋⁺¹,𝑇,(4.4)_‴1𝑋⁻¹,𝑇=𝑌^‴₂𝑋⁺¹,𝑇,(4.5) and the compatibility condition due to the rational flexibility is𝑌^′₂𝑋⁺¹,𝑇−𝑌^′₁𝑋⁻¹,𝑇=𝐸𝐼𝑐𝑌^″₂𝑋⁺¹,𝑇.(4.6) From the above equations, the following quantities are introduced for nondimensional analysis:𝑦=𝑌𝐿,𝑥=𝑋𝐿,𝑡=𝑇,𝑥^𝑖=𝑋^𝑖𝐿,𝑙¹=𝐿¹𝐿,𝑙²=𝐿²𝐿.(4.7) Equation (4.1) can then be expressed in a nondimensional form as𝐸𝐼𝐿⁴𝜕⁴𝑦_𝑖(𝑥,𝑡)𝜕𝑥⁴+𝜌𝐴𝜕²𝑦_𝑖(𝑥,𝑡)𝜕𝑡²=0𝑥_𝑖−1<𝑥<𝑥_𝑖,𝑖=1,2.(4.8) The continuous conditions at the crack position as non-dimensional form are𝑦𝑦₁𝑥⁻¹,𝑡=𝑦₂𝑥⁺¹,𝑡_″1,𝑦𝑥⁻¹,𝑡=𝑦^″₂𝑥⁺¹,𝑡_‴1,𝑦𝑥⁻¹,𝑡=𝑦^‴₂𝑥⁺¹,𝑡_′2,𝑥⁺¹,𝑡−𝑦^′₁𝑥⁻¹,𝑡=𝑐^∗𝑦^″₂𝑥⁺¹,𝑡,(4.9) where 𝑐^∗ is the non-dimensional cracked section flexibility as (3.8).

Figure 2 

Schematic diagram of a simply supported cracked beam.

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, 𝑦_𝑖(𝑥,𝑡)=𝑤_𝑖(𝑥)𝑒^𝑗𝜔𝑡, in (4.8), the differential equation for free vibration can be written as𝑤^⁗_𝑖(𝑥)−𝜆⁴𝑤_𝑖(𝑥)=0,𝑥_𝑖−1<𝑥<𝑥_𝑖,𝑖=1,2,(5.1) where𝜆⁴=𝜌𝐴𝜔²𝐿⁴𝐸𝐼.(5.2) From (4.9), the continuous conditions at the crack position are𝑤𝑤_1″1𝑥⁻¹=𝑤₂𝑥⁺¹,𝑤_‴1𝑥⁻¹=𝑤^″₂𝑥⁺¹,𝑤_′2𝑥⁻¹=𝑤^‴₂𝑥⁺¹,𝑥⁺¹−𝑤^′₁𝑥⁻¹=𝑐^∗𝑤^″₂𝑥⁺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, is𝑤_𝑖(𝑥)=+𝐴𝐶_𝑖s_𝑖in𝜆𝑥−𝑥_𝑖−1+𝐵_𝑖cos𝜆𝑥−𝑥_𝑖−1sinh𝜆𝑥−𝑥_𝑖−1+𝐷_𝑖cosh𝜆𝑥−𝑥_𝑖−1,𝑥_𝑖−1<𝑥<𝑥_𝑖,𝑖=1,2,(5.4) where 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, and 𝐷_𝑖 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 as⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₂𝐵₂𝐶₂𝐷₂⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑡₁₁𝑡₁₂𝑡₁₃𝑡₁₄𝑡₂₁𝑡₂₂𝑡₂₃𝑡₂₄𝑡₃₁𝑡₃₂𝑡₃₃𝑡₃₄𝑡₄₁𝑡₄₂𝑡₄₃𝑡₄₄⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₁𝐵₁𝐶₁𝐷₁⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭=𝑇_4×4⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₁𝐵₁𝐶₁𝐷₁⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,(5.5) where 𝑇_4×4 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 as𝑌𝑌𝑌((𝐿0,,𝑇𝑇))==00⟶⟶𝑤𝑤((01))==00,,((55..67))_″𝑌(0,𝑇)=0⟶𝑤^″(0)=0,(5.8)_″(𝐿,𝑇)=0⟶𝑤^″(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:−𝐴𝐴₂₂sin𝜆𝑙₂+𝐵₂cos𝜆𝑙₂+𝐶₂sinh𝜆𝑙₂+𝐷₂cosh𝜆𝑙₂=0,sin𝜆𝑙₂−𝐵₂cos𝜆𝑙₂+𝐶₂sinh𝜆𝑙₂+𝐷₂cosh𝜆𝑙₂=0,(5.11) which can be expressed in matrix form as⎧⎪⎨⎪⎩00⎫⎪⎬⎪⎭=⎡⎢⎢⎢⎣sin𝜆𝑙₂cos𝜆𝑙₂sinh𝜆𝑙₂cosh𝜆𝑙₂−sin𝜆𝑙₂−cos𝜆𝑙₂sinh𝜆𝑙₂cosh𝜆𝑙₂⎤⎥⎥⎥⎦⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴𝐵²₂𝐶₂𝐷₂⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭=𝐵_2×4⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₂𝐵₂𝐶₂𝐷₂⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,(5.12) where𝐵_2×4=⎡⎢⎢⎢⎣sin𝜆𝑙²cos𝜆𝑙₂sinh𝜆𝑙₂cosh𝜆𝑙₂−sin𝜆𝑙₂−cos𝜆𝑙₂sinh𝜆𝑙₂cosh𝜆𝑙₂⎤⎥⎥⎥⎦.(5.13) Substituting (5.5) into (5.12) and applying (5.10), one obtains⎧⎪⎨⎪⎩00⎫⎪⎬⎪⎭=𝐵_2×4⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₂𝐵₂𝐶₂𝐷₂⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭=𝐵_2×4⋅𝑇_4×4⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₁𝐵₁𝐶₁𝐷₁⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭=𝑅_2×4⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐴₁𝐵₁𝐶₁𝐷₁⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,(5.14) where𝑅_2×4=𝐵_2×4⋅𝑇_4×4=⎡⎢⎢⎢⎣𝑟¹¹𝑟₁₂𝑟₁₃𝑟₁₄𝑟₂₁𝑟₂₂𝑟₂₃𝑟₂₄⎤⎥⎥⎥⎦.(5.15) A nontrivial solution of the simply supported beam requiresdet⎡⎢⎢⎢⎣𝑟¹¹𝑟(𝜆)𝑟₁₃(𝜆)₂₁(𝜆)𝑟₂₃(𝜆)⎤⎥⎥⎥⎦=0.(5.16) The characteristic equation of the cracked simply supported beam can be obtained as𝑐^∗𝜆_𝑛+sin2hsin𝜆h_𝑛𝑙₁𝜆_𝑛sinh𝜆_𝑛𝑙₂sin𝜆_𝑛𝑙₁+𝑙₂−𝑐^∗𝜆_𝑛sin𝜆_𝑛𝑙₁sin𝜆_𝑛𝑙₂sinh𝜆_𝑛𝑙₁+𝑙₂𝑙₁+𝑙₂sin𝜆_𝑛𝑙₁+𝑙₂=0,(5.17) where 𝜆_𝑛 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 𝐾_𝐼 in (2.11) is expressed in this paper. The function 𝐹(𝑎/ℎ) in (2.12) can be compared with the expression given by [2]𝐾_𝐼=6𝑀𝑡ℎ^3/2𝐹₂𝑎𝐹₂ℎ,ℎ𝑎=√𝜋𝑎ℎ2ℎ𝜋𝑎tan𝜋2ℎ𝑎0.923+0.c1o9s9((1𝜋𝑎−/2sℎin)(𝜋𝑎/2ℎ))⁴.(6.1) Another stress intensity factor is [12]𝐾_𝐼=6𝑀𝑡ℎ^3/2𝐹₃𝑎𝐹₃ℎ,ℎ𝑎=√𝜋𝑎ℎ1.122−1.44ℎ𝑎+7.33ℎ𝑎²−13.08𝑎ℎ³+14𝑎ℎ⁴.(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/ℎ.

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=6𝜋1−𝜐²⋅ℎΦ𝑎𝐿⋅Φℎ𝑎,ℎ=0.6272ℎ𝑎²−1.04533𝑎ℎ³+4.5948𝑎ℎ⁴−9.9736𝑎ℎ⁵+20.2948𝑎ℎ⁶−33.0351𝑎ℎ⁷+47.1063𝑎ℎ⁸−40.7556𝑎ℎ⁹+19.6𝑎ℎ¹⁰.(6.3) Another non-dimensional cracked section flexibility is [7]𝑐₃=6𝜋𝑎ℎ²⋅ℎ𝑓_𝐽𝐿⋅𝑓^𝐽ℎ𝑎,ℎ𝑎=0.6384−1.035ℎ𝑎+3.7201ℎ𝑎²−5.1773ℎ𝑎³+7.553𝑎ℎ⁴−7.332𝑎ℎ⁵+2.4909𝑎ℎ⁶.(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/ℎ. A cracked beam with 𝑎/ℎ greater than 0.5 is already severely damaged and is not suitable for applying this sectional flexibility model.

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.

Table 1 

Natural frequency ratio with the crack depth at location 0.8.

(a)

(b)

(c)

(d)

(e)

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.

Table 2 

Natural frequency ratio with the crack depth at location 0.5.

(a)

(b)

(c)

(d)

(e)

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/ℎ=0.4, and normalized amplitude 𝑌/𝑌_max. It is obvious that the mode shapes all show turning point at crack location 𝐿₁/𝐿=0.8.

(a)

(b)

(c)

(d)

(e)

Figure 8 

(a) Normalized mode shape of the first mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/ℎ=0.4. (b) Normalized mode shape of the second mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/ℎ=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/ℎ=0.4. (e) Normalized mode shape of the fifth mode with a crack location 𝐿1/𝐿=0.8 and crack depth ratio 𝑎/ℎ=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/ℎ is reduced, because, with a crack depth ratio 𝑎/ℎ<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/ℎ 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 𝜔_𝑖/𝜔_0𝑖=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.