楊鄂川，李映輝，趙 翔，李 亮

1.西南交通大學(xué)力學(xué)與工程學(xué)院，成都 610031;

2.重慶理工大學(xué)機(jī)械工程學(xué)院，重慶 400054

1.Introduction

Beams are widely used in various engineering fields.Damages of beams will appear due to high load or other abominable working environment.Usually，the damages are marked firstly by the occurrence and extension of surface cracks.The cracks are important reason for structural failure.The research of vibration characteristics of cracked beams is the most important basic theory of the damage detection technology［1］.It need be analyzed quantitatively that how cracks will affect the vibration of a beam.At present，a number of efforts have been made to analyze how will crack parameters influence on the vibration characteristics of a beam［2-10］.

Crack modeling is always the key to study the vibration of a cracked beam，there are two widely used methods to model opening cracks in beams:the weaken section method［3-4］and the local flexibility method［5-7］.Both of the methods are to simulate the stiffness just at the section where the cracks occur.However，in fact，the crack will cause continuous stiffness attenuations near the crack region.Swamidas and Yang established the continuous bending stiffness and shear stiffness of a cracked beam based on energy theory of fracture mechanics［8］.This model reflects the stress concentration of crack area，while taking the stiffness attenuation nearby the crack section into account.In this paper，based on the continuous stiffness model and matrix transfer method，vibration characteristics of cracked Timoshenko beam were calculated.Comparing the results with those obtained by the finite-element method，the validity of the method used in this paper was verified.Meanwhile，by examples in this paper，natural frequency attenuation of the first three orders，which fluctuates with the variation of crack parameters such as depth and location of a crack，was analyzed.

2.Energy expressions of cracked beams

Suppose a cracked beam is subjected to a con-stant bending moment，if the crack has extended for Δa，the bending moment will perform some work，that will not only makes the crack growth，but also increases the strain energy of the beam:

Where，S is the energy for crack growth，ΔU is the elastic strain energy increment of a cracked beam.

Using the Clapeyron’s theorem，it can be inferred that for a cracked beam the work(W)done by stable external force or moment was twice the elastic strain energy increment(ΔU)［9］:

It can be obtained that:

3.The continuous stiffness of cracked beams

3.1.The energy for crack growth

When a crack is formed in one side of the beam，suppose that crack depth increases from 0 to a under constant bending moment，according to the theory of fracture mechanics，the energy needed for crack growth is:

Where，b is the width of beam，a is the crack depth，G is strain energy release rate.Under the bending moment G can be expressed as:

Where，KIis stress intensity factor for the first mode crack(opening crack)，E is Young Modulus.

Figure 1.Diagrammatic sketch of a cracked beam

A beam with a rectangular section and an opening crack is shown in Figure 1，KIcan be expressed as［10］:

Where，F(xiàn)(a/h)is shape coefficient associated with a/h.If a/h≤0.6，a tabular format is given as

Substitution of Eq.(6)and(5)into(4)yields:

3.2.Theelasticstrain energy incrementof cracked beams

If the ΔU in Eq.(2)is considered as a continuous function along the beam，such that:

From fracture mechanics，the stress is highly concentrated around the crack region，so we can assume that the elastic strain energy increment of a cracked beam(ΔU)is also concentrated mainly around the crack region too.The distribution of ΔU (x)along the length of a beam can be expressed as［8］:

Where，Q(a，c)and k(a)are terms to be determined，c is the distance to the crack iocation from one end of the beam.

3.3.Continuous stiffness model

For a beam with an opening crack，Introducing EIC(x)as continuous bending stiffness，that means the bending stiffness of cross-section changes continuously along length direction of the beam.So，the total strain energy of the cracked beam can be expressed as:

Prior to crack occurrence，for a beam under a constant moment M，the bending stiffness is EI，strain energy of the beam can be expressed as:

Therefore，total strain energy of cracked beam can be expressed as:

Substitution of Eq.(8)，Eq.(9)，Eq.(11)and (12)into(13)yields:

From Eq.(14)，the continuous bending stiffness of cracked beam is obtained as［8］:

Where，EI is bending stiffness of the beam ignoring cracks，x is the longitudinal coordinates along the beam，EIR(a，c)and k(a)are terms related to the depth and position of the crack.For rectangular section beams the expressions can be determined as:

In the equation(16)and(17)，l is the length of the beam and h for height of beam section.

By the same derivation processes，continuous shear stiffness of cracked beam can obtained as:

Where，GA is shear stiffness of the beam ignoring cracks.

For a rectangular section beams，take the parameters as l=1 000 mm，h=100 mm，c=l/2，a= h/2，the continuous bending stiffness(EIC(x))and continuous shear stiffness(GAC(x))were calculated from Eq.(15)and(18)，and shown in Figure 2.

Figure 2.Continuous bending and shearing stiffness

As Figure 2 shows，for a beam with a crack occurs in the midpoint of the beam and crack depth a= h/2，stiffness attenuation is very significant in the crack section.However，it is worth noting that the stiffness attenuation is also obvious at the region near the crack location，about 1/6 length of the beam.

4.Examples

Based on the continuous conditions(continuous displacement，rotation，bending moment and shear force)，the relationships of the mode function between two adjacent segments can be established.U-sing matrix transfer method，the relationships of the mode function between the first and the end segment can also be derived.With given boundary conditions of end points of a beam，the characteristic equations can be built，and the natural frequency can be calculated by using Newton-Raphson method［11-12］.

4.1.The verification of validity

To verify the validity of the method used in this paper，the first three natural frequencies of a simply supported Timoshenko beam(a rectangular cross section beam，length l=1 000 mm，width b=100 mm，height h=100 mm)with an opening crack was calculated and the results were compared with those obtained from finite element method.The ANSYS software was selected for finite element calculation，using the element type of SOLID 65 for simulation.

The first three natural frequencies，with different crack depth(a/h=0，a/h=0.3 and a/h=0.5)，obtained from two methods are listed in Table 1，which are in good agreement with one another.When the crack depth is relatively shallow，the maximum error between the two methods is very small.When the crack depth is deeper，the error value increases，but the maximum error in Table 1is only 3.81%.With the error analyses above，the conclusion can be drawn that the method used in this paper is accurate enough to model the beams with opening cracks.

Table 1.Natural frequencies for a cracked Timoshenko beam(l/h=10)

4.2.The effect of crack depth on natural frequencies

Also take the rectangular section beam for example，which has been used above.Suppose the crack occurs in the midpoint of the beam.To analyze the effect of crack depth on natural frequency，the first three natural frequencies of the beam were calculated，as crack depth(a/h)change continuously from 0 to 0.5.ωciand ωirespectively means the ith natural frequency of the cracked beam and uncracked beam.In Figures 3 to 5，the ωci/ωiis taken to express the frequency attenuation caused by cracks.

Figure 3.The first frequency attenuation(c=l/2)

Figure 4.The second frequency attenuation(c=l/2)

Figure 5.The third frequency attenuation(c=l/2)

The attenuation of the first three natural frequencies，caused by cracks with different crack depths are shown in Figures 3～5.It can be seen that the frequency attenuation can be sorted as:the first order＞the third order＞the second order.It is because the crack occurs in the midpoint of the beam(c=l/ 2).In this case，the crack has little impact on second frequency，even crack depth reaches half the height of the cross section(a/h=0.5)，ωc2/ω2is 0.979 6，the frequency attenuation is about 2%and is not obvious.

When the crack depth is shallow(a/h≤0.2)，the attenuations of first and third frequency are also not obvious.The maximum attenuation rate of first frequency is 2.63%.The frequency attenuations become gradually obvious while the crack be deeper.When the crack depth increases(such as a/h= 0.4)，the maximum attenuation rate of first frequency reaches 12.03%，while that of third frequency is 6.54%.When crack depth reaches half the height of the cross section(a/h=0.5)，the frequency attenuation of the cracked beam will become more significant，the maximum attenuation rate of first order frequency reaches 21.0%，while that of third order frequency is 9.99%.

4.3.The effect of crack location on natural frequencies

Assuming that the crack varies continuously from C=0 to C=L，crack depth is fixed(a/h=0.5)，other parameters of the beam are the same to the previous example.To analyze how the crack position will affect on the natural frequency，the attenuation curves of the first three natural frequencies are shown in Figures 6～8.

According to the results，for each frequency，when the crack is close to the position of a modal node，crack has tiny impact on the frequencies.If the crack is just overlap the position of a modal node，the attenuation of the second frequency is 3.2%，while 4.0%for the third frequency(see Figure 7 and Figure 8).When the crack is close to the position，where occurs the biggest mode displacement，it has more significant impact on the frequency.However，the sensitivity of each frequency is not the same.It can be found that，as the modal order increasing，the frequency becomes less sensitive to the crack.For example here(a/h=0.5，l/h=10)，the maximum attenuation of the first frequency is 21.7%，while 14.8%and 10.8%for the second and third frequency(as shown in Figures 6～8).

Figure 6.The first frequency attenuation(a/h=0.5)

Figure 7.The second frequency attenuation(a/h=0.5)

Figure 8.The third frequency attenuation(a/h=0.5)

5.Conclusion

In this paper，using energy method，continuous bending and shear stiffness model of cracked beams were obtained.By means of transfer matrix method and Newton-Raphson method，first three natural frequencies of cracked Timoshenko beams were calculated.By comparing the results with those obtained by finite element software，it was proved that the method used in this paper is valid with good accuracy.

At the same time，through the analysis of examples，this paper summarized the influence of crack depth and crack position on the first three natural frequencies.It concludes that，frequency attenuation is not obvious when the crack depth is shallow(a/h≤0.2).However，as the crack deepening，frequency attenuation becomes more obvious.In the examples given in this paper，as the crack depth increases，or a/h varies from 0.2 to 0.5，the maximum attenuation of the first order frequency increases from about 3.0%to about 20%.In addition，the location of cracks also impacts on the frequency attenuation of a cracked beam.When the crack position is close to a modal node，the influence is very weak.Inversely，when the crack close to the position，where occurs the biggest mode displacement，it has more significant impact on the frequency.Moreover，the Sensitivity of each order frequency to crack is not the same，as the modal order increasing，the sensitivity of the frequency to cracks presents a declining trend.

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