Stress intensity factors evaluation for through-transverse crack

Stress intensity factors evaluation for through-transverse crack in slab track system under vehicle dynamic

load

Shengyang Zhu a ,b ,Chengbiao Cai a ,?

a State Key Laboratory of Traction Power,Southwest Jiaotong University,No.111First Section,North of Second Ring Road,Chengdu 610031,PR China b

Department of Civil and Environmental Engineering,Rice University,6100Main Street,Houston,TX 77005,USA

a r t i c l e i n f o Article history:

Received 12May 2013

Accepted 11September 2014

Available online 22September 2014Keywords:

Slab track,Crack

Extended ?nite element method Vehicle–track coupled dynamics Stress intensity factors

a b s t r a c t

The stress intensity factors (SIFs)for through-transverse crack in the China Railway Track System (CRTS II)slab track system under vehicle dynamic load are evaluated in this paper.A coupled dynamic model of a half-vehicle and the slab track is presented in which the half-vehicle is treated as a 18-degree-of-freedom multi-body system.The slab track is modeled as two continuous Bernoulli–Euler beams supported by a series of elastic rectan-gle plates on a viscoelastic foundation.The model is applied to calculate the vertical and lateral dynamic wheel–rail forces.A three-dimensional ?nite element model of the slab track system is then established in which the through-transverse crack at the bottom of concrete base is created by using extended ?nite element method (XFEM).The wheel–rail forces obtained by the vehicle-track dynamics calculation are utilized as the inputs to ?nite element model,and then the values of dynamic SIFs at the crack-tip are extracted from the XFEM solution by domain based interaction integral approach.The in?uences of subgrade modulus,crack length,crack angle,friction coef?cient between cracked surfaces,and friction coef?cient between faces of concrete base and subgrade on dynamic SIFs are investigated in detail.The analysis indicates that the subgrade modulus,crack length and crack angle have great effects on dynamic SIFs at the crack-tip,while both of the friction coef?cients have negligible in?uences on variations of dynamic SIFs.Also the statistical characteristics of varying SIFs due to random wheel–rail forces are studied and results reveal that the distributions of dynamic SIFs follow an approximately Gaussian distribution with different mean values and standard deviations.The numerical results obtained are very useful in the maintenance of the slab track system.

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1.Introduction

High-speed railway lines in the form of slab tracks have played an important role in ensuring the riding comfort and derailment safety of high-speed trains,owing to their advantages of better stiffness uniformity,higher running stability,and lower maintenance cost [1–3].Fig.1shows the CRTS II slab track and its components.The primary components of the track system are the rails,the rail pads,the concrete slab,the CA mortar layer,the concrete base,and the subgrade.Clearly,the slab track is a layered elastic system with the elastic modulus and material strength decreasing from the top

https://www.sodocs.net/doc/cc15790652.html,/10.1016/j.engfailanal.2014.09.0041350-6307/ó2014Elsevier Ltd.All rights reserved.

?Corresponding author.Tel.:+862886465537.

E-mail addresses:syzhu@https://www.sodocs.net/doc/cc15790652.html, (S.Zhu),cbcai@https://www.sodocs.net/doc/cc15790652.html, (C.Cai).

220S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

Fig.2.Through-transverse crack of concrete base in CRTS II slab track.

layer to bottom layer.To enhance the structural integrity and improve the riding comfort of high-speed trains,the track-lay-ers are connected into a whole through friction forces,bonding forces or force transition components between the track-layers.

Due to the fact that the application time of slab tracks is still short in the development of high-speed railways,the knowl-edge and experience of the design,the construction,and the operation and maintenance of the slab tracks are often quite limited among railway engineers.Therefore,premature structural damage can be found in nearly all kinds of slab tracks caused by a variety of factors.On the Chinese high-speed railway lines,the damage or deterioration of the concrete base with through-transverse cracks in the slab tracks are recognized as one of the most critical problems after a short time operation, as shown in Fig.2.Such damage modes not only increase the cost of railway operation,but also signi?cantly weaken the structural integrity of the slab track system,and contribute to the potential for failure of the slab tracks.During the service life of the CA mortar layers,the through-transverse cracks could be induced by several mechanisms.The vehicle dynamic load is one of the most dominant causes for this premature damage of the concrete base in the slab tracks.Although a nota-ble loss of the bearing capacity of the slab tracks as a direct result of the cracks of the concrete base has rarely been reported, the cracks have a signi?cant effect on the durability and service life of the concrete base because it can cause the rainwater to permeate into the cracks,accelerating the propagation of the cracks to the concrete slabs.Motivated by the desire to min-imize damage of track structures and to ensure the running safety,modern railway engineers require a fundamental under-standing of the damage or failure mechanisms under train and track interactions.However,most studies have focused on rail surface cracks in the ballast track system[4–9],and studied show the SIFs at crack-tip are related to the crack length,the crack angle,the contact patch size,and the friction coef?cient between cracked faces when a contact load moves over the crack.Whereas quite a few studies have been published concerning the damage or fracture mechanisms of slab track com-ponents.Takahashi[10]evaluated the factors contributing to the deterioration of track slabs in cold areas.He found that deterioration occurred more frequently on south facing sides of track slabs through visual inspections,and the track slab deterioration was caused by an alkali–silica reaction and frost damage.Muramoto[11]carried out tests with a full-scale bal-lastless track model laid on a saturated clayey roadbed.He clari?ed the defect mechanism of clayey roadbeds under ballast-less tracks and proposed two countermeasures against failures by means of the full-scale tests.Zhu and Cai[12]employed a cohesive zone model to model the interface damage between the CA mortar layer and the concrete slab,and investigated the

vertical dynamic coupled model of a vehicle and the double-block ballastless track with a track-bed crack modeled with3D singular elements,and studied the dynamic SIFs at the crack-tip due to the vehicle passing.Although this model is physically meaningful and illustrates the characteristics of the SIFs at crack-tip,it is limited in that the model made some simpli?ca-tions and the effect of lateral dynamic loads of the system was not considered.

Stress intensity factors(SIFs)at a crack-tip play an important role in linear elastic fracture mechanics as they govern crack propagation rate and trajectory.The objective of the present paper is to evaluate the SIFs for through-transverse crack at the bottom of concrete base in CRTS II slab track under vehicle dynamic load.The mechanisms of how the subgrade modulus, crack length,crack angle,friction coef?cient between cracked surfaces,and friction coef?cient between faces of concrete base and subgrade affect the dynamic SIFs are investigated in detail by using a calculation model which considers a combi-nation of three dimensional vehicle-track coupled dynamic theory and XFEM solution,and also the statistical characteristics of varying SIFs due to random wheel–rail forces are studied.The numerical results obtained are very useful in the design and maintenance of slab track system.

2.Finite element modeling

2.1.XFEM approximation for the crack

The extended?nite element method,a classic?nite element approximation enriched by additional special terms using the framework of partition of unity[14],has gained considerable attention in fracture mechanics to compute the SIFs for cracked structures.The crack is commonly presented by using the level set method.The introduction of the enriched discon-tinuous solutions into the?nite element basic functions allows for the simulation of crack growth or complicated crack geometries without remeshing and the?nite elements is independent of the crack topology.A literature review of the XFEM method for computational fracture mechanics and its advantages compared with the traditional?nite element method can be found in Yazid[15].The?rst publications of the XFEM models for solving dynamic problems were presented in Belytschko [16,17],and Zi[18].Recently,Motamedi and Mohammadi[19–21]extended the standard XFEM models for the orthotropic materials to investigate the dynamic crack propagation in composites.

For the purpose of fracture analysis,the crack-tip enrichment functions typically consist of the near-tip asymptotic func-tions that capture the singularity around the crack-tip and the Heaviside function that represents the jump in displacement across the cracked surfaces,and they were de?ned in the crack domain X=X1+X2as shown in Fig.3.For simplicity,illus-trations are given in a two-dimensional cracked problem but the generalization for three-dimensional problems is straight-forward.Note that in Fig.3,the circle symbols represent the elements enriched by the Heaviside function and the square symbols show the elements enriched by singular functions.The approximation for a displacement vector u with the partition of unity enrichment is expressed as follow[22]:

u?

X N

I?1N IexTu ItHexTa It

X4

a?1

F aexTb a

I

"#

e1T

where N I(x)are the usual nodal shape functions;the?rst term on the right-hand side of the above equation,u I,is the usual nodal displacement vector associated with the continuous part of the?nite element solution,and it is applicable to all the nodes in the model;the second term is the product of the nodal enriched degree of freedom vector,a I,and the associated Heaviside function H(x)across the cracked surfaces(Heaviside is+1on one side of the discontinuity andà1on other side of the discontinuity),noted that this term is valid for nodes whose shape function support is cut by the crack interior X1;

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237221

and the third term used only for nodes whose shape function support is cut by the crack-tip X2,is the product of the nodal enriched degree of freedom vector,b a

I

,and the associated elastic asymptotic crack-tip functions,F a(x).

Since the crack is not explicitly meshed,the location of the crack is no more included in the mesh.This method therefore requires another way to simulate the crack.In the XFEM model,a powerful tool,the level set method,is used to describe the through-transverse crack,and the crack is treated as a single surface.In level set approach,a crack is described by crack-tip and level set function[23–25],and two level sets are necessary to represent the crack in the level set function,i.e.normal level set(a signed distance function,W which is the closest point to the crack surface)and tangent level set(tangent to the crack surface at the closest point,U).For simplicity,Fig.4illustrates the use of both level set functions to de?ne the geom-etry of a2D crack.

The crack-tip enrichment function,which de?nes the discontinuity near the crack-tip,takes the form of four functions as follow:

F aexT?

???

r

p

sin

h

2

;

???

r

p

cos

h

2

;

???

r

p

sin h sin

h

2

;

???

r

p

sin h cos

h

2

e2T

where r and h are polar coordinates in the local crack-tip coordinate system.

The stress intensity factors K I,K II,and K III play an important role in linear elastic fracture mechanics.They characterize the in?uence of load or deformation on the magnitude of the crack-tip stress and strain?elds and measure the propensity for crack propagation or the crack driving forces.Furthermore,the stress intensity can be related to the energy release rate (the J-integral)for a linear elastic material through.

J?

1

8p

K TáBà1áKe3T

where K=[K I,K II,K III]T and B is called the pre-logarithmic energy factor matrix[26,27].For homogeneous,isotropic materials B is diagonal and the above equation simpli?es to:

J?1

E K

2

I

tK2

II

t

1

2G

K2

III

e4T

where E?E for plane stress and E?E=e1àm2Tfor plane strain,axisymmetry,and three dimensions.And then the domain based interaction integral approach[27]is used to extract the stress intensity factors for the crack under mixed-mode loading.

2.2.Finite element model of the slab track

In order to evaluate the stress intensity factors for through-transverse crack of concrete base under vehicle dynamic load, a three-dimensional?nite element model of the CRTS II slab track system with through-transverse crack at the bottom of concrete base is built up based on the commercial software ABAQUS,as shown in Fig.5.

In this study,series of spring elements are employed to represent rail fastening system at each node and they are parallel at both sides of the rail foot.The nodes of these spring elements can move in the vertical and lateral directions with corre-sponding stiffness.Hinged boundaries are adopted for the rails to prevent free body motions.Linear line elements of type B31are used for the rails,and linear hexahedral elements of type C3D8R are applied for the other components of the slab track system,all the elements in the present study are assumed to behave pure elastically.The slab track system is repre-sented by10,935solid elements and15,800degrees of freedom,and the mesh has a total length of10m and is long enough to be considered as an in?nite slab track compared to the size of the crack.In the current investigation,a through-transverse crack at the bottom of concrete base is assumed to be located right below the rail fastening system,as shown in Fig.5b.A small-sliding formulation have been employed between cracked surfaces to model crack closure,and a?nite-sliding formu-lation have been applied between faces of concrete base and subgrade to consider the friction effect.It should be emphasized

222S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

that the present study deals only with stationary crack subjected to dynamic loads,it does not need to model crack propa-gation.To achieve more accurate results,a relatively ?ner mesh of elements is used close to the crack but larger elements are employed elsewhere.All the displacements of the nodes at the bottom of the slab track are restricted,and all the nodes at the both ends are applied symmetry constraints.In the present FE model,the XFEM described above is utilized to investigate the dynamic SIFs for the through-transverse crack under vehicle dynamic load.

The rail support pitch is 0.65m.The coef?cients of equivalent spring stiffness in the vertical and lateral directions are both selected to be 2.5?107N/m.The coef?cients of equivalent damper in the vertical and lateral directions are selected to be 2.5?104N s/m,respectively.The concrete slab has a Young’s modulus of 3.6?1010Pa,a Poisson ratio of 0.2,and a (a)

Through-transverse

crack

Crack angle θ

(b)

Rail fastener

Concrete slab

CA mortar layer

Concrete base

Subgrade

Rail

(c)

model:(a)The entire ?nite element mesh,(b)through-transverse crack at the bottom of concrete base,and S.Zhu,C.Cai /Engineering Failure Analysis 46(2014)219–237223

224S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

2400kg/m3.The concrete base has a Young’s modulus of2.55?1010Pa,a Poisson ratio of0.2,and a density of2400kg/m3. The subgrade has a Poisson ratio of0.25and a density of1950kg/m3.In order to investigate the in?uences of subgrade mod-ulus E s,crack length l,crack angle h,friction coef?cient k1between cracked surfaces,and friction coef?cient k2between faces of concrete base and subgrade on dynamic SIFs at the crack-tip,we select subgrade modulus E s=1.0?108,2.0?108, 3.0?108Pa,crack length l=0.1,0.15,0.2m,crack angle h=30°,60°,90°,friction coef?cient k1=0.0,0.15,0.3,and friction coef?cient k2=0.0,0.15,0.3,respectively.

3.Coupled dynamic model of a vehicle and the slab track

Fig.6illustrates the three-dimensional coupled dynamic model of a half-vehicle and the CRTS II slab track which is used to predict the dynamic wheel–rail forces when a vehicle moves along the slab track.

3.1.Motion equations of vehicle

The half vehicle is considered as a rigid multi-body model in which the car body is supported on two double-axle bogies with the primary and the secondary suspension systems.It should be noted that since the half vehicle is considered in the dynamics analysis as shown in Fig.1,it cannot be balanced when it runs on the track.Therefore,the pitching and yaw

motions of the half car body have to be neglected[28,29].Therefore the total number of degrees of freedom of the vehicle model is18,as listed in Table1.Based on the system of coordinates moving along the track at a constant speed of the vehicle, the motion equations of the half-vehicle can be easily described in the form of second-order differential equations in the time domain:

M V A VtC VeV VTV VtK VeX VTX V?F VeX V;V V;X T;V TTe5Twhere M V is the mass matrix of the vehicle;C V and K V are the damping and the stiffness matrices,depending on the current state of the vehicle subsystem to describe nonlinearities within the suspension;X V,V V and A V are the vectors of displace-ments,velocities and accelerations of the vehicle subsystem,respectively;X T and V T are the vectors of displacements and velocities of the track subsystem;F V(X V,V V,X T,V T)is the system load vector representing the nonlinear wheel/rail contact forces which is the function of the motions X V and V V of the vehicle and X T and V T of the track.

3.2.Motion equations of slab track

For the modeling of the CRTS II slab track,each rail is treated as continuous Bernoulli–Euler beam resting on railpads,and the vertical,lateral,and torsion motions of the rails are simultaneously taken into account.According to the Ritz’s method the fourth-order partial differential equations of the rails can be given in form of second-order ordinary differential equa-tions:for the vertical vibration:

€q V k etTt

EI Y

m r

k p

l

4

q

V k

etT?à

X N

i?1

P r

Vi

Z kex siTt

X4

j?1

P j Z kex wjTk?1$N Ve6T

for the lateral vibration:

€q L k etTt

EI Z

r

k p

4

q

L k

etT?à

X N

i?1

P r

Li

Y kex siTt

X4

j?1

Q j Y kex wjTk?1$N Le7T

and for the torsional vibration:

€q T k etTt

GK

q I0

k p

l

2

q

T k

etT?à

X N

i?1

M si H kex siTt

X4

j?1

M wj H kex wjTk?1$N Te8T

where q V k(t),q L k(t)and q T k(t)are the generalized coordinates,describing the vertical,lateral and torsional motions of the rail, respectively;EI Y,EI Z and GK are the vertical bending,lateral bending and torsional stiffness of the rail;m r is the mass per unit longitudinal length;I0is the torsional inertia of the rail;q is the density of the rail;Z k,Y k and H k are the k th mode functions of vertical bending,lateral bending,and torsion of the rail,respectively;x si are the coordinates of the rail-supporting points, and x wj are the coordinates of the contact points of the wheel and rail;N V,N L and N T are the total numbers of the rail mode functions which are all selected to be100in the calculation;P j and Q j are the vertical and lateral wheel/rail forces;M si and

M wj are the equivalent calculation moments on the rail;P r

Vi and P r

Li

are the vertical and lateral rail-supporting forces.When

q V k(t),q L k(t)and q T k(t)are obtained through the dynamic calculation,the vertical,lateral and torsional displacements of the rail can be written as:

Z rex;tT?

X N V

k?1

Z kexTq V ketTe9TY rex;tT?

X N L

k?1

Y kexTq L ketTe10T

/rex;tT?

X N T

k?1H kexTq

T k

etTe11T

Table1

Degrees of freedom of the half-vehicle model.

Vehicle component Type of motion

Lateral Vertical Roll Yaw Pitch Wheelset1Y w1Z w1u w1w w1b w1

Wheelset2Y w2Z w2u w2w w2b w2

Bogie frame Y t Z t u t w t b t

Car body Y c Z c u c––

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237225

The concrete slab and concrete base of the slab track are described as elastic rectangle plates supported on viscoelastic foundation.As shown in Fig.6,the three layers of discrete springs and dampers represent the elasticity and damping effects of the rail fastener,the CA mortar layer and the subgrade,respectively.By using the Ritz’s method,the vertical vibrations of the slab can be expressed as a series of second-order ordinary differential equations in terms of the generalized coordinates, as expressed by Eq.(13),which can be solved with the time-stepping integration method.

€T mn etTtC s q

s

h s

_T

mn

etTtD s q

s

h s

B3B2t2B4B5tB1B6

B1B2

T mnetT

?1

q s h s B1B2

X N p

i?1P rVietTX mex piTY ney piTà

X N b

j?1

F sVjetTX mex biTY ney biT

"#

e12Tsurroundings of the through-transverse crack in the concrete base at deformed FEM

226S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

where

B1?R L s

X2

m

exTdx

B2?R W s

Y2

n

eyTdy

B3?R L s

X0000

m

exTX mexTdx

B4?R L s

X00

m

exTX mexTdx

B5?R W s

Y00

n

eyTY neyTdy

B6?R L s

Y0000

n

eyTY neyTdy

8

>>>

>>>

>>>

>>>

><

>>>

>>>

>>>

>>>

>:

e13T

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237227

According to the elastic thin slab theory,the solution of the vertical motions of the slab can be obtained as:

wex;y;tT?

X N x

m?1X N y

n?1

X mexTY neyTT mnetTe14T

Regarding the lateral motions,the slab is considered as a rigid body.Thus,the lateral vibrations are written as:

q s L s W s h s€yetT?

X N p

i?1

P rLietTà

X N b

i?1

F sLjetTe15T

in Eqs.(12)–(15),X m(x)and Y n(y)are the mode functions of the slab with x and y coordinates respectively;T mn(t)are the generalized coordinates,and describes the vertical motion of the slab;m and n are the mode numbers of X m(x)and Y n(y), respectively,and N x=N y=5;C s is the damping of the slab D s is the vertical bending stiffness of the slab;q s,L s,W and h s

228S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237229 are the density,length,width and thickness of the slab;F sVj(t)and F sLj(t)are the vertical and lateral dynamic forces at the j th supporting point under the slab;N p and N b are,respectively,the total number of rail fasteners on the slab and the total num-ber of discrete supporting points under one slab used in the calculation.

Similarly,the equations of motions of the concrete base can be easily obtained and are omitted in the paper for brevity. Detailed descriptions of the equations of the vehicle-track coupled dynamic system can be found in Refs.[30,31].

3.3.Model of wheel/rail contact

As shown in Fig.6,the vehicle and the slab track are spatially coupled by the wheel–rail interface.When calculating the dynamic response of the vehicle/track,the nonlinear Hertzian elastic contact theory is used to calculate the wheel/rail nor-mal contact forces which depends on the displacements of wheels and rails,and the tangential wheel/rail creep forces are calculated?rst by using the Kalker’s linear creep theory and then modi?ed by the Shen–Hedrick–Elkins nonlinear model

230S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

[32].If rail surface irregularities are known,the dynamic wheel–rail forces can be obtained with the help of the coupled vehi-cle-track dynamic model solved by means of a new explicit integration method[33].

4.Results and discussion

4.1.Variations of SIFs

In this paper,the axle load of the vehicle is155kN,and the vehicle speed is300km/h.Typical rail irregularities charac-terized by wavelengths between1and30m are selected to excite the vehicle/track coupled system.

Fig.7a and b shows the time histories of vertical and lateral wheel–rail forces when the vehicle moves along the CRTS II slab track with above rail irregularities,respectively.They are utilized as the inputs to?nite element model in order to inves-tigate the variations of dynamic SIF values when the vehicle moves over the crack.Under the condition of crack length l=0.15m,crack angle h=90°,subgrade modulus E s=200MPa,friction coef?cient k1=0.3,and friction coef?cient k2=0.3, dynamic analysis was conducted by using the ABAQUS dynamic implicit code.

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237231 Fig.8shows the von Mises stress state in the surroundings of the through-transverse crack for the concrete base at deformed FEM mesh.Fig.9shows the time histories of the SIFs at the crack-tip due to the vertical and lateral wheel–rail forces induced by vehicle and track interaction,respectively.It is worth noticing that the value of stress intensity factor K I increases dramatically and reaches the peak value of0.131MPa m1/2when the axle of vehicle is above the crack,and it decreases quickly to small positive magnitude as the vehicle moves away from the crack.It can also be seen from Fig.9that the value of stress intensity factor K II and K III take on positive and negative alternatively while the vehicle passes through the crack,and the peak values of K II and K III are found to beà0.019,à0.014MPa m1/2,respectively.Clearly,the peak value of K I is approximately7times that of K II,and is almost9times that of K III.Therefore it is mainly the presence of K I rather than K II or K III that drives the crack to propagate in this case.

Since the SIFs varied from point to point along the through-transverse crack,it is proper to show the changes at each point along the crack-tip under the vehicle dynamic load.Fig.10shows the absolute values of peak SIFs distribution along the crack-tip in normalized coordinates L P/L B due to the vehicle passing.It is found that the stress intensity factors K I and K II have their maximum value at the point B,below the rail,and the peak value of stress intensity factor K III occurs at location of point

232S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

A(Point A,B are indicated in Fig.1),therefore,the crack would initially start to propagate closer to these points.Hence,for clarity,the following analysis will focus only on the two critical points.

4.2.Effect of subgrade modulus on SIFs

To investigate the effect of subgrade modulus on the dynamic SIFs,three different subgrade modulus(100,200,300MPa) are investigated with the same crack length l=0.15m,crack angle h=90°,friction coef?cient k1=0.3,and friction coef?cient k2=0.3.It is clearly shown in Fig.11that the value of K I decreases with the increase of the Young’s modulus of subgrade due to the fact that lower modulus of the subgrade result in higher bending moment of concrete base,and thus makes more bending displacement occur and increases the associated K I.The peak value of K I for the model with subgrade modulus 100MPa is found to be0.176MPa m1/2,which is larger than that for the model with subgrade modulus200MPa by34%, and that for the model with subgrade modulus300MPa by59%.Therefore,lower subgrade modulus might increase the chance of crack formation.This fact makes clear why many cracks are found where the concrete base could occur relatively

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237233 large deformation due to weak subgrade.However,subgrade modulus has a negligible contribution to the value of K II and K III because the bending moment would closes the crack and cannot affect K II and K III signi?cantly.

4.3.Effect of crack length on SIFs

To study how the variation of crack length affect dynamic SIFs,three different crack lengths(0.10,0.15,0.20m)are inves-tigated with the same subgrade modulus E s=200MPa,crack angle h=90°,friction coef?cient k1=0.3,and friction coef?cient k2=0.3.As shown in Fig.12,the values of K I,K II,and K III for models with crack length0.10and0.15m are very close.When the crack length increases to0.20m,the peak values of K I,K II and K III improve considerably and are found to be around0.183,à0.060,à0.017MPa m1/2,respectively.This is because the crack length0.20m is closer to the critical length which has the highest SIFs.Thus,compared to the model with relatively small length(0.10,0.15m),it can be predicted that large crack length(0.20m)with higher SIFs will lead to higher crack propagation rate.

234S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

4.4.Effect of crack angle on SIFs

Fig.13shows the in?uence of the crack angle on dynamic SIFs for three different crack angles(30°,60°,90°)with the same subgrade modulus E s=200MPa,crack length l=0.15m,friction coef?cient k1=0.3,and friction coef?cient k2=0.3. It can be found from Fig.6that K II is quite sensitive to the crack angle.It can change totally and switches from negative to positive when the angle of the crack decreases to30°.In higher angles(i.e.crack angle90°)the bending stress cannot affect the K II and K III signi?cantly as it closes the crack.With this angle,negative shear occurs when the vehicle pass over the crack.While with a low angle(crack angle30°),the bending stress has more contribution in shear mode(K II and K III), and this low angle cannot cancel bending effects and increases K I signi?cantly.In this case the peak value of K I is found to be0.283MPa m1/2which is much larger than that of models with90°angle by about116%.And for the K II,its value switches to positive when the vehicle pass over the crack,and it starts and ends with a small positive magnitude,but has a signi?cantly high peak value of0.449MPa m1/2at the crack-tip during loading.In regard to K III,it increases when crack angle decreases as illustrated in Fig.13c and has its maximum value of0.030MPa m1/2in the crack with30°angle.

4.5.Effect of friction coef?cients on SIFs

The in?uences of friction coef?cient between cracked surfaces,and friction coef?cient between faces of concrete base and subgrade on dynamic SIFs are investigated in this analysis,where three different friction coef?cients k1(0,0.15and0.3)and k2(0,0.15and0.3)are selected respectively with the same subgrade modulus E s=200MPa,crack length l=0.15m,crack angle h=90°.It can be seen from Figs.14and15that by increasing the friction coef?cients,all the SIFs cannot be effected signi?cantly and they keep nearly unchanged,This occurs because the friction forces occur at those surfaces are not large enough to signi?cantly reduce the opening and shearing displacements of the crack during loading.Therefore,the dynamic SIFs when the vehicle passes over the crack are not sensitive to those friction coef?cients.

4.6.Statistical characteristics of SIFs under vehicle dynamic load

As the rail irregularities at the location where the through-transverse crack of concrete base occurs are found to be ran-dom,the corresponding wheel–rail forces vary with the irregularities.In order to investigate statistical characteristics of SIFs under vehicle dynamic load,the rail irregularities are assumed to move backward at the speed of the vehicle,and the vehicle is static with respect to the slab track system.Therefore,the varying vertical and lateral wheel–rail forces can be obtain with

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237235 the coupled dynamic model,as shown in Fig.16.Noted that the sample size of wheel–rail forces here is large enough to make sure the statistical analysis is statistically signi?cant.

In the current analysis,the leading wheel of the bogie is assumed to be above the crack,and the second wheel of a bogie is almost2.5m from the leading wheel.The vertical and lateral wheel–rail forces due to the rail irregularities are utilized as the inputs to?nite element model.Under the condition of crack length l=0.15m,crack angle h=30°,subgrade modulus E s=200MPa,friction coef?cient k1=0.3,and friction coef?cient k2=0.3,with which the values of SIFs are the highest among the above sensitivity analysis,the varying SIFs under vehicle dynamic load can be obtained with the dynamic implicit anal-ysis,as plotted in Fig.17.The statistical characteristic values of dynamic SIFs are illustrated in Table.2.It can be clearly seen that in this case the mean value,maximum value,minimum value and standard deviation of K II are all found to be the high-est compared to K I and K III.Fig.18shows the frequency histogram of the dynamic SIFs and a Gaussian distribution?tted to the same dynamic SIFs,it can be concluded that the distribution of K I follows an approximately Gaussian distribution with a mean of0.239MPa m1/2and a standard deviation of0.032MPa m1/2,the distribution of K II can be well represented by a

Table2

Statistical characteristic values of dynamic SIFs due to random wheel–rail forces(unit:MPa m1/2).

Mean Maximum Minimum Standard deviation

K I0.2390.3810.1060.032

K II0.3710.5630.1710.045

K III0.0190.0440.0020.005

236S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237

Gaussian distribution with a mean of0.371MPa m1/2and a standard deviation of0.045MPa m1/2,and the distribution of K III is approximately subject to a Gaussian distribution with a mean of0.019MPa m1/2and a standard deviation of 0.005MPa m1/2.

5.Conclusions

In this paper,a three-dimensional?nite element model of the CRTS II slab track system is built up to evaluate dynamic SIFs for through-transverse crack of concrete base created by using the XFEM.A coupled dynamic model of a half-vehicle and the slab track is applied to calculate the dynamic wheel–rail forces which are utilized as the inputs to?nite element model. The values of SIFs at the crack-tip under vehicle dynamic load are extracted from the XFEM solution by domain based inter-action integral approach.Sensitivity analysis using parameters such as subgrade modulus,crack length,crack angle,friction coef?cient between cracked surfaces,and friction coef?cient between faces of concrete base and subgrade are performed in detail.Also the statistical characteristics of varying SIFs due to random wheel–rail forces are analyzed.The numerical results

S.Zhu,C.Cai/Engineering Failure Analysis46(2014)219–237237

(1)The subgrade modulus substantially effects the K I at crack-tip.The subgrade modulus of100MPa can increase K I up to

59%compared with that of higher subgrade modulus of300MPa,but it has a negligible contribution to the value of K II and K III.Thus lower subgrade modulus might increase the chance of crack formation.

(2)Compared to the model with relatively small crack length,large crack length(0.20m)results in higher SIFs and con-

sequently lead to higher crack growth rate.Therefore large crack length might speed up the destruction of the structure.

(3)The dynamic SIFs are quite sensitive to the crack angle.With a low angle(crack angle30°),the peak values of SIFs at

crack-tip are all found to be much larger that of cracks with60°and90°angles during loading,especially for K II,it can change totally and switches from negative to positive when the angle of the crack decreases to30°.

(4)Friction coef?cients have negligible effect on variations of dynamic SIFs.

(5)The distributions of the varying SIFs due to random dynamic wheel–rail forces are subject to an approximately Gauss-

ian distribution with different mean values and standard deviations.

Acknowledgements

This work was supported by the National Basic Research Program of China(‘‘973’’Program)(Grant No.2013CB036202 and No.2013CB036206),the National Natural Science Foundation of China(Grant No.51478397),the Independent Research Project of State Key Laboratory of Traction Power(Grant No.2014TPL_T01),the2013Cultivation Program for the Excellent Doctoral Dissertation of Southwest Jiaotong University,and the Funds from China Scholarship Council.

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小学英语多媒体课件的设计思路

小学英语多媒体课件的设计思路 一、多媒体辅助教学的优势和特点 多媒体辅计算机助教学具有生动形象、信息储存量大、交互性强、适合个别化学习等特点。在多媒体环境中，学生可以根据自己和学习水平与需要选择学习内容，高度体现了学生的主体作用。多媒体辅助教学的应用非常灵活，可以进行课堂计算机辅助教学，也可以进行网络教学。它不同于传统教学，其教学质量与教学效率远高于传统教学。特别是用多媒体计算机辅助英语教学，可以提供形象逼真的情景，正确标准的发音，生动有趣的练习游戏，从而大大地激发了小学生学习英语的兴趣，提高教学效率。然而，在多媒体辅助教学条件下，学校育人环境的改善、教学过程的优化、教学质量的提高，都离不开教学课件的质量支持和保证。这不仅取决于计算机硬件的装备和决策水平，而且取决于教学课件的设计水平。因此，制作高质量的多媒体辅助教学课件，首先要设计好课件的脚本。 二、教学课件设计的指导思想和遵循的原则 课件设计的总指导思想是现代教育理论。“现代教育”认为在科学技术飞速发展的今天，人们开始极为重视人的智能开发。“传统教育”这中再现型教学，不利于发挥学生的独创性和知识的综合利用；过分地不恰当地强调教师的主导作用，也会约束以至压抑学生主体作用的发挥。因此，在教

学中必须以学生为中心，尊重学生的需要，培养有个性的学生，强调独立自主地学习，从而形成学生多方面的能力，特别是学生主动的学习能力和学习态度。现代教育的教育教学观，为开发多媒体辅助教学课件的主旨是不谋而合的。因此，课件的设计思路是以现代教育理论为指导思想，同时必须遵循以下几个原则： 1、课件设计的科学性。因为科学设计教学课件是多媒体辅助教学的核心和关键，它决定整个教学系统的成败。优秀的多媒体辅助教学课件应围绕教学大纲，将教学目标、教学内容和教学方法巧妙地贯穿在整个课件的设计制作中，并通过与计算机硬件相结合充分发挥计算机高密度、大容量、迅速准确、灵活高效的优点，调动学习者各种感官，激活其思维，消除其心理障碍，使其生动地去活学活用。具体表现在：紧扣教学大纲，量化教学目标；运用外语教学原则，开发配套课件；具备优良的教学方法。 2、要适应小学生的个性特点。小学阶段的学生年龄一般为6—12周岁。根据皮亚杰的儿童认识发展理论，该年龄阶段的儿童跨越了前运算阶段、具体运算阶段，向形式运算阶段过渡，因此在设计课件时必须使之符合小学生的年龄特征。英语课件中可能受学生年龄特征影响的组合因素有：课件界面、人机交互控制的难易程度、教学信息内容、文本、图形图像、声音、动画、视频、色彩搭配等。因此，课件界