Deformation behaviour of railway embankment ... - Liikennevirasto

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72 5 MODELLING OF RESILIENT DEFORMATION BEHAVIOUR 5.1 Introduction Despite the fact that the resilient behaviour of granular materials is affected by several factors, the effect of stress parameters is the most significant. It is therefore essential that the stress-strain relationship be modelled as accurately as possible with constitutive laws. Over the years, many constitutive models have been developed to express the resilient response of granular materials. The complexity of the behaviour of unbound granular materials has made it a very difficult task for researchers to combine the theoretical principles of soil mechanics with the simplicity required in procedures for routine analysis of the material characteristics. In this chapter, the models found in the literature are presented and some are reviewed in more detail. Modelling is an important necessity for an analytical approach in describing material performances. Many researchers have outlined different procedures for predicting the resilient and permanent strain responses of granular materials. However, the great number of models available is per se further evidence of the complexities that overshadow this research area. In spite of the significant progress made over the years in understanding the resilient behaviour of granular materials, there is still a great need for further research into developing more general models and procedures which have a sound theoretical basis and wide applicability. In general, two approaches are employed for mathematical modelling of the resilient behaviour of granular materials. In the first approach, the stress-strain relationship is given by a stress-dependent resilient modulus and a constant or stress-dependent Poisson's ratio. This type of modelling is widely used and several numerical formulations, using different stress components, are found in the literature. The K-θ model (Hicks 1970, Seed et al. 1967) and the Uzan model (Uzan 1985) are examples of this type of modelling. In the second approach, the stress-strain relationship is characterised by decomposing both stresses and strains into volumetric and shear components. The resilient response of the material is then defined using bulk and shear moduli instead of resilient modulus and Poisson's ratio. Models of this kind are usually more complex in nature, and the parametric values are more difficult to determine from collected test data. Examples of the models using the shear/volumetric approach are the Boyce model (Boyce 1980), the contour model (Brown and Pappin 1985), and the modified inelastic and anisotropic versions of the Boyce model suggested by Sweere (1990) and Hornych et al. (1998), respectively. The models analysed in this chapter are based on the results obtained by repeated loading triaxial tests. In order to provide a reasonable simulation of traffic-type loading using triaxial equipment, the loading system should be able to cycle both the vertical (deviator) stress and the confining pressure in phase, and at levels and frequencies corresponding to the actual field conditions.
73 There are two test methodologies for conducting repeated load triaxial tests: the constant confining pressure test (CCP) and the variable confining pressure test (VCP). Both the CCP test and the VCP test are described into more detail in Chapter 8. 5.2 Models based on resilient modulus and Poisson’s ratio 5.2.1 Definition of resilient modulus and Poisson’s ratio In the traditional theories of elasticity, the elastic properties of a material are defined by the modulus of elasticity (E) and Poisson's ratio (ν), which are material constants. A similar approach has been widely used in dealing with granular materials, but the modulus of elasticity is replaced with the resilient modulus to indicate the nonlinearity, i.e. the dependence on the stress level, of the behaviour. For repeated load triaxial tests with constant confining pressure, the resilient modulus and Poisson's ratio are defined by M r v ( ) ∆ σ − σ 1 3 = , (Eq. 5.2.1:1) ε1, r ε 3, r = − , (Eq. 5.2.1:2) where ε 1, r M r = resilient modulus, ν = resilient Poisson's ratio, ∆ = indicates "change in", σ 1 , σ 3 = major and minor principal stress, ε 1,r , ε 3,r = recoverable axial and horizontal strain. This method of calculating the resilient parameters is the same as would apply to an isotropic, linear-elastic material under uniaxial stress conditions. When cyclic confining pressure is applied, the generalized Hooke's law is employed for 3- dimensional stress-strain relationships of an isotropic, linear-elastic material. The resilient modulus and Poisson's ratio are then derived from 1, r ( σ 1 −σ 3 ) ⋅ ∆( σ 1 + 2σ 3 ) ⋅ ∆( σ 1 + σ 3 ) ⋅ 2ε 3, r ⋅ ∆σ 3 ∆ M r = , (Eq. 5.2.1:3) ε ν = ∆σ ε 3 1 3, r 3 1, r . (Eq. 5.2.1:4) 3, r − ∆σ ε 1, r ( σ + ) 2∆σ ε − ε ∆ σ 1 3 Many researchers have tried to outline mathematical procedures for describing the stress dependence of the resilient modulus using various stress variables. The great majority of the models found in the literature are based on simple curve fitting procedures using the data from laboratory triaxial testing.
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A 5/2006 Deformation behaviour of r
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Finnish Rail Administration Publica
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4 Past investigations have shown cl
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6 Brecciaroli, Fabrizio - Kolisoja,
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8 kuvataan jakamalla jännitykset j
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10 TABLE OF CONTENTS ABSTRACT .....
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12 8.9 Trondheim NTNU/SINTEF’s re
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14 combination of resilient strains
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16 Modelling is an important necess
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18 Figure 2.1:2 Two-dimensional sta
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20 σ xx − σ 3 2 det τ yx σ yy
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Page 28 and 29: 26 The reason was argued to be that
Page 30 and 31: 28 term matric suction implies the
Page 32 and 33: 30 Figure 3.2.1:1 Idealisation of t
Page 34 and 35: 32 ε ε 2 ν = , (Eq. 3.2.2:4) whe
Page 36 and 37: 34 where sij = σ −σ , (Eq. 3.2.
Page 38 and 39: 36 addition, when both the complexi
Page 40 and 41: 38 supplement continuum mechanics m
Page 42 and 43: 40 Figure 4.2:1 A typical variation
Page 44 and 45: 42 Figure 4.2:3 Modulus of resilien
Page 46 and 47: 44 provided that the applied stress
Page 48 and 49: 46 material (Uzan 1985). Dilation i
Page 50 and 51: 48 Figure 4.5:1 Resilient modulus o
Page 52 and 53: 50 The grain size distribution of g
Page 54 and 55: 52 The grain sizes that determine t
Page 56 and 57: 54 4.6.3 Fines content A simple cha
Page 58 and 59: 56 grained than for fine-grained ag
Page 60 and 61: 58 and the shape of the particles a
Page 62 and 63: 60 Figure 4.7:1 Effect of the wetti
Page 64 and 65: 62 Figure 4.7:3 Modulus of resilien
Page 66 and 67: 64 Figure 4.7:6 Dry -density as a f
Page 68 and 69: 66 Figure 4.7:7 Resilient response
Page 70 and 71: 68 The form index can be measured u
Page 72 and 73: 70 Hovinmoen are gravel materials.
Page 76 and 77: 74 5.2.2 Resilient modulus as a fun
Page 78 and 79: 76 M r k2 k3 ⎛ θ ⎞ ⎛ q ⎞ =
Page 80 and 81: 78 the model showed good representa
Page 82 and 83: 80 discussed. However it can be exp
Page 84 and 85: 82 et al. (2003) calculated the res
Page 86 and 87: 84 p u = unit pressure (1 kPa), δp
Page 88 and 89: 86 Figure 5.2.5:2 Vertical resilien
Page 90 and 91: 88 2 1 ⎛ ⎞ = A q ε ν p ⎜1
Page 92 and 93: 90 Figure 5.3:2 An example of the c
Page 94 and 95: 92 by incorporating a coefficient o
Page 96 and 97: 94 6 FACTORS AFFECTING PERMANENT DE
Page 98 and 99: 96 cycles. He drew the general conc
Page 100 and 101: 98 accumulated strain, ε p , is co
Page 102 and 103: 100 Figure 6.3:1 Effect of loading
Page 104 and 105: 102 load” is not equal to the fai
Page 106 and 107: 104 Chan (1990) noticed, in his stu
Page 108 and 109: 106 Youd (1972) studied the behavio
Page 110 and 111: 108 Figure 6.5:2 Comparison of plas
Page 112 and 113: 110 ample fine-grained fractions th
Page 114 and 115: 112 of a dolomitic limestone with a
Page 116 and 117: 114 Fuller curve (Equation 4.6.2:3)
Page 118 and 119: 116 Figure 6.6:7 Elastic and plasti
Page 120 and 121: 118 Figure 6.7:1 Dependence of the
Page 122 and 123: 120 0,7 0,6 Elastic limit Failure s
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122 Figure 6.8.1:1 Influence of fin
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124 explain why the differences in
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126 was then argued that the variat
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128 7 MODELLING OF PERMANENT DEFORM
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130 Figure 7.2:1 Mohr-Coulomb repre
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132 where K p (N) = bulk modulus wi
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134 1993)) have reported that at le
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136 ε 1,p = accumulated permanent
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138 Barksdale (1972) conducted repe
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140 Figure 7.6:2 Relationship betwe
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142 also be employed for pavement d
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144 repeated loads, although adapta
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146 series of specimens (or in a mu
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148 Figure 7.7:5 S-N design models
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150 In conclusion, both Huurman (19
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152 aim to simulate as closely as p
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154 by the American Association of
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156 Confining pressure is applied t
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158 particles. During the test, the
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160 method takes the average value
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162 Figure 8.7.2:2 Schematic illust
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164 through each platen is provided
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166 measured for the 600 mm central
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168 displayed on the monitor in gra
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170 unbound granular materials. The
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172 Figure 8.8.2:2 Permanent axial
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174 performance-related (functional
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176 the six supports of the vertica
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178 under the same isotropic stress
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180 9 CONCLUSIONS 9.1 Resilient def
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182 − Permanent deformation resis
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184 Balay J., Gomes Correia A., Jou
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186 Cheung, L.W. (1994). Laboratory
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188 El abd, A., Hornych, P., Breyss
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190 Hoff, I., Nordal, S., and Norda
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192 Kolisoja, P. (1996b) Large Scal
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194 Mitry, F.G. (1964). Determinati
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196 Plaistow, N.C. (1994). Non-Line
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198 Tam, W.A. and Brown, S.F. (1988
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200 Wellner, F. (1996). Influence o
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