ESTIMATION OF HYPERBOLIC STRESS-STRAIN PARAMETERS FOR GYPSEOUS SOILS

The hyperbolic model is a simple stress-strain relationship based on the concept of incrementally nonlinear elastic behavior. The hyperbolic stress-strain relationship was developed for use in finite element analysis of stresses and movements in earth masses. To estimate hyperbolic parameter values required for nonlinear finite element analysis, data used from the triaxial compression tests for the gypseous soils exposed to the effect of drying and wetting cycles carried out by (Mohammed, 1993). From these data, the parameters (C, φ, K, n, Rf), which are required by Duncan-Chang model, 1970 can obtained for analyses of dams, excavations and various types of soil-structure interaction problems. In addition, it can be found that the primary loading modulus, K, the exponent number, n, and the failure ratio, Rf, have random values during rewetting cycles for CU and UU triaxial compression tests. ا : م+,-./ 0,12 3,4567 ل9,6.:;او د9,-?;ا @A,B CDAE,B C,FG2 @,2 ةر9,J2 +,ه 3,LاMNا O,DP4Nا وذ جذ+,4SNا نا نU,,4Nا V,,DWGNا ي3,,YاM5Nا فU[,,5Nا . ةد3,,\4Nا U,,]9S6Nا ^,,A1\7 C,,PYU_ V,,` م3a5E,,7 V,,آ c,,.dو C,,FG6Nا ef,,ه C,ABاU5Nا ^5gNا ت9آUiو تاد9-?GN . / jA,F @A,4a5N 5 Nا 3,LاMNا O,DP4Nا تاU,Ak U,]9S61N V,DWGNا ^,A1\51N C,B+1D4 تارود UAlm,5N C,nU64Nا CAE,JoNا بU,51N رو9,\4Nا V,lGl ط9kr,:;ا ت9,]+\` @,/ ت9,/+16/ تf,Wا ةد3,\4Nا ^,JF @,/ ةM,oS4Nاو s,A_U5Nاو t,A.o5Nا (Mohammed, 1993) . 0,12 ل+[,\Nا j,7 ، ت9,/+164Nا ef,ه @,/و تاU,,Ak54Nا (C, φ, K, n, Rf) جذ+,,4SN C,,B+1D4Nا Duncan-Chang model, 1970 دو3E,,Nا ^,,A1\5N mvS4Nاو CBU5Nا @AB ^Wا35Nا ^آ9v/ عا+:أ @/ 3Y36Nاو ت9YU.\Nاو . Dr. Ahmed H. Abdul-kareem Lecturer Dep. of Civil Engineering University of Anbar Ahmed Helal Assistant Lecturer Dep. of Civil Engineering University of Anbar Created by Neevia Personal Converter trial version http://www.neevia.com IJCE-7 th ISSUE FEBRUARY-2007 2 VLا35B;ا ^A4\5Nا ^/96/ نا 3?و yNذ 0Nا C`9nz9B ) K ( V1/9g5Nا د36Nا ، ) n ( ^v.Nا CJE:و ، ) Rf ( jA,F 9,-N و9,\4Nا V,lGl ط9kr,:;ا ت9,]+\` @4n sA_U5Nا ةد92ا تارود لGW CALا+v2 U,A{ C,A2+SB لوM,J/ U,AkNا ر jrS4Nا ) UU ( jrS4Nاو ) CU .( Introduction: The hyperbolic stress-strain relationship was developed for use in finite element analyses of stresses and movements in earth masses. In the ten years since its development, the model has been used in analyses of a large number of dams, braced and open excavations, and a variety of types of soil-structure interaction problems. In its original form, as described by Duncan and Chang (1970)(1), the hyperbolic model employed tangent values of Young's modulus (Et), which varied with the magnitudes of the stresses, and constant values of Poisson's ratio. The Young's modulus relationships remain the same as described by Duncan and Chang (1970)(1). The principal advantage of the hyperbolic model is its generality. It can be used to represent the stress-strain behavior of soil ranging from clays and silts through sands, gravels and rockfills. It can be used for partly saturated or fully saturated soils, and for either drained or undrained loading conditions in compacted earth material or naturallyoccurring soils. Experience with treating these various types of problems, and the accumulated background of stress-strain parameter values for a wide variety of soils, provide a useful base for further applications. To estimate hyperbolic stress-strain parameters required for finite element analysis, the data collected into triaxial compression tests of gypseous soils carried out by (Mohammed, 1993)(2). Stress-Strain Relationships: The hyperbolic stress-strain relationships are developed for incremental analyses of soil deformations where nonlinear behavior is modeled by a series of linear increments. The relationship between stress and strain is assumed to be governed by the generalized Hook's Law of elastic deformations. For plane strain conditions this relationship may be expressed as follows (3): Created by Neevia Personal Converter trial version http://www.neevia.com IJCE-7 th ISSUE FEBRUARY-2007


Introduction:
The hyperbolic stress-strain relationship was developed for use in finite element analyses of stresses and movements in earth masses.In the ten years since its development, the model has been used in analyses of a large number of dams, braced and open excavations, and a variety of types of soil-structure interaction problems.
In its original form, as described by Duncan and Chang (1970) (1), the hyperbolic model employed tangent values of Young's modulus (E t ), which varied with the magnitudes of the stresses, and constant values of Poisson's ratio.The Young's modulus relationships remain the same as described by Duncan and Chang (1970) (1).
The principal advantage of the hyperbolic model is its generality.It can be used to represent the stress-strain behavior of soil ranging from clays and silts through sands, gravels and rockfills.It can be used for partly saturated or fully saturated soils, and for either drained or undrained loading conditions in compacted earth material or naturally-occurring soils.Experience with treating these various types of problems, and the accumulated background of stress-strain parameter values for a wide variety of soils, provide a useful base for further applications.To estimate hyperbolic stress-strain parameters required for finite element analysis, the data collected into triaxial compression tests of gypseous soils carried out by (Mohammed, 1993)(2).

Stress-Strain Relationships:
The hyperbolic stress-strain relationships are developed for incremental analyses of soil deformations where nonlinear behavior is modeled by a series of linear increments.The relationship between stress and strain is assumed to be governed by the generalized Hook's Law of elastic deformations.For plane strain conditions this relationship may be expressed as follows (3): Created by Neevia Personal Converter trial version http://www.neevia.com(1) where: ∆σ x , ∆σ y and ∆τ xy = are the increments of stress during a step of analysis.∆ε x , ∆ε y and ∆γ xy = are the corresponding increments of strain.E t = is the tangent value of Young's modulus.
The value of both E t and υ t in each element change during each increment of loading in accordance with the calculated stresses in that element, in order to account for three important characteristics of the stress-strain behavior of soil, namely non linearity, stress-dependency, and inelasticity.The procedures used to account for these characteristics are described in the following paragraphs.

Nonlinear Stress-Strain Curves Represented By Hyperbolas:
Kondner (1963) (4) showed that the stress-strain curves for many of soils, both clay and sand, can be approximated reasonably accurate by hyperbolas like the one shown in figure (1-A).This hyperbola can be represented by an equation of the form: These hyperbolas have two characteristics which make their use convenient: 1.The parameters which appear in the hyperbolic equation have physical significance.E i is the initial tangent modulus or initial slope of the stressstrain curve, and (σ 1 -σ 3 ) ult is the asymptotic value of stress difference which is related closely to the strength of the soil.The value of (σ 1 -σ 3 ) ult is always greater than the compressive strength of the soils.
2. The values of E i and (σ 1 -σ 3 ) ult for a given stress-strain curve can be determined easily.If the hyperbolic equation is transformed as shown in figure (1-B), it represents a linear relationship between [ε/ (σ 1 -σ 3 )] and ε.
Created by Neevia Personal Converter trial version http://www.neevia.comThus, to determine the best-fit hyperbola for the stress-strain curve, values of [ε/ (σ 1 -σ 3 )] calculated from the test data are plotted against ε.The best-fit straight line on this transformed plot corresponds to the best-fit hyperbola on the stressstrain plot.This research, data from the triaxial compression tests of gypseous soils exposed to rewetting cycles presented by researcher (Mohammed, 1993)(2) is used to re-plot the stress-strain relations to calculate the required nonlinearity parameters.Figures (2) to (7) show stress-strain relationships for triaxial compression tests of gypseous soils exposed to rewetting cycles carried out by the researcher (Mohammed, 1993) (2).
For each of these plotted curves the values of (a) and (b) are obtained, where (a) is the value of the (y-axis) intercept and (b) is the slope of the curve as explained in figure (1-B).
The value of the initial modulus of elasticity (E i ) is obtained as: and the value of asymptotic ultimate deviator stress (σ 1 -σ 3 ) ult is obtained as:b ult The failure ratio (R f ) is evaluated as: The variation of (σ 1 -σ 3 ) f with σ 3 is represented by the familiar More-Coulomb strength relationship, which can be expressed as follows: Mohr-Coulomb failure criterion, then: In which c and φ are the cohesion intercept and the friction angle.
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Stress-Dependent Stress-Strain Behavior:
For all soils, except fully saturated soils tested under unconsolidatedundrained conditions, an increase in confining pressure will result in a steeper stress-strain curve and a higher strength and the values of E i and (σ 1 -σ 3 ) ult therefore increase with increasing confining pressure.This stress-dependency is taken into account by using empirical equations to represent the variations of E i and (σ 1 -σ 3 ) ult with confining pressure (1),( 3).
The variation of E i with σ 3 is represented by an equation suggested to be Janbu (1963)(5) of the form:- The variation of E i with σ 3 corresponding to this equation is shown in figure (14).The parameter K in equation ( 7) is the modulus number, and n is the modulus exponent.Both factors are dimensionless numbers.P a is atmospheric pressure, introduced into the equation to make conversion from one system of units to another more convenient system.The values of K and n are the same for any system of units, and the units of E i are the same as the units of P a .Data of figures (8) to (13) are used to plot the relations between the values of E i and σ 3 for each series of tests as shown in figures (15) to (20).
The value of K is obtained by taking the value of initial modulus corresponding to one unit of confining pressure, while n is evaluated to be the slope of the ( E i -σ 3 ) relation, which is a straight line on a log-log scale.

Relationship Between E t and The Stresses:
The instantaneous slope of the stress-strain curve is the tangent modulus, E t .
By differentiating equation ( 2) with respect to ε and substituting the expressions of equations ( 5), ( 6), (7) into the resulting expression for E t , the following equation can be derived: This section includes the results of hyperbolic stress-strain parameters extracted from the triaxial compression tests of different gypseous soils exposed to the effect of wetting and drying cycles and summarized in the tables (1) and ( 2).The variation of the Duncan-Chang model parameters with wetting and drying cycles is presented in figures (21) to (23).
From figures (21) to (23), it can be observed that the primary loading modulus (K), the exponent number (n) and failure ratio (R f ) have random values during rewetting cycles for both tests.

Conclusions:
The hyperbolic model is a simple stress-strain relationship based on the concept of incrementally nonlinear elastic behavior.It is applicable to virtually any type of soil and to drained or undrained conditions.Experience in applying the hyperbolic model to analyses of dams, excavations and various types of soilstructure interaction problems has shown that it is useful for calculation movements in stable earth masses, and is not suitable for predicting instability or collapse loads.Like any theory hypothesis of soil behavior, its successful application requires the exercise of engineering judgment.In addition, it can be concluded that the primary loading modulus, K, the exponent number, n, and the failure ratio, R f , have random values during rewetting cycles for both tests.

( 8 )
Created by Neevia Personal Converter trial version http://www.neevia.comThis equation can be used to calculate the appropriate value of tangent modulus for any stress conditions [σ 3 and (σ 1 -σ 3 )] if the values of the parameters K, n, c, φ, and R f are known.Results And Discussion of Hyperbolic Stress-Strain Parameters:

Figure
Figure (21): Variation of primary loading modulus number with rewetting cycles.