Indian Geotechnical Conference – 2010, GEOtrendz December 16–18, 2010 IGS Mumbai Chapter & IIT Bombay
A Realistic Way to Obtain Equivalent Young’s Modulus of Layered Soil Brahma, P.
Mukherjee, S.P. 1
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Fargo Consultants Pvt. Ltd., Kolkata Department of Civil Engine ering, Ja davpur University, Unive rsity, Kolkata
1
ABSTRACT For estimation of immediate settlement of layered soil deposits the current practice uses thickness weighted average of Young’ Young’ss Modulus. This method appears to be acceptable if the soils of different layers have strength of similar order. order. However, However, this method tends to conc eal the influence influen ce of weaker soils in presence of a stronger layer as has been shown with typical examples in this paper. A method has also been outlined in this paper to estimate an equivalent modulus of elasticity for layered soil using weighted harmonic mean in respect of thickness.
1. INTR INTROD ODUC UCTI TION ON
Immediate settlement is caused by lateral strain due to applied load. The formula for immediate settlement is based on lateral str ain and is mean t for homogeneous soil. But But in Nature homogeneous soil deposit is very rare. Therefore, the geotechnical engineers often encounter the problem of determining immediate settlement in layered soil particularly for a large size foundation when the pressure bulb goes far deep into the soil. Immediate settlement calculations for layered soils are not clearly addressed in the available literature compared to homogeneous layers. Correlations for estimation of modulus of elasticity for cohesive soils with respect to undrained shear strength values (Bowles 1997) and for non-cohesive soils with respect to N (SPT) values is well documented (Som and Das 2003) Current practice uses thickness weighted weighted average for estimation of equivalent modulus of elasticity (Som and Das 2003). This method appears to be acceptable acceptable if the soils of different layers having strength of similar order. However, However, this method tends to mask the influence of weaker soils in presence of a stronger layer as has been shown with typical examples in this paper. A method has also been outlined in this paper to estimate an equivalent modulus of elasticity for layered soil using thickness weighted harmonic mean. It is shown with typical examples that the weighted harmonic mean yields better results than the direct weighted average method particularly when the strength strengt h of successive soil soil layers vary in a very wide range.
The method has been proved reasonable with the help of application of stress-strain relationship for composite bodies. The paper highlights the usefulness of weighted harmonic mean method for estimation of equivalent Young’s Modulus in case of layered soil deposit. 2. ELASTIC ELASTIC SETTLEMENT SETTLEMENT ESTIMATION ESTIMATION
Estimation of foundation settlement for structures is of major importance to limit the settlement of structures within tolerance levels, which affects the allowable bearing capacity. Settlement comprises of two components i.e. immediate (elastic) settlement and long-term (consolidation) settlement. The calculation for immediate settlement for homogeneous soil layer is carried out using the following formula (Terzaghi 1943): Si = (1−µ2) Ι Β σ / Ε
(1)
where, Si = Immediate settlement µ = Poisson’s ratio Ι = Influence factor Β = Width Width of foundation σ = Applied pressure Ε = = Modulus of elasticity Correlation for estimation of modulus of elasticity/Young’s elasticity/Young’s Modulus (E) for cohesive soils and non-cohesive soils is available in existing literature Young’s Modulus may be obtained from the following correlations (Bowles 1997)
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E = 600 cu (2) where, cu in the undrained cohesion. And for non-cohesive soils modulus of elasticity can be estimated from the correlation provided below: (3)
E = 750 + 80 N t/m2 where, N is the SPT value
3. CURRENT PRACTICE FOR EVALUATION OF EQUIVALENT MODULUS OF ELASTICITY
For large foundations the pressure bulb extends deep into the sub-soil hence the estimation of equivalent E for the various soil layers into which the pressure bulb extends into is needed. The normal practice is to use a weighted average of the modulus of elasticity of the various layers encountered within the depth of influence Eeq = Σ Hi Ei / Σ Hi (4) where, Eeq= Equivalent modulus of elasticity, Hi = Thickness of layer, E i = Modulus of elasticity of layer This method of obtaining equivalent modulus of elasticity is acceptable as long as the variation of individual layers is comparable. However, if there is wide variation in the values, the evaluated modulus of elasticity is not appropriate, since the soils with h igher modulus of elasticity tend to mask the effect of the weaker soil layer. This can lead to under estimation of immediate settlement values. 4. BASIC CONCEPTS OF EQUIVALENT MODULUS OF ELASTICITY
An attempt has been made to determine equivalent modulus of elasticity for a series of welded circular rods of different materials. Although the method takes care of only longitudinal strain but for lateral strain also variation of E appears to be of similar trend since the term comes in denominator. This method has been based on the theory of elasticity. In Fig.1 a few metal circular rods ar e welded end to end and a weight is hung from the last piece of rod. The top of the system is fixed. Length = L
1
: M o d u lu s = E
1
Length = L
2
: M o d u lu s = E
2
Length = L
3
: M o d u lu s = E
3
Length = L
4
: M o d u lu s = E
4
Length = L
5
: M o d u lu s = E
5
W e ig h t = W
Fig. 1: Illustration for Formulation of Equivalent Modulus of Elasticity
The elongation of each metal rod is estimated using the following equation, E = σ/ε (5) σ being the stress developed in each of the metal r ods = W/ A where, W is the weight an d A is the cross-sectional a rea and ε is the strain. Elongation (∆L) is given by ∆L = ε L = σ L /E (6) Hence, elongation of each rod section can be represented by ∆Li = σ Li / E i (7) th Where ∆Li = Elongation of i rod, Li = Length of ith rod, σ= Stress in the rod Total elongation (∆L) for the all the rods is calcula ted as Σ ∆Li. Therefore, total elongation = σ Σ (L /E ) (8) i i Let Eeq be the equivalent E for the composite rod. Hence elongation for the composite rod can be expressed as ∆L = σ Σ ?L /E i eq. Since the total elongation is constant, ΣL /E = Σ (L /E ) i eq i i
(9)
Therefore, Eeq = ΣL / Σ (L /E ) (10) i i i Thus, the equivalent modulus of elasticity is the weighted harmonic mean of the individual rod’s modulus of elasticity. This concept may be equally considered rational in case where the lateral strain is causing the deformation. 5. COMPARISON OF THE TWO METHODS OF CALCULATION
In the first example it will be demonstrated that if the underlying layers are of comparable strength then the estimated immediate settlement using the current practice of calculating equivalent modulus of elasticity and the proposed method of calculating modulus of elasticity will yield approximately results of similar order. Whereas in the second example the strength of the underlying layers are not comparable and the immediate settlement calculated from th e two methods vary widely. A square footing with dimension of 4.0m is placed at a depth of 1.5m below ground level. Bearing pressure on the footing is 15t/m2. The Poisson’s ratio = 0.3 and Influence factor for centre of square footing has been taken as 1.12 (Som and Das, 2003). The following examples demonstrate typical cases for non-cohesive soil. Non-cohesive soils have been chosen for the study since immediate settlement is predominant for such type of soils.
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A Realistic Way to Obtain Equivalent Young’s Modu lus of Layered Soil
Example 1: The degree of compactness of sand layers increases with depth. Fig. 2 presents corrected N value and thickness of the sand layers.
Layer 1: Thickness of layer within zone of influence = 2.5m E = 750 + 80 x 10 = 1550 t/m2 Layer 2: Thickness of layer within zone of influence = 2.0m E = 750 + 80 x 40 = 3950 t/m2
Fig. 2: Illustration of Example 1.
Layer 1: Thickness of layer within zone of influence = 2.5m E = 750 + 80 x 35 = 3550 t/m2 Layer 2: Thickness of layer within zone of influence = 2.0m E = 750 + 80 x 40 = 3950 t/m2 Layer 3: Thickness of layer within zone of influence = 3.5m E = 750 + 80 x 50 = 4750 t/m2 As per current practice: Equivalent E = (2.5 x 3550 + 2.0 x 3950 + 3.5 x 4750) / (2.5 + 2.0 + 3.5) = 4175 t/m2 After substituting values into equation (1) immediate settlement is calculated to be 14.6mm As per suggested method: Equivalent E = (2.5+2.0+3.5)/[ (2.5/3550) + (2.0/3950) + (3.5/4750)] = 4108 t/m2 After substituting values into eqn (1) imm ediate settlement is calculated to be 14.9mm. Thus the settlements calculated are almost identical. Example 2: The properties for Layer-1 of Example 1 have been changed from a dense to loose sand.(Fig 3). Properties of Layer 2 and Layer 3 are same as in Example 1. The size of footing and depth of foundation has not been altered in order have the same depth of influence for both the examples. Properties of Layer 1 have been altered to demonstrate that influence of a weaker layer is masked by the presence of strong layers within the zone of influence.
Layer 3: Thickness of layer within zone of influence = 3.5m E = 750 + 80 x 50 = 4750 t/m2 As per current practice: Equivalent E= (2.5 x 1550 + 2.0 x 3950 + 3.5 x 4750) / (2.5 + 2.0 + 3.5) = 3550 t/m2 After substituting values into eqn (1) immedia te settlement is calculated to be 17.2mm As per suggested method: Equivalent E= (2.5+2.0+3.5)/[ (2.5/1550) + (2.0/3950) + (3.5/4750)] = 2801 t/m2 After substituting values into eqn (1) immedia te settlement is calculated to be 21.8mm. Thus the settlement calculated using current practice is lower than the suggested method by almost 26.7%. 6. RESULTS AND DISCUSSION
Comparison of the equivalent modulus of elasticity using the two methods were carried out for various ratios of modulus of elasticity and various ratio of thickness for a two layered system. The ratio of E obtained by proposed method to that obtained by current practice (Eproposed method / E current practice) has been plotted against ra tio of thickness (H2 / H1) for different values of E 2 /E1 The results are plotted in Fig 4. Here E 1 =Modulus of elasticity of Layer 1, H1=Thickness of Layer 1, E2= Modulus of elasticity of Layer 2 and H2= Thickness of Layer 2. e c i t c a r p t n e r r u c
1.00
E /
d 0.75 o h t e m d e s o p o r p 0.50
E
0.0
1.0
2.0
3.0
4.0
5.0
H2/H1
E2/E1=0. 2
E2/E1=0.25
E 2/ E1= 0. 3333
E 2/ E1= 0. 5
E2/E1=1
Fig. 4: Variation of (E proposed method /E current practice ) vs. (H 2 /H1) for Various (E 2 /E 1) Ratios
Fig. 3: Illustration of Example 2.
It may be observed from Fig. 4 that the ratio of modulus of elasticity calculated from the two methods is at its lowest
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when the thickn ess of the two layers is same. The value of Eproposed method /E current practice decreases when the H2 /H1 ratio approaches the value of 1 and increases when the H2 /H1 ratio increased beyond the value of 1. The Fig. 4 also suggests that for a given value of H2 /H1 the ratio E proposed /Ecurrent practice increaseswith increase of E2 /E1 ratio. The method graph also illustrates that the current method of estimation of equivalent E, always overestimates the equivalent modulus of elasticity for layered soil deposits, which will in turn lead to underestimation of immediate settlement of foundations. This may overestimate the bearing capacity of the foundation. 7. CONCLUSIONS
1. Determination of Equivalent Modulus of Elasticity of layered soil deposits appears to be rational if thickness weighted harmonic mean of individual modulus of elasticity is obtained instead of arithmetic mean.
2. Thickness weighted arithmetic mean of modulus of elasticity always underestima tes the immediate settlement in comparison to settlement obtained from thickness weighted harmonic mean of modulus of elasticity. Underestimation of settlement may lead to overestimation of bearing capacity of foundations. REFERENCES
Bowles, J. (1997). Foundation Analysis and Design. 5th Ed., The McGraw-Hill Compan ies, Inc., New York, 308. Som, N.N., and S. C. Das (2003). Theory and Practice of Foundation Design. PHI Learning Pvt. Ltd., New Delhi, 43. Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons Ltd., 510.
