τ Small strain WEEK 8 Soil Behaviour at Small Strains: Part Strain levels and soil behaviour

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1 WEEK 8 Soil Behaviour at Small Strains: Part Strain levels and soil behaviour τ Soil s shear stress-strain relationships exhibit many typologies. For example, some of them are ductile with continuing strain-hardening, and some of them are brittle with significant post-peak strain-softening. The objective of Week 8-13 is to understand what sort of soil behaviour we should expect from a given soil at a given condition, and consider what impact the observed features have on engineering problems. In doing so, it is convenient to look at soil behaviour at different strain levels. This will allow us to focus on stiffness at small strains, yield characteristics at medium strains and strength at large strains. τ Small strain Shear Large strain γ This view is applicable in principle for compression behaviour that we have studied last week. The only difference is that we normally do not invoke a notion of strength in compression. Medium strain p p Compression ε v 1

2 1. Small-strain stiffness and non-linearity 1-1. Definitions of soil stiffness τ - Tangent stiffness (G tan, E tan, etc.) - Secant stiffness (G sec, E sec, etc.) - Initial (elastic) stiffness (G 0, E 0, etc.) (Equivalent to tangent stiffness at very small strains) Upon unloading and reloading, elastic stiffness is normally observed (but not necessarily identical to the initial stiffness). Normally, soils stiffness is largest at very small strains, exhibiting gradual degradation as the strain becomes larger (due to plastic straining). G 0 G sec G 0 G tan G sec Unloading & reloading γ How small is small? There is no formal definition or consensus on small strain, but when we say small strains, usually we talk about strains smaller than order of 10-4 (imagine, 1 µm over 10 mm). Up to order of 10-4 (0.01% strain) logγ 1-. Some history: Background to recognition of small-strain stiffness Importance of the stiffness non-linearity at small strains started to be recognised mainly after the 1970s. This development had two technical factors in its background; sophistication in laboratory tools and the advent of personal computers. New laboratory tools allowed resolving ever smaller strains with higher accuracy. The computer allowed non-linear numerical analyses, which provided a way to utilise the new laboratory findings on small-strain stiffness for practical problems. Without PCs, prediction needs to be based on analytical solutions, which normally exist for very simple, linearly elastic stress-strain relationships. So in many senses, general recognition of the stiffness non-linearity at small strains coincided with the turning point of soil mechanics from the classical era to the modern.

3 1-3. Testing techniques for measuring small-strain stiffness (i) Laboratory: Static tests Triaxial apparatus with local instrumentation is most commonly used for both research and practice. Hollow cylinder apparatus and plane strain apparatus are also used, but mainly for research purposes. Here we limit the scope to triaxial apparatus. However, the principle itself of local instrumentation is same in any apparatus. Global instrumentation is erroneous due to - Bedding errors - Non-parallel ends - Load cell and system compliance Local instrumentation is capable of avoiding these errors, providing more accurate strain and hence stiffness measurement. Load cell Bender element system (also in other side of soil specimen) LVDTs Tie rod Perspex wall Ram Bearing Ram pressure chamber filled with oil Suction cap Mid-height PWP transducer Radial belt Soil specimen Porous stone Drainage (Global) displacement transducer To oil/air interface or CRS-pump Why not abolish all global instrumentation and just use local one then? It is easier said than done; local transducers are expensive and requires expertise in handling. Example of triaxial apparatus with local instrumentation (Nishimura, 006) d external From external (global) instrumentation: ε axial_external d = H external 0 E external σ = ε axial axial_external Load cell From internal (local) instrumentation: Specimen ε axial_internal = d H internal 0 E internal = σ ε axial axial_internal H 0 H 0 d internal 3

4 Examples of local transducers These devices have very high resolutions in displacement measurement. Consider how high the resolution needs to be to measure, say, Young s modulus for strain of 10-5 (0.001%)? Local Displacement Transducer (LDT; Goto et al., 1991) Axial displacement transducer using inclinometer (Burland & Symes, 198) Linear Variable Differential Transformer (LVDT) for axial displacement (Cuccovillo&Coop, 1997) LVDT for radial displacement (Drawing provided by Prof. Matthew Coop) 4

5 Example of measurements Note how different the magnitudes of stiffness are when measured externally and internally. Triaxial compression on soft mudstone (Goto et al., 1991) This is a typical result; you can find numerous similar comparisons in literature for sands, silts, soft clays, etc. However, the error involved in global measurement of strains is more significant for stiffer soils. The same problems of bedding and system compliance are encountered in oedometer tests too. Another example: Lightly over-consolidated North Sea Clay (Jardine et al., 1984) 5

6 (ii) Laboratory: Dynamic tests Most of the dynamic tests are based on elastic or visco-elastic wave theory. The magnitude of strain is associated to the magnitude of oscillation amplitude. The strain levels involved are normally very small (<10-5 ), in many cases small enough to regard the obtained stiffness as the initial elastic stiffness. One dimensional wave equation is x u t 3 G u µ u = + ρ x ρ x t u(x) where G is the shear modulus, m the viscosity and r the mass density of soil. If the viscosity is disregarded, u t = V s u x Case of one-dimensional shear wave Where V s = G / ρ is the shear wave velocity. Soil Specimen Bender element tests: A bender element is made up of piezo-ceramic semiconductors. It generates shear waves when energised, and conversely, it sends electric signals when receiving shear waves. So by installing a couple of them as transmitter and receiver, and measuring the travel time between a given distance, V s and then G can be calculated. Bender elements hv hh v (or z) h (or r) A caution is required; soil stiffness is anisotropic (the topic of next week), and you need to know which shear modulus you are measuring; G vh G hv or G hh? Amplitude of signals in arbitrary units Beginning of signal Input Output First arrival t = msec Time [msec] TE4: After consolidation f = 9 khz, vh-direction Example of London Clay (Nishimura, 006) 6

7 Resonant Column test In contrast to bender element tests, in which typically a pulse wave is transmitted to monitor its velocity, a sample is put in steady state oscillations in resonant column tests. By gradually changing the input frequency at a constant input force (or torque) amplitude, the frequency at which the oscillation becomes maximum is sought (i.e. the resonance frequency is sought). From the resonance frequency, the sample s stiffness is obtained. If the oscillation is compression extension, E is obtained (E or E?) If the oscillation is cyclic torsional, G is obtained. The resonant column apparatus is normally purpose-built, unlike auxiliary tools such as bender elements. This poses some inconveniences. However, it has a big advantage; by changing the input force, the oscillation amplitude (hence strain amplitude) can be changed. This is a useful feature for estabilishing G γcurves over a wider strain range. F K a F Active Active Active Various types of resonant column Passive (a) Fixed-free (b) Fixed-base-spring top (c) Free-free C a K a F C a Shear modulus measured in crag and Tertiary soils (LC: London Clay, TC: Thanet Sand; Hight et al., 1997) 7

8 (iii) Field Shear wave velocity measurement: Cross-hole and down-hole methods The principle of these field methods is same as that of bender element tests. A receiver (and transmitter in down-hole methods) is placed inside a borehole, or if the soil is soft, it may be installed in a penetration cone (seismic cone penetration test; SCPT). These method measures shear wave velocity, which is a body wave. There are also techniques which use surface wave (Reighley wave). Making waves above a seismic cone Cross-hole measurement (Hight et al., 1997) Down-hole measurement (Hight et al., 1997) 8

9 Example of comparison between different method: G vh [MPa] Down-hole (BH407, North)* Down-hole (BH407, East)* Resonant column (rot. core) Bender element Resonant column (range for blocks) C *Shear wave was transmitted from two sides of borehole 0 10 Biii 0 Depth below GL [m] 30 Bii Bi 0 Elevation [m OD] B Lithological unit: A3 Shear modulus G vh of natural London Clay measured by different laboratory and field methods (Nishimura, 006) Finally In old days, the stiffness moduli measured in dynamic and static tests used to be considered two fundamentally different things due to the strain-rate effects, because the dynamic moduli were always far larger than the static ones. After it was found that the static moduli had been underestimated by global measurement, the agreement of the moduli between dynamic and static tests has been seen (Tatsuoka & Shibuya, 1991). One problem solved? 9

10 1-4. Importance of small-strain stiffness non-linearity: Case studies (i) Excavation: Simpson et al. (1979) One of the early examples of geotechnical non-linear finite element analysis is on construction of an underground car park in front of the Palace of Westminster in the 1970s. To avoid affecting the historic building, the ground deformation caused by the excavation needed to be predicted with high accuracy. A Class A prediction had been given by elastic analysis by Ward and Burland (1973). The problem was revisited by Simpson et al. (1979) by non-linear analysis. Palace of Westminster with Big Ben Clock Tower Cross-section 10

11 (Continued; Simpson et al., 1979) The non-linear analysis was capable of simulating the observed ground movements with good accuracy. An interesting episode is that the linear elastic and non-linear analyses predicted the tower s leaning towards opposite directions. Modelling of stress-strain relationships Predicted ground movements 11

12 (ii) Shallow foundation: Jardine et al. (1995) Experiments at Bothkennar site, Scotland Loading on a.4m x.4m footing on soft silty clay. Analysis with a non-linear model predicted better the observed settlement than with linear elasticity. The elastic analysis predicts that the influence of the footing settlements reaches very far. In reality, it does not, as the non-linear analysis indicates. Testing pad D r δ c δ r Predicting and observed settlements 1

13 (iii) Shallow - deep foundation: Izumi et al. (1997) Rainbow Bridge, Tokyo (Construction work: ) 140,000 tf anchorages built on Tertiary Mudstone ( Google 011) Cross-sections 13

14 (Continued: Izumi et al., 1997) Proper consideration of stress-strain non-linearity at small strains led to significant improvement in settlement prediction. Note how conventional testing methods underestimating the small-strain stiffness led to over-estimation of the settlement. 3-D FEM mesh Non-linear stiffness Settlement: Predictions and observations Simulation cases 14

15 (iv) Tunnelling: Addenbrooke et al. (1997) Jubilee Line Extension Project, London Prediction of settlement troughs with non-linear numerical models Jubilee Line (Grey-coloured) Cross-section Model L4&J4: Non-linear models fitted to locally instrumented triaxial extension tests Stiffness non-linearity from experiments and models 15

16 (Continued; Addenbrooke et al., 1997) Linear elasticity is useless in predicting the settlement trough, which is deeper and narrower than linear elasticity predicts. However, even the non-linear stress-strain models do not do a perfect job. Research is going on to see any other factor is being missed, such as anisotropy and the influence of loading histories. Settlement trough Tunnel excavated D FEM mesh Settlement at the ground surface due to excavation of first (west-bound) tunnel 16

17 References Addenbrooke, T.I., Potts, D.M. and Puzrin, A.M. (1997) The influence of pre-failure soil stiffness on the numerical analysis of tunnel construction, Geotechnique 47(3) Burland, J.B. and Hancock, R.J.R. (1977) Underground car park at the House of Commons, London: Geotechnical aspects The Structural Engineer, The Journal of The Institution of Structural Engineers Burland, J.B. and Symes, M. (198) A simple axial displacement gauge for use in the triaxial apparatus, Geotechnique 3(1) Cuccovillo, T. and Coop, M.R. (1997) The measurement of local axial strains in triaxial tests using LVDTs, Geotechnique 47(1) Goto, S., Tatuoka, F., Shibuya, S. Kim, Y.-S. and Sato, T. (1991) A simple gauge for local small strain measurements in the laboratory, Soils and Foundations Hight, D.W., Bennell, J.D., Chana, B., Davis, P.D., Jardine, R.J. and Porovic, E. (1997) Wave velocity and stiffness measurements of the Crag and Lower London Tertiaries at Sizewell, Geotechnique 47(3) Izumi, K., Ogihara, M., and Kameya, H. (1997) Displacement of bridge foundations on sedimentary softrock; a case study on small strain stiffness, Geotechnique 47(3) Jardine,R.J., Symes, M.J., and Burland, J.B. (1984) The measurement of soil stiffness in the triaxial apparatus, Geotechnique 34(3) Jardine, R J, Lehane, B M, Smith, P,R and Gildea, P A (1995) Vertical loading experiments on rigid pad foundations at Bothkennar, Geotechnique 45(4) Nishimura, S. (006) Laboratory study on anisotropy of natural London Clay, PhD Thesis, Imperial College London. Simpson, B., O Riordan, N.J. and Croft, O.D. (1979) A computer model for the analysis of ground movements in London clay, Goetechnique 9() Tatsuoka, F. and Shibuya, S. (1991) Deformation characteristics of soil and rocks from field and laboratory tests, the 9th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Vol.1, Ward, W. H. & Burland, J. B. (1973). The use of ground strain measurements in civil engineering. Phil. Trans. R. Sot. A