Simple Cyclic loading model based on Modified Cam Clay

Transcription

1 Simle li loading model based on Modified am la Imlemented in RISP main rogram version 00. and higher B Amir Rahim, The RISP onsortium Ltd Introdution This reort resents a simle soil model whih rovides a reasonable redition of behaviour of las under reeated loading. The model is based on the work of arter, Booker and Wroth 1. The model has been imlemented into RISP FE ode. The arameters used are the same as in the existing Modified am la with the addition of one more arameter whih haraterizes li behaviour. This arameter ma be determined b erforming li triaxial tests under undrained onditions. The Modified am la model This model is based on ritial State Soil Mehanis. The main variables in SSM theor are The effetive mean stress, the deviatori stress q and the voids ratio e. These are defined as follows 1 3 ( σ + σ + σ ) 1 3 Eq. 1 [( σ ) + ( ) + ( ) ] 1 σ σ σ 3 σ 3 σ 1 1 q Eq. de dv Eq e where de is the hange in voids ratio, and dv is the inremental volume strain The Modified am la (M) model requires the seifiation of five arameters, values of whih an be obtained from standard oedometer or triaxial omression tests. These arameters are: λ the gradient of the normal onsolidation line in e-ln( ) sae κ the gradient of the swelling and reomression lines in e-ln( ) sae e s the ritial voids ration whih loates the onsolidation lines in e-ln( ) sae. This is taken as the voids ratio at unit on the ritial state line M the value of the stress ratio q/ at ritial state. This is related to the angle of frition 6sin( φ ) though the relationshi M 3 sin( φ ) G the elasti shear modulus When a stress state is elasti, the following relationshi is used: d K 0 dv dq 0 3G dε where K is the bulk modulus defines as Eq. 4 1 arter, J.P, Booker, J.R., Wroth,.P. A ritial State Soil Model for li Loading ublished in Soil Mehanis Transient and li Loads, 198, John Wile & Sons Ltd 1

2 ( 1+ e) K Eq. 5 κ q ritial state line Yield surfae Loading surfae q P P s P P P Isotroi omression Swelling ritial state P P s Figure 1 The Modified am la model P P The funtion for the ield surfae of the M model is defined as { ( )} 0 q M Eq. 6 P is the reonsolidation stress whih ats as a hardening arameter. When a stress states satisfies the ield surfae funtion, lasti deformation takes lae, whih is governed through an assoiated flow rule. The ermanent hange in inremental volumetri strain is given as

3 dv λ κ d 1+ e Eah stress state has a loading surfae, whih is defined as: Eq. 7 q 1 + Eq. 8 M The following onditions al when a material is loaded: Hardening when d d > 0 Softening when d d < 0 Neutral loading when the ield lous does not hange while lasti behaviour ours, d d 0 During lasti behaviour the ield lous hanges aording to the law: d d Eq. 9 The inremental stress-strain law during ielding is defines as dv dε d dq ( λ κ ) a κ 1 + ( 1+ e) 1+ e ( λ κ ) 1 a 1 ( 1+ e) ( λ κ ) b 1 ( 1+ e) 3G 1 + Eq. 10 M a M 4 η 4 + η 4 4η b M 4 4 η η the stress ratio q/ 3

4 li Loading Model Reeated loading usuall auses the ermanent strain to our earlier than revious loading les. This imlies that the ield stress limit is dereasing. In the onventional Modified am la model this annot be rerodued as the ield surfae is unaffeted b ativit in the elasti zone when the material is unloaded. In the new model it is assumed that the size of the ield surfae redues graduall with elasti unloading. The following relationshi is assumed when the loading surfae and is reduing (ie elasti unloading is taking lae) d is within the ield surfae d ϑ Eq. 11 If ϑ is zero, then the model behaves exatl in the same was as the onventional M. If ϑ is unit then the ield surfae would ontrat so that the stress state remains on it. It is exeted that the ield surfae will ontrat onl a small amount in whih ase ϑ is a small value. When the material is elasti and remains unhanged, ie d 0 is inreasing (ie elasti loading), then the urrent ield surfae Eq. 1 The figure below shows the distintion between these two tes of behaviour. q ritial state line I II Path I reresents elasti loading in whih ase is ket onstant Path II reresent elasti unloading in whih ase ontrats. Figure The new li Loading model P P P The new model is ver similar to the onventional M. The riterion for ielding is the same, the flow rule and the hardening law are the same, and the inremental elasti and elasto-lasti stressstrain relations are the same. The onl differene is the modifiation to the ield surfae assoiated 4

5 with elasti unloading. This feature has imortant onsequenes to the reeated loading roblem, whih would be useful for undrained as well as drained onditions. Validation Examles on Normall onsolidated la The following examles will be arried out using the following arameters: o λ 0.5 κ 0. 5 M 1.forφ 30 G 00uo where uo is the initial undrained shear strength defined as κ / λ 4 o o o M Eq. 13 uo The initial voids ratio is taken as e o 0. This gives a ritial voids ratio of: e s e + κ ln( ) + ( λ κ )ln( ) Eq. 14 o o so Model rearations This examle reresents a triaxial test with li axial load at onstant ell ressure. In eah ase loading is alied so that the deviator stress is varied ontinuousl between zero and a seified limit, ie one wa omression where σ z σ r with σ r onstant. q We use ϑ 0.1 and uo The FE model onsists of LST axi-smmetri elements, whih will reresent one quadrant of the triaxial samle due to smmetr. An initial stress of 150Ka is alied to the sides as shown in order to satisf equilibrium. The samle is assumed to be weightless (ie γ bulk 0 ). The initial fixities are as shown below. Figure 3 Starting mesh in RISPD The in-situ stress is set as 150Ka as shown below. As this is an undrained normall onsolidated samle, o o 150KPa 5

6 Figure 4 In-situ stresses The material roerties are entered as shown Figure 5 Material Proerties Stress-ontrolled loading With this test we al a ressure to the to edge whih is reversed (ie removed) for eah le. A number of load bloks were reated in RISP. Eah load blok reresents half a le (ie either loading or unloading). Eah le would have a number of inrements. It is also advised to use the Define otion and var the inremental load frations in the load otions menu as follows: Eah Halves for the loading art of the le, ie inrements reduing in size towards the end of the load blok when stress ath aroahes the ield surfae. Eah Doubles for the unloading art of the le ie inrements inreasing in size as stress ath moves awa from ield surfae into the elasti zone 6

7 Figure 6 Load Bloks The first load blok is then seleted an the to edge is seleted. A ressure load q is then alied. This orresonds to q uo where 0.75 uo M 4 o o o κ / λ Figure 7 Alied stress for a loading ase The roess is reeated for the next load blok, but this time with a ressure of q, ie unloading. 7

8 The following grahs are the results of the stress ontrolled li test; Figure 8 Variation of q v P for stress ontrolled one-wa li loading of undrained normall onsolidated roblem showing initial and final ield surfaes. Normilised q against normalised P/uo Normalised q against vertial strain Vertial Strain Normilised U against normalised Normalised U against vertial strain P/uo Vertial Strain Figure 9 Results of one wa stress ontrolled, undrained triaxial test with OR1, ϑ 0.1 Strain ontrolled Loading The same FE model above was used with a loading of axial strain. Similar ell ressure was used as above. The axial strain loading reresent a two wa test in whih the samle was subjeted to omression as well as tension. 8

9 Figure 10 Inut for edge fixities orresonding to a omressive axial strain of 01 The inremental frations were again adjusted so that smaller inrement are used when the stress ath is aroahing the ield surfae and larger inrements are used when the stress ath is going awa from the ield surfae. The degradation arameter ϑ was set to 0.1, and the material was assumed to be isotroi (k o 1.0). Onl 10 les were used. Figure 11 Stress ath for two wa axial strain test In this te of strain ontrolled test the stress ath moves steadil towards the ritial state ondition, osillating between omression and tension, with the mean effetive stress graduall reduing to zero. This behaviour imlies that the soil is liquefing. Normilised q* against normalised Vertial strain against normalised q* P/uo - V. Strain 9

10 Normilised U against normalised Vertial strain against normalised U P/uo V. Strain * Figure 1 Results for two wa axial strain using OR1, ϑ 0.1 and G00uo. q ( σ σ ) It an be seen from the grahs above that a gradual inrease in ore ressure auses a redution of effetive mean stress as the soil slowl moves to liquefation state. 1 Exerimental determination of the model arameters The arameters λ and κ an be obtained from oedometer or drained triaxial tests. The fritional arameter M is diretl related to φ whih ma also be obtained from drained triaxial tests. The elasti arameter G an either be obtained from bender element tests or it an be estimated as one third of the gradient of the deviator stress-axial strain urve on an unloading ortion of an undrained triaxial test. It is also ossible to us the effetive Poisson s ratio ν instead of using G. Refer to SAGE RISP Tehnial referene manual for further details. It is ossible to obtain a value for the arameter ϑ from the results of one unloading-reloading le in a onsolidation test. However, better estimates an be made using large number of les q for undrained li tests. Knowing the ratio, and the number of les for a artiular soil, it uo is ossible to obtain ϑ from the hart obtained for that te of soil. See figure 9.1 of the referene of arter, Booker and Wroth The referene above also rovides other alternatives on estimating ϑ (eg from the number of les required to generate a given exess ore ressure). omarison of FE results with exerimental results. The normalised effetive mean stress dereases as the number of load les inreases until the effetive mean stress reahes zero (ie liquefation state). Exeriments have been done b Talor and Bahus in whih sinusoidal strain-ontrolled les were alied to artifiiall reared saturated la samles. omarison has been made in the referene b arter, Booker and Wroth. onlusions A soil model aable of roduing the resonse of la when subjeted to reeated loading has been imlemented into RISP. The new model is based on the Modified am la model but with the addition of a degradation arameter to aount for redution in size of ield surfae during li loading. This new arameter ma be determined from undrained li triaxial tests. The model is aable to rodue man of the trends that have been observed in laborator tests. 10

11 The model has the following limitations: A large number of inrements is needed in order to rodue an aurate stress ath. The inrements must be suffiientl small when the stress ath is on or near the ield surfae. The model annot rodue kinemati hardening in whih the ield surfae shifts during li loading. Referenes: 1. arter, J.P, Booker, J.R., Wroth,.P. A ritial State Soil Model for li Loading ublished in Soil Mehanis Transient and li Loads, 198, John Wile & Sons Ltd. Talor, P.W., and Bahus, D.R., Dnami li strain tests on a la, Pro. 7 th Int. onf. Soil Mehs. Found. Engng., Mexio, , RISP Tehnial Referene Manual, b Rik Woods and Amir Rahim. The RISP onsortium Ltd,