Shakedown analysis of soil materials based on an incremental approach

Transcription

1 icccbe 2010 Nottingham University Press Proceedings of the International Conference on Comuting in Civil and Building Engineering W Tizani (Editor) Shakedown analysis of soil materials based on an incremental aroach J. Wang, H.S. Yu & H.X. Li University of Nottingham, UK Abstract In this aer, an incremental technique based on cyclic loading is develoed to imlement shakedown analysis for soils in the framework of elastic-lastic analysis without utilization of the classical shakedown theorems. A finite element model together with the use of ABAQUS is established for soil structures subjected to cyclic loads, where the Mohr-Coulomb yield criterion with associated lastic flow is assumed to cature the lastic behavior of soil materials. A criterion based on the lastic dissiation energy is suggested to distinguish shakedown and non-shakedown status. Meanwhile, the evolution of residual stresses during the whole rocess is examined for the better understanding of shakedown henomenon. The investigation of the lastic dissiation energy and the residual stresses via elastic-lastic analysis rovides a dee insight to the classical kinematic/static shakedown theorems. The effect of soil arameters on shakedown limit is studied with the alication to both one-layered and two-layered soil foundations. Some ossible failure modes of this soil structure under cyclic loads have also been found. The roosed aroach can bridge the ga between the traditional elastic-lastic incremental analysis and the classical shakedown analysis and also rovide the better understanding of shakedown henomenon. Moreover, this technique can be easily extended for nonstandard materials such as non-associated or strain-hardening materials. Keywords: shakedown analysis, lastic dissiation energy, ABAQUS, Mohr-Coulomb criterion 1 Introduction When an elastolastic structure is subjected to cyclic loads, the limit rovided by limit analysis, beyond which the instantaneous load carrying caacity of the material becomes exhausted, is by no means sufficient to revent the failure of the material. Although the alied reeated loads may not cause instantaneous collase of the structure, they ossibly induce the lastic deformation in the material in every load cycle and finally results in structural failure in such a way of either alternating lasticity or unlimited incremental lasticity (ratchetting). It is observed in exeriments that if the load level is lower than a critical limit, the material will have no further develoment of ermanent strains after a number of load cycles and resond urely elastically to the subsequent load cycles. This henomenon is termed as shakedown and the critical limit is regarded as the shakedown limit. The urose of shakedown analysis is to find the critical status of bearing caacity of a structure under cyclic loads. Due to the comlexity of the lastic behaviour of soil materials, the resonse of soils under cyclic loads becomes much more comlicated and therefore the alication of shakedown analysis to geotechnical engineering has been drawn much attention in the last two decades.

2 Shakedown analysis is usually based on two fundamental theorems: static or lower bound shakedown theorem (Melan, 1938), and kinematic or uer bound shakedown theorem (Koiter, 1960). Pande et al. (1980), and Shar and Booker (1984) were among the first to aly shakedown theory to the stability analysis of soil structures. Following their work, more robust aroaches for avements under traffic loads (Ponter, 1985; Yu and Hossain, 1998; Yu, 2005; Li and Yu, 2006) and offshore foundations under wave loads (Haldar et al., 1990; Faria, 2002) have been resented. All these works are based on either static or kinematic shakedown aroaches. Due to the fundamental base of static/kinematic shakedown theorems, these aroaches are only limited to elastic-erfectly lastic materials. However, it is well known that soil materials normally exhibit a non-standard lastic flow and hence the current methods may not obtain the shakedown limit for real soil materials. One ossible solution is to make use of the traditional elastic-lastic analysis together with the recognition criterion of shakedown modes. The urose of this aer is to imlement shakedown analysis for soil materials by means of an elastic-lastic incremental aroach without utilization of the classical shakedown theorems so that both erfectly lastic and non-standard lastic flow can be investigated. The arbitrary variation of cyclic loads is formulated by a secial function and a recognition criterion of shakedown mode based on lastic dissiation energy is roosed. Numerical alications to soil foundations further show the validity and efficiency of the develoed aroach. 2 A shakedown criterion When the amlitudes of cyclic loads are larger than the shakedown limit, the structure will be in a non-shakedown status of either alternating lasticity or ratchetting. In the first non-shakedown status of alternating lasticity, new lastic deformation occurs within each cycle of loading but the total lastic deformation over one cycle of loading is zero. This can be exressed as: T ε & t 0, t [0, T ]; ε& dt = 0. (1a, b) 0 where ε& is the lastic strain-rate and T is the eriod of cyclic loading. If the second non-shakedown status of ratchetting occurs, an unlimited incremental lastic flow will evolve in the structure and can be formulated in terms of strain rate as: T ε & t 0, t [0, T ]; ε& dt 0; Δε at t +. (2a, b, c) 0 It is quite hard to recognize both of alternating lasticity and ratchetting if only one kind of variable, e.g. either lastic strain-rate or stress, is used. However, the common feature in both of nonshakedown modes is that the rate of lastic dissiation energy is larger than zero and that the total lastic dissiation energy over one cycle of loading is growing with load-cycling. Then, it can be found that if shakedown occurs, the total lastic dissiation energy over the whole structure does not have an unlimited increase with the time and can become stable. Therefore, a general shakedown criterion based on the evolution of lastic dissiation energy can be roosed as follows: shakedown can haen in a structure subject to cyclic loads if the increment of lastic dissiation energy over one cycle of loading does not exceed a very small value. It can be formulated as: T ΔE = σ : ε& dvdt Δδ. (3) P 0 V where Δδ is the critical value of the increment of lastic dissiation energy over one load cycle. It can be emirically determined by the evolution of lastic dissiation energy after a few simulations. To imlement the shakedown criterion Eq. (3) for a structure subject to cyclic loads, different load levels with various amlitudes of the cyclic loads will be investigated, so that a critical load level can be found beyond which the shakedown condition Eq. (3) is violated.

3 3 Finite element model Finite element (FE) analysis of a soil foundation, under cyclic loading in the lain-strain condition is imlemented using the commercial finite element software ABAQUS. The size of the simulated region is chosen to be satisfied with the requirement of half-sace. Numerical study reveals that with the increase of the friction angle of materials, the affected soil region exands as well. A tyical finite element mesh is resented in Figure 1, where the structure is discretized by the eight-node reducedintegration quadrilateral elements. The mesh density in the vicinity area of loading is relatively high, to obtain the lastic strains in a reasonable accuracy. The material in this simulation is assumed to be elastic-erfectly lastic. The elastic resonse of materials is linear and isotroic, and a range of Young s modulus E has been investigated. Poisson s ratio is given as relatively high such as 0.49 in case of the Tresca-tye soil (ϕ = 0) while for frictional soils, Poisson s ratio is chosen as a lower value 0.2. Figure 1. A tyical FE mesh and cyclic loads. Both the Mohr-Coulomb and the Drucker-Prager yield criteria are used. In order to comare them, the arameters of the linear Drucker-Prager model are calculated to rovide a reasonable match to the Mohr-Coulomb arameters. In the lane strain model, the relation between (β, d) and (ϕ, c) is: 2 2 tanβ 3(9 tan ψ ) 3(9 tan ψ ) sinϕ =, ccosϕ = d. (4a, b) 9 tanβ tanψ 9 tanβ tanψ 4 Numerical results 4.1 Limit analysis Firstly, in order to verify the validity of the roosed numerical aroach, limit analysis, as a tyical case of shakedown analysis when cyclic loading is reduced to monotonic loading, is executed and also comared with analytical solutions. A steady uniform ressure is alied to a soil foundation, and the critical limit is obtained following the work of Sloan and Randolh (1982). The load-deflection resonse for the case of ϕ=0⁰ is lotted in Figure 2, where the value of ressure/c finally converged to a limit of 5.15 for both the Mohr-Coulomb and Drucker-Prager materials. It agrees well with the analytical solution More results for different friction angles are numerically obtained and they have a very good agreement with Prandtl s static collase limit, as shown in Figure 3. It should be ointed out that in ABAQUS, it is quite hard to be numerically convergent for the Mohr-Coulomb materials using the roosed aroach when the friction angle is larger than 30. This may be because the yield surface of the Mohr-Coulomb criterion is not smooth, which leads to a numerical difficulty in finding the direction of the lastic flow near the vertices of the yield surface.

4 Figure 2. Load-dislacement under monotonic loading. Figure 3. Comarison of limit loads. 4.2 Shakedown analysis for homogeneous soil foundations A set of eriodic loading, vertical ressure and horizontal traction q, of which the amlitudes vary from zero to maximum, as shown in Figure 1, is alied to a homogeneous soil foundation. The relation between these two loads is defined as μ = q/. Shakedown Incremental collase Figure 4. Evolution of ALLPD. Figure 5. Effects of friction angle and surface traction. Figure 4 resents the evolution of the total accumulative lastic dissiation energy (recognized as ALLPD in ABAQUS) for the case of ϕ = 30⁰ and q/ = 0.3. Different load magnitudes are investigated and as a result, different curves of ALLPD are obtained. For the lower load magnitude, ALLPD has a romt increase at the beginning of loading, and then gets to a relatively stable level. However, for the higher load magnitude, ALLPD kees increasing during the whole load history and has no sign of ceasing the growth. Therefore, the latter tye could lead to incremental collase, while the former tye can be recognized as shakedown status. And the shakedown limit should be between the two load magnitudes. It can be seen in Figure 4 that 50 loading cycles are sufficient to get a steady state resonse. Then, the shakedown condition Eq. (3) can be further exressed for this simulation as: ALLPD 50 ALLPD 40 if 0.1%, then ' Shakedown ' (5) ALLPD 40 where ALLPD (i) is the amount of the accumulative lastic dissiation energy at the ith load cycle. Therefore, the maximum load magnitude with the shakedown resonse can be regarded as the shakedown limit.

5 More numerical simulations are erformed to investigate how the frictional angle of materials and the surface traction affect the shakedown limit of soil foundations. The numerical results for this study are resented in Figure 5. It can be seen in Figure 5 that the shakedown limit of a soil foundation increases with rising of the frictional angle of materials but decreases with rising of μ = q/. To further investigate the shakedown modes and the evolution of key variables during cyclic loading, both the equivalent lastic strain and the residual equivalent stress are examined, and the numerical results for the case of ϕ = 30⁰ and q/=0.3 are resented in Figures 6 and 7. The evolution of the equivalent lastic strain at a oint in the lastic region has the similar trend with that of the accumulative lastic dissiation energy as shown in Figure 4. Meanwhile, the residual equivalent stress kees increasing in the case of non-shakedown, as shown in Figure 7. The creation of the residual stress field is one key technique in the static/lower bound shakedown analysis. Therefore, the numerical finding about the evolution the residual stress here may be helful to the classical shakedown analysis and the creation of the residual stress field. Figure 6. Evolution of equivalent lastic strain. Figure 7. Evolution of equivalent residual stress. 4.3 Shakedown analysis for layered soil foundations Figure 8. A two-layered soil foundation. Figure 9. Effects of strength ratio and stiffness ratio. Further numerical simulations are imlemented for a multi-layered soil foundation under a set of eriodic Hertz loads (, q) as shown in Figure 8, where the first layer is the Mohr-Coulomb tye material with the Poisson s ratio 0.3 and the second layer is the Tresca tye material with Poisson s ratio 0.4. The numerical results about the influence of the stiffness ratio and strength ratio uon

6 shakedown limit are resented in Figure 9 for the case of q/=0.4. It can be seen that as a given value of strength ratio, there exists an otimum stiffness ratio at which the shakedown limit is maximized. Further study on the location of the lastic region reveals that the change of stiffness ratio may make the location of the maximum lastic strain move from the to/first layer to the second layer, which can be distinguished non-shakedown in soil foundations as surface failure, subsurface failure and multi-layer interface failure. When the failure haens in the first layer of the soil foundation, the second weak layer has only a little effect on the shakedown limit. 5 Conclusion An incremental aroach has been develoed in this aer to imlement shakedown analysis based on the evolution of accumulative lastic dissiation energy. Numerical imlementation is executed in a commercial finite element software ABAQUS. The full history of stress and strains of soil materials subjected to cyclic loads have been found in shakedown resonse. This rovides a dee insight to the classical shakedown study. Comared with the classical shakedown analysis based on the static/kinematic shakedown theorems, this aroach can be easily extended to more comlicated or non-standard materials, e.g. non-associated lastic flow or strain-hardening materials. The numerical simulations found that the shakedown limit of a homogenous soil material increases with rising of the friction angle of soils but decreases with rising of horizontal traction. Moreover, for a layered soil foundation, there exists an otimum stiffness ratio of layers that allows for the shakedown limit. The evolution of the lastic strain and residual stress are found to have the similar trend with the accumulative lastic dissiation energy. Considering that a cyclic load varies along an arbitrary ath, an investigation of the critical loading aths may be necessary for formulating a cyclic load and this will be studied in the future work. References FARIA, P.O., Shakedown analysis in structural and geotechnical engineering. PhD Thesis, University of Wales. HALDAR, A.K., REDDY, D.V. and AROCKIASAMY, M., Foundation shakedown of offshore latforms. Comuters and Geotechnics, 10, KOITER, W.T., General theorems for elastic-lastic solids. In: Progress in Solid Mechanics (eds Sneddon, I.N. and Hill, R.), 1, LI, H.X. and YU, H.S., A nonlinear rogramming aroach to kinematic shakedown analysis of frictional materials. International Journal of Solids and Structures, 43, MELAN, E., Der sannungsgudstand eines Henky-Mises schen Kontinuums bei Verlandicher Belastung. Sitzungberichte der Ak Wissenschaften Wie (Ser. 2A), 147, 73. PANDE, G.N., ABDULLAH, W.S. and DAVIS, E.H., Shakedown of elasto-lastic continua with secial reference to soil-rock strucutres. In: International Symosium on Soils under Cyclic and Transient loading PONTER, A.R.S., HEARLE, A.D. and JOHNSON, K.L., Alication of the kinematical shakedown theorem to rolling and sliding oint contacts. Journal of Mechanics and Physics of Solids, 33, SHARP, R.W. and BOOKER, J.R., Shakedown of avements under moving surface loads. Journal of Transortation engineering, ASCE, 110, SLOAN, S.W. and RANDOLPH, M.F., Numerical rediction of collase loads using finite element methods. International Journal for Numerical and Analytical Method in Geomechanics, 6, YU, H.S., Three-dimensional analytical solutions for shakedown of cohesive-frictional materials under moving surface loads. Proceedings of the Royal Society A, 461, YU, H.S. and HOSSAIN, M.Z., Lower bound analysis of layered avements using discontinuous stress fields. Comuter Methods in Alied Mechanics and Engineering, 167,