VALIDATION AND VERIFICATION TABLE OF CONTENTS

Transcription

1 VALIDATION AND VEIFICATION TABLE OF CONTENTS 1 Intoduction Elasticity poblems with known theoetical solutions Smooth igid stip footing on elastic soil Stip load on elastic Gibson oil Bending of beams Bending of plates Pefomance of shell elements Updated mesh analysis of a cantileve Plasticity poblems with theoetical collapse loads Beaing capacity of cicula footing Beaing capacity of stip footing Sliding block fo testing intefaces Cylindical cavity expansion Consolidation and goundwate flow One-dimensional consolidation Unconfined flow though a sand laye Confined flow aound an impemeable wall efeences III

2 VALIDATION AND VEIFICATION ELASTICITY POBLEMS WITH KNOWN THEOETICAL SOLUTIONS A seies of elastic benchmak calculations is descibed in this Chapte. In each case the analytical solutions may be found in many of the vaious textbooks on elasticity solutions, fo example Gioud (197) and Poulos & Davis (1974)..1 SMOOTH IGID STIP FOOTING ON ELASTIC SOIL Input: The poblem of a smooth stip footing on an elastic soil laye with depth H is shown in Fig..1. This figue also shows elevant soil data and the finite element mesh used in the calculation. A unifom vetical displacement of 10 mm is pescibed to the footing and the indentation foce, F, is calculated fom the esults of the finite element calculation. Since the poblem is symmetic it is possible to model only one half of the situation as shown in Fig..1. Figue.1 Poblem geomety Output: The scaled defomation of the finite element mesh at the end of the elastic analysis is shown in Fig... The footing foce esulting fom a igid indentation of 10 mm is calculated to be F=15.6 kn. (Note that when only half of the elastic halfspace is modelled the foce calculated by PLAXIS will be exactly one half of this value). Settlement = F δ (1+ ν ) G H with δ = 0.88 fo = 4 1 B Veification: Gioud (197) gives the analytical solution to this poblem in the fomula above, whee H is the depth of the laye, B is the total width of the footing and δ is a constant. Fo the dimensions and mateial popeties used in the finite element analysis this solution gives a footing foce of kn. The eo in the numeical solution is theefoe about 0.7%. -1

3 PLAXIS Fig..3 gives both the analytical and numeical esults fo the pessue distibution undeneath the footing. This figue shows that the numeical esults agee vey well with the analytical solution. Figue. Defomed mesh Figue.3 Pessue distibution at footing -

4 VALIDATION AND VEIFICATION. STIP LOAD ON ELASTIC GIBSON SOIL Input: Fig..4 shows the mesh and the soil data fo a plane stain calculation of the settlement of a stip load on Gibson soil. (Gibson soil is an elastic laye in which the shea modulus inceases linealy with depth). Using z to denote depth, the shea modulus, G, used in the calculation is given by: G = α.z = 100.z. With a Poisson's atio of 0.495, the Young's modulus vaies by: E = 99 z. In ode to pescibe this vaiation of Young's modulus in the mateial popeties window the efeence value of Young's modulus, E ef, is taken vey small and the Advanced option is selected fom the Paametes tab sheet. The incease of Young's modulus E incement is set to 99 and the efeence level y ef is enteed as 4.0 m, being the top of the geomety. Figue.4 Poblem geomety Output: An exact solution to this poblem is only available fo the case of a Poisson's atio of 0.5; in the PLAXIS calculation a value of is used fo the Poisson's atio in ode to appoximate this incompessibility condition. The numeical esults show an almost unifom settlement of the soil suface undeneath the stip load as can be seen fom the velocity contou plot in Fig..5. The computed settlement is m at the cente of the stip load. Figue.5 Absolute displacement contous -3

5 PLAXIS Figue.6 Total stesses in soil Veification: The analytic solution is exact only fo an infinite half-space, wheeas the PLAXIS solution is obtained fo a laye of finite depth. Howeve, the effect of a shea modulus that inceases linealy with depth is to localise the defomations nea the suface; it would theefoe be expected that the finite thickness of the laye will only have a small effect on the esults. The exact solution fo this paticula poblem, as given by Gibson (1967), gives a unifom settlement beneath the load of magnitude: Settlement = p α In this case the exact solution gives a settlement of 0.05 m. The numeical solution is 6% lowe than the exact solution. -4

6 VALIDATION AND VEIFICATION.3 BENDING OF BEAMS Input: Fo the veification of beams two poblems ae consideed. These poblems involve a single point load and a unifomly distibuted load on a beam espectively, as indicated in Fig..7. Fo these poblems the chaacteistics of a HEB 00 steel beam have been adopted. In a plane stain model, the beam is in fact a plate of 1 m width in the out of plane diection. The popeties, the dimensions and the load of the beam ae: EA = kn EI = 100 knm ν = 0.0 l = m F = 100 kn q = 100 kn/m Beams cannot be used individually. A single block cluste may be used to ceate the geomety. The two beams ae added to the bottom line with a spacing in between. Use point fixities on the end points of the beam. A vey coase mesh is sufficient to model the situation. In the Initial conditions mode the soil cluste can be deactivated so that only the beams emain. Figue.7 Loading scheme fo testing beams Output: The esults of the two calculations ae plotted in Figs.8,.9 and.10. Fo the exteme moments and displacements we find: Point load: Distibuted load: M max = 50.0 knm u max = 14.0 mm M max = 50.0 knm u max = 17.4 mm Figue.8 Computed distibution of moments -5

7 PLAXIS Figue.9 Shea foces Figue 1.10 Computed displacements Veification: As a fist veification, it is obseved fom Figs.8a and.8b that PLAXIS yields the coect distibution of moments. Fo futhe veification we conside the well-know fomulas as listed below. These fomulas give appoximately the values as obtained fom the PLAXIS analysis. 1 Point load: M max = Fl = 50 knm 4 u max = F l E I = 13.9 mm Distibuted load: M max = 1 ql 8 = 50 knm u max = q l E I = mm -6

8 VALIDATION AND VEIFICATION.4 BENDING OF PLATES Input: In an axisymmetic analysis, beams may be used as cicula plates. The latte two veification examples involve a unifomly distibuted load p on a cicula plate. In one example the plate can otate feely at the bounday and in the othe example the plate is clamped, as indicated in Fig..11. Figue.11 Loading scheme fo testing axisymmetic plates Solution: Fo the situation of a cicula plate with a unifomly distibuted load one can elaboate and solve a diffeential equation. The analytical solutions fo this equation depend on the bounday conditions. Fo the plate with fee otation at the bounday one finds: Settlement: p 4 5+ ν 6 +ν 4 w = - + EI plaxis D = 64 D 1+ ν 1+ ν 4 1 -ν Moments: p m = p (3 + ν ) - (3+ ν ) m 16 tt = (3 + ν ) 16 - (1+3ν ) Using = 1 m, d = 0.1 m, p = 1 kn/m, ν = 0 and EI = 1 knm /m, this gives: In the cente: w = m Numeical: w = m ( = 0) m = knm/m Numeical: m = knm/m At = /: w = m Numeical: w = m m = knm/m Numeical: m = knm/m Fo the plate with a clamped bounday one finds: Settlement: p 4 w = 1 64 D - -7

9 PLAXIS Moments: m = p 16 (1+ ν ) - (3+ ν ) m tt = p 16 (1+ ν ) - (1+3ν ) Using = 1 m, d = 0.1 m, p = 1 kn/m, ν = 0 and EI = 1 knm /m, this gives: In the cente: w = m Numeical: w = m ( = 0) m = knm/m Numeical: m = knm/m At = /: w = m Numeical: w = m m = knm/m Numeical: m = knm/m At = : m = knm/m Numeical: m = knm/m Figue.1 Computed distibution of moments Figue.13 Computed displacements Veification: The settlement diffeence is mainly due to shea defomation, which is included in the numeical solution but not in the analytical solution. Apat fom this, the numeical esults ae vey close to the analytical solution. -8

10 VALIDATION AND VEIFICATION.5 PEFOMANCE OF SHELL ELEMENTS A beam in PLAXIS can be applied as a tunnel lining. By using this element, 3 types of defomations ae taken into account: shea defomation, compession due to nomal foces and obviously bending. Input: A ing with a adius of =5 m is consideed. The Young's modulus and the Poisson's atio of the mateial ae taken espectively as E=10 6 kpa and ν=0. Fo the thickness of the ing coss section, H, seveal diffeent values ae taken so that we have ings anging fom vey thin to vey thick. In ode to model such a ing the bottom point of the ing is fixed with espect to tanslation and the top point is allowed to move only in the vetical diection. Then the load F=0. kn/m is applied only at the top point. Geometic non-lineaity is not taken into account. Output: The calculated vetical deflections at the top point ae pesented in Fig..14. The defomed shape of the ing is also shown in Fig..14. The calculated nomal foce at the belly of the ing is 0.50 fo all diffeent values of ing thickness. The calculated bending moment at the belly vaying fom 0.18 to as the ing changes fom thin to thick. Typical gaphs of the bending moment and nomal foce ae shown in Figs..15a and.15b. Figue.14 Calculated deflections compaed with analytical solutions -9

11 PLAXIS Figue.15a Nomal foces Figue.15b Bending moments Veification: The analytical solution fo the deflection of the ing is given by Blake (1959), and the analytical solution fo the bending moment and the nomal foce can be found fom oak (1965). The vetical displacement at the top of the ing is given by the following fomula: δ = F λ E λ λ with λ = H The solid cuve in Fig..14 is plotted accoding to this fomula. It can be seen that the deflections calculated by PLAXIS fit the theoetical solutions vey well. Only fo a vey thick ing some eos ae obseved, which is about 7 pe cent fo H/=0.5. But fo thin ings the eo is nealy zeo. The analytical solution fo the bending moment and nomal foce at the belly is 0.18 and 0.5 espectively. Thus even fo vey thick ings the eo in the bending moment is just 4 pe cent, and the eo in the nomal foce is only 0. pecent. -10

12 VALIDATION AND VEIFICATION.6 UPDATED MESH ANALYSIS OF A CANTILEVE The ange of poblems with known solutions involving lage displacement effects that may be used to test the lage displacement options in PLAXIS is vey limited. The lage displacement elastic bending of a cantileve beam, howeve, is one poblem which is well suited as a lage displacement benchmak poblem since a known analytical solution exists, Mattiasson (1981). Geomety non-lineaity is of majo impotance in poblems involving slende stuctual membes like beams, plates and shells. Indeed, phenomena like buckling and bulging cannot be descibed without consideing geomety changes. Soil bodies, howeve, ae fa fom slende and consequently, most finite element fomulations tacitly disegad changes in geomety. This also applies to conventional PLAXIS calculations. Uses should check such esults by consideing the tuly defomed mesh. In most pactical cases this will indicate vey little change of geomety. In some paticula cases, howeve, it may be significant. Fo special poblems of exteme lage defomation an Updated mesh analysis is needed. Fo this eason PLAXIS involves a special module. Fo details on the implementation the eade is efeed the PhD thesis by Van Langen (1991). This module was pogammed using the Updated Lagangian fomulation as descibed by McMeeking and ice (1975). Analysis: The analysis elates to the calculation of the hoizontal and vetical tip displacement fo the cantileve beam shown in Fig..16. The mesh used in the PLAXIS analysis is shown in Fig..17. (Note that it is necessay to use a finite depth of beam in the numeical calculation in contast to the analytical solution which is based on a beam of zeo geometic thickness). Figue.16 Tue defomation of elastic cantileve -11

13 PLAXIS esults: The computed load-displacement cuves ae plotted in Fig..18. The numeical esults ae clealy in close ageement with the analytical solution. Figue.17 Significant geometic non-lineaity Figue.18 Numeical esults agee with theoy -1