Failure mechanisms and corresponding shape factors of shallow foundations

Transcription

1 -4 June 06, Near East University, Nicosia, North Cyrus Failure mechanisms and corresonding shae factors of shallow foundations Stefan Van Baars University of Luxembourg KEYWORDS: Footings; Shallow Foundations; Failure Mechanism; Shae factor; Bearing caacity. ABSTRACT: In 90 Prandtl ublished an analytical solution for the bearing caacity of a maximum stri load on a weightless infinite half-sace. This solution was extended by Reissner in 94 with a surrounding surcharge. In the 940s, Keverling Buisman and Terzaghi extended the Prandtl-Reissner formula for the soil weight. Since then several eole roosed equations for the soil-weight bearing caacity factor. In 96 Meyerhof was the first to write the formula for the (vertical) bearing caacity of shallow foundations with both inclination factors and shae factors. The failure mechanisms belonging to the cohesion bearing caacity factor and the surcharge bearing caacity factor is for an infinite (D) stri footing a Prandtl-wedge failure mechanism, but according to Finite Element Modelling (FEM) the failure mechanism belonging to the soil-weight bearing caacity factor is not. It looks more like a global failure mechanism. This means that the assumed suerosition in the Terzaghi equation, and in the Meyerhof equation, is not automatically allowed. Additional FEM calculations show that in the case of a finite stri footing, and esecially of round footings, the failure mechanism is again very different, and leads to much lower shae factors as factors based on a Prandtl-wedge failure mechanism. In fact the third direction, i.e. the tangential direction, which lays no imortant role in the failure mechanism for infinite stri footings, starts to lay a major role in the failure mechanism and in the magnitude of the bearing caacity of the stri footing. THE FAILUE MECHANISM AND BEARING CAPACITY OF A FOOTING In 90, Ludwig Prandtl ublished an analytical solution for the bearing caacity of a soil under a load,, causing kinematic failure of the weightless infinite half-sace underneath. Figure. Failure mechanism (Left: original drawing of Prandtl; Right: Lab test of Muhs and Weiß).

2 -4 June 06, Near East University, Nicosia, North Cyrus The strength of the half-sace is given by the angle of internal friction,, and the cohesion, c. The failure mechanism roosed by Prandtl (see original drawing of Prandtl in Figure on the left) is indeed the same as laboratory tests erformed by Muhs and Weiß (97) (see Figure on the right). r e r tan tan e r tan tan e e r r 4 4 Figure. The Prandtl-wedge with zones. r tan K q K e q tan qnq Nq K e Prandtl subdivided the sliding soil art into three zones (see Figure ):. Zone : A triangular zone below the stri load. Since there is no friction on the ground surface, the directions of the rincial stresses are horizontal and vertical; the largest rincial stress is in the vertical direction.. Zone : A wedge with the shae of a logarithmic siral, in which the rincial stresses rotate through from Zone to Zone. The itch of the sliding surface equals the angle of internal friction;, creating a smooth transition between Zone and Zone.. Zone : A triangular zone adjacent to the stri load. Since there is no friction on the surface of the ground, the directions of rincial stress are horizontal and vertical; the largest rincial stress is in the horizontal direction. The failure mechanism does not have to be symmetrical, with sliding in two directions, as roosed by Prandtl in Figure, but can also be unsymmetrical, with sliding in one direction as shown in Figure and Figure. This last figure is a lot of the incremental dislacements of a calculation with the finite element model Plaxis, showing how close the Prandtl-wedge theory is to the outcome of finite element calculations. Figure. Plot of incremental strains: The Prandtl-wedge in only one direction (FE calculation).

3 Failure mechanisms and corresonding shae factors of shallow foundations Van Baars, S. The solution of Prandtl was extended by Reissner in 94 with a surrounding surcharge, q, and was based on the same failure mechanism. Keverling Buisman (940) and Terzaghi (94) extended the Prandtl-Reissner formula for the soil weight,. And in 95 Meyerhof was the first to roose equations for inclined loads. He was also the first in 96 to write the formula for the (vertical) bearing caacity v with bearing caacity factors (N), inclination factors (i) and shae factors (s), for the three indeendent bearing comonents; cohesion (c), surcharge (q) and soil-weight (), in a way it is still used nowadays: i s cn i s qn i s BN () v c c c q q q. Prandtl and Reissner solved resectively the first two bearing caacity factors: r Nq K K e r N K e tan c tan cot with: K sin () sin Keverling Buisman (940), Terzaghi (94), Caquot and Kérisel (95), Meyerhof (95; 95; 96; 965), Brinch Hansen (970), Vesic (97, 975), Chen (975) and Van Baars (05) subsequently roosed different equations for the soil-weight bearing caacity factor N. Therefore the following equations for the soil-weight bearing caacity factor can be found in the literature: tan tan tan tan N K e tan.4 (Meyerhof '6), N.5 K e tan (Brinch Hansen, 970), N K e tan (Vesic, 97), N K e tan (Chen, 975) N tan e tan (smooth late) (Van Baars, 05). According to Finite Element calculation, only the last equation really reresents the Prandtl boundary conditions; no shear stresses at the surface and no dilatant soil behaviour (Van Baars, 05). The roblem with the Meyerhof equation (Equation ) is that it is robably incorrect to aly suerosition in this equation, because the failure mechanism belonging to the soil-weight failure bearing caacity ( N ) is not the same as the failure mechanism belonging to both the cohesion bearing caacity (N c ) and the surcharge bearing caacity ( N q ). By using the inclination factors on to of the shae factors, the risk of errors is even larger. () THE FAILURE MECHANISM FOR THE SOIL-WEIGHT BEARING CAPACITY The effects of the cohesion are taken into account by the cohesion bearing caacity factor N c and the effects of a surcharge are taken into account by the surcharge bearing caacity factor N q. So, the soil-weight bearing caacity factor N regards the (remaining) effect of the gravity on a frictional, but cohesionless material without a surcharge. Finite element calculations indicate clearly that the failure mechanism for the soil-weight bearing caacity is different from the bearing caacity for the cohesion and for the surcharge (Figure 4). This fact is already known for some time. Caquot and Kerisel ublished in 966 the circular or ellitical failure mechanism (Figure 5 on the left). The same failure mechanism can be observed in the hoto of a laboratory test (Figure 5 on the right), which was ublished by Lambe & Whitman in 969.

4 -4 June 06, Near East University, Nicosia, North Cyrus Figure 4. Circular failure zone under a footing on non-cohesive soil (FE calculation). Figure 5. Circular wedge under a footing on non-cohesive soil (Laboratory test). A constant, or rectangular shaed, bearing caacity is imossible because just next to the late there is no strength for a cohesionless material without a surcharge. So, unlike the effect of the cohesion and the effect of the surcharge, the effect of the soil-weight does not result in a constant maximum bearing caacity. Therefore the shae of the maximum load is more like a triangle (or even hyerbolic!) because the maximum load will reach a maximum in the middle and becomes zero at the edges next to the load, where the shear and normal stresses go to zero too. This exlains why in the figure of Caquot & Kerisel, not a rectangular load, but a triangle shaed load is shown. Figure 6. Shoe rint in sand: bearing caacity in the middle, but not at the edges. 4

5 Failure mechanisms and corresonding shae factors of shallow foundations Van Baars, S. The effect of the condition of zero bearing caacity at the edges, but not in the middle, exlains the shae of a shoe rint in the sand (see Figure 6), since for a shoe on the sand there is no bearing caacity due to a surcharge or cohesion, but only due to the soil-weight. The reresentation of this satially variable load as a constant bearing caacity factor means that only an average comonent can be calculated for the soil-weight bearing caacity factor. Although the use of the soil-weight bearing caacity factor, in the same way as the other two the bearing caacity factors, suggests a constant bearing caacity, we should be aware that this is definitely not the case. The visible failure mechanism, i.e. the zone showing dislacements, doesn t necessarily have to be the same as the lastic zone. Figure 7 shows the relative shear stress rel. This relative shear stress is defined as the radius of the circle of Mohr divided by the radius of a circle touching the Coulomb line. This means lasticity or failure for soil having a relative shear stress of. The lastic zone in this figure is clearly not like the Prandtl-wedge. There is in this case a circular failure mechanism and a dee lastic zone with another shae. Remarkable is how dee this lastic zone reaches. Figure 7. Plastic zone below infinite stri footing on non-cohesive soil (FE calculation). THE FAILURE MECHANISM FOR ROUND FOOTINGS The revious chater discussed the failure mechanisms of infinite footings. In this chater the effect of finite footings, and in articular round footings, is discussed. If a Prandtl shaed failure mechanism is also assumed for round footings, then the area at the surface (see horizontal grey zone in Figure 8) which collects the surcharge q surrounding the circular footings with load, is far bigger than the area which suorts the cone below the circular footing (see small diagonal grey zone). This means that the cone would have a huge suort, i.e. the bearing caacity factor N q would be extreme high; too high to be realistic (Van Baars, 04). This means that for circular footings, the Prandtl-wedge failure mechanism is certainly not normative. In fact, due to the ushing out of the wedge, the third direction, i.e. the tangential direction, which lays no imortant role in the failure mechanism for infinite stri footings, starts to lay a major role in the failure mechanism and in the magnitude of the bearing caacity of the stri footing, leading to another failure mechanism. 5

6 -4 June 06, Near East University, Nicosia, North Cyrus Figure 8. Unrealistic Prandtl-wedge failure mechanism for circular footings. The question now arises from which width over length ratio B / L of a footing, the Prandtl-wedge failure mechanism does not occur anymore, and also which failure mechanism, or even mechanisms, relaces the Prandtl-wedge failure mechanism? First investigations into this with Finite Element calculations show in this case always a lastic zone, just as in Figure 5. The same calculations rove that the currently used shae factors of Meyerhof (96) and of De Beer (970) are too high, and should be more like (Van Baars, 05): s s q c B s 0.55, L B 0.. L The use of shae factors in the Meyerhof equation (Equation ), suggests that only a small correction is needed to find the bearing caacity of a round footing, or another finite footing, in comarison to an infinite footing, but this is not really the case; since at a certain width over length ratio B / L, the whole failure mechanism is changing. Nevertheless, the use of shae factors functions well for the calculation of the bearing caacity. (4) 4 SUPERPOSITION AND ITS CONSEQUENCES The question arises how big the consequences are of the use of suerosition. It looks that there is a huge difference in failure mechanisms, but as noticed from the shae factors; there seems to be, no discontinuities in the behaviour of the bearing caacity. Figure 9. Plot of incremental strains: New failure mechanism due to suerosition. 6

7 Failure mechanisms and corresonding shae factors of shallow foundations Van Baars, S. Figure 9 shows for examle the failure mechanism of a stri footing loading, calculated with the Finite Element Model Plaxis, on a non-cohesive, but frictional soil, with a surcharge load. This is a very interesting case, since it combines both the surcharge loading (with the Prandtl-wedge failure mechanism of Figures and ) and the cohesion less material loading (with the circular wedge failure mechanism of Figures 4, 5 and 6). Figure 9 shows that this leads to a new failure mechanism, which is clearly a sort of mixture of the two basic failure mechanisms. 5 CONCLUSIONS The failure mechanisms of an infinite stri footing belonging to the cohesion bearing caacity factor and the surcharge bearing caacity factor is a Prandtl-wedge failure mechanism, but the failure mechanism belonging to the soil-weight bearing caacity factor is a circular failure mechanism. This means that the assumed suerosition in the Terzaghi equation, and in the Meyerhof equation, is not automatically allowed. Therefore more research is needed her. In case of a finite stri footing, and esecially round footings, the failure mechanism is again very different, and leads to much lower shae factors as factors based on a Prandtl-wedge failure mechanism. In fact the third direction, i.e. the tangential direction, which lays no imortant role in the failure mechanism for infinite stri footings, starts to lay a major role in the failure mechanism and in the magnitude of the bearing caacity of the stri footing, leading to another failure mechanism. REFERENCES Brinch Hansen, J. A (970) Revised and extended formula for bearing caacity, Bulletin No 8, Danish Geotechnical Institute Coenhagen : 5-. Caquot, A. and Kérisel, J. (95). Sur le terme de surface dans le calcul des fondations en milieu ulvérulents, Third International conference on Soil Mechanics and Foundation Engineering, Zürich : 6 7. Chen, W. F. (975) Limit analysis and soil lasticity, Elsevier. De Beer, E. E. ( 970). Exerimental determination of the shae factors and the bearing caacity factors of sand. Geotechnique, 0 : Keverling Buisman, A. S. (940). Grondmechanica, Waltman, Delft, the Netherlands : 4 Lambe,T. W. and Whitman, R.V, (969) Soil Mechanics, John Wiley & Sons, ISBN : 97 Meyerhof, G. G. (95) The ultimate bearing caacity of foundations, Géotechnique, : 0- Meyerhof, G. G. (95) The bearing caacity of foundations under eccentric and inclined loads, in Proc. III intl. Conf. on Soil Mechanics Found. Eng., Zürich, Switzerland, : Meyerhof, G. G. (96) Some recent research on the bearing caacity of foundations, Canadian Geotech. J., () : 6-6 Meyerhof, G. G. (965) Shallow foundations, Journal of the Soil Mechanics and Foundations Division ASCE, Vol. 9, No., March/Aril : - Muhs, H. and Weiß, K. (97): Versuche über die Standsicheheit flach gegründeter Einzelfundamente in nichtbindigem Boden, Mitteilungen der Deutschen Forschungsgesellschaft für Bodenmechanik (Degebo) an der Technischen Universität Berlin, Heft 8 : Prandtl, L. ( 9 0) Über die Härte lastischer Körer. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch.-hysikalische Klasse : Reissner, H. ( 9 4) Zum Erddruckroblem. Proc., st Int. Congress for Alied Mechanics, C. B. Biezeno and J. M. Burgers, eds., Delft, The Netherlands : 95. Terzaghi, K. (94) Theoretical soil mechanics, J. Wiley, New York. Van Baars, S. (04) The inclination and shae factors for the bearing caacity of footings, Soils and Foundations, ISSN: , Vol.54, No.5, October Van Baars, S. (05) The bearing caacity of footings on cohesionless soils, send for ublication to The electronic journal of geotechnical engineering Vesic, A. S. (97) Analysis of ultimate loads of shallow foundations. J. Soil Mech. Found. Div., 99() : 5. Vesic, A. S. (975) Bearing caacity of shallow foundations, H.F. Winterkorn, H.Y. Fang (Eds.), Foundation Engineering Handbook, Van Nostrand Reinhold, New York : 47 7