horizontal angle into45???2, Called passive Rankine District.
For the above-mentioned circumstances,The Prang Del draw the ultimate bearing capacity of the theoretical solution
Pu=cNc Ne - bearing capacity;
? Nc?cot?[exp(?tan?)tan2(45??)?1]2
c，?——Cohesive force and internal friction Angle of soil kPa。
Figure 5.31
Actually,Base usually has a certain depth d,The weight of the soil on both sides of the underside of the foundation with uniform overload the q=γod instead.,Shown in Figure 5.31.Reissene(Reissene, 1924) is put forward the foundation ultimate bearing capacity formula which based on Prang del theoretical solution .
pu=cNc +qNq
?Besides Nq?exp(?tan?)tan2(45??)
2 Nc?(Nq?1)cot?
Nc、Nq—— Cutting force coefficient
The theoretical solution obtained from Prang del and Reissene is under special conditions,The actual foundation soil is not without weight,Nor there is no friction between soil and base,, also do not take into account of the influence of shear strength of lateral soil in the base depth of range，There is a wide gap between its results and the actual engineering，therefore，Many scholars make the modification and development on the basis of the Prang del and Reissene theoretical solution，and put forward various approximate calculation method of ultimate bearing capacity,One of the most typical is the ultimate bearing capacity of shallow foundation formulas that Terzaghi formula As the
representative and the ultimate bearing capacity of deep foundation formula proposed by Meyer Hove . each of these two categories of limit bearing capacity formula to be introduced
5．9．2 The theoretical formula of Terzaghi formula
1) Terzaghi formula
In the derivation of the formula,The Terzaghi consider:①the foundation soil has weight;②Basal coarse;③Regardless of the shear strength of the fill above the substrate,It is seen as only an overload acting in the horizontal plane of the substrate;④Base will occur overall shear failure under the action of ultimate load;⑤Assumed the shape of the sliding surface in the foundation as shown in Figure 5.32.
Figure 5.32
Area I As the friction between the base and the soil to prevent the occurrence of shear displacement,therefore，When the foundation is to destroy and occur continuous slip surface,part of the soil Under basal will move down t with the base ogether and in the elastic equilibrium state.The part of the soil is called elastic wedge ，area I as shown in figure 5.32.The soil in this area is less prone to shear displacement,but in a compacted state.Boundary αb (or) and the horizontal angle depends on the substrate roughness.
When is assumed that the substrate is completely rough,???In general,?
Area II Sliding surface is assumed by the logarithmic spiral and straight.
Area III The sliding surface the cd (OR d)in a straight line,With
?the horizontal 45??This area is for the passive Rankine District.Take
2the elastic wedge for out of the body,Its effect on the force
(1) the Total ultimate load Pu=pub；
(2) Elastic wedge aba? body weight W??b2tan?
(3) the component of cohesive force in the vertical direction on the Elastic wedge slope of ab anda?b Its value, respectively, C?ctan?
(4) Area II and Area III when Soil sliding, it occurs passive earth pressure Ep on the slope ab and a?b , the EP’s the direction of the slope normal at ?, the force component in the vertical direction is .EP?cos(?-?)
The forces in the vertical direction above on the static equilibrium equation is
?pub?2Epcos(???)?cbtan??b2tan?4
Terzaghi assumes that Ep form the static equilibrium equation above is Epc that is caused by the soil cohesion c when ??q?0.the foundation overload q On both sides cause Epq when ??c?0 . The Epr is Only caused by soil heavy when q?c?0 ,These three parts can be expressed as Ep?Epc?Epq?Epr
Formula (5.36) into (5.35) and
after finishing the foundation ultimate bearing capacity is
Nr、Nq、Nc——Bearing capacity factor,Can be calculated as the following equation or look-up table 5.3
Nc?(Nq?1)cot?
e3(???)tan?214b2Pu?cNc?qNq?1?bN?2
Nq?2cos2(45???2
)
N??4Ep
?b2cos(???)
Because Nγ value is affected by the ψ values , and the ψ value is
associated with basal roughness, making Nγ calculation is quite complicated, thus only the numerical solution of the Nγ directly is
listed in table 5.3, for the ease of use.
Formula (5.37) is obtained in the overall shear failure conditions and suitable for smaller compression of the soil.As Loose or larger compression of the soil, local shear failure may occur.
Terzaghi is recommended to use the method of reducing the shear strength indicators to amend formula(5.37) This is to take
c??c
Revisions to the foundation ultimate bearing capacity formula
1??qNq???bN??pu?c?Nc2
tan???2tan? 323?、N?? is equal to the coefficient of bearing capacity of Where Nc? 、Nqlocal shear failure situation.According to the reduced internal friction
Angle, still can look up table 5.3.
The above formula (5.37) and (5.38) applies only to strip foundation.For square and round basis, Terzaghi suggested the ultimate bearing capacity of the foundation by the following formula: Square base (width b)
The overall shear failure
pu?1.2cNc?qNq?0.4?bN? （5.39） The Local shear failure
??0.4?bN?? （5.40） pu?0.8cNc??qNqCircular base (radius b)
The overall shear failure
pu?1.2cNc?qNq?0.6?bN? （5.41） The Local shear failure
??0.6?bN?? （5.42） pu?0.8cNc??qNqFor the rectangular foundation (width b, length l), approximate value
according to the b/l,is obtained by interpolation method between the bearing capacity of the strip foundation (b/l = 0) and the square foundation (b/l = 1) 2)Vesic formula
In the 70 s ,Vesic,on the basis of the Prang del and Reisner solution , considering the impact of soil self-weight, got the ultimate bearing capacity of strip foundation under the center loads formula form with the Terzaghi formulas (5.37)the same form as the ultimate bearing capacity
formula. According to influence factors of bearing capacity ,Vesic modified the formula (5.37),Formulas take into account the shape of the underside of the foundation, load tilt and eccentricity, basement and the ground tilt,the shearing strength effect based on both sides of the cover layer , etc., and introduce the corresponding correction coefficient, You come to the following formula
pu?cNcScicbcgcdc?qNqSqiqgqdq?1?bN?S?i?b?g?d?2
Nc、Nq、Nγ——Bearing capacity factor, as shown in Table 5-4;
Sc、Sq、Sγ——The base shape correction coefficient, as shown in Table 5-7;
ic、iq、iγ—— Tilt correction coefficient of load, as shown in table 5-7;
dc、dq、dγ——The foundation depth correction factor, as shown in Table 5-7;
gc、gq、gγ——Tilt correction factor on the ground, as shown in Table 5-7;
bs、bq、bγ——The substrate tilt correction coefficient, as shown in Table 5-7
Analysis showed that the inclined or eccentric loads, will significantly impact the overall shear failure shapes, reducing the ultimate bearing capacity.When the strip footing eccentricity e > b / 6, soil could swell at the other side of the eccentric loading,In order to avoid such a situation, it should limit the eccentricity e of the b / 6 range.
For eccentric loading, such as for strip foundation, effective widthb??b?2e is used instead of the original width b,Such as rectangular foundation, effective area A??b??l? is used instead of the original area of A.Wherein b??b?2eb,l??l?2e1,eb、el respectively, the eccentricity of the load on the short side and the long side direction Besides the above effects, Vesic also considered the compressibility of soil in the ultimate bearing capacity formulas, omitted here (3)Hansen formula Hansen formula is the improvement and development of the Terzaghi formula, Its expression is the same as Vesic formula,, but the coefficient of bearing capacity and the rest of the correction coefficient expression is unlike Vesic formula,In order to facilitate readers to use and contrast,and also put them in with those listed in table 5.7.And the
Nc、Nq、Nγ bearing capacity factor can seek from Table 5.5.
In the actual engineering, the soil is often uneven, it have an impact on the bearing capacity,Hansen recommends that when the difference is