stresses and displacements for shallow foundations

[...]... the v e r t i c a l stress distribution σ ζ in depth Curve 1 denotes the stresses on the v e r t i c a l axis passing through point A(x=0) and D curve 2 on the axis passing through point In Fig 1.4 are shown coefficients I C(x=-2=b) z for v e r t i c a l stresses σ on the axis ζ x=B and x=2B, and in Fig 1.5 for x=3B and x=4B For i l l u s t r a t i o n , i n Fig 1.6 is shown the e f f e c t of superposition... with dt = 0.01 and for ι m=10 The second integral has also been solved numerically, Introducing 1? and a f t e r d i f f e r e n t i a t i o n one gets: dt = - — ds s 2 For t=m one obtains s= and for t=°° the value s is zero, m W have now: e 00 / m 1-cos ( f t - ^ - c o s ^ d t t t x 3 (1.34) H ο l-cos(sTf) m -/ —ι — cos(-^-)ds = / 1-cos(^-)s cos(-*-)ds 1 -4 s ο m * S H S H (1.35) S H For m=10 one gets:... E(b-b') L V ( - ) i:l " b w where I,, and I ' w w (1.41) w are dimensionless coefficients which correspond to the r a t i o τ b K and -^-.respectively The values of the coefficients I w are given in Tables 1.8- -1.11 1.3 FLEXIBLE STRIP FOUNDATION; HORIZONTAL UNIFORM LOAD The problem of determination of horizontal uniform load p^ been studied by Kolosov (1935) stresses and displacements, produced by a over... " | χ 2 b x Sili^l 2b ln I2b-xI 2 b-ττΕ b(b+x) (1.22) Consequently, the displacements of points 0 ( 0 , 0 ) , B(b,0) and c(2b,0) are given by: W &ί±νϊ1 (0,0) = (1.23) Ε2π w(b,0) • P B ( 1 V ) ( 1 + I n 2 ) Επ (1.24) w(2b,0) (1.25) =ΜίΗί!ΐ 2TTE The dimensionless coefficients I , z σ , σ ζ χ and τ I and Ι χ Ζ for the componental stresses are given in Tables 1,2-1.4 χ ζ TABLE 1.2 Coefficients I ; f l... integration of Flamant's expressions for componental stresses, Giroud has calculated the very simple form: coefficients for stresses, which can now be expressed in a 12 o z « τ = Ρ I Ν = where: (1.26) z Ρ ΐ " ( 1 ρ = maximum normal stress on the loaded surface, · 2 8 ) b = half-width of the loaded s t r i p , I = dimensionless c o e f f i c i e n t s The expressions for these coefficients are: I =- [c(arctg... 1.5 Vertical stresses σ on Fig 1.6 Superposition of σ ζ outside the loaded area; f l e x i b l e s t r i p foundati­ stresses, (a) Single s t r i p , (b) Superposition 2 where: ρ = the uniform v e r t i c a l load; Β = the width of the s t r i p ; Ε = the modu­ lus of deformation of s o i l ; I w = the dimensionless coefficient for the s e t t l e ­ ment In Fig 1.8 are shown 1^ curves for settlement... c a l load over f l e x i b l e 08 0.6 Q5B 20 22 r w Fig 1.10 Coefficients I w for points inside and outside the loaded area 9 1.2 FLEXIBLE STRIP FOUNDATION; VERTICAL TRIANGULAR O TRAPEZOIDAL LOAD R Polshin (1933) and Gray (1935) have studied the stresses produced by a triangular vertical load over a flexible and strip displacements foundation of width B = 2 b , as shown in F i g 1 1 1 Fig 1.11... has been solved numerically with ds=0.001 The dimensionless coefficients I t l w for the settlement calculation have been c a l - culated for several values of the r a t i o H/b, compressible layer and b is the half-width of where Η is the thickness of the the s t r i p for the settlement is given in the following simple form: Now, the expression 15 where ρ = maximum normal contact stress; Ε = Young's... stress; Ε = Young's modulus of s o i l ; Coefficients I w I w b = half-width of the loaded s t r i p ; = dimensionless c o e f f i c i e n t for the Poisson's r a t i o μ=0; 0.30 and 0.50 have been calcu­ lated by Giroud and Rabatel (1971), and f o r y=0.20; 0.40 and 0.48 by Milovic In Tables 1.8-1.11 are given the values of coefficients I TABLE 1.6 Coefficients Ι ; symmetrical triangular load χ ξ 0... B Vertical stresses σ ζ at several depths ζ are shown in Fig 1.7 Vertical displacement of the surface, produced by a vertical concentrated force P, is given by: w(x,0) = - 2 ( ~ ) P1n[x] + C πΕ 1 (1.16) μ 2 Replacing |x| by |χ-ξ| and dP by pdξ one obtains a f t e r integration: ,0) = - M I V ) / + w ( x πΕ b -b l n M d C = 2POV) πΕ 2 b + m ( 1 |x+b| X + D The simple form of the expression for the settlement