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<strong>BUITEMS</strong> Quality & Excellence in Education Rational Design of Retaining Walls 2 Ø j + a ø l = Œtg 45 - + tga cos a 2 œ º Ł ł ß The range of the angle a lies in the limit: - j < a £ 0 , (1.6) The limits set by the physical meaning of the problem. Next, we introduce the notation: j y = 45 o - 2 Ø a ø l = Œtg y - + tga 2 œ º Ł ł ß , 2 cosa Using the known trigonometric relationship: cos 1 1+ tg a – 2 = , Для дальнейших выкладок For further calculations need to choose the sign before the radical sign. So much so cos α is an even function we take: cos 1 1+ tg a 2 = . a a Then: Ø a ø Œ tgy - tg 2 œ 1 l = Œ Ł ł + tga œ Œ a œ 1+ tg Œ 1+ gy tg 2 œ º Ł ł ß Use the trigonometric relation: tg 2 Figure 1.2. Search of geometry of wall. 2 a a 1- cos a = – 2 1 + cos a (1.6) we see that the angle α - is negative, then the function a 1- cosa tg = - 2 1 + cos a , where is also necessary to choose the appropriate sign. From α tg 2 Ø 1 Œ 1- Œ 1+ tg 2 a tgy + 2 Œ Ø 1 - cosa ø 1 tgy Œ 1+ Œ + œ 2 a a l 1 cos 1 Œ 1+ tg = Œ + - tga œ = 2 1 cosa 1 tg a Œ Œ - œ + 1 gy Œ 1 Œ - 1- º 1+ cosa œ ß Œ 1+ tg 2 a Œ1- gy Œ 1 1+ Œ 1 tg 2 º + a From the geometric meaning of the first derivative, it follows that is also negative. In this connection: ø 2 œ œ œ œ œ - tga œ œ œ œ œ œ ß 1 1+ tg 2 a 96
<strong>BUITEMS</strong> Quality & Excellence in Education Rational Design of Retaining Walls dy y¢ = = tga , (1.7) dz where у=у(z) – the function, which describe the geometry of the back surface of the retaining wall. In view of (1.7) we have: 2 Ø 1 ø Œ 1- 2 œ Œ 1+ y¢ tgy + œ Œ 1 œ Œ 1+ œ 2 Œ 1+ y¢ œ 1 l = Œ - tga œ ; 2 1 1+ y¢ Œ 1- œ 2 Œ 1+ y¢ œ Œ1- gy œ Œ 1 1+ œ Œ 2 1 y¢ œ º + ß 1 Substituting of variable.: f = . We express out y¢ : 2 1+ y¢ f 2 = 1 2 1 + y ; 1 1 2 - f 2 ¢ 2 = + y¢ ; y¢ 2 = 1 1- f 2 ; y¢ = tga = – 2 2 f f f 2 Ø 1- f ø Œ tgy + œ Œ 1+ f l = - tga œ f ; Œ 1- f œ Œ1- gy œ Œº 1+ f œß s From (1.5): l = g (z + z ) , where: z=z o +z 1 – current depth (Figure 1.2); finally: l = s ( z) o 1 z g as a known function of the depth σ=σ(z), is permissible to write the following equation: 2 Ø 1- f ø Œ tgy + 2 œ Œ 1+ f 1- f œ s ( z) - f = ; (1.8) 2 Œ 1- f f œ z g Œ1- gy œ Œº 1+ f œß s ( z) We make one more substitution of variable, F( z) = F 2 (z) = s ( z) z g z g , then: 2 Ø 1- f ø Œ tgy + 2 œ Œ 1+ f 1- f 2 - œ f = F ( z) . Further, let that: k 2 = 1- f 2 Œ 1- f f œ 1+ f Œ1- gy œ Œº 1+ f œß We express f = f (k) : 1- f = (1 + f ) k 2 ; 1- f = k 2 + f k 2 ; f k 2 + f =1- k 2 ; 2 1- k f ( k 1) 2 + = 1- k 2 ; f = 2 1+ k , Given that the values of γ, z o , and φ are known, and the magnitude of the intensity of normal pressure can be represented 97
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