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Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960 965
o I=60 B’ B 953 960 965 977 R’ I Find the radius, R prime, in feet. [pause] In this problem there are two curves, --- O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/1/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. I=60. B’ B R’ I. Find the radius, R prime, in feet. [pause] In this problem there are two curves, --- O’ I. R. O.")
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Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960 965
o I=60 B’ B 953 960 965 977 R’ I AB and AB’. Both curves share the same internal angle, I, which is --- O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/2/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. I=60. B’ B R’ I. AB and AB’. Both curves share the same internal angle, I, which is --- O’ I. R. O.")
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Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960 965
o I=60 B’ B 953 960 965 977 R’ I 60 degrees. [pause] Chord AB is --- O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/3/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. I=60. B’ B R’ I. 60 degrees. [pause] Chord AB is --- O’ I. R. O.")
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Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960 965
o I=60 B’ B 953 960 965 977 R’ I 1,000 feet long, and the distance between points --- O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/4/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. I=60. B’ B R’ I. 1,000 feet long, and the distance between points --- O’ I. R. O.")
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Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960 965
o I=60 B’ B 953 960 965 977 R’ I V prime and V is 20 feet. [pause] O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/5/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. I=60. B’ B R’ I. V prime and V is 20 feet. [pause] O’ I. R. O.")
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? Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B 953 960
965 977 o R’ I=60 To find R prime, the radius of the smaller curve, --- O’ I R O
![Find: R’ [ft] A V’ V CAB=1,000 [ft] LV’V=20 [ft] I=60 B’ B](http://slideplayer.com./slide/15442558/93/images/6/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+CAB%3D1%2C000+%5Bft%5D+LV%E2%80%99V%3D20+%5Bft%5D+I%3D60+B%E2%80%99+B.jpg "o. R’ I=60. To find R prime, the radius of the smaller curve, --- O’ I. R. O.")
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? Find: R’ [ft] A V’ V T’ R’= tan(0.5*I) B’ B R’ I=60 O’ I R
We’ll use the equation for radius, which involves the ---- O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
![Find: R’ [ft] A V’ V T’ R’= tan(0.5*I) B’ B R’ I=60 O’ I R](http://slideplayer.com./slide/15442558/93/images/7/Find%3A+R%E2%80%99+%5Bft%5D+A+V%E2%80%99+V+T%E2%80%99+R%E2%80%99%3D+tan%280.5%2AI%29+B%E2%80%99+B+R%E2%80%99+I%3D60+O%E2%80%99+I+R.jpg "We’ll use the equation for radius, which involves the ---- O’ I. R. LV’V=20 [ft] O. CAB=1,000 [ft]")
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? Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior
angle ? B’ B o R’ I=60 tangent distance, T prime, and the interior angle, I. The interior angle I equals --- O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
![Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior](http://slideplayer.com./slide/15442558/93/images/8/Find%3A+R%E2%80%99+%5Bft%5D+tangent+distance+A+V%E2%80%99+V+T%E2%80%99+R%E2%80%99%3D+tan%280.5%2AI%29+interior.jpg "angle. B’ B. o. R’ I=60. tangent distance, T prime, and the interior angle, I. The interior angle I equals --- O’ I. R. LV’V=20 [ft] O. CAB=1,000 [ft]")
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? Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior
angle ? B’ B o R’ I=60 60 degrees, and the tangent distance, T prime, equals --- O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
![Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior](http://slideplayer.com./slide/15442558/93/images/9/Find%3A+R%E2%80%99+%5Bft%5D+tangent+distance+A+V%E2%80%99+V+T%E2%80%99+R%E2%80%99%3D+tan%280.5%2AI%29+interior.jpg "angle. B’ B. o. R’ I= degrees, and the tangent distance, T prime, equals --- O’ I. R. LV’V=20 [ft] O. CAB=1,000 [ft]")
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? Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior
angle ? B’ B T’=LB’V’ o R’ I=60 the distance between B prime and V prime. [pause] Now the equation for R prime simplifies to ---- O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
![Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior](http://slideplayer.com./slide/15442558/93/images/10/Find%3A+R%E2%80%99+%5Bft%5D+tangent+distance+A+V%E2%80%99+V+T%E2%80%99+R%E2%80%99%3D+tan%280.5%2AI%29+interior.jpg "angle. B’ B. T’=LB’V’ o. R’ I=60. the distance between B prime and V prime. [pause] Now the equation for R prime simplifies to ---- O’ I. R. LV’V=20 [ft] O. CAB=1,000 [ft]")
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? Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior
angle ? B’ B T’=LB’V’ o R’ I=60 R’=1.732*T’ 1.732, times, T prime. If we define the variable T to equal --- O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
![Find: R’ [ft] tangent distance A V’ V T’ R’= tan(0.5*I) interior](http://slideplayer.com./slide/15442558/93/images/11/Find%3A+R%E2%80%99+%5Bft%5D+tangent+distance+A+V%E2%80%99+V+T%E2%80%99+R%E2%80%99%3D+tan%280.5%2AI%29+interior.jpg "angle. B’ B. T’=LB’V’ o. R’ I=60. R’=1.732*T’ 1.732, times, T prime. If we define the variable T to equal --- O’ I. R. LV’V=20 [ft] O. CAB=1,000 [ft]")
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? tangent T distance T’ A V’ V T’ R’= tan(0.5*I) interior angle B’ B
T’=LB’V’ o R’ I=60 R’=1.732*T’ The length from A to V, then T prime equals, --- T=LAV O’ I R LV’V=20 [ft] O CAB=1,000 [ft]
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? tangent T distance T’ A V’ V T’ R’= tan(0.5*I) interior angle B’ B
T’=LB’V’ o R’ I=60 R’=1.732*T’ T minus the length between V prime and V. [pause] The problem states the length between V and --- T=LAV O’ I R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? tangent T distance T’ A V’ V T’ R’= tan(0.5*I) interior angle B’ B
T’=LB’V’ o R’ I=60 R’=1.732*T’ V prime equals 20 feet. [pause] T=LAV O’ I R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V B’ B R’ I=60 R’=1.732*T’ O’ I R
The length of tangent distance T can be calculated by knowing the radius, ---- R’=1.732*T’ ? O’ I R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V interior radius angle B’ B R’ I=60
R, and the internal angle, I. As before, the interior angle, I, equals ---- R’=1.732*T’ ? O’ I R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V interior radius angle B’ B R’ I=60
60 degrees, and the radius of the larger curve can be calculated --- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V interior radius angle C R=
2*sin(0.5*I) ? B’ B o R’ I=60 in terms of the chord length, C, and the ---- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V chord interior length angle C R=
2*sin(0.5*I) ? B’ B o R’ I=60 interior angle, I. For the larger curve, the chord length AB equals --- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V chord interior length angle C R=
2*sin(0.5*I) ? B’ B o R’ I=60 the chord distance from A to B, or 1,000 feet, and the interior angle, is again, --- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V chord interior length angle C R=
2*sin(0.5*I) ? B’ B o R’ I=60 60 degrees. This makes the radius of the larger curve equal to ---- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V chord interior length angle C R=
2*sin(0.5*I) ? B’ B R=1,000 [ft] o R’ I=60 1,000 feet. [pause] Plugging in the values for the radius, and for the ----- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V chord interior length angle C R=
2*sin(0.5*I) ? B’ B R=1,000 [ft] o R’ I=60 interior angle, into our equation for T, we find the tangent distance equals ---- R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
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? ? T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’
R=1,000 [ft] o R’ I=60 577.4 feet. Next, we’ll plug in our value for T, to find T prime. R’=1.732*T’ ? o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
![T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’](http://slideplayer.com./slide/15442558/93/images/24/T+T%E2%80%99+T%3DR+%2A+tan+%280.5+%2A+I%29+A+V%E2%80%99+V+T%3D577.4+%5Bft%5D+C+R%3D+2%2Asin%280.5%2AI%29+B%E2%80%99.jpg "R=1,000 [ft] o. R’ I= feet. Next, we’ll plug in our value for T, to find T prime. R’=1.732*T’ o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft]")
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? T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B
R=1,000 [ft] o R’ I=60 Where feet minus 20 feet equals, --- R’=1.732*T’ o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft]
![T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B](http://slideplayer.com./slide/15442558/93/images/25/T+T%E2%80%99+T%3DR+%2A+tan+%280.5+%2A+I%29+A+V%E2%80%99+V+T%3D577.4+%5Bft%5D+C+R%3D+2%2Asin%280.5%2AI%29+B%E2%80%99+B.jpg "R=1,000 [ft] o. R’ I=60. Where feet minus 20 feet equals, --- R’=1.732*T’ o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft]")
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? T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B
R=1,000 [ft] o R’ I=60 557.4 feet. And finally, the value of R prime, is calculated by multiplying ---- R’=1.732*T’ o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft] T’=557.4 [ft]
![T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B](http://slideplayer.com./slide/15442558/93/images/26/T+T%E2%80%99+T%3DR+%2A+tan+%280.5+%2A+I%29+A+V%E2%80%99+V+T%3D577.4+%5Bft%5D+C+R%3D+2%2Asin%280.5%2AI%29+B%E2%80%99+B.jpg "R=1,000 [ft] o. R’ I= feet. And finally, the value of R prime, is calculated by multiplying ---- R’=1.732*T’ o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft] T’=557.4 [ft]")
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? T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B
R=1,000 [ft] o R’ I=60 1.732 by feet, which equals, --- R’=1.732*T’ o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft] T’=557.4 [ft]
![T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B](http://slideplayer.com./slide/15442558/93/images/27/T+T%E2%80%99+T%3DR+%2A+tan+%280.5+%2A+I%29+A+V%E2%80%99+V+T%3D577.4+%5Bft%5D+C+R%3D+2%2Asin%280.5%2AI%29+B%E2%80%99+B.jpg "R=1,000 [ft] o. R’ I= by feet, which equals, --- R’=1.732*T’ o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft] T’=557.4 [ft]")
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? T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B
R=1,000 [ft] o R’ I=60 965.4 feet. [pause] R’=1.732*T’ R’=965.4 [ft] o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft] T’=557.4 [ft]
![T T’ T=R * tan (0.5 * I) A V’ V T=577.4 [ft] C R= 2*sin(0.5*I) B’ B](http://slideplayer.com./slide/15442558/93/images/28/T+T%E2%80%99+T%3DR+%2A+tan+%280.5+%2A+I%29+A+V%E2%80%99+V+T%3D577.4+%5Bft%5D+C+R%3D+2%2Asin%280.5%2AI%29+B%E2%80%99+B.jpg "R=1,000 [ft] o. R’ I= feet. [pause] R’=1.732*T’ R’=965.4 [ft] o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft] T’=557.4 [ft]")
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? T T’ 953 960 965 977 A V’ V B’ B R’ I=60 R’=1.732*T’ R’=965.4 [ft]
o R’ I=60 When reviewing the possible solutions, --- R’=1.732*T’ R’=965.4 [ft] o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft] T’=557.4 [ft]
![T T’ A V’ V B’ B R’ I=60 R’=1.732*T’ R’=965.4 [ft]](http://slideplayer.com./slide/15442558/93/images/29/T+T%E2%80%99+A+V%E2%80%99+V+B%E2%80%99+B+R%E2%80%99+I%3D60+R%E2%80%99%3D1.732%2AT%E2%80%99+R%E2%80%99%3D965.4+%5Bft%5D.jpg "o. R’ I=60. When reviewing the possible solutions, --- R’=1.732*T’ R’=965.4 [ft] o. O’ I=60. R. LV’V=20 [ft] T’=T-LV’V. O. CAB=1,000 [ft] T’=557.4 [ft]")
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? T T’ 953 960 965 977 A V’ V AnswerC B’ B R’ I=60 R’=1.732*T’
o R’ I=60 the answer is C. R’=1.732*T’ R’=965.4 [ft] o O’ I=60 R LV’V=20 [ft] T’=T-LV’V O CAB=1,000 [ft] T’=557.4 [ft]
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? Index σ’v = Σ γ d γT=100 [lb/ft3] +γclay dclay 1
Find: σ’v at d = 30 feet (1+wc)*γw wc+(1/SG) σ’v = Σ γ d d Sand 10 ft γT=100 [lb/ft3] 100 [lb/ft3] 10 [ft] 20 ft Clay = γsand dsand +γclay dclay A W S V [ft3] W [lb] 40 ft text wc = 37% ? Δh 20 [ft] (5 [cm])2 * π/4
![Index σ’v = Σ γ d γT=100 [lb/ft3] +γclay dclay 1](http://slideplayer.com./slide/15442558/93/images/31/Index+%CF%83%E2%80%99v+%3D+%CE%A3+%CE%B3+d+%CE%B3T%3D100+%5Blb%2Fft3%5D+%2B%CE%B3clay+dclay+1.jpg "Find: σ’v at d = 30 feet. (1+wc)*γw. wc+(1/SG) σ’v = Σ γ d. d. Sand. 10 ft. γT=100 [lb/ft3] 100 [lb/ft3] 10 [ft] 20 ft. Clay. = γsand dsand. +γclay dclay. A. W. S. V [ft3] W [lb] 40 ft. text. wc = 37% Δh. 20 [ft] (5 [cm])2 * π/4.")
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