Ce3201 l3-06

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PE Review Civil

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<ul><li> 1. Vertical CurvesDavid Levinson</li></ul> <p> 2. Vertical Curves Vertical curves provide a " In highway design, the grades ofmeans to smoothly shift from the disjointed segments ofone tangent grade to another.roadway are normally knownThey are usually parabolic in before any vertical curveshape. They are classied intocrest vertical curves and sagcalculations are initiated.vertical curves." In addition, the design speed of the roadway, the stopping sight In highway design, mostdistance, and the decision sightvertical curves are equal- distance are also well established.tangent curves, which means" The rst step in the design of athat the horizontal distancefrom the center of the curve vertical curve is the calculationto the end of the curve is of the curve length, which is theidentical in both directions. length of the curve as it would appear when projected on the x- (Unequal-tangent verticalaxis.curves, which are simply equal-tangent curves that have beenattached to one another, areseldom used.) 3. Stopping Sight Distance and Vertical Curvature Because the stopping sight " Lets assume that you havedistance (S or ds) should alwaysalready calculated the appropriatebe adequate, the length of thelength (L) for your curve.curve depends on the stopping" The rst step in developing thesight distance.prole for your curve is to ndthe center of your curve.The Occasionally, as with any other location of the center-point issection of a highway, the where the disjointed segments ofdecision sight distance is athe roadway would havemore appropriate sightintersected, had they beendistance.In these instances, allowed to do so.In other words,the decision sight distance draw lines tangent to yourgoverns the length of the roadway segments and see wherevertical curve.those lines intersect.Thisintersection is normally called the The curve length calculations vertical point of intersectionare slightly different for sag(PVI).and crest vertical curves, andare covered separately. 4. PVC and PVT The vertical point of curvature (PVC ) and the vertical point of tangency (PVT) are located a horizontal distance of L/2 from the PVI.The PVC is generally designated as the origin for the curve and is located on the approaching roadway segment. The PVT serves as the end of the vertical curve and is located at the point where the vertical curve connects with the departing roadway segment. In other words, the PVC and PVT are the points along the roadway where the vertical curve begins and ends. One you have located the PVI, PVC, and PVT, you are ready to develop the shape of your curve.The equation that calculates the elevation at every point along an equal-tangent parabolic vertical curve is shown below. PVI B PVTPVC L/2L/2 L 5. Equation of Elevation y = ax 2 + bx + cWhere:y = Elevation of the curve at a distance x metersfrom the PVC (m)c = elevation of PVC (m)b=G1a=(G2-G1)/2LL = Length of the curve (m)x = Horizontal distance from the PVC (m) (Varied from0 to L for graphing.) 6. Example 1A 150 m equal tangent crest-vertical curve hasthe PVC at station 8+500 and elevation of 250m.The initial grade is +2.6% and the nal grade is-0.5%.Determine the elevation and stationing of thePVI, PVT and the highest point of the curve. 7. Solution 1Because the curve is equal tangent, " Substitutingthe PVI will be 75m from the PVC,and the PVT will be 150 m from thePVC. 0=2(-0.0001)x+0.026The stationing is therefore at:PVI: 8+575 &amp; Stationing of max = 8+500 + 0+125.8 =PVT: 8+650 8+625.8The grade = +2.6% = .026 m/m " Elevation =y=ax2+bx+cPVI elevation is thus 250 + 0.026*75= 251.95m" =(-0.0001)*125.82 +PVT elevation is 251.95 - 0.005*75 =0.026*125.8+250=-1.583+3.27+250=251251.575m.69Maximum occurs when rst derivative=0 " 251.69 &lt; 251.95 (PVI) checkdy/dx = 2ax+b = 0b= G1= 0.026a=(G2-G1)/2L =(-0.005-0.026)/(2*150)=-0.0001033x=125.8m 8. Suppose G1=0Suppose G1= 0Maximum occurs when rst derivative = 0dy/dx = 2ax+b = 0b= G1= 0a=(G2-G1)/2L =(-0.005)/(2*150)=-0.0000167x=0mSo maximum is at start of curve (x=0)Suppose G2 =0b=G1=0.026a=(G2-G1)/2L=-0.026/300=0.0000867dy/dx = 2ax+b = 0=-0.000173x+0.026=0x=150 mSo maximum is at end of curve (x=150) 9. Crest Vertical Curves: Design Crest vertical curves connect inclined sections of roadway, forming a crest. Objective: nd an appropriate length for the curve that will accommodate the correct stopping sight distance.Sight Distance &gt;h1 Curve Lengthh2 LSight Distance </p>