ARCH PIPE LAYOUT

In order to layout arch pipe in three dimensions you will need to understand how to find the circle intersection points.  The chart below t Start and t Stop for each radius of the arch pipe were calculated using the intersection point formula.  All pipe dimensions are from ASTM C506 for Reinforced Concrete Arch Pipe.

b= Manhole Radius

ArchPipe-ASTMC506

Arch PipeModified to Intersectt Startt Stopt Startt Stopt Startt Stopt Startt Stop
EquivalentTRiseSpanABCR1R2R3R1R1R2R2R3-RightR3-RightR3-LeftR3-Left
152.2511180.3754.68754.96922.87510.6254.054.42240099027825.002376970491180.736915636232422.404677017357374.847105400208920.6567838190577652.484808834532034.57767256056046
182.513.522-0.2565.75027.50013.755.264.434690820621834.990087140147550.8424721419153832.299120511674414.888017936966530.7866926625479072.354899991041894.53676002380285
2431828.53.43755.906259.65640.68814.56254.5964.435826814643824.988951146125560.2549389749405652.886653678649234.932232973804690.2356083886876672.905984264902134.49254498696469
303.522.536.253.757.687512.093755118.756.0324.43756180198914.987216158780280.3202505061103842.821342147479414.963187788390420.2976828395707242.843909814019074.46159017237896
36426.62543.3754.1258.562515.50062.00022.56.384.421470187810415.003307772958970.2885737457847392.853018907805054.91095767576210.2444707306096192.897121922980174.51382028500728
424.531.312551.1255.062510.062518.00073.00026.257.574.423943318246995.000834642522390.2778455862871762.863747067302624.895495744711410.2470224568521482.894570196737654.52928221605797
4853658.5611.5937520.50084.000308.764.430638315506524.994139645262860.282510833646752.859081819943044.932198998355490.2111162509102112.930476402679584.49257896241389
545.540656.6251322.68892.50033.3759.834.431237401178314.993540559591070.2969073801345332.844685273455264.960604827831920.1958359451107722.945756708479024.46417313293746
60645737.514.687525.281105.00037.511.218754.436821525693344.987956435076040.2834962096670752.858096443922724.960782239602090.2552541734298112.886338480159984.46399572116729
727548891731.438126.0004512.56254.42848034593494.996297614834480.2549426747553072.886649978834494.961971789419030.228555688633762.913036964956034.46280617135035

 

We have to think of the unit circle when drawing arch pipe and the t Start value of R2 is never negative so we will start by drawing R2 and work counter clockwise.  We draw this using the same formula as a regular pipe or circle but instead of a t range from 0 to 2*pi it will be from t Start to t Stop.  For this example we will use 24″ Equivalent Arch Pipe.

\[.2549<=t<=2.8867\]

\[\begin{pmatrix} ((R2 * \cos(t) + Offset)*(cos(360-Angle))) -(\sqrt{b^2-[(R2 * cos(t))+Offset]^2} *sin(360-Angle))\\(R2 * \cos(t) + Offset )*sin(360-Angle)+(\sqrt{b^2-[(R2* cos(t))+Offset]^2}*cos(360-Angle)) \\R2 * \sin(t)+Elevation+ A \end{pmatrix}\]

Working in the counter clockwise direction the next radius we will plot is R3-Left.  This radius is not centered on the y axis so we need to treat the x dimension like an offset hole.

\[2.90598<=t<=4.49254\]

\[\begin{pmatrix} ((R3 * \cos(t) + Offset-C)*(cos(360-Angle))) -(\sqrt{b^2-[(R3 * cos(t))+Offset-C]^2} *sin(360-Angle))\\(R3 * \cos(t) + Offset-C )*sin(360-Angle)+(\sqrt{b^2-[(R3* cos(t))+Offset-C]^2}*cos(360-Angle)) \\R3 * \sin(t)+Elevation+ B \end{pmatrix}\]

Working in the counter clockwise direction the next radius we will plot is R1.  This radius is centered and will be similar to the calculations for R2.

\[4.4358<=t<=4.9889\]

\[\begin{pmatrix} ((R1 * \cos(t) + Offset)*(cos(360-Angle))) -(\sqrt{b^2-[(R1 * cos(t))+Offset]^2} *sin(360-Angle))\\(R1 * \cos(t) + Offset )*sin(360-Angle)+(\sqrt{b^2-[(R1* cos(t))+Offset]^2}*cos(360-Angle)) \\R1 * \sin(t)+Elevation+ R1 \end{pmatrix}\]

The final radius to plot is R3 Right.

\[4.9322<=t<=2*\pi+0.29768\]

\[\begin{pmatrix} ((R3 * \cos(t) + Offset+C)*(cos(360-Angle))) -(\sqrt{b^2-[(R3 * cos(t))+Offset+C]^2} *sin(360-Angle))\\(R3 * \cos(t) + Offset+C )*sin(360-Angle)+(\sqrt{b^2-[(R3* cos(t))+Offset+C]^2}*cos(360-Angle)) \\R3 * \sin(t)+Elevation+ B \end{pmatrix}\]

Next you can follow the instructions for flat surface projection for each radius to get a layout as shown below.

Arch Pipe Hole Layout