# 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.

 Arch Pipe Modified to Intersect t Start t Stop t Start t Stop t Start t Stop t Start t Stop Equivalent T Rise Span A B C R1 R2 R3 R1 R1 R2 R2 R3-Right R3-Right R3-Left R3-Left 15 2.25 11 18 0.375 4.6875 4.969 22.875 10.625 4.05 4.4224009902782 5.00237697049118 0.73691563623242 2.40467701735737 4.84710540020892 0.656783819057765 2.48480883453203 4.57767256056046 18 2.5 13.5 22 -0.25 6 5.750 27.500 13.75 5.26 4.43469082062183 4.99008714014755 0.842472141915383 2.29912051167441 4.88801793696653 0.786692662547907 2.35489999104189 4.53676002380285 24 3 18 28.5 3.4375 5.90625 9.656 40.688 14.5625 4.596 4.43582681464382 4.98895114612556 0.254938974940565 2.88665367864923 4.93223297380469 0.235608388687667 2.90598426490213 4.49254498696469 30 3.5 22.5 36.25 3.75 7.6875 12.09375 51 18.75 6.032 4.4375618019891 4.98721615878028 0.320250506110384 2.82134214747941 4.96318778839042 0.297682839570724 2.84390981401907 4.46159017237896 36 4 26.625 43.375 4.125 8.5625 15.500 62.000 22.5 6.38 4.42147018781041 5.00330777295897 0.288573745784739 2.85301890780505 4.9109576757621 0.244470730609619 2.89712192298017 4.51382028500728 42 4.5 31.3125 51.125 5.0625 10.0625 18.000 73.000 26.25 7.57 4.42394331824699 5.00083464252239 0.277845586287176 2.86374706730262 4.89549574471141 0.247022456852148 2.89457019673765 4.52928221605797 48 5 36 58.5 6 11.59375 20.500 84.000 30 8.76 4.43063831550652 4.99413964526286 0.28251083364675 2.85908181994304 4.93219899835549 0.211116250910211 2.93047640267958 4.49257896241389 54 5.5 40 65 6.625 13 22.688 92.500 33.375 9.83 4.43123740117831 4.99354055959107 0.296907380134533 2.84468527345526 4.96060482783192 0.195835945110772 2.94575670847902 4.46417313293746 60 6 45 73 7.5 14.6875 25.281 105.000 37.5 11.21875 4.43682152569334 4.98795643507604 0.283496209667075 2.85809644392272 4.96078223960209 0.255254173429811 2.88633848015998 4.46399572116729 72 7 54 88 9 17 31.438 126.000 45 12.5625 4.4284803459349 4.99629761483448 0.254942674755307 2.88664997883449 4.96197178941903 0.22855568863376 2.91303696495603 4.46280617135035

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.