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
| 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.
