Formula to develop unequal lateral tee

To develop an unequal lateral tee, you typically need to create patterns for two parts: the branch pipe (which forms the "saddle" or "crotch" piece) and the hole in the main pipe. The most complex part, and where a specific formula is used, is the development of the branch pipe pattern.

Here is the formula for calculating the lengths of the generators (ordinates) for the branch pipe pattern:

$$L(\phi) = H_{ref} + \frac{\sqrt{R_M^2 - (R_B \sin(\phi))^2}}{\sin(\alpha)} - R_B \cos(\phi) \cot(\alpha)$$

Where:

• $L(\phi)$ is the length of a generator on the developed branch pattern at a specific angle $\phi$.
• $H_{ref}$ is a reference height. This constant is chosen to ensure all calculated lengths $L(\phi)$ are positive and to define the overall height of the branch pattern. It can be set to the length of the shortest generator or a suitable constant.
• $R_M$ is the radius of the main pipe.
• $R_B$ is the radius of the branch pipe.
• $\phi$ is the angle around the circumference of the branch pipe, measured from the top center line. It typically ranges from $0^\circ$ to $360^\circ$ (or 0 to $2\pi$ radians). $\phi=0^\circ$ corresponds to the top of the branch, and $\phi=180^\circ$ to the bottom.
• $\alpha$ is the angle of the branch pipe's axis with respect to the main pipe's axis. For a perpendicular branch, $\alpha = 90^\circ$.

Steps to use the formula for the branch pattern:

1. Define parameters: Determine the radii of the main pipe ($R_M$) and branch pipe ($R_B$), and the angle of the lateral branch ($\alpha$).
2. Choose $H_{ref}$: Select a suitable reference height. A common approach is to calculate the minimum value of the expression $\frac{\sqrt{R_M^2 - (R_B \sin(\phi))^2}}{\sin(\alpha)} - R_B \cos(\phi) \cot(\alpha)$ and set $H_{ref}$ to its absolute value (or slightly more) to ensure all $L(\phi)$ are positive.
3. Calculate generator lengths: Calculate $L(\phi)$ for various values of $\phi$ (e.g., every $15^\circ$ or $30^\circ$) around the branch circumference.
4. Plot the pattern: On a rectangular grid, plot the calculated lengths $L(\phi)$ against the corresponding arc lengths along the branch circumference ($R_B \times \phi$). Connect the points to form the cutting line for the branch pattern.

For the main pipe hole:
The shape of the hole in the main pipe is typically developed by plotting the intersection points on the unrolled surface of the main pipe. This is usually done graphically or by calculating coordinates for various points along the intersection curve.