The diagonal, not the width

A package traveling through a curve occupies more belt than the same package on a straight run. Watch its corners: on the outside of the turn they swing outward, on the inside they swing inward, and the box sweeps a wider path than its own width. It's the diagonal of the package, not the width, that decides how much belt it needs to keep every corner inside the footprint through the turn. Get that one idea and the rest of curve design follows from it.

The Curve Formula turns that geometry into a number. Give it the inside radius of the curve and the width of the largest package in the mix, and it returns the minimum belt width that keeps that package's corners inside the belt all the way around. You run it on the Product Spec Calc, and the Calc Logic Guide is the authority for the arithmetic behind it. Here in solutioning you run the requirement to size the system; the locked, final-engineering precision on that between-frame width is the gap-and-capacity work, and that's Lesson 25. What you need now is the requirement and what it means.

What it means is that the curve output is almost always wider than the straight sections need. That isn't a preference, it's geometry, and it flips the usual order of design. The curve becomes the specification for the system, and the straight runs follow the curve, not the other way around. So you run the curve requirement first, before you commit any belt width. A curve whose width got guessed at is a curve that produces field problems, and by then the whole run is drawn around the guess.

Skew, the companion geometry

Curves have a cousin worth naming here: the skew. A skewed-roller section sets its rollers at an angle so product drifts against one rail and lines up across the belt, which is how you square cartons up before a scan or a merge. It's sized from the same envelope you just used for the curve, the widest and the longest package deciding how much skewed length it takes to align the narrow one without letting the long one hang off the end. The Calc Logic Guide carries that skew length; run it to confirm the section fits, and hold the tight number for Lesson 25.

The guardrail taper: rigid only

Now the trade. A wide curve can drive a wide run for a long way downstream, and wide conveyor costs more per foot than narrow. Sometimes you can claw that back with a guardrail taper: after the product clears the curve, guardrails funnel it from the wide belt down to a narrower one. It works, but only when every product in the mix is rigid. A rigid case rides the taper and never notices. A poly bag, a padded mailer, anything non-rigid catches at the transition and bunches, and you've traded a wider belt for a jam every shift. If you use a taper, it's a documented aesthetic and product-handling compromise, and you confirm the customer accepts it before it goes in the design. The priced version of that trade, wide belt against tapered belt in dollars per foot, is Lesson 29.

The static angle, and the two forces it can't see

The Box Tumbling calculation gives you the static tumble angle: the angle at which a stationary package on a stationary incline tips over with nothing but gravity acting on it. A tall box on a narrow base tips at a shallow angle; a long, low box holds on much steeper. The Calc Logic Guide is the authority for that angle, and, like the curve, the locked spec is Lesson 25. Design to the worst-case tumble angle in the mix with margin under it, and you've cleared the static check. You haven't cleared the field.

Two forces the static number never included. The first is inertial energy. When the belt starts or stops, the acceleration adds energy the static picture left out, so a package that's dead stable at steady speed can tip at the start or at an abrupt stop. The second is load shift. A carton with contents that can move isn't a uniform block; if the load slides to one end, the center of mass goes with it, and a package that passed the static check with a centered load can tip with a shifted one.

Direction of travel decides where that risk lands. On an incline, the dangerous moment is startup: inertial energy and any load shift both push toward the trailing edge, so the box wants to tip backward. On a decline, the dangerous moment is the stop: both push toward the leading edge, so the box wants to tip forward. The design response is the same on both, margin below the static limit plus controlled ramp rates, and you always check the worst-case load distribution, not the average.

The thirds method: validate it where you can see it

Here's a validation you can run in CAD and in front of a customer. Draw the carton at the incline or decline angle. Divide its length into three equal segments with two lines, a leading third, a middle third, a trailing third. Then drop a line straight down from the center of mass to the incline surface. Land it in the middle third and the package is generally stable; land it in the leading third and it tips forward; land it in the trailing third and it tips backward. For a 12-inch carton the segments are 4 inches each, and the center-of-mass line has to fall inside the middle 4 inches.

The catch is the load. For an evenly loaded carton the center of mass sits at the geometric center. For an unevenly loaded one it shifts toward the heavier end, so you draw the worst case, not the centered case. The thirds method complements the Box Tumbling calculation, it doesn't replace it: when the two agree, confidence is high; when they disagree, the design needs more scrutiny. Its real power is the conversation. A customer who watches the center-of-mass line fall into the leading third understands instantly why the angle has to come down, faster and more convincingly than any tumble-angle number ever gets there.

Gravity pitch, spirals, lifts, and mezzanines

Gravity pitch is the slope of a non-powered roller conveyor, the minimum slope that lets product travel under its own weight. It isn't a fixed number. A light empty carton needs more pitch than a heavy full one, because friction weighs more heavily against the little bit of gravity a light box has to work with. Gravity roller conveyors are typically pitched 2 to 8 inches of fall per 10-foot section, depending on the unit load, the roller style and spacing, and the lubrication, with a minimum of three rollers under the product at all times. Confirm the pitch against the actual product mix, not a generic guideline: too little and product stalls partway down, too much and it runs away at the end of the run.

Above that sits the elevation-change toolkit, and at this stage you meet it as a set of categories with the package question attached to each. A spiral changes elevation in a small footprint. A vertical lift moves product straight up or down between levels. Mezzanine integration is where a powered decline brings product off an upper deck down to the ground-floor system. Choosing a spiral or a vertical lift against the alternatives is specialty elevation-equipment selection, and that's Lesson 17. The structural design of the mezzanine deck itself sits with the building, not the conveyance, so it's out of scope here. What's in scope is the package: what every one of these transitions asks it to survive.