The resolution of suspension is the static principle that splits a hanging weight between two supporting ropes by resolving the load into components along each rope's line of action. A single 1000 N weight hung from two ropes at 30° above horizontal pulls each rope at roughly 1000 N — the tension can be larger than the load itself. The principle lets riggers, theatre flymen, and crane crews predict cable tensions before lifting. You see it every time a sign hangs between two posts or a chandelier swings from twin chains.
Resolution of Suspension Interactive Calculator
Vary suspended load and included sling angle to see the tension, load factor, and force components in each rope.
Equation Used
This calculator uses the symmetric suspension force triangle. For a centered load W carried by two equal ropes with included angle theta, each rope tension is W divided by twice the cosine of half the included angle. As theta opens toward horizontal, the cosine term shrinks and rope tension rises rapidly.
- Two identical ropes share a centered load symmetrically.
- theta is the included angle between the two ropes.
- Static equilibrium with no shock loading or rope stretch effects.
- Rope self-weight and hardware friction are neglected.
The The Resolution of Suspension in Action
A weight hanging from two ropes is the textbook concurrent force system. Three forces meet at one point — gravity pulling straight down, and two rope tensions pulling along their respective rope axes. For the load to hang still, those three vectors must close into a triangle. That is the whole game. Resolve the rope tensions into horizontal and vertical components, balance the verticals against gravity, balance the horizontals against each other, and you have your answer.
The geometry is unforgiving. As the angle between the two ropes opens up — as the ropes become more horizontal — the tension in each rope climbs fast. At 60° between ropes (each 60° from vertical), each rope carries the full weight of the load. At 120° between ropes, each rope carries the load. Open it to 170° and you are looking at tensions of roughly 5.7 times the load. This is why a sling angle factor chart sits in every rigger's pocket. If you assume the rope tension equals half the load just because there are two ropes, you will overload your slings the first time the angle exceeds 60° from vertical.
What goes wrong in practice? The two-rope suspension assumes the load hangs at the geometric centre between the anchors. If the load shifts off-centre, one rope picks up more than its share — sometimes nearly all of it. Stretchy ropes mask the problem because they redistribute, but steel cable does not. A 5% length mismatch between two stiff slings can put 80% of the load on the shorter one. Always assume worst-case load distribution when sizing cable, and never trust that two slings means half tension each.
Key Components
- Suspended Load: The mass hanging from the rigging point. Its weight acts vertically downward through its centre of gravity. If the CG sits off-centre between the two anchors, the tension distribution becomes asymmetric and you must solve each rope independently rather than assume symmetry.
- Two Suspension Ropes or Slings: The pair of tension members carrying the load up to the anchors. Each rope's tension acts along its own axis. Practical slings include 8 mm wire rope, 12 mm polyester round slings, or galvanised chain — the resolution math is identical, only the working load limits differ.
- Anchor Points: The two fixed points where the ropes terminate above. Their horizontal separation and height relative to the load set the rope angles. A spreader beam is often used to control the included angle below 90° and keep tensions sensible.
- Force Triangle: The graphical closure of the three vectors — load weight plus two rope tensions — into a closed triangle. If you draw the triangle to scale with the load weight as the vertical side and the rope angles as the other two sides, the side lengths read directly as rope tensions.
- Included Angle θ: The angle between the two ropes at the load. This is the single parameter that drives tension amplification. At θ = 0° (ropes parallel) tension is W/2 per rope. At θ = 120° each rope carries W. Above 150° tensions climb past 2W and rigging hardware becomes the limiting factor.
Who Uses the The Resolution of Suspension
The resolution of suspension shows up anywhere a load hangs from two or more lines. Riggers use it daily, but so do bridge designers, theatre crews, sign installers, and arborists. Every time you see a banner strung between two lamp posts or a basket lift hanging on twin cables, somebody resolved the suspension to size the lines.
- Theatre Rigging: Counterweighted fly systems at venues like the Royal Opera House use two-line pickups on heavy battens — a 400 kg lighting bar suspended from two purchase lines requires resolved tension calculation per line before the loft block is sized.
- Crane and Hoisting: Two-leg chain slings on Liebherr LTM mobile cranes carry asymmetric loads like generator skids; the rigger calculates the tension in each leg from the included angle before selecting Grade 80 or Grade 100 chain.
- Aerial Performance and Circus: A Cirque du Soleil aerial silk rigging point with two anchor lines splits the performer's dynamic load between two ceiling beams — peak loads during drops can hit 6× static weight, so the resolved tension calculation is the governing design case.
- Suspension Bridges and Footbridges: Cable-stayed pedestrian bridges like the Kurilpa Bridge in Brisbane resolve deck weight into multiple stay cables at varied angles, each tension calculated from the geometry of its anchor and the deck segment it carries.
- Sign and Banner Installation: A 50 kg illuminated sign hung between two storefront awning brackets at a 160° included angle pulls each bracket with roughly 144 N horizontal and 250 N vertical — sufficient to tear out an undersized lag bolt if the installer assumed half-load per side.
- Arborist Rigging: When a tree crew uses a two-point speed line to lower large limbs, the included angle between the rigging points sets the tension in each line. A 200 kg log on a 120° included rigging produces 200 kg of tension in each line — not 100 kg.
The Formula Behind the The Resolution of Suspension
The formula gives the tension in each rope as a function of the load weight and the angle each rope makes with vertical. At small angles (ropes nearly vertical) tension approaches W/2 per rope — the intuitive answer. As the angle opens, the cosine in the denominator shrinks and tension climbs sharply. The sweet spot for rigging design sits between 0° and 45° from vertical, where tension stays under 0.71W per rope. Above 60° from vertical you cross into territory where each rope carries more than the full load — that is where most rigging failures happen because the operator assumed two ropes meant half the load.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| T | Tension in each rope (assuming symmetric load and equal angles) | N | lbf |
| W | Weight of the suspended load | N | lbf |
| θ | Angle each rope makes with the vertical axis | degrees or radians | degrees |
Worked Example: The Resolution of Suspension in a Museum HVAC Unit Lift
A mechanical contractor in Halifax is hoisting a 320 kg rooftop air-handling unit onto a museum mezzanine using a two-leg wire rope sling rigged off a single overhead beam clamp. The unit is 1.8 m wide and the lift point sits 1.2 m above the unit. The contractor needs to verify each leg of the sling — 10 mm 6×19 IWRC wire rope rated 1140 kgf working load — is not overloaded.
Given
- m = 320 kg
- W = 3140 N (320 × 9.81)
- Sling leg geometry = 0.9 m horizontal, 1.2 m vertical per leg m
- θnom = 36.87 (arctan 0.9/1.2) degrees from vertical
Solution
Step 1 — at the nominal geometry, each sling leg makes 36.87° with vertical. Compute the tension per leg:
That is roughly 200 kgf per leg — well under the 1140 kgf WLL of the 10 mm wire rope. The lift is safe at the planned geometry.
Step 2 — at the low end of the practical range, suppose the contractor uses a longer 2.0 m sling that brings each leg to only 24° from vertical:
About 175 kgf per leg. Lower tension, but the sling now hangs further below the beam and the headroom under the museum ceiling may not allow it. This is the classic tradeoff — longer slings are kinder to the rope but eat vertical clearance.
Step 3 — at the high end, suppose the contractor shortens the rig and each leg now sits at 65° from vertical (a flat, wide rig):
That is 379 kgf per leg — still under WLL, but now you have lost half your safety margin. Push the angle to 75° and tension jumps to 6065 N (618 kgf), and at 80° you are at 9039 N (922 kgf) — uncomfortably close to the rope's WLL. This is exactly why riggers fight to keep included angles under 90°.
Result
Each sling leg sees 1963 N of tension at the planned 36. 87° geometry — about 17% of the wire rope's working load limit, comfortable for a museum lift with sensitive load. The full operating window runs from 1718 N at a tall narrow rig (24° per leg) up to 3713 N at a flat wide rig (65° per leg), so the angle choice more than doubles the rope tension for the same payload. If a load cell reads higher than the predicted 1963 N during the lift, the most likely causes are: (1) the centre of gravity of the AHU sitting off-centre — common with HVAC units that have heavy compressors on one side, throwing 70-80% of the load onto the short leg; (2) one sling leg longer than the other by even 25 mm at this geometry, which shifts load onto the shorter leg; or (3) the lift point not directly above the load CG, producing horizontal swing that adds dynamic tension on the leading leg.
When to Use a The Resolution of Suspension and When Not To
The resolution of suspension is one analysis tool among several for predicting rigging loads. The right tool depends on how many lines you have, whether the load is symmetric, and how much accuracy you need.
| Property | Resolution of Suspension (2-rope) | Lami's Theorem | Finite Element Cable Analysis |
|---|---|---|---|
| Accuracy for symmetric 2-rope rig | Exact | Exact | Exact but overkill |
| Handles >2 ropes | No — needs FEA or matrix method | No — limited to 3 forces | Yes — any number |
| Calculation time | Under 1 minute on paper | Under 1 minute on paper | Hours of model setup |
| Software cost | None — pencil and calculator | None | $2,000-$15,000 per seat |
| Handles asymmetric load CG | Yes with separate equations per rope | Yes | Yes natively |
| Practical use case | Daily field rigging, theatre, sign hanging | Statics coursework, 3-force problems | Stay-cable bridges, complex multi-point lifts |
| Field-verifiable | Yes — check angle with inclinometer | Yes | Requires instrumented load cells |
Frequently Asked Questions About The Resolution of Suspension
Asymmetric load distribution is the usual culprit. The resolution formula assumes the centre of gravity sits exactly on the vertical line through the lift point. If the CG is offset by even 10% of the sling spread, the tension shift to the near leg is non-linear and can easily hit 30%.
Drop a plumb line from the hook to the load before lifting and check it lands on the marked CG. For HVAC units, transformers, and skid-mounted gear, the CG is rarely at geometric centre — manufacturer drawings usually mark it.
120° between the two ropes — meaning each rope sits 60° from vertical. At that geometry cos 60° = 0.5, so T = W/(2 × 0.5) = W. Each rope carries the full load weight.
This is the angle riggers memorise as the limit. Most rigging standards including ASME B30.9 require derating slings sharply above this point. If you find yourself rigging above 120° included, add a spreader beam — it is cheaper than a sling failure.
Longer slings reduce angle and therefore tension, but they need vertical clearance you may not have indoors. A spreader beam decouples the sling angle from the load width — the slings drop straight down from the beam ends to the load lift points, putting each sling near 0° from vertical and tension near W/2 per leg.
Rule of thumb: if your headroom forces an included angle above 90°, a spreader beam pays for itself in rope life and safety margin. Below 90° included, longer slings are simpler and cheaper.
Steel wire rope is stiff. A 25 mm length difference in a 3 m sling leg — well within sling manufacturing tolerance — produces enough geometric mismatch that the shorter leg picks up most of the load before the longer one comes tight. Soft slings stretch and equalise; wire rope does not.
For critical lifts with multiple steel legs, use a load equalising shackle or spec matched-length slings from the same fabricator. Better yet, instrument both legs during the first test lift so you know which is tight.
Yes, but the symmetric formula T = W/(2 cos θ) does not. With anchors at different heights, each rope has a different angle from vertical, so you solve the two equilibrium equations separately — sum of horizontal forces = 0, sum of vertical forces = 0 — and get two different tensions.
The rope at the steeper angle (closer to vertical) always carries more vertical load. The rope at the shallower angle carries more horizontal load. Run the numbers on paper first; mismatched-anchor rigs catch a lot of crews out.
A sudden stop or a snatch pickup can multiply static tension by 2× to 6× depending on rope stretch and stop time. The geometry stays the same, but W in the equation effectively becomes the dynamic load, not the dead weight.
For aerial performance rigging or any lift where the load may swing or drop, design to a dynamic factor of at least 5× — that is why a 100 kg performer needs rigging rated for 500 kg static at every leg. Wire rope is unforgiving here because it has almost no stretch to absorb the impulse.
References & Further Reading
- Wikipedia contributors. Statics. Wikipedia
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