Beam-String Structure Calculation: Prestress Design and a 40 m Worked Example

Between the spans where a simple beam is economic (up to roughly 15–20 m) and the spans where a full space frame takes over (beyond 60 m), there is a range of 20–60 m where roof structures must be engineered aggressively to compete on steel tonnage. A beam-string structure (BS) answers this challenge with an elegant mechanism: a rigid top beam, a slack-before-tensioning bottom cable, and compression struts between them. Tensioning the cable raises the struts, which lift the beam through upward forces that cancel a large share of the gravity load. The beam stops being a beam and starts being an arch-inverted system, with bending moments dramatically reduced and the cable carrying the load in pure tension.

This article explains the mechanics and the design quantities an engineer must control — prestress level, sag, cable force, beam stability and vibration — and completes the full calculation of a 40 m span beam-string roof so that owners and contractors can check the numbers that appear in a tender or a shop drawing review. The design follows JGJ 7-2010 (Technical Specification for Space Frame Structures), which covers beam-string systems, together with JGJ/T 351-2015 (Technical Specification for Building Cable Structures) and GB 50017-2017 for the steel members. Cable-stiffened lattice shells of the same family have been built to record spans in China — including the 229 m span enclosed coal shed our group delivered at Guodian Ningxia Fangjiazhuang — so the principles here scale well beyond the worked example.

The 40 m example is deliberately simple: a single beam-string frame, three struts, one cable. The method extends unchanged to beam-string trusses, to multi-frame roofs, and to chord-stiffened shells.

1. How a Beam-String Structure Works

A beam-string system has three load-carrying components. The top member (H-section beam or a light truss) spans the full length and receives the cladding, purlins and snow load. The bottom cable runs from end anchorage to end anchorage with a downward curvature (sag). Between them, vertical compression struts connect the cable low points to the beam's underside at internal stations.

The force flow is the reverse of common intuition. When the cable is tensioned, it pulls its two ends inward (a horizontal force H transmitted into the beam ends through the anchorage nodes) and it pulls down on each strut with a vertical component T·sinθ. The strut, being a compression member, pushes this force upward against the beam. The result is a set of concentrated upward forces at the strut stations that partially or fully offset the downward distributed loads. The beam therefore sees a much reduced net moment, while it also carries a modest axial compression from the cable's horizontal component — exactly the behaviour of an inverted, tied arch.

Two structural facts follow immediately. First, the system only works if the cable stays in tension under every load combination — including wind suction. If the design prestress is too low, the cable goes slack, the struts lose their function and the beam must suddenly carry the full gravity moment. Second, the end anchorages are simultaneously the beam's moment-resisting supports and the cable's anchor blocks; they are the most heavily loaded nodes in the structure and deserve the most detailed design attention.

2. Prestress Equilibrium and the Cable Force–Sag Relation

For a shallow cable (sag f less than about 1/10 of the span) carrying a uniform equivalent load q, the horizontal cable force follows the classic parabola relation:

H = q · L² / (8 · f)

The same geometry gives the equivalent upward load a cable system applies to the beam: an upward distributed force q_eq = 8 · H · f / L². Setting the dead-load cancellation target q_dead equal to q_eq yields the required prestress:

H_prestress = q_dead · L² / (8 · f)

Two design observations come out of these equations. The relation between cable force and sag is inverse: halving the sag doubles the cable force, the anchor forces and the beam-end compression, while gaining headroom. Practical beam-string roofs use a sag-to-span ratio f/L of 1/12 to 1/16, balancing headroom against anchorage economy. And because the cable is a non-linear member — its stiffness grows with tension — the classic mistake is sizing the system from the final state only. The analysis must include the construction sequence: cable initially slack, progressive tensioning through the struts, then application of cladding and live load.

The Sag–Force Trade-Off in One Equation

Because the cable force is approximately H = M₀/f (the moment to be balanced divided by the sag), every centimetre of sag is real money in cable, strut and anchor costs, while every additional metre of sag reduces the usable building height. Doubling the sag from 2.0 m to 4.0 m on the worked example below halves the cable force, but costs the architect 2 m of roof void. The design conversation is therefore always a three-way trade between depth, cable size and internal clear height, and it should happen before member sizes are locked.

H ≈ M₀ / f with M₀ = q·s·L²/8 → at L = 40 m, q · s = 10.5 kN/m per cable line: f = 2.0 m → H ≈ 1,050 kN; f = 2.8 m → H ≈ 750 kN; f = 4.0 m → H ≈ 525 kN

3. Worked Example: 40 m Span Beam-String Roof

The example is a single roof bay of 40 m span with 6 m frame spacing, three struts at quarter points, and roof build-up consisting of metal cladding, insulation, purlins and installation loads.

• Geometry: span L = 40.0 m, sag f = 2.8 m (f/L = 1/14.3), three struts at L/4, L/2 and 3L/4.
• Loads: dead load 0.90 kN/m² and imposed + snow 0.50 kN/m² per GB 50009-2012; tributary width 6.0 m. ULS line load q_d = (1.3 × 0.90 + 1.5 × 0.50) × 6.0 = 11.5 kN/m. SLS line load = (0.90 + 0.50) × 6.0 = 8.4 kN/m.
• Reference simple-beam moment: M₀ = q_d · L²/8 = 11.5 × 1,600 / 8 = 2,300 kN·m. With struts and cable engaged, the beam carries approximately 45% of this (M_beam ≈ 1,035 kN·m); the remaining 55% is delivered by the cable system.
• Main beam: welded H-section H 700 × 300 × 14 × 20 in Q355C. Section modulus W_x ≈ 4.9 × 10⁶ mm³; bending stress = 1,035 × 10⁶ / 4.9 × 10⁶ ≈ 211 N/mm² against f = 310 N/mm² (t > 16 mm, Q355) — utilisation 0.68, with the remaining capacity reserved for cable horizontal force and lateral-torsional stability.
• Cable force: required prestress to cancel the dead load: H_p = 0.90 × 6.0 × 1,600 / (8 × 2.8) ≈ 385 kN. Final cable force at ULS, including live-load increments and second-order effects: T ≈ 750 kN.
• Cable: 12 strands of φ15.2 mm, 7-wire strand to GB/T 5224, 1860 MPa low-relaxation. Total steel area = 12 × 139 = 1,668 mm²; characteristic breaking load ≈ 3,100 kN; design resistance with partial factor 2.5 ≈ 1,240 kN. Utilisation at 750 kN = 0.60, leaving margin for anchor losses and future re-tensioning.
• Struts: CHS φ140 × 8 in Q355B; mid-span strut carries N ≈ 85 kN axial (ULS) with slenderness λ ≈ 60 — sized by slenderness and ductility, not strength.
• Deflection: SLS mid-span deflection from the full non-linear model ≈ 115 mm = L/350, comfortably below the L/250 roof limit of JGJ 7-2010. No camber required for serviceability, though specification of a small pre-camber is recommended to drain the roof positively.

Span (m)	Sag f (f/L)	Design cable force	Typical main member
20	1.4 m (1/14)	250 – 400 kN	H 500 × 250 × 12 × 16
40	2.8 m (1/14)	600 – 900 kN	H 700 × 300 × 14 × 20
60	4.0 m (1/15)	1,200 – 1,800 kN	Truss 1,200 mm deep / box girder

Strand Specification and Anchorages

The 12 × φ15.2 strand bundle at the 750 kN design force uses low-relaxation seven-wire strand with a guaranteed breaking load of 260.7 kN per strand per GB/T 5224, giving a total breaking load of 3,128 kN and a utilisation of roughly 30% — a margin consistent with the fatigue endurance expected of tension members that see live load cycles. The end anchorages are cold-formed wedge sockets to GB/T 14370 Class II, and each anchorage zone is detailed with a doubler plate so the local bearing stress under the socket stays within the edge-distance and projection rules of the design code.

Construction Sequence and Camber Strategy

The roof is erected on temporary propping, the main beam is welded with a precamber equal to the dead load plus 60% of the prestress deflection, and the cable is stressed in two stages: 70% of the final force once all dead load is in place, then the remaining 30% after cladding and installations. Each stage is verified by a load cell reading paired with a measured deck elevation, and the final geometry is surveyed against a design camber tolerance of ±10 mm at mid-span. This staged approach prevents the two classic failures — stressing against propping that has not been removed, and losing prestress to elastic shortening of a beam that was not yet fully loaded.

4. Stability: Lateral Buckling and Strut Design

The slender top beam of a beam-string system is protected in-plane by the cable, but out-of-plane it is only as strong as its lateral bracing. The compression carried from the cable ends makes lateral-torsional buckling the controlling check for spans beyond about 30 m. The standard solution is lateral restraint at each strut station — the struts themselves brace the beam to the cable plane, and cross ties or a light secondary frame restrain the beam between stations. In every twin-frame roof we build, the two beam-string frames brace each other through valley ties, which must be designed for a notional lateral force of about 2% of the vertical load per GB 50017-2017 rather than zero, as some tender models assume.

Struts are compression members with a wide safety band, but they must be checked for the reversed case: wind suction can momentarily reduce the upward force they receive, and under combined suction plus prestress the strut load should never reverse sign to tension without a designed tie. Slenderness of the strut is limited to 180 per JGJ 7-2010; keeping it below 100 makes erection handling and connection detailing simpler.

Durability of the Cable System

The cable is the one member of the beam-string system that cannot be inspected from a walkway, and its corrosion protection is decided in the specification, not on site. Seven-wire strand for indoor roofs is supplied hot-dip galvanised per GB/T 5224 (zinc coating class A, roughly 300 g/m²), with the anchorage zone wrapped in a grease-packed corrosion barrier and the full cable run sleeved in PE where the atmosphere is aggressive. The fire resistance of an open cable is a separate design item: the R90–R120 requirement of the roof system is carried by the main beam, and the cable anchorage is protected with the same intumescent or board system as the beam ends where the building code demands it. A durability table stating the galvanising class, the PE sleeve thickness and the anchorage protection for each project closes the loop between the stress design in this article and the 50-year service life the owner expects.

5. Vibration and Dynamic Behaviour

Beam-string roofs are light and flexible, which makes them sensitive to three dynamic actions: vertical pedestrian or machinery-induced motion, wind buffeting on the roof plane, and wind-rain-induced cable oscillation. The governing design quantity is the first vertical natural frequency. For the 40 m example, the fundamental frequency lands in the range of 2.0–2.5 Hz with realistic mass assumptions; the code practice for roofs is that the first frequency should stay above 3 Hz where people gather beneath, and where it cannot, the design must add damping or mass. We meet this by one of two measures: tuned mass dampers at mid-span (typically a 2–3 t damper per frame) or viscous dampers at the cable anchorage, which have the advantage of also controlling wind-rain vibration.

Cable vibration deserves a dedicated paragraph because it is the failure mode most often missed in review. A bare strand with low sag and low damping can gallop under wind-rain excitation at wind speeds of 8–15 m/s. Countermeasures, in order of cost: (1) cable surface treatment to break the water rivulet (small-diameter spiral wire or dimpled sheath); (2) cross ties between adjacent cables; (3) viscous dampers near the anchorages. For all roofs with cable spans above 2 × 15 m we include cross ties as standard and list dampers as an optional add-on value-engineered against exposure conditions.

Damping and the Human Comfort Check

Long-span roofs with large voids are sensitive to rhythmic crowd loading, and the beam-string system responds with a low first-mode frequency in the 1.5–2.5 Hz band — exactly the walking and marching band. The design compares the computed vertical acceleration against the comfort limit for the occupancy (for a gymnasium or assembly roof, on the order of 0.5–1.0% of g under the applicable dynamic load provisions) and adds damping where the check fails: cable-mounted viscous dampers at the quarter points, or tuned-mass dampers on the main beam, both of which we have detailed and supplied where the architect insisted on a light, long, unsupported span.

Because prestress raises the natural frequency through tension stiffening, the cable force is itself part of the vibration tuning. The frequency check must run twice: at the final prestress and at a 50% loss scenario, because a partially de-tensioned cable lowers the frequency and can move the mode directly into the critical band.

6. Span Applicability and System Variants

The beam-string family scales across three applications. From 20 to 40 m, a single beam-string frame (Zhang Xian Liang) is efficient for gymnasiums, aircraft hangar doors areas and platform canopies. From 40 to 60 m, the beam is usually replaced by a light truss (beam-string truss), trading fabrication simplicity for a two to three times higher stiffness-to-weight ratio. Beyond 60 m, the family continues as chord-stiffened domes and cable-stiffened lattice shells; our 229 m span enclosed coal shed at Guodian Ningxia Fangjiazhuang is a member of this family, where a prestressed cable-and-strut system stiffens a lattice shell against buckling with a fraction of the steel a conventional grid would consume.

For the owner, the economic message is consistent across the range: a beam-string roof saves 15–30% of structural steel compared with a simply supported beam or truss at the same span, at the price of more engineering, a qualified tensioning contractor, and a slightly longer erection critical path for the cable tensioning sequence. The decision is therefore an engineering-dollar trade, not a structural one — and the numbers above are the ones that should appear in every comparison.

Selection Shortcuts for Preliminary Design

For a preliminary scheme, three rules of thumb size a beam-string roof within 20–30 minutes: the main beam depth is L/35 to L/45 of the span; the sag is 1/14 to 1/18 of the span; and the cable force is 60–85% of the force that balances the full design moment at that sag. Applying these to the 40 m example gives a 0.9–1.1 m beam depth, a 2.2–2.9 m sag, and a cable force in the 600–850 kN band — matching the detailed example above and giving the pursuing design team a target to converge to. The numbers are a start point, not a substitute for the staged analysis, but they keep the scheme competitive before the modelling begins.

Key Takeaways

• The mechanism is simple: a tensioned cable pulls down through compression struts to lift the top beam, converting bending into an axial couple — beam moment dropping to roughly 45–55% of the equivalent simply supported beam.
• Prestress is a design decision, not an analysis output: H_p = q_dead · L²/(8f) sets the tension that cancels dead load; the final cable force must be re-checked at every load combination, including wind suction.
• Cable force and sag are inversely related: f/L of 1/12 to 1/16 balances headroom against anchorage cost; the construction sequence (slack → tensioned → loaded) must be modelled.
• Out-of-plane stability controls the beam: brace each strut station to the cable plane and design lateral ties for the notional 2% force, not zero.
• Control vibration by design: target a first frequency above 3 Hz where people gather; add cross ties as standard for long cables and consider dampers where wind-rain excitation is credible.
• The 20–60 m band is where beam-string pays: 15–30% steel saving versus simply supported alternatives, with the anchorages and tensioning sequence as the schedule-critical items.

If a 20–60 m roof in your programme is being compared against space frames or trusses, send us the span, load data and frame spacing. Our engineering team will provide a beam-string concept with member sizes, cable force, sag, natural frequency and indicative tonnage for the tender comparison. Contact our engineering team for the preliminary study.