Strut-and-Tie Models: Designing Concrete by Following the Forces

B-regions, D-regions, and why the flexure formula breaks

Reinforced-concrete members split into two zones. In a B-region (B for Bernoulli, or beam) the strain across a section varies linearly, so σ = Mc/I and the standard flexure/shear equations apply. A D-region (D for discontinuity, or disturbed) is where that assumption collapses: within roughly one member depth d of any concentrated load, reaction, sharp geometry change, or opening, the strain field becomes strongly nonlinear.

St. Venant's principle sets the extent: a static disturbance dies out over a length comparable to the depth of the member, so the rule of thumb is that the D-region extends about a distance d either side of the discontinuity. In a deep beam with a shear-span-to-depth ratio a/d ≤ ~2, the two end D-regions overlap and the entire member is a D-region — there is no B-region left, and a conventional Vc + Vs shear calculation has no theoretical basis.

• Deep beams: transfer girders, wall beams over openings.
• Discontinuities: dapped beam ends, corbels, brackets, ledges, frame corners.
• Concentrated introduction of force: pile caps, post-tensioning anchorage zones, bearing regions.

The STM idea, first formalized as the truss analogy by Ritter (1899) and Mörsch (~1908) and generalized by Schlaich, Schäfer and Jennewein (1987), is to replace the messy continuous stress field with an idealized pin-jointed truss that carries the same loads to the same supports.

The three ingredients: struts, ties, and nodes

Every strut-and-tie model is assembled from exactly three components, and the design reduces to checking one inequality — φFₙ ≥ Fᵤ — at each of them, with the strength-reduction factor φ = 0.75 for all STM elements in ACI 318.

• Struts carry compression through the concrete. Idealized as prismatic or, more realistically, bottle-shaped — the compression spreads laterally as it flows between nodes (roughly a 2:1 longitudinal:transverse slope), which induces transverse tension that must be reinforced or the strut splits.
• Ties carry tension and are almost always steel reinforcement (or prestressing tendons). A tie's capacity is simply Fₙₜ = Aₜₛ fy (plus Aₚ fₚ for tendons); the steel must be developed for its full yield force at the node.
• Nodes are the joints where struts, ties, and applied forces meet. The finite concrete volume around a node is the nodal zone, and it is frequently the true bottleneck.

Nodes are classified by the sign of the forces meeting there: C-C-C (three compressive struts, e.g. under a bearing plate), C-C-T (one tie), and C-T-T (two ties). More ties crossing a node lowers its concrete strength, because anchored bars disrupt the compression field. Equilibrium of the whole assembly must be satisfied — STM is fundamentally a hand statics problem, one free-body at a time.

Governing strength equations and the ACI numbers

Strut strength. The effective compressive strength of a strut or nodal zone is fce = 0.85 βₛ f′c, and the nominal strut force is Fₙₛ = fce · Aₛ, where Aₛ is the cross-sectional area at the strut end. The strut coefficient βₛ penalizes disturbed geometry:

• βₛ = 1.00 for a prismatic strut in uniform compression;
• βₛ = 0.75 for a bottle-shaped strut with adequate crack-control reinforcement;
• βₛ = 0.40–0.60 for bottle-shaped struts without it, or struts in tension members.

Nodal-zone strength. Fₙₙ = fce · Aₙz with fce = 0.85 βₙ f′c, and βₙ = 1.0 (C-C-C), 0.80 (C-C-T), 0.60 (C-T-T). The area Aₙz is measured on the face perpendicular to the strut/tie line of action.

Tie strength. Fₙₜ = Aₜₛ fy. For Grade 420 (60 ksi) bars, fy = 420 MPa.

Put in numbers: for f′c = 35 MPa, a C-C-T node face has fce = 0.85 × 0.80 × 35 ≈ 23.8 MPa. A 300 mm × 300 mm bearing pushing a strut into that node can resist Fₙₙ ≈ 23.8 MPa × 0.09 m² × 1000 = 2140 kN nominal, so φFₙₙ ≈ 1600 kN. If the factored reaction Fᵤ exceeds that, you enlarge the bearing, raise f′c, or reduce βₙ's penalty by re-routing ties — the model tells you exactly which lever to pull.

Why it is safe: the lower-bound theorem

STM is not a heuristic — it inherits a theorem. The lower-bound (static) theorem of plasticity states that if you can find any stress field that (1) satisfies equilibrium with the applied loads and (2) nowhere violates the material yield condition, then the structure will carry at least that load. Concrete's limited ductility is the fine print: it only holds if the structure can redistribute forces to the chosen path, which is why STM caps concrete stresses well below f′c (the 0.85βₛ factors) and demands minimum distributed crack-control steel.

The practical consequence is enormous freedom: any equilibrium-satisfying truss is a safe design, so two engineers can pick different models for the same corbel and both be conservative. The catch is efficiency and serviceability. A tie that yields is a crack that opens; steel strains far exceed concrete strains, so the model that most resembles the elastic stress trajectories (compression along principal-compression lines, ties along principal-tension lines) cracks least at service load. Schlaich's guidance — minimize the total strain energy Σ Fᵢ Lᵢ εᵢ, which for stiff ties reduces to minimizing tie length — steers you to the best of the admissible trusses.

A useful discipline: keep strut angles in the window 25° ≤ θ ≤ 65°. Shallower than ~25° and the strut and tie nearly coincide, demanding huge, incompatible strains; the model becomes both inefficient and unserviceable.

A design procedure you can run by hand

The workflow is disciplined statics. For a typical D-region:

• 1. Isolate the D-region and compute boundary forces. Extend it about d past each discontinuity; find the sectional stresses at the B/D interface and resolve them into resultant strut/tie forces at the boundary.
• 2. Sketch the truss. Place struts along compression flow, ties where reinforcement will go. An elastic FE stress plot (principal-stress vectors) is the best guide for the first sketch; align struts within ±15° of the elastic compression trajectories.
• 3. Solve for member forces by method of joints/sections — the truss is usually statically determinate by design.
• 4. Size ties: Aₜₛ = Fᵤ / (φ fy). Choose bars; verify they fit within the tie width and can be developed (hooks, headed bars) beyond the node.
• 5. Check struts and nodes: compute required node-face and strut areas from φfce·A ≥ Fᵤ; confirm bearing plates, member depth, and cover geometry provide them.
• 6. Add crack-control reinforcement: ACI requires a distributed orthogonal grid (historically ~0.003 in each direction) to justify βₛ = 0.75 for bottle-shaped struts and to control service cracks.

Worked scale: a two-pile cap carrying a 3000 kN factored column with piles at ±0.9 m and an effective depth d ≈ 0.8 m gives strut angle θ = atan(0.8/0.9) ≈ 42°. Each pile takes 1500 kN; the bottom tie force is Fᵤ = 1500/tan42° ≈ 1670 kN. Required tie steel: Aₜₛ = 1670×10³/(0.75×420) ≈ 5300 mm² — about seven 32 mm bars, fully hooked over the piles.

Real hardware: where STM actually governs

STM is the design method of record for a specific family of stubborn details:

• Pile caps. Deep, rigid, and squarely a D-region. STM replaced the fictitious 'beam bending' check and typically yields more, better-anchored bottom steel spread over the piles.
• Transfer/deep beams. A 2 m-deep beam carrying columns above a lobby: a single diagonal strut from load to support plus a bottom tie, with the tie developed past the support centerline.
• Corbels and brackets (a/d ≤ 1). The classic C-C-T model: a compression strut from the bearing to the column, a horizontal tie at the top, and a shear-friction-consistent check.
• Dapped beam ends — reentrant corners with a stress concentration; STM places a hanger tie and diagonal strut to catch the load and hang it back into the full section.
• Post-tensioning anchorage zones. A 5000 kN tendon anchor introduces enormous concentrated compression; STM sizes the bursting (transverse tension) reinforcement behind the anchor plate — this is where AASHTO LRFD leans on STM hardest.
• Beam-column joints and offshore ledges, including the very cell-wall shear regions implicated in the 1991 Sleipner A collapse, where an under-designed tricell region failed under hydrostatic load.

Codes now embed all of this: ACI 318 Chapter 23, AASHTO LRFD Article 5.8, Eurocode 2 §6.5, and fib Model Code 2010 each provide the βₛ/βₙ factors and detailing rules.

Failure modes, limits, and best practice

STM controls strength, but three failure modes remind you it is a lower bound, not a prediction of the real crack pattern:

• Strut splitting. Bottle-shaped struts crack longitudinally from the transverse tension of spreading compression. Without the orthogonal crack-control grid, βₛ drops to 0.60 and the strut can fail brittly at loads below the ideal-truss capacity — this is why the grid is non-negotiable.
• Node crushing. The concrete at a C-C-T or C-T-T node is triaxially disturbed and often the weakest link; underestimating the tie geometry (bar layers, cover) shrinks the node face and crushes it.
• Anchorage/bond failure. A tie that isn't developed past the node simply pulls out — real failures cluster here. Headed bars and hooks bought inside the nodal zone are the fix.

Limits worth stating plainly: STM gives no direct deflection or crack-width output (it's a strength model), so serviceability must be checked separately, and the answer depends on the model you draw — a poorly chosen truss can be safe yet crack badly at service load. Best practice is to (a) seed the model from an elastic FE principal-stress field, (b) keep 25°–65° strut angles, (c) minimize tie length/strain energy, (d) always include distributed reinforcement, and (e) detail ties for full development. Done that way, STM turns an intractable nonlinear stress problem into a truss a graduate engineer can solve, check, and defend.

Sectional (flexure/shear) design vs. strut-and-tie design

Attribute	Sectional design (B-region)	Strut-and-tie (D-region)
Assumption	Plane sections remain plane (Bernoulli)	Nonlinear strain; discrete load path
Governing model	σ = Mc/I, Vc + Vs shear	Truss of struts + ties + nodes
Where valid	Away from loads/supports (>~d)	Within ~d of loads, supports, openings
Strength basis	Empirical/mechanics per section	Lower-bound plasticity theorem
Strength check	φMₙ ≥ Mᵤ, φVₙ ≥ Vᵤ	φFₙₛ, φFₙₜ, φFₙₙ ≥ Fᵤ (φ = 0.75)
Typical members	Slabs, ordinary beams, columns	Deep beams, pile caps, corbels, anchor zones