Strut-and-Tie Model — Rigid-Plastic Analysis of D-Regions
Represent a corbel D-region with a simple truss of compression struts and tension ties. Change the projection, effective depth and load to see how the strut angle, member forces and required reinforcement vary.
Parameters
Concentrated load P
kN
Projection a
mm
Effective depth d
mm
Strut strength f_cd
MPa
Assumed: internal lever arm z ≈ 0.85·d, tie design yield strength f_yd = 500 MPa, strut effective thickness b = 300 mm.
Results
—
Strut angle θ
—
Tie tension T
—
Strut compression C
—
Required tie steel A_s
—
Strut width b_strut
Live loading state
0
Load P [kN]
0
Tie tension T [kN]
0
Strut comp. C [kN]
0.00
Node utilization U
Node OK
Node check
Corbel strut-and-tie model (load-path animation)
Blue solid = tie (tension) / Red solid = strut (compression) / White dots = nodes / Yellow arrow = load P. Line thickness is proportional to member force.
Theory & Key Formulas
For a corbel (short bracket) loaded by a concentrated force P, the force flow is represented by a simple truss made of a top tie T and an inclined compression strut C.
Internal lever arm z (simplified, d is the effective depth):
$$z \approx 0.85\,d$$
Strut angle θ (a is the projection):
$$\theta = \arctan\!\left(\frac{z}{a}\right)$$
From nodal equilibrium, the tie tension T and strut compression C:
$$T = \frac{P\,a}{z} = P\cot\theta, \qquad C = \frac{P}{\sin\theta}$$
Required tie reinforcement A_s and required strut width b_strut (constant thickness b, f_yd and f_cd are design strengths):
My RC textbook says "design D-regions with a strut-and-tie model." What exactly is a D-region?
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The D stands for "Discontinuity." Roughly, it's the region where the plane-section assumption of beam theory breaks down. Concentrated loads, supports, corbels, beam-column joints and the area around openings are all D-regions. As a rule of thumb, the disturbed zone is about one section height h. Outside of those zones, you're in a B-region (Bernoulli region) and ordinary beam theory works fine.
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So beam theory just doesn't work in a D-region?
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Right, you can't design with something like $M = \sigma\cdot Z$ here. Instead you use the strut-and-tie model (STM) like the one drawn in the simulator above. You imagine a "reasonable truss" between the loads and supports, separate the members into compression struts (red) and tension ties (blue), and balance forces at each node. Concrete carries the struts, steel carries the ties.
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When I increase the projection a, the strut angle θ drops fast. A larger a makes the strut lay flatter.
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That's the crux of corbel design. A larger a means a smaller θ. Since $T = P\cot\theta$, the tie tension grows quickly — for the same load, the required steel jumps up. Conversely, a deep corbel with small a keeps θ steep and T low. That's why the standard advice "make corbels deep and keep the projection short" comes straight out of this equation.
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And what about the strut compression C?
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$C = P/\sin\theta$, so it also blows up as θ gets small. You can always add more steel for the tie, but the strut is limited by the local concrete crushing strength. So STM design checks both "tie reinforcement" and "strut width (nodal compression)." If you push f_cd down and a out in the simulator, the required strut width will eventually exceed the section thickness — that's the design saying "no good."
FAQ
Because STM is based on the lower bound theorem, any layout that satisfies equilibrium gives a safe-side answer. In practice, however, choosing a truss that follows the principal stress trajectories from an elastic analysis gives better serviceability (crack width). For corbels the standard pattern is "tie along the top, strut connecting load to support." For complex cases such as openings, compare several layouts and pick the one with smaller member forces.
Transverse tensile strain (from the tie steel) crossing the strut causes "cracked concrete softening," which lowers the compressive strength. ACI 318 captures this with the strut efficiency factor beta_s = 0.4 to 1.0; Eurocode 2 uses a nu prime factor (around 0.6 when transverse tension is present). The allowable compressive stress also depends on the nodal type (CCC = three faces in compression, CCT = two compression plus one tension, CTT).
The tie steel must develop its full anchorage length beyond each end node. At a corbel, that typically requires bending the bar horizontally, welding to an anchor plate, or using U-bar hooks at the outer end. Insufficient anchorage is the most common practical failure mode in STM design — calculations look fine, but on site you get cracks and pullout.
In deep beams (a/d ≤ 2), the main load is carried directly by the strut, so stirrups (shear reinforcement) play a limited role. Codes still require minimum shear reinforcement and "distributed reinforcement (a horizontal-vertical grid)" for crack control. The role of this mesh is to convert a brittle strut-crushing failure into a more ductile response and to control crack widths under service loads.
Real-world applications
Corbel and bracket design: Corbels supporting precast beams on columns, crane runway brackets, and bridge bearing pedestals are typical D-region members. STM brings local bearing under the load, tie tension, and inclined strut shear into the column into a single consistent truss model — that integration is its main strength.
Deep beams and transfer girders: Transfer girders that offset columns in low-rise buildings, basement foundation beams, and the wall beams at the tops of bridge piers are deep beams with small shear span ratios. STM is essentially mandatory: a single compression strut from the load to the support plus a tension tie at the bottom main steel captures the behavior elegantly.
Beam-column joints and openings: Beam-column joints subjected to large seismic shear are explicitly covered by STM provisions in ACI 318 and Japanese RC codes. When an opening is placed at mid-span of a beam, the surrounding zone becomes a D-region and a truss model is used to route the compression and tension flows around the opening.
Prestressed concrete anchorage zones: Behind a PC anchorage, a huge concentrated compression acts locally and a transverse "bursting" force develops behind it. This is a textbook D-region; STM is used to design the bursting reinforcement (a tie) behind the bearing plate. Eurocode 2 gives detailed provisions for these zones.
Common misconceptions and cautions
The most common mistake is believing that "if the member forces from STM are satisfied, the design is safe." STM is a rigid-plastic, lower-bound method, so strength (ultimate limit state) is guaranteed — but serviceability (cracking, deflection) must be checked separately. In particular, if the truss layout deviates strongly from the elastic principal stress trajectories, the ultimate capacity may be fine while large cracks open under service loads. The practical rule is: "choose a truss that follows the elastic stress flow."
The next pitfall is using the bare concrete strength f_c for the strut. In reality, transverse tensile strains from crossing tie reinforcement reduce the compressive strength of concrete by 40 to 60%. The beta_s factor in ACI 318 and the nu prime factor in Eurocode 2 capture this softening. In the simulator, lowering f_cd and increasing a quickly drives the required strut width up — that increase reflects exactly this softening effect.
Finally, do not assume that "as long as you place the tie steel, the corbel will not fail before yield." In practice, insufficient anchorage of the tie can cause a "pullout failure" at the outer end of the corbel before the calculated tension is reached. An STM design is not complete until you have checked, as a set, member forces, anchorage length at both end nodes, internal node stresses, and nodal compression strength. This simulator gives you the member forces and required steel; in real design, do not forget anchorage and nodal checks alongside them.
Set corbel depth D (mm) and load position A (mm from support face) using sliders slDVal and slAVal.
Input applied point load P (kN) via slPVal; adjust concrete compressive strength εp (MPa) with slEpsPVal to match your material grade (typical: 25–50 MPa for corbels).
Read output strut angle θ, tie tension T (kN), strut compression C (kN), and required tie steel area A_s (mm²); verify that C does not exceed 0.6·f_c·A_strut (ACI 318 nodal zone limit).
Worked Example
RC corbel with D = 450 mm, A = 150 mm, applied load P = 80 kN, concrete f_c' = 30 MPa. Simulator computes strut angle θ ≈ 35°, tie tension T ≈ 95 kN, strut compression C ≈ 116 kN. Tie requires A_s = 1050 mm² (Grade 500 steel: 3×φ12 = 339 mm², insufficient; use 4×φ13 = 1327 mm²). Strut width b_s = C / (0.6·f_c·sin θ) ≈ 340 mm confirms feasibility within corbel geometry.
Practical Notes
For brackets with A/D < 0.5, tie angle approaches 45° and strut compression dominates; increase concrete grade or corbel depth to reduce required reinforcement.
Verify nodal zone crushing: C / A_nodal ≤ 0.85·f_c (bearing stress); for confined nodes (three or more members), use 0.75·f_c as safer limit.
HSC (f_c' ≥ 60 MPa) reduces tie area by ~40% but requires careful detailing of anchorage and bearing plate; check code-specific strut efficiency factors (ACI β_s ranges 0.60–0.75).