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CFRP Laminate Tsai-Wu Failure Criterion Simulator
Evaluate the Tsai-Wu failure criterion of a carbon fiber reinforced plastic (CFRP) laminate including the F12 interaction term. Adjust the fiber system (T800/T1000/M55J/E-Glass), fiber volume fraction, principal stresses σ11/σ22/τ12 and primary ply orientation to see the Tsai-Wu failure index, maximum-stress index, strength ratio R and safety factor update together.
Parameters
Fiber system
Sets baseline UD ply strengths (X_t/X_c/Y_t/Y_c/S)
Fiber volume fraction V_f
Typical 0.55–0.65 for prepreg / autoclave parts
Axial stress σ11
MPa
Stress along the fiber direction. Positive = tension
Transverse stress σ22
MPa
Stress transverse to the fibers; matrix-dominated and low allowable
Shear stress τ12
MPa
In-plane shear at the fiber / matrix interface
Primary ply orientation
A simple correction factor is applied to TWI for each orientation
Shows each ply orientation (0°/±45°/90°), the current σ11·σ22·τ12 vector and a simplified Tsai-Wu envelope with the current stress point. If the point sits outside the envelope, TWI > 1 (failure).
σ11 sweep — Tsai-Wu vs Max Stress envelope
Strength comparison by fiber system — X_t / Y_t / S
The Tsai-Wu failure criterion. F_i are linear terms, F_ii are quadratic. X_t / X_c are fiber-direction tensile and compressive strength, Y_t / Y_c the transverse values, and S the in-plane shear strength.
The interaction term F_12 is best measured by a biaxial test; this tool uses Tsai's recommended approximation. R is the multiplier of the stress vector that makes TWI = 1.
I keep hearing about Tsai-Wu. Why isn't the maximum stress criterion enough for CFRP? For metals it's just compare with σ_y and you're done, right?
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Good question. Metals are isotropic and roughly symmetric in tension and compression, so a single allowable works. A unidirectional CFRP ply, however, may have X_t = 2800 MPa along the fibers but only Y_t ≈ 70 MPa transverse, and even then 70 MPa in tension versus 250 MPa in compression — strongly asymmetric. On top of that, the failure strength in biaxial states depends on the signs of σ1 and σ2. A criterion that checks each component independently misses all of that. Tsai-Wu rolls every component into one polynomial.
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How do you pick the interaction coefficient F12? The formula shows -0.5√(F11·F22) but I've also read it should be measured.
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Strictly you measure it by back-calculating from an equibiaxial (σ1 = σ2) test, but that needs a cruciform specimen or a pressure-vessel test and is expensive. So Tsai himself recommended F12 = -0.5·√(F11·F22), which has become the de-facto industry default and is even called out in aerospace certification documents. Setting F12 to zero or to a negative value rotates the long axis of the failure ellipse, and the allowable can differ by 10–30 % — it is not a term you can ignore.
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I'm a bit lost on the strength ratio R. How is it different from a safety factor?
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Crucial point. A naive safety factor 1/TWI tells you the present margin, but because Tsai-Wu is non-linear (quadratic) 1/TWI is not strictly the multiplier that takes the load to failure. R is defined so that scaling the entire stress vector by R lands you exactly on TWI = 1 — i.e. it really is "how many times the present load can the structure still take". CFRP FEM post-processors output R for exactly this reason. R = 1 is failure, R = 2 means the load could double.
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Got it. So what is the benefit of a quasi-isotropic [0/±45/90]s lay-up? UD looks much stronger on paper.
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UD is unbeatable along the fibers, but if you don't know which way the load will come — an aircraft skin, a rocket body, an automotive monocoque — you need decent strength in every in-plane direction. A quasi-isotropic stack has equal amounts of 0°/±45°/90° plies, so the in-plane stiffness and strength become almost isotropic and it tolerates notches well. The 787 (about 50 % CFRP) and A350 (about 53 %) are quasi-isotropic dominated. When stiffness in one direction matters more — wind-turbine spar caps, yacht masts — you go the other way and keep a strong UD lay-up.
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Last question: pushing V_f higher always raises strength in the simulator. Where is the real limit?
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By the rule of mixtures, X_t scales with V_f. In practice the ceiling is around V_f = 0.65 — beyond that the resin can't fully wet the fiber bundles, voids appear and fibers go wavy, so usable strength actually drops. Aerospace prepreg + autoclave parts sit at V_f = 0.58–0.63, filament-wound pressure vessels around 0.65, RTM about 0.55. That is why the slider here is capped at 0.7.
Frequently Asked Questions
The Maximum Stress criterion checks each component σ1, σ2, τ12 independently against its own allowable, so it ignores stress-component interaction. Tsai-Wu is a polynomial criterion that adds an F12·σ1·σ2 coupling term and so captures biaxial tension / compression / tension-plus-shear interaction. Because CFRP strength varies strongly with the signs and combination of σ1 and σ2, there are regions where Max Stress says safe but Tsai-Wu says failed; this tool plots both for direct comparison.
The strength ratio R is the proportional multiplier of the current stress vector that makes Tsai-Wu equal exactly 1. R = 1 means failure, R > 1 means margin remains, R < 1 means the state is already beyond the failure surface. Unlike a simple safety factor, R directly answers "how many times the present load can the structure still take" for a non-linear (quadratic) Tsai-Wu surface, which is why CFRP FEM post-processors usually output R. This tool solves the Tsai-Wu quadratic analytically.
Fiber-direction strengths X_t and X_c scale roughly linearly with V_f (rule of mixtures) because the fibers carry the load. Transverse strengths Y_t, Y_c and in-plane shear S are matrix-dominated; their V_f sensitivity is much weaker, and the tool approximates them with √(V_f/0.6). Production CFRP is typically V_f = 0.55–0.65; pushing higher causes resin starvation, voids and fiber misalignment, which actually lowers usable strength.
A unidirectional (UD) ply is extremely strong along the fibers but very weak transverse, so the in-plane response is sharply anisotropic. A quasi-isotropic lay-up contains equal amounts of 0°/±45°/90° plies, giving nearly isotropic in-plane stiffness and strength and a much milder reaction to notches. That makes it the default for aircraft skins, automotive monocoques and pressure vessels where the load direction is not fully known. This tool uses 1.0 as the 0° UD reference and 0.6 for quasi-isotropic to compare the effect of orientation.
Real-World Applications
Aircraft structure: The Boeing 787 is about 50 % CFRP by structural weight and the Airbus A350 about 53 %, with fuselage, wings and empennage built from quasi-isotropic and pseudo-quasi-isotropic stacks. Demonstrating that the Tsai-Wu criterion is satisfied at limit and ultimate loads is part of the FAA / EASA certification basis. CFRP saves roughly 20 % over aluminium alloy, while lightning-strike paths and post-strike repair become CFRP-specific design issues.
Pressure vessels and hydrogen tanks: The 70 MPa hydrogen tanks on fuel-cell vehicles (Toyota Mirai, Hyundai Nexo) are Type IV with a PE or aluminium liner overwound with T700/T1000 CFRP by filament winding. The hoop and helical winding produces a biaxial stress state where the Tsai-Wu interaction term F12 directly drives the safety-factor calculation. Solid rocket motor cases and drone oxygen tanks follow the same design logic.
Motorsport and production vehicles: Formula 1 monocoques use sandwich CFRP shells, and the FIA crash-test rules embed Tsai-Wu / Hashin-style allowables. Production examples include the BMW i3 LifeModule and the Lexus LFA cabin. Because V_f and strength scatter depend on the process (autoclave vs RTM vs press forming), designers add margin to the Tsai-Wu safety factor accordingly.
Large wind-turbine blades and sporting goods: Onshore turbines now use blades over 100 m long with CFRP spar caps and GFRP shells (quasi-isotropic). Tsai-Wu is combined with fatigue analysis to set the allowable stress that guarantees a 25-year life. Golf-shaft, tennis-racket and road-bike-frame designers (Trek, Specialized and others) routinely use Tsai-Wu to optimise lay-up sequence.
Common Misconceptions and Pitfalls
The most common pitfall is setting F12 = 0. It is numerically convenient, but it collapses Tsai-Wu toward Tsai-Hill and throws away the biaxial interaction that is the whole point of Tsai-Wu. In particular, in σ1-tension / σ2-tension biaxial states, F12 = 0 versus F12 = -0.5·√(F11·F22) can rotate the failure ellipse enough to change the allowable by 10–30 %. Always document the F12 value on the drawing and, where possible, back-calculate it from a biaxial test. This tool defaults to Tsai's recommended F12 = -0.5·√(F11·F22).
Next, treating the ply strength as the laminate strength. The TWI returned here is for a UD ply; the real laminate strength also depends on the stacking sequence, ply variability, voids, free-edge effects and inter-laminar shear (ILSS). At free edges in particular, τ23 and σ33 spike locally and trigger delamination that Tsai-Wu cannot capture. Holes, edges and bolt joints need a dedicated treatment (Hashin criteria, characteristic-distance models, empirical allowables).
Finally, "a large R means a good design" is not true. Pushing R above 2 in a CFRP part rapidly inflates ply count, weight, thickness and cost. Aerospace practice fixes R at about 1.5 and then checks buckling, compression after impact (CAI) and fatigue (CCAR 25), letting whichever criterion is most critical drive the design. The Tsai-Wu R is a first-order screen; it is not a substitute for the full set of failure modes. FEM models normally output both Tsai-Wu and Hashin (which separates fiber and matrix failure) and require both to pass.
How to Use
Enter fiber volume fraction (typically 0.55–0.65 for aerospace-grade UD CFRP) in the dedicated field
Input in-plane stresses: longitudinal σ₁₁ (MPa), transverse σ₂₂ (MPa), and shear τ₁₂ (MPa) from FEA or test data
Click Calculate to generate Tsai-Wu index (TWI), fiber-direction strength X_t, max stress index, strength ratio R, lay-up corrected TWI, and safety factor SF
Worked Example
For a unidirectional AS4/3506 CFRP ply with fiber volume fraction 0.60: apply σ₁₁ = 800 MPa (longitudinal tension), σ₂₂ = 30 MPa (transverse), τ₁₂ = 15 MPa (shear). X_t calculates to approximately 2290 MPa. Tsai-Wu index yields 0.38, indicating safe margin. With lay-up correction (±45° plies reduce effective stiffness), corrected TWI rises to 0.52. Safety factor SF = 1.92 confirms adequate reserve for fatigue and environmental knockdown.