Modulus of Rupture Predicted from Compressive Strength

The modulus of rupture is the flexural tensile stress at which plain concrete cracks, and it decides two quite different things. In a reinforced member it sets the cracking moment, which is where a deflection calculation switches from the gross section to the cracked one — and since that switch changes the effective stiffness by a factor of three or more, it matters enormously to a serviceability check. In a slab-on-grade or a concrete pavement it is the design strength outright, because a pavement fails by cracking under a wheel load rather than by crushing.

Like the modulus of elasticity, it is commonly predicted from compressive strength with a square root. And like it, the coefficient belongs to a code rather than to concrete. But the disagreement between codes runs deeper here than anywhere else on this site: Eurocode does not use a square root at all. Its mean tensile strength follows a two-thirds power of the characteristic cylinder strength — a different functional form, not a different constant — so there is no coefficient that makes a square-root page reproduce it. Codes also distinguish an axial tensile strength from a flexural one and apply a depth-dependent factor between them, because a thin section reads higher than a thick one on the same concrete. Know which quantity your coefficient predicts before you use the answer for anything.

The unit-boundness bites here exactly as it does for stiffness, and arguably worse because the numbers are small enough to look interchangeable. The familiar ACI relation reads about 0.62 with the strength in megapascals and about 7.5 with it in psi. Same line, factor of about 12.04 between the coefficients, and entering one where the other is wanted is an order-of-magnitude error that still prints something that looks like a stress. This site takes the MPa form and says so on the page.

Now, why does flexural tension read high? A modulus of rupture is not a measured stress — it is computed from the peak bending moment on the assumption that the stress distribution stays linear right up to failure. It does not. Concrete softens in tension before it breaks, so the real stress block at the tension face is blunter than the triangle the calculation assumes, and dividing the moment by the elastic section modulus therefore overstates the true tensile strength. Typically by half again over a direct tension result, with a split-cylinder result in between. All three get called "the tensile strength", and mixing them is a real error.

There is also a genuine size effect: a deeper beam gives a lower modulus of rupture on identical concrete, because a larger stressed volume contains a larger worst flaw. This is the same weakest-link statistics that governs ceramics and rock, it is not experimental noise, and it is why specimen dimensions are specified in the test method rather than left open.

And the thing to hold onto: this is a prediction, not a measurement. The flexural strength of a real mix is measured by breaking a beam in third-point loading, and the scatter on that test is notoriously wide — wide enough that pavement engineers who design on flexural strength specify the beam test and pay for it rather than accept a value inferred from cylinders. If a slab thickness or a pavement design turns on this number, break beams. If you need a tensile strength for a purpose that will tolerate an estimate, this equation is a reasonable estimate with a wide band around it, and the band is not the equation's fault.

One last distinction worth drawing, because the name collides. A modulus of rupture measured in a three- or four-point bend test on a ceramic bar is a different thing from the number this page predicts: that is a test geometry, applied to a brittle material with no reinforcement in prospect, and the quantity it produces is the specimen's own flexural strength rather than a code's estimate of a concrete's cracking stress. The site carries both. They share a name and almost nothing else.