Concrete Modulus of Rupture Equations and Calculator

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Concrete Modulus of Rupture Equations and Calculator

Civil Engineering and Design
Strength of Materials Basics and Equations | Mechanics of Materials

The tensile strength of concrete in flexure is known as the modulus of rupture, f_r , and is an important parameter for evaluating cracking and deflection in beams. The tensile strength of concrete is relatively low, about strength. ASTM C78 gives the details of beam testing using third-point loading. The modulus of rupture is calculated from Eq. 1.

Eq. 1
f_r = M c / I

Equation 1 gives higher values for tensile strength than the splitting tensile strength test because the stress distribution in concrete is not linear.

For normal weight concrete, ACI 318 prescribes that Eq. 2 & 3 should be used for modulus of rupture calculations. For all-lightweight concrete, the modulus of rupture is taken as 75% of the calculated values. Other special rules for lightweight concrete may apply [ACI 318 Sec. 9.5.2.3 and Sec. 8.6.1].

Eq. 2, SI Units
f_r = 0.62· λ · f '_c^0.5

Eq.3, U.S.
f_r = 7.5 · λ · f '_c^0.5

Where:

f '_c = compressive strength of concrete, lbf/in² (MPa)
λ = Lightweight Aggregate Factors, see table 1
f_r = Concrete Modulus of Rupture, lbf/in² (MPa)

Lightweight concrete has a lower tensile strength than normalweight concrete, even if both have the same compressive strength. The lightweight aggregate factor, λ, is used to account for this lower tensile strength and is determined from Table 1. For concrete using a blend of lightweight and normalweight aggregates, ACI 318 Sec. 8.6.1 allows for linear interpolation to determine the lightweight aggregate factor. ACI 318 also permits the use of laboratory tests to correlate splitting tensile strength with compressive strength.

Table 1 Lightweight Aggregate Factors, λ

normal weight concrete = 1.0
sand-lightweight concrete = 0.85
all-lightweight concrete = 0.75

Reference

Civil Engineering Reference Manual, Fifteenth Edition, Michael R. Lindeburg, PE