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### Concrete Modulus of Rupture Equations and Calculator

Civil Engineering and Design

Strength of Materials Basics and Equations | Mechanics of Materials

Concrete Modulus of Rupture Equations and Calculator

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/in

^{2}(MPa)

*λ*= Lightweight Aggregate Factors, see table 1

*f*Concrete Modulus of Rupture, lbf/in

_{r}=^{2}(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

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