Is there a way to derive a formula of modulus of rupture (flexural strength) vs compressive strength?
I want to know is there any mathematical reason for the $\sqrt{f_{c} ^{'}}$ placed somewhere in the formula?
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$\begingroup$ I can't answer authoritatively, so I'll just put this as a comment. No, you can't. I believe the relationships between flexural, tensile and compressive strength in concrete are in large part empirical. A good indication of this is the fact that a dimensional analysis of the equations doesn't work (such as the $\sqrt{f_c'}$). $\endgroup$– Wasabi ♦May 7 '16 at 11:45
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$\begingroup$ While the formula is largely empirical, as to your second question (ie mathematical reason), have a look at the Mohr's circle which will lead you to a square root relationship. $\endgroup$– AsymLabsJun 3 '16 at 8:43
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Short answer: No.
Longer answer:
The code equations (e.g. ACI) relating modulus of rupture with compressive strength cannot be mathematically derived. It is an empirical relationship based on curve fitting to extensive experimental data.
The ASTM test for flexural strength (i.e. Modulus of rupture) is ASTM C78/C78M.
In essence, the most common test consists of loading an unreinforced simply-supported concrete beam at third points and determining the load at which fracture occurs. From this data, the modulus of rupture can be calculated.
There is generally a large amount of scatter in modulus of rupture data because a number of factors affect $f_r$ (mix design, aggregate size, rate of loading, specimen size, etc.)
Many researchers have studied the relationship between modulus of rupture and compressive strength. Relating $f_r$ to $\sqrt f'_c$ seems to provide good correlation. Data from one such researcher is shown below. (Admittedly, it's not the definitive reference, but even Google has its limits when it comes to academic publications and paywalls.) I think some of the more recent research has proposed a 2/3 power relationship, though that may only be for high strength concrete.
