Cracking Moment Equation
Cracking moment equation is a famous formula that gives you the value of tensile bending stress at which the concrete of the beam starts to crack. This concept is important to understand the stress distribution of the beam across the depth. It will also help you to determine the deflection in the concrete at the point of failure.
But for deflection calculation you can’t use the gross moment of Inertia (Ig) because of cracking effect. So, instead you have to use the effective moment of inertia (Ie) which provides transition between the upper and lower bounds of Ig and Icr. However, for Ie you have to determine the Mcr which is the cracking moment.
Establishing the Cracking moment Equation
Let’s now derive its formula
Step No. 1 – finding modulus of rupture
But first you have to establish an equation for the modulus of rupture (MOR) of concrete. Modulus of rupture is the tensile flexural strength of the beam and it shows the amount of stress and force an unreinforced concrete can resist before bending failure. It is written as fr and has been established by ACI 318-14 in equation 184.108.40.206 as:
fr = 7.5l√fc’ (in psi )
fr = 0.7l√fc’ (in MPa)
where fc’ is the compressive strength of the concrete while l is the factor for the light or normal weight concrete. For light weight concrete, the modulus of rupture is further reduced as you have to multiply the above equation with 0.75.
Step No. 2 – Finding the distance of extreme fiber of the tension side
From stress distribution of the rectangular beam we know the maximum bending stress is at the extreme fiber from the Neutral Axis (NA). So, for rectangular section, neglecting the effect of reinforcement we can say
Yc = h/2
Step No. 3 – Finding Gross moment of Inertia
The gross moment of inertia is for the rectangular section bh3/ 12 hence,
Ig = bh3/12 where b is the width of the concrete beam and h is the height.
Step No. 6 – finding cracking moment
Now by using the formula of bending stress from Neutral axis as per the elastic beam theory we can write:
f = My / I
f is the bending stress at fiber of y distance
M is the moment applied
y is the distance from the neutral axis
I is the moment of inertia for beam
Now using below calculated values we can write as:
M = (f x I )/ y
Mcr = (fcr x Ig ) / yc
Hence the above the is cracking moment equation.