Reinforced Concrete Design ku Value

Given a bending moment, M, and the properties of a concrete section, it is often necessary to find the neutral axis factor KU to ensure that it is less than the ductility limit of 0.4. The moment capacity is FI*C*JD where FI is the capacity factor (0.8), C is the concrete stress block and JD the distance from centroid of reinforcement to centroid of C.

From clause 8.1.2.2 of AS 3600, C is .85*FC*B*GAMMA*KU*D, where GAMMA is:
0.85 - 0.007*(FC-28) within the limts of 0.65 to 0.85. The following is based on a cue card that can be processed directly in any word processor, using MATHSERV.

M = FI*.85*FC*B*GAMMA*KU*D*(D-GAMMA*KU*D/2) with KU the unknown X.
**** capacity factor FI = .8
**** concrete strength in MPa, FC = 32
**** width in mm, B = 300
**** depth to centroid of steel in mm, D = 450
**** bending moment in kNm, M = 265
GAMMA = max(.65, .85 - .007*(FC-28)*(FC > 28)) := 0.822
FX = "FI*.85*FC*B*GAMMA*X*D*(D-GAMMA*X*D/2) - M*10^6"
KU = bisect(FX, 0, 1, .001) := 0.2744

You can easily modify this to use the online solver. For example, if your values are as above, you would simply punch into your calculator:
.8*.85*32*300*.822*450 = 2414707.2 and set up the equation:
2414707.2*x*(450-.822*x*450/2)-265000000
in the f(x) box.

This quadratic equation can be solved explicitly, but it is much more convenient to solve it with the online solver. For xmin you would enter 0, for xmax .4 because, if there is no solution within this range, the section is not ductile. Of course, if you enter 1 for xmax, you may see the degree of deficiency. For example, change the moment to 365000000 - if xmax is 0.4, there will be no solution. Change xmax to 0.6 and you will see that the section is just outside the requirements.

It doesn't matter much what you enter for ymin (<0) and ymax (>0), but values like 100000000 (- and +) will produce a nice graph.