Hello,
I am wondering how RAM Steel computes the I_{tr} for composite beams. For example, I have a composite beam with the following properties as computed by RAM:
But by hand, I am computing I_{tr} = 941.86, slightly less than RAM. I am guessing that this discrepancy arises from the way in which the transformed width of the concrete, b_{tr}, is computed. In my calcs, I used the modular ratio multiplied by the effective width, i.e.,
b_{tr} = b_{e} * E_{c} / E_{s}
where the concrete Young's modulus I compute from AISC 360-10, Section I2.1b : E_{c} = w_{c}^{1.5} * sqrt{f'_{c}}. Then, with the steel and transformed concrete properties, I find the elastic neutral axis, and then use the parallel axis theorem to compute I_{tr} with {b_tr}.
Where exactly am I, and RAM source code, diverging? The discrepancy is only about 10-in^4, but this actually makes a significant difference once you compute I_{eff} and then use that to compute the post-composite deflections. This couldn't be a numerical error for such a simple calculation. There is clearly a difference between what RAM is calculating and what I am.
This is a quote from an old tech note regarding the effect of light-weight concrete. I think it still applies.
“For stress computations, the compression area of light-weight or normal weight concrete shall be treated as an equivalent area of steel by dividing it by the modular ratio n for normal weight concrete of the strength specified when determining the section properties. For deflection calculations, the transformed section properties shall be based on the appropriate modular ratio n for the strength and weight concrete specified, where n = E/Ec.”
Okay, I agree on this! So perhaps the discrepancy arises from the Ec calculation? Do you know how RAM calculates Ec?
Or maybe I_tr calculation with the parallel-axis theorem differs... I am applying it in the following way (matlab):
Ec = wc^(3/2) * sqrt(fc) ; btr = be * Ec / Es ; Ixc = 1 / 12 * btr * tc^3 ; % transformed concrete MOI Ac = tc * btr ; % transformed concrete area ys = d / 2 ; yc = d + hr + tc / 2 ; % steel and concrete centroids y_bar = (As * ys + Ac * yc) / (As + Ac) ; % elastic neutral axis Itr = Ixs + Ixc + As * (ys - y_bar)^2 + Ac * (yc - y_bar)^2 ; % parallel axis theroem
There is a verification example here: https://communities.bentley.com/products/ram-staad/m/structural_analysis_and_design_gallery/273520
It uses the same Ec equation you noted above.
Ahhh! I see now where the difference is. Thank you for providing this info!
They compute Ec using the PSI form, where I was using the KSI form. This results in a different modular ratio. I changed my formula and now I get exactly what RAM was getting.