Centroid and second moment of area
A_tf:=b_tf·t_tf=0.04 ft^2
A_w:=t_w·h_w=0.04 ft^2
A_bf:=b_bf·t_bf=0.03 ft^2
I_tf:=(b_tf·t_tf^3)/(12)+A_tf·(y_tf-y_bar)^2=8.12e-05 m^4
I_w:=(t_w·h_w^3)/(12)+A_w·(y_w-y_bar)^2=2.81e-05 m^4
Web contribution about NA
I_bf:=(b_bf·t_bf^3)/(12)+A_bf·(y_bf-y_bar)^2=8.89e-05 m^4
I_xx:=I_tf+I_w+I_bf=1.98e-04 m^4
Centroid and Second Moment of Area
Composite cross-section built from a top flange, web, and bottom flange (I-section).
Inputs — Top Flange
b_tf := 200 mm=7.87 in
Top flange width
t_tf := 20 mm=0.79 in
Top flange thickness
Inputs — Web
h_w := 300 mm=11.81 in
Web height (clear distance between flanges)
t_w := 12 mm=0.47 in
Web thickness
Inputs — Bottom Flange
b_bf := 180 mm=7.09 in
Bottom flange width
t_bf := 16 mm=0.63 in
Bottom flange thickness
Calculations — Component Areas
Areas of each rectangular component.
Top flange area
Web area
Bottom flange area
A_total := A_tf + A_w + A_bf=0.11 ft^2
Total cross-section area
Calculations — Component Centroids
Centroid y-coordinates measured from the bottom of the section.
- Bottom flange centroid: t_bf / 2
- Web centroid: t_bf + h_w / 2
- Top flange centroid: t_bf + h_w + t_tf / 2
y_bf := t_bf / 2=0.31 in
Bottom flange centroid from bottom
y_w := t_bf + h_w / 2=6.54 in
Web centroid from bottom
y_tf := t_bf + h_w + t_tf / 2=1.07 ft
Top flange centroid from bottom
Calculations — Centroid of Composite Section
Neutral axis location (y-bar) from the bottom of the section using first moments of area.
y_bar := (A_tf · y_tf + A_w · y_w + A_bf · y_bf) / A_total=7.23 in
Neutral axis from bottom of section
Calculations — Second Moment of Area
Each component: local centroidal I plus parallel axis transfer term A·d².
Top flange contribution about NA
Bottom flange contribution about NA
Total second moment of area about NA
Results — Section Moduli
Elastic section moduli for extreme fibres (top and bottom).
d_total := t_bf + h_w + t_tf=1.1 ft
Total section depth
y_top := d_total - y_bar=6 in
Distance from NA to top fibre
Z_top := I_xx / y_top=0.05 ft^3
Elastic section modulus — top fibre
Z_bot := I_xx / y_bar=0.04 ft^3
Elastic section modulus — bottom fibre
r_gyr := sqrt(I_xx / A_total)=5.41 in
Radius of gyration about NA
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