Sectorial Properties of a C-shaped Thin-walled Section

Aim: To check the accuracy of the geometric properties calculation for a C-shaped thin-walled section.

Name of a file with the initial data: СSection.tns

Formulation: Check the accuracy of the geometric properties calculation for a thin-walled C-shaped rod cross-section.

References: Young W.C., Budynas R.G., Roark's Formulas for Stress and Strain, New York, McGraw-Hill,  New York, 2002.

Initial data:

Geometric dimensions of a section:

b = 100 cm,

b₁ = 30 cm,

h = 120 cm,

t = 3 cm.

Results Obtained in Tonus:

Design model, coordinate and principal axes, center of mass, ellipse of inertia, core of the section

Comparison of results:

Parameter	Theory	TONUS	Deviation, %
Cross-sectional area, A cm2	1140	1140	0
Conventional shear area along the principal U-axis, Av,y cm2	600	600	0
Conventional shear area along the principal V-axis, Av,z cm2	540	540	0
Torsional moment of inertia, It cm4	3420	3420	0
Sectorial moment of inertia, Iw cm6	8024714070,28	8024727272,72	0.00016
Y-coordinate of the shear center, yb cm	-46,364	-46,364	0
Z-coordinate of the shear center, zb cm	60	60	0

Sectorial Coordinate Diagrams

Notes: Geometric properties can be determined analytically by the following formulas:

$$A=(2b+2b_{1} +h)t;$$ $$A_{v,y} =2bt;$$ $$A_{v,z} =t(h+2b_{1} );$$ $$I_{t} =\frac{t^{3}}{3}\left( {h+2b+2b_{1} } \right);$$ $$e=b\frac{3h^{2}b+6h^{2}b_{1} -8b_{1}^{3} }{h^{3}+6h^{2}b+6h^{2}b_{1} +8b_{1}^{3} -12hb_{1}^{2} };$$ $$I_{\omega } =t\left[ {\frac{h^{2}b^{2}}{2}\left( {b_{1} +\frac{b}{3}-e-\frac{2eb_{1} }{b}+\frac{2b_{1}^{2} }{h}} \right)+\frac{h^{2}e^{2}}{2}\left( {b+b_{1} +\frac{h}{6}-\frac{2b_{1}^{2} }{h}} \right)+\frac{2b_{1}^{3} }{3}(b+e)^{2}} \right].$$