Cross

on . Posted in Plane Geometry

Cross is a type of structural beam with a cross shaped cross-sectional profile.  The cross has a central vertical web and horizontal flanges at the center of the vertical web.  The web and flange can have varying dimensions and thicknesses, depending on the specific load bearing requirements and design considerations of the structure.

area of a Cross formula

$$\large{ A = l\;t + s \; \left( w - t \right) }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ l }$$ = height  $$\large{ in }$$  $$\large{ mm }$$
$$\large{ s }$$ = thickness $$\large{ in }$$ $$\large{ mm }$$
$$\large{ t }$$ = thickness  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

Distance from Centroid of a Cross formulas

$$\large{ C_x = \frac{ w }{ 2 } }$$

$$\large{ C_y = \frac{ l }{ 2 } }$$

Symbol English Metric
$$\large{ C }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$
$$\large{ l }$$ = height  $$\large{ in }$$  $$\large{ mm }$$
$$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

Elastic Section Modulus of a Cross formulas

$$\large{ S_{x} = \frac{ I_{x} }{ C_{y} } }$$

$$\large{ S_{y} = \frac{ I_{y} }{ C_{x} } }$$

Symbol English Metric
$$\large{ S }$$ = elastic section modulus $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ C }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$

Perimeter of a Cross formula

$$\large{ A = 2 \; \left( w + l \right) }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ l }$$ = height  $$\large{ in }$$  $$\large{ mm }$$
$$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

Polar Moment of Inertia of a Cross formulas

$$\large{ J_{z} = I_{x} + I_{y} }$$

$$\large{ J_{z1} = I_{x1} + I_{y1} }$$

Symbol English Metric
$$\large{ J }$$ = torsional constant $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$

Radius of Gyration of a Cross formulas

$$\large{ k_{x} = \sqrt{ \frac{ t\;l^3 \;+\; s^3 \; \left( w \;-\; t \right) }{ 12 \; \left[ l\;t \;+\; s \; \left( w \;-\; t \right) \right] } } }$$

$$\large{ k_{y} = \sqrt{ \frac{ s\;w^3 \;+\; t^3 \; \left( l \;-\; s \right) }{ 12 \; \left[ l\;t \;+\; s \; \left( w \;-\; t \right) \right] } } }$$

$$\large{ k_{z} = \sqrt{ k_{x}{^2} + k_{y}{^2} } }$$

$$\large{ k_{x1} = \sqrt{ \frac { I_{x1} }{ A } } }$$

$$\large{ k_{y1} = \sqrt{ \frac { I_{y1} }{ A } } }$$

$$\large{ k_{z1} = \sqrt{ k_{x1}{^2} + k_{y1}{^2} } }$$

Symbol English Metric
$$\large{ k }$$ = radius of gyration $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ l }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ s }$$ = thickness $$\large{ in }$$ $$\large{ mm }$$
$$\large{ t }$$ = thickness $$\large{ in }$$ $$\large{ mm }$$
$$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

Second Moment of Area of a Cross formulas

$$\large{ I_{x} = \frac{ t\;l^3 \;+\; s^3 \; \left( w \;-\; t \right) }{12} }$$

$$\large{ I_{y} = \frac{ s\;w^3 \;+\; t^3 \; \left( l \;-\; s \right) }{12} }$$

$$\large{ I_{x1} = I_{x} + A\;C_{y}{^2} }$$

$$\large{ I_{y1} = I_{y} + A\;C_{x}{^2} }$$

Symbol English Metric
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ C }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$
$$\large{ l }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ s }$$ = thickness $$\large{ in }$$ $$\large{ mm }$$
$$\large{ t }$$ = thickness $$\large{ in }$$ $$\large{ mm }$$
$$\large{ w }$$ = width $$\large{ in }$$ $$\large{ mm }$$

Tags: Structural Steel