# Cross

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Two rectangles that intersect perpendicular at a center point.
• A cross is a structural shape used in construction.

### area of a Cross formula

$$\large{ A = l\;t + s \; \left( w - t \right) }$$

Where:

$$\large{ A }$$ = area

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

### Distance from Centroid of a Cross formula

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

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

Where:

$$\large{ C }$$ = distance from centroid

$$\large{ l }$$ = height

$$\large{ w }$$ = width

### Elastic Section Modulus of a Cross formula

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

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

Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ C }$$ = distance from centroid

$$\large{ I }$$ = moment of inertia

### Perimeter of a Cross formula

$$\large{ A = 2 \; \left( w + l \right) }$$

Where:

$$\large{ A }$$ = area

$$\large{ l }$$ = height

$$\large{ w }$$ = width

### Polar Moment of Inertia of a Cross formula

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

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

Where:

$$\large{ J }$$ = torsional constant

$$\large{ I }$$ = moment of inertia

### Radius of Gyration of a Cross formula

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

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

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

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

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

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

Where:

$$\large{ k }$$ = radius of gyration

$$\large{ A }$$ = area

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

### Second Moment of Area of a Cross formula

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

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

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

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

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ A }$$ = area

$$\large{ C }$$ = distance from centroid

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width