Cross

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • cross beam 1Two rectangles that intersect perpendicular at a center point.
  • A cross is a structural shape used in construction.

Structural Shapes

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

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

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

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

\(\large{ k_{x1} =  \sqrt  {  \frac { I_{x1} }  { A  }   }   }\)

\(\large{ k_{y1} =  \sqrt  {  \frac { I_{y1} }  { A  }   }   }\)

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

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

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

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

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

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

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

 

Tags: Equations for Moment of Inertia Equations for Structural Steel Equations for Modulus