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

 

formulas that use area of a Cross

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

Where:

\(\large{ A }\) = area

\(\large{ l }\) = height

\(\large{ s }\) = thickness

\(\large{ t }\) = thickness

\(\large{ w }\) = width

 

formulas that use Distance from Centroid of a Cross

\(\large{ C_x =  \frac{ w }{ 2  }  }\)   
\(\large{ C_y =  \frac{ l }{ 2  }  }\)   

Where:

\(\large{ C }\) = distance from centroid

\(\large{ l }\) = height

\(\large{ w }\) = width

 

formulas that use Elastic Section Modulus of a Cross

\(\large{ S_{x} =  \frac{ I_{x} }{ C_{y}   } }\)   
\(\large{ S_{y} =  \frac{ I_{y} }{ C_{x}   } }\)   

Where:

\(\large{ S }\) = elastic section modulus

\(\large{ C }\) = distance from centroid

\(\large{ I }\) = moment of inertia

 

formulas that use Perimeter of a Cross

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

Where:

\(\large{ A }\) = area

\(\large{ l }\) = height

\(\large{ w }\) = width

 

formulas that use Polar Moment of Inertia of a Cross

\(\large{ J_{z} =  I_{x}  +  I_{y} }\)  
\(\large{ J_{z1} =  I_{x1}  +  I_{y1} }\)   

Where:

\(\large{ J }\) = torsional constant

\(\large{ I }\) = moment of inertia

 

formulas that use Radius of Gyration of a Cross

\(\large{ k_{x} =  \sqrt{  \frac{  t\;l^3 \;+\;  s^3 \; \left( w \;-\; t \right)       }{ 12 \; \left[  l\;t \;+\;  s \; \left( w \;-\; t \right) \right]    }   }   }\)   
\(\large{ k_{y} =  \sqrt{  \frac{  s\;w^3 \;+\;  t^3 \; \left( l \;-\; s \right)       }{ 12 \; \left[  l\;t \;+\;  s \; \left( w \;-\; t \right) \right]    }   }   }\)   
\(\large{ k_{z} =    \sqrt{  k_{x}{^2} + k_{y}{^2}  } }\)   
\(\large{ k_{x1} =  \sqrt{  \frac { I_{x1} }{ A  }   }   }\)  
\(\large{ k_{y1} =  \sqrt{  \frac { I_{y1} }{ A  }   }   }\)  
\(\large{ k_{z1} =    \sqrt{  k_{x1}{^2} + k_{y1}{^2}  } }\)  

Where:

\(\large{ k }\) = radius of gyration

\(\large{ A }\) = area

\(\large{ l }\) = height

\(\large{ s }\) = thickness

\(\large{ t }\) = thickness

\(\large{ w }\) = width

 

formulas that use Second Moment of Area of a Cross

\(\large{ I_{x} =   \frac{  t\;l^3 \;+\;  s^3 \;  \left( w \;-\; t \right)   }{12}   }\)   
\(\large{ I_{y} =   \frac{  s\;w^3 \;+\;  t^3 \;  \left( l \;-\; s \right)   }{12}   }\)   
\(\large{ I_{x1} =  I_{x}  +  A\;C_{y}{^2} }\)   
\(\large{ I_{y1} =  I_{y}  +  A\;C_{x}{^2} }\)  

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

 

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