Thin Wall Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Thin Wall Rectangle - Geometric Propertieshollow thin wall rectangle 2

  • Rectangle is a quadrilateral with two pair of parallel edges.
  • Interior angles are 90°
  • Exterior angles are 90°
  • Angle \(\;A = B = C = D\)
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Area of a Thin Wall Rectangle formula

\( \large{ A = 2t  \left(  b + a \right)     }\)

Perimeter of a Thin Wall Rectangle formula

\( \large{ P= 2 \left( a + b     \right)  }\)   ( Outside )

\( \large{ P= 2 \left( a + b  - 4t  \right)  }\)   ( Inside )

Side of a Thin Wall Rectangle formula

\( \large{ a = \frac {P} {2} - b   }\)

\( \large{ b = \frac {P} {2} - a  }\)

Distance from Centroid of a Thin Wall Rectangle formula

\( \large{ C_x =  \frac { b }  { 2 }  }\)

\( \large{ C_y =  \frac { a }  { 2}  }\)

Elastic Section Modulus of a Thin Wall Rectangle formula

\( \large{ S_x =  \frac {  2abt }  { 3  }  }\)

\( \large{ S_y =  \frac { 2abt }  { 3  }  }\)

Plastic Section Modulus of a Thin Wall Rectangle formula

\( \large{ Z_x =  2  \left(    bt    \left(  \frac {a}{2} - \frac {t}{2}  \right)     + t   \left(  \frac {a}{2} - t  \right)^2       \right)    }\)

\( \large{ Z_y =     2t   \left(  \frac {a}{2} - t  \right)       \left(  \frac {b}{2} - t  \right)    + 2bt   \left(  \frac {b}{2} - \frac {t}{2}  \right)    }\)

Moment of Inertia about Axis of a Thin Wall Rectangle formula

\(\large{ I_{x} =  \frac {1} {3}    b a^2 t     }\)

\(\large{ I_{y} =  \frac {1} {3}    b^2 at      }\)

\(\large{ I_{x1} =   \left(  \frac {5} {6}  b  +   \frac {1} {2}  a    \right)    a^2 t      }\)

\(\large{ I_{y1} =  \left(  \frac {1} {2}  b  +   \frac {5} {6}  a    \right)    b^2 t      }\)

Polar Moment of Inertia about Axis of a Thin Wall Rectangle formula

\(\large{ J_{z} =   \frac {abt} {3}    \left( a + b  \right)      }\)

\(\large{ J_{z1} =   \left(    \frac {1}{2}  \left(  b^3 + a^3  \right)        +  \frac {5}{6}  ba  \left(  b + a  \right)       \right)  t    }\)

Radius of Gyration about Axis of a Thin Wall Rectangle formula

\(\large{ k_{x} =   \sqrt {  \frac {b} {6  \left(  b + a  \right)   }   }  a   }\)

\(\large{ k_{y} =    \sqrt {  \frac {a} {6  \left(  b + a  \right)   }   }  b   }\)

\(\large{ k_{z} =   \sqrt {  \frac {ab}{6}   }   }\)

\(\large{ k_{x1} =   \sqrt {  \frac {5b + 3a}  {12  \left(  b + a  \right)   }   }  a    }\)

\(\large{ k_{y1} =  \sqrt {  \frac {3b + 5a}  {12  \left(  b + a  \right)   }   }  b    }\)

\(\large{ k_{z1} =   \sqrt {   \frac{  3  \left(  b^3 + a^3  \right)   +  5ba  \left(  b + a  \right)  }    {  12  \left(  b + a  \right)   }     }   }\)

Torsional Constant of a Thin Wall Rectangle formula

\( \large{ J =      \frac  { 2t^2   \left(  b - 2  \right)^2   \left(  a - t  \right)^2 }   { at + bt - 2t^2 }       }\)

 

Where:

\(\large{ A }\) = area

\(\large{ a, b }\) = side

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

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

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

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

\(\large{ P }\) = perimeter

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

 

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