Thin Wall Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • hollow thin wall rectangle 3A two-dimensional figure that is a quadrilateral with two pair of parallel edges.
  • A thin wall rectangle is a structural shape used in construction.
  • Interior angles are 90°
  • Exterior angles are 90°
  • Angle \(\;A = B = C = D\)
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Structural Shapes

Area of a Thin Wall Rectangle formula

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

Where:

\(\large{ A }\) = area

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

\(\large{ t }\) = thickness

Distance from Centroid of a Thin Wall Rectangle formula

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

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

Where:

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

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

Elastic Section Modulus of a Thin Wall Rectangle formula

\( \large{ S_x =  \frac{  2\;a\;b\;t }{ 3  }  }\)

\( \large{ S_y =  \frac{ 2\;a\;b\;t }{ 3  }  }\)

Where:

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

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

\(\large{ t }\) = thickness

Perimeter of a Thin Wall Rectangle formula

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

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

Where:

\(\large{ P }\) = perimeter

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

\(\large{ t }\) = thickness

Plastic Section Modulus of a Thin Wall Rectangle formula

\( \large{ Z_x =  2 \; \left[    b\;t  \;  \left(  \frac{a}{2} - \frac{t}{2}  \right)     + t  \; \left(  \frac{a}{2} - t  \right)^2       \right]    }\)

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

Where:

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

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

\(\large{ t }\) = thickness

Polar Moment of Inertia of a Thin Wall Rectangle formula

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

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

Where:

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

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

\(\large{ t }\) = thickness

Radius of Gyration of a Thin Wall Rectangle formula

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

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

\(\large{ k_{z} =   \sqrt{  \frac{a\;b}{6}   }   }\)

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

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

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

Where:

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

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

Second Moment of Area of a Thin Wall Rectangle formula

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

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

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

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

Where:

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

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

\(\large{ t }\) = thickness

Side of a Thin Wall Rectangle formula

\( \large{ a = \frac{P}{2} - b   }\)

\( \large{ b = \frac{P}{2} - a  }\)

Where:

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

\(\large{ P }\) = perimeter

Torsional Constant of a Thin Wall Rectangle formula

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

Where:

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

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

\(\large{ t }\) = thickness

 

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