Hollow Circle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • circle hollow 4Two circles each having all points on each circle at a fixed equal distance from a center point.
  • Center of a circle having all points on the line circumference are at equal distance from the center point.
  • A hollow circle is a structural shape used in construction.

 

Structural Shapes

 

Formulas that use area of a Hollow Circle

\(\large{ A_{area} =   \pi \; \left(  R^2  - r^2  \right)    }\)   

Where:

\(\large{ A_{area} }\) = area

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Circumference of a Hollow Circle

\(\large{ C =  2 \; \pi \; R   }\)  (outside) 
\(\large{ C =   2 \; \pi \; r   }\)  (inside) 

Where:

\(\large{ C }\) = circumference

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Distance from Centroid of a Hollow Circle

\(\large{ C_x =  r}\)    
\(\large{ C_y =  r}\)  

Where:

\(\large{ C_x, C_y }\) = distance from centroid

\(\large{ r }\) = inside radius

 

Formulas that use Elastic Section Modulus of a Hollow Circle

\(\large{ S =      \frac{  \pi \; \left(  R^4  \;-\; r^4  \right) }{ 4\;R }     }\)   

Where:

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

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Plastic Section Modulus of a Hollow Circle

\(\large{ Z =       \frac   { 4 \; \left(  R^3  \;-\; r^3  \right) }  { 3 }    }\)   

Where:

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

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

 

Formulas that use Polar Moment of Inertia of a Hollow Circle

\(\large{ J_{z} =  \frac { \pi }{2}  \;  \left(  R^4  - r^4  \right)   }\)   
\(\large{ J_{z1} =  \frac { \pi }{2}  \;  \left(  R^4  - r^4  \right)  +     2\; \pi \; R^2    \left(  R^2  - r^2  \right)     }\)  

Where:

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

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Radius of a Hollow Circle

\(\large{ r = \sqrt   {\frac {2 \; A_{area}} {\pi} }   }\)  

Where:

\(\large{ r }\) = inside radius

\(\large{ A_{area} }\) = area

\(\large{ r }\) = inside radius

 

Formulas that use Radius of Gyration of a Hollow Circle

\(\large{ k_{x} =    \frac {1}{2}  \;  \sqrt {   R^2  + r^2   }     }\)   
\(\large{ k_{y} =   \frac {1}{2}  \;  \sqrt {   R^2  + r^2   }       }\)   
\(\large{ k_{z} =   \frac { \sqrt { 2 } }{2}  \;  \sqrt {   R^2  + r^2   }     }\)   
\(\large{ k_{x1} =  \frac {1}{2}  \;  \sqrt {  5 \; R^2  + r^2   }    }\)  
\(\large{ k_{y1} =  \frac {1}{2}  \;  \sqrt {   5 \; R^2  + r^2   }   }\)  
\(\large{ k_{z1} =   \frac { \sqrt { 2 } }{2}  \;  \sqrt {  5 \; R^2  + r^2   }   }\)  

Where:

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

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

 

Formulas that use Second Moment of Area of a Hollow circle

\(\large{ I_{x} =   \frac { \pi }{4} \;   \left(  R^4  - r^4  \right)     }\)   
\(\large{ I_{y} = \frac { \pi }{4} \;   \left(  R^4  - r^4  \right)    }\)   
\(\large{ I_{x1} =   \frac { \pi }{4}  \;  \left(  R^4  - r^4  \right)  +     \pi \; R^2    \left(  R^2  - r^2  \right) }\)   
\(\large{ I_{y1} =  \frac { \pi }{4}  \;  \left(  R^4  - r^4  \right)  +     \pi \; R^2    \left(  R^2  - r^2  \right) }\)  

Where:

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

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Sector of a Hollow Circle

\(\large{ A =  \frac{\pi \; \theta \; \left( r^2 \;-\; R^2 \right)  }{360}  }\)   

Where:

\(\large{ A }\) = sector area

\(\large{ \theta }\) = angle

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

Formulas that use Torsional Constant of a Hollow Circle

\(\large{ J  =  \frac {  \pi \; \left( R^4 \;-\;  r^4 \right)  } {  2  }   }\)   
\(\large{ J  =  \frac {  \pi \; \left( D^4 \;-\;  d^4 \right)  } {  32  }   }\)   

Where:

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

\(\large{ d }\) =  inside diameter

\(\large{ D }\) =  outside diameter

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ \pi }\) = Pi

 

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