# Thin Wall Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Thin Wall Rectangle - Geometric Properties

• 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

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

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

### Polar Moment of Inertia 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 of a Thin Wall Rectangle formula

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

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

$$\large{ k_{z} = \sqrt { \frac {ab}{6} } }$$

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

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

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

### Second Moment of Area of a Thin Wall Rectangle formula

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

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

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

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

### 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