# Thin Wall Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A 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

## formulas that use Area of a Thin Wall Rectangle

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

### Where:

$$\large{ A }$$ = area

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Distance from Centroid of a Thin Wall Rectangle

 $$\large{ C_x = \frac{ b }{ 2 } }$$ $$\large{ C_y = \frac{ a }{ 2} }$$

### Where:

$$\large{ C }$$ = distance from centroid

$$\large{ a, b }$$ = side

## formulas that use Elastic Section Modulus of a Thin Wall Rectangle

 $$\large{ S_x = \frac{ 2\;a\;b\;t }{ 3 } }$$ $$\large{ S_y = \frac{ 2\;a\;b\;t }{ 3 } }$$

### Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Perimeter of a Thin Wall Rectangle

 $$\large{ P_o = 2\; \left( a + b \right) }$$ (outside) $$\large{ P_i = 2\; \left( a + b - 4\;t \right) }$$ (inside)

### Where:

$$\large{ P }$$ = perimeter

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Plastic Section Modulus of a Thin Wall Rectangle

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

### Where:

$$\large{ Z }$$ = elastic section modulus

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Polar Moment of Inertia of a Thin Wall Rectangle

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

### Where:

$$\large{ J }$$ = tortional constant

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Radius of Gyration of a Thin Wall Rectangle

 $$\large{ k_{x} = \sqrt{ \frac{b}{6 \; \left( b \;+\; a \right) } } \; a }$$ $$\large{ k_{y} = \sqrt{ \frac{a}{6 \; \left( b \;+\; a \right) } } \; b }$$ $$\large{ k_{z} = \sqrt{ \frac{a\;b}{6} } }$$ $$\large{ k_{x1} = \sqrt{ \frac{5\;b \;+\; 3\;a} {12 \; \left( b \;+\; a \right) } } \;a }$$ $$\large{ k_{y1} = \sqrt{ \frac{3\;b \;+\; 5\;a}{12 \; \left( b \;+\; a \right) } } \;b }$$ $$\large{ k_{z1} = \sqrt{ \frac{ 3 \; \left( b^3 \;+\; a^3 \right) \;+\; 5\;b\;a \; \left( b \;+\; a \right) }{ 12 \; \left( b \;+\; a \right) } } }$$

### Where:

$$\large{ k }$$ = radius of gyration

$$\large{ a, b }$$ = side

## formulas that use Second Moment of Area of a Thin Wall Rectangle

 $$\large{ I_{x} = \frac{1}{3} \; b\;a^2\; t }$$ $$\large{ I_{y} = \frac{1}{3} \; b^2\; a\;t }$$ $$\large{ I_{x1} = \left( \frac{5}{6} \; b + \frac{1}{2} \; a \right) \; a^2\; t }$$ $$\large{ I_{y1} = \left( \frac{1}{2} \; b + \frac{5}{6} \; a \right) \; b^2\; t }$$

### Where:

$$\large{ I }$$ = moment of inertia

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness

## formulas that use Side of a Thin Wall Rectangle

 $$\large{ a = \frac{P}{2} - b }$$ $$\large{ b = \frac{P}{2} - a }$$

### Where:

$$\large{ a, b }$$ = side

$$\large{ P }$$ = perimeter

## formulas that use Torsional Constant of a Thin Wall Rectangle

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

### Where:

$$\large{ J }$$ = tortional constant

$$\large{ a, b }$$ = side

$$\large{ t }$$ = thickness