# 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.
• See Geometric Properties of Structural Shapes
• 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

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

Where:

$$\large{ A }$$ = area

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

$$\large{ t }$$ = thickness

### Perimeter of a Thin Wall Rectangle formula

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

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

Where:

$$\large{ P }$$ = perimeter

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

### 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

### 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

### 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