A square tube is a common structural member used in engineering designs for things like bridges or buildings. Such a member is a special form of a beam, and the general beam deflection formulas can be used when calculating the deflection of a square tube.

## Components Of The Square Tube Deflection Calculators

To calculate the deflection of any structural member, there are some values that are needed, regardless of the specific engineering situation. These values include the material properties of the member, such as the modulus of elasticity, which is dependent on the material that is used, and the shape and dimensions of the member. The shape and dimensions are used to calculate the moment of inertia. In the case of a square tube, which is a specific case of a rectangular tube, the equation for the moment of inertia is:

where:

*W*is the exterior width of the tube, with SI units of m*H*is the exterior height of the tube, with SI units of m*w*is the interior width of the tube, with SI units of m*h*is the interior height of the tube, with SI units of m

An example of how these values can be determined is shown in the following figure:

In most cases, a tube will have a common thickness on all sides, but the equation for the moment of inertia can be used for different thicknesses on the top and bottom than on the sides.

Using the moment of inertia *I* and the modulus of elasticity *E*, which has SI units of GPa, the deflection of any square tube can be calculated using established equations or forms, which are often found in a lookup table.

## Square Tube Deflection Equations

A square tube used as a structural member can be treated as a square beam when determining the deflection. Depending on the boundary conditions on the square tube and load(s) applied, the square tube will deflect in different ways. The three common boundary conditions that can be applied for a beam are cantilever, simply supported, and fixed-fixed.

### Cantilever Square Tube Deflection

When calculating the deflection of a cantilever square tube, there are five common load cases, which are introduced in the following.

#### Cantilever Square Tube Deflection With End Load

The deflection *ฮด* of an end-loaded cantilever square tube is calculated using the following equation:

where:

*ฮด*is the deflection, with SI units of m*F*is the applied load, with SI units of N*x*is the location where the deflection is being calculated, with SI units of m*L*is the length of the square tube, with SI units of m*E*is the elastic modulus of the material, with SI units of GPa*I*is the moment of inertia of the square tube, with SI units of m^{4}

#### Cantilever Square Tube Deflection With Intermediate Load

The deflection of a cantilever square tube with a load at some point along its length is calculated using the following equations:

where *a* is the distance from the fixed end where the load is applied, with SI units of m.

#### Cantilever Square Tube Deflection With Uniform Distributed Load

The deflection of a cantilever square tube with a uniform load along its length is calculated using the following equation:

where *w* is the uniform load, with SI units of N.

#### Cantilever Square Tube Deflection With Triangular Distributed Load

The deflection of a cantilever square tube with a triangular load along its length is calculated using the following equation:

#### Cantilever Square Tube Deflection With End Moment

The deflection of a cantilever square tube with an end moment is calculated using the following equation:

where *M _{0}* is the moment applied at the end of the square tube, with SI units of Nโm.

### Simply Supported Square Tube

When calculating the deflection of a simply supported square tube, there are seven common load cases, which are introduced in the following.

#### Simply Supported Square Tube Deflection With Intermediate Load

The deflection of a simply supported square tube with an intermediate load is calculated using the following equation:

#### Simply Supported Square Tube Deflection With Center Load

The deflection of a simply supported square tube with a load at its center is calculated using the following equation:

#### Simply Supported Square Tube Deflection With Moment At Each Support

The deflection of a simply supported square tube with a moment at each support is calculated using the following equation:

#### Simply Supported Square Tube Deflection With Moment At One Support

The deflection of a simply supported square tube with a moment at one support is calculated using the following equation:

#### Simply Supported Square Tube Deflection With Center Moment

The deflection of a simply supported square tube with a moment at its center is calculated using the following equation:

### Fixed-Fixed Square Tube Deflection

When calculating the deflection of a square tube fixed at both ends, there are two common load cases, which are introduced in the following.

#### Fixed-Fixed Square Tube Deflection With Center Load

The deflection of a fixed-fixed square tube with a load at its center is calculated using the following equation:

#### Fixed-Fixed Square Tube Deflection With Uniform Distributed Load

The deflection of a fixed-fixed square tube with a uniform load along its length is calculated using the following equation:

## Example Calculation

As an example, the deflection of a cantilever square tube with the following cross section subject to an end load of 50 N is calculated. The tube is made of steel with an elastic modulus of 200 GPa and is 10 m long. To find the deflection at a point 2 m from the fixed end, the following steps are taken.

- First, the moment of inertia of the tube is calculated:

- Second, the deflection equation is used:

## Applications Of Square Tube Deflection

As stated earlier, a square tube is a specific type of rectangular tube, and is a common structural member used when weight needs to be conserved. By optimizing the dimensions of a square tube for the expected loads, it can be determined whether the structure will deflect within certain tolerances. By using a square tube, it is possible to reduce the weight of a structure that does not necessarily need a solid beam for support.