# What is a Reasonable Span for a Carrier made of a Certain Material

#### General Arrangement and Use of a Carrier

In general a carrier is a bulk body with length much bigger than its crossection. Carriers are used for various type of constructions such as houses (roofing and floor), bridges, cars etc. One of the most common arrangement of a carrier is the one seen in figure 1 where the carrier is supported at both ends, in this case through a brick wall and a load is acting at its middle, in this case through a sandbag tied at the middle of the carrier. Also notice that in the present case the carrier has a circular cross-section.

As a next step in this article a simple method is presented for finding out if a carrier similar to the one of figure 1 has a reasonable span with respect to its loading (mass of sandbag), cross-section geometry (round cross-section) and material (wood). For this purpose some simple calculations have to be performed.

Figure 1: Common use of a carrier having circular cross section and being supported from brick walls at its ends. The loading is taking place through the mass of the tied sandbag at its middle

As a next step figure 1 is simplified so that figure 2 results. This allows to make the problem more abstract so that one can better concentrate on the factors having an influence on the span. The carrier is now a thin line and the supported edges are drawn as triangles on the brick wall. The loading of the sandbag is drawn as an arrow.

Figure 2: Simplified carrier arrangement

So finally what are the factors that have a big influence on the span of a carrier? This factors are always three

## The Three Main factors influencing the allowable span of a carrier

The three factor are:

+ The geometry factor and hence the cross-section geometry of the carrier (see examples of cross-sections in figure 3 below)

+ The material factor and thous the material of the carrier (for instance steel, wood or reinforced concrete) Remember: NEVER use for a carrier concrete or similar brittle materials without reinforcement! The possibility that it will collapse is almost certain!

+ The load application factor and hence at which point and how the load is acting on the carrier (for the case of figure 1 it is acting in the middle of the carrier. A load in the middle of the carrier gives the lowest factor and hence can be used for all the other cases too )

### The geometry factor

Using the dimensions of the cross section geometry one gets a single value which forms the geometry factor (In engineering this factor is called section modulus and is also related to the area moment of inertia). For instance 120 cm^3 in which cm^3 is a unit of measure for this factor (like meters or hours would be for length and time). Since the calculation of this factor is difficult and time consuming tables are given for the most common cross-section geometries as can be seen in table 1 highlighted with green. The detailed explanation on how the geometry factor is calculated can be found on Wikipedia: area moment of inertia.

Figure 3: An overview of the most common carrier cross-sections
Table 1: Some examples for the geometry factor highlighted with green (in cm^3)

Notice that from the above mentioned cross-section geometries the best choice concerning its wheight and the allowable span would be to use the "H-shaped" cross-section as for instance one can find in railroad tracks. Never the less the availability of tree stems or wooden carriers with round or rectangular cross-section is higher but they also do the job very good.

### The Material factor

As you may have experienced yourself every material has a certain strength. For instance if you take two rods of the same diameter one of wood and one of metal you will see that to brake the wooden rod in half would need much less effort than the metal one. For the same reason a steel carrier would have a much higher allowable span than a wooden one. This phenomenon is described using a certain number for each material (called yield stress). For instance for steel a value of 200 MPa is a common value. MPa is a unit to measure this phenomenon (like meters or hours would be for length and time). A perfect reference for the values of the different carrier materials can be found under the site Wolfram aplha

For instance for all kind of wooden carriers: values of the material factor (tensile strength) for wood. For Steel as mentioned one can take a value of 200 MPa for cast Iron and aluminum a value of 100 MPa. Notice that the calculation of a reinforced concred carrier is more complicated this is why it is omitted at this point.

### The Load Application factor

As we have shown in the above example the load is applied in the middle of the carrier. Never the less if it would be possible to apply the load instead of one point in the middle at several point of the carrier there would be a big benefit. Since distributed loading leads to smaller loads in the inside of the carrier and allows so higher spans. For the case of the sandsack this would mean that more ropes have to be applied or for the case of a roofing several vertical links would be needed. For most cases the load factor can be taken as 0.31

## Is finally the span reasonable or not?

To answer the question if finally the span of the carrier is reasonable the following calculation has to be done using following values: The geometry factor, the material factor, the loadfactor and finally the value of the load through the use of the mass in kg. Using these values leads to the following expression: $L_{S}= \frac{{F_L} \cdot M_F \cdot G_F}{m}$

The following symbols are used:

$L_{S}$ Allowable Span in meters. Never exceed this length!!!

${F_L}$ Load factor.

${M_F}$ Material factor.

${G_F}$ Geometry factor.

$m$ Mass of the load in Kg.

Remember: A calculation is made so that some addtional safty for the construction is provided. This does not exclude collapse due to other external factors. Its is therefore of big importance to alway keep on eye on the construction and its state.

## Examples of reasonable spans for different Carriers

In the following table some examples of different carriers loaded as seen in figure 1 are presented.

Figure 5: Calculation of a reasonable span for a H-shaped cross section geometry

From figure 5 it results that a H-shaped carrier having the given geometry and made of steel should not exceed 3.98 meter when loaded with a sandbag of 300Kg at its middle.

Figure 6: Calculation of a reasonable span for a round cross section geometry

From figure 6 it results that a round carrier having the given geometry and made of would should not exceed 1.99 meters when loaded with a sandbag of 30Kg at its middle.

Notice: The author accepts no responsibility for the safety of a construction or the correctness of the article