## Calculation of roof members

It is important to calculate the dimension of the true length of a roof member. These dimensions are essential to determine the accurate span of rafter, overhang, underpurlins, fan struts etc.
You can also use the Roof Multiplier Table, which you can print from the web page.

Example

Find the span of a rafter.

The AS 1684.2 – 2006 Residential timber-framing construction clearly differentiate between spacing and span (see Fig. 2.18 and Paragraph 2.7.5.

2.7.5.2 Spacing   the centre-to-centre distance between structural members, unless otherwise indicated.

2.7.5.3 Span        the face-to-face distance between points capable of giving full support to structural members or assemblies. In particular, rafter spans are measured as the distance between points of support along the length of the rafter and not as the horizontal projection of this distance.

2.7.5.5 Single span        the span of a member supported at or near both ends with no intermediate supports.

2.7.5.5 Continuous span        the term applied to members supported at or near both ends and at one or more intermediate points such that no span is greater than twice another.

Span/length calculation for roof members.

As can be seen the span of the rafter in the figure below is not quite in agreement with the Code but we will use the figures for the rafter span as calculated below. Figure 1

We use trigonometric functions for the calculation

The angle will be the roof pitch
The opposite site is the rise of the roof
The adjacent side is the rafter run
The hypotenuse is the true length of the rafter

The example shows brick veneer construction (see Figure 1) the roof pitch is 20°, the rafter run is 3.340 metres and for an eaves width of 0.6 metres, the actual measurement to calculate the overhang is in this case 0.76 metres (0.60 + 0.16 face brick + cavity).  Therefore the span of the rafter is 3.554 metre.

Please remember that the rafter span according to the Code is the face-to-face distance and that our calculated distance is 96 mm longer in this case (see Figure 2). This provides a safety margin, and in a border case, where the span is just a couple of mm too short, you may use the face-to-face distance for the span.

Calculation of roof overhang

You need to distinguish between clad timber framing and a brick-veneer construction calculate the eaves overhang. Look also at Figure 2.18 (b) in Section 2 of the code regarding the length of the overhang.

Let's assume an eaves width of 600 mm. The horizontal dimension for the overhang is then 600 mm plus the 110 mm brick wall plus the 50 mm cavity, which equals 760 mm.

Now do the same as for the rafter (the overhang is in this case the Hypotenuse).  The eaves overhang for the brick veneer building with 600 mm eaves width is 0.809 metre

Find the length of a strut

If a rafter is supported on to points only (single span) and an appropriate size cannot be found in the tables then an additional support (underpurlin) is needed. Underpurlin must be supported by struts.

Struts may be arranged vertically as shown in Figure 4  (a) or perpendicular to the rafter (b). The position of the underpurlin must be determined before you can calculate the length of the strut. To utilise the continuous span of a rafter the position of the underpulin must be in the middle one-third of the rafter as shown in the Figure 5. Remember that continuous span member is a member whereby no span is greater than twice another (Section 2, Paragraph 2.7.5.5). As soon as you have determined the position of the underpurlin you can calculate the length of the vertical strut using sin, cos or tan. However, the underpurlin in class examples is usually placed at midspan, because that simplifies the calculation for all roof-members of the Roof Framing Section.

Strutting

Roof struts can be applied in various ways, some examples are: Verticle struts                                           Perpendicular to the roof

Roof struts 7.2.15.1

Where necessary, struts shall be provided to support roof members, such as underpurlins, ridgeboards and hip and valley rafters.

Ridge strut

 This strut support the ridge down the center of the roof. If a ridge is strutted, then you need to find the length of the ridge strut. The length of the strut is found by using the opposite calculation: Vertical strut

If the underpurlin is positioned at midspan, then the vertical underpurlin strut length equals half of the length of the ridge strut (½ x 1216 = 608). Alternatively,by calculation ½ of the rafter run (½ x 3340 = 1670) multiplied by the tangent of the roof pitch. Therefore the length of the vertical strut is: Perpendicular strut

To calculate the length of a strut perpendicular to the rafter you need to calculate the rafter span (Hypotenuse) first. The rafter span (strut at midspan) is is half of the rafter run (3340/2) divided by cos 20°. Now you can use the tan-function again to calculate the strut perpendicular to the rafter. The length of the perpendicular strut is therefore  Fan strut

The span of an underurlin can be reduced if a fan-strut is used instead of a single strut. Paragraph 7.2.15.3 in AS 1684.2 - 2006 stipulates that the angle of a fan-strut should not exceed 45°.
(A single strut should not be less than 30° from the vertical.)

Fan struts are more effective with steeper roof pitches where the length of the strut is notable.

To reduce the span of an underpurlin effectively the fan strut should have an angle of 45° because this results in a maximum spread of the fan strut.

Find the spread of a fan strut.

Geometrically the fan strut should consists of two isosceles right angle triangles as shown in Figure 6. As both angles are the same therefore both sides must be the same. Check on your calculator the sin of 45° (= 0,707) and cos 45° (= 0.707) and you will see that you will get equal figure for both (sin and cos). In the previous calculation the length of the vertical strut is 0.608 m (at midspan) and the length of the perpendicular strut 0.647 m (at midspan)

The spread of a vertical fan strut is therefore                   0.608 x 2 = 1.216 m
and
the spread of a strut perpendicular to the rafter fan strut is therefore
0.647 x 2 = 1.294 m

The calculation of load width and roof area supported is easily understood if you consider the load on a structural member. Ask yourself what load is going onto a member. Study Section 2 of the code and look at the Figure 2.10 & 2.11 Floor Load Width (FLW), Figure 2.12 Ceiling Load Width (CLW) and Figure 2.13 - 2.16 Roof Load Width (RWL) and make sure you understand the significance of the load width.

Roof area supported (RAS)

For dimensioning of strutting beams (7.3.11), combined counter-strutting beams (7.3.10) or combined strutting/hanging beams (7.3.9) you need to know the roof area supported. The area can be easily found by be multiplying the RLW with the length of the underpurlin (½ x span 1 + ½ x span 2) that the strut supports. Usually we select only one size for the underpurlin and therefore only the worst case need to be considered. Find out how many strutting beams are needed. As soon you have determined how many strutting beams are required identify the worst case. This situation will be used for the size selection of the underpurlin.

Figure 7 below is an example that illustrate the process to find RAS. RAS equals the RLW of the underpurlin multiplied by the longest span of the underpurlin.

Refer to Figure 7 to find the worst case in the roof structure (longest underpurlin span). The strut on the left is vertical because the span on the left side of the strut is less then the reduced span 1. The span between the struts supported on walls is excessive and a strutting beam or combined counter-strutting beam is required. A fan struts have been chosen (see Figure 7) to reduce the span of the underpurlin even more (span u/p 1 and span u/p 2). As can be seen span u/p 1 has been reduced by ½ spread of the fan strut resulting in a reduce span 1. Span u/p 2 is reduced by the spread of the fan struts (left and right side) i.e. reduce span = span u/p 2 minus strut height ×2 ). Figure 7

Alternative strutting system

Where it is not possible to support underpurlins off walls or struts some alternatives can be applied as shown in Figure 8. Often underpurlins are projected (cantilevered) more than 25% of the maximum allowable span then you may reinforce the hiprafter with a tie-bolt truss system. The hiprafter in this case will support the underpurlin. Figure 8

Span and spacing

The Figure 8 shows you the difference between span and spacing of members and the load width (e.g. FLW in this case) for the middle bearer. The load area supported by the middle stump would be the floor load width (L1/2+L2 /2) × Bearer span (i.e. half of the bearer span to the left and half of the bearer span to the right, as indicated by the blue area). Figure 9

### Calculation set out

All calculations should be done on a separate A4 sheet . Make sure it's logical set out because you may need to refer to previous calculation figures. Write all dimensions down as well as you calculated figures. Follow a similar procedure as shown below:
1. Rafter run = external width between the wall plates divided by two.
2. Rafter span = rafter run divided by cos °.
3. Overhang = eaves width divided by cos ° (add dimensions for brick veneer).
4. Ridge strut = rafter run times tan °.
5. Decide whether an underpurlin is needed; if it is place it at mid-span.
6. New rafter span = rafter span found in 2) divided by two.
7. Vertical strut to underpurlin = ridge strut length divided by 2 (if u/p positioned at midspan).
8. Strut perpendicular to rafter = rafter span time tan °.
9. Determine the position of struts (usually on supporting walls).
10. If the distance between supporting walls is excessive a strutting beam may be needed.
11. Span of underpurlin can also be reduced if fan-strut is used.
12. Determine the length of the strut and the dimensions between the struts (or fan-struts).
13. Roof load width (RLW) = rafter span (if placed at midspan) otherwise ½ span1 + ½ span2.
14. Roof load area = RLW × (½ u/p span left + ½ u/p span right) or with fan struts
RLW × (½ u/p span left + ½ u/p span right + spread of fan strut).
15. Hanging beams are required if ceiling joist span is excessive.
16. Place hanging beams in center of room or if needed divide room length/width by 3 (4) and space them equally.
 Click here for a Calculation Template that you can print and use

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