 # How to Calculate Road Offset Profiles

## Why we need to use Profile Boards for Road Setting Out?

When constructing a new, or even an existing road we need to be able to translate design information into practical information that can be used throughout the road construction process. We do this by providing profile boards at a set distance off to the side of the road and at a set height above the proposed road.  These profile boards are placed on either side of the road usually at ten metre intervals along the road.  By using these profile boards along with a “Traveller” (a move-able profile board that is set at a desired height, for example at formation level) the site operatives can achieve the desired profile of the road to the required tolerance.

One of the first jobs a Site Engineer will have to undertake on a new Road Construction Job is to work out the positions and levels of the profiles that will be used to control the line and level of the proposed road. This of course is presuming that the site is not using Machine Control for this construction activity. In the example shown in the image below, we can see a cross-section through a typical proposed road. What the site engineer needs to do is calculate the plane of the road and offset it vertically so the site operatives can see the “traveller”.

The profiles also need to be set at a distance beyond the edge of the proposed edge of the road construction. This adds a further complication to the calculations that need to be done. So to offset the plane of the LHS (Left Hand Side) of the carriageway, the site engineer would need to work out points 1 and 3 in the below diagram, and for this the site engineer would need to calculate the easting, northing and elevation for both points.

At this point the site engineer would only have information (easting, northing and elevation) for the centreline point and the channel (Face of the Kerb) point from which to calculate these offset points. The site engineer would also have to calculate the elevations for the profile boards at positions 2 and 4, and check the easting and northing position for the these profile boards too, these should be the same as the easting and northing positions for points 1 and 3.

## How does a Site Engineer Calculate Positions and Levels for Road Offset Profiles?

The calculation of the positions and levels of the profiles is complicated and time consuming with many points being needed to be worked out even before you set foot out on site.  But a lot of this time can be saved through automation and using a programme to do the numerous calculations for you.  At the end of this page you can find out how to get a spreadsheet programme that calculates all of the offset points and levels that we are going to go through now.

#### Keep reading for a discount code for the brilliant spreadsheet that does all this work for you at the end of this article.

First you need to look through the (supplied from the designer) proposed road data. This should be in the format of cross-sectional information. That is, it should be presented in a format of point name, easting, northing, elevation, offset from the Centerline, for every 10m (minimum) Chainage intervals and every tangent point.

#### Cross-sectional information should look something like this:-

CHAINAGE      40.000

OFFSET         R.L.           LABEL CUT               EASTING                 NORTHING

1      -10.010      136.931          V001                   394499.950            295852.514

2       -8.955       136.934          F001                    394499.177            295853.232

3       -5.941       136.723          CL01                    394496.968            295855.282

4        0.000       136.787          M001                  394492.613            295859.323

5        6.876       136.861          CR01                    394487.573            295864.000

6        9.583       137.072          V002                    394485.589            295865.841

The negative offset denotes that point is to the left of the centerline and the convention (when considering roads that is) is to consider everything as we look up chainage.

From the above information a Site Engineer would need to use trigonometry to work out the Whole Circle Bearing (WCB) and the distance between the Centerline and the two channels.  If their calculations are correct, they should find the distances calculated to be the same as the offset distance (or a sum of these).

Using the information above, we first need to calculate the WCB and distance between points M001 and CL01.  This is done by using trigonometry and Pythagoras Theorem.

## To calculate the distance between two co-ordinated points we need to use the Pythagoras Theorem.

a² + b² = c²

(394496.968-394492.613)² + ( 295855.282-295859.323)²=c²

4.355² + (-4.041)² = c²

18.966 + 16.330 = c²

(Square Root) 35.296 = c

5.941 = c

As you can see this is the same dimension as the OFFSET distance from M001 to CL01 in the table above.

Now we need to work out the WCB (Whole Circle Bearing) from M001 to CL01.  To calculate the WCB we need to use trigonometry.

## What is the Whole Circle Bearing?

Plotting the relative co-ordinated points will help identify the WCB value.  The Whole Circle Bearing is the total angle from North that is the direction of travel from the points being considered.  This is measured in a clockwise direction from North.

From the information already calculated above, we have the lengths of all three sides of the right angled triangle to use for working out the bearing.  That is the Hypotenuse (5.491m), difference in Eastings (4.355m) and the difference in Northings (4.041). Plotting the two co-ordinated points relative to Easting and Northing Position the WCB becomes easy to see.

So, using trigonometry, we can work out the value for the angle x for a right angled triangle.

SOHCAHTOA

Using the Tan function we get

Tan (x) = -4.041/4.355

x= 42 Degrees 51 Minutes 30 Seconds

As we have drawn out the direction between M001 and CL01 above, we know that we now need to add 90 Degrees to the angle x we have calculated to give a WCB of 132 Degrees 51 Minutes 30 Seconds.

The above calculations only so far calculate information that we need in order to provide the information that we need in order to calculate the offset points we need to set out on site.

#### Get a free spreadsheet to work out the Whole Circle Bearings and Distances from Co-ordinates.

If you have followed the above calculation through and worked out your own WCB and Distance and have a load more points that you need to work out then you may find this excel spreadsheet available for free from Lichfield Survey Supplies, on their website.  It uses the same calculations as described above to calculate the WCB and distance between points given in Easting and Northing format.

## How to calculate the co-ordinates of the offset point.

Now that we know the WCB between M001 and CL01, we can use this information to calculate the position of the offset point that we need to mark out on site. This offset wants to be consistent throughout the length of the road construction.  Usually when doing this in road construction we choose 1m behind the face of kerb or channel.  The below diagram illustrates the point that we need to calculate.

So, we need to find the Eastings and Northings of the (1m) offset point by projecting the line from CL01 along the path of the WCB we calculated from M001 to CL01 and for this we need to use trigonometry again.

SOHCAHTOA

To do this we need to find out the distance in the Eastings and the distance in the Northings and add these dimensions (of the eastings and northings) to the Co-ordinates of CL01.

From the right-angled triangle, we know the hypotenuse length to be 1m, and our angle is going to be x (WCB of 132 Degrees 51 Minutes 30 Seconds) as we are traversing the same direction.  This time for the calculations of the Eastings we will need to use Cos and for the Northings we will need to use Sin.

So, for the Eastings we have (Hypotenuse is 1, as we are wanting a 1m offset in this case).

Cos (42-51-30) = (E)/1

E = 0.733m

And for the Northings we have (Hypotenuse is 1, as we are wanting a 1m offset in this case).

Sin x = Opposite/Hypotenuse

Sin (42-51-30) = (N)/1

N = 0.680m

Now that we know the individual easting and northing lengths we need to add these dimensions to the co-ordinates of point CL01.  But we need to be mindful of the direction that the offset point to the point CL01.  Looking at the diagram above we can see that the Eastings are going to increase and so we will need to add the dimension to the easting value of point CL01, but the Northings are going to decrease so we will need to subtract the dimension from the northing value of point CL01.  So, the Co-ordinates of the (1m) offset point would be: –

Easitngs: –

394496.968 + 0.733 = 394497.701

Northings: –

295855.282 – 0.680 = 295854.602

## Get a free spreadsheet to work out the co-ordinates from the Whole Circle Bearing and Distance.

If you have followed the above calculation through and worked out your own WCB and Distance and have a load more points that you need to work out then you may find this excel spreadsheet available for free from Lichfield Survey Supplies, on their website.  It uses the same calculations as described above to calculate the WCB and distance between points given in Easting and Northing format. Please note that this spreadsheet requires the WCB is entered in a decimal format, not in the format of degress, minutes and seconds.

## How to Calculate the Levels for the Offset Points.

So now that we have the position of the offset profile (for a 1m offset to the left of the kerbline) in eastings and northings, we now need to calculate the desired level for the profile board to be erected at.  To do this we need to work out the fall across the road from the centerline to the channel, that is the difference in height from M001 to CL01 divided by the distance from M001 to CL01.

We want to work out the fall across the road for the left hand side carriageway and express the answer in millimetres per metre.  We have the elevation of the centerline and the elevation of the left hand channel and we have the width of the lane from our calculations done earlier.

For the Left Hand Side of the road we have

M001 Elevation – CL01 Elevation = Fall from the centreline to the channel.

136.787 – 136.723 = 0.060m Fall from the centreline to the channel.

We can now calculate the rate of fall from the centreline to the channel.

Height of Fall / Length of Fall = Rate of Fall.

0.060 / 5.941 = 0.011m/m or 11mm/m fall

So, for the profile level for our offset point we would now need to do the following calculation, assuming we are offsetting 1m and raising the profile 1m Above FRL (Finished Road Level), as is common on site.

For the left hand Side of the road we have: –

CL001 Elevation – (Offset Distance * Rate of Fall) + Offset Height = Profile Height

136.723 – (1 * 0.011) + 1 = 137.712m AOD.

How to Achieve Accurate Levels.

How to Check a Total Station.

Network Rover GPS Accuracy.

## Now to work out the rest of the Profile Board Points.

At this point, we only have one offset position and one profile board level worked out.  This information, by itself is useless to the operative out on site.  They need a further profile board on the opposite side of the road for them to “bone” too.

We need this other profile board to be outside the road construction area where it will be safe for the duration of the works to be carried out.  To do this we need to start these calculations again, paying attention to the fact that our offset will be bigger this time.  So, for the LHS carriageway we need a profile board offset behind the RHS Kerbline, this should be a constant offset too, again 1m behind the face of kerb is ideal.

In order to ensure that the position we calculate for our profile board on the RHS is in a position that is satisfactory we need to check for any glaring errors in the data provided.  To do this we once again turn to the Pythagoras Theorem and Trigonometry.  We need to check that the offset distance from M001 to CR01 is correct and that the WCB from M001 to CR01 is 180 degrees to the WCB from M001 to CL01.  We calculated the WCB from M001 to CL01 earlier and it was 132 Degrees 51 Minutes 30 Seconds.

To calculate the distance between two co-ordinated points we need to use the Pythagoras Theorem.

a² + b² = c²

(394487.573-394492.613-)² + (295864.000-295859.323)²=c²

(-5.040)² + (4.677)² = c²

25.402 + 21.874 = c²

(Square Root) 47.276 = c

6.876 = c

As you can see, this is the same dimension as the OFFSET distance from M001 to CR01 in the table above, thus confirming our first check.

Now we need to work out the WCB (Whole Circle Bearing) from M001 to CR01.  To calculate the WCB we need to use trigonometry.

From the information already calculated above, we have the lengths of all three sides of the right angled triangle to use for working out the bearing.  That is the Hypotenuse (6.876m), difference in Eastings (5.040m) and the difference in Northings (4.677m).

So, going back to trigonometry, we can work out the value for the angle x for a right angled triangle, and therefore the WCB from M001 to CR01

SOHCAHTOA

Using the Tan function we get

Tan (x) = (4.677)/(-5.040)                              x= 42 Degrees 51 Minutes 38 Seconds

The angle from M001 to CR01 we have this time is just 8 seconds different from the angle we calculated from M001 to CL01, and over the distance we using we can ignore this difference.

As we have drawn out the direction between M001 and CR01 above, we know that we now need to add 270 Degrees to the angle x we have calculated to give a WCB of 312 Degrees 51 Minutes 38 Seconds.  We can see that this WCB is in effect 180 degrees to the WCB for M001 to CL01.

So, we need to find the Eastings and Northings of the (1m) offset point past the RHS Channel Line (CR02) by projecting the line from M001 along the path of the WCB we calculated from M001 to CR01 and for this we need to use trigonometry again.  Bear in mind that the total offset from M001 will be 7.876m.

SOHCAHTOA

To do this we need to find out the distance in the Eastings and the distance in the Northings and add these dimensions (of the eastings and northings) to the Co-ordinates of M001.

From the right-angled triangle, we know the hypotenuse length to be 1m plus the RHS Carriageway width of 6.876m, hence our offset of 7.876m from M001, and our angle is going to be x (WCB of 312 Degrees 51 Minutes 30 Seconds) as we are traversing across the road.  This time for the calculations of the Eastings we will need to use Cos and for the Northings we will need to use Sin.

So, for the Eastings we have (Hypotenuse is 1, as we are wanting a 1m offset in this case).

Cos (42-51-30) = (E)/7.876

E = 5.773m

And for the Northings we have (Hypotenuse is 1, as we are wanting a 1m offset in this case).

Sin x = Opposite/Hypotenuse

Sin (42-51-30) = (N)/7.876

N = 5.357m

Now that we know the individual easting and northing lengths we need to add these dimensions to the co-ordinates of point M001.  But we need to be mindful of the direction that the offset point to the point M001.  Looking at the diagram above we can see that the Eastings are going to decrease and so we will need to subtract the dimension from the easting value of point M001, but the Northings are going to increase so we will need to add the dimension to the northing value of point M001.  So, the Co-ordinates of the (1m past the RHS Kerbline) offset point would be: –

Eastings: –

394492.613 – 5.773 = 394486.840

Northings: –

295859.323 + 5.357 = 295864.680

We now have a position for our other profile board location.  We now need to work out the level that we are going to put the profile board on at.  This would be point No. 3 in this example.  To do this we need to use the rate of fall for the LHS carriageway (11mm/m) that we calculated earlier and project this fall from the M001 point.

The fall that we have this time is going up, so we need to add height to the M001 elevation at a rate of 11mm/m over the distance to the profile board and also add 1m to the height so that we have the same offset height on both profile boards.  The calculation would be as follows: –

Elevation = M001 Elevation + (Rate of Fall * Distance to profile)

Elevation = 136.787 + (0.011 * 7.876) + 1

Elevation = 136.787 + 0.087 + 1

Elevation = 137.874m AOD.

We now have the data required to set out on site just two profiles that can be used across the road.  These points are

Offset Profile 1.

394497.701m Eastings

295854.602m Northings

137.712m Elevation

Offset Profile 3.

394486.840m Eastings

295864.680m Northings

137.874m Elevation

At the moment we have just two profile boards worked out (1 and 3) for use on the LHS of the road.  It can be seen that we to also work out levels for the profile boards (2 and 4) for the RHS of the road.  The position of the offset points should be the same and so we only need to calculate the elevations for the profile boards using the methods detailed above.