Baseline for Setting Out Theodolite

Topic 7: Setting out

Aims -Understand the roles of the various different types of personnel who are involved in the setting out process -Understand the aims of setting out -Refer to the different types of plans that may be used in the setting out process -Appreciate the good working practices that should be undertaken in order that the aims of setting out can be achieved -Understand the procedures required to ensure that the horizontal and vertical control requirements of setting out operations can be met -Set out design points on site by a number of methods -Apply horizontal and vertical control techniques to second-stage setting out operations -Appreciate the application of laser instruments in surveying and setting out

What is setting out? A definition of setting out, often used, is that it is the reverse of surveying. Whereas surveying is a process for forming maps and plans of a particular site or area, setting out begins with plans and ends with the various elements of a particular plan correctly positioned on site. However most techniques and equipment used in surveying are also used in setting out i.e. while surveying may be the opposite of setting out, the processes and instruments are almost identical.

The International Organisation for Standardisation (ISO) define setting out as: Setting out is the establishment of the marks and lines to define the position and level of the elements for the construction work so that works may proceed with reference to them. This process may be contrasted with the purpose of surveying which is to determine by measurement the position of existing features.

-Setting out is one application of surveying -Most of the techniques and equipment used in surveying are also used in setting out -Mistakes in setting out can be costly -For setting out to be undertaken successfully good work practices should be employed -There are three parties involved in the construction procedures: the employer, the engineer and the contractor -Although the engineer checks the work, the setting out is the responsibility of the contractor -The cost of correcting any errors in the setting out has to be paid for by the Contractor, provided the engineer has supplies reliable information in writing

The aims of setting out There are two main aims when undertaking setting out operations:

-The various elements of the scheme must be correct in all three dimensions both relatively and absolutely, that is each must be its correct size, in its correct plan position and correct reduced level

-Once setting out begins it must proceed quickly with little or no delay in order that the works can proceed smoothly and the cost can be minimised. It must always be remembered that the contractors main commercial purpose is to make a profit – therefore setting out needs to be done efficiently.

Principles of setting out The main aim of setting out is to ensure that the various elements of the scheme are positioned correctly in all three dimensions.

Horizontal control techniques In order that the design of the scheme can be correctly fixed in position, it is necessary to establish points on the site which the E, N coordinates are known. These are horizontal control points and, once they have been located they can be used with a positioning technique to set out E, N coordinates of the design points. Two factors need to be taken into account when establishing horizontal control points. 1. The control points should be located throughout the site in order that all the design points can be fixed from at least two or three of them so that the work can be independently checked. 2. The design points must be set out to the accuracy stated in the specifications

The accuracy must be obtained throughout the whole network and this can be achieved by establishing different levels of control based on one of the fundamental tenets of surveying: working from the whole to the part. In practice, this normally involves starting with a small number of very accurately measured control points (known as first level or primary control) which enclose the area in question and then using these to establish second level or secondary control points near the site. When establishing the control network care needs to be taken that the tolerances specified are met. An example if working from the whole to the part using two different levels of control are shown in the next diagram. In this, the first level of control is provided by a traverse which is run through the site in question to provide a number of well positioned primary control points. These in turn are used to establish a second level of control, in this case secondary site points at each of a series of baselines which define important elements of the scheme.

On some schemes the same control points that were used in the production of the site plan prior to design work are used for setting out. These muse be remeasured before setting out – as positions may have changed for a number of reasons. Horizontal control points should be located as near as possible to the site in open positions for ease of working, but well away from the construction area and traffic routes to avoid them being disturbed.

The construction and protection of control points is very important. Wooden pegs are often used for non-permanent stations. For permanent control points it is recommended that they be constructed with concrete – as shown below.

Baselines A baseline is a line running between two points of a known position. Any baselines required to set out a project should be specified on the setting out plan by the designer and included in the contract. Baselines can take many forms: they can be simply two specified points joined, they can run between two buildings, they can mark the boundary with an existing building/development or they can mark the centre line for a new road. Baselines can be used in a number of different ways: - Where a baseline is specified to run between two points then once the points have been established on site, the design points can be set out from the baseline by offsetting using tapes (as seen below).

A design point D is to be set out at right angles to a baseline AB from point C which lies at a distance y from point A. The required offset distance from C to D is x. Distances x and y will be given by the designer and will usually be horizontal distances.

- Primary site control points, such as traverse stations E & F in the figure below can be use to establish a baseline AB by angle α and distance l values.

Subsidiary offset lines can then be set off at right angles from each end of the baseline to fix two corners R and S of building Z. Once R and S have been pegged out, the horizontal length of RS is measured and checked against its designed value. If it is within the required tolerance, points R and S can be used as a baseline to set out the corners T and U.

- Design points can be set out by taping known as distances from each end of a baseline as shown below.

At point A on building X is set out by taping dimensions 1 and 2 from the baseline and point B by taping dimensions 3 and 4. As before, the set out lengths of AB is then checked against its designed value and within tolerance, it can be used as a baseline to set out corners C and D.

-In some cases, the designer may specify a baseline that runs between points on two existing buildings. Design points are then set out from this line either by offsetting at right angles or by measuring distances from points on the line. The accuracy of this method depends upon how well the baseline can be established and how the dimensions required to set out the design points are known.

The accuracy of the baselines method increases if two baselines at right angles to each other are used. Design points can be established either by measuring and offsetting from both lines, or a grid system can be set up to provide additional control points in the area enclosed by the baselines.

Reference grids A control grid enables points to be set over a large area. Several different grids can be used in setting out -Survey grid: is drawn on the survey plan from the original traverse or network. The grid points have known eastings and northings related either to some arbitrary origin or to the national grid. -Site grid: is used by the designer. It is usually related in some way to the survey grid and should, if possible, actually be the survey grid, the advantage of this being that if the original control stations have been permanently marked then the design points will be on the same coordinate system and setting out is greatly simplified.

- The structural grid is established around a particular building or structure which contains much detail such as columns, which cannot be set out with sufficient accuracy from the grid site. -The secondary grid is established inside the structure from the structural grid when it is no longer possible to use the structural grid to establish internal features of the building – as the vision becomes obscured.

Offset pegs Whether used in the form of a baseline or a grid, the horizontal control points are used to establish design points on the proposed structure. Once excavations for foundations begin, the corner pegs will be lost. To avoid this extra pegs called offset pegs are used

Vertical control techniques In order that design points on the works can be positioned at their correct levels, vertical control points of known elevation relative to some specified vertical datum are established. To ordnance datum is commonly used and levels on the site are reduced to a nearby OS benchmark. Transferred or temporary benchmarks The positions of TBMs should be fixed during the initial reconnaissance so that their construction can be completed in good time and they can be allowed to settle before levelling them in. In practice, 20mm diameter steel bolts and 100mm long, driven into existing steps, ledges, footpaths etc are ideal.

If TBM are constructed at ground level on site, a design to that shown below should be used.

There should never be more that 80m between TBMs on site and the accuracy of levelling should be within the following limits: Site TBM relative to the MBM ± 0.005m Spot levels on soft surfaces relative to a TMB ± 0.010m Spot levels on hard surfaces relative to a TBM ± 0.005m

Sight Rails These consist of a horizontal timber cross piece nailed to a single upright or a pair of uprights driven into the ground (see below)

The upper edge of the cross piece is set to a convenient height above the required plane of the structure, usually to the nearest 100mm, and should be a height above ground to ensure convenient alignment by eye with the upper edge.

Sight rails are usually offset 2 or 3 metres at right angles to construction lines to avoid them being damaged as excavations proceed.

Travellers and boning rods A traveller is similar in appearance to a sight rail on a single support and is portable. The length of the upper edge to its base should be a convenient dimension to the nearest half metre.

Travellers are used in conjunction with sight rails. The sight rails are set some convenient value above the required plane and the travellers are constructed so that their length is equal to this value.

As excavation works proceeds, the traveller is sighted in between the sight rails and used to monitor the cutting and filling.

Slope rails or batter boards For controlling side slopes on embankments and cuttings slope rails are used. For an embankment the slope rails usually define a plane parallel to the slope of the embankment offset by a convenient distance:

For a cutting the slope rails can either be used to define the actual plane of the slope or an offset plane as shown below:

The advantage of the above method being that additional slope rails may be added as excavation proceeds.

The advantage of this method being that the slope rail can be lower in height and may make it easier to sight along than the example above.

Positioning Slope Rails In order to position slope rails we must first locate the toe of the embankment. Consider the embankment below, which runs from A to B with a width of 12m. Point C is on the existing ground level. The sides of the embankment are to slope at 1 in s. the procedure is as follows:

1.From the Road Design / Plans obtain the reduced level of A. 2. Peg out point C by measuring a distance 6m horizontally from F at right angles to the centreline. 3.Peg out points at 5m intervals from point C towards and beyond T. 4. Measure the reduced level on the ground surface at the first 5m peg 5.Calculate the proposed reduced level of on the embankment slope above this point from:

6. Compare the measure and calculated values at the 5m point, if the ground level measured is lower than the calculated slope level, the toe is located a further 5m away from C. 7. Repeat the procedure for the 10m peg, the calculation becomes:

Once the Toe has been located the wooden uprights of the slope rails can be hammered in at some offset from the embankment/cutting. The next stage is to calculate the required reduced levels at which the tope edges of the slope rails must be fixed to the wooden uprights.

For an embankment, assuming that a 1.5m traveller is to be used as shown, the reduced levels of P and Q should be obtained using (it is assumed that the RL at the toe is known):

For a cutting the reduced levels of R and S should be obtained using (it is assumed that the RL at edge of the embankment is known):

Profile boards These are similar to sight rails but are used to determine the corners and sides of buildings. Offset pegs are normally used to enable building corners to be relocated after foundation excavation. Profile boards are normally erected near each offset peg and used in the same way as a sight rail.

A variation on corner profiles is to use a continuous profile all around the building ser to a particular level above the required structural plane. The advantage of a continuous profile is that the lines of the internal walls can be marked on the profile and strung across to guide construction.

Coordinate positioning techniques For setting out by coordinates to be possible, a control network consisting of coordinated points (with heights) must be established on site. These are obtained by using theodolites, tapes, GPS and total station. Setting out using a theodolite and tape To set out using coordinates by theodolite and tape, one of the following procedures is used: 1. Angle and distance from two control points e.g. from point A below, can be set out from a control point S using one of two methods: Using the inverse calculation, determine the horizontal length l (SA) and the whole circle bearings of ST and SA.

With the theodolite set up at S, sight T and set the horizontal circle to read zero along this direction. Then the telescope is rotated through angle α to fix the direction to A and measure l along this direction to fix the position of A. This is known as setting out by angle and distance.

An alternative method would be to: compute l, WCB (ST) and WCB(SA) as per the first method. Sight T from S and set the horizontal circle of the theodolite to read the WCB of ST. Rotate the telescope towards point A until the WCB of SA is read on the horizontal circle.

The telescope line of sight is no defining the direction of A and the exact position of A can be fixed by measuring a horizontal distance l along this direction. This is setting out by bearing and distance.

2. Intersection with two theodolites, from four control points using angles or bearings only. Intersection is shown below.

When setting out using coordinate-based methods with theodolites and tapes, the situation may arise where there are no nearby control points available for this. This is overcome by establishing a free station at any convenient place for setting out. This is shown in the next FIG and it is essentially a resection. Free station points are particularly applicable to large sites where the coordinates of prominent features and targets on nearby buildings or parts of the construction are known.

The following steps are used when setting up a free station point: -The theodolite is set up at some suitable place in the vicinity of the points which are to be set out – hence the title free station as the choice of the instrument position is arbitrary. -Any angular resection is carried out to fix the position of the free station point. -The coordinates of the free station are calculated Following this, setting out continues as before and the required design points are ser out using the theodolite at the free station point.

Although setting out can be conducted using theodolites, tapes (and levels) in what might be sometimes called traditional methods, a lot of work on site is done using total stations and GPS equipment.

When setting out by so-called traditional methods, direct methods of angle and distance are taken to position structures and other works from nearby control points or from baselines.

Following this, offsets and profiles are put in place to define the main lines of a building and provide vertical control for second stage setting out.

Despite their popularity on site, these well-established methods have the disadvantages that the horizontal and vertical components of setting out have to be done separately (levelling must be used for any heighting), they can be time consuming if a lot of points have to be set out, and they require at least two people to do the setting out.

Setting out by total station To use a total station for setting out, it must be levelled and then centred over a control point in the same way as for a theodolite. As before this must be done correctly otherwise the subsequent readings taken with the instrument will not give the correct results.

Having set up the total station, it has to be orientated horizontally to the site coordinate system and it may also have to be orientated vertically. For horizontal orientation, the coordinates of the control point at which the instrument is set up are entered into the total station.

An adjacent control point is then chosen as a reference point (reference object) and the coordinates for this site are also keyed in. To orientate the total station, the RO is sighted and the horizontal circle orientation programme automatically computes the bearing from the total station to the RO.

For vertical orientation, the height of collimation of the total station has to be determined. If the height of the control point at which the total station is known, this is entered into the instrument or is already stored in the control point data.

Once the total station has been orientated it can be used for setting out horizontal positions either using the coordinates of the points to be set out directly or using bearing and distance values calculated from these coordinates. Two approaches can be used.

-When the coordinates of the point to be set out are used, these are usually contained in the file together with the coordinates of the control points for the project, and this is downloaded to the total station before work commences.

-If the bearing and distance to be set out are known, these can also be used for setting out. They are entered into the total station and, as soon as the appropriate key(s) are pressed to activate this is setting out mode, the instrument once again displays the difference between the entered and measured bearing values.

Setting out by GPS For setting out by GPS, an RTK system is required consisting of two geodetic receivers working in precise relative mode. One of these will be permanently located at a base station and the other (the rover) will move around the site and take the measurements needed for positioning design points. In common with all other setting out methods, GPS is based on a control network, which must be in place before any work can start. Control points with positions defined on the site grid are needed for base stations, for determining transformation parameters when deriving site coordinates from GPS coordinates. Depending on the site, control can be local and based on an arbitrary coordinate system or it can be connected to a national system.

For small local sites a control network consisting of at least three but preferably five points with known site coordinates and heights is required for determining transformation parameters. This can be surveyed using a total station and traverse methods.

On large sites, whether they cover an extensive area or are long linear sites such as those occurring on road and railway projects, site control is often based on national control.

Applying the principles of setting out Stages in setting out As the works proceed, the setting out falls into two broad stages.

First stage setting out In practice, first stage setting out involves the use of many of the horizontal and vertical control methods and positioning techniques . The purpose of this stage is to locate the boundaries of the works in their correct position on the ground surface and to define the major elements. In order to do this, horizontal and vertical control points must be established on or near the site.

Second stage setting out Second stage setting out continues on from the first stage, beginning at the ground floor slab, road sub-base level etc. Up to this point, all the control will be outside the main construction, for example, the pegs defining building corners, centre lines and so on will have been knocked out during the earthmoving work and only the original control will be undisturbed.

Examples of setting out Setting out a pipeline This operation falls into the first category of setting out. General considerations: sewers normally follow the natural fall in the land and are laid at gradients which induce self-cleansing velocity. The figure below shows a sight rail offset at right angles to a pipe line laid in a granular bedding trench.

Horizontal control: the working drawings will show the directions of the sewer pipes and the positions of the manholes. The line of the sewer is normally pegged at 20 to 30m intervals using coordinate methods of positioning from reference points or in relation to existing detail. The direction of the line can be sighted using a theodolite and pegs.

Vertical control: involves the erection of sight rails some convenient height above the invert level of the pipe.

Erection and use of sight rails: the sight rail uprights are hammered firmly into the ground, usually offset from the line rather than straddling it. Using a nearby TBM and levelling equipment, the reduced levels of the tops of the uprights.

Where the natural slope of the ground is not approximately parallel to the proposed pipe gradient, double sight rails can be used as shown in the next fig. Often it is required to lay storm water and foul water sewers in adjacent trenches. Since the storm water pipe is usually at a higher level than the foul water pipe, it is common to dig one trench to two different levels – as shown in fig 2 on the next slide.

Both pipe runs are then controlled using different sight rails nailed to the same uprights. Pipe laying: on completion of the excavation, the sight rail control is transferred to pegs in the bottom of the trench as shown below

Setting out a building to ground-floor level This process falls into the first category of setting out. It must be remembered when setting out that, since dimensions, whether scaled or designed, are almost always horizontal, slope must be allowed for in surface taping on sloping ground. The steps involved in setting out a building are as follows: -Two corners of the building are ser out from a baseline, site grid or control points -From these two corners, the two other corners are ser out using a theodolite to turn off the right angels as shown below -Diagonals are checked -Profile boards are placed at each corner

Setting out bridge abutments Structures such as bridge abutments can be set out by a combination of horizontal control methods and coordinate positioning. The following procedure should be used: -The centre line of the two roads are set out

-The bridge is set out in advance of the road construction -The bridge is set out in advance of the road construction. If GPS techniques are to be used, the abutment points A, B, C and D can be set out directly. -However, if total stations or theodolites and tapes are to be used then it will be necessary to establish secondary site control points around the area containing the abutments. These secondary points could either be in the form of a structural grid -TBMs are set up as separate levelled points or a control point can be levelled and used as a TBM.

If a structural grid in used (as in a), the distances from the secondary site control points to abutment design points A, B, C and D must first be calculated. They are then set out either using a theodolite to establish the directions and steel tapes to measure the distances or by using a total station.

-If coordinates are used as shown (b), the bearings and distances from the secondary site control points to A, B, C and D are calculated from their respective coordinates such that each design point can be established from at least two control points. -Once points A, B, C and D have been set out, their positions should be checked by measuring between them and also measuring to them from control points not used to establish them initially. -Offset pegs are established for each of A, B, C and D to allow excavation and foundation work to proceed and to enable the points to be relocated as and when required. -Once the foundations are established, the formwork, steel or precast units can be positioned with reference to the offset pegs.

Controlling vertically One of the most important second stage setting out operations is to ensure that those elements of the scheme which are designed to be vertical are actually constructed be so, and there are a number of techniques available by which this can be achieved. Particular emphasis is placed on the control verticality in multi-storey structures. In order to avoid repeating information earlier in this chapter, the following assumptions have been made. - Offset pegs have been established to enable the sides of the building to be located as necessary. -The structure being controlled has already had its ground floor slab constructed and the horizontal control lines have already been transferred. Plumb-bob methods The traditional method of controlling verticality is to use plumb-bobs, suspended on piano wire or nylon. A range of weights is available (from 3 kg to 20 kg) and two plumb-bobs are needed in order to provide a reference line from which the upper floors may be controlled.

In an ideal situation, the bob is suspended from an upper floor and moved until it hangs over a datum reference mark on the ground floor slab. If it is impossible or Inconvenient to hang the plumb-bob down the outside of the structure, holes and openings must be provided in the floors to allow the plumbbob to hang through, and some form of centring frame will be necessary to cover the opening to enable the exact point to be fixed.

Theodolite methods These methods assume that the theodolite is in perfect adjustment so that its line of sight will describe a vertical plane when rotated about its tilting axis. Controlling a multi -storey structure using a theodolite and targets A and B are offset pegs. The procedure is as follows. - The theodolite is set over offset peg A, carefully levelled and aligned on the reference line marked on the side of the slab - The line of sight is transferred to the higher floor and a target accurately positioned at point C. - A three-tripod traverse system is used and the target and theodolite are interchanged. The theodolite, now at C, is sighted onto the target at A, transited and used to line in a second target at D. Both faces must be used and the mean position adopted for D. - A three-tripod traverse system is again used between C and D and the theodolite checks the line by sighting down from D to the reference mark at B, again using both faces. - It may be necessary to repeat the process if a slight discrepancy is found. - The procedure is repeated along other sides of the building.

Transferring height from floor to floor Reduced levels must be transferred several times during the second stage setting out operations as the construction proceeds from floor to floor. One method by which this can be done is to use a weighted steel tape to measure from a datum in the base of the structure as shown in FIG A. The base datum levels should be set in the bottom of lift wells, service ducts and so on, such that an unrestricted taping line to roof level is provided. The levels should be transferred to each new floor by always measuring from the datum rather than from the previous floor. Each floor is then provided with TBMs in key positions from which normal levelling methods can be used to transfer levels on each floor. Alternatively, if there are cast-in situ stairs present, a level and staff can be used to level up and down the stairs, as shown in FIG B. Note that both up and down levelling must be done as a check.

Setting Out Example 1 : Setting Out a pipeline using sight rails and a Traveller An existing sewer at P is to be continued to Q and R on a falling gradient of 1 in 150 for plan distances of 27.12m and 54.11m consecutively, where the position of P, Q and R are defined by wooden uprights.

Level reading to staff on TBM (RL 89.52m) = 0.39m Level reading to staff on top of upright at P = 0.16m Level reading to staff on top of upright at Q = 0.35m Level reading to staff on top of upright at R = 1.17m Level reading to staff on invert of existing sewer at P = 2.84m All readings are taken at the same instrument position.

Solution Height of collimation of instrument = 89.52 + 0.39 = 89.91m Invert level at P = 89.91-2.84 = 87.07m This gives: Sight rail top edge level at P = 87.07 +2.5 = 89.57m Level of top of upright at P = 89.91 – 0.16 = 89.75 Hence Upright level – sight rail level = 89.75 – 89.57 = +0.18m Therefore the top edge of the sight rail at P must be fixed 0.18m below the top of the upright. Fall of sewer from P to Q = -27.12 x (1/150) = -0.18m Invert level at Q = 87.07 – 0.18 = 86.89m

Sight rail top edge level at Q = 86.89 +2.50 = 89.39m Level of top of upright at Q = 89.91-0.35=89.56m Upright level – sight rail level = 89.56 – 89.39 = 0.17m

Therefore the top edge of the sight rail must be fixed 0.17m below the top upright at Q.  27.12 + 54.11  −  = −0.54m 150 Fall of sewer from P to R =  

Invert level at R = 87.07 - 0.54 = 86.53m Sight rail level at R = 86.53 + 2.50 = 89.03m Level of top of upright at R = 89.91 -1.17 = 88.74m Upright – sight rail = 88.74 - 89.03 = -0.29m Therefore the top edge of the sight rail must be fixed 0.29m above to the top of the upright at R, i.e. the upright must be extended.

Setting Out Example 2 : Setting Out by intersection A rectangular buildings having plan sides of 75.36 and 23.24m was set out with its major axis aligned precisely east-west. The design of the coordinates of the SE corner were (348.92, 591.76) and this corner was fixed by theodolite intersection from two stations P and Q whose respective coordinate were (296.51, 540.32) and (371.30, 522.22). The other corners were set out by similar methods. When setting out was completed, the sides and the diagonals of the building were measured as a check. To help with this the existing ground levels at the four corners of the proposed structure were determined by levelling: SE(152.86m)

SW(149.73m)

NE(151.45m)

NW(146.53m)

Calculate the respective horizontal angles (to the nearest 20”) that were set off P relative to PQ and at Q relative to QP in order to intersect position SE.

Calculate the surface check measurements that should have been obtained for the four sides and two diagonals (assuming even gradients along the surface). Calculation of α and β Let the corner SE of the building be X: Easting of X Easting of P ∆EPX

348.92 Northing of X 296.51 Northing of P +52.41 ∆NPX

591.76 540.32 +51.44

Therefore by rectangular to polar conversion: Bearing PX = 45o32’07”

Easting of X Easting of Q ∆EQX

348.92 Northing of X 371.30 Northing of Q -22.38 ∆NQX

591.76 522.22 +69.54

Therefore by rectangular to polar conversion: Bearing QX = 342o09’37”

Easting of Q Easting of P ∆EQP

371.30 Northing of Q 296.51 Northing of P +74.79 ∆NQP

522.22 540.32 -18.10

Therefore by rectangular to polar conversion: Bearing PX = 103o36’17”

This gives: Angle α = bearing PQ – bearing PX = 58o04’10” Clockwise angle to be set off P relative to PQ = 360o - 58o04’10” = 301o56’00” Angle β = bearing QX – bearing QP = 58o33’20” Clockwise angle to be set off P relative to PQ = 360o - 58o04’10” = 301o56’00” (angles rounded to nearest 20” as specified)

Calculation of surface checks Recall that slope correction = + (∆h2/2L): From SE to SW, ∆h = 156.82 – 149.73 = 7.09

∆h2 = 50.27

From NE to NW, ∆h = 151.42 – 146.53 = 4.92

∆h2 = 24.21

From SE to NE, ∆h = 156.82 – 151.42 = 5.37

∆h2 = 28.84

From SW to NW, ∆h = 149.73 – 146.53 = 3.20

∆h2 = 10.24

Hence the slope distances for all four sides should have been: SE to SW =

 50.27  75.36 +   = 75.36 + 0.33 = 75.59m  2 × 75.36 

NE to NW =

 24.21  75.36 +   = 75.36 + 0.16 = 75.52m  2 × 75.36 

SE to NE = SW to NW =

 28.24  23.24 +   = 23.24 + 0.62 = 23.86m  2 × 23.24   10.24  23.24 +   = 23.24 + 0.22 = 23.46m  2 × 23.24 

For the diagonals: Horizontal diagonals =

(75.36) 2 + ( 23.24) 2 = 78.86 m

From SE to NW, ∆h = 156.82 – 146.53 = 10.29

∆h2 = 105.88

From SW to NE, ∆h = 151.45 – 149.73 = 1.72

∆h2 = 2.96

Slope distances: SE to NW =

 105.88  78.86 +   = 78.86 + 0.67 = 79.53m  2 × 78.86 

SW to NE =

 2.96  78.86 +   = 78.86 + 0.02 = 78.88m  2 × 78.86 

Setting Out Example 3 : Using Site Rails The six corners of a proposed L shaped excavation shown below have been set out on site and offset pegs haven been established to help define the sides of the excavation.

The proposed formation level of the surface of the excavation at point R is 95.72m. The surface is to fall at 1 in 150 from R to W and is to rise at a slope of 1 in 100 at right angle to the line RW. To help with excavation sight rails are to be erected above the offset pegs for use with a 2m traveller.

Given the reduced levels of the offset pegs calculate the heights of the sight rails to be used at P1, P2, P3 and P4. Solution: for line P1RWP2

Formation level at P1 = 95.72 + (3/150) = 95.74m Formation level at P2 = 95.72 – (48/150) = 95.40m For offset peg P1 Required top of sight rail level = 95.74 + 2.00 = Actual to of peg level Therefore, distance above P1

97.74m = 96.95m = 0.79m

For offset peg P2 Required top of sight rail level = 95.40 + 2.00 = 97.40m Actual to of peg level = 96.45m Therefore, distance above P1 = 0.95m Solution: for line P4UTP3

Formation level at Z = 95.72 - (15/150) = 95.62m Formation level at P3 = 95.62 – (28/100) = 95.90m Formation level at P4 = 95.62 - (3/100) = 95.59m For offset peg P3 Required top of sight rail level = 95.90 + 2.00 = Actual to of peg level Therefore, distance above P1

97.90m = 97.12m = 0.78m

For offset peg P4 Required top of sight rail level = 95.59 + 2.00 = 97.59m Actual to of peg level = 96.75m Therefore, distance above P1 = 0.84m

Setting Out Example 4: Using Slope Rails An embankment was constructed with a formation width of 36m and a formation level of 103.59m. The traverse slope at right angle to the centre line was 1 in 12 and the side slopes 1 in 2. Slope rails were used with a 1.50m traveller held vertically to monitor the formation of the embankment.

The point R (ground level at CL) had a level of 85.08m. the slope rails on either side of the embankment were attached to verticals A and B on the left and C and D on the right. These were positioned as shown above. The tops of the vertical stakes A, B, C and D were levelled as 80.54m, 80.81m, 90.59m and 89.89m respectively. Using this information calculate the slope that were set out along the ground surface from point P at right angle to the centre line to establish the centres of stakes A, B , C and D. Calculate the Vertical distances that were set out from the tops of the stakes A, B, C and D to fix the top edges of the sight rails in their correct positions. Solution The parameters of the embankment are : h = (103.59 - 85.08+ = 18.51m; n = 2 S = 12; b = 18m For a two level cross section :

WL =

s(b + nh) 12(18 + (2)18.51) = = 47.16m ( s + n) 12 + 2

WG =

s (b + nh) 12(18 + (2)18.51) = = 66.02m ( s − n) 12 − 2

Wg = greater side width WL = lesser side width h = depth of cut on the centre line from the existing to the proposed levels 1 in n = side slope 1 in s = ground on the traverse slope b = formation width The slope distances set out were: For stake A = WG + 1.0 + 1.0 = 68.02m For stake B = WG + 1.0 = 67.02m For stake C = WL + 1.0 = 48.16 For Stake D = WL + 1.0 + 1.0 = 49.16m

But the transverse slope = 1 in 12 hence: 1 Transverse slope = tan −1   = cos 04 o 45'49' '  12 

Therefore to the centre of stake A = To the centre of stake B =

68.02 = 68.26m cos 04 o 45'49' '

67.02 = 67.25m cos 04 o 45'49' '

To the centre of stake C = 48.33m and

centre of stake D = 49.33m

Vertical distances: For stake B: RL of the top of the rail = RLP + 1.50 – 0.50 RLP = existing RL on the centreline – (WG/12) RL of the top of rail = 85.08 – (66.02/12) + 1.50 – 0.50 = 80.58m RL of the top of the stake was given as 80.81m

vertical distance = (80.58 - 80.81) = -0.23m. the top edge of the slope rail must be set 0.23m below the top of the vertical stake B. For Stake A: The top of the rail is 0.50m below the top of the rail at stake B, hence: RL of the top of the rail = 80.58 – 0.50 = 80.08m Vertical distance = (80.08 – 80.54(given)) = -0.46m Therefore the top edge of the slope rail at A must be fixed 0.46m below the top of the stake. For stake C: RL of the top of the rail = RLQ +1.50 – 0.50 RLQ = existing RL at R + (WL/12) 85.08 + (47.16/12) + 1.50 – 0.50 = 90.01m Vertical distance = (90.01 – 90.59(given)) = -0.58m Therefore the edge of the slope rail at C must be fixed 0.58m below the top of the stake.

For Stake D: The top of the rail is 0.50m below the top of the rail at stake C, hence: RL of top of the rail = 90.01 – 0.50 = 89.51m Vertical distance = (90.01 – 90.59) = -0.38m