samedi 2 avril 2011

Construction de la Grande Pyramide : une étude de Peter Prevos

Peter Prevos est un ingénieur hollandais qui réside et travaille en Australie, comme Manager Land Development dans la société Coliban Water.        
En 1997, il a réalisé une étude sur la construction de la pyramide de Khéops, que je reproduis ici, avec évidemment son accord, dans son intégralité.
Dans cette étude, Peter Prevos s’inspire de la théorie de Peter Hodges (*) qu’il a “prise comme modèle”, tout en y ajoutant une “quantification”.

(*) sur cet auteur, voir la note de Pyramidales ICI

1. Pyramid Construction : Desk Study by Peter Prevos

During 1995 I was working at the Jamuna Multipurpose Bridge Project in Bangladesh where we were doing so called river training works. The goal of the project was to fix the river between two guidance dikes. These are needed to protect the foundations of the newly built bridge from eroding. One part of the project was to build a dike in a bend of river, this to prevent the river from eroding too far inland and flow in front of the bridge instead of under it.
Because of unavailability of suitable equipment we employed approximately 5,000 local people to do part of the excavation works manually. All work, except the compacting, was done manually using so called 'headpans'. These are small baskets that are filled with soil and then carried on the heads of the workers to remove the material. Seeing these people move all that soil with, from our point of view, primitive tools made me think about how the pyramids Would have been built.
My goal was to analyse how a pyramid of similar size as the Great Pyramid at Giza can be constructed, with resources similar to those of the ancient Egyptians. Many books have been written about this subject, but none of these books have looked at this from a civil engineers point of view. It is not possible to give an detailed account on how the great pyramid has been built since there are no records on pyramid construction written by the Egyptians. The earliest account is written by the Greek historian Herodotus, more than 2,000 years after the actual construction. This study is therefore not about how the pyramid was built, but how it could be built. The goal of this study is to analyse the logistical aspects of pyramid construction.
For this preliminary desk study a simplified model is used. The pyramid shall be approximately of the same dimensions as the Great Pyramid of Khufu on the Giza plateau in Egypt, but shall have a massive core, without any passages or chambers. The building method to be used is described by Peter Hodges, in his excellent book How the Pyramids were built. In his view the problem of the vertical transport of the individual elements that make up the core of the pyramid was solved by using levers.
The basic outline of this report is as follows: First a general description of the object is given, stating the dimensions, materials used, equipment and manpower. Then the building method will be described in some detail, including estimates concerning lifting and transportation speed of the elements. With this data a plan is made, estimating the total construction time and the total use of labour resources. Finally a cost price is estimated, assuming that a pyramid would be built in Bangladesh.
The report is written as a 'project preparation report' as if a new pyramid would be actually built, using only the technology known to the Egyptians of the fourth dynasty.
Therefore, past as well as future tense is used in the text, this to distinguish between facts concerning the original pyramid in Khufu and the new idea's put forward in this study.
I hope this study is a valuable addition to all the previous work that has been done on this subject.

2. General description of the works

2.1 Geometry

The great pyramid was built during the reign of Khufu (Cheops in Greek), second king of the fourth dynasty 2,720-2,560 BC. It stands on the Giza plateau nearby Cairo and is the biggest pyramid in Egypt.
The pyramid itself now stands 137 meters high, its original height of 146.16 meters is indicated by an iron post erected on the apex. Each side originally measured 230.362 meters or 440 royal cubits (1 cubit=0.524 metres). At present the side measures 227 meters, due to the loss of the casing stones. The core masonry consists of large blocks of local limestone taken from the nearby quarries and built around and over a rocky knoll. The size of the knoll cannot be determined, since it is completely covered by the pyramid.
The entrance to the pyramid is in the centre of the northern face. It is located in the thirteenth course of masonry from the base. This entrance has a pointed roof formed of massive slabs of local limestone and opens into a long steeply descending passage. From there a 36 meters long ascending passage leads to a 35 meters long horizontal passage that leads to the so called 'Queen's chamber'. This chamber measures 5.2 by 5.7 meters and the maximum height of its pointed roof is about 15 meters. The north and south walls each have a small hole a few centimetres square about 1 meter from the floor. These lead into narrow channels that originally opened on the exterior of the pyramid. At the juncture of the ascending and horizontal passage is an opening of a shaft which descends to a depth of 60 meters. It opens into the lower part of the descending passage, close to the unfinished, underground chamber, and is believed to have been an escape shaft for the workmen who filed the ascending passage with huge stones after the king's funeral. From the horizontal passage the Grand Gallery, which leads to the king's chamber, starts. It is 47 meters long and 8.5 meters high, and has a corbelled roof. In the centre of the floor is a sunken ramp about 60 centimetres deep. The Grand Gallery ends in a horizontal granite passage which serves as an antechamber. It measures 8.4 meters long and 3.1 meters high, and has slots for three portcullises. Beyond the antechamber is the so-called 'King's Chamber' which is lined, roofed and paved with red granite. It measures 5.2 by 10.8 meters and is 5.8 meters high. Its flat roof is formed of nine monolithic slabs of granite The northern and southern walls each have an 'air channel', one of which is open to the outside. The Pyramid can be seen to have about two hundred level courses of squared stones. The layers all have a different thickness ranging between approximately 50 and 145 centimetres. The average block size is about 1 cubic meter.
On the Khufu pyramid all the casing elements were removed in the 14th century. The few casing stones which do remain in the Great Pyramid all lie in the 1.5 meter thick bottom course and cannot be representative of the stones which would have been used in the higher parts of the construction. The only examples of face work which remains are those on the pyramids at Meidum, Dashur and Giza (Khafre's).

Fig. 1
Figure 1 shows a wire-frame model of the Pyramid. 
The wall around the pyramid and the temple are not included in this report.

A simplified model of the great pyramid will be used for this study. The pyramid has a square base measuring 230 meters in length and the height of the apex will by 146 meters. The foundation is a level surface on which the first layer can be placed. Any settlement due to the pressures during or after construction will be disregarded. The pyramid is solid, without any passages or chambers. Because the volume of the passages and chambers is only 0.07 % of the total volume the passages and chambers can be omitted. The construction will be done in 200 subsequent layers of equal thickness being 0.73 meter. The standard element will be cubical with sides of 1.17 meters. This is a theoretical assumption since blocks of this size will not be able to balance on the already placed steps. To be able to estimate to construction time for the core, the used parameters, such as the number of layers and the element size, have to be as close as possible to those of the actual pyramid. During the actual construction the layer thickness has to be adapted according to the thickness of the strata at the quarry. The block size will have to be adapted to be able to balance on the steps. Each layer will have fitting elements that are slightly smaller or larger than the standard element size. The core elements will thus form a step pyramid. The size of the steps is 0.575 meters. The outer layer will consist of wedge shaped casing elements so the pyramid will have a smooth surface when completed.

2.2 Materials

The core of the great pyramid consists of solid limestone blocks. Limestone is a sedimentary rock with a density between 2.5 and 2.7 tonnes per cubic meter. It is quite sensitive to weathering, therefore the top layer was constructed of a more durable limestone. The Kings Chamber and the Grand Gallery are constructed of red granite. Granite, an igneous rock is more dense than limestone and has better general physical properties and is therefore used in the upper chambers. The bulk of the limestone was quarried on the plateau itself. The red granite had to come from near Aswan about 700 km upstream from Cairo.
The problem of opening and exploiting the quarries needed for the construction is a very interesting and complex problem but is not a part of this study. The pyramid for this study is completely made of limestone, assuming a density of 2,600 kg/m3. The weight of one element is: 1.17 * 1.17 * 0.73 * 2,600 = 2,598 kg.

2.3 Labour and equipment

The ancient Egyptians built the pyramids with the simplest methods. Both in quarrying and building workmen used copper chisels, as well as flint, quartz and diorite pounders. Further they used wooden crowbars, sledges and rollers to transport the elements. Figure 2 is a scene showing the transport of blocks of stone from the quarries in Tura, in which we see oxen dragging the sledges.

Figure 2 : Transportation of stone blocks in a quarry at Tura
This method works very well for transporting over long distances, but usually manpower was used to move building elements. Figure 3 shows how 172 men work to drag an alabaster colossus of the twelfth dynasty monarch, Dhutihotep, from the quarries of Hatnub in Middle Egypt. This statue measured over 6.5 meters high and weighed about 60 tonnes.
The scene also shows men carrying levers and others pouring liquid, presumably water, from pots in order to reduce the friction between the statue and the surface. To transport a 2.6 tonnes element in a similar way, 172 / 60 * 2.6 = 7 to 8 people are needed. To transport the elements on the sledges special roads were constructed. They consisted of a base of rock rubble on which wooden planks where embedded at regular intervals in a layer of clay. The friction was reduced by wetting the clay, as can be seen in figure 3.

Figure 3 : Transporting a statue from the tomb of Dhutihotep, El Bersheh
All lifting work will be done by means of levers. With these levers all elements can be jacked up along the sides of the already constructed part and put into position. The lifting method will be described in more detail in the method statement. Levers can also be used for horizontal transport, but only for short distances or to put an element at its correct place.
The levers are made of wood and have a length of 2 meters. At the fulcrum the lever measures 100*100 mm and is tapered to both ends for easy handling. The maximum lifting force for one lever is determined by the length of the lever, the place of the fulcrum, the section of the lever and the type of wood used for the lever. The section has to be able to resist the bending stresses induced by the levering action. When assuming that the leverage point is at 0.1 meters from the end, and the maximum downward force to be applied by one person is 600 N the upward force on the other end is 1.9 * 600 / 0.1 = 11,400 N. The actual bending stress in the lever will be 6.84 N/mm2 which is allowable for most timber. Three levers would be sufficient to lift a 2.6 tonnes element, but for reasons of stability four levers have to be used.
Work will be done on 350 days per year, taking into account any religious or other holiday that may occur. A working day is from dusk to dawn, since working with artificial light is not possible. Average daylight per day, measured over a year, is 12 hours. Any lunch breaks or otherwise will also be accounted for by means of an efficiency factor of 70%.

3. Method statement

The construction of the pyramid can be divided into four different activities :
  • Preparation of the work area ;
  • Construction of the core ;
  • Casing and trimming ;
  • Production of the elements.

3.1 Preparation of the work area

The work area has to be cleared from sand and weathered rock and the surface has to be smooth to enable the horizontal transport of the elements.
Assuming that it takes 1 man about 2 hours to clear one square meter and that two teams of 115 men will work towards each other it will take 460 hours, or 5.5/0.7=9 weeks to clear the complete area. For each man doing the clearing and cutting a team of four more men have to remove the debris behind them and dispose it outside the work area. The total workforce needed for this activity is 2 * 115 * 5 = 1,150 men for a period of 9 weeks. This activity has to be completed before the construction of the core can commence.

3.2 Construction of the core

Many theories have been put forward concerning core construction. Most of these include the use of ramps to transport the elements to the work platform. There are three possible versions of a ramp. A long ramp or a short steep ramp with horizontal plateau and the spiral ramp. There are several problems with these theories. First the volume of the ramp to be built. In the long or short ramp scenario the volume of these structures would be more than the volume of the pyramid itself. In case of the spiral ramp stability will be a major problem. The slope of the pyramid itself is more than 50° and so the wall of the ramp has to be even steeper. Al this is very well analysed in Hodges.
Core construction is done in two actions. First the element has to be jacked up to the layer under construction after which it can be placed in its final position.
On the sides and the top of the pyramid teams can work in jack-up lines as shown in figure 4.

Figure 4 : Distribution of work teams on the pyramid

Lifting teams can work from only two sides of the pyramid simultaneously. When working from more than two sides the work platform can not be divided into equal square area's which is needed to ensure a smooth flow of elements. Working from one side would not be a very efficient option. When working from three or four sides the placing teams will be in each others way. The remaining two slopes can be used to lift the big slabs needed for the construction of the chamber. For this the geometry can be changed by leaving bigger steps on which the big size elements can be jacked up the sides and placed on the platform. When no more odd size elements are needed, the steps will be filled with standard size elements. Between two 'jack up lines' there must be some space for supervisors etc. There must be working space between the teams in vertical direction also. Figure 5 shows the distribution of the jack up teams along the pyramid side. 
Figure 5 : Spacing between jack-up teams

Because there are blocks being jacked up every four meters there is an almost constant flow of material to the work surface. The horizontal transport on the work platform actually determines the construction time for that particular layer and not the time needed to lift the elements. When an element has reached the level under construction it will be moved to its final place in the pyramid by the same team that has brought it to the top. This will create a continuous flow of elements on the work platform. After a row is placed the teams go down the unused sides of the pyramid and start a new cycle down at the base where they receive a new element to bring to the work platform. At the moment the first two rows of a layer are placed, halfway along the work platform, the maximum number of workers are being employed. This gradually reduces until the last rows are placed. When placing the next layer the number of work teams increases again. This is a complex flow of people and material which needs very good management to make sure there is minimum time loss.
The big problem with this method is the safety while jacking up the elements. Is it quite likely that an element can lose its balance and tumble down the pyramid creating a very dangerous situation. Because of this danger the vertical transport has to be done very carefully. Except for the great time loss in such a situation, many human lives can be lost when workers are hit by a limestone block avalanche.
In the next paragraphs this method is being quantified and the total construction time for the core is estimated.

Vertical transport

The elements will be lifted to the required level by teams of 7 men using 4 levers. Four men operate the levers, two to insert the timber packing and one man to co-ordinate the actions. The jack up procedure is as follows :
  1. The levers are put in position for the first 'jack'. Two on each side, four in all. The element has been delivered on timber packing ;
  2. After the first jack is complete extra packing has to be inserted below the stone at each end ;
  3. An extra packing has to be inserted below the levers ;
  4. Start the next jack
This sequence is repeated until the element reaches the next level.

Figure 6 : Jack-up procedure

According to Hodges, who has conducted full size tests of the above described procedure, it takes 25 seconds to complete one cycle. With every jack the element is lifted 100 mm, taking 8 complete cycles to lift the element to the next level. To move the element onto the next step it has to be moved horizontally. This horizontal transport can be done by using the levers in a paddling movement. Hodges mentions a horizontal travelling speed of 13 meters per hour for this method. A complete cycle of moving a block to the next step (0.73 meter vertical and 0.575 meter horizontal travel) takes 359 seconds. In the calculations 513 seconds (5.12 meter per hour) will be used, taking into account any loss due to resting, human error or otherwise (efficiency 70%).

Placing the elements

This is also done by using the paddling movement used in the lifting procedure. In the calculations a speed of 9.10 meter per hour will be used (70% of 13 meters per hour). The distance between two teams on the platform is the time that one team travels horizontally before the next team reaches the top. In our case this distance is: 9.10 * 4/5.12 = 7.11 meters. This is slightly more than the distance in horizontal direction and should be sufficient.

Calculation of construction time

The construction time for each layer is calculated using parameters listed below .
  • W - Width of the pyramid [230 m]
  • H - Height of the pyramid [146 m]
  • z - Element height [0.73 m]
  • l, b - Element size [1.17 m * 1.17 m]
  • Vv - Vertical speed [5.12 m/h]
  • Vh - Horizontal speed [9.10 m/h]
  • Sv - Vertical spacing [4 m]
  • Sh - Horizontal spacing [7 m]
  • Sp - Spacing on the platform [7.11 m]
  • n - Layer number [1 to 200 ]

For each layer n the height h(n), width w(n) and the number of elements to be placed e(n) are calculated.
The time needed to place all the elements is the time between the moment that the first row of elements arrives on the work platform and the moment that the last row has been placed. After every Sv / Vv = 0.78 hours a new batch arrives. This is repeated until the last row, ½ w(n) / b in number, is placed.
The elements also have to be moved sideways to fill up the rows. All elements have to be moved sideways over an average distance of ½.Sh meter. The above sequence has to be repeated Sh/l times to fill the platform.
In figure 7 the sequence of placing one row of elements on the platform is illustrated.
Figure 7 : Placing sequence

Now the construction time per layer can be calculated by :
The total construction time for the core of the pyramid, 2,556,988 blocks, is 45,684 hours, or 10.88 years. In Hodges the construction time or the core is estimated at 17 years, but with completely different assumptions and method. The vertical speed in his calculations is 9.125 m/h which is to my point of view not possible. Also the efficiently is taken as 100%, which would mean 17 years of work without any mistakes and problems! The horizontal transport on the platform is not taken into account. Also the number of teams to be employed at any time is equal for every layer.
When placing at maximum capacity 56 blocks per hour can be placed on average throughout the almost 11 years the core is under construction. The in the above described way calculated capacity is the maximum capacity. In case the quarry can not supply sufficient elements the placing capacity will be equal to the quarry capacity, leaving no stock. In paragraph 3.4 the quarry production will be looked at in more detail and the placing capacity will be levelled.


At any stage there are teams working on two sides and on the top of the pyramid. By calculating the number of teams that can be employed at one time in horizontal direction Th(n), in vertical direction Tv(n) and on the work platform Tp(n) the total number of teams can be determined.
When a new layer is started only the two sides are occupied, the number of teams being:
2 * ( Th(n) * Tv(n) ). When the first row of elements reaches the middle of the work platform the platform is also occupied.
The maximum number of teams to be employed at one time for the construction of the layer is calculated with :
The actual number of elements being moved on the pyramid at one time lies between 2 * Tv(n) * Th(n) and T(n).
For each layer the maximum number of teams is calculated by using above mentioned formulas. In figure 8 the number of work teams is set out against the layer number. From this it can be seen that the total workforce will be smaller as the pyramid grows. This is because the surface to work on is getting smaller, although the pyramid is getting higher, less teams can be placed on it.
To estimate the total use of labour resources the workforce is multiplied by the construction time per layer and the number of men per jack-up team. This way the total use of labour resources is estimated to be 3,579,745 man weeks, or 71,595 man years. The maximum number of workers is at layer number 11, being 1,135 * 7 = 7,
In figure 8 the number of workers during the construction period are shown.

Figure 8 : Number of labourers used for core construction

à suivre : deuxième partie