lundi 4 avril 2011

Construction de la Grande Pyramide : une étude de Peter Prevos - 2ème partie

 Première partie de l’étude de Peter Prevos

3.3 Casing and trimming

After the highest point is reached the final action, smoothing of the surface, can start. There are two distinctively different methods of shaping the surface of the pyramid, both of which will be analysed. The illustration below shows the principle of the two different methods of pyramid shaping.
Figure 9 : Final shaping of the pyramid

Placing casing elements

The first method described is the placing of ready made casing elements. The wedge shaped elements measure 2 * 0.575 meters at the base and are 0.73 meters high.To cover the core 45,970 elements have to be placed which is calculated with :
The elements have to be elevated to the required level and subsequently placed accurately on the face of the pyramid. Jack up teams can work on all four sides of the pyramid simultaneously using the same method as for the construction of the core. The first elements take 146 / Vv = 29 hours to reach the top. When working 100% efficient this action will already start right after the last core element starts its way up to the top. But when the highest point of the pyramid is reached a two week holiday will be very appropriate so the jack-up lines have to be established again. Every jack-up line has to move Sh / 2 = 3.5 elements to the work area. After every Sv / Vv = 0.78 hours a new element arrives. It takes 3.5 * 0.78 = 2.73 hours to fill one layer. Thus the total jack up time for all the casing elements is 200 * 3.5 * 0.78 = 546 hours, or 6.5 weeks. Placing the elements in the face will also consume a lot of time. It is assumed that it takes twenty minutes to position a casing element into the final position. This is a very crude assumption but no reliable data is available to me at present. The total placing time for all elements is 45,970 * 20/60 = 15,323 hours (3.65 year). Thus the total construction time for the casing action is 29 + 546 + 15,323 = 15,898 hours (3.79 year).
The layer numbers are in reversed order since the casing has to start at the top. For each layer the number of lifting teams is calculated in the same way as with the core construction, but working from four faces instead of two. Then the total construction time for the layer is calculated using 20 minutes placing time per block, adding the 2.73 hours per layer and for the top layer the 29 hours to position the first element. To get the total man weeks the construction time is multiplied by the number of teams working on that layer and the number of workers per team. The total use of labour to place the casing elements of the face of the pyramid is 1,088,028 man weeks, or 21,761 man year.

Trimming the pyramid

In this method the outer layer of the core is constructed of the casing material which is a more durable limestone than the core material. After the last core element has been placed the stone masons start to chip away material so that the final shape can be made.
This method implies that more core material has to be placed. Effectively this means that every layer will be 1.15 meters wider. One extra layer of 230 * 230 meters, consisting of 38,644 elements, has to be constructed. Using the formulas from paragraph 3.2 gives a construction time of 460 hours, using at maximum 1,126 teams (43,198 man weeks). This has to be added to the core construction.
The actual trimming is done at one layer at the time because all the debris has to be removed continuously. The volume of material to be removed can be calculated by :
Assuming that one stone mason can chip away 0.5 m3 per day and that they are positioned every two meters it takes one full day to complete one layer and 200 days to trim the complete pyramid. To remove the material two chains of workers are placed on the side of the pyramid. This method is often used in Bangladesh and is called the 'headpan' method. A basket, containing approximately 20 kg of material is passed on, over the heads of the workers, down the slope of the pyramid, hence the name headpan. The second row is used to bring the empty headpans back to the stone masons. This way a production of 4 m3/hour, or 48 m3/day, can be achieved. One 'chain' can remove the material produced by 96 stone masons.
At the bottom of the pyramid the chain has to continue to be able to deposit the material at its final destination. Behind every mason two labourers work to remove the debris from under his feet and bring it to the removal teams. This way the produced debris can be removed from the face of the pyramid very easily.
For every layer the width is calculated and the number of teams, using above mentioned parameters. The total number of labourers depends on the distance the material has to be transported from the base of the pyramid. In the calculations 100 meters are assumed. (...) In total 348,052 / 0.70 = 497,217 man days or 1,421 man years are used to trim the pyramid. The total time needed to trim the pyramid face is 200/0.70=286 days or 40.8 weeks.


The trimming method is the fastest and most efficient way of giving the pyramid its final shape. The extra material that has to be placed in the core will be added to that activity bringing the total number of core elements to 2,556,988 + 38,644 = 2,595,632 elements. The construction time will increase by 460 hours. The total use of labour will increase by 40,710 man weeks.

3.4 Production of the elements

The production and transportation capacity of core elements has to be high enough to make sure that there is always sufficient stock.
The average placing capacity is calculated per layer. This ranges between 0.4 and 84 elements per hour. The average placing capacity is 56 elements per hour. The quarry capacity is mainly determined by the geometry of the quarries. The thickness and size of the strata to be used determines the number of elements that can be yielded at one time. The distance of the quarries and the number of quarries are also an important factor which is not known to met at this time.
When there is always sufficient stock, the construction time for the core will be as calculated in paragraph 3.2. When there are no more elements to be placed the core construction will be delayed. In this situation the placing capacity will be equal to the quarry capacity.
An important parameter is the number of elements in stock before starting the actual construction of the core. Producing this stock will take extra time, but the construction time for the core will be reduced. The following table shows the relation between material in stock and the total construction time for the core, assuming a production capacity of 56 elements per hour.
Table 1 : Construction time for different stock levels
The figures do not include the extra time needed to place the elements for core trimming as put forward in paragraph 3.3. From this data it can be concluded that no stock is really needed since the longer construction time is compensated by a shorter preparation time. The overall construction time stays more or less the same. Producing more then the 378,000 elements is not effective, because the core construction is on maximum capacity and the total construction time is at its minimum.
In case the quarry capacity is less than the placing capacity the construction time will be negatively effected. When the quarry capacity is equal to the maximum placing capacity (84 pcs/hour) the construction time will be lowest. Table 2 shows the total construction time for different quarry productions. No stock is produced beforehand.
Table 2 : Construction time for different quarry production levels
To make a good estimate of the total construction time the quarry capacity has to be known as exact as possible. Since it is not feasible to give a reliable estimate of the quarry capacity at this time an arbitrary figure is used. For further calculations a capacity of 30 elements per hour is used. A higher quarry capacity is not very likely and a even lower capacity will increase the construction time considerable as can be seen in table 2. Assuming that it takes about 1 day to cut one element from the quarry face, 360 elements have to be in production at one time at the different quarries. There has to be enough space in the quarries to produce all these elements and enough transportation capacity to bring the elements to the construction site. Detailed study of the quarries sites is required to determine the quarry capacity more accurately.
Because of the extended construction time the use of labour will also be different to the situation in which there is always sufficient stock. To make sure there are not too many teams waiting for supply the placing capacity will be reduced. This is done by increasing the vertical spacing of the jack-up teams.
Figure 10: Optimizing the placing capacity
The vertical spacing can be changed in the spreadsheet so that the new maximum capacity can be calculated. In figure 10 the results of the calculations for the core construction time are given for different values of the vertical spacing, using a quarry capacity of 30 elements per hour and no stock produced before construction. The top line shows the use of labour in man-years. At a vertical spacing of 7.5 meters the use of labour is minimal. The other two lines are the construction time in hours, using the maximum placing capacity and the levelled capacity. The waiting time for the jack-up teams is proportional to the difference between the maximum capacity and the levelled capacity. The difference between these two is minimal at a vertical distance of more than 10 meters.


The quarry or production capacity can not be estimated properly since no data about the geometry of the quarries is available to me at this time.
From table 1 can be seen that production of stock before starting the construction of the core will not have a substantial beneficial effect on the total construction time for the core.
Because the placing capacity is different for every layer, but the quarry capacity is taken as a constant, the placing capacity has to levelled for that. Table 2 shows the construction times for the core for different quarry productions. A quarry production of 30 elements per hour is used for further calculations. The validity of this figure can not be properly established at this time.
The quarry capacity is most of the time lower then the placing capacity so there will be a lot of waiting time for the jack-up teams. To reduce this waiting the placing capacity is reduced by increasing the vertical spacing. Figure 10 shows that at a vertical spacing of 7.5 meters the use of labour is minimal.
Using this model the total time needed to produce the 2,595,632 elements is 1,030 weeks. The minimum construction time for the core is 85,233 hours, the levelled construction time is 97,758 hours (1164 weeks), using 128,641 man years to complete. The average placing capacity is equal to the quarry production capacity of 30 elements per hour.

4. Planning

To determine the total construction time for the pyramid all four activities will be summarised. The use of labour will not be levelled, assuming that an unlimited supply of workers is available at any time.

4.1 Preparation of the work area

This is the first activity to start and lasts for 5.5 weeks, using 1,150 workers, 6,325 man weeks.

4.2 Construction of the core

Starts directly after the preparation of the work area is completed. The total time needed to position all the core elements is 97,758 + 460 = 98,218 hour or 1,169 weeks. The total use of labour for this activity, as calculated in paragraph 3.4. is 6,432,039 man weeks. On average 5,500 people will be at work on the pyramid at any time.

4.3 Trimming the pyramid

The trimming method takes 286 days to complete (41 weeks) and will start right after the core construction is completed. 497,217 man days will be used for the trimming.

4.4 Production of the elements

The production of the elements will start the same time as the preparation of the work area. A small stock, of six weeks production, will be created. This will not have a great influence on the core construction. The time needed to produce all the elements has to be less than the core construction time. The total time to produce all the 2,595,632 elements is 86,521 hour or 1,030 weeks.

4.5 Total construction time

The total construction time is determined by adding up all activities as listed above including a two week holiday between the completion of the core and the start of the trimming activities.
9 + 1,169 + 2 + 41 = 1,221 weeks, or 24.4 years are needed to construct the pyramid. Figure 11 shows that the critical path in bold. The only activity with some float is the element production.

Figure 11 : Planning chart
Although the core construction is in the critical path it has to be noted that the placing capacity has been reduced to minimise waiting time. When using the maximum placing capacity the element production will be critical.

5. Cost estimate

To give an impression of the actual volume of the work the costs of the total project are estimated. For this it is assumed that someone would want to built a pyramid, of similar size as the Khufu complex. The construction site will be situated in Bangladesh because of the abundance of cheap labour and the fact that the local people have a lot of experience with moving great volumes of material by hand. The local currency in Bangladesh is the Taka (BTK), all costs will be estimated in the European Currency, the Euro. The following exchange rates will be used:
1 Euro = 51 Taka
1 US Dollar = 40 Taka
At the 1995 price level the average salary for a unskilled labourer in Bangladesh is about 65 Taka for a 8 hour day. Overtime is paid 50% extra per hour. The working days at the pyramid are 12 hours, making the daily rate 114 Taka per day, or 2.23 Euro per day.

5.1 Element production

Because rock of any bigger size is not available in Bangladesh concrete will be used to produce the core and casing elements. Only boulders are found in Bangladesh, they are normally crushed and used as aggregate for concrete. The density of light aggregate concrete is about 2,00 kg/m3, the pressure in the bottom layer becomes 3.4 MPa. This gives an indication of the required compression strength of the core elements. Low grade concrete will be sufficient.
The mineral aggregate can be replaced by brick chips, which is a very common and much cheaper option. The price for concrete with brick chip aggregate, without reinforcement would be approximately 3,000 BTK/m3, including all costs for labour and transport. One core element would cost 2,998 BTK or 58.78 Euro per element.

5.2 Preparation of the work area

In paraghah 3.1 it was estimated that one team of five men will clear 6 m2 per day. The unit rate for the site clearing will be ( 5 * 2.23 ) /6 = 1.87 XUE/m2.

5.3 Core construction

In the used scenario, with a quarry production of 30 pcs/hr the total use of labour resources is 6,432,039 man weeks. The costs for core construction per element are: (6,432,039 * 7 * 2.23) / 2,595,632 = 36.68 XUE per element.

5.4 Trimming the pyramid

The trimming of the pyramid takes 286 days and a total of 497,217 man days. Total costs of this activity is 497,217 * 2.23 = 1,108,794 Euro lump sum.

5.5 Housing and accommodation

Housing and accommodation for all the labourers will cost approximately 350 Taka per man per week. The total number of man weeks over the complete project is :

Table 3 : Total number of man weeks
The occupation of the workers camp will change according to the need of labour. The average number of people to be taken care of at one time is 5,637. The element production is sub contracted, the costs of housing etc. are all included in the unit rate of the subcontractor. One week accommodation for one man will cost 6.86 Euro. The total costs for food and accommodation will be 47,139,217 Euro.

5.6 Total construction costs

In table 3 the costs for each activity are summarised and totalled. All the prices are exclusive of overheads, profit & risk.

Table 4 : Total construction costs

6. Conclusions

From all the calculations made in the previous chapters can be seen that it is very well possible to construct a pyramid of the gigantic size as the one at Giza by merely using the technology and resources from the ancient Egyptians.
The construction requires very good planning and management. The work has to be done very carefully to prevent a block avalanche which will set back the construction for weeks. It is very conceivable that such a disaster will actually happen but no risk analysis are made on that point.
The most important conclusion that can be made from this report is that when trying to analyse this problem the quarry production needs the most attention. All previous reports have focused solely on the problem of vertical transport. I have tried to show that this is not the real problem if all work is organised well. The maximum placing capacity is sufficient to ensure continuous flow of elements to the work platform. The production of the more than 2.5 million elements is the real achievement of the pyramid builders That build the great pyramid. Accommodation and feeding of the labourers is the second major task. Not to mention the great number of supervisors and other staff members to control the construction process.

6.1 Reliability of the assumptions

The reliability of the assumptions made to reach to the conclusions differs per activity. Preparation of the work are is only a very small part of the whole process. Core construction can be estimated quite well since the used data comes from actual experiments as described in Hodges. All figures include an efficiency factor of 70%. There is no actual data available to estimate the element production so an arbitrary figure is used. The total construction time highly depends on this activity so a good estimate would be essential. The figure of 30 element per hour is quite high, the reason I used it is because it leads to a total construction time of about 20 years which is what Herodotus mentions in his book the Histories.

6.2 Further study

There are several points not fully analysed in this study. First the quarry problem. I would need more data about the actual place of the quarries, the maximum yield per face and to get a better insight into the quarrying methods used by the Egyptians. I am planning a trip to Egypt, later this year, to study the possible quarry sites and try to analyse the quarry production process.Also the influence of the construction of the chambers on the construction process needs some further analyses. The lifting of the big size elements will need adaptations to the geometry of the core, which have to be filled up after the construction of the chambers is completed.
This report will be followed by a second one in which a more reliable estimate of the quarry production will be done and possible amendments to this report.
It was Archimedes who first described the mathematical aspects of using levers for moving heavy objects and I would like to end this report with a quote from him :
Give me a firm place to stand and I will lift the world.