Construction of the Great Pyramid of Giza in Egypt. Explore Ancient Stone
Technology and various theories on the movement of heavy stones
which could have been used during construction of the Great Pyramid of Giza in
Egypt, Stonehenge, and similar sites.
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).
Figure 1 shows a wire-frame model of the Pyramid. The wall around the pyramid
and the temple are not included in this report.
Figure 1,
Wire-frame model of the Great Pyramid
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:
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.
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:
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;
After the first jack is complete extra packing has to be inserted below
the stone at each end;
An extra packing has to be inserted below the levers;
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
In Appendix 1 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, as
calculated in Appendix
1 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.
Labour
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 = 7,945.men.
In figure 8 the number of workers during the construction period are
shown.
Figure
8, Number of labourers used for core construction
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 use of labour is calculated in Appendix 2. 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. From Appendix 2 it can be seen
that 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.
The total amount of workers needed per layer is calculated in Appendix 3. 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. The figures from Appendix 3 have to be
corrected for a efficiency factor of 70%. 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.
Conclusion
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.
The production and transportation capacity of core elements has to be high
enough to make sure that there is always sufficient stock.
In Appendix 1 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. In
Appendix 1 the
construction time as well as the stock are levelled for this situation.
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.
In stock
Preparation time
Construction time
TOTAL
0
0
52,429
52,429
100,000
1,786
50,742
52,510
200,000
3,571
48,864
52,435
300,000
5,357
47,079
52,436
378,000
6,750
45,684
52,434
2,556,988
45,684
45,684
91,368
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.
Quarry capacity
Construction time [hr]
Construction time [yr]
84
45,684
10.88
70
47,116
11.22
60
50,384
12.00
56
52,429
12.49
50
56,444
13.44
40
67,369
16.04
30
87,168
20.75
20
128,707
30.64
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.
Summary
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 paragraph 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:
Activity
Man Weeks
Duration
Labour
Preparation
10,350
9
1,150
Core construction
6,432,039
1,169
5,502
Trimming
429,217
41
10,468
TOTAL
6,871,606
1,219
5,637
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.
Activity
Quantity
Unit
Unit rate
Amount
Element production
2,595,632
pcs
58.78
152,576,898
Preparation work area
52,900
m
1.87
98,923
Core construction
2,595,632
pcs
36.68
95,207,782
Trimming
LS
1,108,794
Housing
6,871,606
weeks
6.86
47,139,217
TOTAL
296,131,614
Table 4, Total construction costs
6. Conslusions
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 as described in the next paragraph.
6.3 Herodotus comments
The oldest known document about the construction of the pyramid is written in
Herodotus' book the Histories, part II. He visited visited Egypt in the fifth
century BC. In Appendix
4 the relevant text about pyramid construction is given. Herodotus mentions
several parameters that are also referred to in this document. One furlong, as
mentioned in the text is 1/8 of a mile, 201.163 meters. A fathom measures 6
feet, 1.829 meter.
Line 11 states 'A hundred thousand men laboured constantly ...'
According to the calculations done in this report an average work force of
about 5,500 men would be present at any time, excluding the labour needed to
produce and transport the element to the quarry site. Assuming that it takes one
whole day with two people to cut one element out of the quarry face and that 30
elements are produced per hour, 360 elements have to be in production and in
transportation at the same time. Transportation from the quarries to the pyramid
face is done by oxen sledges as shown in figure 2. To produce and transport 360
elements simultaneously not more then 3,600 men would be needed. The maximum
number of workers employed at any time during the project, including staff,
cooks etc. Would probably not be more then 10,000 at any time.
Line 20 states: 'The pyramid itself was twenty years in building.'
To construct the pyramid in this time frame a quarry production of 30
elements per hour would have to be maintained. It is still questionable if this
is achievable.
Line 27 states ...machines formed of short wooden planks.
This passage could describe the levers that would have been used to transport
the elements along the sides and on top of the pyramid. The lever method is the
only way the vast volume of big size elements can be brought to the required
levels.
Line 38 states ...1,600 talents of silver.
One talent of silver is equal to 26 kg. These were only the costs of food for
the labourers who constructed the pyramid. These days 1,600 talents of silver
would be equivalent to approximately 4,900,000 Euro (118 Euro/kg). Using the
calculations from chapter 5 this would be 1.35 Euro per man per week to feed the
workers.
6.3 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.
