www.cheops-pyramide.ch Copyright 2006 Franz Löhner and Teresa Zuberbühler Calculating the force and kinetic coefficient of frictionThe tables on this page are based on calculations by Dr. Heribert Illig and Prof. Dr. Dipl. Ing. H.U. Niemitz [1]. The results in a nutshellWith the following calculations [1] we provide
evidence, that with the help of the rope roll it is possible to haul a
2.5 tons stone block including sledge up the 52° incline of the flank
of the pyramid. The rope roll redirects the force needed, so you need
only 46 haulers and they are actually walking down the slope, thus using
their own weight in addition to their strength to pull the stone.
For the pyramid of Khufu we compute the following numbers per team (= hauling team for one 2.5 tons stone block):
Detailed calculations how many workers were necessary to build the pyramid Can this model also be used for hauling the 40 to 50 tons granite blocks
required for the King's chamber up the pyramid?
What factors have to be taken in account in calculating the force required?
What is friction and what is a coefficient of frictionThe resistance to lateral motion when one attempts to slide the surface of one object over another surface is called friction or traction. Depending on the type of materials that are in contact, the force one needs to overcome friction can vary. To overcome or lower friction / traction a lubricant is used. The coefficient of friction is a value which describes the ratio of the force of friction between two bodies and the force pressing them together. The coefficient of friction depends on the materials used and if the bodies are moving or not [4].
Friction and lubricationFriction of the rope: Kinetic (or dynamic) friction of the rope roll:
These numbers are not the same as for static friction - those are higher.
For example for wood on wood the static friction is 0,40 to 0,75 (dry)!
Lubricating medium: We use the lower values for calculating: 1. Lubricating the interface between the tracks and the sledge
assembly: 2. Lubricating the bearing of the rope roll:
Force exerted by each haulerWe think, that each hauler was capable of developing a force of 12 kp (= 117.7N / 1kp = 9,80665 g·m/s²). This value once was used to calculate the force exerted by one worker when towing barges on the French canals. This value was also used by Goyon for his calculations [3]. Various publications use values of 10 to 15 kp (see calculations with 15kp instead of 12kp). We use the lower values and see if it still works. Towing and hauling both demand a continuous effort over a long period of time. That is why this value is noticeable lower than the maximum force a human being can apply. The process of hauling should be interrupted as little as possible, because every time you have to start moving the sledge again, you have to overcome static friction (stiction [4]). For this reason, the haulers should be able to walk at a normal pace (= walking 20m per minute) and continuously without stopping until the next team takes over from them for the next leg. The hand off to the next hauling team should be accomplished in such a way, that the sledge doesn't stop its movement. For checking purposes:
F = Force / Gm = Weight / μs = Coefficient of friction (between sledge and tracks)
Force on an inclined plane without a rope rollA ramp is an inclined plane. By changing the angle of the ramp, the force necessary to raise or lower a load is varied.
F = Force (downhill-slope force) If we increase the angle of inclination of the plane (right), the stone will start to slide down faster. As you know from your own experience the steeper a slope, the greater will be your effort to push or haul a load upwards. As we will see further on, at a certain point the angle of inclination is so large, that a stone block will start sliding down. But if we use a rope roll to help, this will happen much later and at a much steeper angle. Without employing a rope roll we use the following equation:
F = Force / Gm = Weight / μs = Coefficient of friction (between sledge and tracks) / α = Angle of inclination
Force dependent on frictionNow the coefficient of friction of the bearing of the rope roll μz
has to be included in the equation. If this coefficient increases, the
force needed becomes distinctly higher (see last column of chart 1).
We use the following values for calculating: CHART 1 (figures [1] page 76)
The value 0,00 (first column) shows which forces accrue without the rope roll. The value 0,04 (third column) assumes you are using the rope roll. The values of the third column are higher, because we haven't included the influence of weight and gravity yet (see further down).
Force dependent on adhesion - at what point do you start slipping?If the friction is smaller than the weight component along the plane, the block starts slipping and slides down with acceleration. We now have to include the coefficient of static friction / adhesion
μo in our calculations. The force used depends on the
adhesion on the inclined plane. For a ramp we can use μo
= 0,2.
If using Gm = 60kg as the weight of one hauler, the stone starts to slip at an angle of inclination of 20° and with better adhesion the stone starts slipping at 30° (Force = negative, shown on the chart as "-"). CHART 2 (figures [1] page 77)
Including the weight of the haulers and the influence of gravityThe steeper the ramp becomes, the better the haulers can use their own weight (60 kg) as a load to counterbalance the stone being carried up. F = Force / Gm = Weight / μo = Coefficient of static friction (between sledge and tracks) / α = Angle of inclination
CHART 3 (figures [1] page 80)
Conclusion: With 90° you would have a hoist, where the haulers can use their full 60kg as ballast. With 52° a hauler can use 54,7 kg of his 60 kg as counterweight. But using the rope roll a hauler is also pulling the stone at the rope, so in addition to his weight he also is using his strength (traction force).
Calculating the required numbers of haulers with and without using the rope rollEgyptologists calculate that ramps of a gradient of 8 to 12° can
be managed by hauling teams. Our calculations (below) show, that slopes
of more than 5° already need large hauling teams and a slope of 10°
needs over 400 men! But with a pair of rope rolls installed directly on
the flank of the pyramid, an angle of inclination of 52° is no problem
at all and you only need 46 haulers! CHART 4 (figures [1] page 80) * Number of men in a team: there are two groups walking down to the left and right side of the track. Furthermore there are always 2 men pulling side by side at the rope. Construction ramps: Considering that you work on a very
narrow ramp, the hauling team should not be too large in numbers. A 50
men-team seems to be realistic and manageable. Anything larger certainly
poses problems, specially considering the length of the whole contraption
of sledge plus hauling teams. Most ramp theories don't consider these
facts and suggest ramps that are definitely too narrow or too steep.
For checking purposes: increasing the coefficient of frictionIf we increase the coefficient of friction (= worse values) our calculations show that the 52° inclination of the pyramid face still can be managed. Using a coefficient of static friction μz of 0,1, a value that we are sure the Egyptians could achieve - we get a hauling team of 51 men instead of one of 46 men. This signifies, that lowering the coefficient of friction from 0,04 to 0,1 (= +150%) we will only need 5 men more (= +11 %). 2781 kp : 54,7 kp ≈ 51 haulers (2781 kp see chart 1, column 5 / 54,7 kp see chart 3, column 1)
For checking purposes: increasing the force exerted by each haulerWe calculate again, but with better values for the force exterted by each hauler. Instead of 12kp we use 15kp per person. We use the same numbers as a well known American archeologist, Mark Lehner [6]. On level surface he assumes, that a 2.5 tons block could be pulled by 7.5 men. If we use the same coefficients of friction as before for our calculations, then we get a force of 15kp per hauler (chart 5, second column). To calculate these kind of numbers, Mark Lehner had to use a coefficient of static friction μo = 0.25 (chart 2, 2nd column). Because Mark Lehner mentions in his text the well known frieze from the tomb of Djehutiotep at Deir El Bersheh, where 172 men haul a statue weighting about 60 tons, we also calculate the coefficient of friction μs and get 0,043 - on this page we did all our calculations with μs= 0,04 (172 x 15kp = 2580kp / 2580 : 60'000 = 0,043). More information. Mark Lehner thinks, that with an angle of inclination of 6° a total of 20 men could haul the same 2.5 tons block, and cover a 333m-distance in 19 minutes. Our calculations on the contrary already show, that with an inclination of 5° you need already 36.5 haulers (chart 2, 2nd column for 15kp). Mark Lehner has, like a lot of other theorists, definitely undervalued the additional force, that is necessary, if you haul on an inclined plane! On a 5° incline, even with a higher tractive force of 12kp instead of 15kp, you still need less haulers when using Löhner's rope rolls than without using them - we calculate only 23 instead of 36 men!
Conclusions
For transporting materials on a slope with an angle of inclination of
5° or more it is expedient to use Löhner's rope roll. This means
that the rope roll can also be used for transporting stones from the harbor
to the building yard close to the pyramid, while building the causeway
and for the transports from the quarries located on the Giza plateau to
the pyramid. The rope roll also is a useful resource in the quarries of
Aswan and can be used to load the granite stones onto the barges.
Lifting the granite blocks with a heavy duty track system with counter weightsCan this method also be used for hauling the 40 to 50 tons granite blocks
required for the King's chamber up the pyramid? Yes - using counterweight
stones and larger hauling teams. We use 8 large stones (5.6 tons each)
as counterweights and calculate, that the difference in weight is distributed
among 2 hauling teams. So the difference is, depending on the size of
the granite beam, from 3.2 to 7.2 tons (48-52t minus 44.8t). This works
out to 1.6 to 3.6 tons for each of the two hauling teams to the left and
right of the main track system. Our calculations show, that we need between
32 and 64 haulers per team.
Sources[1] H. Illig and F. Löhner
Der Bau der Cheops-Pyramide pages 75 sqq. Calculations Dr. H.
Illig and Prof. Dr. Dipl. H.U. Niemitz www.cheops-pyramide.ch Copyright 2006 Franz Löhner and Teresa Zuberbühler |