How to build a Pyramid
Created | Updated Dec 3, 2002
The Mathematics of Pyramid Building
Introduction. Below is the first part of a three part discussion about how pyramids and Stonehenge may have been built and also how obelisks and Easter Island statues could have been raised using simple ropes and wooden levers. Basically, I try to show how it was mathematically possible. The document should have diagrams in it, but I believe that, as it stands, it may be understandable and I am going to leave it here while I work on pasting in the diagrams. Some people may be interested in the following publication, the Handbook and Transactions of the Tennessee Junior Academy of Science, 1993,page 34, The Mystery of the Great Pyramid, Theresa Jane Coombe. My 16 year old daughter raised a 1200 pound block by herself. I would very much appreciate getting feedback. Corrections are welcome.
My e-mail address is [email protected].
The ancient builders of Stonehenge and the Egyptian pyramids moved big stones. There is evidence that the Great Pyramid was built in 30 years so each of the 2 or so million blocks averaging two and a half ton in weight was moved into place every three or so minutes. Since the lower one third of a pyramid contains two thirds of the volume, the use of lower ramps is logical. But, it is probable that other methods were used to convey the stones farther up. In fact, the Greek author Herodotus is reported to have written about the use of wooden mechanical devices. Pulleys and wheels had not been invented or the materials to make them were not available so it is a fair assumption that ropes and wooden poles were employed, the same materials in sailing boats. Herodotus visited the pyramids some 1,000 years after they were built and his brief account was based on conversations with the priests. He mentioned the use of short planks.
We can infer that the short planks were the cribbing, dunnage that could have been used to support the loads when the levers were moved. Essentially all methods, other than ramps, include these concepts.
The poles can be used primarily as levers and secondarily for storage of energy because they are flexible. The storage of energy permits the load to be shared between levers. Ropes, if used in windlasses, can enormously increase the mechanical advantage.
A stout pole used a lever is an obvious choice to begin to move a large object but, in practice, it is clumsy, difficult to hold against a stone and building a stable pivot gets harder as it gets higher. A-frames are popular and easy to mathematically evaluate so it not easy to understand why they have been misused in attempts to raise obelisks, when in one failure the mechanical advantage was around two. A near vertical A-frame has a very high mechanical advantage, for example in a simplified case in which a stone is being lifted, it is equal to the cosine divided by the sine of the vertical angle, for example if the angle is one degree, the MA, mechanical advantage, is 57.
The A-frame is thus a splendid device to turn or move a stone nearly in place so it can be fitted correctly. The Middle American walls famed for their tight clearances could well have used this concept.
A two-rope windlass with two poles, one the lever and one the axis, provides a high MA. For example, if the lever pole is 4 foot long and the axis pole is 2 inches in diameter, the MA is theoretically 12 for a symmetrical lever. The double windlass can be employed in the middle of the ropes between the anchor and the object to be moved or it can be part of an anchor device like a two bar fence.
Poles that are flexible or a fence bar that is flexible enable the sharing of the load. An example of not sharing a load could be a second jack on the same corner of an automobile. However, if each jack had a coil spring on it, the weight of the automobile would be shared.
We have extensively tested this type of windlass. It really can exert a lot of tension between an anchor and an object. The axis can be tethered to hold it straight and it helps to keep the two ropes very symmetrical on the axis. If the diameter of the axis is d and the length of one arm of the lever is l, then the mechanical advantage is l/d.
Actually, the windlasses could be ganged together, especially when they are closely spaced. This concept could keep the axis poles square to the pulling ropes. A simple rope loop parked on the ropes is slid up to hold the lever pole after it has been used to tension the ropes.
Here is a worked example of the mathematics of the A-frame.
If a block size is 3 feet, the length w = 8 feet, say. Assume the A-frame is 20 feet long and rigid, so L = 20. The sine of angle A = w/L = 8/20 = 0.4 so angle A = 24 degrees.
Similarly, using the cosine of angle A, p = 18.3 feet. We chose a horizontal pull, P to simplify the mathematics.
Using the mechanical engineering moment equation,
W * w = P * p
we can substitute the weight of the block, say 5000 pounds and calculate the pull, P pounds, P = W * w/p = 5000 * 8/18.3 = 2186 pound.
The value p/w or 18.3/8 or 2.3 is the mechanical advantage. It is also equal to the ratio of the cosine and sine of the vertical angle.
Mechanical advantage = cos A / sin A.
If heavy block has just to be pick up to adjust its position slightly or rework it, the angle would be small and, for example, if A = 1 degree, then the mechanical advantage is 57!
A 5000 pound block could be lifted with a pull of 87 pound. Inefficiencies like bending of the A-frame and stretching of the rope would lower the theoretical mechanical advantage but you could see how several strong men could raise a 5000 pound block using ropes and poles.
If the angle A is 5°, the mechanical advantage drops to about 11 and the total theoretical pull increases to 440 lbs, which is not out of the question especially with several ropes with or without windlasses and each time the A-frame is straightened to the vertical, the block is raised a little less than 1 (theoretical) inch; whereupon, short planks are inserted underneath the block. If there are 4 ropes hanging from the A-frame, mini-windlasses on each pair, shorten the ropes and the process starts again.
In evaluating the mechanical advantage in other situations, it should be understood that in these cases there has to be a pivot point, from which you helps to draw a line at RIGHT ANGLES to the force like the weight at the center of gravity of the object or the rope, for example which is applying the pull, or in the case of a simple lever, the upward lift applied.
Lifting a Block
Height h, widths b, angle a, weight W, uplift L.
Center of gravity to pivot, w
Uplift to pivot, l
Block diagonal, d = 0.5*(b^2+h^2)^0.5
Diagonal angle, t, tan t = b/h t = tanh(b/h)
W * w = L * l, mechanical moment
w = cos(tanh(b/h)+a) * 0.5 * (b^2+h^2)^0.5
l = h * cos(a)
W * cos(tanh(b/h)+a) * 0.5 * (b^2+h^2)^0.5 = L * h * cos(a)
For example, when the angle a is zero, the block is flat on the ground and L should be half the weight W
W * cos(tanh(b/h)) * 0.5 * (b^2+h^2)^0.5 = L * h * 1.0
b = 2 h = 3
W * cos(tanh(2/3)) * 0.5 * (3.6) = L * 3
W * 0.832 * 0.5 * 3.6 / 3 = L is OK
W * 0.5 = L
If a thin post is lifted, the upward force changes little but lateral movement of the center of gravity in a larger diameter object reduces the upward force like in the familiar example below of a 55 gallon drum full of diesel:
W = 450 lbs, h = 34 ins, w = 22 ins
When it is tilted (90 - 33) or 57 degrees, the weight is over the center of gravity.
Diagonal angle = tanh(22/34) = 33 degrees
In order to get it started tilting, the sideways push at the top has to be
450 * (22/2) / 34 = 146 lbs
Of course, if the drum is on its side, you have to lift half of 450 lbs to start picking it up.
If the shape of the block is made like a pole or post.
Then h = 96 and b = 4
W * cos(tanh(b/h)) * 0.5 * (b^2+h^2)^0.5 = L * h * 1.0
W * cos(tanh(4/96)+ a) * 0.5 * (4^2+96^2)^0.5 = L * 96 * cos(a)
W * cos(2.4 +a) * 0.5 * 96.1 = L * 96 * cos(a)
When a = 60 degrees
W * 0.463 * 0.5 * 96.1 = L * 96 * 0.5
W * 22.25 = L * 48
W * 0.463 = L
L uplift is mainly nearly half the weight
Raising an Obelisk with an A-frame
The techniques learned above can be used to investigate the problems of raising an obelisk with an A-frame.
Distances to pivots:
A-frame, p1 and p2 to the two ropes
Obelisk, center of gravity, w, rope, l.
The pull on the obelisk as a result of the pull, P on the A-frame, which we have assumed is vertical, is equal to P * p1/p2
p2 = l*sin(t) and p1 = l * sin(a+tanh(b/h))
p = (h^2+b^2)^0.5 *sin(a+tanh(b/h)
w = 0.5*(b^2+h^2)*0.5*cos(a+tanh(b/h)
and the mechanical moment equation is
W * w = P * (p1/p2) * p
If the obelisk dimensions are 20 ft by 3 ft, weight 20 tons, the A-frame height is 25 ft,
angles a = 45 degrees and t = 75 degrees
w = 6.0 ft, p2 = 24.1 ft , p1 = 20.1 ft and p = 16.3 ft
P = W * 6.0 / (20.1/24.1) / 16.3 = 0.44 * W tons
P = 0.44 * 20 * 2000 = 17,654 lbs
Divide this weight by 120 men, each would have to pull, 147 lbs, a tough order if the rope is above your head.
But, if the obelisk angle has risen to 75 degrees, then the theoretical pull for each man is only 18 lbs. So one solution is to drag the obelisk up a ramp so that the bottom can be dropped down to provide the practicable angle. This technique, which was used to raise a Stonehenge type sarsen stone in the last few years, is discussed later.
Techniques that may have been used to build the pyramids.
Diagram needed for definitions, ask please!
A 5000 lb stone block in a frame, making an effective side size of 4 foot by 4 foot, would permit the insertion of levers, wooden poles which being flexible share the load. Ropes tied to the ends of each lever go to an anchor fence, where double windlasses can exert an enhanced pull. The frame is symmetrical and the levers are removed and inserted as necessary. Wooden cribbing would be used to stabilize the block during re-arrangement of the levers.
For a simplified situation which doesn't involve steps, we assume that the side of the pyramid is made flat, possibly, say, with wooden inserts.
p = l * cos(38)
The angle of the pyramid is 38 degrees and assume the rope is horizontal,
Assume the length of the lever is 20 ft
p = 20 * 0.79 = 15.8 ft, distance from the block pivot to the rope
Assume a 2.5 ton block being moved is in a 4 foot wooden frame with lever poles
w = 2.82 ft
P * p = W * w
P = 5000 * 2.82 / 15.8 lbs = 892 lbs for one lever.
Depending on the length of the frame, you can use multiple levers, say 4. These levers being wooden poles are flexible and share the load.
P = 223 lbs
If the anchor were fitted with an equal number of double windlasses, with 4 foot arms and 2 inch axes, providing a mechanical advantage of 24, the pull would be reduced to 223/24 = 10 lbs.
Of course, these are theoretical values and there would be an appreciable loss of efficiency due to rope stretch, for example. However, this system would permit the stone to be hauled up the side of the pyramid with perhaps as few as 3 or 4 men. . This technique could even be used for an obelisk in which case there would be 3 or 4 levers for every 2.5 tons of obelisk. Because of the long shape of the obelisk, this technique is ideal. A 40 ton obelisk might have 50 or so levers.
If the frame around the block or obelisk is made as round as possible, it makes the task easier
Let the radius of the roller be r = 3 ft, L the length of the pole = 12 ft. Slope = 38 degrees.
v45, the length of the slope from the point the roller touches the surface of the slope up to the anchor = 50 ft when b=45 degrees. Then, start with b = 24 degrees.
Effective v, as the roller moves = v45-2*pi*r*(90-2*b)/360 = 47.8 ft
n= r/tanb = 6.7 ft
h= (L+n)*sin(2*b) = 13.9 ft
q= (L+n)*cos(180-2*b) = -12.5 ft
p= v*sin(g) = 15.0 ft
g= tanh((h/(q+v+n)) = 18.3 degrees
w= r*sin(a) = 1.85 ft
P = W*w/p Moment equation = 614 lbs for W = 5000 lbs
So with a 24:1 ratio windlass, individual force = 26 lbs
So one man could roll a pyramid type block up the pyramid, although several others would be required to stabilize it while the lever pole is shifted. The individual who is using the windlass would experience that the force needed would change from a theoretical 25 lbs up to 35 lbs as the roller rolls up the slope and the angle of the lever pole changes. The actual force might be twice the theoretical. Also, in practice at least two windlasses would be needed to steer the block.
The block is supported on a number, n, of poles or long flexible planks that are supported by dunnage, short planks, perhaps. Due to the flexibility of the poles, each will support a share of the weight. In a very simplified analysis, we can assume that a pole can be raised at one end and the equivalent shared weight will be raised.
The mechanical moment equation,
(W/n) * w = P * p
Assume, W = 5000 lbs, w = 3ft and p = 9 ft and n = 6
P = (5000/6) * 3 / 9 = 278 lbs
Raising this weight is close to the range of a strong man but it would not be difficult to use a simple lever with a small A-frame.
Cradle Technique to Raise an Obelisk
Some of the Egyptian standing obelisks are made out of one stone, over 250 tons in weight. How did they do that? Of course, we don’t know. But, if you know the mathematics, have some practical experience handling heavy stones using only ropes and wooden levers, we can speculate. Then, we hope that others will critique and improve, maybe, even test the concepts.
Without a diagram, I am going to describe a concept, which I haven’t tested at any scale. Imagine an upside down dinner table with two east legs and two west legs pointing up. Put a row of blocks underneath the east side and a row underneath the west side just inside the line of the legs. Put a 20 gallon drum, 31 inches high and 14 inches in diameter, full of water, it would weigh about 200 pounds, on the table so that it sits above the east blocks. You can see that it would be quite easy to lift the west side of the table and add another row of blocks. Roll the drum up the table to the west side. Raise the east side and add a double row of blocks. Roll the drum back. Repeat this process to continue to raise this heavy drum. Using some of the equations developed above, you can calculate what force is required.
And you can use these mathematics to demonstrate whether it might be possible to raise a 250 ton obelisk, for example.
We used the upside down table as a ‘cradle’. A big cradle would consist of a row of horizontal beams with vertical braced poles, again on the east side and the west side. Blocks again used just inside the line of the vertical poles. The horizontal beams extend well outside the blocks and the triangular bracing is outside the line of the blocks.
An obelisk encased in a round frame would rest on the cradle above the east blocks. (After carving an obelisk, it surely would be very adequately protected against surface damage and breaking in two!)
To raise the west side of the cradle, you might have rope windlasses attached to the top of the east poles. You might have an A-frame or wood lever attached by ropes to the west ends of the horizontal beams for lifting. Here the concept used is that flexible beams share the weight. The long shape of the obelisk means that many beams can be used. After blocking up the west side, the obelisk is fitted with flexible poles, and again windlasses are used to roll the round frame up the slope. The frame accommodates re-insertion of the poles, perhaps, each quarter turn of the frame half the poles are removed and re-inserted.
As, my daughter said when she was lifting a 1200 pounds block with ropes and flexible poles using no more that a 25 pounds force, “This is so complicated, surely they wouldn’t have done this!!” Yes, it’s a mess but have you ever been on a fully rigged sailing ship? This method enables dozens of men to raise a huge obelisk instead of thousands building ramps miles long.
By the way, you continue to build up the blocks to make a wall each side, adequately buttressed to support the wall and to keep the cradle in place. And then, I like the idea of the sand box demonstrated on the American PBS Nova program. With end walls and a middle wall, you have a big old box with one side filled with dry fluid sand on which one end of the obelisk rests. When the sand is allowed to flow out the obelisk tips down with a controlled descent that permits it to accurately hit a turning groove in the intended plinth. Nova, by the way, did not show how they put their obelisk on top of their sand box. I imagine they used a crane to lift their 40 ton obelisk up there.
I will add some mathematics later.
Rollers for moving the large stones, especially an obelisk.
The American PBS Nova program on Easter Island statues comprehensibly demonstrated that rollers to move large stones were impracticable. However, I have always felt that to build the Great Pyramid in 30 years, one stone every 3 or so minutes must have required a very efficient transportation system. I imagine that they started with an upside down wooden railway track with the sleepers (ties) uppermost.
The stones on sleds were slid along the track using water or grease for lubrication. It was not as dry then but there must have been sand present to wear out the tracks. They would have worn to a wedge shape. Some blithering genius came along and when he suggested using rollers on top of the track, was probably told that rollers were unsuccessfully tried centuries ago. He may have persisted and when they tried them, they found that the track kept the rollers square and separated. When each team arrived at the top of the ramp, they took their sled and a bunch of rollers back to the beginning of the ramp.
I tested this idea with a track made out of rather knobbly half slab cut-offs. A sled weighing nearly one ton, was pulled with one windlass, but wouldn’t go over the first wedge step. I asked my petite wife to give it a push and it went up and over. In practice, the potential energy to raise the sled on a step is gained back but I would think that the sled needs to be kept moving for best operation. I later used the track to move a 2.5 tons stone on a sled off a trailer using the rollers. I used a tractor to pull the sled, in this case, but the rollers stayed square and separated.
Next Sections
I elaborate on how obelisks were raised and Stonehenge sarsen stones put in place. I discuss recent attempts especially those shown on Nova television. I further discuss how rollers could be used to move large stones using a wooden track.
Introduction. Below is the first part of a three part discussion about how pyramids and Stonehenge may have been built and also how obelisks and Easter Island statues could have been raised using simple ropes and wooden levers. Basically, I try to show how it was mathematically possible. The document should have diagrams in it, but I believe that, as it stands, it may be understandable and I am going to leave it here while I work on pasting in the diagrams. Some people may be interested in the following publication, the Handbook and Transactions of the Tennessee Junior Academy of Science, 1993,page 34, The Mystery of the Great Pyramid, Theresa Jane Coombe. My 16 year old daughter raised a 1200 pound block by herself. I would very much appreciate getting feedback. Corrections are welcome.
My e-mail address is [email protected].
The ancient builders of Stonehenge and the Egyptian pyramids moved big stones. There is evidence that the Great Pyramid was built in 30 years so each of the 2 or so million blocks averaging two and a half ton in weight was moved into place every three or so minutes. Since the lower one third of a pyramid contains two thirds of the volume, the use of lower ramps is logical. But, it is probable that other methods were used to convey the stones farther up. In fact, the Greek author Herodotus is reported to have written about the use of wooden mechanical devices. Pulleys and wheels had not been invented or the materials to make them were not available so it is a fair assumption that ropes and wooden poles were employed, the same materials in sailing boats. Herodotus visited the pyramids some 1,000 years after they were built and his brief account was based on conversations with the priests. He mentioned the use of short planks.
We can infer that the short planks were the cribbing, dunnage that could have been used to support the loads when the levers were moved. Essentially all methods, other than ramps, include these concepts.
The poles can be used primarily as levers and secondarily for storage of energy because they are flexible. The storage of energy permits the load to be shared between levers. Ropes, if used in windlasses, can enormously increase the mechanical advantage.
A stout pole used a lever is an obvious choice to begin to move a large object but, in practice, it is clumsy, difficult to hold against a stone and building a stable pivot gets harder as it gets higher. A-frames are popular and easy to mathematically evaluate so it not easy to understand why they have been misused in attempts to raise obelisks, when in one failure the mechanical advantage was around two. A near vertical A-frame has a very high mechanical advantage, for example in a simplified case in which a stone is being lifted, it is equal to the cosine divided by the sine of the vertical angle, for example if the angle is one degree, the MA, mechanical advantage, is 57.
The A-frame is thus a splendid device to turn or move a stone nearly in place so it can be fitted correctly. The Middle American walls famed for their tight clearances could well have used this concept.
A two-rope windlass with two poles, one the lever and one the axis, provides a high MA. For example, if the lever pole is 4 foot long and the axis pole is 2 inches in diameter, the MA is theoretically 12 for a symmetrical lever. The double windlass can be employed in the middle of the ropes between the anchor and the object to be moved or it can be part of an anchor device like a two bar fence.
Poles that are flexible or a fence bar that is flexible enable the sharing of the load. An example of not sharing a load could be a second jack on the same corner of an automobile. However, if each jack had a coil spring on it, the weight of the automobile would be shared.
We have extensively tested this type of windlass. It really can exert a lot of tension between an anchor and an object. The axis can be tethered to hold it straight and it helps to keep the two ropes very symmetrical on the axis. If the diameter of the axis is d and the length of one arm of the lever is l, then the mechanical advantage is l/d.
Actually, the windlasses could be ganged together, especially when they are closely spaced. This concept could keep the axis poles square to the pulling ropes. A simple rope loop parked on the ropes is slid up to hold the lever pole after it has been used to tension the ropes.
Here is a worked example of the mathematics of the A-frame.
If a block size is 3 feet, the length w = 8 feet, say. Assume the A-frame is 20 feet long and rigid, so L = 20. The sine of angle A = w/L = 8/20 = 0.4 so angle A = 24 degrees.
Similarly, using the cosine of angle A, p = 18.3 feet. We chose a horizontal pull, P to simplify the mathematics.
Using the mechanical engineering moment equation,
W * w = P * p
we can substitute the weight of the block, say 5000 pounds and calculate the pull, P pounds, P = W * w/p = 5000 * 8/18.3 = 2186 pound.
The value p/w or 18.3/8 or 2.3 is the mechanical advantage. It is also equal to the ratio of the cosine and sine of the vertical angle.
Mechanical advantage = cos A / sin A.
If heavy block has just to be pick up to adjust its position slightly or rework it, the angle would be small and, for example, if A = 1 degree, then the mechanical advantage is 57!
A 5000 pound block could be lifted with a pull of 87 pound. Inefficiencies like bending of the A-frame and stretching of the rope would lower the theoretical mechanical advantage but you could see how several strong men could raise a 5000 pound block using ropes and poles.
If the angle A is 5°, the mechanical advantage drops to about 11 and the total theoretical pull increases to 440 lbs, which is not out of the question especially with several ropes with or without windlasses and each time the A-frame is straightened to the vertical, the block is raised a little less than 1 (theoretical) inch; whereupon, short planks are inserted underneath the block. If there are 4 ropes hanging from the A-frame, mini-windlasses on each pair, shorten the ropes and the process starts again.
In evaluating the mechanical advantage in other situations, it should be understood that in these cases there has to be a pivot point, from which you helps to draw a line at RIGHT ANGLES to the force like the weight at the center of gravity of the object or the rope, for example which is applying the pull, or in the case of a simple lever, the upward lift applied.
Lifting a Block
Height h, widths b, angle a, weight W, uplift L.
Center of gravity to pivot, w
Uplift to pivot, l
Block diagonal, d = 0.5*(b^2+h^2)^0.5
Diagonal angle, t, tan t = b/h t = tanh(b/h)
W * w = L * l, mechanical moment
w = cos(tanh(b/h)+a) * 0.5 * (b^2+h^2)^0.5
l = h * cos(a)
W * cos(tanh(b/h)+a) * 0.5 * (b^2+h^2)^0.5 = L * h * cos(a)
For example, when the angle a is zero, the block is flat on the ground and L should be half the weight W
W * cos(tanh(b/h)) * 0.5 * (b^2+h^2)^0.5 = L * h * 1.0
b = 2 h = 3
W * cos(tanh(2/3)) * 0.5 * (3.6) = L * 3
W * 0.832 * 0.5 * 3.6 / 3 = L is OK
W * 0.5 = L
If a thin post is lifted, the upward force changes little but lateral movement of the center of gravity in a larger diameter object reduces the upward force like in the familiar example below of a 55 gallon drum full of diesel:
W = 450 lbs, h = 34 ins, w = 22 ins
When it is tilted (90 - 33) or 57 degrees, the weight is over the center of gravity.
Diagonal angle = tanh(22/34) = 33 degrees
In order to get it started tilting, the sideways push at the top has to be
450 * (22/2) / 34 = 146 lbs
Of course, if the drum is on its side, you have to lift half of 450 lbs to start picking it up.
If the shape of the block is made like a pole or post.
Then h = 96 and b = 4
W * cos(tanh(b/h)) * 0.5 * (b^2+h^2)^0.5 = L * h * 1.0
W * cos(tanh(4/96)+ a) * 0.5 * (4^2+96^2)^0.5 = L * 96 * cos(a)
W * cos(2.4 +a) * 0.5 * 96.1 = L * 96 * cos(a)
When a = 60 degrees
W * 0.463 * 0.5 * 96.1 = L * 96 * 0.5
W * 22.25 = L * 48
W * 0.463 = L
L uplift is mainly nearly half the weight
Raising an Obelisk with an A-frame
The techniques learned above can be used to investigate the problems of raising an obelisk with an A-frame.
Distances to pivots:
A-frame, p1 and p2 to the two ropes
Obelisk, center of gravity, w, rope, l.
The pull on the obelisk as a result of the pull, P on the A-frame, which we have assumed is vertical, is equal to P * p1/p2
p2 = l*sin(t) and p1 = l * sin(a+tanh(b/h))
p = (h^2+b^2)^0.5 *sin(a+tanh(b/h)
w = 0.5*(b^2+h^2)*0.5*cos(a+tanh(b/h)
and the mechanical moment equation is
W * w = P * (p1/p2) * p
If the obelisk dimensions are 20 ft by 3 ft, weight 20 tons, the A-frame height is 25 ft,
angles a = 45 degrees and t = 75 degrees
w = 6.0 ft, p2 = 24.1 ft , p1 = 20.1 ft and p = 16.3 ft
P = W * 6.0 / (20.1/24.1) / 16.3 = 0.44 * W tons
P = 0.44 * 20 * 2000 = 17,654 lbs
Divide this weight by 120 men, each would have to pull, 147 lbs, a tough order if the rope is above your head.
But, if the obelisk angle has risen to 75 degrees, then the theoretical pull for each man is only 18 lbs. So one solution is to drag the obelisk up a ramp so that the bottom can be dropped down to provide the practicable angle. This technique, which was used to raise a Stonehenge type sarsen stone in the last few years, is discussed later.
Techniques that may have been used to build the pyramids.
Diagram needed for definitions, ask please!
A 5000 lb stone block in a frame, making an effective side size of 4 foot by 4 foot, would permit the insertion of levers, wooden poles which being flexible share the load. Ropes tied to the ends of each lever go to an anchor fence, where double windlasses can exert an enhanced pull. The frame is symmetrical and the levers are removed and inserted as necessary. Wooden cribbing would be used to stabilize the block during re-arrangement of the levers.
For a simplified situation which doesn't involve steps, we assume that the side of the pyramid is made flat, possibly, say, with wooden inserts.
p = l * cos(38)
The angle of the pyramid is 38 degrees and assume the rope is horizontal,
Assume the length of the lever is 20 ft
p = 20 * 0.79 = 15.8 ft, distance from the block pivot to the rope
Assume a 2.5 ton block being moved is in a 4 foot wooden frame with lever poles
w = 2.82 ft
P * p = W * w
P = 5000 * 2.82 / 15.8 lbs = 892 lbs for one lever.
Depending on the length of the frame, you can use multiple levers, say 4. These levers being wooden poles are flexible and share the load.
P = 223 lbs
If the anchor were fitted with an equal number of double windlasses, with 4 foot arms and 2 inch axes, providing a mechanical advantage of 24, the pull would be reduced to 223/24 = 10 lbs.
Of course, these are theoretical values and there would be an appreciable loss of efficiency due to rope stretch, for example. However, this system would permit the stone to be hauled up the side of the pyramid with perhaps as few as 3 or 4 men. . This technique could even be used for an obelisk in which case there would be 3 or 4 levers for every 2.5 tons of obelisk. Because of the long shape of the obelisk, this technique is ideal. A 40 ton obelisk might have 50 or so levers.
If the frame around the block or obelisk is made as round as possible, it makes the task easier
Let the radius of the roller be r = 3 ft, L the length of the pole = 12 ft. Slope = 38 degrees.
v45, the length of the slope from the point the roller touches the surface of the slope up to the anchor = 50 ft when b=45 degrees. Then, start with b = 24 degrees.
Effective v, as the roller moves = v45-2*pi*r*(90-2*b)/360 = 47.8 ft
n= r/tanb = 6.7 ft
h= (L+n)*sin(2*b) = 13.9 ft
q= (L+n)*cos(180-2*b) = -12.5 ft
p= v*sin(g) = 15.0 ft
g= tanh((h/(q+v+n)) = 18.3 degrees
w= r*sin(a) = 1.85 ft
P = W*w/p Moment equation = 614 lbs for W = 5000 lbs
So with a 24:1 ratio windlass, individual force = 26 lbs
So one man could roll a pyramid type block up the pyramid, although several others would be required to stabilize it while the lever pole is shifted. The individual who is using the windlass would experience that the force needed would change from a theoretical 25 lbs up to 35 lbs as the roller rolls up the slope and the angle of the lever pole changes. The actual force might be twice the theoretical. Also, in practice at least two windlasses would be needed to steer the block.
The block is supported on a number, n, of poles or long flexible planks that are supported by dunnage, short planks, perhaps. Due to the flexibility of the poles, each will support a share of the weight. In a very simplified analysis, we can assume that a pole can be raised at one end and the equivalent shared weight will be raised.
The mechanical moment equation,
(W/n) * w = P * p
Assume, W = 5000 lbs, w = 3ft and p = 9 ft and n = 6
P = (5000/6) * 3 / 9 = 278 lbs
Raising this weight is close to the range of a strong man but it would not be difficult to use a simple lever with a small A-frame.
Cradle Technique to Raise an Obelisk
Some of the Egyptian standing obelisks are made out of one stone, over 250 tons in weight. How did they do that? Of course, we don’t know. But, if you know the mathematics, have some practical experience handling heavy stones using only ropes and wooden levers, we can speculate. Then, we hope that others will critique and improve, maybe, even test the concepts.
Without a diagram, I am going to describe a concept, which I haven’t tested at any scale. Imagine an upside down dinner table with two east legs and two west legs pointing up. Put a row of blocks underneath the east side and a row underneath the west side just inside the line of the legs. Put a 20 gallon drum, 31 inches high and 14 inches in diameter, full of water, it would weigh about 200 pounds, on the table so that it sits above the east blocks. You can see that it would be quite easy to lift the west side of the table and add another row of blocks. Roll the drum up the table to the west side. Raise the east side and add a double row of blocks. Roll the drum back. Repeat this process to continue to raise this heavy drum. Using some of the equations developed above, you can calculate what force is required.
And you can use these mathematics to demonstrate whether it might be possible to raise a 250 ton obelisk, for example.
We used the upside down table as a ‘cradle’. A big cradle would consist of a row of horizontal beams with vertical braced poles, again on the east side and the west side. Blocks again used just inside the line of the vertical poles. The horizontal beams extend well outside the blocks and the triangular bracing is outside the line of the blocks.
An obelisk encased in a round frame would rest on the cradle above the east blocks. (After carving an obelisk, it surely would be very adequately protected against surface damage and breaking in two!)
To raise the west side of the cradle, you might have rope windlasses attached to the top of the east poles. You might have an A-frame or wood lever attached by ropes to the west ends of the horizontal beams for lifting. Here the concept used is that flexible beams share the weight. The long shape of the obelisk means that many beams can be used. After blocking up the west side, the obelisk is fitted with flexible poles, and again windlasses are used to roll the round frame up the slope. The frame accommodates re-insertion of the poles, perhaps, each quarter turn of the frame half the poles are removed and re-inserted.
As, my daughter said when she was lifting a 1200 pounds block with ropes and flexible poles using no more that a 25 pounds force, “This is so complicated, surely they wouldn’t have done this!!” Yes, it’s a mess but have you ever been on a fully rigged sailing ship? This method enables dozens of men to raise a huge obelisk instead of thousands building ramps miles long.
By the way, you continue to build up the blocks to make a wall each side, adequately buttressed to support the wall and to keep the cradle in place. And then, I like the idea of the sand box demonstrated on the American PBS Nova program. With end walls and a middle wall, you have a big old box with one side filled with dry fluid sand on which one end of the obelisk rests. When the sand is allowed to flow out the obelisk tips down with a controlled descent that permits it to accurately hit a turning groove in the intended plinth. Nova, by the way, did not show how they put their obelisk on top of their sand box. I imagine they used a crane to lift their 40 ton obelisk up there.
I will add some mathematics later.
Rollers for moving the large stones, especially an obelisk.
The American PBS Nova program on Easter Island statues comprehensibly demonstrated that rollers to move large stones were impracticable. However, I have always felt that to build the Great Pyramid in 30 years, one stone every 3 or so minutes must have required a very efficient transportation system. I imagine that they started with an upside down wooden railway track with the sleepers (ties) uppermost.
The stones on sleds were slid along the track using water or grease for lubrication. It was not as dry then but there must have been sand present to wear out the tracks. They would have worn to a wedge shape. Some blithering genius came along and when he suggested using rollers on top of the track, was probably told that rollers were unsuccessfully tried centuries ago. He may have persisted and when they tried them, they found that the track kept the rollers square and separated. When each team arrived at the top of the ramp, they took their sled and a bunch of rollers back to the beginning of the ramp.
I tested this idea with a track made out of rather knobbly half slab cut-offs. A sled weighing nearly one ton, was pulled with one windlass, but wouldn’t go over the first wedge step. I asked my petite wife to give it a push and it went up and over. In practice, the potential energy to raise the sled on a step is gained back but I would think that the sled needs to be kept moving for best operation. I later used the track to move a 2.5 tons stone on a sled off a trailer using the rollers. I used a tractor to pull the sled, in this case, but the rollers stayed square and separated.
Next Sections
I elaborate on how obelisks were raised and Stonehenge sarsen stones put in place. I discuss recent attempts especially those shown on Nova television. I further discuss how rollers could be used to move large stones using a wooden track.
