Arch includes circular arch, parabolic arch and catenary inverted arch (gravity arch). The vault of catenary inverted arch (gravity arch) is the best, because its gravity pressure curve can be made thin and firm on its gravity arch line, that is, the center line of gravity arch section (curved surface center line).

The top of the dome is shell-shaped, including round shell, parabolic shell and gravity arch shell (gravity round shell). Gravity arch shell is the best shell type of dome, because its gravity pressure curve is at the center line of shell section, which is a very theoretical and complicated work.

I built the gravity arch and gravity dome myself.

For the pouring of the dome, the shape and thickness of the shell should be determined first, and then the dome inner shell-inner cylinder mold should be made according to the shape of the shell. The shell should be made in layers and sections according to the principle of equal thickness, and then poured in layers (cement concrete).

From 1962 to 1968, I have been having difficulties in housing and want to build two houses with less money. And less money can only be saved on the roof, so starting from 1965, I studied the structure and mechanical principles of arched and shell roofs. I wrote two papers on the application of mathematical mechanics and architecture in 1973, Gravity Arch and Gravity Circular Shell. 1975 the mathematical and mechanical principles on which the "gravity arch" roof and "gravity dome" roof of two houses I built are based have been given in the above two papers.

After I wrote two papers, Gravity Arch and Gravity Round Shell, I sent them to the Institute of Mathematics, the Institute of Mechanics, the Institute of Architectural Design and the Design Institute of the Ministry of Grain, hoping to give my opinions. If I think there is a new theory, I hope to publish it in their journal. Finally, the Design Institute of the Ministry of Grain wrote back to me, pointing out two problems: "1, the top of the earth silo is a parabola; 2. The arch pressure curve should be within its cross section. "These two questions are very important, but they didn't understand my two papers. For gravity arch and gravity dome given in those two papers, their gravity arch lines (that is, pressure curves) are not only on the cross section of their graphs, but also on the middle line of their cross sections, that is, their pressure curves change with the trend of their graphs, which can make the graphs ("gravity arch" or "gravity dome") very thin, thus saving materials to the maximum extent.

At that time, when designing "gravity arch" and "gravity round shell shape", I had not studied advanced mathematics systematically, but I used the principle of "replacing curve with straight line" and limit to divide the arch line into many small segments of equal length; On the contrary, each segment can be approximately regarded as a curve. Because they are equal in length, the weight of an arch segment made of the same material should be considered equal under the same width, thickness and length, so every additional unit arch length from the vault to both ends will cause the change of its dead weight pressure direction. According to the principle of statics, a "gravity arch line" (this is my name, that is, the pressure curve of gravity arch) is formed. As mentioned earlier, this "gravity arch line" follows "gravity pressure". The formula I used at that time was as follows:

Let a basic unit of arch length be C, then ci is the arch length of I-shaped arch; The vault point is A, and the end point of the nth segment on one side of the arch is M (symmetrical with the other side), so the arch length on one side of the arch is AM, the end point of the first segment of the arch is Mi, the arch length is AMi, the base angle is θi, that is, ∠AiMiBi, and the tangent of the arch intersection point Mi is AiMi. As shown in the following figure-the basic drawing of I-shaped steel arch; Let the semi-arch span of the nth arch be x and the height be y?

artificial intelligence

Ci pen

Rice? θi？ ai？ bismuth (Bi)

Gravity arch foundation drawing Ⅰ section arch

So there is

c=ci= AiMi=√(ai2+bi2) (G 1)

TGθI = bi/ai = itgθ 1(g 2. 1)

tgθn=bn/an=ntgθ 1 (G2.2)

ai=MiBi=cicosθi (G3)

Bi= Abby =cisinθi (G4)

x =σI = 1 ncicosθI =σI = 1 nai？ (G5)

y =σI = 1 ncisinθI =σI = 1 nbi？ (G6)

A⌒Mi=nc？ (Group of Seven)

The basic mechanical principles of gravity dome and gravity dome are the same, but the difference is that each arch length is based on the basic unit arch length, while gravity dome is based on the fact that the side area of each truncated body of its dome is equal to the basic unit circle area. "Gravity circular shell shape" can be regarded as the continuous connection of equilateral circular platforms, and the tangent of any point on the generatrix of gravity circular shell is consistent with the generatrix, that is to say, the pressure curve of gravity circular shell coincides with the generatrix on the side of gravity circular shell.

The calculation formula used in the design of gravity circular shell is as follows:

The first basic unit at the top of gravity circular shell can be regarded as the side of a cone. Because its height H 1 is very small, its edge can be regarded as a plane circle with radius R 1 and area s 1. As the basic unit area, the subscript can be removed, and its side bus is marked as l 1, recording the initial angle of the bottom angle.

s=s 1=πR 12=πR2？ (Q 1. 1)

l 1≈R 1？ (Q 1.2)

TGθ 1 = b 1/a 1 = h 1/r 1？ (Q 1.3)

air

li- 1？ hi- 1？

hello

Mi- 1 Ri- 1 Oi- 1

ci θi bi Ri Oi

Rice? Aiai

Basic drawing of I-order axial section of gravity circular shell

S 1 and tgθ 1 are basic units, and appropriate small values should be adopted in practical application, which depends on the design size of gravity circular shell (radius span length is Rn, base angle θn) and the number of calculation steps n (generally 200≤n≤ 1000 can meet the requirements).

From the mathematical limit theory, it should be

LIMS 1→0 limtgθ 1→0 LiMn→∞

The radius span length Rn and the bottom angle θn are the design certainty values. This paper only records the actual application, but does not make a deeper theoretical study on this issue.

Then the side area of the basic unit frustum is s, so

s=si？ i= 1，2，3……，n (Q2)

As shown in the basic drawing of the I-axis section of the gravity circular shell.

hi = QiOi = hi- 1+bi(q 3. 1)

hi- 1=QiOi- 1？ (Q3.2)

bi=Mi- 1Ai=cisinθi (Q4)

Ri = MiOi = Ri- 1+ai(q 5. 1)

ri- 1 = Mi- 1Oi- 1？ (Q6)

ai=MiAi=cicosθi (Q7)

tgθi=bi/ai=itgθ 1 (Q8)

Li = Mickey = Li-1+ci (Q9. 1)

Li- 1 = Mickey-1=Ri- 1/cosθi? (Q 10)

ci = Mi- 1Mi = Li-Ri- 1/cosθI(q 1 1)

si =πR2 =πRili-πRi- 1li- 1？ (Q 12. 1)

R2=Rili-Ri- 1li- 1？ (Q 13. 1)

Ri=licosθi (Q5.2)

rili = Li 2 cosθI(q 14. 1)

Ri- 1li- 1 = Ri- 12/cosθI(q 14.2)

R2 = li2 cosθI-Ri- 12/cosθI(q 13.2)

li2 =(R2+Ri- 12/cosθI)/cosθI？ (Q9.2_ 1)

Li2 = R2/ KOS θI+Ri- 12/ KOS 2θi (Q9.2_2)

Li = √ [R2/cosθ i+ri-12/cos2θ i]? (Question 9.3)

In addition, the radius span of the bottom surface of the nth gravity frustum-shaped circular shell is Rn, the height of the circular shell is Hn, the generatrix length of the circular shell is Cn, and the total side area of the circular shell is Sn.

ln =√[R2/cosθn+Rn- 12/cos 2θn](q 15. 1)

Rn=lncosθn？ (Q 15.2)

HN =σI = 1 ncisinθI =σI = 1 nbi(q 15.3)

cn =σI = 1 NCI(q 15.4)

TGθn = bn/an = ntgθ 1(q 15.5)

Sn=ns？ (Q 15.6)

I built the inner tube mold of the gravity dome roof with Amorpha fruticosa (I learned to knit things by groping before, weaving baskets, stores, steaks and other furniture. ), coated with a thin layer of mortar, polished firmly, and became the inner tube mold. After the completion, the tire mold was not removed.

After 1978, I taught myself advanced mathematics (mainly used by physics department of Sichuan University, with other textbooks), and looked back at the gravity arch line-pressure curve of "gravity arch", which turned out to be the inversion of catenary. Isn't it? The mechanical property of catenary is the tension curve produced by the natural sag of uniform catenary, which is consistent with the trend of catenary, or more accurately, coincident; After the catenary is upside down, it forms an arch line, and its stress is pressure, forming a pressure curve-gravity arch line, so the arch line formed by it coincides with its pressure curve.

y

n？ P(x，y)

Answer?

o？ m？ x

Catenary diagram

Catenary equation:

AO=a (X 1)

Y=a(ex/a+e-x/a)/2 (X2. 1)

Y=ach(x/a) (X2.2)

A⌒P=a(ex/a-e-x/a)/2 (X3. 1)

A⌒P=ash(x/a)？ (X3.2)

1975, I built two houses with "gravity arch" roof and "gravity dome" roof on the ruins of two houses destroyed by fire in Dunxiao's house, in order to save money, solve my housing difficulties and build them for scientific experiments. 1979 When we returned to our hometown, we gave the two experimental rooms to Dun School for free. His family was in trouble, so he lived in those two rooms. Until 2009, his superiors granted relief to their family and built three rooms. In order to build these three houses, the foundation was demolished, and the one with the "gravity dome" roof was demolished, while the one with the "gravity dome" roof was still intact.

—— Excerpt from Guo Dunqing's autobiography