Handout Vol.15

Update : October 4th, 2017

Transparent Structures as well as Environmental Filters

Morphogenetic Operations in Structural Design



Dynamics Operations

Approach using transparent glass, acrylic structure

Approach using Manual Form Optimization Algorithm based on safety ratio due to allowable stress, safety ratio due to buckling phenomenon, energy absorption

Approach using steel mesh forms based on welding technique and manipulation of buckling phenomenon


Geometry Operations

Kigumi : Approach using wooden mesh forms developed with traditional connection system

Development of 2D projection method for 3D complicated geometry

Approach using accumulative form

Fuzzy Node Algorithm

1/f fluctuation / 1D Spectrum Analysis : Naturalness, Comfortableness

Komorebi / 2D Spectrum Analysis : Naturalness, Comfortableness, Preference


Workshop Scale Experiments

Composition of Morphogenetic Process

Lightweight and Ductile Structures : preventing death in the event of collapse


Energy consumption experiments

Manipulation of Buckling Phenomenon

Reciprocal System


Structural Tips


Little by little, learning Great Nature

Soul of Engineering



October 27th, 2015 : Lecture for Rhode Island School of Design RISD

October 26th, 2015 : Lecture for Harvard GSD

April 30th, 2015 : Archi-Neering Design AND in Nanjing

October 27th, 2014 : Lecture for University of Oregon

September 19th, 2014 : Keynote Speech for Smart Geometry in Hong Kong



Transparent Structures as well as Environmental Filters


Dynamics Operations

Approach using

Park Groot Vijversburg

Extreme Nature

Stained Glass

Iz House


Glass Welding

Approach using

Community Center

Approach using

Research Building



Approach using




3D Diagonal Joinery

Flexible Angle Joinery



Approach using














































Transparent Structures as well as Environmental Filters


Jun Sato

Jun Sato Structural Engineers Co., Ltd.

Associate Professor, University of Tokyo

Visiting Professor, Stanford University



佐藤淳構造設計事務所, 東京大学准教授






Thinking of structures that are composed with slight elements, we can notice that transparent / translucent structures have potential to serve as environmental filters to generate Komorebi space.

We should compose a morphogenetic operation for those complicated targets through these practices :

Dynamics Operations

Geometry Operations

Workshop Scale Experiments

Using these operations / manipulations we would be able to develop more morphogenetic forms based on geometry, materials, dynamics, craftsmanship, site matters and the spirit of engineering. These operations are also helpful for us to collaborate with architects.










Dynamics Operations


● Approach using transparent glass, acrylic structure



Park Groot Vijversburg, Netherlands, 2017

Architect : Junya Ishigami, Marieke Kums / studio MAKS

Structural advisor : Jun Sato

Engineering : ABT

Published :

MARK #54

Shinkenchiku, Sepember 2017

Curved glass wall structure generated by manipulating buckling phenomenon and load distribution.


Glass walls work as shear wall to resist wind load.



Buckling strength is manipulated by curvature.



The load distribution is manipulated by arrangement of beams.



Screenshot of form optimization software to manipulate load distribution

Buckling analysis on curved thin wall


Extreme Nature, Venezia Biennale 2008, Italy

Architect : Junya Ishigami

Structure : Jun Sato

Slight, rahmen (rigid jointed frame) structure using ultra high strength steel with glass walls serve as tension bracing.



Greenhouse of 2 m height with 16x16mm steel columns + 8mm glass



Left : Greenhouse of 2 m height

Right : First model


fix-fix         fix-hinge     hinge-hinge   fix-fix+sway   fix-hinge+sway

α=0.5         α=0.708       α=1.0         α=1.0         α=2.0


Buckling strength  :  Pcr =


E  :  Young’s modulus [tf/cm2]

I  :  Moment inertia of section [cm4]

Lk  :  Buckling length Lk =αL [cm]


While developing ideas with Junya Ishigami, we try brainstorming again and again for hours on each day.

Through those detailed thinking of each phenomenon, sometimes a new idea can be generated.




Deformation, Bending stress diagram of rahmen structure due to gravity


Heat cambering of the steel is an important process in the fabrication of the structure as it reduces deformation as well as bending stress.


Process to reduce deformation and bending stress


Heating process by gas burner


Development of Manual Form Optimization Software is contributing for optimised location of columns.




Screenshot of form optimization software


Finally they are composed with columns 16x16mm, 32x32mm, glass wall t = 8mm.

こうして柱は 16x16mm32x32mm、ガラス壁は 8mm と絞られた。



Left : Greenhouse of 2 m height with 16x16mm columns

Right : Greenhouse of 6 m height with 32x32mm columns


Stained Glass Structure

Design & Research : Jun Sato Laboratory, University of Tokyo

Structure in Architecture is appearing diverse forms composed of diverse materials, constructed by diverse methods, and exposed to diverse impacts.

The stained glass panels are made by fixing glass in a slight metal frame, which represents the diverse forms in the manner mentioned below.

As we can see it is sufficiently complex composite to develop a dynamics operation, when completed, it can be adopted for many of other structures.




Left, Middle : Stained Glass Structure test specimen

Right : Pop-up Stained Glass using brass frames, Workshop 2012, Jun Sato Lab




As the cushioning materials inserted between the glass and metal frame are required to be resistant to UV damage, we are using tin plates which we found to be effective.




Joint Detail

Buckling analysis for 3D grid frame


Perfectly Plastic Surface Diagram

Axial force – Bending moment interaction curve considering buckling phenomenon





This structure is representing a design that can take on diverse forms and are subject to the diverse phenomena :


Composed of multiple materials.

Composed of bar and plate elements.

The optimized framework pattern existent to make it stronger using curved line elements.

Elastic behaviour of the tin plate while in a plastic state.

Algorithm to describe plastic hinges, which also describes buckling behavior.

Mutual buckling resistance between the glass plates and the steel frames.








Apartment House in Wasedatsurumaki, 2015, Architect : Yoshio Sakurai

Published : Nikkei, 2017

早稲田鶴巻町の集合住宅 (2015年,建築家:櫻井義夫) で実現


Iz House, Shizuoka, Japan, 2005

Architect : Sou Fujimoto

Structure : Jun Sato

Stacked structure of glass walls and acrylic resin walls.

A simple structural analysis model was developed for seismic response analysis.



Exterior and interior under construction, Structural Test


Structural Analysis Model


Section Drawing





Load displacement curve, Seismic response analysis


● Condensation of the Eigenvalue Equation of Buckling






Buckling analysis for 3D frame

Buckling strength distribution of dimpled wall calculated by condensation method

Published in JIA paper, 2017




[KE] {Ui} = λi [KG] {Ui}  ←  6n行の方程式






まず j 番目の行と列を消去する縮約作業をするために、ガウスの消去法に似た手順を行う。

縮約された剛性マトリックス、幾何剛性マトリックスは係数行列 [Cj][Dj]を用いて

[KE’] = [Cj] [KE] [Dj]

[KG’] = [Cj] [KG] [Dj]


最終的に残る剛性マトリックスを[KE”]、幾何剛性マトリックスを[KE”] と書くと、

 [KE”] {Ui”} = λi [KG”] {Ui”}


この係数行列 [Cj][Dj] の設定方法に改善の余地があるので、今後その研究を進める。



● Glass Welding






Published : JIA journal, 2015 & 2016








2015年度、目地部1000、周辺温度 120 長さ30cmのガラス同士の溶着に成功した。
















この卒業制作「GLASS CUBE」により、滝口雅之が「中村達太郎賞」を受賞。






● Approach using the Manual Form Optimization Algorithm



Community Centre in Kawatana Onsen, Shimonoseki, Japan, 2010

Architect : Kengo Kuma

Structure : Jun Sato

Polyhedral form generated by adjusted position of nodes.


建築家:隈 研吾



川棚温泉交流センター (建築家:隈研吾) の山並みのような多面体





Development of algorithm for form finding under several load scenarios such as :


Gravity + Seismic load X / Gravity + Seismic load Y

Gravity + Wind load X / Gravity + Wind load Y

Self weight + Snow

The parameter to be focused will be such as :

Safety ratio due to allowable stress

Safety ratio due to buckling phenomenon

Energy absorption

This software has automatic execution option which will substitute for Macro or Plug-in of some specific 3D modeling software.




許容応力度に対する「安全率(検定比)」を指標にする場合、各荷重パターンから算出される検定比のうち、材毎に最大値のみを表示します。全ての部材で検定比が 1.0 を超えない形状を見つけることが目標になります。重力や地震、積雪、風など複数の荷重を総合的に評価して形状を最適化することになります。




Polyhedral form for “ Community Centre, Kawatana Onsen”, architect : Kengo Kuma

Safety ratio color chart is displayed while morphing the shape by mouse


This form finding software is applicable to various shapes such as Free Surface, Stacked Clusters, Branched Tree, Randomly Located Columns.




Polyhedral mesh for “ Naoshima Pavilion ”, Kagawa, Architect : Sou Fujimoto

直島パビリオン (建築家:藤本壮介) の溶接金網による透明感ある多面体



Free surface for “ House of Pease HOPE ”, Copenhagen, Architect : Junya Ishigami



Twisted ribbon for “ Cloud Arch ”, Sydney, Architect : Junya Ishigami


Form Optimization Software Component : Hogan + Rhinoceros + Grasshopper



Rhinoceros + Grasshopper + Python + Hogan によるソフトウェアコンポーネント



Stacking free curved wall, architect Sou Fujimoto : Genus of topology is also optimized in this case

アメーバ形ボリュームの積層形態 (建築家:藤本壮介)



Free level floors for “ House NA ”, Architect : Sou Fujimoto

House NA (建築家:藤本壮介) の多段スラブ



Branched tree, Architect : Sou Fujimoto

Randomly located columns for “ Extreme Nature”, Architect : Junya Ishigami

樹状形態 (建築家:藤本壮介) Extreme Nature の林立する柱 (建築家:石上純也)


Algorithm of operations :

Topological Operation  : Manipulate stiffness and stress by morphing the shape. Genus, which can be related to environment matters or space continuity, should be also manipulated.

Density Operation     : Manipulate stiffness and stress by density / porosity of arrangement of elements.

Stress Based Operation : Arrange strong components onto stressful area.


Global / Local shape :

Global Optimization : Optimization of global shape.

Local Optimization  : Optimization of local shape such as drapes, wrinkle, dimples, groove.


03DSCN0039 DVC00195 

Dimple / Groove / Wrinkle to increase buckling strength of thin plate

Water Pavilion, World Expo in Aichi, 2005, Architect : Naomi Sakuragi + Students, University of Tokyo

Pulp Mold Screen, Tokyo University of Science, 20

COCOON, Seminar in Stanford University, 2016




COCOON, Seminar in Stanford University, 2016

We can recognize catenary arch or ellipse dome, in which overall form results, is generated by Global Optimization of stress and stiffness. Also we can find the wrinkle generated by Local Optimization.



Energy absorption diagram for “ New Hakushima Station ”, 2015, Architect : Kazuhiro Kojima / CAt

白島新駅 (建築家:小嶋一浩+赤松佳珠子)







Published in JIA paper, 2017


● Approach using Steel Mesh Forms based on welding technique and manipulation of buckling phenomenon



Research Building, Hakodate Future University, 2005

Architect : Riken Yamamoto

Structure :            Jun Sato

Steel mesh structure composed of vertical and diagonal elements.

Mesh tectonics shows the craftsmanship is necessary to generate these structural, environmental elements.


建築家:山本 理顕






Welding tecnique and reforming tecnique is necessary to fabricate these mesh.








Tsuda Veterinary Clinic, 2003

Architect : Kazuhiro Kojima / CAt

Structure : Jun Sato

Shelf shaped sturcture with 6mm plates, without backboard by controlling 3 dimensional buckling.


建築家:小嶋 一浩/CAt







When flat bar columns are located in radial arrangement, buckling strength can be found 4 times bigger than parallel arrangement.


Buckling control of flat bar columns : Radial, Polygonal, Parallel


Elevation of each grid structure based on buckling phenomena


In some case, forms generated by the optimization of buckling appears not visualizing the stress flow.








その場での簡素な計算 MOOM


Geometry Operations


● Approach using Wooden Mesh Forms



Prostho Museum Research Center, 2010

Architect : Kengo Kuma

Structure : Jun Sato

1st examination of Kigumi with Kengo Kuma

Timber 3D grid structure without metal fixings at joint.




Starbucks Coffee in Dazaifu, 2011

2nd examination of Kigumi with Kengo Kuma

3D diagonal grid acting as a hunched portal frame.




Sunny Hills in Aoyama Tokyo, 2013

Architect : Kengo Kuma

Structure : Jun Sato

Published : MARK #54

3rd Examination of Kigumi – timber joints without metal fixings with Kengo Kuma



A mesh structure can serve as a filter of light, sight, air, heat, sound, water and ecosystem.

Inner space will be filled up with Komorebi – sunlight through leaves.




Joint zoom in / Processed timbers / 2D projection of overlapping timbers

We should develop a suitable way of projecting onto a 2D display.




Discussion with Kengo Kuma is always just a brief moment. It is necessary to develop some imaginations from his few words such as “ scattering a lot of particles ”.




Structural analysis model / Detailed buckling analysis model





Considering the geometry operation, this system can be interpreted as follows :



Local State (shape of components and connection type) :

Component with complicated but singular shape, with no parameters.

The connection type is singular with no parameters.

Component         Growth Process

Growth process is similar to stacking boxels.



As the growth process is easy, random operation can be composed as follows :

Growth process can generate many random global shapes, and each shape is evaluated individually. Finally a single shape is then decided upon.

Estimation values :

Structural dynamics values such as safety ratio, strain energy / Environmental factors / Space volume etc.


Random operation



In this case also a feedback operation is easy.

It indicates when we have a target global shape, we already have a way to compose the local state.

This process can also be understood as the topological optimization.

Thickness of the structure generates gradation under a condition of constant porosity.


                    Target Global Shape

フィードバックも容易で、Topological Optimization を適用しやすい


3D Diagonal Kigumi Joinery



Library and Social Care Center, Yusuhara, 2017, Architect : Kengo Kuma

Under Construction




Flexible Angle Kigumi Joinery




Pergola in the bottom of Carved Tower, Vancouver, 2017, Architect : Kengo Kuma

Published : “ Lixil eye 13 ”, 2017





¿ - cube, 2013

Design & Construction : Ken Yokogawa Laboratory, Nihon University

Structural Adviser : Jun Sato

Like particles gathering into a protein molecule, 60 mm cubes made of hemlock spruce are connected by “ ¿ - inverted question” mark shaped eye bolts.

The structure gradually changes from a hard structure at the base to a soft membrane-like structure on the roof.

The distance between nodes should be the dimension of the cube with factors of x 1, x , x .






1, ,  の長さのみで形成されるとも理解できる。





In this case, the growth process and the feedback process are complicated.


Local State :

Particle with simple and singular shape with no parameters.

The connection type is simple but the angle coordination of the particle may be the parameter.


Growth Process

Adding a single particle : difficult but appears hard and strong

Adding multiple particles : easy but appears soft and weak


頂点を繋いでいくので、1, ,  の長さのみで形成されると考えてしまうとつなぎ方が限定される。


When we want to add a cube, the distance between the nodes might become a limiting dimensions.

As the compromised operation, we can add multiple cubes to span that distance but it appears to be soft and weak.



Fuzzy Node Algorithm

It seems we don’t have so many choice to compose the global / whole shape with only the 3 distances, but we could feel at the construction site, it is not so hard to add a cube.

It has been found because of the flexibility of the connection which can be called “ Fuzzy Node ”.

接合部にわずかな「自由度」を許すと格段に組みやすくなることが体感できる。このぼんやりとした節点は Fuzzy Node と呼べる。

Fuzzy Node

Fuzzy Node の分かりやすい例。


Growth Process with Fuzzy Node

ぼんやりとした節点 Fuzzy node


An algorithm to describe this fuzzy node is necessary to develop these operations.

機構解析において、ぼんやりとした節点 fuzzy node をアルゴリズム化することがひとつの手段。

Soft Computing の分野に通ずる。


● Approach using accumulative form


Different Brick, Exhibition Real Size Competition 2013

Design & Construction : Yusuke Obuchi Lab, the University of Tokyo

Structural Adviser : Jun Sato Lab

Masonry structure composed of ellipse shaped bricks.

The bricks are cast using cone shaped moulds. The moulds were soft enough to be deformed, so different ellipses could be generated from the same mould.

Ellipse packing is a very complicated geometric problem which is solved by finding the solution of simultaneous quartic equations. These are developed using the conditions that the length of circumference must be identical and every adjacent 2 ellipses should have single intersection. Here we proposed an approximate solution.








Not every global shape can be composed. This limitation should be considered as a characteristic of this system.

Another estimation other than those mentioned above will be the compression state, which necessitates that the final shape should be developed with no tension arising.

In this case a feedback operation is complicated to compose, but as we could develop an approximate algorithm of the relationship between the local state and the target curvature of the global shape, we could develop the growth , feedback and iteration processes.


● 1D Spectrum Analysis 1次元スペクトル解析


Cafeteria in Chiba University of Commerce, 2015

Architect : Kazumi Kudo + Hiroshi Horiba / Coelacanth K&H

Structure : Jun Sato

Geometry Advisor : Takashi Chiba

Arranging thin LVL beams in a 1/f fluctuation pattern of spacings.

The 1/f fluctuation makes musics or visual patterns to be comfortable and natural.




薄っぺらいLVL梁を斜め格子状に並べ、そのピッチに1/f ゆらぎ」のリズムを持たせることが千葉氏より提案された。





Roof pattern 屋根伏図


Section 長手の軸組図


LVL屋根を支持する柱は φ141.3x30 で柱頭ピン接合。

鉄骨ラーメン構造部分の柱は H-125x125x6.5x9 柱頭剛接合。



Safety ratio diagram 安全率の色表示図


The location of columns was optimized due to the pattern of beams.

ルーバー状のLVL梁のピッチを並べた数列を波形と見なして 1/f ゆらぎの模様を描く。



Original wave                                       Fluctuation

Progression of spacings


Assume the number of data = N.

波のデータが N 個あるとする。(N は偶数とするのがよい)

am = a0, a1, a2,… aN-1 (m = 0N-1)

Assume the interval of data = Δt, total period Td results in as follows.

データ取得の間隔を Δt とすると、継続時間 Td は、

Td = N Δt

Assume Ck (k = 0N-1) as the factors of Complex Fourier Transform, Ck and the amplitudes Xk are expressed as follows.

複素フーリエ係数を Ck (k = 0N-1)とすると、フーリエ変換の式は、






Power Spectrum by Fourier Transform, logarithm scale

Horizontal axis : f = frequensy of wave

“ power ” can be understood as similar as “ amplitude “.


When the logarithm scale graph with the horizontal axis “ f ” shows a distribution of -1 gradient, it indicates 1/f distribution.



Spectrum analysis is expected to be useful for manipulating environmental factors. There might be some other formulae or parameters existent which are related to environment elements.

1/f ゆらぎ」は他にも多様な形態に適用できそうです。

建築家が「ランダム」と言うのは「心地よいランダム」であり、それは「1/f ゆらぎ」のことかもしれません。




● 2D Spectrum Analysis



The 1/f fluctuation of 1D spectrum makes musics or visual patterns to be comfortable and natural.

Spectrum analysis is applicable to 2D phenomena.



Komorebi  :  Sunlight through leaves, known as a term of which unnoticed in other language

Sazanami  :  Ocean ripple, containing visual scene, sound and atmosphere

Seseragi  :  River stream, containing visual scene, sound and atmosphere





Using 2D Fourier Transform for 2D image resuts in a spectrum diagram as follows.

Values of R/G/B of the pixels are interpreted as 2D wave.

Published in 2017

画像処理の手法に「2次元スペクトル解析」があります。画像のピクセルの色の値は、X, Y方向に並ぶ数列と見ることができますが、それを波形と見なし、その「ゆらぎ成分」をフーリエ変換し、「パワースペクトル」を描きます。2次元画像に描く方法と、1次元グラフに描く方法があります。


1次元スペクトルを指標にすると、1/f ゆらぎ」がひとつのターゲットになることが知られています。



Original image / Power spectrum image drawn in 2D gray scale

White = high power, Dark gray = low power, Navy = 0.0



Power spectrum drawn in 1D graph

The gradient of the distribution is around -1 to -2 ( = 1/f to 1/f2 ).



Original image / Monochrome / 2D power spectrum / 1D power spectrum


Komorebi (sunlight through leaves), Full wave



Komorebi (sunlight through leaves), Fluctuation wave





Komorebi (Sunlight through leaves) / Japanese Pampas Grass / Weird Ground


From these spectra, we can see some contrast of density will be interpreted into naturalness.

Weird Ground is showing similarity with White Noise.





Glass Structure in Stanford / Nebuta Tectonics in Structural studio / Kigumi structure of Sunny Hills Japan

上段:Sunny Hills in Aoyama は「木漏れ日」「紅葉の森」「すすき野原」に近い。




The spectrum image of Glass structure looks quite similar to Komorebi scene.

Nebuta Tectonics looks similar to Japanese Pampas grass scene.

Kigumi structure looks in between Komorebi scene and Japanese Pampas Grass scene.

These are indicating naturalness.





There are some options for this method :

Use color / monochrome image

Use full / fluctuation wave

Filter by some functions before Fourier Transform


Finally we can notice some categories of spectra, for example,

Natural / Artificial / Comfortable / Color oriented preference





Foliage : Gray and RGB respectively

Color preference will be analyzed by spectra of such as RGB, CYM, HSV.





COCOON, Stanford University, 2016


It seems possible to manipulate the 2D spectra by mesh pattern.






Flower / Ocean Ripple / Rough wave / Cherry Blossoms / Komorebi (Sunlight through leaves)

草花 さざなみ 荒波 木漏れ日





Vegetated Cliff / Fleecy Cloud / Pampas Grass Field / Cirrostratus Cloud / Cove /

崖の植生 わた雲 すすき野原 すじ雲


Workshop Scale Experiments



Through performing design-build process in workshops or ephemeral installations, we can compose the process as a morphogenetic operations.

In this process we use these operations in parallel.

Learn material properties / Form study / Structural Experiments / Structural Calculations / Construction

Transparent / translucent structure which works as an environmental filter to generate Komorebi space

Lightweight and ductile structure preventing deth in the event of collapse


Komorebi  :  Sunlight through leaves, known as a term of which unnoticed in other language

Seseragi  :  River stream, containing visual scene, sound and atmosphere

Sazanami  :  Ocean ripple, containing visual scene, sound and atmosphere







Wire works sample

Traditional wire knitting in Okinawa, Japan

Gecko and Cicada by wire artist Hironori Hashi






● Composition of Morphogenetic Process 形態生成アルゴリズム/設計法の構築

Lightweight and ductile structures preventing death in the event of collapse


Nebuta Tectonic – preventing death in the event of collapse

Structural Design Studio, IEDP Integrated Environmental Design Program, the University of Tokyo, 2014 & 2015

Operations will also result in developing a lightweight and ductile structure which will prevent death in the event of collapse.

From this studio we proposed Nebuta Tectonics composed of steel wire frame covered with Washi paper.

Published in GA JAPAN, 2015 / Shinkenchiku, 2015

 “ねぶた構造”− 壊れても死なない構造

建築構造デザインスタジオ, 東京大学環境デザイン統合プログラム 2014 & 2015






Japanese traditional Washi papers are made from fibers of “ Kozo ” or “ Mitsumata ” plants. It is an organic material, made of only the fibers of plants, without chemical glue. It is strong as the fibers are longer than other papers.



(No Photo) (No Photo)

Nebuta ねぶた




Nebuta floats in Nebuta Festival, Aomori, Japan are made of Washi paper, steel strings.




Japanese traditional umbrella Wagasa, representing a lightweight structure,

made of Washi paper coated with linseed oil or perilla oil for waterproof.

The frame is slight and woven with colorful string to prevent buckling.







To resist against the first blow in spring (February or March) called “ Haru Ichi-ban ”, imagining the wind speed 20 m/sec, we practiced a materially nonlinear analysis, concerning the Washi papers as tension elements and controlling the buckling phenomenon and plastic state of 3 mm steel wires.

In this case the buckling length was found to be manipulated to less than 40cm.

The shape was decided through structural analyses, material strength tests, drag coefficient tests, anchor strength tests.

「春一番」を想定して風速 20 m/sec に耐える形態を目指します。3 mm の針金の「座屈長さ」 40cm 以下に制御する必要がありました。和紙の材料試験、抗力係数の解析と模型による計測、アンカー用スクリューペグの引張試験、風荷重に対する材料非線形解析を経て、空気抵抗の少ない形状を決定しました。



Captured model



Drag coefficient analysis by Flow Design (Autodesk), experimentation

抗力係数を計測し、Flow Desgin (Autodesk) の解析と比較する



Washi paper tensile test / Screw peg plucking test



Materially Nonlinear Analysis








Nebuta Tree House, 2015 (Photo by Ying Xu)



When the Washi papers are coated with oil, they turn into translucent material. They will work not only for bracing but also serve as Filters for environmental elements.

Lightweight and ductile structure will be applicable also for Lunar Base and Mars Base.






To save more people, little by little, we are learning Great Nature.



DFL Pavilion 2013

Collaboration with Obuchi Lab ( G30 Digital Fabrication Laboratory )

Tensegrity structure composed of X shaped components work as compression element and cables work as tension element.

計画  :東京大学小渕祐介研究室 (G30 Program)






Geometric non-linear analysis is necessary for this structure because it stabilizes after large deformation such as 47 cm at its center.


極端に剛性の異なる部材が混在する場合の非線形解析は収束しにくい。この解析により、φ3mmのケーブルがピンと張ると概ねφ6mmの丸鋼と同等の硬さの部材と解釈すれば計算が収束しやすく、収束回数が少なく済むことが分かった。コンポーネントに発生する圧縮は 0.4 tf 程度、ケーブルに発生する張力は 0.3 tf 程度。



Compression test for a X shaped component, Tensile test for a cable joint


The X shaped component consists of 3 stainless steel plate of 0.3 + 0.8 + 0.3mm thick.

Ultimate compression strength of this component has found 0.41 tf.

コンポーネントの圧縮試験は 200 tf 試験機を使用、ケーブルの引張試験はテコを応用してバネ秤で載荷した。

圧縮試験で得られた最大荷重は例えば 0.41 tf など。

これを再現できるよう解析モデルの設定を調整する。板厚は実際と同じ 0.3 + 0.8 + 0.3 mm とし、部材幅を 40 mm と設定すれば座屈荷重が 0.29 tf となり、概ね再現しながら少し安全側の算定ができることが分かった。

Once it is found possible to use 40mm wide elements for these grid model, we can find the strength of other options only by the analysis without loading tests.

モデル化の方針が分かれば他の形状の強度や、強度不足のコンポーネントの対処法も解析のみで見つけることができる。板厚を 0.5 + 1.2 + 0.5 mm とすれば座屈荷重が 0.758 tf となり十分なことなどが分かる。













Poured Sticks Structure in DFL Pavilion 2014

Obuchi Lab, G30 Digital Fabrication Laboratory

Sticks are connected in simple way just touching. But they have the parameter of 3D angle and connection point, which makes the geometry complicated.

To make the problem simple, poured sticks are interpreted as the porous volume material.

計画  :東京大学小渕祐介研究室 (G30 Program)




Zoom out from detail

 Finally the way of calculation was found different from Sunny Hills, structure of complicated Kigumi joint.



Sunny Hills Japan, 2013, Architect : Kengo Kuma

Structural analysis model for Sunny Hills, composed with every timbers modeled into bar elements.


Compression Test

Left : Initial 3 specimens

Middle : Improved 2, provided 2 by Shimizu Co.


Specific Gravity ρ= 0.06 tf/m3

Young’s Modulus E = 1.0 kgf/cm2

Yield Stress σy = 0.108 kgf/cm2


Bending Test

Specific Gravity ρ= 0.04 tf/m3

Young’s Modulus E = 0.6 kgf/cm2

Yield Stress σy = 0.209 kgf/cm2


Representing values for structural analysis :



Specific Gravity

Young’s Modulus

Yield Stress





Urethane Sponge




Poured sticks




Styrene foam




Balsa wood




Cedar wood









Even the material was found soft and weak, still we can use it for structure with thick volume.



 Structural analysis for the mock up, gravity and wind speed 10 m/sec loaded.

Operated by Mika Araki, Jun Sato Lab



Structural analysis for the mock up, gravity and wind speed 20 m/sec loaded.

Red elements indicating the lack of strength.

Operated by Mika Araki, Jun Sato Lab




Structural analysis and Optimization progress for final shape.

Operated by Mika Araki


Interface for adjustment of Karamba+Grasshopper to Hogan

Developed by Masaaki Miki



Stainless cables inserted in the wall for extra safety.

They work as tension rings.


Transparent Structure as Perceptual Filter

Stanford University seminar and workshop : Winter semester January ~ March, 2015

Tokyo session : June 13 ~ 14th, 2015

Lecturer : Beverly Choe, Jun Sato

Published in GA JAPAN, 2015

Using 1.3 mm thick engineered, high strength glass panels, joined by an aluminum clamp system, the installation was formed into a triangulated matrix resembling a 3 dimensional truss, reciprocal compositions, or polyhedral shapes.

これは、1.3mmという薄い超強化ガラス Leoflex, Dragontrail を使用したガラス構造です。ReciprocalLamellendach などと呼ばれる相互依存で安定する集積構造にも近い形式です。




Glass : “Leoflex” and “Dragontrail”, 1.3mm thick, with holes, with safety film,

size = 600x600mm, 600x440mm, 300x300mm

Connection : Aluminium straps, rubber washers, metal washers, glazing tape, bolts & nuts






Leoflex and Dragontrail are ultra high-strength and elastic/flexible glass products manufactured by Asahi Glass Company.

It is an alkali-aluminosilicate sheet glass, chemically strengthened and therefore much stronger (6 to 8 times the strength of normal glass), and thinner than conventional tempered glass.

As it is chemically strengthened, it can also be drilled or notched after the strengthening process.





1.3 mm thick glass panels with dimensions of 600x600mm, 600x440mm, 300x300mm were provided.


パネルは 600x600mm, 600x440mm, 300x300mm の3種類



Local geometry to reduce buckling length into 40cm out of 60cm panel




Structural analysis was resulted in the manipulation of buckling length to be less than 400mm.


These forms are generated by controlling the buckling phenomenon through the geometrical configuration and optimization method.







Vault shape with buttress was developed in Stanford University.






Branching dome was developed in Tokyo.






Inthis case we might also apply the fuzzy node algorithm related to Soft Computing.


ですが、アルミストラップを曲げ、ひねって孔に馴染ませる接合具のおかげで接点がぼんやりとした節点 fuzzy node となり、そのおかげで格段に全体形を形成しやすくなりました。形状が位置によって異なる接合具を製造するのはそう不経済でなくなりつつあり、あらかじめ設計しておくことができます。


Fuzzy Node のイメージ





Komorebi Workshop in Harvard GSD, 2016-2017


Komorebi Pavilion in Gund Hall, May 2017



Transparent / Liquid / Shaggy









2D power spectrum of some configurations which students have developed :

Similar as Komorebi spectrum



2D power spectrum of Komorebi


Final shape of flakes and configuration example



Algorithm to convert the complicated configuration typology into structural analysis model


Software component Hogan + Spectrum + Grasshopper + Python


Software component Hogan + Excel



Set up / Heat bending mold / Formed base



Snowflakes / Start assemblage



Build up walls / Engage



Interlock + Slot / Curved wall



Reflection / Silhouette


Hull shaped inner surface / Transparency


Komorebi Pavilion in Autodesk Build Space, January 2017


Creative Structures : art4d workshop in Bangkok, 2012

Using local materials, 4 teams constructed pavilions of 4 to 8 m spans, in 2 days.







French Braid Tectonics

State of the Community 2016 / CITIZENS, COMMUNITIES and MULTILAYERED IDENTITIES, Dhillon Marty Foundaion

Jun Sato Lab, University of Tokyo

Marc Dilet and students, Ecole Nationale Superieure d’Architecture Paris - Val de Seine




Transparent / translucent structure which works as an environmental filter to generate Komorebi space

Lightweight and ductile structure preventing deth in the event of collapse


Komorebi  :  Sunlight through leaves, known as a term of which unnoticed in other language

Seseragi  :  River stream, containing visual scene, sound and atmosphere

Sazanami  :  Ocean ripple, containing visual scene, sound and atmosphere


The form was developed using copper wire, referring to French Braid Hair geometry. The structure was composed of slight elements of 3mm diameter by manipulating geometry, buckling phenomenon and plastic state of material.


To optimize the shape to have target naturalness / comfortableness, one strategy using 2D spectrum analysis will be shown at the end of this essay.

Note that the lightweight, ductile and natural structure will also be applicable for Lunar Base and Mars Base.



French Braid Tectonic in response to the State of Community 2016 conference


Preview : Metal Wire


Metal wires are useful to compose quite ductile structure. Even in the event of collapse, it won’t fall down suddenly, rather makes a soft landing, because of the ductility.

Referring to a following Nebuta Tectonic project, when the slight wires are composed into 3 dimensional framework, the buckling length can be reduced and the wires are reinforced a lot. In this case Washi paper will work as tensile bracing.

When the wires are braided as a following image, they will be reinforced much more.



Nebuta Tectonic : Steel wire + Washi paper, Jun Sato studio 2015 (Photo : Ying Xu)

Traditional Pampas grass braiding, metal wire braiding in Okinawa, Japan


Metal wires are also useful for developing fastener pieces. It is easy to develop the shape of interlocking, slotting,  hooking or bouncing as following images.



Carbon Cable Pavilion for the Housevision Exhibition 2016, Architect : Kengo Kuma

Custom formed pad and fastener


View : French Braid


There are some options to strengthen slight metal wires. One is composing bunches and another is letting them cross at many points. Finally I could imagine that the shape referring to French Braid Hair will be the target.



French braid formed wires, Samples of braiding tecnique


For the material, I have selected copper. Even though copper is much weaker than steel, it still has attractive characters. Copper is one of the materials which we can enjoy the aged surface turned into dark brown or green. Copper has enough ductility which means ability to deform and absorb energy of wind or earthquake. We can think optimistically that because of the weakness we can manipulate geometry and dynamics seriously and an attractive form will be generated by those manipulations.



Overall form resulted in a ellipse dome composed of mingling arches.

Dimensions of the dome part of the pavilion : 3m long×2m wide×1.5m high


Review : Naturalness


We can recognize the ellipse dome, which overall form had been resulted in, is generated by Global Optimization of stress and stiffness. Also we can find the density contrast of braid generated by Local Optimization.

Moreover, we can recognize not only dynamical optimization but Optical Optimization had been manipulated as follows.


Spectrum Analysis :

There is a theory of “1/f fluctuation”  which makes musics or visual patterns to be comfortable and natural. Many natural phenomena have this 1/f fluctuation character. And so it is expected to be useful for manipulating environmental matters.

This character can be recognized when we see the 1D (single dimension) power spectrum of a wave calculated by Fourier Transform. This analysis is applicable to 2D phenomena using 2D Fourier Transform. When that idea was applied for 2D visual image, we can find the 2D power spectrum figure as follows.



Original image / 2D Power spectrum in 2D gray scale

Values of R/G/B of the pixels are interpreted as 2D wave.

White = high power, Dark gray = low power, Navy = 0.0


There are some options for this method :

Use color / monochrome image. Color preference will be analyzed by spectra of such as RGB, CYM, HSV.

Use full / fluctuation wave

Filter by some functions before Fourier Transform

Finally we can notice some categories of spectra, for example :

Natural / Artificial / Comfortable / Color oriented preference


To analyze something designed pattern, we should know samples of 2D spectrum figures as follows :




Ocean Ripple / Cherry Blossoms / Komorebi (Sunlight through leaves) / Vegetated Cliff / Fleecy Cloud





Pampas Grass Field / Cirrostratus Cloud / Cove / Ridge / Braided Copper Wire


Looking at these 2D spectra, it can be mentioned that 2D spectrum of Braided Copper Wire has some similarities with Ocean Wave and Vegetated Cliff.

For example, a couple of other projects are analyzed as follows :




Glass Pavilion at Stanford University Seminar & Workshop

Kigumi (timber joint) structure in “ Sunny Hills in Aoyama ”

Overlapping Glass Structure shows a spectrum quite similar to Komorebi scene.

Kigumi structure can be described that it has naturalness between Komorebi and Pampas Grass scene.


2D power spectrum can be one of the evaluation values for the form optimization. It will be applicable to environmental elements such as heat, acoustics, airflow, waterflow or ecosystem as well as optical matters.


When someone preferred to live in an environment like a tree house in woods, we will be able to provide not only the structural design but everything we design which have the Komorebi spectrum. When someone preferred to sit on an environment similar to a hot beach and read a book, we will be abe to provide the Sazanami spectrum.


To save more people, little by little, we are learning Great Nature.


● Manipulation of Buckling Phenomenon


NYH, 2006

Architect : Makoto Yokomizo

Cylindrical steel plate wall structure stacked up to 4 strories.

NYH : 2006年,建築家=ヨコミゾマコト


円形壁は、9mmの鉄板にリブ FB-16x38 @300mm をつけて作っている。

PICT3279 PICT3273 八木04-Pict2887

Steel plate 9mm thick with reinforcing flat bar 16x38@300mm



PICT2986 PICT3018 八木06-Pict3022


Buckling strength of thin plate comes bigger when the curvature got bigger.



Structural analysis model, Buckling strength - Curvature diagram



Water Pavilion, World Exhibition in Aichi 2005

Design & Build : Naomi Sakuragi + Postgraduates of the University of Tokyo

Booth composed of dimpled acrylic resin walls. Thin plate wall comes 4 to 6 times stronger when dimpled.

愛知万博2005展示ブース : 2005年,建築家=櫻木直美+東京大学大学院生







03DSCN0039 04PICT2707 05PICT2712

Dimpled acrylic resin wall, Forming process using Chinese pan

Physical model made of aluminum, Buckling analysis



Moulded Pulp Installations : Seminar for master course of Tokyo University of Science

パルプモールド : 東京理科大学大学院講義, 2010




DVC00200 DVC00195

高さ1.8mの「シェルター」, 窓枠に取り付けられる「障子」


Rest house in Zoorasia

Architect : Rikan Yamamoto

When flat bar columns are located in radial arrangement, buckling strength can be found 4 times bigger than parallel arrangement.


Buckling control of flat bar columns : Radial, Polygonal, Parallel



● Reciprocal System


Rest house in the Forest

Seminar at Keio University, 2006

Free form shell structure composed of 19x140mm timbers. Each element is leaning on another element, mutally continuing. This system is called “ lamellendach ” or reciprocal system”.

森の休憩所, 2006, 慶應義塾大学大学院生



細かな材が並んだ状態をラメラ(Lamella, Lamellen)と呼び、短い材がお互いに支えあうことで成立するという意図からも、これはラメラ架構の一種と言える。



● Tensegrity テンセグリティ




Kenneth SnelsonBuckminster Fuller により広められた。







3D Tensegrity Experiments on Geometries and Dynamics : workshop at Stanford University


Experiments on Geometries and Dynamics : workshop at Stanford University, 2014


Beverly Choe, Architect, BACH architects / Stanford University

Jun Sato, Structural Engineer, Jun Sato Structural Engineers / University of Tokyo

Pop Up Structure and 3 Dimensional Tensegrity Structure were studied in 2 days and constructed in 2 days.





Category 1, Tensegrity Volume : Tensegrity to have 3D volume.



3D Tensegrity Volume composed of 18 galvanized bars and lengths of stainless cables (Photo by Nick Xu)

Tensegrity model, Pop-up Tectonics model




Dimensionality of Tensegrity

It is hard for a basic tensegrity to find a stable shape as a “3 dimensional” volume with not a modular system.





Category 2, Pop-up Tectonics : Foldable structure like a pop-up book, composed of 22 panels made of washi, traditional Japanese paper, and timber frames.

Washi paper provided by Takeo Co., Ltd.

Echizen Washi production : Shimizu Washi Co., Ltd.





Raising process of Pop Up Tectonics,

(Top right, bottom left : photo by Nick Xu)

(Top left, bottom right : photo by Jun Sato)


It is hard to find an exactly foldable shape when using thick plates.

Extensions of sides should cross at the same focus point.

Panels belonging to the same layer should not be overlapped when they are folded down and the total angle of the sets of panels, which coupled, should be same.




Geometrical conditions can be recognized by studying the model.

For example : from the top view, a ridge line or thalweg line should be seen to lie on a straight line.

When the loop is connected, panels have twisted shape like a Mobius loop and it is hard to find the focus point.




Community Week 2014

Dhillon Marty Foundation international workshop in Punjab, India

Schools       : The University of Tokyo, Stanford University, The University of Oregon, Rhode Island School of Design, Guru Nanak Dev University

Students from : Japan, U.S.A., India, China, Greece, Columbia, Indonesia


Public Toilet Design Competition in 3 days

5 clusters of students proposed the public toilet design.

Public toilet represents the social problems in India as follows,

Sanitation on water, foods, streets

Gender problem such as safety against crimes for ladies

Gap between rich and poor




Design Build Workshop in 2.5 days

Design build team was composed with 2 or 3 spies from each 5 clusters of students.

A kind of private space, also imagining the public toilet, was designed with some elements extracted from those design proposal of 5 clusters. The spies had to bring those informations from each clusters.

We can design structural elements which also work as environmental elements by designing filter for light, heat, air, water, sight, insects, person.

Keywords Delivered : water filtering

air ventilation

use waste for fertilizing

natural material

lift up the floor

These can be not yet actual solutions but indicating what we should think.



Shopping for materials and tools

Materials : local fabrics, bamboo, metal wire, strings, metal bars, plywoods, screws

Tools : saws, pliers, hand drills, hammers, needles, screwdrivers



Studies on bamboo frames

Brick and timber for lifted platform.



Instable frames stabilized by fabrics

Mesh structure with semi-transparent fabrics for filtering light and sight

Cellular spaces by branching membrane



Final shape with 15m length, indicating a gate, lifted private room covered with layered filters,

rest space, air ventilater.


Big Art, Exhibition Archi-neering

Tensegrity structure composed of a membrane supported by carbon (CFRP) pipes.

Designed and Constructed by about 15 students

Structure : Jun Sato et al.

アーキニアリング展 “ Big Art ”, 2008






DVC00279 DVC00278 


MOOM (Membrane Oom) :

Design & Construction : Kazuhiro Kojima Laboratory, Tokyo University of Science

Structure : Jun Sato

Membrane tensegric structure composed of membrane and aluminum pipes.

Membrane work as tension wires and pipes are working as compression particles.

Length 26 m, Span 8 m.



スパン 8m,全長 26m,総重量 600kgf で、40人程度で持ち上げることができる。




Structural calculations which I have provided were only these written on this paper at a meeting.

“ Omission ” is one of the tecniques of “ Engineering ”.



Not all the phenomena have been clarified, and as the “ project ” never have enough time and money we can not check all the matters which we want to check, but we engineers have the tecnique to imagine a simple model and find several critical matters to be checked, and finally we can find with just simple calculations if it can be built.








重さを w = 5 kgf/m2 とする。

スパン L = 8m、高さ H = 3m のアーチとする。

棒のピッチを 1m とする。

アーチを放物線で y = αx2 として y = 3 x = 8/2 = 4 を代入すると、

α= 0.1875

アーチの足元での傾きは y’= 2αx = 1.5 なので傾きは 1 : 1.5 だと分かる。


5 kgf/m2 × 1 m × 12 m ÷ 2 = 30 kgf


N =  × 30 kgf = 37 kgf



A = 4.91 cm2

生まれるデプスが 100mm とする。膜の断面積は不明だが、木材の棒と同じとしてみると、断面2次モーメントは、

I = 4.91 cm2 × 52 cm × 2 = 245 cm4

ヤング率は硬質樹脂程度で E = 20 tf/cm2 とする。

アーチの座屈長さ Lk = 0.4L0.5L 程度と知っておくとよい。Lk = 0.5L = 0.5 × 800cm = 400cm とする。座屈荷重は、

Pcr =  =  = 0.302 tf

これは N = 37 kgf に対して8倍の余裕があるので問題ないと分かる。




● Other Forms and Materials



Aluminium lattice endoskeleton for “ Balloon ”, architect Junya Ishigami

Copper Shell for “Earth: Material for Design”, Jun Sato Lab., University of Tokyo



Pop Up Structure, Jun Sato Lab., University of Tokyo

Woven timbers for “Ashikita Community Hall”, architect : Akiko Takahashi & Hiroshi Takahashi / Workstation

Accumulated cubes for “ ¿ - cube ”, Design & Construction : Ken Yokogawa Laboratory, Nihon University



Copper Shell, “Earth : Material for Design” by The National Museum of Emerging Science and Innovation, 2010

Design & Build : Jun Sato Laboratory, University of Tokyo

Energy consumption experiments

A copper shell structure with 8m long fabricated by only hammering from a flat plate, with 40 students.

Considering the total energy consumption for this structure to appear, we discovered the energy consumption in processing the copper shell by hammering was only 7 %, of the total energy, while 93 % for manufacturing copper plate.




Balloon, 2007

Architect : Junya Ishigami

Structure : Jun Sato

Aluminium “balloon” of 14m height, weighing roughly 1 tonne.

The balloon with aluminium lattice endoskeleton, filled with helium gas.



Architecture as air, Venezia Biennale 2010

Architect : Junya Ishigami

Structure : Jun Sato

Rigid frame structure composed of 0.9 mm CFRP columns, 1.2 mm CFRP beams, and invisible braces made of 0.02mm polyalyrate fibers.


(No photo)

Left : http://contessanally.blogspot.jp/2010/08/venice-12th-biennale-of-architecture_26.html


Ashikita Community Hall - Communication Center of Local-Resource-Utilization, 2010

Architect : Akiko Takahashi, Hiroshi Takahashi / Workstation

Structure : Jun Sato

Woven-like interlocked thin laminated timber structure inspired by bamboo baskets.

The timber bands can be woven in various directions and the members follow a geodesic line of surface.





● Developable Surface




Venezia Biennale 2018 のパビリオンへ応用する。




● Workshops





“ Digital Fabrication Laboratory Pavilion” , 2015, 東京大学小渕研究室(G30)への協力





Rhinoceros + Grasshopper Hogan (Released by Jun Sato)



・日本建築学会 “ Student Summer Seminar 2015 ”

カーボンケーブル(CFRP ケーブル)による繭のようなドーム構造の提案。


「ゆらぐフレーム」, 考案:隈 太一 澁谷 達典,協力:佐藤研究室学生一同



● Exhibitions



Real Size Competition 出展作品への協力




GA Gallery 展覧会

建築設計系の雑誌「GA JAPAN」主催の展覧会。

Leoflex を使ったガラス構造の部分モックアップ、Venezia Biennale 2016 のための可展面のスタディ模型を出展中。


Leoflex による構造モックアップ


Venezia Biennale 2018 への準備


GA gallery 展覧会出展中。






● Structural Tips


Material properties 材料特性


比重 unit weight (ρ)

剛性 stiffness (K),ヤング係数(ヤング率,弾性率)young’s modulus (E)

ポアソン比 Poisson’s ratio (ν)

弾性 elastic,塑性 plastic,粘弾性 viscoelastic

降伏 yield

強度(引張,圧縮,曲げ,せん断)strength (tensile, compression, bending, shear)

終局強度 ultimate stress

線形 linear,非線形 nonlinear

伸び性能 ability of elongation,延性 ductility,靱性 toughness,脆性 brittleness

線膨張係数 linear expansion coefficient


Stress - Strain Curve 応力ひずみ関係

鋼材 SS400 の引張試験

Tensile Test of Steel SS400


Load - Displacement Curve 荷重変形関係


Typical Load-Displacement Curve       ガラス板(普通強度)の曲げ試験


Bending Test of Glass Plate

Glass has no ductility


Section Properties 断面性能


断面積 Area                           A [cm2]

断面2次モーメント Moment Inertia     I [cm4]

ねじり定数 St. Venant’s Tortion Factor   J [cm4]

断面係数 Section Modulus              Z [cm3]


Example of rectangular section B x D

A = BD

I = BD3/12

Z = BD2/6



Structural Calculation 構造計算





Basic formulae

These formulae will be also used for simplified calculations, understanding structural test, etc.


・バネの荷重と変形 Load – Displacement on spring

F = k x


F : 荷重 force (load)

k : バネ定数 factor (stiffness) of spring

x : 変位 displacement


・弾性曲げモーメント Bending moment in elastic level

M = σ Ze


M  : 曲げモーメント bending moment

σ : 縁部の応力度 stress on edge

Ze : 断面係数 elastic section modulus, for rectangle section Ze = BD2/6


・全塑性モーメント Bending moment in plastic hinge level (ultimate level)

Mp = σy Zp


Mp  : 全塑性モーメント plastic hinge bending moment

σy : 降伏応力度 yield stress

Zp  : 塑性断面係数 plastic section modulus, for rectangle section Zp = BD2/4


・集中荷重を受ける単純梁 Simple beam with concentrated load


bending moment   displacement


・等分布荷重を受ける単純梁 Simple beam with uniform distribution load


bending moment   displacement


・等分布荷重を受ける両端固定梁 Fixed beam with uniform distribution load


bending moment   displacement


・集中荷重を受ける片持梁 Cantilever beam with concentrated load


bending moment   displacement


・等分布荷重を受ける片持梁 Cantilever beam with uniform distribution load


bending moment   displacement


オイラー座屈荷重 Euler’s Buckling Load

fix-fix         fix-hinge     hinge-hinge   fix-fix+sway   fix-hinge+sway

α=0.5         α=0.708       α=1.0         α=1.0         α=2.0


Buckling strength  :  Pcr =


E  :  Young’s modulus [cm2]

I  :  Moment inertia of section [cm4]

Lk  :  Buckling length Lk =αL



アーチの形状 Arch Shape

等分布荷重に対しては放物線を描く Parabola for uniform distribution load

自重に対してはカテナリーを描く Catenary for self weight load


アーチの略算 approximate calculation

Approximate Buckling Length Lk = 0.4 0.5 L

放物線  で近似する。

αは、スパン L と高さ H により、 で求められる

傾きは微分して、 なので、端部での傾きは、1.0 :  = 1.0 :  となる。

鉛直荷重を w [単位 tf/m など] とすると、支点の鉛直反力

端部の軸力 N は、傾き方向なので、

これに対し、アーチの座屈荷重は、座屈長さを 0.4 L などとしてオイラー座屈の式に代入して求める。



Basic Process of Structural Analysis



(1) 材料と形状の想定 Assumption of Material Property, Shape of Members and Shape of Frame

(2) 荷重の想定 Decision of Loads as its Type and Level

Load Type : Gravity, Earthquake, Wind, Temperature

Load Level : Elastic / Ultimate


Diagram of diverse loads


荷重 Concerning Load

重力 Gravity          : 重力加速度 1G = 980 cm/sec2

静止している人 Stationary person = 80 kgf

階段を降りる人 Person descending stairs = 200 kgf

地震荷重 Seismic Load : 振動を静的荷重に換算する方法がある。数十年に1度の地震で重量の 0.2 倍など。

Seismic vibration load considering a couple of decades in Tokyo can be interpreted to 0.2 of factor of weight.

風荷重 Wind load      : 東京の建物では、数十年に1度の風速として 34 m/sec を想定し、概ね 100 kgf/m2 となる。


数ヶ月では 20 25 m/sec

数週間では 10 15 m/sec


Wind pressure considering a couple of decades in Tokyo can be estimated to be 100kgf/m2 due to the wind speed 34m/sec (10min. average speed).

Wind pressure is proportional to the square of wind speed.

20 25 m/sec for several months

10 15 m/sec for several weeks



 (3) モデル化 Modeling for Analysis

Finite Element Method (FEM)

Element Type : Bar element (Wireframe), Plate element, Solid element

Joint Type : Rigid, Hinge (Pin), Half rigid (Spring)

(4) 構造解析 Calculation

Results       : Deformation, Stress

Bending Stress Diagram, Deformation Diagram


(5) 強度のチェック(部材断面設計) Analysis of Stress


Basic Case

Safety Ratio = N/Na + M/Ma 1.0

N : axial force

M : bending moment

a : allowable


Structural analysis software HOGAN

Safety Ratio = M/Ma / (1.0 - N/Na) 1.0


● Little by little, learning Great Nature


We are learning little by little about Great Nature such as ground vibration, water flow, air flow, optical permeability of vegetation, porosity of insect bodies, buckling phenomena and the elastic / plastic state of material.


Everytime a disaster happens, we engineers feel

it is impossible to know everything about great nature,

it is impossible to control great nature,

but even a little the more we could learn the vibration of earthquake, the flow of water, a little the more people we could save.

Our mission and dream.