![Page 1: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/1.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
Lower-bound Analysis of Masonry Vaulted Structures
Philippe BlockProf. John Ochsendorf
MIT
![Page 2: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/2.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
![Page 3: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/3.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
t/R = .08 t/R = .12
![Page 4: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/4.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
(Föppl, 1892)
![Page 5: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/5.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
![Page 6: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/6.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
![Page 7: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/7.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
KEY REFERENCE
Force Network Method ( D. O’Dwyer, 1999) > Analysis of 3-D Masonry Structures
![Page 8: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/8.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
INTRODUCTION TO METHODOLOGY
Assumptions for the analysis of masonry structures
– Limit analysis (lower bound theorem)“the vault will stand as long as a thrust network can be found that fits within its section”
– Three main assumptions (introduced by Heyman)• No tensile strength• Infinite compressive strength (rigid)• Sliding does not occur
![Page 9: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/9.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
INTRODUCTION TO METHODOLOGY
General assumptions
– Parallel, such as gravitational, loading cases only– Vault does not curl back onto itself– Funicular solution (= compression-only or tension-only) is
assumed
(Henrici, 1903)
![Page 10: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/10.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
GEOMETRY (Γ) ↔ FORCES (Γ*)
: THRUST NETWORK
THRUST NETWORK METHOD
![Page 11: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/11.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
THRUST NETWORK METHOD
RECIPROCAL RELATIONSHIP
![Page 12: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/12.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
Maxwell (1864). Two plane figures are reciprocal when they consist of an equal number of lines, so that corresponding lines in the two figures are parallel, and corresponding lines which converge to a point in one figure form a closed polygon in the other.
RECIPROCAL DIAGRAMS
(Maxwell, 1864) (Williams, 1986)
![Page 13: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/13.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY (OVERVIEW)1. Choose a possible thrust network topology Γ2. Attribute the correct weights per node3. Compute reciprocal diagram Γ* from Γ4. Formulate problem as linear optimization problem to solve for G
![Page 14: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/14.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY
1. Nodal equilibrium in thrust network
2. Using reciprocal grid Γ*
iV
liVki
Vji PFFF =++
( )( ) ( )
( )( ) ( )
( )( ) ( ) i
lili
liHli
kiki
kiHki
jiji
jiHji P
yyxx
zzFyyxx
zzFyyxx
zzF =
−+−
−⋅+−+−
−⋅+−+−
−⋅
222222
*, ji
HF Hji ⋅= ζ *
,kiHF H
ki ⋅= ζ *,li
Hli HF ⋅= ζ
![Page 15: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/15.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY
1. Formulation3.
0,
*,
,
*,
,
*,
,
*,
,
*,
,
*, =⋅−⋅−⋅−⋅−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛++ rPz
HH
zHH
zHH
zHH
HH
HH
illi
lik
ki
kij
ji
jii
li
li
ki
ki
ji
ji
*, jiH
jiH ,
ζ1=r
where : length of the branch i,j in the primal grid Γ*
: length of the branch i,j in the dual / reciprocal grid Γ*
: the scale of the dual grid Γ* (force magnitude)
0=⋅−⋅+⋅+⋅+⋅ rPzCzCzCzC illkkjjii LINEAR
![Page 16: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/16.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY
1. Nodal height constraints
> Envelope of actual vault (or middle third)
4.
Eii
Ii zzz ≤≤
![Page 17: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/17.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY
1. Objective function5.
ζ1=rmin / max
![Page 18: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/18.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
METHODOLOGY
1. Overview formulation
– Linear constraints:
– Linear objective function: minimizing / maximizing
– Solve using linear optimization (Simplex method)
7.
iV
liVki
Vji PFFF =++ *
, jiHF H
ji ⋅= ζ Eii
Ii zzz ≤≤
0=⋅−⋅+⋅+⋅+⋅ rPzCzCzCzC illkkjjii
ζ1=r
Iii zz −≤− & i
Eizz ≤
![Page 19: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/19.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
ΔZmax 5 7.5 10
ζ 16.643 11.095 8.321
INDETERMINACY / PARAMETERS (1)
![Page 20: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/20.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
ΔZmax 5 7.5 10
ζ 16.643 11.095 8.321
INDETERMINACY / PARAMETERS (1)
![Page 21: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/21.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
ΔZmax 5 7.5 10
ζ 16.643 11.095 8.321
INDETERMINACY / PARAMETERS (1)
![Page 22: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/22.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
INDETERMINACY / PARAMETERS (2)
One parameter (i.e. scale ζ ) controls infinite number of funicular solutions (equivalent to horizontal thrust in 2-D)
![Page 23: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/23.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
INDETERMINACY / PARAMETERS (3)
![Page 24: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/24.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
FORCE PATTERNS Cross vaults
(Ungewitter, 1890) (Rave, 1939)
![Page 25: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/25.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
FORCE PATTERNS Cross vaults
![Page 26: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/26.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
Hthrust
Hthrust (*)
(*) as percentage of theweight of the vault
23% (min)
32% (max)
EXAMPLES ANALYSIS Groin vault (pseudo 3-D)
![Page 27: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/27.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
EXAMPLES ANALYSIS Groin vault (full 3-D)
![Page 28: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/28.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
FAN VAULTS Kings College Chapel, Cambridge, UK
![Page 29: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/29.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
FAN VAULTS Kings College Chapel, Cambridge, UK
![Page 30: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/30.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
FAN VAULTS Kings College Chapel, Cambridge, UK
![Page 31: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/31.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
Historical spiral staircases in thin brick/tile by the Guastavino Company
SPIRAL STAIRCASES by the Guastavino Company
![Page 32: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/32.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
SPIRAL STAIRCASES by the Guastavino Company
![Page 33: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/33.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
ROSE WINDOWS Mantes-la-Jolie Cathedral, France
![Page 34: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/34.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
ROSE WINDOWS Mantes-la-Jolie Cathedral, France
![Page 35: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/35.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
• Three-dimensional equivalent for graphic statics
– clear graphical feedback of the forces in the system– high level of control of different equilibrium solutions
• Fully 3-D method to understand the stability of vaults with complex geometries in unreinforced masonry
KEY CONTRIBUTIONS
![Page 36: Lower-bound Analysis of Masonry Vaulted Structuresweb.mit.edu/masonry/thrustNetwork/SAHC08_ppt_block-ochsendorf.pdfLOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURES Philippe Block,](https://reader034.vdocuments.us/reader034/viewer/2022051808/6009f68b34c75514606f0158/html5/thumbnails/36.jpg)
SAHC’08 6th International Conference
STRUCTURAL ANALYSIS OF HISTORIC CONSTRUCTION
Bath, UK 2-4 July 2008 http://web.mit.edu/masonry/
LOWER-BOUND ANALYSIS OF VAULTED MASONRY STRUCTURESPhilippe Block, John Ochsendorf
Thank you… Prof. John Ochsendorf, MITProf. Chris Williams, Bath University
Xuan Vinh Doan, MIT
http://web.mit.edu/masonry/