Lower-bound Analysis of Masonry Vaulted...
Transcript of Lower-bound Analysis of Masonry Vaulted...
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
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
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
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)
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
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
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
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
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)
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
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
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)
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
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
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
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 ≤≤
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
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 ≤
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)
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)
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)
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)
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)
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)
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
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)
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)
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
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
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
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
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
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
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
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
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/