structural analysis of masonry vaults - mit opencourseware · pdf filelimit analysis of...
TRANSCRIPT
![Page 1: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/1.jpg)
Lecture 3
Analysis of Masonry Structures: Analysis of Masonry Structures: Arches, Vaults, and ButtressesArches, Vaults, and Buttresses
![Page 2: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/2.jpg)
Limit Analysis of MasonryLimit Analysis of Masonry
Lower Bound Theorem– Seeks permissible line of compressive force
for the given loading
Upper Bound Theorem– Seeks critical load which results in a failure
mechanism
![Page 3: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/3.jpg)
PoleniPoleni (1748) applies lower bound (1748) applies lower bound to St. Peter’s of Rometo St. Peter’s of Rome
![Page 4: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/4.jpg)
Selby Abbey, 12Selby Abbey, 12thth C, EnglandC, England
Tower construction “punches” through
Arches deform to accommodate support movements
Stable because a line of thrust can be found within the deformed arch
![Page 5: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/5.jpg)
Experiments by Experiments by DanyzyDanyzy (1732)(1732)
Collapse occurs byhinging between blocks, when a load path can not be contained within themasonry
Safety is a question of geometry and stability, not crushing of stone
![Page 6: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/6.jpg)
Cathedral in Palma de Cathedral in Palma de MallorcaMallorca, , Analysis by Joan Rubio (1912)Analysis by Joan Rubio (1912)
![Page 7: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/7.jpg)
Analysis of a Flat ArchAnalysis of a Flat Arch
Image courtesy of Denis Y. Yu, structurae.de
![Page 8: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/8.jpg)
Analysis of a Flat ArchAnalysis of a Flat Arch
Thrust of a flat arch?
![Page 9: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/9.jpg)
Safety of an Arch on ButtressesSafety of an Arch on Buttresses
Must solve three problems:
1. Load capacity of buttress (and influence of lean)2. Collapse state of arch on spreading supports
3. Analysis of arch supported on leaning buttresses
![Page 10: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/10.jpg)
Buttress Analysis, Buttress Analysis, DuPuitDuPuit (1870)(1870)
![Page 11: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/11.jpg)
HeymanHeyman on Leaning Walls (1992)on Leaning Walls (1992)
![Page 12: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/12.jpg)
Buttress CollapseButtress Collapse
Hs
V V
Wb+V Wb+V-WcWb+V
Wb
Wc
Hu
Hu
Possible fracture
HsHs
Hs (a) (c)(b)
Wb-Wc
Locus of pressure points
nm
V
p
![Page 13: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/13.jpg)
Assumed Compressive Stress Assumed Compressive Stress DistributionDistribution
P P
Ineffective area Ineffective area
Opening between stones
Linear compressive stress distribution
Locus of pressure points (line of internal forces)
m
2m
Interface between stones
![Page 14: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/14.jpg)
Determine Shape of FractureDetermine Shape of Fracture
dy
dx
H
Wo
b
b/3
y
W + dW
dW
W
dy
x/3
H
HP
dx
x
![Page 15: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/15.jpg)
Assumed Buttress Stress Assumed Buttress Stress Distribution at CollapseDistribution at Collapse
Plan view
V
H
A
B
A
B
C C
A-A: Locus of pressure points acts near the centroid and the entire section is in compression.
B-B: Locus of pressure points reaches kern point of rectangular section (1/3 point).
C-C: Section properties change as fracture occurs and locus moves to 1/3 of the new section.
Locus of pressure points
Surface of fracture
Fracture reduces thrust capacity by >30% in many cases
![Page 16: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/16.jpg)
VicatVicat Experiments on Suspension Experiments on Suspension Bridge Towers (1832)Bridge Towers (1832)
![Page 17: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/17.jpg)
VicatVicat Experiments on Suspension Experiments on Suspension Bridge Towers (1832)Bridge Towers (1832)
![Page 18: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/18.jpg)
Model Buttress ExperimentsModel Buttress Experiments
Fracture reduces thrust capacity by 20% to 30%
![Page 19: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/19.jpg)
Gothic Buttress Failure StateGothic Buttress Failure State
(b)
VHa
(a)
VHb
![Page 20: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/20.jpg)
Leaning ButtressLeaning Buttress
(b)
Wb - Wc
Wb - Wc + V Wb - Wc + V
Wb - 2Wc + V
Wc
Wb
V
Hφ
b
(a) (c)
V
Hφ
φ
φ
φφ
Hφ Hφ
Hφ
y
xO
![Page 21: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/21.jpg)
Leaning Buttress CapacityLeaning Buttress Capacity
Angle of lean, φ (radians)Hor
izon
tal t
hrus
t cap
acity
of b
uttr
ess,
Hφ
1
Vertical buttress
Maximum lean
⎟⎟⎠
⎞⎜⎜⎝
⎛+
µψγ
21
bbh
![Page 22: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/22.jpg)
Load Capacity of a ButtressLoad Capacity of a Buttress
• A masonry buttress will fracture at collapse, reducing its load capacity.
• A leaning buttress has a linear reduction in capacity, based on a small angle approximation as the centroid shifts horizontally.
![Page 23: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/23.jpg)
Church in Church in GuimareiGuimarei, Spain, Spain
Geometry changes may threaten stability of the structure
Huerta and Lopez (1997)
![Page 24: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/24.jpg)
Church in Church in GuimareiGuimarei, Spain, Spain
1. Buttress leans outward (e.g. foundation deforms)
2. Arch deforms and thrust increases
3. Buttress leans further and thrust increases further.
![Page 25: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/25.jpg)
Church in Church in GuimareiGuimarei, Spain, Spain
Must solve three problems:
1. Load capacity of buttress (and influence of lean)
2. Collapse state of arch on spreading supports
3. Analysis of arch supported on leaning buttresses
![Page 26: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/26.jpg)
Spreading Arches:Spreading Arches:Viollet’sViollet’s study of study of VezelayVezelay (1854)(1854)
![Page 27: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/27.jpg)
ViolletViollet--lele--DucDuc on Spreading Archeson Spreading Arches
![Page 28: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/28.jpg)
Arch on Spreading SupportsArch on Spreading Supports
![Page 29: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/29.jpg)
Amiens Possible Collapse ModeAmiens Possible Collapse Mode
![Page 30: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/30.jpg)
Flying Buttresses at Palma de Flying Buttresses at Palma de MallorcaMallorca
![Page 31: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/31.jpg)
ConclusionsConclusions
Unreinforced masonry structures have very low stress levels: stability, not strength, governs the safety
Limit analysis can be used to determine collapse states based on thrust line analysis
Capacity for displacements may be more important than load capacity (particularly for historic buildings)
For high vaulted buildings, the arch will collapse and the buttress will remain standing in most cases.
![Page 32: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/32.jpg)
Research Papers on MasonryResearch Papers on Masonry
Comparative studies – Arches– Vaulting – Buttresses– Individual structures
![Page 33: Structural Analysis of Masonry Vaults - MIT OpenCourseWare · PDF fileLimit Analysis of Masonry ... Geometry changes may threaten stability of the structure Huerta and Lopez ... zUnreinforced](https://reader031.vdocument.in/reader031/viewer/2022020213/5a74b97a7f8b9ad22a8c067c/html5/thumbnails/33.jpg)
Research Papers on MasonryResearch Papers on Masonry
– Tile vaulting (Guastavino)
– Gothic– Romanesque– Mamluk– Maya/Aztec– Mycenaean tholos
tombs– Individual structures