analysis of masonry structures
TRANSCRIPT
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Lecture 3
Analysis of Masonry Structures:Analysis of Masonry Structures:
Arches, Vaults, and ButtressesArches, Vaults, and Buttresses
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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
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PoleniPoleni (1748) applies lower bound(1748) applies lower bound
to St. Peters of Rometo St. Peters of Rome
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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 befound within the deformed arch
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Experiments byExperiments by DanyzyDanyzy (1732)(1732)
Collapse occurs byhinging between blocks,
when a load path can not
be contained within the
masonry
Safety is a question ofgeometry and stability,
not crushing of stone
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Cathedral in Palma deCathedral in Palma de MallorcaMallorca,,
Analysis by Joan Rubio (1912)Analysis by Joan Rubio (1912)
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Analysis of a Flat ArchAnalysis of a Flat Arch
Image courtesy of Denis Y. Yu, structurae.de
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Analysis of a Flat ArchAnalysis of a Flat Arch
Thrust of a flat arch?
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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
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Buttress Analysis,Buttress Analysis, DuPuitDuPuit (1870)(1870)
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HeymanHeyman on Leaning Walls (1992)on Leaning Walls (1992)
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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
A d C i StA d C i St
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Assumed Compressive StressAssumed 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
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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
A d B tt StA d B tt St
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Assumed Buttress StressAssumed 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
Vi tVi t E i t S iE i t S i
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VicatVicat Experiments on SuspensionExperiments on Suspension
Bridge Towers (1832)Bridge Towers (1832)
Vi tVi t E i t S iE i t S i
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VicatVicat Experiments on SuspensionExperiments on Suspension
Bridge Towers (1832)Bridge Towers (1832)
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Model Buttress ExperimentsModel Buttress Experiments
Fracture reduces thrust capacity by 20% to 30%
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Gothic Buttress Failure StateGothic Buttress Failure State
(b)
VHa
(a)
VHb
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Leaning ButtressLeaning Buttress
(b)
Wb - Wc
Wb
- Wc
+ V Wb
- Wc
+ V
Wb - 2Wc + V
Wc
Wb
V
H
b
(a) (c)
V
H
H H
H
y
xO
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Leaning Buttress CapacityLeaning Buttress Capacity
Angle of lean, (radians)Horizontalthrustcapacityo
fbuttress,
H
1
Vertical buttress
Maximum lean
+
2
1b
bh
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Load Capacity of a ButtressLoad Capacity of a Buttress
A masonry buttress will
fracture at collapse, reducing itsload capacity.
A leaning buttress has a linear
reduction in capacity, based on
a small angle approximation as
the centroid shifts horizontally.
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Church inChurch in GuimareiGuimarei, Spain, Spain
Geometry changes may threaten stability of the structure
Huerta and Lopez (1997)
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Church inChurch 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.
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Church inChurch in GuimareiGuimarei, Spain, Spain
Must solve three problems:
1. Load capacity of buttress (and influence of lean)
2. Collapse state of arch on spreading supports3. Analysis of arch supported on leaning buttresses
Spreading Arches:Spreading Arches:
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Spreading Arches:Spreading Arches:
ViolletsViollets study ofstudy ofVezelayVezelay (1854)(1854)
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ViolletViollet--lele--DucDuc on Spreading Archeson Spreading Arches
A h S di S tA h S di S t
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Arch on Spreading SupportsArch on Spreading Supports
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Amiens Possible Collapse ModeAmiens Possible Collapse Mode
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Flying Buttresses at Palma deFlying Buttresses at Palma de MallorcaMallorca
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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.
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Research Papers on MasonryResearch Papers on Masonry
Comparative studies
Arches
Vaulting
Buttresses Individual structures
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Research Papers on MasonryResearch Papers on Masonry
Tile vaulting(Guastavino)
Gothic
Romanesque
Mamluk
Maya/Aztec
Mycenaean tholostombs
Individual structures