analysis of masonry structures

8/2/2019 Analysis of Masonry Structures

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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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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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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

analysis of masonry structures

Documents