Download - Introduction to Analysis of Masonry Arches
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Analysis of Historic StructuresAnalysis of Historic Structures
Lecture 2: Intro to Masonry Structure
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Review of Last MeetingReview of Last Meeting
z Principles of historic structure
EQUILIBRIUM
z Lower bound and upper boundtheorems
z Possible research topics
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Lower Bound TheoremLower Bound Theorem
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Upper Bound TheoremUpper Bound Theorem
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Theorems of Limit AnalysisTheorems of Limit Analysis
1. Lower Bound: If you can demonstrate atleast one possible equilibrium state, thenthe structure can also find at least one
possible stable state
2. Upper Bound: When the load path can nolonger be contained within the structure,and it is the unique and largest possible
load, then it is the collapse load
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HookesHookes 22ndnd Law (1675) Law (1675)
ut pendet continuumflexile, sic stabit
contiguum rigidum
inversum
As hangs the flexible
line, so but inverted
will stand the rigid
arch.
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Compression vs. TensionCompression vs. Tension
Compression Tension
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Equilibrium at a PointEquilibrium at a Point
F = 0 (sum of forces is zero)
Compression
Tension
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Structural EquationsStructural Equations
Only three types of equations:
1) Equilibrium
2) Material properties (elasticity, etc)3) Compatibility (geometry)
We will focus on equilibrium equations because
they are the most important.
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GraphicGraphic StaticsStatics
Applet by Simon Greenwold
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HookesHookes Hanging ChainHanging Chain
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Design of MasonryDesign of Masonry
z Main principle: must be
kept in compression
z Also applies to cast iron,
unreinforced concrete, and
other brittle materials
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3D Vaults: Slice into arches3D Vaults: Slice into arches
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Structural Analysis of MasonryStructural Analysis of Masonry
z The Stone Skeleton
by Jacques Heyman
z Three main assumptions:
No tensile strength
Infinite compressive strength (rigid) Sliding does not occur
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Arch on Spreading SupportsArch on Spreading Supports
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Range of Arch ThrustRange of Arch Thrust
Internal thrust lines due to self weight of arch
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Range of Arch ThrustRange of Arch Thrust
TENSION
Hmax
Hmin
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Range of Arch ThrustRange of Arch Thrust
COMPRESSION
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Model Arch ExperimentModel Arch Experiment
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Model Arch at Collapse StateModel Arch at Collapse State
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Understanding cracks in masonryUnderstanding cracks in masonry
1. Why do cracks occur?
2. What do they tell us?
3. Are they a cause for concern?
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Understanding cracks in masonryUnderstanding cracks in masonry
1. Why do cracks occur?
-Small movements of supports
2. What do they tell us?
-Where forces are NOT acting
3. Are they a cause for concern?
-Usually not, but they can be
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Design and Analysis ofDesign and Analysis of
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Design and Analysis ofDesign and Analysis of
UnreinforcedUnreinforced MasonryMasonry
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Stability rather thanfailure of the material is
the dominant concern
z Collapse occurs when the
load path can no longer
be contained within the
masonry
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Model Arch at Collapse StateModel Arch at Collapse State
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Single Span Stone ArchSingle Span Stone Arch
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Single Span ArchSingle Span Arch
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Single Span ArchSingle Span Arch
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Single Span ArchSingle Span Arch
D bl S S A hD bl S St A h
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Double Span Stone ArchDouble Span Stone Arch
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Double Span Arch with FillDouble Span Arch with Fill
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Double Span Arch with FillDouble Span Arch with Fill
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Double Span Arch with FillDouble Span Arch with Fill
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Double Span Arch with FillDouble Span Arch with Fill
This makes sense for bridges, but buildings dont usually
have trucks driving on top of the vaults
Deformation over time can cause collapse in buildings
ConclusionsConclusions
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ConclusionsConclusions
z Unreinforced masonry structures have very low
stress levels: stability, not strength, governs thesafety
z Determine collapse states based on thrust line
analysis using graphic statics
z Equilibrium equations are most important when
analyzing historical structures