Introduction to Analysis of Masonry Arches

7/27/2019 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

P


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Upper Bound TheoremUpper Bound Theorem

P


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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

P


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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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TENSION

Hmax

Hmin


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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

z

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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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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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

Introduction to Analysis of Masonry Arches

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