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Basis of Structural Design

Course 2

Structural action: cables and arches

Course notes are available for download athttp://www.ct.upt.ro/users/AurelStratan/

Structural action

� Structural action: the way in which a structure of a given type and configuration resists the loads acting on it

� Types of structures:

– Cables

– Arches

– Trusses

– Beams

– Plates and shells

– Frames

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Cable / chain structures

� Cable and chains:

– excellent tensile strength

– no strength/stiffness in compression

– no strength/stiffness in bending

� Cable and chain structures exploit the benefits of high tensile strength of natural fibres and steel

� Especially useful in large-span structures

Cable / chain structures

� The form of a chain under its own weight?

� The form of a chain under equal loads applied in the pins?

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A chain subjected to a single force

� The simplest chain structure:

– links connected by pins

– load W acts on the central pin

� Solution (equilibrium of node C):

– the pin C is acted by three forces: load W, and two tensile internal

forces T

– the vectors representing the three forces can be represented as a

a triangle of forces 012 (W=12, T=20, T=01)

– length of lines 20 and 01 gives the tensions in the chain

A chain carrying two vertical forces

� Weights W1 and W2 attached to pins D and E

� Tensions T1, T2 and T3 will be set up in three parts of the chain

� Problem: determine magnitudes of T1, T2 and T3 if deformed shape is known

� Solution (equilibrium of nodes D and E)

� Node D

– node D is acted by three forces:

load W1, and to tensile internal

forces T1 and T2

– the vectors representing the

three forces can be represented

as a a triangle of forces 012

(W1=12, T1=20, T2=01)

– length of lines 20 and 01 gives

the tensions in the chain

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A chain carrying two vertical forces

� Node E

– node E is acted by three forces:

load W2, and to tensile internal

forces T2 and T3

– the vectors representing the

three forces can be represented

as a a triangle of forces 023

(W2=23, T2=02, T3=30)

– length of lines 02 and 30 gives

the tensions in the chain

� The two triangles can be combined to get a force diagram

A chain carrying four vertical forces

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A chain carrying equal weight at each pin

� The chain hangs symmetrically about point C

� Each inclined line in the force diagram gives the magnitude and inclination of the force in the corresponding link

� Starting from the midspan, the slope of the links increases in proportion to the horizontal distance from

the midspan ⇒⇒⇒⇒ parabola

A chain carrying equal weight at each pin

� The slope at the sides: twice the average slope ⇒⇒⇒⇒tangents at the ends A and B will intersect at point F (GF=2GC)

� Considering the equilibrium of the chain as a whole, the chain is acted by the tensions T1, T16 and the total weight W.

� Provided the chain sag is known (GC), end tensions can be determined from triangle of forces 120

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Deformed shape of a cable / chain

� Actual deformed shape of a cable or chain hanging under

its own weight: catenary (slightly ≠≠≠≠ from parabola)

� Parabola: the shape of a chain carrying uniform loads for each horizontal span

� Catenary:

– the shape of a chain hanging under its own weight

– weight of the chain per unit horizontal span increases toward the

sides due to increasing slope of the chain

� Parabola:

– easier to calculate

– differences between parabola and catenary negligible for small

spans

Arches

� The simplest chain structure (material working in tension):

� If the load direction is reversed (material working in compression)

⇒⇒⇒⇒ an arch is obtained

� Internal forces are the same in the two structures, but are compressive in the arch

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Three-bar linear arch

Three-bar chain Three-bar arch

� Internal forces are the same in the two structures, but are compressive in the arch

� Linear arch (funicular shape) - the shape for which under loads acting on it (including its own weight), the thrust in the arch acts along the axis of members at all points

Three-bar linear arch

� The forces in an arch can be deduced from those in a chain of the same shape (first to be realised by Robert Hooke)

� An essential difference between a chain and an arch:

– a change in the relative values of loads W1 and W2 in a chain leads

to a new position of equilibrium

– a change in the relative values of loads W1 and W2 in a hinged

arch leads to collapse of the structure

� Collapse of the arch due to small changes of loading can be avoided by connecting the bars rigidly together

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Arches: line of thrust

� Linear arch gives the smallest stresses

� Shape of the arch is not important for small arches: own weight has a small contribution to stresses in comparison with imposed (traffic) loads

� Shape of the arch is very important for large arches: own weight has a major contribution to stresses

Arches: forms

� Perfect arch: shape of catenary (example: Taq-e KisraPalace, Ctesiphon, Iraq - built 220 B.C.)

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Arches: forms

� The first civilisation to make extensive use of arches: Romans

� Shape of Roman arches: semicircular

why?

� Circle - the easiest way to set out

Semicircular arch

� A cable takes a circular form when subjected to a uniform radial load

� A linear semicircular arch: loaded by uniform radial pressure

� Loading in bridges and buildings quite different from the condition above

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Romanesque semi-circular arches and vaults

� Semi-circular arch used extensively in the Romanesque period

� Severe architectural restrictions:

– Romanesque barrel vault

requires continuous support

and makes the interior dark

when used for roofs

– groined arch: enables light to

enter from all sides but allows

only square bays to be covered

Gothic arches

� Gothic period - pointed arches

� Rectangular spans can be covered by varying the ratio of rise to span

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Gothic arches

� A kink in an weightless cable implies a concentrated force at the kink, as well as a distributed load along the

two sides ⇒⇒⇒⇒ corresponding shape of linear Gothic arch

� This condition is not present in almost all Gothic arches, which requires support from the adjoining masonry

Gothic arches

� Correct use of pointed arch: Font Pedrouse viaduct in France

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Arches: design

� A stone arch (no strength in tension) will fail when the thrust line reaches the extrados and intrados in four points, becoming a mechanism

Arches: design

� 19th century approach - avoid cracking (tensile stresses) under service loads - keep the thrust line within the middle third of the arch cross-section

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Arches: design

� Thrusts at springings(reactions at supports) are inclined:

– vertical component

– horizontal

component

� Horizontal reactions tend to spread the

supports apart ⇒⇒⇒⇒buttresses can be used, especially for arches/vaults on high walls

Arches: buttresses