Course 2
Structural action: cables and arches
Course notes are available for download at http://www.ct.upt.ro/users/AurelStratan/
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
1

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
The form of a chain under its own weight?
The form of a chain under equal loads applied in the pins?
2

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
Weights W
1 and W
2 attached to pins D and E
Tensions T
1
, T
2 chain and T
3 will be set up in three parts of the
Problem: determine magnitudes of T
1 deformed shape is known
, T
2 and T
3 if
Solution (equilibrium of nodes D and E)
Node D
– node D is acted by three forces: load W
1
, and to tensile internal forces T
1 and T
2
– the vectors representing the three forces can be represented as a a triangle of forces 012
(W
1
=12, T
1
=20, T
2
=01)
– length of lines 20 and 01 gives the tensions in the chain
3

Node E
– node E is acted by three forces: load W
2
, and to tensile internal forces T
2 and T
3
– the vectors representing the three forces can be represented as a a triangle of forces 023
(W
2
=23, T
2
=02, T
3
=30)
– length of lines 02 and 30 gives the tensions in the chain
The two triangles can be combined to get a force diagram
4

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
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 T
W.
1
, T
16 and the total weight
Provided the chain sag is known (GC), end tensions can be determined from triangle of forces 120
5

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
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
6

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
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 W
1 to a new position of equilibrium and W
2 in a chain leads
– a change in the relative values of loads W
1 arch leads to collapse of the structure and W
2 in a hinged
Collapse of the arch due to small changes of loading can be avoided by connecting the bars rigidly together
7

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
Perfect arch: shape of catenary (example: Taq-e Kisra
Palace, Ctesiphon, Iraq - built 220 B.C.)
8

The first civilisation to make extensive use of arches: Romans
Shape of Roman arches: semicircular why?
Circle - the easiest way to set out
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
9

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 period - pointed arches
Rectangular spans can be covered by varying the ratio of rise to span
10

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
Correct use of pointed arch: Font Pedrouse viaduct in
France
11

A stone arch
(no strength in tension) will fail when the thrust line reaches the extrados and intrados in four points, becoming a mechanism
19th century approach - avoid cracking (tensile stresses) under service loads - keep the thrust line within the middle third of the arch cross-section
12

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
13