Upload
ngocong
View
251
Download
3
Embed Size (px)
Citation preview
1
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
2
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?
3
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
4
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
5
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
6
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
7
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
8
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.)
9
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
10
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
11
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
12
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
13
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