Basis of Structural Design

Size: px

Start display at page:

Download "Basis of Structural Design"

Clementine Harrington
6 years ago
Views:

Basis of Structural Design Course 2 Structural action: cables and arches Course notes are available for download at http://www.ct.upt.

1 Basis of Structural Design Course 2 Structural action: cables and arches Course notes are available for download at 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 1

2 Cable and chains: Cable / chain structures 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? 2

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 W 1 and W 2 attached to pins D and E Tensions T 1, T 2 and T 3 will be set up in three parts of the chain Problem: determine magnitudes of T 1, T 2 and T 3 if deformed shape is known 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

4 Node E A chain carrying two vertical forces 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 A chain carrying four vertical forces 4

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

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 6

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 W 1 and W 2 in a chain leads to a new position of equilibrium a change in the relative values of loads W 1 and W 2 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 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

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

9 The first civilisation to make extensive use of arches: Romans Shape of Roman arches: semicircular Arches: forms 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 Semicircular arch 9

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

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

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

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 11

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 12

tend to spread the supports apart buttresses can be used,

13 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: design Arches: buttresses 13

CHAPTER The linear arch

CHAPTER The linear arch CHAPTER 6 The Romans were the first to use arches as major structural elements, employing them, mainly in semicircular form, in bridge and aqueduct construction and for roof supports, particularly the