Basis of Structural Design · prestress is retained due to the bond between the concrete and the steel wires. Problems related to prestressing: –When the concrete sets up, it shrinks,

Basis of Structural Design

Course 4

Structural action:

- prestressing

- plate and shell structures

Course notes are available for download athttps://www.ct.upt.ro/studenti/cursuri/stratan/bsd.htm

Prestressing

Prestressing: setting up an initial state of stress, that makes the structure work better than without it

Examples:– wall plugs

– spider's web

– bicycle wheel

Main use in structural engineering: prestressed concrete

Prestressing examples: wall plug

A hole in the wall is filled with a wooden or plastic plug

The screw driven into the plug squeezes the plug against the sides of the hole, generating compressive stresses in the plug and in the wall around it

Compressive prestressing generates frictional resistance to pulling out the screw

Prestressing examples: spider's web

Spider's web threads: high tensile, but no compressive resistance

Spider pulls its threads tight, creating a tensile prestressing

A load in the centre of the web produces compressive forces in the threads below it

Without the tensile prestress, the lower part of the web would go slack, being more prone to collapse

Prestressing examples: bicycle wheel

Wire spokes are strong in tension but weak in compression (due to buckling)

Spokes must be kept in tension

When the wheel is assembled, spokes are tightened up uniformly by the turnbuckles at the rim

Under a downward load on the wheel, the spokes in the lower part of the wheel tend to be subjected to compression

Tensile prestress in the spokes must be higher than the compression force to keep all the spokes in tension


Other types of loading on the wheel: due to braking and due to taking a sharp corner

Forces due to braking:– could not be resisted if the spokes were arranged radiating from

the centre of the hub

– spokes are set at an angle to the radii, each pair forming a triangulated system which is able to generate tensile and compression forces which oppose the braking force

– tensile prestress ensures that all spokes are in tension and active


Forces due to cornering:– force is imposed on the wheel at right

angles to its plane

– the spokes are inclined with respect to the plane of the wheel, forming a triangulated system, which resists the forces due to cornering

– tensile prestress ensures that all spokes are in tension and active

Other prestressing examples

Pneumatic tire of cycle wheel

Inflated membranes for storage spaces and sport halls– air pressure inside is maintained above the atmospheric pressure

by blowers

– fabric of the membrane permanently in tension

Other prestressing examples

A set of books: no tensile resistance between the volumes

The books can be moved if a pressure is applied at the middepth:– the row of books act as a simply

supported beam

– the pressure overcomes the tensile stress in the lower part due to own weight of the books, enabling them to act as a unit

The books can be moved with lower pressure if it is applied somewhat lower than the middepth: an upward moment is introduced, which counteracts the downward moment due to own weight of the books

Reinforced concrete beams

Concrete: weak in tension

When loading is applied on a simply supported beam, the concrete cracks at the tension side:– Concrete active in compression

– Steel reinforcement active in tension

– Only a small part of the concrete cross-section resists the applied loading

Prestressed concrete beams

Concrete is kept in compression by cables or rods

The whole concrete cross-section can be considered in design

Substantial economy in material

If prestressing is applied in the centroid of the cross-section:– by choosing correctly the

prestressing force, the entire cross-section can be kept in compression

– a large stress is present at the compression side


Position of prestressing force: important

If prestressing is applied at 1/3 of the beam depth from the bottom face:– a negative moment due to eccentric

prestressing counteracts the positive bending moment due to applied moment

– the pestressing force needed to keep the entire cross-section in compression can be reduced

– the stress at the compression side is reduced the required concrete strength can be reduced


Bending moment due to dead weight in a simply supported beam: parabolic shape

The best arrangement of the prestressing tendons?

a parabolic shape along the beam, in order to generate bending moment M=Fe counteracting the bending moment due to dead load

Prestressed concrete

Type of prestress:– Posttensioning: the prestressing force is applied after concrete

has been cast and has set, through tendons located in holes left in concrete elements. The prestress is retained due to anchorage of steel tendons at the end of the element.

– Pretensioning: prestressing wires are stretched over a long length and the concrete is cast around them in steel forms. The prestress is retained due to the bond between the concrete and the steel wires.

Problems related to prestressing:– When the concrete sets up, it shrinks, leading to loss of

prestressing (in the case of pretensioning)

– Concrete shortens in time (creep) after it sets up due to compression acting on it, leading to loss of compression

– High strength steel required for prestressing, in order to reduce the loss of prestress due to shrinkage and creep

– Higher strength concrete is needed to resist higher compression and to reduce the contraction due to creep and shrinkage

Plates

Plates: a flat surface element that acts in bending in order to resist out of plane loading

The simplest plate: a flat slab spanning between two supports

It may appear to behave like a wide beam, but it is not as simple as that

One-way plates

When a narrow beam bends, the material in the lower half of the beam extends longitudinally it contracts in the transversal direction due to Poisson effect ( times the longitudinal strain)

The material in the upper half of the beam contractslongitudinally it expands in the transversal direction

An anticlastic curvature of the beam in the transversal direction equal with times the longitudinal curvature

One-way plates

In plates the anticlastic curvature is suppressed due to large dimension in the transversal direction (the deflected shape is almost cylindrical, except near the free edges)

At any point of the beam there is a transverse bending moment equal to times the spanwise bending moment

Suppression of the transverse curvature induces an additional spanwise curvature

In one-way plates reinforcement is needed in both spanwise and transverse direction

Two-way plates

Two-way plates simply supported on all four sides: complicated interaction between the two ways in which a load is supported

If a slab is more than about 4 times as long as it is wide, the bending moment at the center of the plate is almost the same as in a one-way plate supported on longer edges. Why?

Stiffer structural action (bending in the short direction) attracts larger forces

Stiffness in structural action

A straight bar of length L and rectangular cross-section can support a concentrated force P in two ways:– as a column acting in compression

– as a cantilever acting in bending

In the column the stress 1 is axial and uniform

In the cantilever the stress 2 has a linear variation along the bar and across the cross-section the material is far less efficient

Stiffness in structural action

Column is much stronger than the beam: 2/1 = 6(L/h)for L/h=20 2/1 = 120

Column is much stiffer than the beam: 2/1 = 4(L/h)2

for L/h=20 2/ 1 = 1600 (P=k∙ k1/k2 = 1600)

If the beam and the column are used in conjunction to support the load P:– the two members deflect by the

same ammount

– P=k∙ P1=k1∙1; P2=k2∙2. If the deflection is the same for the two members 1=2 P1/k1 = P2/k2; P1/P2=k1/k2 = 1600

– the column carries a load of (1600/1601)P

– the beam carries a load of (1/1601)P

Of the two alternative modes of action open to this structure, it chooses the column compression, because it is stiffer

Membrane action

Some structures can support loads only in bending.Example: simply supported beam

Uniform loading:– the neutral axis becomes curved

– roller support moves slightly toward the other end of the beam

Membrane action

A beam pinned at both ends

Uniform loading:– the neutral axis becomes curved

– horizontal movement of the support is prevented longitudinal tension H develops the beam begins to support load as a slightly curved cable or catenary

Membrane action

The catenary action is much stiffer than bending

Beam action: stiffness remains constant

Catenary action: stiffness increases with the square of the deflection

As the load increases, the portion of the load carried axially (w1), as catenary, increases rapidly

It can be shown that w1/w2 = 3.33(/h)2

w2 - the portion of the loading carried through bending.When the deflection ammounts to twice the depth of the beam, w1/w2 = 13.33, so that the catenary action ammounts to 13.33/14.33 = 0.93 of the total resistance to load

Membranes: surface elements in which loading is resisted through direct (axial) stresses

Shells

Shells: surface elements resisting loading through bending and membrane action

Examples:– dome

– human skull

– turtle's armour

– bird egg

Shells

Bird's egg: weak under a concentrated loading (breaking against a cup's rim) but strong under distributed loading (squeezing between ends with palms)– distributed loading resisted through membrane action (stronger)

– concentrated loading resisted through bending action (weaker)

Domes: – used since ancient times

– capable of resisting through membrane action a variety of distributed loading

Dome: structural action

The shape of a cable changes as the shape of the applied loading changes

The same behaviour if a set of cables are hanged around a circular perimeter – uniform loading: "bowl" shape

– larger loading toward the supports: the "bowl" bulges toward supports and the bottom rises slightly

– a different shape of the cable is needed in order to resist the applied loading through axial action only


If a series of circumferential cables are added, capable of resisting both tension and compression

When the load changes, the circumferential cables prevent the dome from changing its shape:– circumferential cables near the rim are

put into tension

– those near the bottom are put into compression


A system formed by using enough cables in order to obtain a surface approximates a thin-shelled dome

Such a structures is capable of carrying a variety of distributed loading through membrane action (stresses which are uniformly distributed over the thickness of the shell)

A shell is capable of resisting loads either through bending stresses or direct (membrane) stresses

Membrane action is "preferred" by the dome, as it is much stiffer for this action

Ideally, for a membrane action to take place in a shell, it must be thin and its shape should be similar to that assumed by a flexible membrane under the same loading


The heaviest load in many domes is their own weight

In a hemispherical dome of a uniform thickness,– the stresses 1 in the direction of meridians are compressive

throughout

– the circumferential stresses 2 are tensile near the rim: tensile reinforcement needed to resist them

Shells: hyperbolic paraboloid

Rectangular area to be covered: (a) taking a portion of a sphere and arching it between supports

Rectangular area to be covered: (b) hyperbolic paraboloid - can be obtained by taking a rectangular grid of straight lines and lifting one of the corners, so that the lines would remain straight

A flat surface becomes a curved one, known as hyperbolic paraboloid

Lines drawn diagonally are parabolas, humped in one direction and sagging in the other direction


Constructional advantage that elaborate formwork is not needed

Hyperbolic paraboloid supports loads by tension/compression, as opposed to a plate, acting in bending

Given the opportunity, a structure will support loads by direct tension and compression rather than bending

Basis of Structural Design · prestress is retained due to the bond between the concrete and the steel wires. Problems related to prestressing: –When the concrete sets up, it shrinks,

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