Basis of Structural Design
Course 4
Structural action:
- prestressing
- plate and shell structures
Course notes are available for download at
http://www.ct.upt.ro/users/AurelStratan/
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
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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
2
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
Prestressing examples: bicycle wheel
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
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Prestressing examples: bicycle wheel
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
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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
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Prestressed concrete beams
Concrete is kept in compression
by cables or rods
The whole concrete crosssection 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 crosssection can be kept in compression
– a large stress is present at the
compression side
Prestressed concrete beams
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
6
Prestressed concrete beams
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=F⋅⋅e counteracting the bending
moment due to dead load
Prestressed concrete beams
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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
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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 (µ times the
longitudinal strain)
The material in
the upper half of
the beam contracts
longitudinally ⇒
it expands in the
transversal direction
An anticlastic
curvature of the beam in the
transversal direction equal
with µ 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 µ
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
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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 σ1 is axial and uniform
In the cantilever the
stress σ2 has a linear
variation along the
bar and across the
cross-section ⇒
the material is
far less efficient
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Stiffness in structural action
Column is much stronger than the beam: σ2/σ1 = 6(L/h)
for L/h=20 ⇒ σ2/σ1 = 120
Column is much stiffer than the beam: δ2/δ1 = 4(L/h)2
for L/h=20 ⇒ δ2/δ 1 = 1600 (P=k∙δ
δ ⇒ 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 δ
– P=k∙δ
δ ⇒ P1=k1∙δ
δ1; P2=k2∙δ
δ2. If the deflection
is the same for the two members δ1=δ
δ2 ⇒
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
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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(δ/h)2
w2 - the portion of the loading carried through bending.
When the deflection δ 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
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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
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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
Dome: structural action
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
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Dome: structural action
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
Dome: structural action
The heaviest load in many domes is their own weight
In a hemispherical dome of a uniform thickness,
– the stresses σ1 in the direction of meridians are compressive
throughout
– the circumferential stresses σ2 are tensile near the rim: tensile
reinforcement needed to resist them
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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
Shells: hyperbolic paraboloid
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
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Shells: hyperbolic paraboloid
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