Upload
kumarjr86
View
229
Download
2
Tags:
Embed Size (px)
DESCRIPTION
cable prestressing
Citation preview
Topic 2(ii)
Concept of Prestressing
Concept of Prestressing
Prestressing the concrete is to transfer precompression
(compressive stress) to the concrete
How the prestressing force transmitted to concrete can be
explained by concept of prestressing
Degree of Prestressing
This classification introduced depend on the level of prestress
introduced in the structural element to nullify the stress due
to external load.
Fully prestressed : all cracking should be avoided restricted
by no tensile stress allowed under service load, the whole
section in permanent state of compression
Limited prestressing : tensile stresses do not exceed the
cracking stresses of concrete
Partial prestressing : combination of tensioned and
untensioned steel. Represent form of construction which is
intermediate between reinforced and prestressed concrete
Concept of Prestressing
Concept of
Prestressing Stress
concept Force
concept
Load
balancing
concept
Stress Concept The concept that considering prestressing force transmitted to
concrete as initial internal stress to counteract the internal stress developed due to external loads is known as stress concept
The combination of the effect of external loads and prestressing are studied together as equivalent stresses and compared with permissible levels of stresses in the material
The procedures of this concept can be divided into two stages: stress at transfer/stage 1 and stress at service/stage 2
For stage 1, the stresses across cross section due to self weight and prestressing are taken into account
For stage 2, the stresses caused by prestressing, dead and live loads and other external loads are calculated together through the depth of cross section
The stresses should be within the permissible limits
Permissible stress
Stresses at transfer
+ + =
Axial stress Stress due to eccentricity of
prestressing force
Stress due to dead load
+
+
- +
-
L
NA
P e
P
Stresses at transfer
Stress at top fibre :
Stress at bottom fibre :
(Allowable tensile stress
at transfer, Clause 4.3.5
BS8110)
(Allowable compressive
stress at transfer)
Stresses at service
+ + =
Axial stress Stress due to
dead load
Stress due to
eccentricity of
prestressing force
+
Stress due to
external load
Stresses at service
Stress at top fibre :
Stress at bottom fibre :
(Allowable compressive
stress at service, Clause
4.3.4 BS8110)
(Allowable tensile stress
at service)
Stress concept
In stress concept, we used theory of bending throughout the
analysis where:
- it is assumed that plane sections remain plane before or
after the moments are applied
- the top and the bottom fibre of the structural elements are
subjected to maximum stresses
The permissible/allowable streses under compression and
tension in the materials concrete and steel do have a major
role to play in analysis and design of prestressed concrete
structure based on stress concept
Example 1
A simply supported prestressed concrete beam of cross section
400mm x 600mm has a span of 10m. It is subjected to an
uniformly distributed load of 30kN/m in addition to its self-
weight and is prestressed with a force of 1740kN with a
prestressing able of parabolic profile. The cable is anchored at
the center of gravity of the cross section at support and has an
eccentricity of 160mm below NA at the mid span cross section.
Analyze the beam for the effects of prestressing and the loads at
mid cross section using the philosophy of stress concept.
Solution
Span of the beam = 10 m
Cross section = 400mm x 600mm
External load = 30 kN/m
Unit weight of concrete = 24 kN/m3
Prestressing force = 1740 kN
Cable profile = parabolic
Eccentricity of mid cross-section = 160mm (below NA)
Eccentricity at support section = 0 mm
Properties of section
Area of cross section, A = 0.4 x 0.6 = 0.24 m2
Moment of inertia, I =
Modulus of section, Zt = Zb =
Selfweight of the beam, w/m = 24kN/m3 x 0.24m2 = 5.76kN/m
Calculate stress due to axial load, moment from
eccentricity of prestress force, bending moment from
selfweight and external load
Forces Axial force, P = 1740 kN
Moment due to eccentricity of prestressing force = P x e
Pe = 1740 kN x 0.16 m = 278.4 kNm
Bending moment due to :
selfweight = wl2/8 = 5.76 x 102/8 = 72 kNm
external load = 30 x 102/8 = 375 kNm
All causes and effects are converted to stresses in stress concept
for further evaluation.
At transfer
Stresses at top fibre :
Stresses at bottom fibre :
+ + =
Axial stress Stress due to eccentricity of
prestressing force
Stress due to dead load
At service
Stress at top fibre :
Stress at bottom fibre :
+ + =
Axial stress Stress due to
dead load
Stress due to
eccentricity of
prestressing force
+
Stress due to
external load
Force Concept
In this approach the structural element is considered as if it is
a reinforced concrete element
The total prestressing force is taken tensile force and the
stresses generated in concrete will produced compression
force of an equal value. The forces are collinear to keep the
element in equilibrium if only prestressing force is
considered
Hence, the structural element at any cross-section will be
subjected to tensile force in the prestressing element and a
compressive force in the concrete which is the resultant force
of all compressive stresses acting on that cross-section
Force Concept
If any additional load (say dead load) is considered, the tensile force in
prestress element will be modified and the center of compression will
also be shifted.
For a case of positive sagging bending moment applied on the structure
due to external loads the tensile force in the prestressed steel element is
marginally increased and the compressive force which is the resultant
stresses caused by the prestressing and by the loading will be shifted
upwards from the line of action of tension.
The tensile force or the compressive force multiplied by the shift
between these two forces will be the external moment.
This concept is used to design the structures and to get the moment
resisting capacity of the cross section
Since the capacity of the section is decided based on the total tension
and compression it carries, this approach is called the force approach
Example
Analyze the beam in Example 1 using force concept.
Solution
In force concept all causes and effects are considered as forces
for evaluation.
Bending moment at mid span due to :
a) Selfweight = 72 kNm (top – comp, bottom –tension)
b) External load = 375 (top – comp, bottom –tension)
Total bending moment = 447 kNm
Prestressing force = 1470 kN
When the prestressing force of 1740 kN (tensile) in the cables alone is acting, the stresses generated in concrete will lead to a resultant compression of equal value (1740 kN) and the compression also acts at the same level of prestressing force. The forces are collinear.
Stage 1/at transfer
When selfweight starts acting (which is immediately after prestress) there will be a small increase in the tensile forc in the cables. But this is neglected.
The total tensile force in the cable = 1740 kN
Total compressive force = 1740 kN (to keep the section in equilibrium)
But the resultant compression will act at a different level, so that the compression and the tension will form a couple to resist dead load bending moment
Dead load bending moment = 72 kNm
Distance between the tensile force (cable position) and the
center of resultant compression, a
a = M/P = 72/1740 = 0.04138 m
Distance of compression from the NA of cross section
= 0.16 – 0.04138 m = 118.6 mm
15.85
1.35
T
a = 41.38mm
NA 118.6mm
160mm
The resultant compression will act at 118.6mm from NA only for
a given stress distribution.
The stress distribution can be evaluated as detailed below :
Stress at top =
Stress at the bottom = 7250 + 8599.95 = 15849.95 N/mm2
Stage II/ at service
When the external load also starts acting the resultant (final)
bending moment shall be resisted by the total compression and
total tension with a lever arm.
Total tension = 1740 kN
Total moment to be resisted = dead load + bending moment
due to other loads
= 72 + 375 = 447 kNm
Lever arm required, a = M/P = 447/1740 = 0.2569 m
Resultant center of compression will be located at 256.9mm
from the center of tension cable position
Hence position of compression will be located at 256.9mm from the center of
tension cable position.
Hence position of center of compression from center of NA
= 256.9 -160 = - 96.9mm (upward)
This resultant compression (1740kN) will act at 96.9mm above NA only for one
particular stress distribution across the section.
The stress distribution is evaluated as follows
Stress at top =
Stress at bottom =
The stresses are the same as we obtained in the stress concept.
Load balancing concept
Opposite type of loads in structural element (opposite in nature to
the external loads)
If the external loads cause a sagging curvature in the beam, any load
which introduces the hogging curvature on to the beam, equal and
opposite in nature to that caused by external loads is also called
prestressing and this method of prestressing is recognized as load
balancing concept.
The external loads are treated only as loads and not converted as
stress on the structure
Prestressing also converted as equivalent load and this equivalent load
must counteract or balance the external loads
The load balancing concept is used for analysis of indeterminate
prestressed concrete structures and complicated analysis where the
effect of prestressing cannot easily depicted
For example the parabolic profile of prestressing cable with prestressing force, P can be considered equivalent to the upward force of
Example 3
Analyze the prestressed concrete beam described in Example 1
using load balancing concept.
Solution
In this concept all the causes and the effects will be considered
as loads and the member will be analyzed
Total downward load = 30 + 5.76 = 35.76 kN/m
The equivalent upward uniformly distributed load provided by
prestress =
Net downward on the beam = 35.76 – 22.272 = 13.488 kN/m
The bending moment caused by resultant downward force at
center section
Stresses at mid span caused by this moment
Stresses at mid span caused by pretensioning force that acting at
the centroid of the section
(Compression at top, tension
at bottom)
(compression)
Hence net stresses :
At top fibre = stress due to prestress + stress due to downward
force
At bottom fibre
The stresses are the same as obtained in stress concept and
force concept at service