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