Prestressed Concrete
Design for Serviceability
Lecture 9
Dr Ahsan Parvez
Prestressed Concrete Outline
Stress Distribution
Loss of Prestress
Determination of Prestress and Eccentricity
Stress Limits
Examples
Prestressed Concrete
 The fundamental aim of prestressed concrete is to
limit tensile stresses, and hence flexural cracking , in
the concrete under working condition.
 Design is therefore based initially on the requirements
of the serviceability limit state.
 Subsequently considered are limit state criteria for
bending and shear.
 In addition to concrete stresses under working loads,
deflection also must be checked.
Serviceability Design
 Serviceability design of prestressed members includes
ensuring that the appropriate deflection and crackwidth criteria are satisfied.
 Serviceability requirements determine the magnitude
of prestressing force required and the shape of an
appropriate tendon profile.
 The prestressing force is based upon the size and
number of prestressing tendons necessary to provide
a satisfactory level of axial precompression in the
member.
Serviceability Design
Two critical stages in the design of Prestressed Concrete
beams for serviceability:
 Stage 1 : Immediately after the prestressing force is
transferred to the concrete. At initial transfer of prestress
to the concrete, the prestress force will be considerably
higher than the long-term value as a result of subsequent
loses
 Stage 2: After time-dependent losses have occurred and
the member is under full service loads.
 At both these stages serviceability requirements of the
members must be satisfied.
Stress Distribution
Elastic analysis of a rectangular uncracked section
Stress Distribution
Elastic analysis of a rectangular cracked section
Stress Distribution
(a) The stress due to prestress varies from a maximum
compression in the bottom fibre to a small tension in
the top fibre and causes the beam to deflect upwards,
i.e. to camber.
(b) When design loads, w are applied, the resulting
bending moment near midspan produces a stress
distribution as shown.
(c) The stress in (b) combines with the stress due to
prestress (a) and produces a state of stress which varies
from a maximum compression in the top fibre to a
minimum compression (or possibly small tension) in the
bottom fibre.
Stress Distribution
(d & e) As the load increases, tensile stresses in
the lower fibre increase until the tensile strength
of the concrete is reached and cracking occurs.
Therefore, the bending moment is resisted by
internal stresses in a similar fashion to the
reinforced concrete beam.
Loss of Prestress
Loss of Prestress – Immediate Losses
During the stressing operation, immediate losses
can occur by:
 Elastic deformation of the concrete
 Slip and deformation in the end anchors
 Friction along the length of members, between
tendons and duct
Loss of Prestress – Time Dependent
Losses
 Stress relaxation of prestressing steel
 Shrinkage deformation of concrete
 Creep deformation of concrete
Loss of Prestress – Time Dependent
Losses
Stress relaxation
• The initial stress level in prestressing steel after transfer is
usually high, often in the range 60-75% of the tensile
strength of the material.
• At such stress levels, high-strength steel creeps.
• If a tendon is stretched and held at a constant length
(constant strain), the development of creep strain in the
steel is exhibited as a loss of elastic strain, and hence a loss
of stress.
• This loss of stress in a specimen subjected to constant
strain is known as relaxation.
• Relaxation in steel is highly dependent on the stress level
and increases at an increasing rate as the stress level
increases.
Determination of Prestress and
eccentricity in Flexural members
 A number of possible starting points exist for the
determination of the prestressing force P and
eccentricity e required at a particular cross-section.
 The quantities of P and e are often determined to
satisfy preselected stress limits.
 Other serviceability limits include: -Deflection
-Camber
-Axial shortening
Prestressing force and cable profile may be selected
using a load-balancing approach to design.
Satisfaction of Stress Limit
 If a member is to remain uncracked throughout,
suitable stress limits should be selected for the tensile
strength, 𝑭𝑭𝒕𝒕𝒕𝒕 at transfer and the tensile stress at full
load, 𝑭𝑭𝒕𝒕 .
 In addition, limits should also be placed on the
concrete compressive stress at transfer, 𝑭𝑭𝒄𝒄𝒊𝒊 and under
full load 𝑭𝑭𝒄𝒄 .
 If cracking under the full load is permitted, the stress
limit 𝑭𝑭𝒕𝒕 is relaxed and remaining three limits are
enforced.
Satisfaction of Stress Limit
Transfer Condition
Concrete Stress at Transfer Condition
Concrete Stress at Transfer Condition
Full Loading Condition
R = reduction factor
Full Loading Condition
Full Loading Condition
Satisfaction of Limits
Satisfaction of Limits
Magnel’s Design Diagram
Magnel’s Design Diagram
 Magnel diagram is a powerful design tool as it covers
all possible solutions of the inequality equations and
enables a range of prestress force and eccentricity
values to be investigated.
 The diagram shows that for a minimum prestress force
(the largest value of 1/P) corresponds to the value of
greatest eccentricity and as eccentricity is reduced,
the prestress force must be increased to compensate.
Stress Limits
Prestressed beams at Transfer (Cl. 8.1.6.2)
The maximum compressive stress in the concrete under
the design loads at transfer shall not exceed
′
𝐹𝐹𝑐𝑐𝑐𝑐 ≤ 0.5 𝑓𝑓𝑐𝑐𝑐𝑐
′ = characteristics strength of concrete at transfer
𝑓𝑓𝑐𝑐𝑐𝑐
Crack Control for Flexure Under Service Loading
(Cl. 8.6.3)
Flexural cracking shall be deemed to be controlled if,
under the short-tern service loading, the maximum
tensile stress in the concrete
𝐹𝐹𝑡𝑡 ≤ 0.25 𝑓𝑓𝑐𝑐′
Stress Limits
Limits at Transfer
′
𝐹𝐹𝑐𝑐𝑐𝑐 ≤ 0.5𝑓𝑓𝑐𝑐𝑐𝑐
′
𝐹𝐹𝑡𝑡𝑡𝑡 ≤ 0.25 𝑓𝑓𝑐𝑐𝑝𝑝
Limits under Service Loading
𝐹𝐹𝑐𝑐 ≤ 0.5𝑓𝑓𝑐𝑐′
𝐹𝐹𝑡𝑡 ≤ 0.25 𝑓𝑓𝑐𝑐′
Example
A one-way slab is simply supported over a span of 12m
and is to be designed to carry a service load of 7 kPa
(𝑘𝑘𝑘𝑘/𝑚𝑚2 ) in addition to its own self-weight. The slab is
post-tensioned using flat duct by regularly spaced
tendons with parabolic profiles. The slab thickness
D=300 mm with cover and the strand diameter of 12.7
mm. Assuming instantaneous and time dependant loss
is 8 and 16% respectively.
The material properties are:
′ = 25 𝑀𝑀𝑀𝑀𝑀𝑀
𝑓𝑓𝑐𝑐𝑐𝑐
𝑓𝑓𝑐𝑐′ = 32 𝑀𝑀𝑀𝑀𝑀𝑀
𝑓𝑓𝑝𝑝 = 1840 𝑀𝑀𝑀𝑀𝑀𝑀
𝐸𝐸𝑐𝑐 = 28600 𝑀𝑀𝑀𝑀𝑀𝑀
𝐸𝐸𝑐𝑐𝑐𝑐 = 25300 𝑀𝑀𝑀𝑀𝑀𝑀
Example
Step 1
The prestressing force and eccentricity are to be
determined to satisfy the following concrete
stress limits:
Example
Step 2
Slab self-weight
(which is the only load other than the prestress
at transfer):
Example
Step 3
The moments at mid-span both at transfer and
under the full service load are:
Example
Step 4
Calculate Cross-section Properties:
Given: b=1000 mm , D=300 mm and y=150 mm
Example
Step 5
Calculate and draw Magnel’s Equation
Example
Example
Example
Step 6
Calculate the emax value from Magnel’s Diagram
Example
Example
Step 7
From Magnel’s Diagram get the stress values:
Example
Step 8
Calculate the prestressing force
Example
Step 9
Calculate the number of cable required
Example
Example
The minimum number of cables required in each metre
width of slab is therefore: