Stress Analysis

Prestress Effects
Stress in the beam is caused by two principal sources: applied loads and prestress. At 100ᵗʰ points
along the beam, the stress at the top of the beam and bottom of the beam are evaluated. For
composite sections, the stress at the top of the deck and top of the overlay (if present) are also
computed.

Stress due to applied loads is computed in accordance with the following:
S
f = M
where,
f = stress at extreme fiber in beam M = applied moment at section
S = section modulus
Stress due to prestress is computed using the formula:
S
Pe
A
f = P ±

where,
f = stress in beam
P = applied effective prestress force
e = eccentricity of the applied prestress force S = section modulus
Prestress Losses
The level of prestress in a prestressed concrete beam does not remain constant throughout the life
of the beam. The prestress level varies during the various stages of the loading of the girder as a
function of time and the material properties of both the concrete and the prestressing steel. In
accordance with the AASHTO Specifications, several components of prestress loss are evaluated:
• Elastic shortening (instantaneous effect)
• Anchor set (post-tensioning tendons only)
• Friction (post-tensioning tendons only)
• Relaxation of prestressing steel (time-dependent)
• Shrinkage of concrete (time-dependent)
• Creep of concrete (time-dependent)
Of these four components, elastic shortening, anchor set, and friction loss are the only non-time-
dependent components.
Elastic Shortening
Elastic shortening (ES) loss occurs as the girder shortens when the strands are detensioned (i.e.,
cut) after the concrete has reached its specified minimum release strength. As the strands are
detensioned, either by being individually cut, detensioned together as a gang, or by some other
method, the prestress is imparted into the girder. When this force acts on the girder, the girder
itself shortens a small amount and the prestressing steel embedded in the girder shortens with it,
reducing the stress level in the strands. The amount of prestress loss attributed to this elastic
shortening is a function of the level of prestress, area of the concrete cross section, and the
modulus of elasticity of
both the concrete and the steel.

Note that if the strands are specified to be transformed for “all stresses’ or ‘all stresses except
live load” in the Analysis Options dialog box, the elastic shortening component of prestress loss
is automatically accounted for through the transformed properties, and ES will not appear in the
results (internally ES will be set to zero). This will happen only when the LRFD 4ᵗʰ Edition
approximate losses are selected.

Relaxation of Prestressing Steel
If a prestressing strand were stretched some distance and its ends were anchored a fixed distance
apart, over time the steel would gradually relax and the stress level in the strand would decrease.
This phenomenon is called steel relaxation.
Anchor Set
After a post-tensioning tendon has been stressed to its required level, each strand must be
anchored using a chuck. Typically, for a strand to fully anchor, it shifts by an amount equal to
the anchor set.
Friction
During the stressing operation of post-tensioning tendons, friction causes some portion of the prestress to
be lost. This loss occurs as a function of the wobble coefficient (K), the fiction coefficient (μ),
the length of the tendon, and the total angle change of the tendon.

Shrinkage of Concrete
After casting, over time, concrete gradually shrinks due to water loss. The amount of shrinkage is
a function of the ambient relative humidity at the project site.
Creep of Concrete
When a load is applied to concrete, it gradually moves, or creeps. In the case of a prestressed
concrete girder, under permanent loads (i.e., dead load and prestress), there is a component of
creep in the direction of the prestressing strands, which causes the strands to shorten over time.
This, in turn, results in a loss of prestress.

Eriksson Girder supports four different prestress loss procedures (note that the first two procedures yield
similar results):

1. Standard Specs

2. LRFD 3ʳᵈ Edition Specs

3. LRFD 5ᵗʰ Edition and later - Detailed

4. LRFD 5ᵗʰ Edition and later - Approximate

See the Eriksson Girder Quality Control Document for a detailed discussion of the above loss computation
procedures and example calculations.
Special Notes on 2005 LRFD Prestress Loss Procedure

The prestress loss procedure introduced in the 2005 Interim Revisions of the LRFD Specifications
differs significantly in several respects from the previous methods of the LRFD Specifications and
the Standard Specifications. As noted above, complete details on each of these procedures are given
in the Eriksson Girder QC Document. However, the main differences are discussed below:

• Transformed Properties – When the 2005 LRFD Loss Method is selected, the option of whether to
transform the strands is not available to the user. This is due to the need to use specific section
properties (either gross or transformed) at certain points in the analysis.
• Elastic Shortening - LRFD C5.9.5.2.3a states, “When calculating concrete stresses using
transformed section properties, the effects of losses and gains due to elastic deformations are
implicitly accounted for…” This means that, although the phenomenon of elastic shortening does
indeed occur, a separate calculation for elastic shortening is not made; the

effects of elastic shortening are implicitly included when the jacking force is applied to the
transformed section. However, for comparison purposes, the elastic shortening is calculated and
displayed.
• Steel Relaxation – Prestress loss due to steel relaxation at release although small, was,
nevertheless, computed with the previous method. Now, however, it is ignored with the new
procedure.
• Timeframes for Evaluating Loss – In the previous method, losses were computed at release and at
final conditions. With the new method, however, it is not necessary to compute the loss at release
since the elastic shortening component need not be computed separately and the loss due to steel
relaxation at release is ignored. With the new method, losses are evaluated at the time of deck
placement (or placement of barriers for non-composite systems) and at final time.
• Effective Prestress Force – With the previous prestress loss method, losses were factored into
the stress calculations by reducing the effective prestress force by the total loss and applying
that force to the bare precast concrete cross section to obtain the stresses in the concrete. With
the new procedure, losses are factored into the calculations by converting the total prestress
losses between transfer and deck placement and between deck placement and final into forces. The
net change in concrete stress is then computed by applying the force computed for the time frame
between transfer and deck placement to the bare precast section and the force computed between deck
placement and time final to the composite section. These stress components appear in the detailed
text report as “PLoss T->D” and “PLoss D->F”, respectively.
Loss of Prestress in Spliced Girders

When post-tensioning is introduced into a pretensioned girder, there is interaction between the PT
and the pretensioning. Essentially this means that the losses in the pretensioned strands are
increased due to the extra force applied to the girder during the post-tensioning operation. The
method of computing losses described in NCHRP Report 517 (Castrodale 2004) is incorporated into
Eriksson Girder.

Strand Transfer and Development
The effects of strand transfer and development are computed in accordance with each respective
specification, but both specs use a substantially similar treatment. The transfer distance, Lt, is
the distance over which the force in a strand is transferred to the beam concrete. It is assumed to
be directly proportional to the diameter of the prestressing strand. With respect to the Standard
Specifications, the transfer distance is assumed to be 50 strand diameters (Std Art. 9.20.2.4), and
for the LRFD Specifications the transfer distance is assumed to be 60 strand diameters (LRFD Art
5.11.4.1/5.9.4.3.1).

The development length of the strands is the distance, measured from the end of the strand (or
terminal point of debonding), that is required for the strand to develop its full strength at
ultimate strength or at the strength limit state. The following equation is used to predict Ld:

Ld = K(fps

− 2 f
3 pe

)db

(LRFD Eq. 5.11.4.2-1/5.9.4.3.2-1)
(Std Eq. 9-42)

where:
K = 1.6 for precast, prestressed concrete beams
2.0 for debonded strands
fpe = effective stress in strand after prestress losses (ksi)
fps = average prestress in prestressing steel at the time for which nominal resistance of the
member is required (ksi)

db = nominal diameter of strand (in)
The value of K is set to 1.6 for default. This may be overridden by the user. For debonded strands,
K is automatically internally set to 2.0, and is applied only to those strands that are debonded.

For sections near the ends of the beam, if debonding is specified, strands might be completely
ineffective, partially effective, or fully effective. When checking stresses or strength, the curve
of strand stress versus distance from the end of the beam (or terminal point of the debonding) is
idealized as a trilinear curve, as shown in the figure below:

Figure 5-1: Plot of strand stress versus bond length.

Debonding
As an alternative to draping strands to control the stresses in the ends of the beam at release,
debonding can be used. Debonding consists of wrapping or sleeving individual strands to break the
bond between the strand and the concrete for a specified distance. Typically, debonding starts at
the end of the girder and runs along the strand towards the interior of the girder. Some precast
fabricators favor this method of controlling girder end stresses in girders because it eliminates
the safety risks and expenses associated with draping strands.

Debonding is covered in LRFD Art. 5.11.4.3/5.9.4.3.3. In summary: The number of debonded strands
should not exceed 25 percent of the total number of strands in the strand pattern. For any given
row of strands, no more than 40 percent of the strands should be debonded. These upper limits were
changed in the 9ᵗʰ edition to 45% in each row and no upper limit on the total, although Eriksson Girder will
flag an attempt to debond more than 45% of the total strand. Debonding patterns should be symmetric
and the end strands of each row should not be debonded.
A debonding pattern can be determined using Eriksson Girder’s fast and efficient design algorithm, or the
pattern can be manually entered. In either case, the Debonding Dialog Box is for reviewing and, if
necessary, editing the debonding pattern.

Two types of debonding can be specified in Eriksson Girder: end debonding and midspan debonding (Fig. 2).
End debonding is the most common and consists of debonding strands at the ends of the girder,
typically in the lowermost levels of the girder. In contrast, midspan debonding is debonding in the
midspan region, typically near the top of the girder. End debonding is used to control end
stresses, as discussed earlier. Midspan debonding is discussed below.

Figure 5-2: Types of debonding.
Midspan debonding is generally used infrequently. It is typically specified in cases where it is
desirable to include fully stressed strands in the top of the girder (to, for example, reduce the
amount of end debonding) but where those strands have a detrimental effect on the midspan design of
the girder. In such cases, debonding can be applied to the top strands near the midspan region to
disable the effect of those strands in that zone. Note that if midspan debonding is used, then a
pocket must be cast in the top of the girder that can be used to access the strands so they can be
detensioned (i.e., cut).

The contributions of debonded strands to concrete stresses, camber, shear, and the flexural
strength of a section are determined in accordance with the curve given in Fig. 5-1. Lt, the
transfer length, and Ld, the development length, are computed as discussed above. Note that the K
factor in Eq. 5.11.4.2- 1/5.9.4.3.2-1 for debonded strands is 2.0. Eriksson Girder treats each strand
separately and integrates the
contributions of partially transferred or partially developed strands accordingly.