Matthew Stuart, P.E., S.E., P.Eng.

Course Outline

This one-hour online course will enable you to obtain a general understanding of the most commonly used structural design philosophy and analysis of post-tensioned concrete in the industry today. This course will define the difference between pretensioned and post-tensioned concrete, and will establish the basis of design for post-tensioned concrete. A description of the methods of design including primary and secondary moment effects,load-balancing, preliminary sizing of members, tendon arrangement, losses and service and ultimate strength analysis will be provided. This course includes a multiple choice quiz at the end.

Learning Objective

At the conclusion of this course, the student will:

• Be able to define the difference between pretensioned and post-tensioned
• Be familiar with the basis of design for post-tensioned concrete
• Be knowledgeable in:

primary and secondary moment effects
load-balancing
preliminary sizing of members
tendon arrangement

Course Introduction

The structural design of reinforced concrete can be divided into two categories;

1. conventionally reinforced
2. prestressed

Prestressed concrete design can be further subdivided into pretensioned and post-tensioned reinforcement. This subject of this course covers the design of post-tensioned concrete only.

This course deals primarily with the design philosophy and analysis of post-tensioned concrete. For detailed design and computational aspects of post-tensioned concrete structures, the reader should refer to PDHonline Course s133: Post-Tensioned Concrete Design Spreadsheet Program.

Course Content

Definitions

Prestressed Concrete: member is stressed via tensioned tendons prior to application of external loads.

Types;

Pretensioned - tendons are stressed prior to casting of concrete; strands anchored to external abutments or self-stressing form prior to transfer of prestressed force to hardened concrete. Strands are typically bonded (i.e. force transfer to concrete via mechanical bond between stranded wire and surrounding concrete)

Post-Tensioned - tendons are stressed after concrete is cast and hardened; strands are anchored against concrete member. Strands are typically unbonded (i.e. anchored only at the ends via anchorage assembly) but can be bonded (i.e. stressed in ducts and grouted in place in addition to end anchorages)

Design Philosophy

Eccentricity of tensioned cables produces internal moments that act in opposition to moments induced by external loading. Precompression of concrete (P/A) also benefits crack control and other serviceability issues.

Typically the required prestressing force (i.e. number, size and profile of tendons) is determined by service stress conditions.

fb = (P/A ± Mnet/S) < or = fallowable
Where: Mnet = [(MDL+LL) – Mbalancing]

The ultimate flexural and shear capacity of the section are then checked at the required critical points.

Analysis:

Primary & Secondary Moments due to P/T;

In simple span beams the primary P/T moments induced by the prestressing force are directly proportional to the eccentricity of the tendons with respect to the neutral axis of the member (i.e. Pe). In continuous or indeterminate post-tensioned structures the moments due to the prestressing force are typically not directly proportional to the tendon eccentricity. This condition occurs because the deformation (i.e. camber) of the member imposed by the P/T force is restrained where it is continuous over other supporting members within the structure. This restraint modifies the reactions and therefore affects the elastic moments and shears resulting from the P/T force. The moments resulting from these restraints are called secondary moments. This term refers to the fact that these moments are induced by the primary Pe and not because they are negligible or necessarily smaller than the primary moment. It is important to note that secondary moments are functions of the reactions and therefore vary linearly between supports. In addition, the total P/T moment is equal to the super-position of the Pe and secondary moments.

In most continuous structures secondary moments have the effect of increasing the magnitude of the positive P/T moment at interior supports and reducing the negative P/T moment between supports. ACI-318 18.10.3 requires that the secondary moments (with a load factor of 1.0) be included in the strength design of a member. Secondary effects are typically not, however, included in the service stress analysis.

Methods:

1. Area Moment
2. Equivalent Load
3. Load-Balancing - Introduced by T.Y. Lin in June 1963. The basic concept of load-balancing is also a representation of the influence of tendons by using equivalent loads. This method is by far the most convenient method and recommended by PTI.

Load-Balancing:

A magnitude of prestressing force is selected to “balance” or counteract some portion of the load. A theoretically perfectly “balanced” structure would result in no deflection and only axial compression forces (P/A) from the tendons. The net moment at any point within a beam is therefore that moment that results from that portion of a load that is not balanced. This concept helps visualize the effects of post-tensioning on any structure and greatly simplifies the calculations. In addition, secondary moments are easily obtained by subtracting the primary Pe from the moments caused by the balancing load at any location along the beam.

This initial portion of the analysis is very iterative and “trial and error” in nature. Because of this there are a number of different approaches to establishing a starting point. A lot of engineers like to think in terms of a percentage of dead or live load as basis for starting the analysis. From my experience, however, particularly with structures having highly variable spans and loading conditions I like to start with a tendon profile based on experience and simply run the numbers (friction, wedge set & other losses and initial service stress analysis). From these results I then make adjustments to the strand drape and jacking sequence as necessary.

It is also important to note that the load-balancing method assumes a sharp bend in the tendon geometry over the supports. In reality the tendons are laid over supports with a reverse curvature to help minimize frictional losses during the stressing operation. Tests have shown however that for practical tendon geometries (in particular with flat plate construction) the effects of the actual tendon profile over the supports are only in the order of between 5% and 10% error. As the calculated load-balancing moments only directly effects service stress calculations more so than ultimate strength the load-balancing method is therefore sufficiently accurate in most cases without consideration of the reverse tendon curvature over the supports.

Design

Preliminary Sizing of Members:

Recommended Span-to-Depth Ratios:

Tendons:

Typically tendons are located near the bottom fiber at positive moment regions and near the top fiber at negative moment regions with the intent to install the cable with the maximum total drape. Exceptions include the need to anchor at the neutral access of an exterior end support condition which can be particularly limiting at a flat plate structure. In addition, the variability of adjacent spans lengths or loading conditions will also have an impact on the final tendon geometry.

Type & Arrangement;

1. Bonded
2. Unbonded
3. Parabolic Drape
4. Straight Line (typically only used in pretensioned member)
5. Horizontal Sweep

Placement & Details

A.Tendons at the “high points” that join adjacent draped strand profiles exert downward reactions. The tendons should be laid out so that these reactions occur and can in turn be resisted by columns, walls and/or “upward” tendon loads. Therefore in any structure (beam and one-way slab/joist or two-way flat plate) all tendons in one direction should be placed through or immediately adjacent to a column while the tendons in the other perpendicular direction should be spaced uniformly across the bay width. This requirement to band the strands in one direction and uniformly distribute them in another for the above statically rational reasons also has obvious advantages in the field in that this arrangement simplifies the construction sequence.

B. At least two of the uniformly distributed tendons should be placed through the column reinforcing cage in a two-way flat plate.

C. Provide conventional bonded reinforcement in the non-compressed zones along the slab edge between the diffusion areas of the end anchorages.

D. Account for volume change (i.e. P/A elastic shortening)and/or avoid restraints where possible.

E. Review live and dead end anchorage arrangements and availability of space as well as confinement reinforcement requirements.

Prestress Losses:

Refer to attached Table (does not include friction or wedge set losses)

Actual losses, whether greater or smaller than computed values, have little effect on the design strength of the member but do affect service load conditions (i.e. deflections, camber, etc.)

Friction & Wedge Set Losses:

Service Stresses:

Initial - stresses immediately after transfer of P/T force, before long-term losses or superimposed loading occurs. (ACI-318 18.4.1)

Final - stresses at service loading including long-term losses.(ACI-318 18.4.2)

Typically maximum fiber stress in tension controls design. It is my personal experience to design up to the rupture modulus of the concrete thereby avoiding the need to analyze the member as a transformed cracked section (i.e. use Ig).

Other considerations;

Minimum P/A = 125 psi for slabs (ACI-318 18.12.4)
[ACI 314 recommends 200 psi for parking garages]

Maximum P/A = 500 psi for slabs
[ACI 314 recommendation to avoid excessive shortening]

Ultimate Shear Strength:

Shear in both statically determinate and continuous P/T members is affected by the shear carried by the tendons. Essentially the “balancing” load reduces the design shear in a manner similar to that associated with the design moments.

It has been my personal experience to conservatively ignore this contribution of the P/T effects. My rational for this is as follows. First, the allowable shear contribution of the concrete, Vc, permitted by ACI for pretensioned members already accounts for the enhanced characteristics of precompressed concrete. Secondly, from a practical standpoint, more minimum stirrup reinforcement is required in P/T concrete members than conventionally reinforced beams because of the need to provide adequate means of supporting the tendon drape through out the entire length of the beam. Finally, with two-way flat plate construction, a little conservatism never hurts when it comes to punching shear capacity particularly when you never know when some trade will form or cut a slab opening directly adjacent to the column without first checking with the structural engineer.

Ultimate Flexural Strength:

f_ps - dependent on whether bonded or unbonded tendons. Value is lower for unbonded strand.

Bonded conventional reinforcement;

required for unbonded tendons, intended to provide
control and distribution of cracking at high load levels. Also insures that structure will behave as a flexural element rather than a shallow tied arch.

In almost all cases, the most economical design for flexural strength will utilize the maximum permissible tensile stresses for prestressed concrete.

ACI code does not provide guidelines for the effective flange width of cast-in-place P/T concrete T-beams. Recommendations are available from PTI and ADAPT. It has been my experience to use the ACI requirements for conventionally reinforced T-beams for the calculation of section modulus for the purposes of service stress analysis with the exception that the gross area of the member be used for the determination of P/A values. In addition, it has been my experience to use the same ACI effective flange width criteria for the calculation of ultimate strength capacities.

Course Summary

Related Links

Once you finish studying the above course content, you need to take a quiz to obtain the PDH credits.