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Re: Balanced Moment vs PT Moment
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 Subject: Re: Balanced Moment vs PT Moment
 From: GSKWY(nospamat)aol.com
 Date: Mon, 6 Jun 2005 10:03:09 EDT
Oh good. This question is not about codes. You are not a code monkey.
The balanced moment is, in fact, the PT moment. The words balanced and balancing do not have any technical meaning. They are simply one way of visualizing the effects of posttensioning. The terms are not always used consistently.
The "loadbalancing" design method was introduced by T.Y. Lin in a paper titled "LoadBalancing Method for Design and Analysis of Prestressed Concrete Structures" in the June, 1963 issue of the ACI Journal. Although the idea of looking at the prestressing as a negative load was not new, T.Y. Lin was the first to formalize the concept and present a detailed treatment of the method.
The concept behind loadbalancing is that the prestressing can be expressed as equivalent loads. The equivalent loads consist of axial compression and a flexural load that counteracts or "balances" a portion of the load on the structure, hence the term loadbalancing. The member can thus be viewed as a nonprestressed member with a reduced loading, to which an axial precompression is superimposed. The advantage of the loadbalancing method is that the moments are only added once for each load pattern. The stresses at any section can then be checked.
Some writers use the term "balanced condition" to refer to a member where 100% of the load under a given loading condition is balanced. The member will have no bending or deflection under that load condition and will only be subjected to axial load. The stresses at any point in the beam will be pure compression and will be equal to P/A where P is the applied prestressing force and A is the crosssectional area of the member.
This is also called a concordant tendon profile. It is not something that is typically done, however. It is not an efficient profile because it is not possible to use the full tendon drape. If you use the maximum possible high point, you end up with too much positive moment at the supports. It is also not necessary, even if you have a design requirement of zero tension, because you also have axial compression.
If only a portion of the load is balanced by posttensioning, the stress at any point will be equal to P/A + Mc/I where M is the moment caused by the amount of unbalanced load.
Theoretically, if 100% dead load plus sustained live load is balanced, there will be no longterm deflection because the beam will be under pure compression most of the time. Typical practice, however, is to balance between 60 and 80% of the dead load. This results in an economical, satisfactory design in most cases.
The posttensioning (balancing) moment consists of a primary moment and a secondary moment, where at any point along the span, the primary moment is equal to P*e. The secondary moments are produced by the support reactions due to the balancing load. Despite their name, the secondary moments are often larger than the primary moments.
Loadbalancing is one of three concepts that can be used to visualize the effects of the prestressing. The other two are superposition of stresses (elastic theory) and ultimate strength design.
Superposition of stresses is typically used in the design of statically determinate members such as simplespan pretensioned members. Superposition of stresses is also used for posttensioned slabs on ground, where the tendons run flat (without profile). The prestressing will cause a uniform compression of P/A and a moment of P*e where e is the eccentricity of the tendon with respect to the centroid of the section. This results in stresses of +Pec/I at the top and bottom of the section.
If the tendon is below the centroid of the section at midspan, the moment due to the eccentricity will cause a compressive stress at the bottom of the section and a tensile stress at the top of the section. The superposition of the stresses at the bottom of the section at midspan will thus consist of the compressive stress P/A , a compressive stress due to the moment Pe and a tensile stress due to the dead and live load moment. At the top of the section there will be the compressive stress P/A, a tensile stress due to the moment Pe, and a compressive stress of Mc/I due to the dead and live load moment. The force in the tendon and the eccentricity can be adjusted as necessary to keep the stresses at the top and bottom of the section within acceptable limits under different loadings.
There is really no advantage to using the loadbalancing method in a statically determinate design. The midspan section will usually control the design; at the most, it will be necessary to investigate the midspan section and a section at each end of the member.
Most posttensioned members are continuous, though, and are therefore statically indeterminate. The moment due to the prestressing is not simply the force in the tendon times the eccentricity. Different sections may be critical under different load patterns and changes to the tendon force or eccentricity to adjust the stresses at one section will affect other sections.
It is worth reading T.Y. Lin's original paper. It wanders a bit and doesn't do a good job of explaining some things. But it is better than trying to understand some of the textbooks (Solomon and Wang for example), which completely mangle the interpretation. There were some criticisms to T.Y. Lin's suggestions, printed in a later ACI Journal. I think most of the criticisms were pointing out the method is an approximation and relies on some of the typical geometrical assumptions about small angles. But most design theories involve approximations.
And, in closing, whoever wants to is free to criticize my posts. This one in particular is just a collection of random statements.
Gail Kelley

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