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load method. Once a load is selected to be balanced, and the tendon forces and
drapes are chosen, it is a then a matter of elastic structural analysis, which is well suited
to the computer, to find the moments due to the balanced load. The analysis for
balanced loads is just another load case as far as the computer is concerned. In the
balanced load method, the balanced load moments are used to determine the
hyperstatic moments, which we will cover next. But first, we shall establish the sign
conventions that we are using throughout this course. www.SunCam.com Copyright 2010 John P. Miller Page 19 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course A Word About Sign Conventions
In this course, moments causing tension in the bottom fiber are considered positive.
Moments causing tension in the top fiber are negative. Eccentricities below the neutral
axis are negative and above the neutral axis they are positive. These sign conventions
are illustrated below. +M (Tension in Bottom) Positive Moment ‐M (Tension in Top) Negative Moment P +e above NA +e Neutral Axis P ‐e below NA ‐e Eccentricity www.SunCam.com Copyright 2010 John P. Miller Page 20 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Hyperstatic (Secondary) Forces
Hyperstatic, or secondary, forces are the forces generated in a statically indeterminate
structure by the action of the post-tensioning. Generally, hyperstatic forces are
generated due to support restraint. Hyperstatic forces are not generated in a statically
determinate structure. It is important to isolate the hyperstatic forces as they are treated
with a separate load factor when considering ultimate strength design.
Let's consider the two span post-tensioned beam in the following illustration. We know
from previous examples that the tendon force will create an upward uniformly distributed
load acting on the beam as shown in the figure. If the center support were not there,
the beam would deflect upward due to the post-tensioning force. Since the center
support is there, and it is assumed that it resists the upward deflection of the beam, a
downward reaction is induced at the center support solely due to the post tensioning
force. Balanced Uniform Load Hyperstatic Reaction Post-Tensioning Induced Reaction
The next figure shows the deflected shape of the two-span beam due to the posttensioning force (ignoring self-weight) as if the center support were not there. In order
to theoretically bring the beam back down to the center support, a force equal to the
center reaction would have to be applied to the beam. This induces a hyperstatic
moment in the beam. The hyperstatic moment diagram is illustrated below for the twospan beam in this example. Note that the hyperstatic moment varies linearly from
support to support. An example of a three-span hyperstatic moment diagram is also
www.SunCam.com Copyright 2010 John P. Miller Page 21 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Hyperstatic Reaction Theoretical Deflected Shape Hyperstatic Moment Diagram for a 2-Span Beam Hyperstatic Moment Diagram for a 3-Span Beam
Note that the above example assumes that the beam is supported by frictionless pin
supports and therefore no moments can be transferred into the supports by the beam.
However, in real structures, the post-tensioned beam or slab is normally built integrally
with the supports such that hypersatic moments are also induced in support columns or
walls. Hyperstatic moments in support members are normally ignored in hand and
approximate calculations, but can and should be accounted for in post-tensioning
computer software. Thus, when designing supporting columns or walls in posttensioned structures, it is important to take into account hyperstatic moments induced
by post-tensioning forces. For purposes of this course, we will ignore hyperstatic
moments that are induced in supports. We will assume our beams and slabs are
supported by frictionless pin supports and deal only with the hyperstatic moments
induced in the beam or slab members being considered.
www.SunCam.com Copyright 2010 John P. Miller Page 22 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course The above illustrations serve to introduce you to the concept and source of hypersatic
forces in continuous post-tensioned structures. The hyperstatic moment at a particular
section of a member is defined as the difference between the balanced load moment
and the primary moment. We refer to the primary moment as M1 and this is the
moment due to the eccentricity of the post-tensioning force with respect to the neutral
axis of the member. In equation form, Consider the two-span post-tensioned beam shown below. As we have seen
previously, the beam can be analyzed with equivalent loads due to the tendon force.
The draped tendons with force P may be replaced with an equivalent upward acting
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This document was uploaded on 01/28/2014.
- Spring '14