Post-Tensioned Concrete Fundamentals

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Unformatted text preview: nced 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 illustrated below. 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 uniform load...
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This document was uploaded on 01/28/2014.

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