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Unformatted text preview: of 8 / . When the two-span beam is analyzed using this load, a
moment diagram is developed as illustrated below. This is called the balanced moment
diagram. Recall that the sign convention results in a negative moment when tension is
in the top.
8 / 2 P P Drape = a e1 Drape = a e2 Equivalent Loading Due to Post-Tensioning / / Moment Diagram Due to
Balanced Loads www.SunCam.com Copyright 2010 John P. Miller Page 23 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Using our sign conventions, we can construct a primary moment diagram for moments
M1. The primary moment is obtained from the product of the post tensioning force times
its eccentricity with respect to the neutral axis of the beam. Thus, referring to the
diagram above, the primary moment at the center support is P x e1 and the primary
moment at the mid-span of each span is P x e2. According to our sign conventions, the
primary moment at the center support is positive. Primary Moment
Diagram Now we can determine the hyperstatic moments using the equation At the center support, we obtain Near the mid-span of each span we have 4
7 Hyperstatic Moment Diagram www.SunCam.com Copyright 2010 John P. Miller Page 24 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Example
The two-span post-tensioned beam shown below.
Fe 7" Fe Drape = 26.5" 23" 60'‐0" 60'‐0" Beam size 14 x 36
WDead = 1.84 kips/ft
Determine the hyperstatic moments due to post-tensioning resulting from balancing
80% of the dead load.
Determine Balanced Load Moments 0.8 1.84 1.47 / Using well known formulas for a beam with two equal spans subjected to a uniformly
distributed load, we find the balanced moment diagram to be: Balanced Load Moment
Diagram Next we can compute the required effective post-tensioning force:
www.SunCam.com Copyright 2010 John P. Miller Page 25 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course 1.47 60 300 8
8 26.5 /12 Referring to the beam diagram above, we see that the primary moments are 300 x 7"
and 300 x 23" at the center support and at mid-span, respectively. Using our sign
convention, we find the primary moment diagram to be: Primary Moment
Diagram Therefore, using the relationship of At Center Support: 662 Near Mid-Span: 378 , 175 we find: 487 575 197 Hyperstatic Moment Diagram
Discussion of Results
It is important to isolate the hyperstatic moments because there is a load factor of 1.0
applied to these moments in factored load combinations as we will see in Part Two of
this course. Since we know that the hyperstatic moment diagram is linear, the location
of the +197 ft-k hyperstatic moment is not at the center of the span – by similar
triangles, it is actually 24.3 feet from the exterior support. This is usually considered to
be close enough to the theoretical location of the maximum moment under uniform load
to be able to algebraically add the factored moments at "mid-span" of the beam for
design purposes. It is interesting to note that if we had included 18 inch square
columns, which are fixed at the bottom, below the beam in our frame analysis, the
hyperstatic moment diagram for the above example would look like this:
www.SunCam.com Copyright 2010 John P. Miller Page 26 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course ft‐k ft‐k Hyperstatic Moment Diagram Including Columns The above hyperstatic moment diagram is much closer to realistic conditions when a
post-tensioned beam is built integrally with the columns below. Note that the
hyperstatic moment in the beam at the center support is significantly less when the
columns are included in the analysis, and that there is a substantial hyperstatic moment
induced into the exterior columns. The center column has no hyperstatic moment, only
because of the perfect geometrical symmetry, and because the tendon drapes and
balancing loads in both spans are identical.
When an unbonded tendon is stressed, the final force in the tendon is less than the
initial jacking force due to a number of factors collectively referred to as pre-stress
losses. When a hydraulic jack stresses an unbonded tendon, it literally grabs the end of
the tendon and stretches the tendon by five or six or more inches. As the tendon is
stretched to its scheduled length, and before the jack releases the tendon, there is an
initial tension the tendon. However, after the tendon is released from the jack and the
wedges are seated, the tendon loses some of its tension, both immediately and over
time, due to a combination of factors, such as seating loss, friction, concrete strain,
concrete shrinkage, concrete creep, and tendon relaxation.
Prior to the 1983 ACI 318, pre-stress losses were estimated using lump sum values.
For example, the loss due to co...
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This document was uploaded on 01/28/2014.
- Spring '14