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Unformatted text preview: load balancing method, a portion of the design load is selected to be "balanced", or
carried, by the action of the tendons. The balanced load is commonly taken as 80% of
the dead load. The required force in the tendons to carry the balanced load is easily
calculated using statics. The concrete member is then analyzed using conventional
structural analysis techniques with the equivalent set of tendon loads acting on the
member in combination with other externally applied design loads, such as dead load
and live load.
Let's consider the free body diagram at mid-span of the following simple span beam
with a draped tendon with force P. Note that the common simplifying assumption made
in post-tensioned concrete analysis is that the tendon force acts in the horizontal
direction at the ends of the member and the small vertical component, if any, can be
ignored or is transferred directly to the support. Note that the shear force at the right
side of the free body diagram is zero since this occurs at the mid-span of a simple span
beam with a uniformly distributed load.
cl P a = drape P
L/2 L/2 If we sum the moments about the force P at the left support, we get:
4 Copyright 2010 John P. Miller Page 8 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course 8 The load balancing concept is further illustrated in the figure below, which shows a
simply supported beam and a tendon with a parabolic profile. The beam shown in the
first figure below may be analyzed with an equivalent set of tendon loads acting on the
member as shown in the second figure. Thus, the equivalent loads acting on the beam
consist of the axial force P, an upward uniform load of w, and a clockwise moment M at
the left end due to the eccentricity of the tendon with respect to the neutral axis of the
P P e Neutral Axis a = drape Cross‐Section at Mid‐Span Span Length = L Parabolic Tendon Drape 8 M Pe P Neutral Axis 4 P 4 Equivalent Tendon Loads Applied to Beam www.SunCam.com Copyright 2010 John P. Miller Page 9 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Note that reactions are induced at both ends to keep the system in equilibrium. If we
sum the moments about the left support, we get:
8 2 0 4 0 And if we sum the vertical forces we find the left reaction is:
4 Note that the vertical component of the applied pre-stressing force is neglected. This is
practical since the tendons are customarily horizontal, or very nearly horizontal, at the
end of the members, and the vertical component is usually small.
As we have seen, a draped tendon profile supports, or balances, a uniformly distributed
load. Now let's consider a beam that is required to support a concentrated load. In the
case of a concentrated load on a beam, a concentrated balancing load would be ideal.
This can be achieved by placing the pre-stressing tendons in a harped profile. This
concept using harped tendons is illustrated in the figure below.
External Applied Concentrated loads P P e Neutral Axis a = drape cL cL
Span Length = L Harped Tendon Drape
www.SunCam.com Copyright 2010 John P. Miller Page 10 of 49 Fundamentals of Post‐Tensioned Concrete Design for Buildings – Part One A SunCam online continuing education course Harped tendons can be used in pre-tensioned, precast concrete members, but this is
not common. In precast members, since the tendons are tensioned in the forms prior to
concrete placement, it is challenging to hold the pre-tensioned tendons in the proper
position within the formwork until the concrete is placed and has reached a sufficient
strength to release the tension. Harped tendons are easily accommodated in posttensioned construction since the tendons may be positioned within the formwork in the
field in virtually any arrangement because they are not tensioned until after the concrete
is placed and hardened. Harped tendons in post-tensioned construction are commonly
used, for example, in a transfer beam that carries the concentrated load of a
discontinuous column. The tendons in a transfer beam are sometimes stressed in
stages to balance a certain portion of the column dead load as the construction
progresses. Harped tendons are also commonly used in the repair or strengthening of
existing beams where the post tensioning forces are applied externally.
The following figure illustrates the equivalent set of loads due to the harped tendons
acting on the member shown above. The upward component of the tendon force P at
the point where its direction changes from sloped to horizontal, B, is a function of the
drape, a, and the distance from the end of the member, cL, and is equal to Pa/cL.
Since the harped tendons are symmetrical in this case, the reactions in the free body
diagram are equal. For unsymmetrical harped tendons, the upward component, B, and
the reactions are found using statics. Thus, the equivalent loads acting on the beam
consist of the axial force P, upward conc...
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