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Chapter 1 tells the history of post-tensioned concrete as only Ken Bondy can tell it. The hyperstatic moment diagram is illustrated below for the two- span 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. 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 post- tensioned 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.
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. As we have seen previously, the beam can be analyzed with equivalent loads due to the tendon force.
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. 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. 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. 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. Pre-stress Losses 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 in 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 ACI , pre-stress losses were estimated using lump sum values. For example, the loss due to concrete strain, concrete creep, concrete shrinkage, and tendon relaxation in normal weight concrete was assumed to be 25, psi for post- tensioned concrete. This did not include friction and tendon seating losses. This was a generalized method and it was subsequently discovered that this method could not cover all situations adequately.
Since the ACI , each type of pre-stress loss must be calculated separately. The effective pre-stress force, that is the force in the tendon after all losses, is given on the design documents and it is customary for the tendon supplier to calculate all pre-stress losses so that the number of tendons can be www.
Even though most design offices do not normally calculate pre-stress losses, it is informative and relevant to understand how losses are calculated. Therefore, each type of pre-stress loss is discussed in more detail below.
The anchorage hardware tends to deform slightly, which allows the tendon to relax slightly. The friction wedges deform slightly, allowing the tendon to slip slightly before the wires are firmly gripped. Minimizing the wedge seating loss is a function of the skill of the operator. The average slippage for wedge type anchors is approximately 0. There are various types of anchorage devices and methods, so the calculation of seating losses is dependent on the particular system used.
Friction also occurs at the anchoring hardware where the tendon passes through. This is small, however, in comparison to the friction between the tendon and the sheathing or duct throughout its length. This friction can be thought of as two parts; the length effect and the curvature effect. The length effect is the amount of friction that would occur in a straight tendon - that is the amount of friction between the tendon and its surrounding material. In reality, a tendon cannot be perfectly straight and so there will be slight "wobbles" throughout its length.
This so called wobble effect is rather small compared to the curvature effect. The amount of loss due to the wobble effect depends mainly on the coefficient of friction between the contact materials, the amount of care and accuracy used in physically laying out and securing the tendon against displacement, and the length of the tendon.
The loss in the pre-stressing tendons due to the curvature effect is a result of the friction between the tendon and its surrounding material as it passes though an intentional curve, such as drape, or a change in direction, such as a harped tendon. The amount of loss due to the curvature effect depends on the coefficient of friction between the contact materials, the length of the tendon, and the pressure exerted by the tendon on its surrounding material as it passes through a change in direction.
If this compressive force were removed, the member would return to its original length. Although ACI does not specifically give procedures or requirements for calculating losses due to elastic strain, there are references available that provide some guidance.
In general, the elastic strain shortening is a simplified computation involving the average net compressive stress in the concrete due to pre-stressing and the moduli of elasticity of the pre-stressing steel and the concrete at the time of stressing. The creep rate diminishes over time. Although ACI does not specifically give procedures or requirements for calculating losses due to creep, there are references available that provide some guidance.
In general, the creep shortening is a simplified computation involving the average net compressive stress in the concrete due to pre-stressing and the moduli of elasticity of the pre-stressing steel and the day concrete strength. For post-tensioned members with unbonded tendons, for example, creep strain amounts to approximately 1.
The amount of water used in a batch of concrete to make it workable far exceeds the amount of water necessary for the chemical reaction of hydration. Therefore, only a small portion of the water in a typical concrete mix is consumed in the chemical reaction and most of the water needs to evaporate from the hardened concrete. When the excess mix water evaporates from a particular concrete member, it loses volume and therefore tends to shrink. Reinforcing steel and surrounding construction can minimize concrete shrinkage to some extent, but nonetheless shrinkage stresses are developed.
Factors that influence concrete shrinkage include the volume to surface ratio of the member, the timing of the application of pre-stressing force after concrete curing, and the relative humidity surrounding the member. The concrete member shortens due to the above three sources — elastic strain, creep, and shrinkage. Thus, the total tendon relaxation is a summation of these three concrete shortening sources.
There may also be some relaxation in the pre-stressing steel over time, similar to concrete creep. This steel relaxation is a function of the type of steel used and on the ratio of actual steel stress to the specified steel stress. As mentioned earlier, most design offices only show the required effective pre-stress force and the location of the center of gravity of the tendons on the construction documents.
Remember that the effective pre-stress force is the force in the tendons after all losses have been accounted for. Calculating pre-stress losses for the design office can be very tedious and may not be exact, and therefore it is customary for the tendon supplier to calculate all pre-stress losses based on their experience and the specific tendon layout. Then, the number of tendons are determined to satisfy the given effective pre-stress force.
We will be focusing on Chapter 18, Pre-stressed Concrete. The requirements in Chapter 18 have changed very little over the last several editions of the ACI ACI places limits on the allowable extreme fiber tension stress at service loads according to the classification of a structure. Class U members are assumed to behave as uncracked sections and therefore gross section properties may be used in service load analysis and deflection calculations.
Class C members are assumed to be cracked and therefore cracked section properties must be used in service load analysis, and deflection calculations must be based on an effective moment of inertia or on a bilinear moment-deflection relationship.
Class T members are assumed to be in a transition state between cracked and uncracked, and the Code specifies that gross section properties may be used for analysis at service loads, but deflection calculations must be based on an effective moment of inertia or on a bilinear moment-deflection relationship.
The first is a check of the concrete tension and compression stresses immediately after transfer of pre-stress. The concrete stresses at this stage are caused by the pre-stress force after all short term losses, not including long term losses such as concrete creep and shrinkage, and due to the dead load of the member. These limits are placed on the design to ensure that no significant cracks occur at the very beginning of the life of the structure. The second serviceability check is a check of the concrete tension and compression stresses at sustained service loads sustained live load, dead load, superimposed dead load, and pre-stress and a check at total service loads live load, dead load, www.
These checks are to preclude excessive creep deflection and to keep stresses low enough to improve long term behavior. Note that the specified day concrete compressive strength is used for these stress checks. Also note that the above stress checks only address serviceability. Permissible stresses do not ensure adequate structural strength. Permissible stresses for other type of pre-stressing steel, including deformed bars, vary slightly and can be found in the ACI commentary.
Since the scope of this article is limited to unbonded post-tensioned systems, and we will only be considering Grade low-relaxation steel, the permissible stresses for other steels are not given here. Minimum Bonded Reinforcing All flexural members with unbonded tendons require some amount of bonded reinforcing.
For beams and one-way slabs, this bonded reinforcing is required www. For two-way slabs, the requirement for this bonded reinforcing depends on the services load stresses. This bonded reinforcing is intended to limit crack width and spacing in case the concrete tensile stress exceeds the concrete's tensile capacity at service loads.
ACI requires that this bonded reinforcement be uniformly distributed and located as close as possible to the tension face. The bonded reinforcing required in negative moment regions at columns shall be distributed between lines that are 1.
ACI also requires a minimum length of bonded reinforcing in positive and negative moment regions. It should be evident that the minimum lengths and areas of bonded reinforcement required by the ACI may be exceeded by structural demand, as we will see later on.
The minimum length of bonded reinforcing is shown below. Solution: Let's begin by finding lG , the area of that part of the cross section between the flexural tension face and the center of gravity of the cross-section for the top and bottom of the beam.
The neutral axis has already been determined to be 10 inches from the top of the beam. Use 3 6 Bottom The ACI states that this minimum bonded reinforcement shall be uniformly distributed as close as close as possible to the tension face. Therefore, the bottom www. This seems like an appropriate spacing, but in all likelihood, flexural demands will require a larger area of steel and possibly more than three bars.
The minimum bonded reinforcing in the top of the beam will be placed at both ends, where negative bending moments will occur due to support fixity. The 12 5s result in a spacing of approximately 10 inches on center, which seems to be better than almost 18 inches on center had 12 6s been selected.
However, either spacing would probably meet the intent of the ACI It is supported by 16" wide concrete beams that are at 22'-0" on center. Find: Determine the minimum area and length of bonded reinforcing required.
Solution: Since the concrete slab has a symmetric cross section, the lG for the and bottom will be the same and equal to 12x5. The minimum length of the bottom bars is one-third the clear span, or The results are illustrated below. It is supported by 16" x 16" columns that are at 20'-0" on center both ways. The extreme fiber tension at service loads in positive moment areas in the bottom of the slab , after all pre-stress losses, is psi.
Find: Determine the minimum area, length, and distribution of bonded reinforcing required. Solution: Because the bays are equal in both directions, we need only investigate one direction and apply the results to both. From chapter 13 of ACI , the middle strip width is defined as being bounded by two column strips, which is www.
Therefore, the total area of reinforcing is 15 x 0. Recall that the minimum length of the bottom bonded reinforcing is one-third the clear span, or So, we will use 14 5 bottom bars, 6'-3" long, equally spaced in a foot wide middle strip, both ways. For columns at the edge of the floor plate, only half of the larger bay is used. The width over which this area of bonded reinforcing is distributed is 1. The minimum length extended beyond the column face is one-sixth the clear span, or So, we will use 5 5 top bars, 7'-7" long, equally spaced in a inch wide strip centered on the column, both ways.
The design moment strength may be computed using the same methodology used for non pre-stressed members. That is, a force couple is generated between a simplified rectangular compression block and the equal and opposite tensile force generated in the reinforcing steel, which is in our case pre-stressing steel.
For purposes of this course, we will narrow our focus to post-tensioned members with unbonded tendons, and we will use the approximate values for the nominal stress in the pre-stressing steel, fps, instead of using strain compatibility. For more accurate determinations of the nominal stress in the pre-stressing steel, and for flexural members with a high percentage of bonded reinforcement, and for flexural members with pre- stressing steel located in the compression zone, strain compatibility should be used.
Assuming the bonded reinforcement is at the same depth as the pre-stressing steel in the beam a slightly conservative assumption since the center of gravity of the mild reinforcing is often closer to the face of the beam, i. In addition to the post-tensioning tendons, the beam has 3 10 bars in the bottom with a yield strength of 60 ksi.
ACI offers a simplified approach to computing the shear capacity of a pre-stressed member and this is what will be used here. Otherwise, the minimum amount of shear reinforcement for pre-stressed www. The design of concrete members for torsional forces is outside the scope of Part One of this course. Likewise, the investigation of the shear strength of two-way slabs is not covered here in Part One but is covered in Part Two.
This is the maximum nominal shear strength that can be provided by the concrete alone. Therefore, 5 18 " o. Most designers would space the stirrups in convenient groups of incrementally larger spacing away from the support until they are no longer required by structural demand or minimum reinforcing requirements. However, it is common practice to use stirrups throughout the entire span to have something to which to tie the tendon bundle support bars. With a good understanding of the material in Part One of this course, you should know something about the historical background of post-tensioned concrete and the difference between post-tensioned members and pre-tensioned members.
You should also understand the load balancing concept, hyperstatic moments, pre-stress losses, and the basic requirements of ACI Building Code Requirements for Structural Concrete. We also covered nominal flexure and shear capacities of post-tensioned members, including a few examples. In Part Two, we will use the material learned in Part One to work design examples of two common structural systems used in buildings and parking structures, namely a one-way continuous slab and a continuous beam.
Then in Part Three, the more advanced topic of two-way slabs is covered, including the Equivalent Frame Method, punching shear, and moment transfer at columns, with several examples. Design of Prestressed Concrete Structures, T. Lin and Ned H.
Miller Page 49 of Post tensioning in Building Structures By chintal patel. Louter , and Frederic Veer. Download PDF.
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