Basic Concepts

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BASIC CONCEPTS AND HISTORICAL BACKGROUND Reading: Nawy, E.G. (2000) Prestressed Concrete – A Fundamental Approach, 3rd Edition, Prentice Hall, Inc., Chapter 1, pp. 1-30. WHAT IS PRESTRESSING AND WHY USE IT? Prestressing is the imposition of permanent internal stresses into a structural member to neutralize the undesireable stresses caused by applied loading. The concept of prestressing is NOT new as the barrel below will attest.

Wooden Barrel Figure 1: Early Prestressing. We all should appreciate that concrete is very strong in compression, but rather weak in tension. Thus, a very useful thing to do is to prestress the concrete (i.e. provide initial compression) to overcome undesirable tension stresses in the concrete. This is the basis for prestressed concrete.

PRESTRESSED CONCRETE vs. REINFORCED CONCRETE The main difference between normal reinforced concrete and prestressed concrete is the fact that the steel reinforcement in the prestressed member is active. In a reinforced concrete member, the reinforcement is passive. The reinforced concrete member contains tensile reinforcement that must react to the applied loading (be it beam self-weight or superimposed live loading). In other words, the reinforcement contains no stress or no loading until loads are applied to the beam. The prestressed concrete member, on the other hand, has steel reinforcement that contains significant force that is present prior to any superimposed loading being applied to the member

2 in which it resides. The effect of prestressing, from an internal force point of view, is shown in Figure 2 below. w=0

w=0

C=0 a

e

T=0

T=P L/2

L/2 w=w 1

w=w 1

C=C1

C=P

a

a=e

T=P

T=T 1 C=C2

w=w

C=P

w=w

C=P e` e T=P

a T=T2

(a) Non-Prestressed

a

(b) Prestressed

Figure 2: Demonstration of Active versus Passive Internal Reinforcement The “plain” reinforced concrete member is shown above at the left. The prestressed member is shown at the right. One can see the behavioral difference between these members by looking at the location of the compression force within the cross-section When no loading is applied to the presetressed concrete beam, _________ ____________________________________________________________ ____________________________________________________________ In the case of the reinforced concrete member, there is not tension nor compression force until loading is applied. As the loading is applied to the member(s), the reinforced concrete member has a variable compression and tension force. The prestressed concrete member includes ____________________________ tension force. In BOTH cases, the distance separating the couple increases to ensure equilibrium of the member’s section.

3 ECONOMY OF PRESTRESSED CONCRETE There are several advantages to prestressed concrete members. These advantages include: 1. Prestressed concrete members can generally be shallower than the corresponding nonprestressed member. 2. A prestressed concrete member is generally stronger. The ______________________ of a prestressed member is greater than the non-prestressed member. 3. There is the possibility to _____________________________ in prestressed concrete members. 4. Fabrication of members in the plant allows superior quality control. This translates into higher reliability with respect to member strength. There is really a single disadvantages to prestressed concrete. Construction of prestressed members can be costly (especially post-tensioned members). When tendon ducts are required in post-tensioning applications, ensuring that adequate support is provided can become critical to ensure that the ducts remain where intended during concrete placement. Furthermore, stressing tendons in the field can be a very dangerous proposition. Therefore, the disadvantage arises due to added construction cost.

BRIEF HISTORY OF PRESTRESSED CONCRETE The following is a highly abbreviated history of prestressed concrete: 1872 - P.H. Jackson of California: Improved the load carrying capacity of individual cement blocks using bar with threaded ends.

Figure 3: Beam Constructed Using Individual Blocks. In Wisconsin, there was a product similar to this: DOX Plank.

4 1888 – C.W. Doering: Patented prestressing slabs with wires. This was the precursor to hollow core slab technology. Problems in the Beginning: While definitely useful, these early efforts at prestressing failed due to the inevitable loss of prestress with time. Consider the following comparison. Low Strength Steel: (18,000 psi allowable stress) With σ s = Esε s , the maximum strain that can be generated in the steel prestressed reinforcement is,

Creep and shrinkage strain (as we will see) can account for as much as 0.0006 in/in. Thus, whatever strain (and therefore, stress) was put in the steel initially, was soon lost as a result of the concrete creeping and shrinking with time. High Strength Steel: (150,000 “+” psi allowable stress)

If we assume 0.0006 in/in loss due to creep and shrinkage, there is still,

ε s = 0.005 − 0.0006 = 0.0044 in in in the reinforcement. With Es = 30e6 psi , this translates to 132,000 psi left after losses. 1928 – Freyssinet: First to use high strength steels to overcome prestress force losses. 1940 – Freyssinet: Introduced conical wedge system to anchor 12-wire tendons. This became known as the Freyssinet system.

5 1949 – First Use in United States: The Walnut Lane Bridge in Philadelphia (Dr. Zitomer is PROUD) was the first wide-spread use of linear prestressing. This is really the first bridge that foretold the future of prestressed concrete in transportation structures. (also found on the web-page)

Figure 4: Walnut Lane Bridge in Philadelphia, PA (still beautiful today).

PRESTRESSED CONCRETE IN BUILDINGS There are many, many applications for prestressed concrete in buildings. We will look at a few of the more common uses of prestressed concrete in building construction. Multi-Story Beam-Column Construction: This use is very common for parking structures.

Figure 5: Low-Rise Precast Beam and Column Structural System

6 Exterior Bearing Wall Construction: This system is very effective for multi-story constructions in regions of low-seismic risk.

Figure 6: Precast, Prestressed Bearing Wall Construction Wall units are cast in one- or multi-story segments. These wall elements bear on one another as the building rises. Precast, prestressed concrete wall cladding can also be hung on a reinforced concrete or structural steel skeleton.

7 Shearwall Systems: Precast wall elements make very efficient lateral load resisting elements when included in a building structure. Shearwalls are used to resist lateral shear that results from wind or earthquake forces.

Interior Shearwall Elements

Exterior Shearwall Elements

Figure 7: Shearwall Framing Systems. Lateral loading is transmitted through diaphragms to the interior and exterior shear walls. Lateral loading is then carried down to the foundation. The interior shear walls can also be tie together to form tubes. After the 1994 Northridge earthquake, these systems were called into question due to the large forces that must be transferred through the diaphragms to the interior and exterior shearwalls. Rigid (Moment Resisting) Frame System: Lateral loads are transferred through the diaphragm to the MRF’s.

Figure 8: Rigid Frame Structural System (Precast Beams and Columns).

8 Other Uses: There are a myriad of uses for precast, prestressed concrete. Some of the sections that are commonly constructed are shown below.

Figure 9: Precast, Prestressed Concrete Sections. Prestressed precast concrete is also used for, •

Storage Tanks

•

Sound Barriers

•

Transmission Line Poles

•

Ship Docks

•

Railroad Ties

CLASSIFICATION AND TYPES OF PRESTRESSING Prestressing is classified in a number of ways. The classifications often used and a brief description of each is contained in the following. External vs. Internal Prestressing: Internal prestressing contains tendons within the member. External prestressing has tendons lying outside the member. Linear vs. Circular Prestressing: Circular prestressing is often applied to round tanks or silos. Prestressing tendons in this case are most often used to “wrap” the structure (e.g. the

9 tank). Linear prestressing may include ____________________ or ________________________ tendons. Pretensioning vs. Post-Tensioning: Pretensioning describes tensioning of the prestressed reinforcement ___________ to the ______________________________. Posttensioning describes tensioning of the tendons _______________ the concrete has cured. End-Anchored vs. Non-End Anchored: End anchoring refers to the anchoring of the tendons via mechanical devices at the ends of the member. Non-End Anchored members may have anchoring points along the length (e.g. continuous beam). Bonded vs. Un-Bonded Tendons: A bonded tendon is “bonded” throughout their length to the surrounding concrete or tendon duct (in the case of a post-tensioned member). Pretensioned members are considered as “bonded”. One can create an unbonded member by “greasing” a tendon to inhibit bonding to the surrounding concrete. Precast, Cast-In-Place (CIP) and Composite Construction: Precast members are constructed off-site at a production facility. CIP members are made on-site. CIP members require assembly of formwork on-site, which can be labor intensive. However, there is little or no transportation and handling cost as with precast members. Composite members involve portions of a structural member that are precast and other portions that are CIP.

STAGES IN THE LIFE OF A PRESTRESSED MEMBER Prestressed concrete members undergo three stages in their life: (a) the initial stage, (b) the intermediate stage, and (c) the final stage. Each of these stages will be discussed in some detail in the following.

10 Initial Stage: During this stage in the prestressed members life, the member is subjected to _______________ __________________ and ___________________________, but no external loading. There are several stages within the initial stage: 1. Before Prestressing: In a pretensioned member, the concrete has not cured. 2. During Prestressing: During this stage, the tendons are undergoing very large stresses and strains. Bearing areas and tendon hold-downs within the member are undergoing very large stresses as well. 3. Transfer of Prestress: In a pretension member, the transfer of the prestressing force occurs very rapidly (i.e. the tendon is cut). The member self-weight comes into “play” in the initial stage because as the prestressing force is transferred, the member will often “bow” upward or camber will be initiated into the member. Members can also be __________________________ at different stages in their lifespan. The stress states during all these stages must then be studied. Intermediate Stage: This stage in the life of the member includes ________________________________ and _______________________ of the member into the structure. ___________________________ ______________________________________________________________________________ ____________________________________________________________________________. Placement of superimposed dead loads during construction is also critical, since the tendon arrangement within the member often assumes a specific SDL position and/or pattern. Final Stage: The anticipated (and sometimes un-anticipated) service loading arrives on the structure during this stage. The design of the member will often require that behavior be studied at a cracking stage as well with service and ultimate (limit) loading applied. There are several loading levels to consider at this stage. 1. Sustained Loading: The designer should always be aware of camber and deflection in the member under sustained loading since creep of concrete is very important. ___________ ________________________________________________________________________ ________________________________________________________________________

11 _______________________________________________________________________. 2. Service (working) Load: This is the every day loading that the prestressed element or member will support. Stresses are the limiting design consideration at this level of loading. 3. Cracking Load: Investigation of the loading level to cause cracking in the member is important (e.g. tanks, pipes, or elements subjected to corrosion). 4. Ultimate Load: A structure designed for only service loading may not possess sufficient capacity for the occasional overload condition. Therefore, prestressed members will often be checked at the ultimate (strength limit state) condition. The designer will essentially ensure that the there is sufficient separation between the “cracking” load level and the “ultimate” load level. This will ensure that there is _______________________ before impending collapse.

ANALYSIS OF PRESTRESSED BEAMS The analysis of prestressed concrete beams can seem very complicated at first. On should always remember that what we will be doing is statics and mechanics of materials. Therefore, the design of prestressed concrete members will be tedious. Basic Concepts and Review We will have to address several basic issues before we use the three main techniques for analysis of prestressed members. Let’s take a look at stress states within members. A. Simply-Supported Beam with Concentric Prestressing Force

The stress state due to the prestressing force is uniform over the entire cross-section. Furthermore, the entire cross-section is in compression. The stress acting on the crosssection is therefore,

f =−

P Ac

12 B. Simply-Supported Beam with Concentric Prestressing Force and Uniformly Distributed Loading

The state of stress in this case varies linearly over the cross-section. The stresses at the top and bottom fibers can be easily computed as,

C. Simply Supported Beam with Eccentric Prestressing Force

In this case, the prestressing force itself causes a linearly varying state of stress over the height of the cross-section. The state of stress, in this case, is, ft =−

P Pect P + =− Ac Ig Ag

 ect  1 − r 2 

fb = −

P Pecb P − =− Ac Ig Ag

 ecb  1 + r 2 

The simple replacement Ac = Ab has been made assuming the gross concrete area is used. D. Simply Supported Beam with Eccentric Prestressing Force and Uniformly Distributed Loading

13 This is the most complicated stress state thus far. Using mechanics of materials, the state of stress can be simply computed as, ft =−

P Pect Mct P + − =− Ac Ig Ig Ag

 ect  Mct 1 − r 2  − I g

fb = −

P Pecb Mcb P − + =− Ac Ig Ig Ag

 ecb  Mcb 1 + r 2  + I g

PRESTRESSING TENDON PROFILES Having a constant depth tendon profile is highly economical, but not very efficient at resisting applied loads (especially in a beam flexure sense). Therefore, tendon arrangements are most often varied within the cross-section. Two examples of tendon profile variation are shown below. Harped Tendon (Single Hold-Down): The harped tendon is very economical because one is not trying to adhere to a “difficult” tendon arrangement (i.e. one that varies throughout the length of the span). In this case, the prescaster merely pulls on the tendon at the ends and “pushes” it down in the center.

The eccentricity varies the ends and center of the beam above. There can also be multiple hold-downs in these members. Draped Tendon (Parabolic Profile) The draped tendon is a bit more difficult to fabricate.

14 INITIAL vs. EFFECTIVE PRESTRESSING FORCE Our initial discussions implied that the prestressing force within a member is NOT constant over its service life. In fact, the largest prestressing force that one will have in a member is on the day the prestress is transferred into the member. The losses are what changes the stress in the tendons. We will define the following: Initial Prestress: __________________________________________________________ Effective Prestress: _______________________________________________________ The initial prestress can be assumed at the time of force transfer for pretensioned members and after the losses due to tendon “seating” and “friction” in post-tensioned members. The effective prestress can be assumed present when the member supports, •

Dead Loading

•

Super-imposed Dead Loading

•

Live Loading

A residual stress factor is often defined to help simplify discussion,

ANALYSIS METHODS Analysis of prestressed concrete members can ALWAYS be done using mechanics of materials procedures (in fact, this is the first method – which we will not discuss). There are two additional methods that give some “physical feel” to the prestressed concrete problem. These methods are the C-Line Method and the Method of Load Balancing. We will look at each in this section of the notes. It should be noted that ALL methods that we will use for analysis assume linearly elastic material behavior. Analysis Using the C-Line Method This technique is based upon the line of pressure or thrust force within the cross-section. Recall our earlier discussions, which included visualization of the compression force within the cross-

15 section rising up through the member as loading is applied. This is the fundamental issue for the C-Line method of analysis. Using this method, the prestressing force is assumed to be an externally applied loading and the tendon force is assumed to be constant. The FBD of the left “chunk” of a prestressed member is shown below.

At the cut made, the bending moment supported by the cross-section is computed as,

The eccentricity of the compression force with respect to the centroid of the concrete crosssection is computed as,

Recognizing that C = T = P gives, ft =−

C Ce′ct P − =− Ac Ig Ag

 e′ct  1 + r 2 

fb = −

C Ce′cb P + =− Ac Ig Ag

 e′cb  1 − r 2 

Analysis Using Load Balancing This method adds a high degree of physical “feel” to the prestressing problem. This allows one to handle some rather complicated structures. T.Y. Lin first discussed the technique in 1963. The basic philosophy of the method is to counteract the forces applied to the structure with

16 forces created by prestressing force. The method has distinct advantages when one is analyzing statically indeterminate structures and computing deflections. It has also been used to analyze highly complex structures such as that shown below,

Figure 10: Arizona State Fair Coliseum with Load Balancing Analytical Model. 380 foot diameter structure with 2 ½” thick lightweight concrete “waffle” slab roof. (taken from Lin and Burns (1981) Design of Prestressed Concrete, 3rd Edition, John Wiley & Sons, Inc. We will look at two basic tendon arrangements in this section: the single hold-down harped tendon and the parabolic tendon. Harped Tendon (single hold-down) Analysis: The basic FBD of the entire beam (only prestressing forces and applied loading considered) is shown below.

F P P

P

P P

A FBD of the central “chunk” of tendon and the moment diagrams due to the prestressing force and applied loading are shown below. M (x) F

FL 4

MF x

MPS Mps (x)

2Psin 2

P 2PLsin2 4

2

P

2

17 Parabolic Tendon Analysis: The beam FBD (exclusive of support reactions) is shown below.

P

P

P

ps

P

The moment diagrams for the prestressing force and the applied uniformly distributed loading are shown below. 2

MF (x)

wL

8

MF x

Mps Mps (x)

Ph

An equivalent uniform loading (resulting from the prestress) can be defined as,

Definition of Net Moment: Once the moment resulting from the prestress and applied loading is defined, one can define a net moment acting on the cross-section. Therefore,

After the net moment is computed, we can make several judgments regarding beam behavior based upon its sign.

18 If M F > M ps ; the beam smiles If M F < M ps ; the beam frowns If M F = M ps ; the beam is flat. These can then be used to define the appropriate signs in the mechanics of materials relationships to compute stresses using the net moment. ft =−

P′ M net ct ± Ac Ig

fb = −

P′ M net cb ± Ac Ig

where P′ is the ___________________________________ of the prestressing force at the section under consideration.

It should be noted that in most examples, the section considered will lie at the beam centerline (for simplicity). For the single hold-down, harped tendon, the horizontal component of the prestressing force will not be the prestressing force (i.e. additional computation must be made). In the case of the parabolic tendon, the horizontal component of the prestressing force (at least at the beam centerline) is the prestressing force. One should be very careful to use the horizontal component of the prestressing force.

19 EXAMPLE 1 – COMPUTATION OF FIBER STRESSES IN PRESTRESSED CONCRETE BEAM MEMBERS

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