Prestressed Concrete Design Lecture Notes PDF

PRESTRESSED CONCRETE DESIGN CHAPTER 1. INTRODUCTION Concrete is strong in compression, but weak in tension. Its tensile strength varies from 8-14 percent of its compressive strength. Due to such a low tensile capacity, flexural cracks develop at early stages of loading. In order to prevent these...

PRESTRESSED CONCRETE DESIGN CHAPTER 1. INTRODUCTION Concrete is strong in compression, but weak in tension. Its tensile strength varies from 8-14 percent of its compressive strength. Due to such a low tensile capacity, flexural cracks develop at early stages of loading. In order to prevent these cracks from developing, a concentric or eccentric force is imposed in the longitudinal direction of the structural element. Tensile stresses at the critical midspan and support sections are considerably reduced. Therefore, almost the full capacity of the concrete in compression can be efficiently utilized across the entire depth of the concrete sections. Pre-stressing – is the preloading of a structure before the application of service loads Prestressing Force - a compressive force that prestresses the secions along the span of the structural element prior to the application of the loads Types of Prestressing  Linear Prestressing - the prestressing force is applied longitudinally along or parallel to the axis of the member - analogous to series of blocks that forms a beam  Circular Prestressing - used in liquid containment tanks, pipes, and pressure reactor vessels - the circumferential hoop neutralizes the tensile stresses at the outer fibers of the curvilinear surface caused by the internal contained pressure - done by applying tensile force on wires wound in circles in circular structures CES5. PRESTRESSED CONCRTE DESIGN AJDM Comparison with Reinforced Concrete Reinforced Concrete (RC) Prestressed Concrete (PSC) Tensile strength is negligible and disregarded. Permanent stresses are created before imposed loads are applied in order to considerably reduce the net tensile stresses. Tensile forces resulting from the bending If the flexural tensile strength of the concrete is moments are resisted by the bond created in the exceeded, the prestressed member starts to act reinforcement. like a reinforced concrete element. Crack widths are roughly proportional to tensile High stresses are not accompanied by wide reinforcement stress. cracks since much of the strain is applied to the steel before it is anchored to the concrete. High stress in concrete would inevitably produce By prestraining the high strength reinforcement, large rotations and deflections. large rotations and deflection is avoided. Also, uncracked members are stiffer. Economics of Prestressed Concrete The depth of a prestressed concrete is usually 65-80% of its equivalent RC member. It requires less concrete and about 20-35% of the amount of reinforcement. However, this saving in material weight is balanced by higher cost of the higher quality materials needed on prestressing. If a large number of precast units are manufactured, the difference between the two is not usually very large. The indirect long-term savings are substantial because less maintenance is needed, a longer working life is possible, and lighter foundations are achieved due to the smaller cumulative weight of the superstructure. CES5. PRESTRESSED CONCRTE DESIGN AJDM Basic Concepts of Prestressing Consider a simply supported rectangular beam subjected to a concentric prestressing force P. The compressive stress on the beam cross section is uniform and has an intensity P f  Ac Where Ac = bh b = width h = total depth Note: ( - ) compression ( + ) tension If external transverse loads are applied to the beam causing maximum moment M at midspan, the resulting stress becomes: P Mc ft   A Ig P Mc fb    A Ig Where ft = stress at top fiber fb = stress at top fiber Ig = gross moment of inertia c = h/2 To induce tensile stresses at the top fibers due to prestressing, the presressing tendon is placed eccentrically below the neutral axis at midspan. If the tendon is placed at at eccentricity, e, from the center of gravity of the concrete (cgc), a moment, Pe, is created and the ensuing stresses at midspan becomes: CES5. PRESTRESSED CONCRTE DESIGN AJDM  P Pec Mc ft    A Ig Ig P Pec Mc fb     A Ig Ig Since the support section of a simply supported beam carries no moment from the external transverse load, high tensile fiber stresses are caused by eccentric prestressing force. To limit such stresses, the eccentricity of the prestressing tendon profile is - made less at the support section - eliminated altogether - a negative eccentricity above the cgc line is used Equivalent Loads The effect of a change in the vertical alignment of a prestressing tendon is to produce a transverse vertical force on the concrete member. That force, together with the prestressing forces acting at the ends of the member through the tendon anchorage, may be looked upon as a system of external forces in studying the effect of prestressing. MEMBER EQUIVALENT LOAD MOMENT CES5. PRESTRESSED CONCRTE DESIGN AJDM  MEMBER EQUIVALENT LOAD MOMENT For any arrangement of applied loads, a tendon profile can be selected such that the equivalent loads acting on the beam from the tendon are just equal and opposite to the applied loads. Note that these equivalent loads would produce a deflection called CAMBER that counters that of the imposed loads. It is also worth mentioning that the equivalent loads and moments produced by the prestressing tendon are self equilibrating. Prestressing Methods A. Pre-tensioning Done at the fabrication plant for production of precast members Tendons are tensioned first before concrete is placed The concrete is cast around the stressed tendon As the fresh concrete hardens, it bonds to the steel When the concrete has reached the required strength, the jacking force is released and the force is transferred by bond from steel to concrete B. Post-tensioning Done at the construction site for the construction of cast-in-place members CES5. PRESTRESSED CONCRTE DESIGN AJDM  Hollow conduits containing the unstressed tendons are placed in the beam forms before pouring the concrete The tendons are tensioned after the concrete has hardened and achieved sufficient strength Changes in Prestress Force The magnitude of the prestressing force in a concrete member is not constant but changes during the life of the member. These changes are due to: - instantaneous losses - time-dependent losses - losses as a function of the superimposed loading Jacking Force – force applied during the jacking operation denoted by Pj Instantaneous Losses in Prestressed Concrete a. anchorage slip – occurs at the moment of transfer of prestress force from the jack to the anchorage fittings that grip the tendon b. elastic shortening – occurs in the concrete as the prestress force is transferred to it (always occurs in pretensioning) c. friction losses – occurs between the steel and the tendon conduit (applicable only to post-tensioned members) The effect of the instantaneous losses is a reduction in the jacking force, Pj, to a lower value, Pi, defined as the initial prestress force. With the passage of time, under sustained compressive stress, the steel stress is further reduced due to time dependent losses such as: - concrete shrinkage - concrete creep Creep – property of materials by which they continue to deform over considerable lengths of time at constant stress or loads The result of all time-dependent effects is a reduction in the initial prestress force termed effective prestress force, Pe CES5. PRESTRESSED CONCRTE DESIGN AJDM The sum of all losses, immediate and time-dependent, maybe of the order of 20 to 35% of the original jacking force Note: All losses must be accounted for in the design of prestressed concrete elements LOADS Loads that act on structures are divided into three broad categories: a.) Dead Loads, DL b.) Live Loads, LL Fixed in location c.) Environmental Loads, EL Constant in magnitude - Snow loads - Wind pressure and suction Concrete density: - Earthquake loads Lightweight concrete - Soil pressure (acting on structure 90 to 120 pcf (14 to 19 KN/m3) subsurface) Normal concrete - Rainwater ponding on flat surfaces 145 pcf (23 KN/m3) - Forces from temperature differential Service Load – sum of calculated DL, LL, EL – estimate of the maximum load that can be expected to act during the service life of the structure Factored Load – failure load that a structure must be capable of resisting to ensure an adequate margin of safety against collapse – load factors, larger than unity, are applied to the calculated DL, LL, EL Factored Load Combinations 1) 1.4D 2) 1.2D + 1.6L + 0.5(Lr or S or R) 3) 1.2D + 1.6(Lr or S or R) + (0.8W or 0.5L) 4) 1.2D + 1.6W + 0.5L + 0.5(Lr or S or R) 5) 1.2D + 1.0E + 0.5L + 0.2S 6) 0.9D + (1.6W or 1.0E) Where: D = Dead Load L = Live Load Lr = Roof Live Load S = Snow Load R = Rain Load W = Wind Load E = Earthquake Load References Nilson, A. H. (1987). Design of Prestressed Concrete. South Tower, Singapore: John Wiley & Sons Singapore Pte. Ltd. Nawy, E.G. (2010). Prestressed Concrete: A Fundamental Approach (5th ed.). Upper Saddle River, N.J. :Prentice Hall. CES5. PRESTRESSED CONCRTE DESIGN AJDM  PRESTRESSED CONCRETE DESIGN CHAPTER 2. MATERIALS IN PRESTRESSED CONCRETE What is Prestressed Concrete? “Prestressed concrete is a structural concrete in which internal stresses has been introduced to reduce potential stresses in concrete resulting from loads” - ACI Pre-stressed concrete is composed of: a) High Strength Concrete b) Tendon/ Strand c) Non-prestressed reinforcement (ordinary bar reinforcement) Concrete -Concrete used in PSC members is of higher strength than that used for RC Steel Tendons -High strength steel used for prestressing Designers must consider strength, differences in ductility, lack of a well-defined yield point, etc. Ordinary Bar Reinforcement -Same type used for ordinary RC structures -Used for web reinforcement, supplemental longitudinal reinforcement The lack of success of most early attempts to PSC was the failure to employ steel at a sufficiently high stress and strain. The time-dependent length changes permitted by shrinkage and creep of the concrete completely relieved the steel of stress. The importance of high initial strain and the corresponding high initial stress in the steel is shown by a simple example in Fig. 2.1 Consider a short concrete member that is to be axially prestressed using a steel tendon. CES5. PRESTRESSED CONCRTE DESIGN AJDM Importance of High Strength Steel In the unstressed state, Length of concrete is lc Unstressed length of steel is ls After tensioning of the steel and transfer of force to the concrete through end anchorages, Length of concrete shortened to lc’ Length of stretched steel is ls’ lc  l s ' ' With the passage of time the concrete experiences a shrinkage strain εsh and a creep strain εcu The total length change in the member is lc   sh   cu lc which may exceed the stretch in the steel that produced the initial stress. Thus, complete loss of prestress force will result. Suppose that the member is prestressed using ordinary reinforcing steel at an initial stress 30 ksi. The modulus of elasticity for all steel is Es = 29000 ksi. The initial strain in the steel is. f si 30  si    1.03  10  3 Es 29 ,000 and the total elongation is  s l s  1.03  10 3 l s A conservative steel estimate of the sum of shrinkage and creep strain in the concrete is  sh   cu  0.90  10 3 The corresponding length change is  lc   sh   cu lc  0.90  10 3 lc The effective steel stress remaining after time-dependent effects would be    f se  1.03  0.90   10 3  29  10 3  4 ksi  Alternatively, suppose that the prestress were applied using high strength at an initial stress of 150 ksi. In this case, the initial strain would be CES5. PRESTRESSED CONCRTE DESIGN AJDM  150  si   5.17  10  3 29 ,000 and the total elongation  s l s  5.17  10 3 l s The length change resulting from the shrinkage and creep effects would be the same as before  sh   cu lc  0.90  10 3 lc and the effective steel stress after losses due to shrinkage and creep would be     f se  5.17  0.90   10 3  29  10 3  124 ksi The loss is about 17% of the initial steel stress in this case compared with 87% loss when mild steel was used. Types of Prestressing Steel There are three common forms in which steel is used for prestressed concrete tendons: a) Cold-drawn round wires b) Stranded cable c) Alloy steel bars Tendons - normally composed of groups of wires - the number of wires in each group depends on the particular system used and the magnitude of prestress force required A. Round Wires - The individual wires are manufactured by hot-rolling steel billets into round rods - After cooling, the rods are passed through dies to reduce their diameter to the required size. - Available in Grades 235 (minimum ultimate strength 235,000 psi) to Grade 250 (minimum ultimate strength 250,000 psi) B. Stranded Cable - Fabricated with six wires wound tightly around a seventh of slightly larger diameter - The pitch of the spiral winding is between 12 to 16 times the nominal diameter of the strand - The same type of cold-drawn stress-relieved wire is used in making stranded cable as is used for individual prestressing wires - Two grades are manufactured: CES5. PRESTRESSED CONCRTE DESIGN AJDM  Grade 250 – minimum ultimate strength of 250,000 psi Grade 270 – minimum ultimate strength of 270,000 psi C. Alloy Steel Bars - The high strength is obtained by introducing certain alloying elements, mainly manganese, silicon and chromium during the manufacture of the steel - In addition, cold work is done in making the bars, further increasing the strength - Available in diameters ranging from 5/8 in to 1 3/8 in , and in Grade 145 (minimum ultimate strength 145,000 psi) and Grade 160 (minimum ultimate strength 160,000 psi) Non-prestressed Reinforcement - Non-prestressed steel consists of bars, wires, and welded wire fabric - Used for web reinforcement for diagonal tensile stress - Used to provide longitudinal bar steel to control shrinkage and temperature cracking - Used to increase the flexural strength of prestressed beams using supplementary longitudinal reinforcement - Available in Grades 40, 50, 60 (minimum yield strengths of 40,000 psi, 50,000 psi, 60,000 psi, respectively) Figure 2.15. Various forms of ASTM-approved deformed bars. (Nawy, Prestressed Concrete: A Fundamental Approach, 5th ed.) Stress-Strain Properties of Steel The figure below shows a stress-strain diagram for a material subjected to a tension test. CES5. PRESTRESSED CONCRTE DESIGN AJDM Proportional Limit - Up to this point, stress is proportional to strain - The point where the material leaves the elastic region and moves into the inelastic (plastic) region of the curve Elastic Limit - stress beyond which the material will not return to its original shape when unloaded but will retain a permanent deformation called permanent set Yield Point - the point at which there is an appreciable elongation or yielding of the material without any corresponding increase of load Yield Stress - closely associated with the yield point - determined by an offset of 0.20% for materials that do not have a well-defined yield point Ultimate Strength -maximum stress the material can sustain corresponds to the highest point on the stress-strain curve Rupture Strength - stress at failure Ductility - the ability of a material to withstand plastic deformation without rupture -may also be thought of in terms of bendability and crushability Figure 2.2. Comparative stress-strain curves for reinforcing steel and prestressing steel (Nilson, Design of Prestressed Concrete, 2nd ed.) CES5. PRESTRESSED CONCRTE DESIGN AJDM Obsevations: PRESTRESSING STEEL ORDINARY REINFORCING STEEL ROUND WIRE ALLOY BARS GRADE 60 BARS GRADE 40 BARS PROPORTIONAL LIMIT YIELD POINT FAILURE STRESS FAILURE STRAIN 1/3 THAT OF ORDINARY RC STEEL DUCTILITY Figure 2.3. Typical stress-strain curves for non-tensioned reinforcing bars (Nilson, Design of Prestressed Concrete, 2nd ed.) Observations on prestressing steel VS ordinary steel - Elastic modulus for all steel is the same, Es = 29,000 ksi - Although Grades 40 and 60 have a well-defined yield point, the same is not true for higher strength steel - For higher strength steel, equivalent yield point is defined as the stress at which the total strain is 0.35 percent - Ductility is significantly less for the higher grades - Elastic modulus Es ≈ 29,000 ksi (round wires) Es ≈ 27,000 ksi (stranded cable) Es ≈ 27,000 ksi (alloy bars) - Yield stress is taken as the stress producing 1% extension (round wires, stranded cable) 0.7% extension (alloy bars) CES5. PRESTRESSED CONCRTE DESIGN AJDM Steel Relaxation Steel relaxation in prestressing steel is the loss in prestress when the wires or strands are subjected to essentially constant strain. It is identical to creep, except that creep is a change in strain whereas steel relaxation is a loss in steel stress. The loss of stress due to relaxation in stress-relieved wires and strands can be evaluated from the expression fp log t  f pi   1  0.55  f pi 10  f py   Where fp final stress after t hours fpi initial stress fpy yield stress defined as the stress at which the total strain is 0.35% log t to the base 10 f pi  0.55 f py The expression for stress relaxation in pretensioned members at any time tn is given by: fp  log t n  log t r  f pi   1    0.55  f pi  10  f py   Where tr time of release Concrete Concrete used for prestressed construction is characterized by a higher strength than that used for ordinary reinforced concrete. Compressive strength between 4,000 and 8,000 psi (28 and 55 Mpa) is commonly specified for PSC members, although strengths as high as 12,000 psi (83 Mpa) have been used. Importance of Using High Strength Concrete PSC members are usually subjected to higher forces, and an increase in concrete quality generally leads to more economical results (dimensions of member cross-sections can be reduced to the minimum) Loss of prestress force resulting from elastic shortening of concrete and creep is reduced Higher bond strength results in a reduction in the development length required to transfer prestress force from the cables to the concrete Concrete of higher compressive strength also has a higher tensile strength, so that formation of flexural and diagonal tension cracks is delayed CES5. PRESTRESSED CONCRTE DESIGN AJDM Properties of Hardened Concrete A. Compressive Strength Concrete is useful mainly in compression and is often subject to a state of uniaxial stress. The figure below shows a typical stress-strain curve obtained from tests using cylindrical concrete specimens loaded in uniaxial compression over several minutes. Figure 2.2. Typical stress-strain curve of concrete (Nawy, Prestressed Concrete: A Fundamental Approach, 5th ed.) Observations: -The first portion of the curve to about 40% of the ultimate strength can be considered linear -At 70% of the ultimate strength, the material losses a large portion of its stiffness thereby increasing the curvilinearity of the diagram -At ultimate load, cracks parallel to the direction of loading become distinctively visible, and most concrete cylinders suddenly fail shortly thereafter The following figures show the stress-strain curves of concrete of various strengths reported by the Portland Cement Association. CES5. PRESTRESSED CONCRTE DESIGN AJDM  Figure 2.3. Stress-strain curves for various concrete strengths. (Nawy, Prestressed Concrete: A Fundamental Approach, 5th ed.) Observations: -The lower the strength of concrete, the higher the failure strain -The length of the initial relatively linear portion increases with the increase in the compressive strength of concrete -There is an apparent reduction in ductility with increased strength -The strength of concrete varies with age, the gain in strength being rapid at first, then much slower. The variation in strength is especially important in the design and fabrication of prestressed concrete members. B. Modulus of Elasticity The modulus of elasticity is the slope of the initial straight part of the stress-strain curve. C. Tensile Strength Cracks in prestressed concrete members may be caused by direct tension, flexure, combined shear and flexure in beam webs and torsion. The behavior of members often changes abruptly when tensile cracks form. Thus, it is important to know the tensile strength of the material. Tensile strength is measured by: 1. Modulus of rupture test 2. Split cylinder test CES5. PRESTRESSED CONCRTE DESIGN AJDM  Figure 2.9. Tests to determine the tensile strength of concrete. (a) Modulus of rupture test. (b) Split cylinder test. Time-dependent Deformation of Concrete Time-dependent deformation of concrete resulting from creep and shrinkage result in a partial loss of prestress force and significant changes in deflection. Creep Creep – property of materials by which they continue to deform over considerable lengths of time at constant stress or loads The initial deformation due to load is the elastic strain, while the additional strain due to the same sustained load is creep strain. Creep strain for concrete has been found experimentally to depend on: - time - mix proportions - humidty - curing conditions - age of the concrete when first loaded Creep strain is nearly linearly related to stress intensity It is therefore possible to relate creep strain to the initial elastic strain by a creep coefficient defined as  cu Cu   ci Where  ci initial elastic strain  cu additional strain in the concrete, after a long period of time, resulting from creep The creep coefficient can also be expressed as Cu   u Ec Where u unit creep coefficient CES5. PRESTRESSED CONCRTE DESIGN AJDM  The creep coefficient at any time ct can be related to the ultimate creep coefficient Cu by the equation: t 0.60 Ct  Cu 10  t 0.60 Shrinkage In a concrete element, shrinkage results to a decrease in volume when the concrete loses moisture by evaporation. References Nilson, A. H. (1987). Design of Prestressed Concrete. South Tower, Singapore: John Wiley & Sons Singapore Pte. Ltd. Nawy, E.G. (2010). Prestressed Concrete: A Fundamental Approach (5th ed.). Upper Saddle River, N.J. :Prentice Hall. CES5. PRESTRESSED CONCRTE DESIGN AJDM  PRESTRESSED CONCRETE DESIGN CHAPTER 3. FLEXURAL ANALYSIS Analysis vs Design Analysis = investigation Given: concrete and steel section, magnitude, and line of action of prestressing force Required: stresses and permissible load Design Given: stresses and permissible load Required: concrete and steel section, magnitude, and line of action of prestressing force Load stages to be considered in the Design and Analysis 1. Initial Prestress Force, Pi 2. Pi + Self-Weight 3. Pe + Full Dead Load 4. Pe + Full Service Loads Notations: Stress and Strain ( - ) Compression ( + ) Tension Strains Subscript 1 - top strain Subscript 2 - bottom strain Eccentricity e ( + ) positive when measured downward from concrete centroid e ( - ) negative when measured upward from concrete centroid PSC can be treated in two ways 1. Elastic 2. Similar to RC - compressive stress is resisted only by concrete and tensile stress is resisted only by steel Elastic Flexural Stresses in Uncracked Beams Behavior of a prestressed beam in the elastic range A simple-span prestressed beam with a curved tendon is shown CES5. PRESTRESSED CONCRTE DESIGN AJDM Where F force acting on the concrete at the tendon anchorage near the end of the member P force at midspan; resultant of all the normal compressive stresses in the concrete at that section N force exerted on the concrete by the tendon due to curvature The forces can be alternatively presented as When a uniformly distributed load of intensity is applied as shown Elastic Stresses If the member is subjected only to the initial prestressing force placed at eccentricity, the ensuing stresses at mid-span are Pi Pi ec1 f1    Ac Ic Pi Pi ec 2 f2    Ac Ic Where fi stress at the top surface of the member f2 stress at the bottom surface of the member c1 distance from the concrete centroid to the top surface of the member c2 distance from the concrete centroid to the bottom surface of the member e tendon eccentricity measured downward from the concrete centroid Ic moment of inertia of the concrete cross-section Ac area of concrete crosssection Substituting the radius of gyration, these equations can be written as: Pi  ec1  f1   1  2  Ac  r  CES5. PRESTRESSED CONCRTE DESIGN AJDM  Pi  ec 2  f2   1  2  Ac  r  The resulting stress distribution is shown The self-weight of the beam will cause moment to be superimposed immediately. Consequently, immediately after transfer of prestress force, the stresses at mid-span in the concrete are Pi  ec1  M o f1   1  2   Ac  r  S1 Pi  ec 2  M o f2   1  2   Ac  r  S2 Where Mo moment resulting from the self-weight of the member Ic S1  section modulus wrt the top surface of the member c1 Ic S2  section modulus wrt the bottom surface of the member c2 The resulting stress distribution is shown When the effective prestress Pe acts with the moments resulting from self-weight (Mo), superimposed dead load (Md), and superimposed live load (Ml), the resulting stresses are CES5. PRESTRESSED CONCRTE DESIGN AJDM  Pe  ec1  M t f1   1  2   Ac  r  S1 Pe  ec 2  M t f2   1  2   Ac  r  S2 Where Mt  Mo  Md  Ml Calculation of Sectional Properties In calculating the properties of the concrete cross-section to be used in the previous equations, the following should be noted (applies to post tensioned PSC): 1. Before tendons are grouted, stresses in the concrete should be calculated using the net section with holes deducted. 2. After grouting, the transformed section is used, holes may be considered filled with concrete and steel replaced with its transformed area of equivalent concrete At  n p  1A p Ep np  ; Ec Where At transformed area of steel Ep modulus of elasticity of prestressing steel Ec modulus of elasticity of concrete Ap area of prestressing steel 3. In practical cases, although the hole deduction may be significant, use of the gross concrete section after grouting rather than the transformed section will normally be satisfactory. 4. In many cases, as in the case of unbonded wrapped tendons with ducts, gross concrete section can be used. Cross Section Kern or Core Kern points – limiting points inside the section to which the prestress force can be applied to cause NO TENSION in the section CES5. PRESTRESSED CONCRTE DESIGN AJDM The lower limit is r2 k2  c1 The upper limit is r2 k1   c2 * The minus sign shows that the limit dimension is measured upward from the concrete centroid. CES5. PRESTRESSED CONCRTE DESIGN AJDM  Where fpu ultimate strength of steel fpy specified yield strength Cracking Load The moment causing cracking may be found for a typical beam by writing the equation for the concrete stress at the bottom face, based on the homogeneous section, and setting it equal to the modulus of rupture: Pe  ec 2  M cr f2   1  2    fr Ac  r  S2 Where Mcr total moment at cracking (including moment due to self-weight, superimposed dead and partial live loads) fr modulus of rupture Rearranging, we obtain M cr Pe  ec 2   fr  1  2  S2 Ac  r  Ic If S 2  , the equation for cracking moment is c2  r2  M cr  f r S 2  Pe   e   c2  The safety factor relative to cracking is expressed as M o  M d  Fcr M l  M cr Factor Fcr is given by M cr  M o  M d Fcr  Ml CES5. PRESTRESSED CONCRTE DESIGN AJDM Flexural Strength Analysis Stress-Strain Curves Where fpe , εpe stress and strain in the steel due to effective prestress force after all losses fpy , εpy yield stress and yield strain for the prestressing steel fpu , εpu ultimate tensile strength and ultimate strain of the steel fps , εps stress and strain in the presstressing steel when the beam fails fc’ , εcu ultimate compressive strength and failure strain Equivalent Rectangular Stress Block Where a   1c CES5. PRESTRESSED CONCRTE DESIGN AJDM  0.85  1  0.65  f c '4000  1  0.85  0.05    1000  Flexural Strength by Strain-Compatibility Analysis At stage (1) where acting alone, the stress in the steel and associated strain are respectively Pe f pe  Ap f pe  1   pe  Ep CES5. PRESTRESSED CONCRTE DESIGN AJDM  an intermediate load stage (2) corresponds to the decompression of the concrete at the level of the steel centroid where Pe  e2  2  1  2  Ac E c  r  at the failure stage (3), the neutral axis is at a distance below the top of the beam and the increment of strain is  dp c   3   cu    c  the total strain at failure is  ps   1   2   3 depth of stress block at failure is obtain from 0.85 f c ' ab  A p f ps solving for the stress block depth gives A p f ps a   1c 0.85 f c ' b Therefore, the nominal flexural strength is  a M n  A p f ps  d p    2 fps is unknown and is determined from iteration described in the following 1. assume a reasonable value of and note the corresponding strain from the stress-strain curve A p f ps 2. calculate c from a    1c 0.85 f c ' b  dp c  3. calculate Ɛ3 from  3   cu   and add this to the prior strains as indicated by  c   ps   1   2   3 4. if the computed strain Ɛps differs significantly with that assumed in step (1), revise that assumption and repeat steps (1) through (3) until satisfactory agreement is obtained 5. with both a = β1c and fps known, calculate the flexural capacity using  a M n  A p f ps  d p    2 CES5. PRESTRESSED CONCRTE DESIGN AJDM Partial Prestressing Full Prestressing – kind of design where the limiting tensile stress in the concrete at full service load is zero Partial Prestressing – an alternative approach in which flexural tension and usually some cracking are permitted in the concrete at normal service load CES5. PRESTRESSED CONCRTE DESIGN AJDM Elastic Flexural Stresses After Cracking and Strength of Partially Prestressed Beam Figure 3.18 Basics for analysis of cracked cross section. (a) Cracked cross section. (b) Concrete and steel strains. At stage 1, the stress at tendons is Pe f p1  f pe  Ap The compressive strain of the bar reinforcement at this stage is f s1   E s  s 2 Consider a fictitious load stage (2), corresponding to complete decompression of the concrete in which there is zero concrete strain through the entire depth, the changes in stress in the tendon and the bar reinforcement are, respectively f p 2  E p p 2 f s 2  E s s 2 At this hypothetical load stage, the stress in the bar reinforcement, neglecting creep and shrinkage, is f s  E s   s 2   s 2   0 The change in strain in the tendon is the same as that in the concrete at that level Pe  e 2   p2  1  2  Ac E c  r  To produce the zero stress state in the concrete, the tendon must be pulled with a fictitious external force (Fig. 3.18c) CES5. PRESTRESSED CONCRTE DESIGN AJDM Figure 3.18 (c) Decomposition force. (d) Forces on cracked section. (e) Resulting stresses. this fictitious force is cancelled by applying an equal and opposite force F as shown in Fig. (3.18d) this force can be represented by a resultant force, R, applied with eccentricity where M t  Fe e R incremental stresses are found using the transformed section concept (Fig. 3.19) * R Re * c1 fc3    Act I ct  R f p3  n p    Re * d p  c1  *    Act I ct   R  ns     Re * d s  c1  *  f s3   Act I ct  Where CES5. PRESTRESSED CONCRTE DESIGN AJDM  Ep np  Ec Es ns  Ec Act effective transformed area Ict moment of inertia of the cracked section about its own centroid from the top surface e* distance of R from the centroid of the cracked section Act is computed based on neutral axis, y, which is determined from equilibrium condition that the moment of all internal forces about the line of action of R is zero. the final stress in the tendon is f p  f p1  f p 2  f p 3 the stress in the bar reinforcement is f s  f s3 the concrete stress at the top surface of the beam is f c  f c3 References Nilson, A. H. (1987). Design of Prestressed Concrete. South Tower, Singapore: John Wiley & Sons Singapore Pte. Ltd. Nawy, E.G. (2010). Prestressed Concrete: A Fundamental Approach (5th ed.). Upper Saddle River, N.J. :Prentice Hall. CES5. PRESTRESSED CONCRTE DESIGN AJDM