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CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 1 Due Thursday, February 3 1) The girder shown in Fig. 1(a) is post-tensioned with two overlapping tendons. Tendon 1 is tensioned from the left end, A, and tendon 2 is from the right end, F. The friction forces between tendon and concrete at points B and D reduce the prestressing force in tendon 1 to 0.9P between points B and D and to 0.8P between points D and E. Similarly, the friction force at point D reduces the prestresing force in tendon 2 to 0.9P between D and C. The corresponding variation of the prestressing forces is shown in Fig. 1(b). Note that interior anchorages of both tendons are placed at an eccentricity of e. (a) Evaluate and plot the equivalent loads (on the tendon from concrete and on the concrete from tendon). Reduce equivalent loads to force/couple systems acting at the member axis in order to ease finding the diagrams of stress resultants. (b) Using these equivalent loads, evaluate and plot the diagrams of the concrete stress resultants, Mp, Vp, Np. (c) Check the results obtained using the “section-by-section” method (self equilibrating state of stress in each section). Assume that sintan=3e/L, cos1; similarly for . Give results in units of Pe for moments, Pe/L for shear forces, and P for axial forces. Overlapping tendons with interior anchorages is generally poor practice, particularly if the interior anchorages are eccentrically located. Explain why! 2) For the cases shown in Fig. 2 below, plot qualitatively the equivalent loads (on tendon from concrete and on concrete from tendon) and the diagrams of the concrete stress resultants Mp, Vp, Np acting on sections normal to the beam axis. If the beam axis is not straight, plot along the actual (non-straight) axis. Assume that the prestressing force P is constant. Express equivalent loads and selected values of Mp, Vp, Np in terms of P, e, , . Clearly indicate whether the equivalent loads due to anchorage are resolved into vertical and horizontal components or components normal and parallel to the beam axis. 3) For cases (a) though (e) below, sketch the tendon profile that is best in the sense of balancing the indicated external loads. Ignore member self weight and the effect of prestress losses. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 2 Due Tuesday, February 15 1) A prestressed simply supported I-beam has a span of 34 ft and is 36 in. deep. The cross section properties are shown below. Ten 0.5 in. diameter seven wire strand tendons are used for prestressing with a total tendon area of Ap=10*0.153=1.53 in2. The tendon profile is parabolic with e=0 at the beam ends and e=13.12 in. at the midspan. The beam is subjected to a uniform live load intensity of WL=3600 plf in addition to self weight of WD=384 plf. Compute the extreme fiber stresses at the midspan using: (a) Basic concept method (b) C-line method (c) Load balancing method For part (a), compute the extreme concrete fiber stresses at the midspan under: (1) initial conditions due to prestressing plus self weight; and (2) service load conditions after prestress losses have taken place. For parts (b) and (c), only the stresses under service load conditions need to be computed. Compare the concrete stresses in parts (a)-(c) with the corresponding allowable stresses. Material properties: fpu=270,000 psi (ultimate strength of prestressing steel) fpy=220,000 psi (yield strength of prestressing steel) fpi=189,000 psi (initial prestress at transfer, before losses) fpe=145,000 psi (effective prestress at service conditions, after losses) f’ci=4800 psi (initial strength of concrete at transfer) f’c=6000 psi (concrete strength at service conditions) Allowable stresses: fci=0.6f’ci (allowable compression stress under initial conditions at transfer) fc=0.45f’c (allowable compression stress at service conditions) fti=3(f’ci)1/2 (allowable tension stress under initial conditions at transfer) ft=12(f’c)1/2 (allowable tension stress at service conditions) CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 3 Due Thursday, March 3 1) A normal-weight steam-cured pre-tensioned simply supported 10DT24 double T-beam (see attached page from the PCI Handbook for section properties) has a span of 64 ft. The beam geometry is shown on the next page. In addition to a self-weight dead load of WD=468 plf, the beam carries a transient live load of WL=300 plf and a 2 in. thick normal-weight topping with a superimposed weight of WSD=250 plf. Ten 0.5 in. diameter stress-relieved seven-wire strand tendons are used for prestressing. The tendon profile is parabolic over the entire beam length with eccentricity, ee=7.77 in. at the beam ends and ec=14.77 in. at the midspan as measured from the cgc. The relative humidity, H=70%. Assume that at time, t=0 → the strands are jacked and the beam is cast t=24 hours → steam curing ends and prestress is transferred to the beam t=35 days → topping is cast t=5 years after topping is placed → service stage after long term losses Compute, by the detailed step-by-step method, the total losses in prestress at the midspan of the beam at: (a) Stage I - at transfer (b) Stage II - immediately after topping is placed (c) Stage III - five years after topping is placed Provide a summary of the losses. Material properties: fpu=270,000 psi (ultimate strength of prestressing steel) fpy=0.85fpu (yield strength of prestressing steel) fpj=0.70fpu (initial jacking stress) Eps=28,500,000 psi f’ci=4,800 psi (initial strength of concrete at transfer) f’c=6,000 psi (concrete strength at service conditions) Section properties: Ac=449 in2 ec=14.77 in Ic=22,469 in4 ee=7.77 in r2=50.04 in2 Sb=1,264 in3 cb=17.77 in St=3,607 in3 ct=6.23 in V/S=1.35 in. 2) Repeat Problem 1 for a post-tensioned beam. Assume that at time, t=0 → the beam is cast t=24 hours → steam curing ends and the beam is post-tensioned t=35 days → topping is cast t=5 years after topping is placed → service stage after long term losses Assume also that: - The anchorage seating is 0.25 in. - The strands are jacked two at a time in a flexible duct. - The total jacking force at the live end prior to the anchorage seating loss is fpj=0.70fpu. Provide a summary of the stress losses. 3) Repeat Problems 1 and 2 by the approximate lump-sum method and compare the results. Use values from Table 3.1. of textbook. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 4 Due Tuesday, March 22 1) A straight prismatic pre-tensioned simply supported beam with straight strands and section properties as shown below is to carry a live load of 1300 plf and a superimposed dead load of 500 plf on a 40 ft span in addition to its own weight of 270 plf. The initial prestressing force is Pi=272 kips at an eccentricity of 8.84 in. Losses are estimated to be 15%. Ultimate strength of the prestressing steel is fpu=270 ksi. Concrete strength is f’c=6000 psi at 28 days and 4200 psi at transfer. The modulus of rupture is fr=7.5(f’c)0.5. Moduli of elasticity are Ep=27000 ksi and Ec=4500 ksi (n=6). (a) Determine the distributed load at which the midspan section cracks. (b) Determine the decompression force, Fd at midspan after losses. (c) Verify that the midspan section is cracked under full service load. (d) Determine the flexural concrete and steel stresses in the cracked midspan section at full service load. Draw the stress distribution over the cross section. Conduct cracked section analysis even if the beam falls in ACI Class T. 2) Modify the formulas for cracked section analysis of rectangular sections to T-sections (for the case where the n.a. is in the web). This may be accomplished by splitting the total concrete compression stress resultant into three components as done in the numerical class example. A simpler approach may be to deduct from the enclosing rectangular section the stress effects of the “missing” rectangular sections on either side of the web. Note that the compressive stress resultant of the “missing rectangular sections” is not at the same level as the original compressive resultant. See figure below for notation. Do not try to further express the new “corrective” terms in terms of powers of (ξ-hf/d) through powers of ξ as the equation for the neutral axis has to be solved numerically (e.g., by iteration) anyhow. Check that for β=1, the new equations reduce to the equations given in the class handouts for rectangular sections. Save this problem for later 3) (a) For the spandrel beam section shown below, determine the normal stresses due to prestressing at points A and B. The prestressing force is P=200 kips and the section is not cracked. (b) The tendon is vertically inclined by α=0.1. Determine the magnitude, direction, and location of the shear force, Vp due to prestressing. (c) Is there a torque, Tp due to prestressing? If so, describe how to calculate it. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 5 Due Tuesday, April 5 1. Determine the nominal flexural strength of the midspan cross-section shown below based on: a. Strain compatibility analysis and finding fps by iteration b. ACI equation for fps, without iteration The beam is prestressed with four strands having a total area of Ap=0.575 in2. Eccentricity of the steel varies parabolically from e=0 at the supports to e=7.58 in. at the center of the 30-ft span. The effective stress in the steel after losses is fpe=132,000 psi. Concrete strength, f’c=4,000 psi and has an ultimate strain capacity of cu=0.003. The idealized stress-strain relationship of the prestressing steel is shown below. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 6 Due Tuesday, April 19 A simply supported post-tensioned beam is to carry a superimposed dead load of 300 plf and service live load of 1000 plf on a 40 ft span. Assume that 40% of the live load is sustained. An un-symmetric I-section (see Table A.10 below) with b=h, b2=0.5b will be used. Flange thickness hf=0.2h, and web width bw=0.3b. The member will be prestressed using ordinary stress relieved Grade 270 ksi strands. Time dependent losses are estimated at 20% of Pi. Normal density concrete will be used with f’c=5000 psi and f’ci=4000 psi. It can be assumed that deflections are not critical and that a tensile stress of 12√f’c is permissible at full service load. Eccentricities may vary along the span; however, the PT strands cannot be placed closer than 3.5 in. from the bottom of the beam. Prestressing should balance self-weight, dead load, and 40% of the live load before losses. (a) Find the concrete dimensions at mid-span to satisfy the ACI allowable stresses. (b) For the mid-span section, draw an e versus 1/Pi plot (Magnel diagram) and indicate the region of permissible combinations of e and Pi. Include allowable stress design Equations (5) through (8) in addition to Equations (1) through (4). What is the minimum and maximum permissible Pi? (c) Select a combination of Pi and e that lies within the feasible region and satisfies the aforementioned load balancing and maximum eccentricity requirements. (d) Plot the permissible range of eccentricities along the span together with a sketch of the PT tendon profile. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 7 Due Thursday, April 28 Consider the I-section below, where b=24 in., h=24 in., b2=12 in., hf=4.8 in., bw=7.2 in., Ac=276.5 in2, c1=10.03 in., and r2=61.2 in2. The unfactored superimposed dead load is 300 plf and the unfactored live load is 1000 plf on a 40 ft span. Self weight of the beam is to be determined based on a weight density of 150 pcf. The beam will be post-tensioned using ordinary stress relieved Grade 270 ksi strands (assume fpy=240 ksi). The PT tendon is comprised of seven 9/16 in. diameter strands placed at an eccentricity of e= 9.33 in. The area of each strand is 0.192 in2, resulting in a total tendon area of 1.344 in2. The initial stress in the strands is fpi=0.70fpu. Time dependent losses are estimated at 20% of fpi. Normal density concrete will be used with f’c=5000 psi. (a) Determine the nominal and design flexural strength of the cross section. You can use the ACI formula for fps. (b) Compare the design strength with the factored moment. Is mild steel required? (c) If mild steel is required, how much? Assume 2.5 in. distance from the bottom of the beam to the centroid of the mild steel. Grade 60 bars will be used. CE 40275/60275-01: Prestressed Concrete Design Spring 2011 Homework No. 8 Due Thursday, May 5 Using the strut-and-tie analysis approach, design the post-tensioned anchorage general zone reinforcement (i.e., closed stirrups) for the beam described in the example problem discussed in class from “Design of Prestressed Concrete” by Nilson. Assume that the post-tensioning anchor depth, a=6 in. Do not use any load factors or capacity reduction factors and compare your solution with the design given in the example from Nilson.