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- 1. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures Homework assignments and solutions, Spring 2013 Janne Hanka 17-Dec-13 Foreword: This educational material includes assignments+solutions of the course named Rak-43.3111 Prestressed and Precast Concrete Structures from the spring term 2013. Course is part of the Master’s degree programme of Structural Engineering and Building Technology in Aalto University. Each assignment has a description of the problem and the model solution by the author. Description of the problems and the solutions are given in Finnish and English. European standards EN 1990 and EN 1992-1-1 are applied in the problems and references are made to course text book Naaman A.E. "Prestressed concrete analysis and design, Fundamentals”. Questions or comments about the assignments or the model solutions can be sent to the author. Author: Place: Year: MSc. Janne Hanka janne.hanka@aalto.fi / janne.hanka@alumni.aalto.fi Finland 2013 Table of contents: Homework 1. Principles Homework 2. Working stress design Homework 3. Ultimate strength of post-tensioned beam with bonded tendons Homework 4. Prestress losses of post-tensioned beam with bonded tendons Homework 5. Composite structures Homework 6. Precast element frame predesign All rights reserved by the author.
- 2. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures Homework 1, Principles 9.2.2013 1(1) a) Explain the meaning of following terms: - Pretensioned prestressed concrete structures - Un-bonded and bonded post-tensioned concrete structures b) What kind of material properties of concrete and pre-stressing steel is beneficial for prestressed concrete? c) What kind of risks related to materials can be identified in prestressed concrete structures? d) The figure below shows alternative methods to execute a foundation bolt connection of an un-braced column. After casting and hardening of grout, nuts on top of the baseplate are tightened with a torque moment MT. Shortly after tightening connection is loaded with high bending moment M, large shear force V and relatively small axial compressive force N≈0. How do the different execution methods (a) and (b) affect behavior of the connection (eg, rotational stiffness), when external loads N, M & V are acting on the connection? Hint: Draw a free body diagram of the baseplate. 6 1=Nuts 2=Grout 3=Foundation 4=Foundation bolt 5=Baseplate 6=Loads N, V &M 1 5 2 3 4 (a) (b) Figure 1. a) Nuts below the baseplate are used to level the baseplate and they are left in place before pouring of grout. b) There are no nuts under the baseplate or they are loosened before pouring of grout. Baseplate is leveled with other means. Return to Optima in PDF-format by Friday 1.2.2013.
- 3. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures Homework 2, Working stress design 16.1.2013 1(1) Slab in figure 1 prestressed with pretensionded bonded tendons. Tendons will be released when the age of concrete is t=28d. At time t=29d dead load (g1) starts to effect the structure. At time t=30d concrete section shrinks shown if figure 1. At time t=31d live load (q1) starts to effect the structure. Modulus of elasticity for different materials and shrinkage of concrete *Concrete C40/50, Ecm=34GPa, Δεsc.top=0,3% , Δεsc.bot=0,1% *Reinforcement A500HW, Es=200 GPa, *Prestressing steel St 1500/1770, Ep=195GPa h=200mm bw=1000mm etop= 35mm ebot= 50mm As.top=260mm2 Ap.bot=750mm2 L=5000mm yc=25kN/m3 g1=15kN/m2 q1=5kN/m2 Figure 1. Calculation model, cross section, shrinkage and geometric properties. a) Calculate the (un-cracked) cross section properties by using method of transformed cross section. b) Calculate the initial value of prestress (σmax=?), so that the bottom fibre of cross section is decompressed (σbot=0) at midspan (x=L/2). Structure is loaded combination of dead load g1+selfweight (g0)+prestress (P) at time t=29d. c) Calculate the change of stresses at top (Δσc.sh.top) and bottom (Δσc.sh.bot) of the cross section due to shrinkage at time t=30d. d) Calculate the total stress at top and bottom fibre of the cross section at time t=31d. Structure is loaded with Live load(q1)+shrinkage+dead load(g1)+selfweight(g0)+esijännitys(P). Additional voluntary task: Does the assumption of uncracked cross section still comply at time t=31d? As.top = area of top reinforcement Ap.bot = area of bottom prestressing steel Return to Optima in PDF-format by Friday 8.2.2013.
- 4. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures 3.2.2013 Homework 3, Ultimate strength of post-tensioned beam with bonded tendons 1(1) Beam in figure 1 is prestressed with bonded post-tensioned tendons when the age of concrete is t=28d. After post-tensioning duct will be injected with grout. After hardening of grout structure is loaded with distributed dead load g1 and distributed live load q1 (in addition to selfweight). L=10m h=450mm bw=300mm ep(x=0)= h/2 ep(x=L/4)= 131,25mm ep(x=L/2) = 100mm Ap=780mm2 g1=5kN/m q1=5kN/m Figure 1. Prestressed beam with bonded post-tensioned tendons. Information * Concrete C40/50, Ecm=35GPa, selfweight of concrete ρc=25 kN/m3 * Parabolic tendon geometry * Prestressing steel: Ep=195GPa, fp0,1k=1500 MPa, fpk=1770 MPa and εuk=3% * Strain hardening of prestressing steel is not taken into account [EN1992-1-1 fig 3.10] * Initial prestress σmax=1000 MPa. Total prestress losses (immediate and timedependant) 20%. * Partial factors for materials γc=1,50; αcc=0,85 ja γs=γp=1,15 [EN 1992-1-1 2.4.2.4(1)] * Partial factor for prestress force γP,fav=0,9 [EN1992-1-1 2.4.2.2(1)] * Partial factor for dead loads γG=1,15 and live loads γQ=1,15. Factor depending on the reliability class KFI=1. [EN1990] * Ultimate compressive strain of concrete εcu=0,0035 [EN1992-1-1 Table 3.1] * Factors used in the figure 2 calculation model λ=0,80; η=1,00 [EN1992-1-1 3.1.7(3)] Additional voluntary task aa) Explain how is the effect of (design value of) prestressing force taken into account in the Ultimate Limit State in bonded post-tensioned concrete structures… - …. when calculating the Resistance of actions MRd at section concerned - …. when calculating the Effect of actions MEd at section concerned Calculate the design value of bending moment resistance MRd and the design value of effects of actions due to bending moment MEd in ultimate limit state…. a)…at section x=L/2 b)…at section x=L/4 c) Draw envelope curves that describe the bending moment capacity and the effects of actions due to bending moment along the beam x-axis. Is the bending moment capacity adequate in all sections? (a) (b) Figure 2. a) Calculation model in ultimate limit state. b) Stress-strain curve of prestressing steel [EC2, fig 3.10] Return to Optima in PDF-format by Friday 15.2.2013.
- 5. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures 6.2.2013 Homework 4, Prestress losses of post-tensioned beam with bonded tendons 1(1) Beam in figure 1 is prestressed with bonded post-tensioned tendons when the age of concrete is t=28d. After post-tensioning duct will be injected with grout. After hardening of grout structure is loaded (t=29…50*365d) with distributed dead load (g1) and distributed live load (q1) (in addition to selfweight). Long term part of the live load is (ψ2). L=18m h=950mm bw=300mm ep(x=L/2)=ep1=100mm Ap=780mm2 1=Stressing end 2=End anchorage g1=5kN/m q1=5kN/m ψ2=0,3 Figure 1. Prestressed beam with bonded post-tensioned tendons. Information: * Concrete C40/50, Ecm=35GPa, selfweight ρc=25 kN/m3, RH=60%, cement type N * Parabolic tendon geometry u(x) = ax2+bx+c * Prestressing steel 1500/1770: Ep=195GPa, fp0,1k=1500 MPa, fpk=1770 MPa, εuk=3% and Relaxation class 2 (small relaxation) ρ1000=2,5%. * Diameter of duct D=60mm * Initial stress (force of jack/area of tendons) σ max=1200 MPa. a) Calculate the stress in tendons and stress distribution of the concrete section at midspan immediately after pre-tensioning. Consider the immediate losses due to friction. Voluntary additional assignment aa) Calculate also the immediate losses due to anchorage set. b) What is the value of distributed load that prestressing force balances after immediate losses? Voluntary additional assignment cc) Calculate the stress in tendons and stress distribution of the concrete section at midspan, at time t=29d, when the quasi permanent combination of actions starts to effect the structure p=∑gi+∑ψ2qi (in addition to prestress force). Tip: Immediate prestress losses due to friction can be calculated with the following information * Losses due to friction in post-tensioned tendons: ΔPμ(x)=P0(1-e-μ(θ+kx)) [EN1992-1-1 5.10.5.2(1) Eq.(5.45)] * θ is the sum of the angular displacements over a distance x * coefficient of friction between the tendon and its duct μ=,25 * unintentional angular displacement for internal tendons (per unit length) k = 0,0150m-1 * slip of tendon δ= 2 mm Tip: Equation that describes the elevation of the tendon along beam x-axis conforming to figure 1. (You can also formulate you own equation to describe the elevation of the tendon to your own coordinate system of choice) u(x)=[(4ep1-2h)/L2]*x2 + [(-4ep1+2h)/L]*x Tip: Loss due to anchorage set is treated in the course textbook [Naaman] chapter 8.17 p.498. Return to Optima in PDF-format by Friday 22.2.2013.
- 6. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures Homework 5, Composite structures 13.2.2013 1(1) Composite structure in figure 1 is prestressed with pre-tensioned bonded tendons (initial prestress σmax=1000MPa). Area of one tendon is 52mm2. Total number of tendons is 6. Span of composite structure is L=5m and supports can be assumed to hinges. Top surface of the hollowcore slab can be assumed to be smooth during casting of topping. Concrete sections near supports may be assumed to remain uncracked until failure. Figure 1. Pre-tensioned hollow core slab and topping. a) Calculate the maximum value of shear stress at the interface between hollow core slab and topping at end of the slab (=at support) due to distributed live load qEd=10 kN/m2 (design value). b) Is the design shear resistance at the interface between hollow core slab and topping adequate? Apply EN 1992-1-1 section 6.2.5 Shear at the interface between concrete cast at different times. Tip: Shear resistance of indented construction joint [EN1992-1-1, §6.2.5) Construction joint in the figure 101 is affected with compressive stress σn and shear stress vEd. Area of reinforcement per unit length that is crossing the interface is ρ. The design value shear resistance at the interface is according to EC2: Reduction factor for concrete cracked in shear (NDP value) • c and μ are factors which depend on the roughness of the interface • fck and fcd are characteristic and design value of concrete correspondingly • fctd and fyd are design values for tension of concrete and reinforcement correspondingly • α is defined in figure 101, and should be limited by 45° < α < 90° • σn is stress per unit area caused by the minimum external normal force across the interface that can act simultaneously with the shear force (positive for compression) Classification of the interface roughness [EN 1992-1-1+AC §6.2.5(2)] c μ Hyvin sileä 0,025-0,10 0,5 Sileä 0,20 0,6 Karhea 0,45 0,7 Vaarnattu 0,50 0,9 Material design values [EN 1992-1-1] Material αcc=0,85 αct=1,0 γc=1,5 γs= γp=1,5 C30/37 fck= 30MPa fcd= 17,0MPa C50/60 fck= 50MPa fcd= 28,3MPa A500HW fyk= 500MPa fyd= 434MPa St1640/1860 fp0,1= 1640MPa fpd= 1426MPa Return to Optima in PDF-format by Friday 1.3.2013. fctd= 1,35MPa fctd= 1,9MPa
- 7. Aalto University Rak-43.3111 Prestressed and Precast Concrete Structures Homework 6, Element frame predesign 13.2.2013 1(1) Plan view of shopping center floor is presented in figure 1. Floor consists of hollow core slabs that are supported by lowbeams (WQ-beam). Lowbeams are supported by columns at the intersections of grid lines. Connection between lowbeams and columns is hinged. Floor is affected by dead (g1) and live (q1) loads: g1=3,0 kN/m2 Topping (design thickness 20mm), partitions and suspended load q1=5,0 kN/m2 Shopping areas [EN 1991-1-1, class D2] Partial factor for dead loads γG=1,15 and live loads γQ=1,5. Consequence class CC2 and factor KFI=1 [EN1990] L1=12m L2=7,2m 1= Columns Figure 1. Plan view and section of lowbeam (WQ-beam). [http://www.elementtisuunnittelu.fi/fi/runkorakenteet/palkit/matalapalkit] a) Pre-design the loadbearing components of floor (choose profiles for lowbeam at module C/2-3 and hollow core slabs), so that the lowbeams are spanning in shorter load carrying direction. b) Calculate the deflection at midspan for lowbeam (pre-designed at (a)) for characteristic combination of actions (pc=Σgi+q1+ Σψ0qi+1). c) Pre-design the loadbearing components of floor (choose profiles for lowbeam at module BC/2 and hollow core slabs), so that the lowbeams are spanning in longer load carrying direction. d) Calculate the deflection at midspan for lowbeam (pre-designed at (c)) for characteristic combination of actions. e) To which spanning direction (shorter or longer) would you put the lowbeams, when you consider the composite action (between the lowbeams and hollowcore slabs) and the calculated deflections? And how would you justify the selection of lowbeam carrying direction from the perspective of material costs? Note a, c) Use manufacturers pre-design curves for predesigning of hollow-core-slabs and WQ-beams. HC-slabs: WQ-Beams: http://fsiviewer.taskut.net/Parma/Parma/ontelolaatat/perustukset_ontelolaatat_suunnittelutohje.html http://www.betonika.lt/en/gaminiai-ir-paslaugos/gaminiai/perdangos-plokstes/ http://www.concast.ie/sites/default/files/pdfs/Hollowcore/hollowcore.pdf http://www.stahlton.co.nz/idc/groups/web_stahlton/documents/webcontent/nz_00016547.pdf http://www.ruukki.fi/~/media/Files/Building-solutions-brochures/Ruukki-WQ-beam-manual.pdf Note b, d) Due to simplification you can estimate the deflections by using only WQ-beams flexural rigidity (without considering the composite action). Thickess of the WQ-beam flanges can be assumed to be 30mm. Thickness of the webs may be assumed to be 10mm correspondingly. Return to Optima in PDF-format by Friday 15.3.2013.

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