Prestressed concrete Course assignments, 2014

Aalto University Janne Hanka

Rak-43.3111 Prestressed and Precast Concrete Structures 22-Feb-14

Homework assignments and solutions, Spring 2014

All rights reserved by the author.

Foreword:

This educational material includes assignments of the course named Rak-43.3111 Prestressed and

Precast Concrete Structures from the spring term 2014. 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: MSc. Janne Hanka

[email protected] / [email protected]

Place: Finland

Year: 2014

Table of contents:

Homework 1. Prestressed foundation bolt connection

Homework 2. Working stress design using stress inequality equations (Magnel’s diagram)

Homework 3. Ultimate strength of post-tensioned beam with unbonded tendons

Homework 4. Prestress losses of post-tensioned beam with unbonded tendons

Homework 5. Composite structure subjected to differential shrinkage

Homework 6. Predesign of prestressed continuous slab using loadbalancing

Aalto University J.Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 29.12.2013 Homework 1, Prestressed foundation bolt connection 1(1)

Return to Optima in PDF-format by 19.01.2014.

Baseplate in figure 1 is prestressed against the foundation by prestressing the foundation bolts. After prestressing of bolts, connection is loaded by an external variable normal force N (upwards). Number of foundation bolts n=4. Foundation bolt material characteristics: yield strenght fy=950MPa, ultimate strength fu=1050MPa, strain at yield εy=0,5%, strain at ultimate εu≈6*εy, modulus of elasticity Ep=190GPa. Diameter and area of single bolt are d=25mm and Ap1=490mm2. Baseplate dimensions a=250mm. Modulus of elasticity of the grout underneath the base plate is Ec=30GPa. Height of the grout under the baseplate is h=50mm. Normal force N is the only load effecting the baseplate. Selfweight is not taken into account.

Figure 1. Prestressed foundation bolt connection. After hardening of grouting foundation bolts are prestressed to value σP=0,5*fy (assumed remaining prestress in bolts after all losses). Thus the total prestress force acting on the connection is P=n*Ap1*σP. a) Determine the stress level in bolts σP,i and the contact stress σC,i between baseplate and grouting when the external load N is zero (N =0)? b) What is the value of normal force N=Nlim that leads to loss of contact between baseplate and grouting (contact stress is zero)? What is the stress level in bolts σP,lim? c) Connection is loaded in such a way that normal force N increases from zero to Nlim, after which load N is removed (decreases back to zero). Determine the stress level in bolts σP and the contact stress σC between baseplate and grouting after removal of the external load N. d) What is the value of ultimate load N=Nmax at which the bolts will fail? Tip: Assume that the weakest link of the connection is the tensile resistance of bolts. Other failure modes are not examined. e) Connection is loaded in such a way that normal force N increases from zero to 0,9999*Nmax after which load N is removed (decreases back to zero). Determine the stress level in bolts σP and the contact stress σC between baseplate and grouting after removal of the external load. What observations can you make about the behavior of prestressed connection when comparing results to (c)? Note: Use characteristic values of loads and materials without partial factors. Tip: Draw the stress-strain diagrams of bolts and bearing (grouting under baseplate). Sketch the calculation model and freebody diagram of the baseplate. Model the bolts and bearing as springs that have rigidities kP=EpnpAp1/Lf and kC=Eca2/h correspondingly.

1=External load 2=Baseplate 3=Prestressed foundation bolts 4=Anchor plate(s) 5=Concrete foundations 6=Bond break over free stressed lenght 7=Grout (bearing) Lf = 20*d = Free stressed lenght of the foundation bolt d=25mm=Diameter of bolt

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 29.12.2013 Homework 2, Working stress design using stress inequality equations 1(1)

Return to Optima in PDF-format by Friday 26.01.2014.

Simply supported beam in figure in 1 is prestressed with posttensionded tendons when the age of concrete is t=28d. After tensioning beam is loaded with permanent load g1=5kN/m and live load qk=10kN/m (EN 1990 Class E). Span of the beam is Leff=20m. * Concrete C40/50; Compressive strenght fck(t=28d)=fck=40MPa Mean flexural tensile strenght fctm(t=28d)=fctm=4,07MPa Density ρc=25kN/m3 * Environmental classes and design working life: XC3, XD1. 50 years * Characteristic combination of actions: pc=∑gj + q1 + ∑ψ2,i+1qi+1 * Assumed prestress losses (immediate and time dependant) Δσ=25%

Figure 1. Graphical presentation of stress inequality equations, cross section and stresses. Table 1. Allowable stresses of concrete in serviceability limit state (SLS). Condition # Combination EN1990 Limitation EC2 Clause

Initi

al I Max tension Initial σct.ini < fctm(t)

II Max compression Initial σcc.ini < 0,6*fck(t) 5.10.2.2(5)

Fina

l III Max tension Characteristic σct.c < fctm IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2) V Max tension Frequent σct.f < 0 MPa (decompression) Table 7.1 VI Max compression Quasi-permanent σcc.qp < 0,45*fck 7.2(3)

a) Calculate the combination of actions and effects of actions (bending moment M at midspan) in serviceability limit state for: - Initial combination pmin, Mmin Tip: Use only beam selfweight - Characteristic combination pc, Mc b) Calculate the cross section properties for the gross-cross section: - Height of centroid, moment of inertia and cross section area ygr, Igr, Agr - Section modulus with respect to the top and bottom fiber Wtop, Wbot (Ztop, Zbot using Naaman’s notations) - Distance from centroid to the upper & lower limit of the central kern kt, kb c) Determine the inequality equations required for the flexural analysis of the section using the stress condition I, II, III and IV given in table 1. Initial value of the tendon force Fi is used in conditions I-II and final value Fiη is used in conditions III-VI. d) Present the equations determined in (c) in the form required for the graphical presentation of stress inequality equations in the coordinate system (1/Fi, e0). Draw the graphical representation (Magnel’s diagram) of the stress inequality conditions where vertical axis is e0 and horizontal axis is 1/Fi (figure 1). e) Choose the value of initial prestress Fi and eccentricity e0 that satisfies the conditions I-IV in table 1. Additional voluntary task) Calculate the bending moments (Mf and Mqp) for frequent (pf=∑gj + 0,9*q1 ) and quasi-permanent combination (pqp=∑gj + 0,8*qi) of actions. Consider conditions V and VI in the Magnel’s diagram in (d) and in the selection of initial prestress and eccentricity in (e). Note: Fi = Initial prestress before losses (conditions I-II) Fiη=Fi(1-Δσ)=Final prestress after all losses (Conditions III-VI) e0= Distance of prestress force from centroid ebot= Smallest allowable distance of tendons from the bottom of the section (practical condition)

Dimensions: beff=800mm hf=200mm bw=300mm hw= 1000mm ebot= 50mm

(Magnel’s diagram, Course textbook, p.158-160)

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 31.12.2013 Homework 3, Ultimate strength of post-tensioned beam with unbonded tendons 1(1)


Beam in figure 1 is prestressed with unbonded post-tensioned tendons when the age of concrete is t=28d. After post-tensioning beam is loaded with distributed live load qk (in addition to selfweight). Beam is also reinforced with normal reinforcement steel.

Figure 1. Prestressed beam with unbonded post-tensioned tendons and reinforcement. Information: * Concrete C40/50, selfweight of concrete ρc=25 kN/m3, strain at ultimate εcu=0,0035 [EC2Table 3.1] * Prestressing steel: Ep=195GPa; fp0,1k=1500 MPa; fpk=1770 MPa; εpuk=3%; parabolic tendon geometry * Reinforcement steel Es=200GPa, fyk=500 MPa * Strain hardening of prestressing nor reinforcing steel is not taken into account [EN1992-1-1 fig 3.10] * Initial prestress σmax=1000 MPa. Assumed total prestress losses (immediate and timedependant) 25%. * Assumed stress increase of unbonded tendons in ultimate limit state Δσp.ULS=50MPa [EN1992-1-1 5.10.8(2)] * 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=1,5. Factor KFI=1. [EN1990] * Factors used in the figure 2 calculation model λ=0,80; η=1,00 [EN1992-1-1 3.1.7(3)] Additional voluntary task) Explain how is the effect of prestressing force taken into account in the Ultimate Limit State in unbonded post-tensioned concrete structures when calculating the effects of actions MEd and resistance of actions MRd at section concerned. a) Calculate the design value of (effects of actions) bending moment MEd in ULS at critical section x=L/2. b) Calculate the design value of (resistance of actions) bending moment resistance MRd in ULS at section x=L/2. c) Is the bending moment capacity adequate in section x=L/2? If not, how the capacity could be improved? Additional voluntary task d) What is maximum value of distributed liveload qk.max that the beam can sustain in ULS? (what is the value of liveload qk.max when the effects of actions equals to bending moment resistance MRd at section x=L/2)

(a) (b) (c) Figure 2. (a) Calculation model in ultimate limit state. (b) Stress-strain curve of prestressing steel [EC2 fig 3.10]. (c) Stress-strain curve of reinforcing steel [EC2 fig 3.8].

L=10m h=450mm bw=300mm ep(x=0)= h/2 ep(x=L/2) = 100mm es=50mm AP=780mm2

AS=402mm2

AP= area of tendons AS=area of reinforcing steel qk=12kN/m Live load

fcd=αccfck/γC

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 31.12.2013 Homework 4, Prestress losses of post-tensioned beam with unbonded tendons 1(1)


Beam in figure 1 is prestressed with unbonded post-tensioned tendons when the age of concrete is t=28d. Number of tendons is 4 and they are stressed one-by-one to initial prestress σmax. Area of one tendon is Ap1 and the total area of tendons is Ap. After prestressing, beam is loaded (t=29…50*365d) by distributed live load qk (in addition to selfweight and prestress). Long term part of the live load is (qk*ψ2).

Figure 1. Prestressed beam with unbonded post-tensioned tendons. Information: * Concrete C35/45, Ecm=34GPa, selfweight ρc=25 kN/m3 * Parabolic tendon geometry: u(x) = ax2+bx+c * Prestressing steel 1500/1770: Ep=195GPa, fp0,1k=1500 MPa, fpk=1770 MPa, εuk=3% * Diameter of duct D=20mm * Initial stress (force of jack/area of tendons) σmax=1350 MPa. a) Calculate the immediate losses due to friction ΔPμ at midspan. Voluntary additional assignment) Calculate also the immediate losses due to anchorage set ΔPsl. b) Calculate the immediate losses due to instantaneous deformation of concrete ΔPel at midspan. c) What would be the loss due to instantaneous deformation of concrete ΔPel if all tendons were stressed simultaneously using 4 jacks? d) Calculate the stress in tendons and stress distribution of the concrete section at midspan immediately after prestressing. Consider losses calculated in (a) and (b). Use net- or gross-cross section properties. e) What is the value of distributed load that prestressing force balances after immediate losses? Voluntary additional assignment) Calculate the stress increase of tendons at midspan at t=29d when the long term part of the liveload ψ2qi starts to effect. 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=0,25 * 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: Losses due to anchorage set and elastic shortening is treated in the course textbook [Naaman] chapters 8.17 and 8.7 respectively.

L=16m h=700mm bw=350mm ep(x=L/2)=ep1=50mm Ap1=150mm2 Ap=4*Ap1=600mm2

1=Stressing end

2=End anchorage qk=10 kN/m ψ2=0,5

Aalto University J. Hanka

Rak-43.3111 Prestressed and Precast Concrete Structures 14.1.2014

Homework 5, Composite structure subjected to differential shrinkage 1(1)


Hollowcore slab structure in figure 1 is prestressed with pre-tensioned bonded tendons (initial prestress

σmax=1000MPa). Area of one tendon is Ap1=52mm2. Total number of tendons is 6. Strenght class of the

slab is C40/50. Span of the slab is L=10m and supports can be assumed to hinged. When the age of the slab

is t=28d topping of C25/30 with thickness 50mm is casted on top of the slab. Topping and the hollowcore

slab are assumed to act together as a composite structure.

Figure 1. Pre-tensioned hollow core slab and topping.

a) Calculate the cross section properties of the composite section using method of transformed section.

b) Calculate the stresses of the composite section due to differential shrinkages of the topping and slab,

when uniform shrinkages are:

* uniform shrinkage of the topping -0,0003=0,3‰

* unifrom shrinkage of the hollow-core slab -0,00005=0,05‰

c) Draw a curve that describes the stress distribution of the composite section due differential shrinkages

calculated in (b).

Note. Calculations are done in serviceability limit state without partial factors.

Table 2. Materials characteristics (EN 1992-1-1 table 3.1).

Part Strenght class Compressive strength Modulus of elasticity

Topping C25/30 fck= 25MPa Ecm=31 GPa

Hollowcore slab C40/50 fck= 40MPa Ecm=35 PGa

Tendons St1640/1860 fp0,1= 1640MPa Ep=190 GPa

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 23.2.2014 Homework 6, Predesign of a prestressed continuous slab using loadbalancing 1(1)


Two-equal-span continuous slab of parking garage displayed in figure 1 will be prestressed with unbonded tendons. Slab is loaded with a distributed liveload q1, which can act on both or one span only. Information:

- Concrete class C35/45, fck=35MPa, Ecm=34GPa, γC=1,5, αcc=0,85 - Unbonded tendons. Grade St1600/1860, fp0,1k=1600 MPa, fpu=1860MPa, Ep=195GPa, γP=1,15 - Area of one tendon Ap1=150mm2 - Smallest allowable distance of tendons from the bottom/top of the section ep=50mm - Total prestress losses (initial & timedependant) are assumed to be Δf=25% in both spans [Pm.t=Pmax(1-Δf)] - Initial prestress losses (friction, slip of anchorage and elastic) are assumed to be Δini=15% [Pm.0=Pmax(1-Δini)] - Reinforcement: A500HW, fyk=500MPa, Es=200GPa, γS=1,15 - Span lengths: L1=L2=10,8m. Liveload qk=5kN/m2; ψ0=0,7; ψ1=0,5; ψ2=0,3 (EN 1990 Class G, garages) - Partial factors for loads in ULS: γG=1,35 ; ξγG=1,15 ; γQ=1,5 ; KFI=1 - Partial factors for tendon force in ULS: γP.fav=0,9 ; γP.unfav=1,1 - Characteristic combination: pc=∑gj + q1 + ∑ψ2,i+1qi+1 Quasi-permanent combination: pc=∑gj + ∑ψiqi

- Allowable deflection for quasi-permanent combination: L/250

- Allowable active deflection (due to change of deflection due to imposing of liveload q1): L/500

- Allowable tensile stress in concrete (during prestressing and after all losses) σct.all=fctm

- Allowable compressive stress in concrete (during prestressing and after all losses) σcc.all=0,6fck

- Allowable tensile stress in tendons (during prestressing) σp.all=0,9fp0,1k Goal of the assignment is to predesign & choose optimal values for the thickness of the slab, tendon force, tendon geometry and total amount of tendons. Assignment is solved by investigating one unit width b=1m of the slab.

Figure 1. Two-span post-tensioned slab with hinged supports. Design in SLS using theory of elasticity: a) Choose the thickness of the slab (h), so that the active deflection due to liveload qk does not exceed the maximum allowable value for active deflection (L/500). Choose the thickness (h) in 20mm increments. b) Choose the tendon geometry (eA, eB, eC) and required value of maximum tendon force (jackingforce Pmax), so that the final value of tendon force [Pm.t=Pmax(1-Δf)] balances the slab selfweight and long term part of the live load q1ψ2. Check that the allowable stresses (σct.max<fctm ; σc.max<0,6fck) in serviceability limit state for the following loadings: c) …when slab is loaded with initial tendon force Pm.0 and slab selfweight (initial situation during prestressing). d) …when slab is loaded with final tendon force Pm.t and characteristic combination of actions pc. e) Choose the total number of tendons (np*Ap1=Ap.tot) and the initial value for prestress (σmax). Draw a schematic drawing (sideview and cross section) of the tendon geometry. Voluntary additional assignment: Design in ULS using theory of plasticity f) Calculate the design value of effects of actions (bending moment MEd) and resistance of actions (MRd) in ultimate limit state at critical sections. Make a note should the slab be strengthened with reinforcing bars and calculate the required amount of reinforcement. Instructions: You can make justified assumptions and simplifications in the calculations. It is not required to round the tendon geometry on top of the support. Use gross-cross section properties in the calculations.

Prestressed concrete Course assignments, 2014

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