Aalto University Janne Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 8-Mar-15 Homework assignments and solutions, Spring 2015

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 2015. 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 janne.hanka@aalto.fi / janne.hanka@alumni.aalto.fi Place: Finland Year: 2015

Table of contents:

Homework 1. Prestressed rock anchors

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

Homework 3. Working stress design using limit core and zone

Homework 4. Analysis of post-tensioned continuous slab using loadbalancing

Homework 5. Prestress losses of precast pretensioned pile

Homework 6. Ultimate strength of composite structure

Aalto University J. Hanka

Rak-43.3111 Prestressed and Precast Concrete Structures 29.12.2014

Homework 1, Prestressed rock anchors 1(1)

Return to Moodle in PDF-format by 18.01.2015.

Figure 1. Prestressed rock bearing footing

Figure 2. Slipping of the anchors.

Rock bearing pad footing shown in figure 1 is loaded by permanent

vertical force N=160 kN and variable horizontal force H=210kN,

which can affect in both directions. The design requirement given

for the structure is that the foundation base has to be fully

stressed (loss of contact between bottom of foundation and

bedrock is not allowed). Stability of the foundation will be improved

by post-tensioned rock anchors that are placed symmetrically about

the centroid of the pad footing.

Foundation

Foundation column: bc*bc= 0,4m * 0,4m.

Foundation column height hc= 1,1m

Foundation slab: bf*bf= 1,2m * 1,2m.

Foundation slab thickness hf= 0,6m.

Unit weight of concrete ρc= 25 kN/m3

Rock anchors:

Area of one anchor: Ap1= 490 mm2 ;

Modulus of elasticity Ep=195GPa.

Yield and ultimate strength fy=950MPa ;

fu=1050MPa

Strain at yield and at ultimate εy=0,5%, ; εu≈6*εy

Slip of the anchors (see figure 2) Δslip=1,5 mm.

a) Calculate the required prestress force Pf.req that fulfills the design requirement. Choose total number of

anchors nP and final prestress σf. Note: The chosen final prestress of the anchors shall not exceed the

allowable stress σf < σf.all=840MPa.

b) Determine the required free stressed length of the rock anchors when the maximum prestress during

jacking is σmax=945 MPa. (Tip: ε=ΔL/L0=σ/Ep)

c) Check the maximum base pressure and safety against sliding of the foundation. Allowable stress of the

rock is σall=3MPa. Friction coefficient between foundation and rock is assumed to be μ=0,2.

d) Execution of the rock anchors will be deviated from the design in such a way that the actual free stressed

length is only half of the required Lfree.act=0,5Lfree. What will be the final prestress and –force in this case?

e) How big prestress would remain in the rock anchors if the free stressed length is zero? What is the

significance of the free stressed length?

Note: Use characteristic values of loads and materials without partial factors.

Tip b) Slipping of the anchors has to

considered in the calculation of the free

stressed length.

Rock

anchors

Found.

column

Found.

slab

Bedrock

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 28.1.2015 Homework 2, Working stress design using stress inequality equations 1(1)

Return to Moodle in PDF-format.

Cross section dimensions h= 0,7 m ; b=0,4 m ; Span lengths: L2=6,0 m ; L1=L3=2,4 m ; L4=12m

You are exploring the feasibility of using a precast pretensioned beam with straight tendons as a double cantilever beam (Fig. 1). Beam is supporting a floor of hollow core slabs. Connection between beam and its support columns is hinged. * Concrete C50/60; Compressive strenght fck(t=28d)=fck=50 MPa Mean flexural tensile strenght fctm(t=28d)=fctm=4,07 MPa 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% * Smallest allowable distance of tendons from top or bottom of the section (practical condition) ebot=etop=50mm

Figure 1. Pretensioned precast beam with straight tendons. Plan and section views. Table 1. Allowable stresses of concrete in serviceability limit state (SLS). Condition # Combination EN1990 Limitation EC2 Clause

Init

ial I Max tension Initial σct.ini < fctm(t=28d)

II Max compression Initial σcc.ini < 0,6*fck(t=28d) 5.10.2.2(5)

Fin

al

III Max tension Characteristic σct.c < fctm IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)

a) Form the calculation model of the beam and calculate the actions affecting the beam. b) Calculate the combination of actions and effects of actions in critical sections (bending moment at midspan and support) in serviceability limit state for: - Initial combination pmin, Mmin - Characteristic combination pc, Mc Tip: Live load can vary from span-to-span c) 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 d) Find the initial prestressing force Pi and its eccentricity e0 that satisfies conditions given in table 1 at critical sections (midspan and support). Tip: Draw the graphical representation of the stress inequality conditions where vertical axis is e0 and horizontal axis is 1/Pi. e) If you were told that the minimum prestressing force for both section A (support) and B (midspan) corresponds to satisfying stress condition IV III, derive the analytical solution for question (c). Note: Use gross-cross section properties in calculations. Pi = Initial prestress before losses (conditions I-II) Pf=Piη=Pi(1-Δσ)=Final prestress after all losses (Conditions III-VI) e0= Distance of prestress force from centroid

Imposed dead and Live arealoads Liveload q1 = 7,5 kN/m2 Screed g2 = 1 kN/m2 Hollowcore slabs: g3 = 4 kN/m2

Double cantilever beam

Wall

Column

HC-slabs

Aalto University J. Hanka

Rak-43.3111 Prestressed and Precast Concrete Structures 13.1.2015

Homework 3, Working stress design using limit core and zone 1(1)

Return to Moodle in PDF-format.

Simply supported post-tensioned beam in figure 1 is prestressed when the age of concrete is t=28d. After tensioning,

beam is loaded with live load qk =7,5 kN/m2. Span of the beam is Leff = 6m. Spacing of the beams is L2=12m.

* Concrete C50/60; Compressive strenght fck(t=28d)=fck=50 MPa

Mean flexural tensile strenght fctm(t=28d)=fctm=4,07 MPa

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) ∆σ=25%

* Initial prestress force obtained from previous design Pi= 2400 kN

*Smallest allowable distance of tendons from top or bottom of the section etop = ebot = 50mm

Figure 1. Post-tensioned beam section and relationship between limit kern & zone along the span [Modified from Naaman A.E. "Prestressed concrete analysis and design, Fundamentals", 2004, p.184]

Table 1. Allowable stresses of concrete in serviceability limit state (SLS).

Condition # Combination EN1990 Limitation EC2 Clause

Init

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

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

Fin

al III Max tension Characteristic σct.c < fctm

IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)

a) Form the calculation model of the beam and calculate the actions affecting the beam. Calculate the combination of

actions and effects of actions along the span:

- Initial combination pmin, Mmin(x)

- Characteristic combination pc, Mc(x)

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) With the given a prestressing force Pi and external moments calculated in (a), find the limiting eccentricities emin(x)

and emax(x) of the prestressing force along the span (limit zone) so that none of the allowable stresses given in table 1

are violated.

d) Draw the limit zone obtained in (c) and propose a tendon profile.

Voluntary additional assignment: e) Does the limit zone restrict the location of the tendons near support? What is the minimum value of distance xD from

the support at which tendons should be draped if maximum allowable eccentricity is used?

Note: Due to simplification it is allowed to analyze the beam section as a rectangular section.

Use gross-cross section properties in calculations.

Pi = Initial prestress before losses (conditions I-II)

Pf=Piη=Pi(1-∆σ)=Final prestress after all losses (Conditions III-VI)

Tip (a), (c) and (d): Run the computations every tenth of the span.

500

70

0

22

0

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 4.2.2015 Homework 4, Analysis of a prestressed continuous slab using loadbalancing 1(1)

Return to Moode in PDF-format.

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.

- Concrete class C35/45, fck=35MPa, Ecm=34GPa, γC=1,5, αcc=0,85, creep factor φ=1,5 - Unbonded tendons. Grade St1600/1860, fp0,1k=1600 MPa, fpu=1860MPa, Ep=195GPa, γP=1,15 - Smallest distance of tendons from the bottom/top of the section ep=50mm - Area of one tendon Ap1=150mm2. Spacing of tendons ccP=125mm. Jacking stress σmax=1300MPa. - Total prestress losses (initial & timedependant) are assumed to be Δf=25% in all sections [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 - Smallest distance of reinforcement from the bottom/top of the section es=40mm - 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 characteristic and quasi-permanent combinations: L/250

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

- Allowable tensile and compressive stress in concrete σct.all=fctm ; σcc.all=0,6fck

- Allowable tensile stress in tendons (during prestressing) σp.all=0,9fp0,1k

Goal of the assignment is to analyze the slab and check does it fulfill the given requirements in serviceability- and ultimate limit states for the given dead loads, live loads and tendon profile. Assignment is solved by investigating one unit width b=1m of the slab.

Figure 1. Two-span post-tensioned slab and tendon profile. a) Form the calculation model of the slab and calculate the actions affecting the slab. Calculate the loadbalancing forces due to initial tendon force Pm.0 and final tendon force Pm.t. Check that the allowable stresses (σct.max<fctm ; σc.max<0,6fck) are not exceeded at middle support in serviceability limit state for the following loading situations: c) …when slab is loaded with initial tendon force Pm.0 and slab selfweight (initial situation during prestressing). b) …when slab is loaded with final tendon force Pm.t and characteristic combination of actions pc. d) Calculate the maximum value of deflection in serviceability limit state when slab is loaded with final tendon force Pm.t and characteristic combination of actions pc. Is the calculated deflection bigger than the allowable deflection? e) Calculate the design value of effects of actions (bending moment MEd) and resistance of actions (MRd) in ultimate limit state at middle support. Is the moment resistance of the slab adequate? Instructions: You can make justified simplifications in the calculations. Use gross-cross section properties in the calculations.

Slab geometry: L=10 800 mm h=220 mm a0=300mm Tendon geometry: L1=4 240 mm L2=9 720 mm eP=50 mm eS=40 mm

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 25.2.2015 Homework 5, Prestress losses of pretensioned precast pile 1(1)

Return to Moodle in PDF-format.

Precast pretensioned pile given in figure 1 has 8 tendons, area of one tendon is 93 mm2. All tendons are prestressed to maximum initial prestress σmax and tendons are released simultaneously. Strength of the concrete at release is fck=50MPa. After stressing pile is installed and it is affected by permanent normal force Nqp=1200kN to the centroid of the cross section, selfweight of concrete can be neglected. Different modules of elasticities should be taken into account in the calculations. Pile is casted, prestressed, installed and loaded according to the timetable below: Time: Action: t=0 d Prestress of tendons σmax and casting of concrete. t=28 d Removal of formworks and release of tendons. t=29 d(…50*365d) Pile is installed and normal force Nqp affects the pile.

Figure 1. Symmetrically pretensioned precast pile. Goal of the assignment is to evaluate the amount of time dependent prestress losses in tendons and calculate the stresses of the concrete section after all losses. a) Calculate the prestress losses due to relaxation ΔPrel of the tendons at time t=28d, before tendons are released. b) Calculate the stress of the concrete section σc0 and stress in tendons σp0, immediately after release of tendons. c) Calculate the stress of the concrete section σc.QP and stress in tendons σP.QP at time t=29d, when the external normal force Nqp starts to effect the pile. d) Calculate the amount of time dependent losses Δσc+s+r, when the age of pile is t=50*365 d. Creepfactor of concrete is assumed to be φ=φ(t=50*365)=1,5 and total shrinkage of concrete is assumed to be εcs=0,05%. e) Calculate the final stresses of the concrete section σc.QP.fi and final stress of tendons σP.QP.fi at time t=50*365, when the pile is affected by normal force Nqp and all losses (immediate and time dependant) are considered. Note (a) & (d): Loss due to relaxation in tendons may be evaluated with EN 1992-1-1 equation (3.28):

3)1(75,0

7,61000 10

100039,5

t

epi

pr

Δσpr = Loss due to relaxation in tendons at time “t” σpi = stress in tendons t = time after tensioning in hours μ = σpi/fpk = ratio of tendon stress with respect to ultimate strenght Note (d): For centric loading time dependent losses may be evaluated with simplified equation:

prpcsQPccm

prscp E

E

E 8,0,,

Δσp,s+s+r = loss in tendons due to relaxation of tendons, shrinkage and creep of concrete Δσpr = Loss due to relaxation in tendons at the time under consideration

Concrete Class: C50/60 Compressive strength fck=fck(t=28d)=50 MPa Modulus of elasticity Ecm=37 GPa Bonded tendons Class: St1640/1860 Initial prestress: σmax=1000 MPa Modulus of elasticity Ep=195GPa Ultimate strength fpk=1860 MPa Relaxation after 1000h ρ1000=8%=0,08 Relaxation class: 1

Aalto University J. Hanka Rak-43.3111 Prestressed and Precast Concrete Structures 21.1.2015 Homework 6, Ultimate strength of composite structure 1(1)

Return to Moodle in PDF-format.

Hollowcore slab figure 1 is prestressed with straight pre-tensioned bonded tendons (initial prestress σmax=1000MPa). Area of one tendon is Ap1=93mm2. Total number of tendons is 8. Strength class of the slab is C40/50. Span of the slab is L=10m and supports are assumed to be hinged. When the age of the slab is t=28d topping of C20/25 with thickness 50mm is casted on top of the slab. After hardening of topping, hollowcore slab and topping are assumed to act together as a composite structure. Information: * Live area load acting on the slab: qk=10kN/m2. * Hollowcore slab concrete C40/50, fck=40MPa, selfweight ρc=25 kN/m3, strain at ultimate εcu=0,0035 [EC2 Table 3.1] * Topping concrete C20/25, fck=20MPa, selfweight ρc=25 kN/m3, strain at ultimate εcu=0,0035 [EC2 Table 3.1] * Prestressing steel: Ep=195GPa; fp0,1k=1640 MPa; fpk=1860 MPa; εpuk=3%; straight tendons * Strain hardening of prestressing nor reinforcing steel is not taken into account [EN 1992-1-1 fig 3.10] * Initial prestress σmax=1000 MPa. Assumed total prestress losses (immediate and timedependant) 15 %. * Material partial factors for concrete γc=1,50; αcc=0,85 and tendons 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 2a calculation model λ=0,80; η=1,00 [EN1992-1-1 3.1.7(3)]

Figure 1. Pre-tensioned hollow core slab and topping. a) Calculate the bending moment resistance of the hollowcore slab MRd.hc without the composite action between topping and hc-slab. b) Calculate the bending moment resistance MRd.comp of the composite structure. c) How much is the bending moment resistance increased when composite action is taken into account? d) Calculate the design value of bending moment MEd in Ultimate Limit State at midspan. Is the bending moment resistance of the composite structure adequate? If not, how the capacity could be improved? e) What kind of requirements utilization of composite action between hollow core slab and topping imposes to the execution of the structure?

(a) (b)

Figure 2. (a) Calculation model in ultimate limit state. (b) Stress-strain curve of tendons [EC2 fig 3.10].

fcd=αccfck/