This homework involves analyzing a composite pretensioned concrete beam with bonded tendons. The beam supports precast plank slabs and a cast-in-place concrete topping. The student is asked to: 1. Develop a calculation model for the beam considering effects of self-weight, plank slab installation, and live loads. 2. Check stresses in the beam and deflections under various load combinations to ensure compliance with design codes. 3. Draw a cross section of the beam showing the placement of tendons. The provided document gives specifications of the beam, slabs, reinforcement, and loads to facilitate the design checks. Composite action between the beam and slabs is to be considered in the analysis

1. Aalto University Janne Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 18-Aug-23
Homework assignments and solutions, 2023
All rights reserved by the author.
Foreword:
This educational material includes assignments of the course named CIV-E4050 Prestressed and
Precast Concrete Structures from the academic year 2023. Course is a 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 in English. European standards EN 1990 and EN 1992-1-1 are
applied in the problems.
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: 2023
Table of contents:
Homework 1. Principles, prestressed bolt
Homework 2. Design of a pretensioned beam
Homework 3. Analysis of a post-tensioned beam
Homework 4. Analysis of a composite beam

2. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 14.8.2023
Homework 1, Prestressed bolt connection 1(1)
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You are investigating a prestressed bolt connection. Anchor plates can be assumed to be rigid, concrete
anchoring capacity is not a limiting factor. Characteristic material properties and the symbols given can be
used. Anchorbolt is free to move inside the hole that has been drilled through the slab.
Partial factors for materials and loads can be neglected in this exercise (yM=yL=1).
- Yield and ultimate strength of the anchor bolt rod fy=950MPa ; fu=1050MPa
- Ultimate strain of the anchor bolt rod material εu=3,0 %
- Anchor rod diameter diam=35mm
- Modulus of elasticity of the bolt rod and anchor plates Ep=195GPa
- Concrete strength of the slab C35/45
- Thickness of the slab hL=900mm
- Dimensions of the anchor rod plates: 300 * 300 * t=50mm
Figure 1. Section of a slab that has been bolted with a Prestressed anchor plate.
Bolt is prestressed in such a way that the remaining prestress force in the bolt after losses is Pm.0 = 500kN
a) What is the total force in the bolt and the clearance between the bottom plate and concrete when the
load is Q1=0 kN ?
b) What is the total force in the bolt and the clearance between the bottom plate and concrete when the
load is Q2=300 kN ?
c) What is the total force in the bolt and the clearance between the bottom plate and concrete when the
load is Q3=600 kN ?
d) What is the contact pressure σc between the top plate and concrete when the load is Q4=400kN? And
what is the contact pressure between bottom plate and concrete for the same load?
e) What is the maximum force Qmax that the bolt allows? (Any partial safety factors for materials and
loads can be assumed to be y=1 due to simplification)
f) What should be the jacking force Pmax of the anchor rod? Slipping of anchor during stressing can be
assumed to be 1,25mm. (Friction and long term losses can be neglected.)
g) External load Q is increased in such a way that the total strain in the bolt is εu=2,999 % after which
the load is removed. What is the remaining prestress force in the bolt after removal of the external
load?

3. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 14.8.2023
Homework 2, Design of precast pretensioned beam 1(2)
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You are designing a precast single-span beam (figure 1 and 2) that will be prestressed with pretensioned bonded
tendons. Beams are supporting a floor made of precast panel slabs (height of panel slabs hKL=100mm) and cast in
place slab (total slab thickness hKL+hCIP=hTOT=200mm). Beams are supported by columns. Connection between
beam and columns may be assumed to be hinged. Composite action between the slab and beam shall be ignored.
- Beam concrete strength at final condition: C50/60
- Beam concrete strength during stressing/release of tendons: C25/30
- Exposure classes XC3, XF1. Design working life: 50 years. Consequence class CC2
- Bonded tendons. Grade St1640/1860. Diameter 12,5mm. Area of one tendon Ap1=93mm2
Tendon geometry is straight.
- Prestress force in tendons at release is σmax= 1350 MPa
- Assumed smallest distance of tendon centroid from the bottom/top of the section ep=120mm
- Total prestress losses (initial & timedependant) are assumed to be Δf=15% [Pm.t=Pmax(1-Δf)]
- Beam span length: L1=15m. Spacing of beams (slab span lengths) L2=5m.
- Beam dimensions are H=880mm and B=580mm.
- Superimposed dead load: gDL= 0,5 kN/m2
. Concrete selfweight ρc=25kN/m3
.
- Liveload qLL=2,5 kN/m2
. Combination factors: ψ0=0,7; ψ1=0,7; ψ2=0,6 (EN 1990 Class F, garages)
The purpose of this HW is to predesign and analyze precast-pretensioned beam. NOTE! Any composite action
between the beam and the slab shall be ignored.
a) Form the calculation model of the beam. Calculate the effect of actions due to selfweight, dead load and live
load at midspan. Calculate also the cross-section properties used in the prestress design:
- Moment of inertia and cross section area IC , AC *
b) Choose the amount of tendons and tendon geometry (distance of tendon centroid from bottom of beam).
Calculate the axial force and bending moment due to prestress at midspan.
c) Check that the allowable stresses given in table 1 are not exceeded in critical section at midspan in SLS.
d) Calculate the beam rotation at supports and deflection for quasi-permanent combination Δqp. Check that
the allowable deflection given in table 1 is not exceeded. Calculate the beam is shortening due to prestress also.
e) Check the flexural resistance in ULS (Ultimate limit stete) at midspan. Calculate the design moment due to
loads MEd and the cross section moment capacity MRd using the chosen amount of tendons. Is the capacity
adequate?
f) Draw a schematic drawing (cross section) of the beam and place the tendons inside the beam.
Table 1. Allowable stresses of concrete in serviceability limit state (SLS) for bonded tendons in XC3.
Condition # Combination EN1990 Limitation EC2 Clause
Initial
I Max tension Initial σct.ini < fctm.i
II Max compression Initial σcc.ini < 0,6*fck.i 5.10.2.2(5)
Final
III Max tension Frequent σct.f < fctm
IIIb Max tension Quasi-permanent σct.qp < fctk.0.05 FI-NA 7.3.1(5)
IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)
IVb Max compression Quasi-permanent σcc.c < 0,45*fck 7.2(3)
Max deflection Quasi-permanent
Creep factor = 2
Δ < Span / 250 7.4.1(4)
Max crack width Frequent wk.max < 0,2mm 7.3.1(5)
* Bonded tendons require decompression (vetojännityksettömyys) for quasi-permanent combination.
However, for tensile stress of fctk.0.05 is allowed for quasi-permanent loads that have a longterm combination
factor bigger than 0,5 according to FI-National annex.
*Note (b): You can use simplified gross-cross section or transformed section properties EP/ECM

4. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 14.8.2023
Homework 2, Design of precast pretensioned beam 2(2)
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Figure 1. Plan view and main section of the floor.
Figure 2. Typical section of middle beam under consideration in this homework.

5. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 10.8.2023
Homework 3, Investigation of a post-tensioned unbonded beam 1(2)
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You are investigating a post-tensioned beam JPV-7 (see figure 1 & 2) that has unbonded tendons.
- Beam concrete strength at final condition: C35/45
- Beam concrete strength during stressing of tendons: C25/30
- Unbonded tendons. Grade: fp0,1=1640MPa ; fpk=1860 MPa. Area of one tendon Ap1=150mm2
. Diameter of tendon
ducts (with the plastic pipe) dP=20mm.
Number of tendons np=24. Anchor type: Monoanchors, 1-tendon for one anchor
- Tendon geometry: See the attached drawings.
- Jacking force for one anchor Pmax= 218 kN / for one anchor
- Tendons allowable stress during jacking σmax.all = min{0,80 fpk ; 0,90 fp0,1 }
- Tendons allowable stress immediately after jacking σpm0.all = min{0,75 fpk ; 0,85 fp0,1 }
- Friction coefficient, wobble coefficient and slipping of anchors myy=0,06 ; beta=0,01/m ; slip=6mm
- Beam is supported by columns. You can assume hinged connection between beam and columns. You can also assume
that the beam is free to deform during stressing works (=>100% of the stressing force is transmitted to the beam).
Drw translations: Punosten tuenta = location of the tendon chair along the beam.
Punoskorot punoksen alapintaan = Distance between bottom of the beam and tendon bottom (tendon chair height)
HA = Korko ankkurin keskelle = Distance between bottom of the beam and centroid of the anchor
Figure 1. Section and side view of the beam. Loading information.
Part I - Calculation of immediate losses and elongations:
a) Form the calculation model of the tendons. Calculate the immediate losses due to friction ΔPμ, anchorage set ΔPsl and instantaneous
deformation of concrete ΔPel for the tendons.
b) Draw a curve that describes the tendon force after initial losses from jacking end to the dead anchorage end for the STAGE 1
tendons. What is the average tendon force after initial losses Pm.0?
c) Is the maximum stress in tendons within allowable limits during and immediately after jacking?
d) Calculate the elongation of the tendons at the stressing end before (∆max) and after locking of tendons (∆m.0).
Part II - Calculation of load balancing forces and effects of actions at midspan between modules PE-PD
e) Calculate the average force in the tendons after all losses Pm.t*. Assume long term losses is Δσp.c.s.r=100MPa. Calculate the load
balancing forces pbal along the beam based on the final tendon force Pm.t (see figure 3 for a tip)
f) Form the calculation model of the beam. Place the calculated load balancing forces to the calculation model with selfweight, dead
load and live load. Calculate the bending moment for the characteristic combination due to post-tensioning (PT), selfweight (SW),
dead load (DL) and live load (LL) at midspan between modules PE-PD. **
Part III - checking of stresses at at midspan between modules PE-PD
g) Check that the allowable stresses are not exceeded at the investigated section for the characteristic combination of actions. Check
the following criterias for the characteristic combination of actions (PT+SW+DL+LL): ***
(I) max tension < fctm (Does the cross section crack?)
(II) max compression < 0,6fck
h) Calculate the maximum deflection at midspan for long-term combination combination of actions (PT+SW+DL+0,3*LL). You can
use un-cracked cross section properties as a simplification.**
* Alternatively you can use value Pm.t = Ap.tot*σm.t = AP.tot * 1259 MPa
** Use of FEM program is allowed and encouraged https://www.dlubal.com/en/education/students/free-structural-analysis-software-for-students
*** “Effective” width beff=3000mm can be assumed in the calculations
Dimensions of the cross section that can be used in the calculations:
bw=1100 mm bf=5100 (spacing of beams) h=900mm hf=250mm
PE-PD span length: 15500mm ; PD-PC span length: 16325mm
Load on floor that can be used in the calculations:
Live load qk=8 kN/m2
; Dead Load gk= 12 kN/m2
; Concrete selfweight pc=25kN/m3

6. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 10.8.2023
Homework 3, Investigation of a post-tensioned unbonded beam 2(2)
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Figure 2. Plan view of the post tensioned floor with the beam marked (for information)
Location of tendon lowpoint L2+L1=Ltot=~0,5*Lspan
where Lspan is the total span length between supports
Figure 3. LOAD BALANCING FORCE FOR HALF-PARABOLAS [2]
[2] Bsc Thesis: Calculation of Instant Losses and Elongation in Post-tensioned Concrete Structures, Konsta Suominen
(2019) https://www.theseus.fi/handle/10024/163639

7. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 29.9.2023
Homework 4, Analysis of pretensioned composite beam 1(2)
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Precast single-span beam (figure 1 and 2) is prestressed with pretensioned bonded tendons. Beam supports a floor made
of precast panel slabs and cast in place slab (total slab thickness hKL+hCIP=hTOT). Beams are supported by columns.
Connection between beam and columns may be assumed to be hinged. Composite action between the slab and beam
shall be considered. Beams are propped during casting of topping, see figure 1 and 2.
- Beam concrete strength at final condition: C55/65 (fck.FINAL=55MPa)
- Beam concrete strength during stressing/release of tendons: C38 (fck.INI=38MPa)
- Plank slabs & Cast-in-place concrete strength at final condition C35/45 (fck.KL=fck.CIP=35MPa)
- Beam height & width: H=500mm Bw=780mm
- Thickness of the plank slabs hKL=120mm. Thickness of the cip-slab hCIP=80mm
- Exposure classes XC3, XF1. Design working life: 50 years. Consequence class CC2
- Bonded tendons. Grade St1640/1860. Diameter 12,5mm. Area of one tendon Ap1=93mm2
Tendon geometry is straight.
- Number of tendons: TOP tendons np.top=4 ; BOTTOM tendons np.bot=32
- Prestress force in tendons TOP TENDONS: σmax.top= 1100 MPa ; BOTTOM TENDONS σmax.top= 1320 MPa
- Centroid distance from the top of the beam for the top strands ep.top=50mm
- Centroid distance from the bottom of the beam for the bottom strands ep.bot=65mm
- Total prestress losses (initial & timedependant) are assumed to be Δf=15% [Pm.t=Pmax(1-Δf)]
- Beam span length: L1=16000 mm. Spacing of beams (slab span lengths) L2=5m and L3=5m
- Concrete selfweight ρc=25kN/m3
.
- Liveload qLL=2,5 kN/m2
. Combination factors: ψ0=0,7; ψ1=0,7; ψ2=0,6 (EN 1990 Class F, garages)
a) Form the calculation model of the beam. Calculate the effect of actions due to selfweight MSW, installation of plank
slabs MKL, and live load MLL at midspan.
b) Calculate the effects of actions due cast in place slab after removal of temporary supports MCIP.
c) Calculate the cross-section properties:
- Effective width of the flange beff
- Neutral axis height, moment of inertia and cross section area of the pretensioned beam yB, IB , AB
- Neutral axis height, moment of inertia and cross section area of the composite beam yC, IC , AC
d) Calculate the stresses for the prestressed beam at midspan in INITIAL condition.
e) Calculate the stresses for the composite section in the final condition for the quasi-permanent combination of
actions. Are the stresses within acceptable limits?
f) Calculate the change of stress in the section due to differential shrinkage at the bottom and top of the composite
section. Total shrinkage of the beam is εcs.b=0,30% and for the slab εcs.slab=0,34%.
g) Calculate the beam deflection at midspan quasi-permanent combination Δqp. Is the deflection acceptable?
h) Check the flexural resistance in ULS (Ultimate limit state) at midspan for the composite section. Calculate the
design moment due to loads MEd and the cross-section moment capacity MRd.comp. Is the capacity adequate?
Table 1. Allowable stresses of concrete in serviceability limit state (SLS) for bonded tendons in XC3.
Condition # Combination EN1990 Limitation EC2 Clause
Initial
I Max tension Initial σct.ini < fctm.i
II Max compression Initial σcc.ini < 0,6*fck.i 5.10.2.2(5)
Final
III Max tension Quasi-permanent** σct.qp < fctk.0.05 FI-NA 7.3.1(5)
IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)
IVb Max compression Quasi-permanent** σcc.c < 0,45*fck 7.2(3)
Max deflection Quasi-permanent
Creep factor = 2
Δ < Span / 250 7.4.1(4)
Max crack width Frequent wk.max < 0,2mm 7.3.1(5)
*Note: Bonded tendons require decompression (vetojännityksettömyys) for quasi-permanent combination.
However, tensile stress of fctk.0.05 is allowed for quasi-permanent loads that have a longterm combination factor bigger than
0,5 according to FI-National annex.
**Long term combination factor ψ2 can be taken as zero for traffic lanes according to FI-NA. Width of the traffic lane is
Leff - 2*5m and it is placed at the center of the beam.

8. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2023 29.9.2023
Homework 4, Analysis of pretensioned composite beam 2(2)
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Figure 1. Plan view and main section of the floor.
Figure 2. Section of middle beam under consideration. Support length for the plank slabs is Akl=100mm.