This document contains homework assignments for a course on prestressed concrete structures. It includes 5 assignments related to the design and analysis of precast pretensioned and post-tensioned beams. The first assignment involves designing a precast pretensioned beam and analyzing stresses, deflections, and tendon layout. The second compares required reinforcement for pretensioned, bonded post-tensioned, and unbonded post-tensioned beams. The third involves designing an unbonded post-tensioned T-beam. The fourth calculates prestress losses and deformations in an unbonded post-tensioned beam. The fifth assignment involves designing a precast pretensioned composite beam.

1. Aalto University Janne Hanka
CIV-E4050 Prestressed concrete structures 20-Jun-18
Homework assignments and solutions, 2017
All rights reserved by the author.
Foreword:
This educational material includes assignments of the course named CIV-E4050 Prestressed
concrete from the 2017. 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 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: 2017
Table of contents:
Homework 1. Design of precast pretensioned beam
Homework 2. ULS Design of prestressed beams
Homework 3. Design of post-tensioned beam
Homework 4. Prestress losses and deformations of a post-tensioned beam
Homework 5. Design of precast pretensioned composite beam

2. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 1, Design of precast pretensioned beam 1(2)
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You are designing a precast single-span rectangular beam (figure 1 and 2) that will be prestressed with
pretensioned bonded tendons in a typical carage structure. Beams are supporting a floor made of precast
panel slabs (height of panel slabs hKL=120mm) and cast in place slab (total slab thickness
hKL+hCIP=hTOT=220mm). 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= 1100 MPa
- Assumed smallest distance of tendon centroid from the bottom/top of the section ep=90mm
- Total prestress losses (initial & timedependant) are assumed to be Δf=20% [Pm.t=Pmax(1-Δf)]
- Beam span length: L1=17m. Spacing of beams (slab span lengths) L2=8,1m.
- Superimposed dead load: gDL= 0,5 kN/m2
. Concrete selfweight ρc=25kN/m3
.
- Liveload qLL=5 kN/m2
. Combination factors: ψ0=0,7; ψ1=0,5; ψ2=0,3 (EN 1990 Class G, garages)
Purpose of this HW is to predesign and analyze precast-pretensioned beam.
a) Form the calculation model of the beam. Choose the beam height H and width Bw. Calculate the effect of
actions due to selfweight, dead load and live load at midspan.
b) Calculate the cross-section properties used in the prestress design:
- Moment of inertia and cross section area IC , AC *
c) 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.
d) Check that the allowable stresses given in table 1 are not exceeded in critical section at midspan.
e) 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.
f) Draw a schematic drawing (cross section) of the beam and place the tendons inside the beam. Assume
cover to stirrups c=40mm. Stirrup diameter 12mm.
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 < 0 * 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.
*Note (b): You can use simplified gross-cross section or more accurate transformed section properties
EP/ECM

3. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 1, 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.

4. Aalto University J. Hanka
Rak-43.3111 Prestressed and Precast Concrete Structures 2016 9.9.2017
Homework 2, ULS Design of prestressed beams 1(1)
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Goal of this HW is compare the effect of prestressing method (pretensioned / post-tensioned) to the required
amounts of rebar in ULS.
- Beam concrete strength at final condition: C35/45
- Consequence class CC2
- Tendons: Grade St1640/1860. Area of tendons AP=3600mm2
.
Centroid of tendons from bottom of beam eP=95mm
- Rebar: A500HW. Centroid of rebar from bottom of beam eS=65mm
- Prestress force in tendons at release/jacking stress σmax= 1400 MPa
- Total prestress losses (initial & timedependant) are assumed to be Δf=15% [Pm.t=Pmax(1-Δf)]
- Stress increase of unbonded tendons in ULS Δσp.ULS=50MPa [EN1992-1-1 5.10.8(2)]
- Beam span length: L1=17m.
- Concrete selfweight ρc=25kN/m3
. Live and dead lineloads to the beam qk=40 kN/m ; gk=27 kN/m
BEAM #1. Bonded pretensioned tendons (tartuntajänteet), straight tendon geometry:
a) Calculate the effect of actions moment MEd at midspan and shear force at support VEd.
b) Calculate the required amount of flexural reinforcement AS at midspan for the bending moment obtained in (a).
Calculate also the required amount of shear reinforcement ASW for the shear force VEd obtained in (a).
BEAM #2. Bonded post-tensioned tendons (injektoidut ankkurijänteet), parabolic tendon geometry:
c) Calculate the effect of actions moment MEd midspan and shear force at support VEd.
d) Calculate the required amount of flexural reinforcement AS at midspan for the bending moment obtained in (c).
Calculate also the required amount of shear reinforcement ASW for the shear force VEd obtained in (c).
BEAM #3. UBbonded post-tensioned tendons (tartunnattomat ankkurijänteet), parabolic tendon geometry:
e) Calculate the effect of actions moment MEd at midspan and shear force at support VEd.
f) Calculate the required amount of flexural reinforcement AS at midspan for the bending moment obtained in (e).
Calculate also the required amount of shear reinforcement ASW for the shear force VEd obtained in (e).
H = 800 mm
Hf = 180 mm
bw = 1100 mm
beff = 3000 mm
AP = 3450 mm2
eP = 95mm
eS = 65mm
H = 800 mm
Hf = 180 mm
bw = 1100 mm
beff = 3000 mm
AP = 3450 mm2
eP = 95mm
eS = 65mm
H = 800 mm
Hf = 180 mm
bw = 1100 mm
beff = 3000 mm
AP = 3450 mm2
eP = 95mm
eS = 65mm

5. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 3, Design of post-tensioned beam 1(2)
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You are designing a cast-on-situ single-span T-beam (figures 1 and 2) that will be prestressed with post-
tensioned unbonded tendons and monoanchors. Beam height is H and width Bw. Slab (beam flange)
thickness is hTOT=180mm. Beams are supported by columns. Connection between beam and columns is
hinged.
- Beam concrete strength at final condition: C35/45
- Beam concrete strength during stressing/release of tendons: C20/25
- Exposure classes XC3, XF1. Design working life: 50 years. Consequence class CC2
- Unbonded tendons and monoanchors. Grade St1640/1860. Diameter 15,7mm. Area of one tendon
Ap1=150mm2
. Tendon geometry is assumed to be parabolic.
- Jacking force for one tendon Pmax= 210 kN
- Assumed smallest distance of tendon centroid from the bottom/top of the section eP=95 mm
- Assumed height of centroid of anchors at beam end is eA=H/2
- Total prestress losses are assumed to be Δf=15% [Pm.t=Pmax(1-Δf)= ~180kN]
- Beam span length: L1=17m. Spacing of beams (slab span lengths) L2=8,1m.
- Superimposed dead load: gDL= 0,5 kN/m2
. Concrete selfweight ρc=25kN/m3
.
- Liveload qLL=5 kN/m2
. Combination factors: ψ0=0,7; ψ1=0,5; ψ2=0,3 (EN 1990 Class G, garages)
a) Form the calculation model of the beam. Choose the beam height H and width Bw. Calculate the effect of
actions due to selfweight, dead load and live load at midspan.
b) Calculate the effective width beff of the flange and cross-section properties used in the prestress design:
- Moment of inertia and cross section area IC , AC *
c) Choose the amount of tendons and tendon geometry (distance of tendon centroid from bottom of
beam). Calculate the load balancing forces along the span and bending moment due to tendon forces at
midspan.
d) Check that the allowable stresses given in table 1 are not exceeded in critical section at midspan.
e) Calculate the beam rotation at support and deflection at midspan for the quasi-permanent
combination. Check that the allowable deflection given in table 1 is not exceeded. Calculate the beam
shortening due to prestress also.
f) Draw a schematic drawing (cross section) of the beam and place the tendons inside the beam. Assume
cover to stirrups c=40mm. Stirrup diameter 12mm.
Table 1. Allowable stresses of concrete in serviceability limit state (SLS) for unbonded 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.c < fctm
IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)
V 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 Quasi-permanent wk.max < 0,3mm 7.3.1(5)
Note (b): You can use simplified gross-cross section properties
Tip for (b): http://www.adaptsoft.com/resources/ADAPT_T901_Effective-Width-PT-beamsr.pdf

6. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 3, Design of post-tensioned beam 2(2)
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Figure 1. Plan view and main section of the floor. Sideview of beam with the parabolic tendon geometry.
Figure 2. Typical section of middle beam under consideration in this homework.

7. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 9.9.2017
Homework 4, Prestress losses and deformations of a post-tensioned beam 1(1)
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T-beam in figure 1 will be prestressed with unbonded tendons. Beam height is H=800mm and width Bw=1100mm. Slab
(beam flange) thickness is hTOT=180mm. Beams are supported by columns. Connection between beam and columns may be
assumed to be hinged and fixed during stressing.
- Beam concrete strength at final condition: C35/45
- Beam concrete strength during stressing of tendons: C25/30
- Unbonded tendons and monoanchors. Grade St1640/1860. Diameter 15,7mm. Relaxation class 2, ρ1000=2,5%.
Tendon geometry is parabolic. Area of one tendon Ap1=150mm2
. Number of tendons np=24
- Jacking force for one tendon Pmax= 210 kN.
- Allowable stress in tendons during stressing σmax.all = min{0,80 fpk ; 0,90 fp0,1 }
- Allowable stress in tendons after stressing and locking of anchors σpm0.all = min{0,75 fpk ; 0,85 fp0,1 }
- Distance of tendon centroid from the bottom/top of the section eP=95 mm
- Assumed height of centroid of anchors at beam end is eA=400 mm
- 2nd
degree parabolic tendon geometry: u(x)=ax2
+bx+c
- Beam span length: L1=17m. Spacing of beams (slab span lengths) L2=8,1m.
Figure 1. Single-span unbonded post-tensioned T-beam with hinged supports.
a) Calculate the immediate losses due to friction ΔPμ, anchorage set ΔPsl and instantaneous deformation of concrete ΔPel.
(Tip: in this case the anchorage set will extend over the beam length).
Immediate prestress losses due to friction can be calculated with the following information
* Losses due to friction in post-tensioned tendons [EC2 Eq.(5.45)]: ΔPμ(x)=PMAX(1-e-μ(θ+kx)
)
* θ is the sum of the angular displacements over a distance x
* Coefficient of friction between the tendon and its duct μ=0,06 (for unbonded tendons)
* Unintentional angular displacement for internal tendons k=0,01/m (for unbonded tendons)
b) Draw a curve that describes the tendon force after initial losses from jacking end (x=0) to the dead anchorage end
(x=L). What is the average tendon force after initial losses Pm.0?
c) Check is the allowable stress in the tendon immediately after tensioning exceeded. If not, how much the jacking force
could be increased?
d) Calculate the long-term losse. What is the average tendon force after all losses Pm.t? How much of the initial jacking
force is lost? Creep factor of concrete is φ=2 and total shrinkage in concrete is εcs=0,030%.
Time dependent losses for unbonded tendons may be evaluated with simplified EC2 equation:
Δσp,c+s+r=EP/ECM*φ*σc,QP + εcs*Ep + 0,8*Δσpr
Δσp,s+s+r = Loss in tendons due to relaxation of tendons, shrinkage and creep of concrete
σc,QP = Average compressive stress in concrete due to precompression after immediate losses = Pm.0/Ac
Δσpr = Loss due to relaxation in tendons
Loss due to relaxation for unbonded tendons may be evaluated with simplified equation [BY69 / NCCI2 3.3.2(5)]:
Δσpr=3*ρ1000*σp.m.0
Δσpr = Loss due to relaxation in tendons
σp.m.0 = average stress in tendons after immediate losses = Pm.0/Ap
e) Calculate the elongation of the tendons at the active end after stressing.
1=Active end 2=Anchorage end
* value for an-
chorage set
slip = 6mm

8. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 5, Design of precast pretensioned composite beam 1(2)
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You are designing a precast single-span rectangular 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=120mm) and cast in place slab (total slab thickness hKL+hCIP=hTOT=220mm). Beams are supported
by columns. Connection between beam and columns may be assumed to be hinged. Composite action
between the slab and beam is to taken into account. Beams are propped during casting of topping, see
figure 1 and 2.
- Beam and panel slab concrete strength at final condition: C50/60
- Beam concrete strength during stressing/release of tendons: C25/30
- Cast-in-place concrete strength at final condition C35/45
- Cover to rebar and stirrups c=40mm
- 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= 1100 MPa
- Assumed smallest distance of tendon centroid from the bottom/top of the section ep=90mm
- Total prestress losses (initial & timedependant) are assumed to be Δf=20% [Pm.t=Pmax(1-Δf)]
- Beam span length: L1=17m. Spacing of beams (slab span lengths) L2=8,1m.
- Superimposed dead load: gDL= 0,5 kN/m2
. Concrete selfweight ρc=25kN/m3
.
- Liveload qLL=5 kN/m2
. Combination factors: ψ0=0,7; ψ1=0,5; ψ2=0,3 (EN 1990 Class G, garages)
a) Form the calculation model of the beam. Choose the beam height H and width Bw. Calculate the effect of
actions due to selfweight, dead load and live load at midspan.
b) Calculate the effects of actions for the beam due to after removal of temporary supports.
c) Calculate the cross-section properties used using method of transformed section properties:
- Effective width of the flange beff
- Moment of inertia and cross section area of the prestressed beam IB , AB
- Moment of inertia and cross section area of the composite beam section IC , AC
d) Choose the amount of tendons and tendon geometry (distance of tendon centroid from bottom of
beam). Check that the allowable stresses given in table 1 are not exceeded in critical section at midspan
with the chosen tendon amounts.
e) 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.
f) Draw a schematic drawing (cross section) of the structure and place the tendons inside the beam and the
rebar over panel slab supports. Assume cover to stirrups c=40mm. Stirrup diameter 12mm.
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 < 0 * 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.

9. Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2017 9.9.2017
Homework 5, Design of precast pretensioned composite beam 2(2)
Return to MyCourses in PDF-format.
Figure 1. Plan view and main section of the floor.
Figure 2. Typical section of middle beam under consideration in this homework.