Prestressed concrete course assignments 2018

Aalto University Janne Hanka
CIV-E4050 Prestressed concrete structures 17-Oct-18
Homework assignments and solutions, 2018
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: 2018
Table of contents:
Homework 1. Free-body-diagrams of PT-structures
Homework 2. ULS Design of anchor area in post-tensioned beam
Homework 3. Design of post-tensioned slab
Homework 4. Prestress losses and deformations of a post-tensioned beam
Homework 5. Design of precast pretensioned composite beam using inequality equations
ANNEX:
Homework formulas

Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2018 11.9.2018
Homework 1, Free body diagrams of post-tensioned structures 1(7)
Return to MyCourses in PDF-format.
Beams in pictures (a…h) are stressed with post-tensioned tendons. Profile of the tendon is shown in the
pictures. Tendon is stressed at the stressing end with force “P”. Curved parts of the tendon profile are
parabolas, which can be assumed to produce an evenly distributed vertical load over the length of each
parabola. Any losses in tendon force can be neglected due to simplification.
Goal of the assignment is to draw and sketch the forces transmitted from the tendon to the concrete as a free
body diagram of the concrete structure (=free-body diagram of the concrete structure after removal of
tendon).
- Load balancing force due to tendon profile
- Point moments due to eccentricity of the anchor at the anchor areas and change of cross section
Tip see also:
https://adaptsolutions.files.wordpress.com/2010/01/load-balancing.pdf

Example2:

Tip: example of coupler anchor (p. 23/42 and 35/42):
https://www.naulankanta.fi/files/BBR_ETA-06-0165_CMM_EN_Rev4_0916.pdf

Tip: example of movable coupling (p. 30/40 and 24/40):
https://www.dywidag-systems.com/fileadmin/downloads/global/construction/approvals/en/dsi-suspa-systems-eta-03-
0036-monostrandsystem-en.pdf

Rak-43.3111 Prestressed and Precast Concrete Structures 2016 12.9.2018
Homework 2, ULS Design of anchor area in post-tensioned beam 1(1)
Goal of this HW is to design the anchor area reinforcement for the given unbonded beam. Beam dimensions:
h=1000mm and b=1200mm.
- Beam concrete strength During stressing of tendons: C25/30
At final condition: C35/45
- Consequence class CC3
- Tendons: Grade St1640/1820. Area of one tendon: AP=165 mm2
.
Unbonded tendons, Multianchors (4-tendons for 1-anchor)
ETA: http://www.bbrnetwork.com/fileadmin/bbr_network/PDFs/Approvals/CMM/BBR_ETA-06-0165_CMM_EN_Rev4_0916.pdf
- Rebar: A500HW.
- Prestress force each anchor: Jacking force: Pmax = 4x234 = 936kN/ANCHOR
Total number of anchors = 8. Layout of anchors see fig.1.
Anchors are stressed one-by-one with one stressing jack.
- Initial losses for each anchor (after locking of tendons) is are assumed to be Δini=10% [Pm.0=Pmax(1-Δini)]
a) Choose and justify your choice for the sequence of jacking of anchors.
a) Form the strut & tie model of the beam end due to jacking forces
b) Calculate the required rebar to be used in the anchorage zone.
d) Draw the reinforcement at the beam end. Tip. You can use the autocad file given in MyCourses
Figure 1. Layout of anchors at beam end.

Homework 3, Design of post-tensioned one-way-slab 1(2)
You are designing a cast-on-situ cantilevering one-way-slab (figures 1 and 2) that will be prestressed with
post-tensioned unbonded tendons and monoanchors. Slab thickness is hL. Slab is supported by walls.
Connection between slab and walls is hinged.
- Concrete strength at final condition: C35/45
- Concrete strength during stressing of tendons: C25/30
- Exposure classes XC1. Design working life: 100 years. Consequence class CC3
- Unbonded tendons and monoanchors. Grade St1640/1860. Diameter 15,7mm. Area of one tendon
Ap1=150mm2
.
- Jacking force for one tendon Pmax= 210 kN
- Assumed smallest distance of tendon centroid from the bottom/top of the section eP=85 mm
- Assumed height of centroid of anchors at beam end is eA=hL/2
- Total prestress losses are assumed to be Δf=15% [Pm.t=Pmax(1-Δf)= ~180kN]
- Span length: L1=18m. Width of slab L2=50m. Cantilever lenght L3=4m
- Superimposed dead load: g1= 1,3 kN/m2
. Concrete selfweight ρc=25kN/m3
.
- Liveload q1=7 kN/m2
. Combination factors: ψ0=0,8; ψ1=0,7; ψ2=0,6 (Warehouses)
a) Form the calculation model of the slab (for the tendons to be designed in Y-direction). Choose the slab
thickness. 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. 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 deflection at (midspan & end of cantilever) 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

Homework 3, Design of post-tensioned one-way-slab 2(2)
Figure 1. Plan view and main section of the floor
Figure 2. Typical section of middle beam under consideration in this homework.

Homework 4, Calculation of prestress losses and elongations 1(1)
You are investigating a post-tensioned beam JPV-2 that has unbonded tendons and a construction joint.
Fixed couplers are used at the construction joint. Time between construction and stressing of stage 1 and 2 is
approximately 3 weeks.
- Beam concrete strength at final condition: C35/45
- Beam concrete strength during stressing of tendons: C25/30
- Unbonded tendons, BBR anchors ETA-06/0165. Grade: fp0,1=1640MPa ; fpk=1820 MPa. Area of one
tendon Ap1=165mm2
. Diameter of tendons (with the plastic pipe) dP=20mm.
Number of tendons np=32 + 32 (stage1 + 2). Anchor type: Multianchors, 4-tendons for one anchor
- Tendon geometry: See the attached JPV-2 Beam drawing.
- Jacking force for one anchor Stage 1 Anchors: Pmax= 940 kN / for one anchor / 4-tendons
Stage 2 Anchors: Pmax= 900 kN / for one anchor / 4-tendons
- 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 }
- Friction coefficient, wobble coefficient and slipping of anchors can be found from ETA-06-0165 [1, s.10-11]
- Allowable force in the construction joint [1, p.8/42, ch.1.2.3]:
Anchor force in Stage 1 anchor (after immediate losses) > Anchor force in Stage 2 coupler (during stressing)
a) Calculate the immediate losses due to friction ΔPμ, anchorage set ΔPsl and instantaneous deformation of concrete ΔPel.
for the tendons in STAGE 1.
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) Calculate the immediate losses due to friction ΔPμ, anchorage set ΔPsl and instantaneous deformation of concrete ΔPel.
for the tendons in STAGE 2.
d) Draw a curve that describes the tendon force after initial losses from jacking end to the dead anchorage end for the
STAGE 2 tendons. What is the average tendon force after initial losses Pm.0?
e) Check is the allowable stress in the tendon immediately after tensioning exceeded in any sections along the span.
f) Calculate the elongation of the tendons at the stressing end after stressing (for both Stage 1 and Stage 2 tendons).
[1] http://www.bbrnetwork.com/fileadmin/bbr_network/PDFs/Approvals/CMM/BBR_ETA-06-0165_CMM_EN_Rev4_0916.pdf
Dimensions of the cross section that can be assumed in the calculation:
bw=1200mm bf=8100 (spacing of beams)
h=800mm hf=300mm

Homework 5, Design composite beam using inequality equations 1(3)
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 be considered. 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=15% [Pm.t=Pmax(1-Δf)]
- Beam span length: L1=17m. Spacing of beams (slab span lengths) L2=8,1m.
- Beam height & width: H=980mm Bw=880mm
- 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. Calculate the effect of actions due to selfweight, dead load and live
load at midspan. Calculate also the combination of actions required for the flexural analysis of the section.
b) 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
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/Pmax, e0). Draw the graphical representation (Magnel’s diagram) of the
stress inequality conditions where vertical axis is e0 and horizontal axis is 1/Pmax (figure 1).
e) Choose the value of initial prestress Pmax (=amount of tendons) and eccentricity e0 that satisfies the conditions
I-IV in table 1.
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
Initia
l
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)
*Note: Bonded tendons require decompression (vetojännityksettömyys) for quasi-permanent combination.

Figure 1. Plan view and main section of the floor.
Figure 2. Section of middle beam under consideration.

Figure 3. Graphical presentation of stress inequality equations, cross section and stresses.

Prestressed concrete course assignments 2018

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