![Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2020 13.9.2020
Homework 2, Stress analysis using inequality equations 1(2)
Return to MyCourses in PDF-format.
You are investigating a prestressed concrete beam and its cross-section. Use linear theory of elasticity and it
can be assumed that the external moment does not change its direction.
- Concrete strength at final condition: C35/45
- Concrete strength during stressing: C25/35
- Live load q=30kN/m
a) Form the calculation model of the beam. Calculate the effect of actions due to selfweight, dead load
and live load at midspan.
b) Determine the inequality equations required for the flexural analysis of the section at midspan. Use
criterias given in table 1.
c) Give the equations in the form required for presenting them graphically in the coordinate syem
(1/Pmax, e0). Where Pmax is the jacking force and its value after all losses is Pm.t=μPmax.
d) Place the given (see figure 1) 1/Pmax end e0 value to the graph. Does the tendon force satisfy all given
criteria’s given in table 1 at midspan?
e) Determine the cross-sections without any axial tension stress along the beam with the given tendon
geometry shown in figure 1 when the live load is affecting.
Figure 1. Prestressed beam. Units in [mm].
f) Determine alternative tendon force, tendon geometry and the location of the tendon along the beam
where all cross-sections remain compressed when the live load is affecting.
Table 1. Allowable stresses of concrete in serviceability limit state (SLS) for unbonded tendons.
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 Characteristic σct.f < fctm
IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2)
Initial combination: SW + PTmax
Characteristic combination: SW + DL + LL + PT
SW = Selfweight ; DL = Imposed Dead Load ; LL = Imposed Live Load
PTmax = Prestressing force ; PT = Prestress Force after all losses
Tendon force
Pmax=600kN
Total losses 10%
b=500mm](https://image.slidesharecdn.com/civ-e4050ppcs20cover-201011115857/85/Prestressed-concrete-course-assignments-2020-3-320.jpg)
![Aalto University J. Hanka
CIV-E4050 Prestressed and Precast Concrete Structures 2020 12.09.2020
Homework 3, Investigation of a post-tensioned unbonded beam 2(3)
Return to MyCourses in PDF-format.
Figure 2. Plan view of the post tensioned floor with the beam marked (for information)
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](https://image.slidesharecdn.com/civ-e4050ppcs20cover-201011115857/85/Prestressed-concrete-course-assignments-2020-5-320.jpg)
Prestressed concrete course assignments 2020
- 1. Aalto University Janne Hanka CIV-E4050 Prestressed concrete structures 11-Oct-20 Homework assignments and solutions, 2020 All rights reserved by the author. Foreword: This educational material includes assignments of the course named CIV-E4050 Prestressed concrete from the 2020. 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: 2020 Table of contents: Homework 1. Prestressed bolt connection Homework 2. Stress analysis using inequality equations Homework 3. Investigation of a post-tensioned unbonded beam (losses, load balancing forces, stresses) Homework 4. Analysis of a Continuous Prestressed composite slab
- 2. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 2020 9.9.2020 Homework 1, Prestressed bolt connection 1(2) Return to MyCourses in PDF-format. 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. Anchor bolt 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. - 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 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 slab when the load is Q4=400kN ? 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.)
- 3. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 2020 13.9.2020 Homework 2, Stress analysis using inequality equations 1(2) Return to MyCourses in PDF-format. You are investigating a prestressed concrete beam and its cross-section. Use linear theory of elasticity and it can be assumed that the external moment does not change its direction. - Concrete strength at final condition: C35/45 - Concrete strength during stressing: C25/35 - Live load q=30kN/m a) Form the calculation model of the beam. Calculate the effect of actions due to selfweight, dead load and live load at midspan. b) Determine the inequality equations required for the flexural analysis of the section at midspan. Use criterias given in table 1. c) Give the equations in the form required for presenting them graphically in the coordinate syem (1/Pmax, e0). Where Pmax is the jacking force and its value after all losses is Pm.t=μPmax. d) Place the given (see figure 1) 1/Pmax end e0 value to the graph. Does the tendon force satisfy all given criteria’s given in table 1 at midspan? e) Determine the cross-sections without any axial tension stress along the beam with the given tendon geometry shown in figure 1 when the live load is affecting. Figure 1. Prestressed beam. Units in [mm]. f) Determine alternative tendon force, tendon geometry and the location of the tendon along the beam where all cross-sections remain compressed when the live load is affecting. Table 1. Allowable stresses of concrete in serviceability limit state (SLS) for unbonded tendons. 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 Characteristic σct.f < fctm IV Max compression Characteristic σcc.c < 0,6*fck 7.2(2) Initial combination: SW + PTmax Characteristic combination: SW + DL + LL + PT SW = Selfweight ; DL = Imposed Dead Load ; LL = Imposed Live Load PTmax = Prestressing force ; PT = Prestress Force after all losses Tendon force Pmax=600kN Total losses 10% b=500mm
- 4. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 2020 12.09.2020 Homework 3, Investigation of a post-tensioned unbonded beam 1(3) Return to MyCourses in PDF-format. 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=1650MPa ; 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 - 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). 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 and 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) 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 C-B d) 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) e) 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 critical section Check that the allowable stresses are not exceeded in critical section for the characteristic combination of actions. Check the following criterias for the characteristic combination of actions (PT+SW+DL+LL): *** f) (I) max tension < fctm (Does the cross section crack?) g) (II) max compression < 0,6fck * Alternatively you can use value Pm.t = Ap.tot*σm.t = AP.tot * 1030MPa ** 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 Load on floor that can be used in the calculations: Live load qk=8 kN/m2 Dead Load gk= 12 kN/m2 Selfweight of concrete pc=25kN/m3
- 5. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 2020 12.09.2020 Homework 3, Investigation of a post-tensioned unbonded beam 2(3) Return to MyCourses in PDF-format. Figure 2. Plan view of the post tensioned floor with the beam marked (for information) 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
- 6. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 12.9.2020 Homework 4, Analysis of a Continuous Prestressed composite slab 1(1) Return to MyCourses in PDF-format. You are investigating a composite slab that shall be casted to be a continuous structure. Prestressed slab is fully propped during casting of topping, see figure 1. Propping is installed after installation of composite slabs to the deformed shape of the composite slabs. Ordinary reinforcement is installed over the intermediate supports. After hardening of surface slab concrete, structure will be loaded with a live load qk . Information: - Prestressed slab concrete strength: C40/50 ; Strength of the prestressed slab during transfer of prestress force: C20/25 - Surface slab concrete strength: C30/37 ; - Bonded tendons. Grade Y1860S7 diameter=9,3mm (fp0,1k/fpk=1640MPa/1860MPa ; Ep=195GPa) - Grade of ordinary reinforcement B500B (fyk=500MPa, ES=200GPa). - Stress of tendons at release σmax=1200MPa. Tendon geometry is straight. - Total losses due to creep, shrinkage and relaxtation of tendons can be assumed Δσ=100MPa - Area of one tendon Ap1=52mm2 . Number of tendons np=22. - Liveload qLL=5 kN/m2 ; Figure 1. Prestressed composite slab. Sideview and section Calculation of the cross section properties a) Calculate the cross section properties of the composite section Calculation for the effects of actions b) Calculate the effects of actions at the edgemost midspan. Consider selfweight (SW), dead load (DL) and liveload (LL) c) Calculate the effects of actions over intermediate support. Calculation of stresses in SLS at the midspan and midsupport d) Calculate TOP and BOTTOM stresses at midspan for the characteristic combination of actions (PT+SW+DL+LL) in SLS. Use the cross section properties of the composite section using method of transformed section in SLS. Does the cross section crack? e) Calculate TOP and BOTTOM stresses at the support for the characteristic combination of actions (PT+SW+DL+LL) in SLS. Does the cross section crack? Calculation of bending moment resistance in ULS at the midspan and midsupport f) Calculate the design bending moment MEd.f in ULS and the (positive) bending moment resistance of the composite structure MRd.f in ULS at the edgemost midspan. Is the bending moment resistance of the structure adequate in ULS? g) Calculate the design bending moment MEd.s in ULS and the (negative) bending moment resistance of the composite structure MRd.s in ULS at the intermediate support. Is the bending moment resistance of the structure adequate in ULS? Geometry: L=7000mm ; bw=1200m ; (span and width of slab) h1=150mm (height of composite precast slab); h2=150mm (height of cast-in-situ slab); ep=45mm (eccentricity of tendons from the bottom of slab) es=50mm (eccentricity of rebar from top of surface slab)
- 7. Aalto University J. Hanka CIV-E4050 Prestressed and Precast Concrete Structures 12.9.2020 Homework 4, Analysis of a Continuous Prestressed composite slab 1(1) Return to MyCourses in PDF-format. Figure 2. Location of prestress strands in h1=150mm composite slab (KL150). (Mitat punoksen alapintaan = dimension to bottom of strand)