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Structural Design Guide for Prestressed Slabs

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Structural Design Guide for Prestressed Slabs

VOL. 02 STRUCTURE INSIGHT

Chapter 1 What is Post-Tensioning

1.1 Advantages & Applications of Post-Tensioning 1

1.1.1 Post-tension type 2

1.2 Strength of Prestressing Steel 3

1.3 Design Steps 4

1.4 Loading Conditions 5

1.5 Definition of Member Forces 6

1.6 Thickness or Depth of Section 7

1.7 Post Tension Slab Procedure 8

Chapter 2 Design Flow for Post-Tension

2.1 Choose Tendon Force and Profile 12

2.1.1 Tendon profile type 12

2.1.2 How to arrange tendon of parabolic shape 13

2.2 Check Final Stresses, Check Initial Stresses 14

2.2.1 Allowable stress for concrete 14

2.2.2 Allowable stress for tendon (Strand) 14

2.2.3 Calculate & check stresses in simple beam 14

2.2.4 Stress distribution according to tendon arrangement 17

2.3 Check Prestress Losses 17

2.3.1 Kind and calculation of loss 18

2.4 Check Flexural Strength 21

2.4.1 Calculation of flexural strength 21

2.5 Check Shear Strength 22

3.1 Tip for Post-Tension Design 25

3.1.1 Assumption in post-tension design 25

3.1.2 Arrangement of tendon 26

3.2 Extra Rebar (Additional Rebar) 27

3.2.1 Extra rebar in unbonded type 27

3.2.2 Extra rebar in bonded type 30

3.3 Building Design Example 30

3.3.1 Building information 30

3.3.2 Service stage 30

3.3.3 Calculation of secondary moment 32

3.3.4 Calculation of design moment 32

3.3.5 Calculation of flexural strength (Mn) 32

3.3.6 Check punching shear 33

Chapter 3 Post-Tension Design Tip and Example

Chapter 4 Tutorial for Post-Tension Beam

4.1 Modeling of Post-Tension Beam 34

4.1.1 Tendon material definition 35

4.1.2 Post-tensioning tendon property 36

4.1.3 Tendon profile definition 36

4.2 Loading of Prestressing Loads 39

4.2.1 Apply prestressing loads to the tendons 39

4.2.2 Activate prestressing loads in construction stage 40

Chapter 1

What is Post-Tensioning

Chapter 1

What is Post-Tensioning

Structural Design Guide for Prestressed Slabs 1

Post-tensioning is a technique for reinforcing concrete. Post-tensioning tendons, which are prestressing steel cables inside plastic ducts or sleeves, are positioned in the forms before the concrete is placed. Afterwards, once the concrete has gained strength but before the service loads are applied, the cables are pulled tight, or tensioned, and anchored against the outer edges of the concrete.

Post-tensioning is a form of prestressing. Prestressing simply means that the steel is stressed (pulled or tensioned) before the concrete has to support the service loads. Most precast, prestressed concrete is actually pre-tensioned-the steel is pulled before the concrete is poured. Post-tensioned concrete means that the concrete is poured and then the tension is applied-but it is still stressed before the loads are applied so it is still prestressed.

Figure 1. Post-Tensioning

1.1. Advantages & Applications of Post-Tensioning


• It reduces or eliminates shrinkage cracking-therefore no joints, or fewer joints, are needed.

• Cracks that do form are held tightly together.

• It allows slabs and other structural members to be thinner.

• It allows us to build slabs on expansive or soft soils.

• It lets us design longer spans in elevated members, like floors or beam.

1.1.1 Post-tension type

Figure 2. Bounded Post-Tensioning & Unbounded Post-Tensioning

(a) Bonded Post-Tensioning

It is used for large structural elements such as beams and transfer girders, design advantages include increased span lengths and load carrying capacity and reduced deflection.

(b) Unbonded Post-Tensioning

It is typically used in new construction for elevated slabs, slabs-on-grade, beams, and transfer girders, joists, shear walls, and mat foundations. Light and flexible, the unbonded mono strand can be easily and rapidly installed – providing an economical solution.


Figure 3-a. Jacking Stage and Service Stage

Step Immediately after

Prestressing After Anchoring

Tension Stress Min[ 0.94 fpy, 0.80 fpu ] 0.70 fpu

Diameter Aps

(mm2) fpu

(Mpa) fpy

(Mpa) fpe

(Mpa) Fst (kN) Fse (kN)

12.7mm 98.71 1,860 1,674 1,200 146 118.4

15.2mm 138.7 1,860 1,674 1,200 205 166.4

Aps: Area of prestressing steel

fps: Stress in prestressed reinforcement at nominal strength

fpu: Specified tensile strength of prestressing steel

fpy: Specified yielding strength of prestressing steel (0.90* fpu= 0.90*1,860Mpa = 1,674Mpa)

fpi: Initial prestress stress of prestressing steel

fpe: Effective prestress stress of prestressing steel (0.65* fpu= 0.65*1,860Mpa = 1,200Mpa)

Fst: Max. prestress force ( Min[ 0.94 fpy, 0.8 fpu ] *Aps = Min[ 1,573 or 1,485 ] *138.7 = 205kN )

Fse: Effective prestress force ( fpe*Aps = 1,200Mpa*138.7 = 166.4kN )

Fps: Ultimate prestress force

1.2. Strength of Prestressing Steel


After Jacking Stage

• Activity of anchorage device

• Tension force loss due to friction

• Tension loss due to elastic shrinkage of concrete

After Transfer Stage

• Tension loss due to creep

• Tension loss due to shrinkage

• Tension loss due to strand relaxation

Figure 3-b. Design Steps

Figure 4. Post-Tensioned Member and Tendon Removed.

Figure 5. Service Condition (SLS) and Strength Condition (ULS)

1.3. Design Steps


Transfer Stage

• Using Prestress Force → Fsi (before short/long-term tension loss occurs, Max. prestress)

• Using Concrete strength: fci (compressive strength of concrete when prestress is introduced, Min.

strength)

• Loading: only Self-weight (before adding a finishing and live loads)

• Checking item: Tensile cracking and crushing of concrete, failure of anchorage part

Service Stage

• Using Prestress Force → Fse (Effective tension after short/long-term tension loss occurs)

• Using Concrete strength: fck (Max. design strength)

• Loading: All design load (Unfactored Load)

• Checking item: Deflection, Crack Width

Ultimate Stage

• Using Prestress Force > Nominal Strength

• Using Concrete strength: fck (Max. design strength)

• Loading: All design load (Factored loads)

• Checking item: Flexural failure or compression stress under flexural behavior

Figure 6. Load Conditions

1.4. Load Conditions


Example 1: Continuous Beam

Figure 7. Primary Moment (a), (b), and (c)

MS: Moment due to own weight of concrete section

MO: Moment due to other applied loads

M1: Primary moment → Moment by the distance between the section center and the prestressing

point

In example 1: M = Prestressing force x Eccentricity distance = P x - e = -Pe

Δ: Deflection by M1 →Deflection at point B when considered as a simple beam

In example 1: ∆ 𝑀 𝑙 8𝐸𝐼⁄ = 𝑃𝑒 2𝑙 8𝐸𝐼⁄ 𝑃𝑒𝑙 2𝐸𝐼⁄

R: Reaction by Δ → Point B should not be deflected by the support, so a reaction force is generated

so that deformation as much as (-) Δ occurs.

In example 1: → 𝑹𝑩𝟑𝑷𝒆

𝒍 , 𝑹𝑨 𝑹𝑩 ∗ 𝟎. 𝟓

𝟑𝑷𝒆

𝟐𝒍

1.5. Definition of Member Forces


Figure 8. Secondary Moment (d) and Total Moment (e)

MSecondary: Secondary Moment → Moment caused by R

* MSecondary means the moment created by the reaction force that restrains the deformation caused by M1.

In example 1: 𝑀 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐵 𝑙

MNet: Net final Moment = M1 + MSecondary: Moment acting on the actual continuous beam by prestressing.

In example1: 𝑀 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐵 𝑃𝑒

Figure 9. Thickness-Span Correlation by Each Slab Systems

RC member

Span/Depth Continuous Span Simple Span

Beam 21 16

2-Way Slab 30 -

1-Way Slab 28 20

1.6. Thickness or Depth of Section


Post-Tension Member

Span/Depth

Continuous Span Simple Span

Roof Not Roof Roof Not Roof

Beam 35 30 30 26

2-Way Slab 48 42 - -

1-Way Slab 50 45 45 40

Reduction Ratio of Thickness or Depth when PT is applied.

Span/Depth(%)

Continuous Span Simple Span

Roof Not Roof Roof Not Roof

Beam 60% 70% 53% 62%

2-Way Slab 63% 71% - -

1-Way Slab 56% 62% 44% 50%

The thickness is reduced by 30-50% compared to the RC member. Alternatively, an economical design can be made by reducing the amount of rebar or tendon required for the same thickness.

Step 1: Installing slab formwork

Step2: Installing slab rebar

Step 3: Placing bar chair by profile height

1.7. Post Tension Slab Procedure


Step 4: Placing tendon on bar chair

Step 5: Add extra rebar

Step 6: Pouring concrete

Step 7: Insert block and jaws bar chair

Step 8: Stressing and measuring elongation


Step 9: Grouting

Chapter 2

Design Flow for Post-Tension System

Chapter 2

Design Flow for Post-Tension


Design Flow Chart for Post-Tension

Figure 10. Design Flow Chart for Post-Tension


2.1.1 Tendon profile type

Figure 11. Tendon Profile Types

2.1. Choose Tendon Force and Profile


2.1.2 How to arrange tendon of parabolic shape

Figure 12. Tendon of Parabolic Shape

After Transfer Stage

a/L = c/L

y3/y1 b/L y0/y1 y4/y1 y5/y3 y6/y3

0.1

0.1 0.730 1.159 0.863 0.630 1.588 0.2 0.669 1.176 0.851 0.698 1.434 0.3 0.630 1.189 0.841 0.730 1.370 0.4 0.600 1.200 0.833 0.750 1.333 0.5 0.576 1.210 0.826 0.764 1.309 0.6 0.556 1.219 0.820 0.775 1.291 0.7 0.540 1.228 0.815 0.783 1.277 0.8 0.525 1.235 0.809 0.790 1.267 0.9 0.512 1.243 0.805 0.795 1.258 1.0 0.500 1.250 0.800 0.800 1.250

0.05

0.1 0.746 1.072 0.933 0.803 1.245 0.2 0.681 1.079 0.927 0.843 1.186 0.3 0.638 1.085 0.922 0.862 1.160 0.4 0.607 1.090 0.918 0.873 1.146 0.5 0.581 1.094 0.914 0.881 1.136 0.6 0.560 1.098 0.911 0.886 1.128 0.7 0.542 1.102 0.908 0.891 1.123 0.8 0.526 1.105 0.905 0.894 1.118 0.9 0.512 1.108 0.902 0.897 1.114 1.0 0.500 1.111 0.900 0.900 1.111

Figure 12. After Transfer Stage Table


2.2.1 Allowable stress for concrete

Checking Stage

Conditions Allowable Stress

Under Initial Service Load (Initial)

1. Extreme fiber stress in compression pci = 0.60 fci

2. Extreme fiber stress in tension pti = 0.25 fci0.5

3. Extreme fiber stress in tension at ends of simply supported members

pti = 0.50 fci0.5

Under Service Load

(Final)

1. Extreme fiber stress in compression (at Long-Term Service Load)

pcf = 0.45 fck

2. Extreme fiber stress in compression (prestress + total load)

pcf = 0.60 fck

3. Extreme fiber stress in tension ptf = 0.50 fck0.5

2.2.2 Allowable stress for tendon (Strand)

Checking Stage Conditions Allowable Stress

During Stressing - Max [0.94 fpy, 0.8fpu]

Immediately after Stressing

- 0.70fpu

2.2.3 Calculate & check stresses in simple beam

Figure 14. Check Stresses in Simple Beam

1. Due to initial prestressing force At Top: σpti = Pi /Ac – Pi e/Zt = (1-eAc /Zt ) (Pi /Ac) = St (Pi /Ac) At Bottom: σpbi = Pi /Ac + Pi e/Zb = (1+eAc /Zb ) (Pi /Ac) = Sb (Pi /Ac)

2. Check initial stress At Top: σti = σpti + M0 /Zt > pti (Tensile stress), if N.G., the area of tendon must be increased. At Bottom: σbi = σpbi - M0 /Zb < pci (Compressive stress), if N.G., the concrete section must be increased.

3. Due to final prestressing force At Top: σptf = St (Pf /Ac) At Bottom: σpbf = Sb (Pf /Ac)

4. Check final stress At Top: σtf = σptf + M0 /Zt < pcf (Compressive stress), if N.G., the concrete section must be increased. At Bottom: σbf = σpbf - M0 /Zb > ptf (Tensile stress), if N.G., the area of tendon must be increased.

5. Check final prestressing force (Pf) Min. Pf: Pf,min = [ptf + (M0+Ms) /Zb ] Ac /Sb

2.2. Check Final Stresses Check Initial Stresses


Max. Pf: Pf,max = Rp (pci + M0 /Zb ] Ac /Sb ** Rp = Pf /Pi (The ratio of final to initial prestressing force) Pf,max > Pf,min If Pf,max > Pf,min, the concrete section in inadequate and it must be increased. Example

Figure 15. Design Loads and Permissible Stresses

Permissible stresses:

Pti = -1.8 N/mm2 Pcf = +13.2 Pci = +12.5 Ptf = -2.3

1. Area and section centroid yt from top

width x depth = area (mm2) x y = Ay (mm2) 1000 x 100 = 110.0 x 103 x 55 = 6.05 X 106 400 x 170 = 680 x 195 = 13.26 (60/2) x 170 = 5.1 x 167 = 0.85

Ae = 183.1 x 103 Acy1 = 20.16 x 106

→ y1 = Acy1/Ac = 110mm

yb = D – y1 = 170

em = yb – 40 = 130

2. Moment of inertia (IC)

A D yt y 110.0 x 103 x [1102/12 + (110 – 55)2] = 443.67 x 106 mm4 68.0 x 103 x [1702/12 + (110 – 195)2] = 655.07 5.1 x 103 x [1702/18 + (110 – 167)2] = 24.76

Moment of inertia Ic = 1123.40 x 106

3. Section module

Zt = Ic/yt = 10.213 x 106 mm3

Zb = Ic/yb = 6.608 x 106

4. St & Sb

St = 1 – emAc/Zt = -1.331

Sb = 1 + emAc/Zb = +4.602

5. Self-weight & moments


Section weight = 24 x 183.1/1000 = 4.39 kN/m2

Moments:

Mo = 4.39 x 102/8 = 54.9 kN.m/m self-weight

Ms = 6.00 x 102/8 = 75.0 other loads

Mo + Ms = 129.9 total

6. Prestressing forces

Check final prestressing force (Pf):

Min. Pf: Pf,min = [ptf + (M0+Ms) /Zb ] Ac /Sb

= [-2.3 + 129.9/6.608] * 183.1/4.602 = 690.6 kN

Max. Pf: Pf,max = Rp (pci + M0 /Zb ] Ac /Sb

= 0.85 * (12.5 + 54.9/6.608) * 183.1/4.602 = 703.7 kN

Check: Pf,max = 690.6 > Pf,min = 703.7 O.K.

Use Pf = 690.6 kN, Pi = Pf / 0.85 = 812 kN

7. Number of strands per rib

Assuming 160kN final force per 15.7mm strand,

the number of strands per rib = 690.6/160 = 4.3... apply 5

8. Check stresses

From 4.

St = 1 – emAc/Zt = -1.331

Sb = 1 + emAc/Zb = +4.602

Stresses:

Pi/Ac = 812.5/183.1 = 4.44 N/mm2

σpti = StPi/Ac = -5.91

σpbi = SbPi/Ac = 20.43

Pf/Ac = 690.6/183.1 = 3.77 N/mm2

σptf = StPf/Ac = -5.02

σpbf = SbPf/Ac = 17.36

Initial Stresses

σtj = -5.91 + 54.9/10.213 = -0.53 > Pti

σbi = +20.43 – 53.4/6.608 =+12.35 < Pci

Final Stresses

σtf = = -0.53 > Pti

σbf = +20.43 – 53.4/6.608 =+12.35 < Pci


2.2.4 Stress distribution according to tendon arrangement

Figure 16. Tendon Arrangement

What is prestress losses?

The prestress losses are defined as the loss of tensile stress in the prestress steel which acts on the concrete component of the prestressed concrete section. In pretensioned concrete, the four major sources of prestress losses are elastic shortening (ES), creep (CR), shrinkage (SH) and relaxation (RE).

2.3. Check Prestress Losses


Figure 17.Correlation between Strand Stress and Time

Figure 18. Length of Tendon and Prestress

2.3.1 Kind and Calculation of Loss

Stress loss due to friction (Curvature and Wobble)

Figure 19. Stress Loss Due to Friction – Curvature and Wobble


Values of μ and K

Duct material BS 8110 ACI 318 Commentary

μ K/m μ K/metre K/ft

Rigid galvanized

0.25 0.0017 0.15-0.25

0.0005-0.0020 0.15-0.61

x 10-3 Greased plastic

0.12 0.0025 0.05-0.15

0.0003-0.0020 0.09-0.61

x 10-3 Non-rigid 0.0033

Percentage can be simplified to:

Px = P0(1- μθ - Kx)

Percentage Loss = 100(μθ + Kx)

P(X) = P0e-(μα+Kx) , where

μ = Curvature Friction Coefficient

α or θ = Sum of the Tendon Angular Change from the Tendon Jacking End to a Distance x

K = Wobble Friction Coefficient (rad/unit length2)

P(x) = Post-Tensioning Force at a Distance x

P0 = Post-Tensioning Force at Stressing

Anchorage Set Slip Losses

At the last stage of the stressing operation, the tendons usually are anchored with two-piece conical wedges. Anchoring operations normally result in an additional prestress loss due to seating of the wedges, considering that the strand retracts when it is released and pulls the wedges into the anchoring device. Calculation of the stress losses is typically performed in an iterative manner. As shown in Figure 19, the distance “c” refers to the extent of influence of an anchor set. Procedurally, anchor set is chosen first (usually about 0.25 to 0.375 in or 6 to 8 mm), then the distance “c” is set, and finally the corresponding stress is computed, with the assumption that the stresses vary linearly from the jacking point.

The seating loss is then calculated using the following equation:

SL 𝑎


Figure 20. Anchor Set Influence Distance Diagram

Elastic Shortening of Concrete

Elastic shortening refers to the shortening of the concrete as the post-tensioning force is applied. As the concrete shortens, the tendon length also shortens, resulting in a loss of prestress. If sequential jacking steps are used, the first tendon jacked and locked off will suffer the maximum amount of loss from elastic shortening. Conversely, there will be no loss because of elastic shortening for the last tendon in a sequence or in a single tendon because the elastic shortening will take place before the tendon is locked into the anchoring device. The user specified amount of prestress loss from elastic shortening is applied uniformly over the entire length of the tendon.

Long-Term Losses

The long-term prestress losses of a member include creep, shrinkage, and steel relaxation effects. Creep, shrinkage, and steel relaxation effects are governed by material properties and, in some cases, other environmental conditions that need to be accounted for when specifying the long-term loss values. Each stress loss is treated separately and then summed up, as follows:

TL = CR + SH + RE

Where TL is the total loss of stress; CR is the stress loss due to creep of the concrete; SH is the stress loss due to shrinkage of the concrete; and RE is the stress loss due to relaxation in the tendon steel. The sum of these losses is applied to the initial (jacking) load of the tendon. All of the long-term losses are uniformly applied over the length of the tendon.


2.4.1 Calculation of flexural strength

The flexural strength review of members follows the same strength installation method as that of reinforced concrete members. However, the tension member stress ( 𝑓 ) when the nominal strength is expressed is calculated according to the strain-appropriate condition.

Figure 21. Compression Block Depth Ratio

Compression Block Depth Ratio, ACI 318 (Metric Units)

ƒ’c 30 35 40 45 50 55 N/mm2

β1 0.85 0.81 0.77 0.73 0.69 0.65

yp = Factor for the type of tendon

= 0.55 for ƒpy/ƒpu not less than 0.80



β1 = dc/dn

ω = Asƒy/(ƒ’cbd)

ω’ = A’sƒy/(ƒ’cbd)

rp = ap/bdp = ratio of tendon area

In case of ƒpe 0.5 ƒpu

Bonded Type: ƒ = ƒ 1 𝑟ƒ

ƒ𝜔 𝜔

Unbonded Type: ƒ = ƒ 70 ƒ

Figure 22. Ultimate State Stress Diagrams

2.4. Check Flexural Strength


C Compression 𝑝 𝑏𝑑

T Tension 𝑓 𝐴 𝑓 𝐴 𝑇 𝑇

Where,

𝑝 𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 𝑎𝑡 𝑓𝑎𝑖𝑙𝑢𝑟𝑒.

𝑇 𝑓 𝐴 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝑖𝑛 𝑝𝑟𝑒𝑠𝑡𝑟𝑒𝑠𝑖𝑛𝑔 𝑡𝑒𝑛𝑑𝑜𝑛𝑠

𝑇 𝑓 𝐴 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝑖𝑛 𝑏𝑜𝑛𝑑𝑒𝑑 𝑟𝑜𝑑 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡

𝑓 𝑆𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑏𝑜𝑛𝑑𝑒𝑑 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡

Taking moments of the tensile forces about the center of compression C gives the ultimate capacity of the section Mu, which, of course, should not be less than the required ultimate moment of resistance Mr.

M 𝑓 𝐴 𝑑 𝑑 𝑓 𝐴 𝑑 𝑑

where d 𝑇ℎ𝑒 𝑑𝑒𝑝𝑡ℎ 𝑓𝑟𝑜𝑚 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑓𝑎𝑐𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒

The procedure for the ultimate state check.

1. Calculate the required moment of resistance Mr.

2. Calculate Tr and Tp, initially Tr may be assumed zero.

3. Calculate dx and dc.

4. Calculate Mu.

5. If Mu < Mr then add bonded reinforcement and repeat from 2.

Figure 23. Effect of Inclined Tendon on Effective Shear Force

VwL2

wx Psinα

The factor of 0.85 is introduced after the calculation of shear strength.

Uncracked section

V β 𝑓 . 0.3𝑝 𝑏 𝑑 𝑉

where, for a one-way spanning member,

β 0.3 𝑁 𝑚𝑚 𝑢𝑛𝑖𝑡𝑠

2.5. Check Shear Strength


The vertical component of the prestres, 𝑉

Cracked section

V 0.05 f . 𝑏 𝑑𝑚 𝑉

𝑀𝑖𝑛 𝑁 𝑚𝑚 𝑢𝑛𝑖𝑡𝑠

or 0.60 f . 𝑏 𝑑𝑚 𝑉

𝑀𝑖𝑛 1𝑏 𝑖𝑛 𝑢𝑛𝑖𝑡𝑠

where, M 𝐿𝑖𝑣𝑒 𝑙𝑜𝑎𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝑐𝑎𝑢𝑠𝑒 𝑐𝑟𝑎𝑐𝑘𝑖𝑛𝑔

𝑍 0.5 𝑓 . 𝑓 𝑁 𝑚𝑚 𝑢𝑛𝑖𝑡𝑠

𝑜𝑟 𝑍 𝑓 . 𝑓 1𝑏 𝑖𝑛 𝑢𝑛𝑖𝑡𝑠

f 𝑆𝑡𝑟𝑒𝑠𝑠 𝑎𝑡 𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑓𝑖𝑏𝑟𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑑𝑒𝑎𝑑 𝑙𝑜𝑎𝑑𝑠

Shear reinforcement for beam and 1-way slab

A /S𝑉 0.85𝑉0.85𝑓 𝑑

where A 𝑇𝑜𝑡𝑎𝑙 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑙𝑒𝑔𝑠 𝑜𝑓 𝑎 𝑙𝑖𝑛𝑘

𝑆 𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 𝑜𝑓 𝑙𝑖𝑛𝑘𝑠

𝑓 𝑌𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑒𝑒𝑙

460𝑁/𝑚𝑚 66.7 𝑘𝑠𝑖 𝑓𝑜𝑟 ℎ𝑖𝑔ℎ 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑒𝑒𝑙

250𝑁/𝑚𝑚 36.3 𝑘𝑠𝑖 𝑓𝑜𝑟 𝑚𝑖𝑙𝑑 𝑠𝑡𝑒𝑒𝑙

The multiplied value of 0.87 is a material partial safety factor for the shear reinforcement.

Min. shear reinforcement: only when 𝐕𝐮 𝟎. 𝟓𝑽𝒄

𝐀𝐬𝐯

𝑺𝒗

𝒃𝒗

𝟑𝒇𝒚𝒗

𝐨𝐫

𝐀𝐬𝐯

𝑺𝒗

𝑨𝒑𝒇𝒑𝒖

𝟖𝟎𝒇𝒚𝒗𝒅𝒅𝒃𝒗

𝟎.𝟓

Example of one-way shear calculation

What is the value of the one-way shear capacity of a solid slab continuous over two equal spans?

The two equal spans are supported on 250mm thick walls. Suppose that the tendon profile is concordant and the tendon reverse curvature over the support is calculated by the parabola

e 0.000375x 65 𝑖𝑛 𝑚𝑚 𝑢𝑛𝑖𝑡𝑠

with a downward eccentricity taken as positive. And it is proportion to an origin at the intersection of the slab and the interior wall centerline.

Here is additional data:

Span 9.0 m

Imposed Live Load 2.5 kN/m2

Slab Depth 200 mm

Concrete Strength 𝒇𝒄𝒖 40 N/mm2 (𝑓 = 32 N/mm2)

Rod Reinforcement 0.2 %

Steel yield Point 𝒇𝒚 460 N/mm2

Tendons 15.7 mm @ 300 mm centers


Tendon Ultimate Stress 1770 N/mm2

Tendon Force, Final 170 kN each

Solution: Using ACI 318.

Factored Load = 24 1.4 0.2 1.7 2.5 10.97kN/m

Reaction at outer support = 0.375 10.97 9.0 37.02kN/m

Moment at critical section = 37.02 8.875 10.97 8.875 2 103.5𝑘𝑁𝑚/𝑚

Shear at critical section = 0.625 10.97 9.0 10.97 . 60.30𝑘𝑁/𝑚

The stress at the extreme fibre stress by prestress = 1 eA /𝑍)

= . 1000 1 .

.

.

From Eq (10.7), M 6.67 10 0.5√32 7.90 71.5𝑘𝑁𝑚/𝑚

The actual moment is over this value. The section is cracked, thus use Equation (10.6)

V 0.05√32 165 71.560.3

103.589.7𝑘𝑁/𝑚

The value which is including a shear strength factor of 0.85 has the shear strength of 76.3kN/m, and it is over the actual shear force of 60.3kN/m. Thus, the shear reinforcement is not needed.

Chapter 3

Post-Tension Design Tip & Example

Post-Tension Design Tip & Example

Chapter 3


3.1.1 Assumption in PT design

Figure 24. (a)Non-Prestressed Beam and (b)Post-Tensioned Beam

Variables in post tension member design

• Average line compression force to be introduced.

• The size of the balance load to be introduced.

• Arrangement shape of tension member.

Determination of average linear compressive stress

• Determine the tendon spacing so that the average linear compressive stress is 0.9 MPa or more

(ACI 318-02).

• Introduction of average linear compressive stress of about 1.0~1.4MPa to the roof slab for

controlling the crack.

• It is economical to introduce the maximum average linear compressive stress at 2.0 MPa for

slabs and 2.5 MPa for beams . Determination of balance load size

• Balanced load: vertical load acting on the structure at the position where the tension member is

placed.

• Calculate the average of the upward force over the entire span and express it as a ratio to self-

weight.

• It is economical to calculate 50~80% of the weight of the slab and 70~100% of the weight of the

beam.

3.1. Tip of Post- Tension Design


3.1.2 Arrangement of tendon

Beam and slab continuous span

Figure 25. Beam and Slab Continuous Span

Slab band

Figure 26. Slab Band

2-Way slab

Figure 27. 2-Way Slab


3.2.1 Extra rebar in unbonded type

Figure 28. (a) Plan of Design Strip and (b) Elevation of Design Strip

• Role of extra rebar (Additional rebar)

Control of cracks under service loading.

Get adequate ductility under ultimate load conditions.

• Minimum rebar in 2-way slab system

Min. rebar area in negative moment section (Not consider the status for service load stress) As,min = 0.00075 Acf

Example 01 (at top of column)

Acf1 =0.5L1*h1+0.5L2*h2,

Acf2 =0.5(a*h1+c*h2) + 0.5(b*h1+d*h2)

As,min=0.00075*Max[Acf1, Acf2]

Example 02 (at slab band)

he=[t*a+b(h-t)]/a

Acf=a*he

As,min=0.00075*a*he

Min. rebar area in positive moment section (For the section where the tensile stress of concrete is

(1/6)√fck or more) As,min = Nc / (0.5*fy)

3.2. Extra Rebar (Additional Rebar)


Figure 29. Uniformed Rebar and Slab Band

• Minimum rebar in 1-way slab system

Placement at the top of the column and at the bottom of the center of the span section. (Not consider the status for service load stress) As,min = 0.00075 Acf

Area A in each moment type

Figure 30. Area A in Each Moment Type

Placement area of top extra rebar

Figure 31. Placement Area of Top Extra Rebar


Placement area of bottom extra rebar

Figure 32. Placement Area of Bottom Extra Rebar

• Minimum length of extra rebars

In the positive moment section, 1/3 of the net span.

1/6 of the net span from the column surface in the negative moment section.

Figure 33. Minimum Length of Extra Rebars

• Extra rebar against shrinkage and temperature change

Figure 34. Arrangement of Rebar and Tension Member against Shrinkage & Temperature

When an average compressive stress of 0.7 MPa or less is introduced to the entire section (P/A) / 0.7 + As / (0.0018*A) >= 1


3.2.2 Extra rebar in bonded type

• No extra rebars are required in the following cases:

When the post tension member satisfies the requirements of the design code under the service load condition.

When the post tension member satisfies the strength requirements.

3.3.1 Building information

Figure 35. Building Design Example

Unbonded System

3.3.2 Service stage

Load balancing and tendon profile

- Influencing factor: Shape of longitudinal section of tension member and prestressing force

- Tendon profile: Single parabola shape

3.3. Building Design Example

- WSD=1.00kPa

- WD=3.90kPa

- WL=2.00kPa

- WW=WD+L=6.90kPa

- WU=1.2WD+1.6WL=9.08kPa

- c1 : 360X300

- c2 : 510X300

- Story heights: 2.67m

- fc’=28MPa

- fci’=21MPa

- fpu=1860MPa

- fps=1280MPa

- fpy=1680MPa

- fpe=1100MPa

- Eps=200000MPa

- fy=400MPa


Figure 36. Load Balancing and Tendon Profile

a Y𝑌 𝑌

2

W8𝐹𝐿

- Average line pressure → The effective prestressing force → The tension water

Figure 37. Average Line Pressure

- Mnet (Moment by Wnet) and the tension stress check

Figure 38. Net Load Moments (Mnet)

- Column B

𝑓𝑃𝐴

𝑀𝑆

𝑓𝑀

𝑆1.63𝑀𝑃𝑎

𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒𝑓 0.5 𝑓 2.62𝑀𝑃𝑎 1.63𝑀𝑃𝑎 𝑂. 𝐾.

- Span BC

𝑓𝑃𝐴

𝑀𝑆

𝑓𝑀

𝑆0.545𝑀𝑃𝑎

𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒𝑓 0.17 𝑓 0.9𝑀𝑃𝑎 0.545𝑀𝑃𝑎 𝑂. 𝐾.


3.3.3 Calculation of secondary moment

Figure 39. Secondary Moments (Msecondary)

- 𝑀 𝑀 𝑀 𝑀 𝑒

𝑀𝑏𝑎𝑙 Balance moment by balance load

𝑀1 Moment by prestressing force

𝑒 Eccentricity between the prestressing force and the neutral axis of the cross section (Upper neutral axis: (+), lower neutral axis: (-))

3.3.4 Calculation of design moment

Figure 40. Design Moments (Mu)

- Moments that take into account the secondary elements of coefficient load and prestressing

𝑀 𝑀 𝑀

𝑀 : Moments by factored loads that do not take into account balanced loads.

ex) 1.2D+1.6L+1.0HYP

- Bonded 2-way slab system: Designed to resist more than 1.2 times the factored of the cracked load.

3.3.5 Calculation of flexural strength (Mn)

- For the unbonded system, the minimum rebar must be arranged.

- At top of column : As=0.00075hLn

column B : As=7.93 cm2

𝑓 𝑓 70𝑓 ′

300𝜌1,236𝑀𝑃𝑎

𝑓 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑓 𝑡𝑒𝑛𝑑𝑜𝑛 𝑖𝑛 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ

𝑀 𝐴 𝑓 𝐴 𝑓 𝑑𝑎2

36.5𝑘𝑁 𝑚/𝑚𝑀𝜑

35𝑘𝑁 𝑚/𝑚


- Positive moment: Additional rebar is required if bending forces due to tendon are exceeded.

- Span BC

𝑀 𝐴 𝑓 𝑑𝑎2

30.2𝑘𝑁 𝑚/𝑚𝑀𝜑

31.7𝑘𝑁 𝑚/𝑚𝐴 𝑓𝑀

𝜑 𝑑 𝑎2

𝐴 𝑓

1.67𝑐𝑚 ∴ 2 𝐻𝐷13

3.3.6 Check punching shear

- 𝑣 𝑐 𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒𝑣 𝛽 𝑓 ′ 0.3𝑓

- 𝛽 min. , . .

Chapter 4

Tutorial for Post-Tension Beam

Chapter 4

Tutorial for Post-Tension Beam


Model shape

Figure 41. Post-Tension Beam

4.1. Modeling of Post-Tension Beam


4.1.1 Tendon material definition

Model > Properties > Material > ‘Add’ Material

Figure 42. Material Data Window

1. Enter material name for tendon (Strand).

2. Enter the values for tendon (Strand).

3. Enter details as shown here and click ‘OK’.

Post-Tension Slab Analysis and Design 36

4.1.2 Post-tensioning tendon property

Loads > Prestress Loads > Tendon Property > ’Add’ Tendon Property

Figure 43. Add/Modify Tendon Property Window

1. Enter the tendon name.

2. Select the tendon type as ‘Internal (Post-tension)”.

3. Enter the tendon material defined previously.

4. Click “...” button.

5. Select the strand diameter and enter No. of strands.

6. Click “O.K.” and then, total tendon area is calculated and enter.

7. Enter the duct diameter value.

8. Enter the Relaxation coefficient as per the code selection for relaxation losses.

9. Enter the ultimate and yield strength of strand.

10. Enter the friction loss parameters.

11. Enter the Anchorage slip values at each end of the tendon.

12. Select the bond type.

13. Click “OK”.

4.1.3 Tendon profile definition

Tendon profile definition in the left span

Figure 44. Tendon Profile Definition in the Left Span


Tendon-1

Point x (mm) y (mm) z (mm)

1 0.0 100.0 750.0

2 10000.0 100.0 -750.0

3 17600.0 100.0 -750.0

4 27600.0 100.0 750.0

Tendon-2

Point x (mm) y (mm) z (mm)

1 0.0 -100.0 750.0

2 10000.0 -100.0 -750.0

3 17600.0 -100.0 -750.0

4 27600.0 -100.0 750.0

• The table shows the tendon coordinates which are prepared in an excel sheet beforehand by the

engineer.

• x, y and z are the element local directions i.e. longitudinal, transverse and vertical directions.

• y and z are measured from the centroid of the section of the element.

Load > Prestress Loads > Tendon Profile > ’Add’

Figure 45. Add/Modify Tendon Profile Window


1. Enter the tendon profile name.

2. Select the tendon property for the post-tensioning tendon.

3. Select elements included to the beam member with post-tensioning tendon.

4. Select the tendon profile input type as 3D and the curve type as Spline.

5. Choose Tendon reference axis as “Straight” type.

6. Input the profile insertion point as center of beam member and select x-axis direction as “Y”(global axis).

7. Copy paste the coordinates x,y,z in the table from the excel sheet.

8. Click on OK after all the details shown in the figure.

Similarly all the tendon profiles in the webs can be defined for all the spans. So finally after inserting all the tendon profiles, this is how the model looks after displaying all the tendon profiles

Figure 46. Final Tendon Model


4.2.1 Apply prestressing loads to the tendons

Load > Prestress Loads > Tendon Prestress Loads

Figure 47. Tendon Prestress Loads

1. Select or Create ‘Prestress’ Load case.

2. Select or Create the load group name.

3. Select the tendons to be prestressed from the left column from the tendon list.

4. Apply prestress in the form of stress.

5. Select the method of 1st Jacking as from “Begin” of the tendon.

6. Enter prestress value at the Begin and End of tendon.

7. Press on ‘Add’ to assign the prestressing to the tendons.

4.2. Loading of Prestressing Loads


4.2.2 Activate prestressing loads in construction stages

In the load section of construction stages activate the prestress load groups as per shown below: So at various stages the post-tensioning can be achieved. The tendons associated with these load groups will be prestressed in these stages. Please study the technical notes on construction stage analysis first before applying prestressing load. Load > Construction Stage Analysis Data > Define Construction Stage > Modify/Show

Figure 48. Compose Construction Stage Window

1. Select the stage to which the prestressing load is applied.

2. Click “Load” tap.

3. Select “Prestressing” in Group and click “Add” in Activation.

4. Click “OK”.

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