Prestressed Concrete Bridge Design. Basic Principles. Emphasizing AASHTO
LRFD Procedures. Praveen Chompreda, Ph. D. MAHIDOL UNIVERSITY | 2009 ...
Prestressed Concrete Bridge Design Basic Principles
Part I: Introduction
Emphasizing AASHTO LRFD Procedures
Praveen Chompreda, Ph. D.
Reinforced vs. Prestressed Concrete Principle of Prestressing H Historical lP Perspective Applications Classification and Types Advantages Design Codes Stages of Loading
MAHIDOL UNIVERSITY | 2009 | EGCE 406 Bridge Design
Reinforced Concrete
Reinforced Concrete
Recall Reinforced Concrete knowledge:
Concrete is strong in compression but C b weakk in tension Steel is strong in tension (as well as compression) Reinforced concrete uses concrete to resist compression and to hold the steel bars in place, and uses steel to resist all of the tension Tensile strength of concrete is neglected (i.e. zero) RC beam always crack under service load Cracking moment of an RC beam is generally much lower than the service moment
Principle of Prestressing
Prestressing is a method in which compression force is applied to the reinforced concrete section. section The effect of prestressing is to reduce the tensile stress i the in h section i to the h point i that h the h tensile il stress is i below b l the cracking stress. Thus, the concrete does not crack! It is then possible to treat concrete as an elastic material The concrete can be visualized to have 2 force systems
Principle of Prestressing
Internal Prestressing Forces External Forces (from DL DL, LL LL, etc…) etc )
These 2 force systems must counteract each other
Principle of Prestressing
Stress in concrete section when the prestressing force is applied at the c.g. c g of the section (simplest case)
Historical Perspective
Stress in concrete section when the prestressing force is applied eccentrically with respect to the cc.g. g of the section (typical case)
The concept of prestressing was invented centuries ago when metal bands were wound around wooden pieces (staves) to form a barrel. barrel
Smaller Compression
c.g.
+
e0
=
F/A CrossSection
+ Fe0y/I
Prestressing Force
MDLy/I
MLLy/I
Stress from DL
Stress from LL
Small Compression Stress Resultant
The metal bands were tighten under tensile stress, which creates compression between the staves – allowing them to resist internal liquid pressure
Historical Perspective
Historical Perspective
The concept of prestressed concrete is also not new. In 1886 a patent was granted for tightening steel tie rods in 1886, concrete blocks. This is analogous to modern day segmental constructions. constructions Early attempts were not very successful due to low strength of steel at that time. time Since we cannot prestress at high stress level, the prestress losses due to creep and shrinkage of concrete quickly reduce the effectiveness of prestressing.
Applications of Prestressed Concrete
Bridges Sl b in Slabs i buildings b ildi Water Tank Concrete Pile Thin Shell Structures Offshore Platform Nuclear Power Plant Repair p and Rehabilitations
Eugene Freyssinet (1879-1962) (1879 1962) was the first to propose that we should use very high strength steel which permit high elongation of steel. steel The high steel elongation would not be entirely offset by the shortening of concrete (prestress loss) due to creep and shrinkage.
First prestressed concrete g in 1941 in France bridge First prestressed concrete bridge in US: Walnut Lane B id iin P Bridge Pennsylvania. l i Built B il in 1949. 47 meter span.
Classification and Types
Pretensioning v.s. Posttensioning External v.s. Internal Linear v.s. v s Circular End-Anchored v.s. Non End-Anchored Bonded v.s. Unbonded Tendon P Precast t v.s. Cast-In-Place C t I Pl v.s. Composite C it Partial v.s. Full Prestressingg
Classification and Types
Classification and Types
Pretensioningg vs. Posttensioningg
In Pretension, the tendons are tensioned against some abutments before the concrete is place place. After the concrete hardened, the tension force is released. The tendon tries to shrink back to the initial length but the concrete resists it through the bond between them, thus, compression force is induced in concrete. concrete Pretension is usually done with precast members. Pretensioned Prestressed Concrete Casting Factory
Classification and Types
In Posttension, the tendons are tensioned after the concrete has hardened. hardened Commonly, Commonly metal or plastic ducts are placed inside the concrete before casting. After the concrete hardened and had enough strength, strength the tendon was placed inside the duct, stressed, and anchored against concrete. concrete Grout may be injected into the duct later. This can be done either as precast or cast-in-place. ti l
Concrete Mixer
Classification and Types Precast Segmental Girder to be Posttensioned In Place
Classification and Types
E External l vs. IInternall Prestressing P
Linear vs. Circular Prestressing
Prestressingg mayy be done inside or outside Prestressing can be done in a straight structure such as beams (linear prestressing) or around a circular structures such as tank or silo (circular prestressing) structures,
Bonded vs. Unbonded Tendon
The tendon may be bonded to concrete (pretensioning or posttensioning with grouting) or unbonded ( (posttensioning i i without ih grouting). i ) B Bonding di hhelps l prevent corrosion of tendon. Unbonding allows readjustment dj t t off prestressing t i force f att later l t times. ti
Classification and Types
Partial vs. Full Prestressing
Prestressing tendon P d may be b used d in combination b withh regular reinforcing steel. Thus, it is something between full prestressed concrete (PC) and reinforced concrete (RC). The goal is to allow some tension and cracking under full service load while ensuring sufficient ultimate strength. We sometimes use partially prestressed concrete (PPC) to control camber and deflection, increase ductility, and save costs.
Classification and Types
End-Anchored vs. Non-End-Anchored tendons
IIn P Pretensioning, tendons d transfer f the h prestress through the bond actions along the tendon; therefore, it is non-end-anchored In Posttensioning, g tendons are anchored at their ends using mechanical devices to transfer the prestress to concrete;; therefore,, it is end-anchored. ((Groutingg or not is irrelevant)
RC vs vs. PPC vs. vs PC
Advantages of PC over RC
Take full advantages of high strength concrete and high strength steel
Need N d less l materials i l Smaller and lighter structure No cracks Use the entire section to resist the load Better corrosion resistance Good for water tanks and nuclear plant
ACI-318 Building Code (Chapter 18) AASHTO LRFD (Chapter 5)
Other institutions
PCI – Precast/Prestressed Concrete Institute PTI – Post Post-Tensioning Tensioning Institute
Very effective for deflection control Better shear resistance
Stages of Loading
Design Codes for PC
Unlike RC where we primarily consider the ultimate lti t lloading di stage, t we mustt consider id multiple lti l stages of construction in Prestressed Concrete The stresses in the concrete section must remain below the maximum limit at all times!!!
Stages of Loading
Typical stages of loading considered are Initial and d Service S i Stages St Initial ((Immediatelyy after Transfer of Prestress))
Full prestress force N MLL (may No ( or may nott have h MDL depending d di on construction type)
Service
Prestress loss has occurred MDL+MLL
Stages of Loading
For precast construction, we have to investigate some intermediate states during transportation and erection
Part II: Materials and Hardwares for Prestressingg Concrete Prestressing Steel Prestressing Hardwares
Concrete
Mechanical properties of concrete that are relevant to the prestressed concrete design:
Compressive Strength M d l off Elasticity Modulus El i i Modulus of Rupture
Concrete: Compressive Strength
AASHTO LRFD For prestressed concrete, the compressive strength should be from 28-70 MPa at 28 days For reinforced concrete,, the compressive strength should be from 16-70 MPa at 28 days Concrete with f’c > 70 MPa can be used when supported by test data
Concrete: Modulus of Elasticity
AASHTO (5.4.2.4) Ec = 0.043γ 0 043γc1.5(f (f’c)0.5 MPa
Concrete: Modulus of Rupture
γc1.5 in kg/m3 ff’c in MPa
For normal weight concrete, we can use Ec =4800(f’c)0.5 MPa
Indicates the tensile capacity of concrete under bendingg Tested simply-supported p concrete beam under 4-point bending configuration fr = My/I = PL/bd2 AASHTO (5.4.2.6)
Concrete : Summary of Properties
fr = 0.63 (f’c)0.5 MPa
Prestressing Tendons
Prestressing tendon may be in the form of strands, t d wires, i round d bar, b or threaded th d d rods d Materials
High Strength Steel Fib R i f Fiber-Reinforced d Composite C it ((glass l or carbon b fib fibers))
Tendons
Prestressing Steel
Common shapes of prestressing tendons
Most Popular Æ ((7-wire Strand))
Prestressing Strands
Prestressing strands have two grades
Grade G d 250 (fpu = 250 ksi k or 1725 MPa) MP ) Grade 270 ((fpu = 270 ksi or 1860 MPa))
Types of strands
SStressed d Relieved R li d SStrand d Low Relaxation Strand (lower prestress loss due to relaxation of strand)
Prestressing Strands
Prestressing Strands
Prestressing Strands
Modulus of Elasticity
Hardwares & Prestressing Equipments
Pretensioned Members
H ld D Hold-Down Devices D
Posttensioned Members
Anchorages
Stressing St i A Anchorage h Dead-End Anchorage
Ducts Posttensioningg Procedures
Pretensioned Beams
197000 MPa for Strand 207000 MPa for Bar
The modulus Th d l off elasticity l i i of strand is lower than that of steel bar because strand is made from twisting of small wires together.
Pretensioning Hardwares
Posttensioned Beams
Hold-Down Devices for Pretensioned Beams
Posttension Hardwares
Posttensioning Hardwares - Anchorages
Stressing St i A Anchorage h Dead-End Anchorage Duct/ Grout Tube
Posttensioning Hardwares - Anchorages
Posttensioning Hardwares - Anchorages
Posttensioning Hardwares - Ducts
Posttensioning Procedures
Posttensioning Procedures
Grouting is optional (depends on y used)) the system
Prestress Losses
Part III: Prestress Losses
Prestress force at any time is less than that during jacking Sources of Prestress Loss
Sources of Prestress Losses Lump Sum Estimation of Prestress Loss
Prestress Losses
Sources of Prestress Loss (cont.)
Friction : Friction in the duct of p posttensioningg system y causes stress at the far end to be less than that at the jacking end. Thus, the average stress is less than the jacking stress
Anchorage Set : The wedge in the anchorage h may set in i slightly li h l to llockk the tendon, causing a loss of stress
Elastic Shortening : Because concrete shortens when the prestressing force is applied to it. The tendon attached to it also shorten, causing stress loss
Prestress Losses
Sources of Prestress Loss (cont.) Shrinkage : Concrete shrinks over time due to the loss of water, leading to stress loss on attached tendons Creep : Concrete shortens over time under compressive stress, stress leading to stress loss on attached tendons
Prestress Losses
Time Line of Prestress Loss Posttensioningg
Sources of Prestress Loss (cont.)
FR
Jacking
AS ES
Initial
fpj
Steell R St Relaxation l ti : Steel loss its stress with time due to constant elongation the elongation, larger the stress, the larger the loss. loss
Pretensioning (AS RE) J ki Jacking (against abutment)
fpj
fpi
Release
ES
( (cutting g strands)
Instantaneous Losses
Prestress Loss – By Types Pretensioned Instantaneous
TimeDependent
SH CR RE
fpe
Initial
SH CR RE
fpi
Elastic Shortening
Friction A Anchorage Set S Elastic Shortening
Shrinkage (Concrete) Creep (Concrete) Relaxation (Steel)
Shrinkage (Concrete) Creep (Concrete) Relaxation (Steel)
Effective fpe
Time-Dependent Losses
Prestress Loss - Pretensioned Posttensioned
Effective
Prestress Loss - Posttensioned
Lump Sum Prestress Loss
Pretress losses can be very complicate to estimate ti t since i it d depends d on so many factors f t In typical yp constructions,, a lump p sum estimation of prestress loss is enough. This may be expressed in terms of:
Lump Sum Prestress Loss
Total stress loss (in unit of stress) Percentage of initial prestress
Lump Sum Prestress Loss
A. E. Naaman (with slight modifications) – not including FR, AS
Start with 240 MPa for Pretensioned Normal Weight Concrete with Low Relaxation Strand Add 35 MPa for Stress-Relieved Strand or for Lightweight Concrete D d Deduct 35 MPa MP ffor Posttension P
Types of Prestress Pretensioned
P t Prestress Loss L (fpi-fpe) (f i f ) (MP (MPa)) Types of Concrete
Stress-Relieved Low Relaxation Strand Strand
Normal Weight Concrete Li ht i ht C Lightweight Concrete t
275 310
240 275
Posttensioned Normal Weight Concrete Lightweight Concrete
240 275
205 240
ACI-ASCE Committee (Zia et al. 1979)
This is the Maximum Loss that you may assumed
T Types off Prestress Pretensioned
Types of Concrete Normal Weight Concrete Lightweight Concrete
Maximum Prestress Loss (fpi fpe) (MPa) (fpi-fpe) Stress-Relieved Low Relaxation Strand Strand 345 380
276 311
Lump Sum Prestress Loss
Lump Sum Prestress Loss
T.Y. Lin & N. H. Burns S Source off Loss L
AASHTO LRFD (for CR CR, SR SR, R2) (5.9.5.3) (5 9 5 3)
P Percentage off Loss L (%) Pretensioned
Posttensioned
Elastic Shortening (ES)
4
1
Creep of Concrete (CR)
6
5
Shrinkage of Concrete (SR)
7
6
Steel Relaxation (R2)
8
8
25
20
Total
Note: Pretension has larger loss because prestressing is usually done when concrete is about 1-2 days old whereas Posttensioning is done at much later time when concrete is stronger.
Lump Sum Prestress Loss
AASHTO LRFD (Cont.)
Partial Prestressing Ratio (PPR) is calculated as:
PPR =
Aps fpy Aps fpy + As fy
PPR = 1.0 1 0 for Prestressed Concrete PPR = 0.0 for Reinforced Concrete
Elastic Shortening Loss (ΔfpES) is calculated as:
ΔfpES =
E ps Eci
Part IV: Allowable Stress Design g
fcgp,Fi +G
E ps ⎡ Fi Fi e02 MG e0 ⎤ = + − ⎢ ⎥ Eci ⎣ A c I I ⎦
Stress of concrete at the c.g. of tendon due to prestressing force and dead load
Stress Inequality Equation Allowable Stress in Concrete Allowable Stress in Prestressing Steel Feasible Domain Method Envelope and Tendon Profile
Basics: Sign Convention
Basics: Section Properties
In this class, the following convention is used:
c.g. g off Prestressingg Tendon Area: Aps
Concrete CrossSectiona Area: Ac
Tensile Stress in concrete is negative g (-) () Compressive Stress in concrete is positive (+) Positive Moment:
yt ((abs))
e ((-)) kt (-)
Positive Shear:
h (abs)
kb (+)
e (+)
Center of Gravity of Concrete Section (c.g.c)
I Kt Kb Zt Zb
yb (abs)
In some books,, the sign g convention for stress mayy be opposite so you need to reverse the signs in some formula!!!!!!!!!
Basics: Section Properties
Moment of Inertia, I
I = ∫ y dA 2
A
Rectangular section about c.g. Ixx = 1/12*bh3 Ix’x’ = Ixx + Ad2
yt and yb are distance from the c.g. of section to top and bottom fibers, respectively Sectional modulus modulus, Z (or S)
Zt = I/yt Zb = I/yb
c.g. of Prestressing Tendon Area: Aps
Basics: Section Properties
Kern of the section, section k, k is the distance from cc.g. g where compression force will not cause any tension i iin the h section i Consider C id T Topp Fib Fiber (Get Bottom Kern, kb)
0=
F Fe0 y t − Ac I
e0 =
I = kb Ac y t
Consider C id Bottom B tt Fiber Fib (Get Top Kern, kt)
0=
F Fe0 y b + Ac I
e0 = −
I = kt Ac y b
Note:Top kern has negative value
Basics: General Design Procedures
Stress in Concrete at Various Stages
Select Girder type, materials to be used, and number b off prestressing t i strands t d Check allowable stresses at various stages g Check ultimate moment strength Check cracking load Check shear Check deflection
Stress Inequality Equations
Allowable Stress in Concrete
We can write four equations based on the stress at the top and bottom of section at initial and service stages
No.
Case
Stress Inequality Equation
I
Initial-Top
F Fe M F ⎛ e ⎞ M σ t = i − i o + min = i ⎜ 1 − o ⎟ + min ≥ σ ti Ac Zt Zt Ac ⎝ kb ⎠ Zt
II
Initial-Bottom
F Fe M F ⎛ e σ b = i + i o − min = i ⎜ 1 − o Ac Zb Zb Ac ⎝ kt
III
Service-Top
IV
! ServiceBottom
F Feo Mmax Fi ⎛ eo ⎞ Mmax σt = − + = ≤ σ cs ⎜1− ⎟ + Ac Zt Zt Ac ⎝ k b ⎠ Zt
⎞ Mmin ≤ σ ci ⎟− ⎠ Zb
F Feo Mmax F ⎛ eo σb = + − = ⎜1− Ac Zb Zb Ac ⎝ kt
⎞ Mmax ≥ σ ts ⎟− Zb ⎠
AASHTO LRFD (5.9.4) provides allowable stress in p strength g at that concrete as functions of compressive time Consider the following limit states:
Immediately after Prestress Transfer (Before Losses)
Compression Tension
Service (After All Losses)
Compression C i Tension
Allowable Stress in Concrete
Immediately after Prestress Transfer (Before Losses)
Using compressive strength at transfer, f’ci
At service (After All Losses) Compressive Stress
Allowable All bl compressive i stress = 00.60 60 f’ci Allowable tensile stress
Allowable Stress in Concrete
Allowable Stress in Concrete
At service (After All Losses) Tensile Stress
Allowable Stress in Concrete - Summary Stage g Initial
Where Tension at Top
Load Fi+MGirder
Compression Fi+MGirder at Bottom Service Compression F+MSustained at Top 0.5(F+MSustained)+MLL+IM Tension at Bottom
Limit
Note
-0.58√f’ci
With bonded reinf…
-0.25√f’ci > -1.38 MPa
Without bonded reinf.
0.60 f’ci 0.45f’c
*
0.40f’c
*
F+MSustained+MLL+IM
0.60Øwf’c
*
F+MSustained+0.8MLL+IM (Service III Limit State)
-0.50√f’c
Normal/ Moderate exposure
-0.25√f’c
Corrosive exposure
0 U b d d tendon Unbonded d * Need to check all of these conditions (cannot select only one)
Allowable Stress in Prestressing Steel
ACI and AASHTO code specify the allowable stress t in i the th prestressing t i steel t l att jacking j ki and d after ft transfer
Allowable Stress in Prestressing Steel
ACI-318 ACI 318 (2002)
Allowable Stress in Prestressing Steel
AASHTO LRFD (5.9.3)
Allowable Stress in Prestressing Steel
Allowable Stress Design
There are many factors affecting the stress in a prestressed girder
For bridges, we generally has a preferred section type for a given range of span length and we can select a girder spacing to be within a reasonable range
The Section used ((dead load)) Girder Spacing (larger spacing Æ larger moment) Slab Thickness (larger spacing Æ thicker slab)
Stages of construction
Sections
Prestressing Force (Fi or F) L Location off prestress tendon d (e0) ( 0) Section Property (A, Zt or Zb, kt or kb) External moment, which depends on
Allowable Stress Design
AASHTO Type I-VI Sections
ft 50 75 100 150
m 15 23 30 46
Sections
AASHTO Type I-VI Sections (continued)
Bridge Girder Sections
Bridge Girder Sections
Allowable Stress Design
Feasible Domain - Equations
For a given section, we need to find the combination bi ti off prestressing t i fforce (Fi or F, F which hi h depends on the number of strands), and the location of strands (in terms of e0) to satisfy these equations Possible methods:
Keep trying some number of strands and locations ((Trial & Error)) We use “Feasible Domain” Method
We can rewrite the stress inequality equations and add one more equation to them
No No.
Case
Stress Inequality Equation
I
Initial-Top
⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠
)
II
Initial-Bottom
⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠
)
III
Service-Topp
IV
ServiceService Bottom
V
P Practical i l Li Limit i
(
(
(
)
(
)
⎛ 1⎞ e0 ≥ k b + ⎜ ⎟ Mmax − σ cs Zt ⎝F ⎠
⎛ 1⎞ e0 ≥ kt + ⎜ ⎟ Mmax + σ ts Zb ⎝F ⎠
e0 ≤ ( e0 )mp = y b − dc ,min = y b − 7.5 cm
!
Feasible Domain – Graphical Interpretation
Feasible Domain
Envelope - Equations
I II III
Envelope - Equations
We use the same equation as the feasible domain, except that we’ve already known the F or Fi and want to find e0 at different points along the beam
No No.
Case Initial-Top Initial-Bottom Service-Top p
IV
ServiceService Bottom
V
P Practical i l Li Limit i
Feasible domain tells you the possible location and prestressing force at a given section to satisfy the stress inequality equation We usually use feasible domain to determine location and d prestressing i force f at the h most critical i i l section i (e.g. ( midspan of simply-supported beams) After we get the prestressing force at the critical section, section we need to find the location for the tendon at other points to satisfy stress inequalities We use the prestressing envelope to determine the g of the beam (tendon ( location of tendon alongg the length profile)
Stress Inequality Equation ⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠
)
⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠
)
(
(
(
⎛ 1⎞ e0 ≥ k b + ⎜ ⎟ Mmax − σ cs Zt ⎝F ⎠
(
⎛ 1⎞ e0 ≥ kt + ⎜ ⎟ Mmax + σ ts Zb ⎝F ⎠
We then have 5 main equations
I & II provide the lower bound of e0 (use minimum of the two) III and d IV provide id the th upper bound b d off e0 (use ( maximum i of the two)
)
)
e0 ≤ ( e0 )mp = y b − dc ,min = y b − 7.5 cm
!
IIIa uses F+MSustained III IIIb uses 0.5(F+MSustained)+MLL+IM IIIc uses F+MSustained+MLL+IM IV uses F+MSustained+0.8MLL+IM
V is a practical limit of the e0 (it is also the absolute lower bound)
Envelope & Tendon Profile
Envelope & Tendon Profile
Envelope & Tendon Profile
Envelope & Tendon Profile
Note
The tendon Th d profile f l off pretensioned d members b are either straight or consisting of straight segments The tendon profile of posttensioned member may be g tendon or smooth curved, but no sharpp one straight corners
There is an alternative to draping the strands in pretensioned t i d member b We pput pplastic sleeves around some strands at supports to prevent the bond transfer so the prestress force will be less at that section
Load – Deflection – Concrete Stress
Part II: Ultimate Strength g Design g Concrete and Prestressing Steel Stresses Cracking Moment Failure Types A l i ffor Mn – Rectangular Analysis R t l Section S ti T-Section A l i ffor Mn – TAnalysis T Section S i
Load - Deflection
1 & 2: Theoretical camber (upward deflection) of prestressed beam 3: Self weight + Prestressing force 4: Zero deflection ppoint (Balanced ( ppoint)) with uniform stress across section 5: Decompression point where tension is zero at the b bottom fiber fb 6: Cracking point where cracking moment is reached 7: End of elastic range (the service load will not be larger than this) 8 Yielding 8: Yi ldi off prestressing i steell 9: Ultimate strength (usually by crushing of concrete)
Prestressing Steel Stress
Prestressing Steel Stress
Cracking Moment
The pprestressingg steel stress increases as the load increases Crackingg of beam causes a jump j p in stress as additional tension force is transferred from concrete (now cracked) to prestressing steel At ultimate of prestressed concrete beam, the stress in g fpy and steel is somewhere between yyield strength ultimate strength fpu Stress is lower for unbonded tendon because stress is distributed throughout the length of the beam instead of just one section as in the case of bonded tendon At ultimate, the effect of prestressing is lost and the section behaves jjust like an RC beam
Cracking Moment
F Feo Mcr F ⎛ eo + − = ⎜1− Ac Zb Zb Ac ⎝ kt
Concrete cracks when bottom fiber reaches the tensile capacity (modulus of rupture)
fr = -0.63 (f’c)0.5 MPa (5.4.2.6)
Failure Types
The moment at this stage is called “cracking moment” which depends on the geometry of the section and prestressing force σb =
⎞ Mcr = fr ⎟− ⎠ Zb
Solve the above equation to get Mcr
Mcr = F (eo − kt ) − fr Zb Note: Need to input fr and kt as negative values !!!
This is similar to RC Fracture of steel after concrete cracking. This is a sudden failure and occurred because the beam has too little reinforcement Crushing of concrete after some yielding of steel. This is called tension tension-controlled. controlled. Crushing of concrete before yielding of steel. This is a brittle failure due to too much reinforcement. reinforcement It is called overreinforced or compression-controlled.
Failure Types
Analysis for Ultimate Moment Capacity
Analysis assumptions
Analysis for Ultimate Moment Capacity
Recall from RC Design that the followings must b satisfy be ti f att allll times ti no matter tt what h t happens: h
Analysis for Ultimate Moment Capacity
For equilibrium, there are commonly 4 forces
EQUILIBRIUM
Compression in concrete C Compression p in Nonprestressed p reinforcement Tension in Nonprestressed reinforcement Tension in Prestressed reinforcement
STRAIN COMPATIBILITY
Plane section remains plane Pl l after f bending b d (linear (l strain distribution) Perfect bond between steel and concrete (strain p y) compatibility) Concrete fails when the strain is equal to 0.003 Tensile strength stren th off concrete c ncrete is neglected ne lected at ultimate ltimate Use rectangular stress block to approximate concrete stress distribution
They also hold in Prestressed Concrete!
For concrete compression, we still use the ACI’s rectangular stress block
Rectangular Stress Block
Rectangular Stress Block 0.85 f 'c ≤ 28 MPa ⎧ ⎪ ⎛ f ' − 28 ⎞ ⎛ 1 ⎞ ⎪ β1 = ⎨0.85 − 0.05 ⎜ c ⎟ ⎜ 1 ⎟ 28 ≤ f 'c ≤ 56 MPa 7 ⎝ ⎠⎝ ⎠ ⎪ f 'c ≥ 56 MPa 0.65 ⎩⎪
β1 is equal to 0.85 0 85 for f ’c < 28 MPa It decreases 0.05 for everyy 7 MPa increases in f ’c Until it reaches 0.65 at f ’c > 56 MPa
Analysis for Ultimate Moment Capacity
For tension and compression in nonprestressed reinforcement, i f t we d do th the same thing thi as in i RC design:
Assume that the steel yield first; i.e. Ts = Asfy or Cs = As’ffy’ Check the strain in reinforcement to see if they actually yield or not, not if not not, calculate the stress based on the strain at that level & revise the analysis to find new value of neutral axis depth, depth c Ts = Asfs = AsEsεs = AsEs· 0.003(c-d)/c
Analysis for Ultimate Moment Capacity
For tension in prestressing steel steel, we observe that we cannot assume the behavior of prestressing steel (which is high strength steel) t l) tto bbe elasticl ti perfectly plastic as in the h case off steell reinforcement in RC
Analysis for Ultimate Moment Capacity
At ultimate of prestressed concrete beam, the stress in steel is clearly not the yield strength but somewhere between yield strength fpy and ultimate strength fpu W called We ll d iit fps The true value of stress is difficult to calculate (generally requires nonlinear moment-curvature analysis) so we ggenerallyy estimate it usingg semi-empirical p formula
Ultimate Stress in Steel: fps
⎛ c fps = fpu ⎜ 1 − k ⎜ dp ⎝
ACI Æ Bonded Tendon or Unbonded Tendon AASHTO Æ Bonded Tendon or Unbonded Tendon
Ultimate Stress in Steel: fps
AASHTO LRFD Specifications For Bonded tendon only (5.7.3.1.1) (5 7 3 1 1) and for fpe 0 5fpu p > 0.5f p
⎞ ⎛ fpy ; k = 2 ⎟⎟ ⎜⎜ 1.04 − fppu ⎠ ⎝
⎞ ⎟⎟ ⎠
Note: for ppreliminaryy design, g we mayy conservativelyy assume fps=fpy (5.7.3.3.1)
For Unbonded tendon, see 5.7.3.1.2
Analysis for Ultimate Moment Capacity
Notes on Strain Compatibility The strain in top of concrete at ultimate is 0.003 We can use similar triangle to find the strains in concrete or reinforcingg steel at anyy levels from the topp strain We need to add the tensile strain due to prestressing (occurred before casting of concrete in pretensioned or before grouting in posttensioned) to the strain in concrete at that level to get the true strain of the prestressing steel
Resistance Factor φ
Maximum & Minimum Reinforcement
M i Maximum Reinforcement R i f t (5 (5.7.3.3.1) 7 3 3 1)
The maximum of nonprestressed and prestressed reinforcement shall be such that c/de ≤ 0.42 c/de = ratio between neutral axis depth (c) and the centroid depth of the tensile force (de)
Minimum Reinforcement (5.7.3.3.2)
The minimum Th i i off nonprestressed t d and d prestressed t d reinforcement shall be such that ØMn > 1.2M 1 2Mcr (Mcr = cracking ki moment), ) or ØMn > 1.33Mu (Mu from Strength Load Combinations)
R it Resistance Factor F t Ø Section Type
RC and PPC w/ PPR < 0.5
PPC with 0.5< PPR < 1
(PPR = 1.0)
Under-Reinforced Section c/de ≤ 0.42
0 90 0.90
0 90 0.90
1 00 1.00
Over-Reinforced O R i f d SSection ti c/de > 0.42
Nott N Permitted
0 70 0.70
0 70 0.70
Rectangular vs. vs T T-Section Section
Most prestressed concrete p or Tbeams are either I-Shaped shaped (rarely rectangular) so they have larger compression flange If the neutral axis is in the flange we called it rectangular flange, section behavior. But if the g neutral axis is below the flange of the section, we call it Tsection behavior This has nothing to do with the overall shape of the section !!!
PC
Note: if c/de > 0.42 the member is now considered a compression member and different resistance factor applies (see 5.5.4.2) AASHTO does d es not n t permit ermit the use se off over-reinforced er reinf rced RC (defined as sections with PPC < 0.5) sections
Rectangular vs. vs T T-Section Section
If it is i a T-Section T S ti bbehavior, h i th there are now two t value l off widths, idth namely b (for the top flange), and bw (web width) We need to consider nonuniform width of rectangular stress block
Rectangular vs. vs T T-Section Section
T-Section T Section Analysis
We divide the compression side into 2 parts
Overhanging O h portion off flange fl (width ( d h = b-b b bw ) Web ppart (width ( = bw )
We generally assume that the section is rectangular first and check if the neutral axis depth (c) is above or below the flange thickness, thickness hf Note:ACI method checks a=ß1c with hf, which may give slightly li htl diff differentt result lt when h a < hf but b t c > hf
T-Section T Section Analysis
T-Section T Section Analysis
From equilibrium
0.85f 'c bw β1c + 0.85f 'c (b − bw )β1hf = Aps fps + As fy − As ' fy '
0.85f 'c bw β1c + 0.85f 'c (b − bw )β1hf = Aps fps + As fy − As ' fy '
For preliminary analysis, or first iteration, we may assume fps = fpy and solve for c
c=
Aps fy + As fy − As ' fy '− 0.85f 'c (b − bw )β1hf 0 85f 'c bw β1 0.85
For a more detailed approach, we recall the equilibrium
⎛ ⎜ ⎝
Substitute fps = fpu ⎜ 1 − k
c=
c dp
⎞ ⎟⎟ , Rearrange and solve for c ⎠
Aps fpu + As fy − As ' fy '− 0.85 0 85f 'c (b − bw )β1hf 0 85f 'c bw β1 + kAps fpu / d p 0.85
T-Section T Section Analysis
T-Section T Section Analysis Flowchart
Moment Capacity (about a/2) a⎞ a⎞ a⎞ ⎛ ⎛ ⎛ Mn = Aps fps ⎜ d p − ⎟ + As fy ⎜ ds − ⎟ − As ' fy ' ⎜ ds '− ⎟ 2⎠ 2⎠ 2⎠ ⎝ ⎝ ⎝ h ⎞ ⎛ +0.85 0 85f 'c (b − bw )β1hf ⎜ a − f ⎟ 2⎠ ⎝
T-Section T Section Analysis Flowchart
T Section T-Section
In actual structures, the section is pperfect T or I shapes p there are some tapering flanges and fillets. Therefore, we need to idealized the true section to simplify the analysis. Little accuracy may be lost.
We need this for ultimate analysis only. We should use the true section property for the allowable stress analysis/ design
Composite
Part III: Composite p Beam
Composite generally means the use of two diff different t materials t i l iin a structural t t l elements l t Example: p Reinforced Concrete
Typical Composite Section Composite p Section Properties p Actual, Effective, and Transformed Widths Allowable Stress Design Stress Inequality Equation, Feasible Domain, and Envelope Cracking Moment Ul i Ultimate M Moment C Capacity i
Composite Beam
In the context of bridge design, the word composite it bbeam means the th use off ttwo diff differentt materials between the beam and the slab
Steel Beam + Concrete Slab
Steel beam carries tension Concrete in slab carries compression
Prestressed Concrete Beam (high-strength (high strength concrete) + Concrete Slab (normal-strength concrete)
Prestressed P d Concrete C beam b carries i tension i Concrete in slab carries compression
Concrete – carry compression St l R Steel Reinforcement i f t – carry tension t i
Example: p Carbon Fiber Composite p
Carbon Fiber – carry tension E Epoxy Resin Matrix Matri – hold h ld the fibers in place lace
Typical Composite Sections
Typical Composite Sections
Slab may be cast:
Why Composite?
EEntirely l cast-in-place l with removable formwork Using precast panel as a formwork formwork, the pour the concrete topping
There are some benefits of using precast elements l t
There are some benefits of putting the composite slab
Particular Design Aspects
There are 3 more things we need to consider specially for composite section (on top of stuffs we need to consider for noncomposite sections) T Transformation f i off SSection i
Loadingg Stages g
Actual width vs. Effective width vs. Transformed width Composite Section Properties Allowable Stress Design SShored o vs. U Unshored s o Beams a s
Horizontal Shear Transfer
Save Time Better Quality Control Cheaper
Provide continuity between elements Quality control is not that important in slabs
Composite Section Properties
There are 3 value of widths we will use:
Actuall width A d h off the h composite section (b): (b) This Th is equal to the girder spacing Effective width of the composite section (be) Transformed width of the composite section (btr)
Composite Section Properties
Effective Width
Composite Section Properties
The stress distribution across the width are not uniform – the farther it is from the center, the lesser the stress. To simplify the analysis, we assume an effective width where the stress are constant throughout We also assume the effective width to be constant along the span.
Effective Width (AASHTO LRFD - 4.6.2.6.1)
boverhang ⎧ b b 'w = max ⎨ w ⎩bf / 2
Exterior Girder
Exterior Beam
be,ext
Composite Section Properties
Typically the concrete used for slab has lower strength than h concrete used d for f precast section i Lower strength Æ Lower modulus of elasticity Thus, we need to use the concept of transformed section to transform the slab material to the precast material
btr = be nc = be
Ec ,CIPC Ec ,PPC
≅ be
Modular Ratio, usually < 1.0
f 'c ,CIPC f 'c ,PPC
⎧b 'w / 2 + 6ts ⎪ = + min ⎨ boverhang 2 ⎪ L/8 ⎩ be,int
Transformed Width
ts
bw Interior Girder
Interior Beam
⎧b 'w + 12ts ⎪ be = min ⎨ s ⎪ L/4 ⎩
Composite Section Properties
Transformed Width
s be
be
bf
Composite Section Properties
Composite Section Properties
Summary of steps for Width calculations
Actual Width b Equals to girder p g spacing
Effective Width be Accounts for nonuniform stress distribution
Composite CrossSectiona Area: Acc
Precast Cross CrossSectiona Area: Ac
ytc ((abs) b)
yt (abs)
yy’tc (abs)
dp
Aps
dpc
Most of the theories learned previously for the noncomposite it section ti still till hold h ld but b t with ith some modifications We will discuss two design limit states
Aps
Precast vs. Composite
btr
ybc (abs)
yb (abs)
Acc = Ac + tsbtr ytc, ytb Igc Ztc, Zbc dpc
Design of Composite Section
c.g. p Composite c.g. Precast
h (abs)
Transformed Width btr Accounts for dissimilar material properties
Composite Section Properties
After we get the transformed section, we can th calculate then l l t other th section ti properties ti
Allowable All bl St Stress D Design i Ultimate Strength Design
Allowable Stress Design - Composite
OUTLINE Shored vs. Unshored Stress Inequality Equation Feasible Domain & Envelope
Allowable Stress Design - Composite
In allowable stress design, we need to consider two loading stages as pprevious; however, the initial moment ((immediatelyy after transfer) is resisted by the precast section whereas the service moment (after the bridge is finished) is resisted by the composite section (precast section and slab acting together as one member) We need to consider two cases of composite construction methods: h d
Shored vs vs. Unshored
Shored – beam is supported by temporary falsework when the slab is cast The falsework is removed when the slab hardens cast. hardens. Unshored – beam is not supported when the slab is cast.
Shored vs vs. Unshored
Moments resisted by the precast and composite sections are different in the two cases Fully Shored
Precast: Girder Weight Composite: Slab Weight, Superimposed Loads (such as asphalt ) and Live Load surface),
Unshored
Precast: Girder Weight and Slab Weight Composite: Superimposed Loads (such as asphalt surface), and Live Load
Shored vs vs. Unshored
FULLY SHORED Consider, as example, the top of precast beam σt =
Shored vs vs. Unshored
Top of precast, not top of composite i
F Feo (MGirder ) (MSlab + MSD + MLL +IM )y 'tc − + + ≤ σ cs Ac Zt Zt Igc
σt =
Shored vs vs. Unshored
UNSHORED Consider, as example, the top of precast beam F Feo (MGirder + MSlab ) (MSD + MLL +IM )y 'tc − + + ≤ σ cs Ac Zt Zt Igc
Stress Inequality Equations
From both case we can rewrite the stress equation q as:
Case σt =
Mp = Moment resisted by the precast section (use Zt, Zb)
F Feo (MP ) (MC ) − + + ≤ σ cs Ac Zt Zt Z 'tc
Fully Shored: Mp = Mgirder Unshored: Mp = Mgirder + Mslab
Mc = M Moment resisted i d bby the h composite i section i ((use Z’tc, Zbc)
Fully Shored: Mc = Mslab + MSD + MLL+IM Unshored: Mc = MSD + MLL+IM
I
Initial-Top
II
I ii lB Initial-Bottom
III Service-Top ! IV Service-Bottom VI Service-Top Slab
We can also write similar equation for stress at the bottom of composite beam
Top p of precast, p not top of composite
Stress Inequality Equation σt =
Fi Fi eo Mmin Fi ⎛ eo ⎞ Mmin − + = ≥ σ ti ⎜1− ⎟ + Ac Zt Zt Ac ⎝ kb ⎠ Zt
σb =
Fi Fi eo Mmin Fi ⎛ eo ⎞ Mmin + − = ≤ σ ci ⎜1− ⎟ − Ac Zb Zb Ac ⎝ k t ⎠ Zb
σt =
F Feo M p Mc F ⎛ eo ⎞ M p Mc − + + = + ≤ σ cs ⎜1− ⎟ + Ac Zt Zt Ztc Ac ⎝ k b ⎠ Zt Z 'tc
σb =
F Feo M p Mc F ⎛ eo + − − = ⎜1 − Ac Zb Zb Zbc Ac ⎝ kt
σ t ,slab =
⎞ M p Mc − ≥ σ ts ⎟− ⎠ Zb Zbc
Mc M E nc = c c ,CIPC ≤ σ cs,Slab Ztc Ztc Ec ,PPC
Stress at the top of the slab must also be less than the allowable compressive stress
Feasible Domain & Envelope
Top of precast
We can rewrite the stress equations and add practical limit equation
No. I
Case
Stress Inequality Equation
Initial-Top p
II
Initial Bottom Initial-Bottom
III
S Service-Top T
⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠ ⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠ ⎞ Z ⎛ 1 ⎞⎛ e0 ≥ k b + ⎜ ⎟ ⎜ M p + Mc t − σ cs Zt ⎟ Z 'tc ⎝ F ⎠⎝ ⎠
(
)
(
)
IV
ServiceBottom
⎞ Z ⎛ 1 ⎞⎛ e0 ≥ kt + ⎜ ⎟ ⎜ M p + Mc b + σ ts Zb ⎟ Zbc ⎝ F ⎠⎝ ⎠
V
Practical Limit
e0 ≤ ( e0 )mp = y b − dc ,min
VI
Service-Top Slab
σ t ,slab =
!
F Feo M p ΔMcr F ⎛ eo + − − = ⎜1− Ac Zb Zb Zbc Ac ⎝ kt
ΔMcr =
⎞ M p ΔMcr − ≥ σ ts ⎟− Z Z b bc ⎠
Zbc ⎡F (eo − kt ) − M p ⎤⎦ − fr Zbc Zb ⎣
Mcr = ΔMcr + M p
σb =
F Feo Mcr F ⎛ eo ⎞ Mcr + − = = fr ⎜1− ⎟ − Ac Zb Zb Ac ⎝ k t ⎠ Zb
Mcr = F (eo − kt ) − fr Zb
Ultimate Strength Design - Composite
Cracking C k occurs in the h composite section We find ∆Mcr ((moment in addition to Mp) σb =
Cracking occurs in the precast section The equation is the same as noncomposite section
Mc M E nc = c c ,CIPC ≤ σ cs,Slab Ztc Ztc Ec ,PPC
II. Cracking moment is greater Mp
We consider 2 cases 1. Cracking moment is less than Mp
Cracking Moment - Composite
Cracking Moment - Composite
Ultimate strength of composite section follows similar procedure to the T-section. T-section Some analysis tips are:
When the neutral axis is in the slab, we can use a composite Tsection with flange width equals to Effective Width and using ff’c of the slab When the neutral axis is in the precast section section, we may use a Transformed Section and f’c of the precast section - This is an pp value but the errors to the ultimate moment approximate capacity is small.
Shear Transfer Mechanisms
Shear Transfer
To get the p composite behavior, it is very important that the slab and girder must not slip past each other
The key parameter that determines whether these two parts will slip past each other or not is the shear strength at the interface of slab and girder This interfacial shear strength comes from:
Shear Transfer – Cohesion & Friction
Shear Transfer - Formula
Cohesion is the chemical bonding of the two materials. It depends on the cohesion factor ((c)) and the contact area. The ggreater the area, the larger the cohesion force.
Friction (F = μN) Cohesion
AASHTO LRFD (5.8.4) The nominal shear resistance at the interface between two concretes cast at different times is taken as: Friction Factor
Friction is due to the roughness of the surface. It depends on the friction factor or coefficient of friction (μ) and the normal force (N). To increase friction, we either make the surface rougher (increase μ) or increase the normal force.
Area of Concrete Transfering Shear Cohesion
Area of shear reinforcement crossing the shear plane Compressive force normal to shear plane
⎧≤ 0.2f 'c Acv Vnh = cAcv + μ( Avf fy + Pc ) ⎨ 5 5 Acv ⎩ ≤ 5.5 COHESION FRICTION
N
Vhu ΦVhn =ΦμN
Shear Transfer – Cohesion & Friction
Shear Transfer – Cohesion & Friction
AASHTO LRFD (5.8.4.2) (5 8 4 2)
Minimum Shear Reinforcement
For Vn/Acv > 0.7 MPa, the cross-sectional area of shear reinforcement crossing the interface per unit length of beam must not be less than
0.35bv Avf ≥ fy
If less, then we cannot use any Avffy in the nominal shear strength The spacing of shear reinforcement must be ≤ 600 mm Possible reinforcements are: Single S l bbar
Width of the interface (generally equals to the width of top flange of girder)
Stirrups (multiple legs) W ld d wire Welded i fabric f bi
Reinforcement must be anchored properly (bends, hooks, etc…)
The normal force in the friction formula comes from two parts Yielding of shear reinforcement If cracking k occurs at the h interface, there will be tension in the steel reinforcement crossing the interface. This tension force in steel is balanced b the by h compressive i force f in i concrete at that interface; thus, creating normal “clamping” clamping force. Permanent compressive force at the interface Dead Weight of the slab and wearing earin surface s rface (asphalt) (as halt) Cannot rely on Live Loads
Avf
N=Avf fy
Ultimate Shear Force at Interface
There are two methods for calculating shear force per unit length at the interface ((the values mayy be different))
Using Classical Elastic Strength of Materials
Vuh =
( ΔVu ) Q Igc
Moment of Inertia of the composite section
Factored shear force acting on the composite section only (SDL +LL+IM) Moment of Area above the shear plane about the centroid of composite section
Using Approximate Formula (C5.8.4.1-1)
Vuh =
Vu de
Total Factored vertical shear at the section Distance from centroid of tension p of the deck steel to mid-depth
Ultimate Shear Force at Interface
The critical section for shear at the interface is generally the g section where vertical shear is the greatest
Some Design Tips
First critical section: h/2 from the face of support May calculate at some additional sections away from the support (which has lower shear) to reduce the shear reinforcement accordingly
Critical Section For Shear
h h/2
h/2
Resistance Factor (Φ) for shear in normal weight concrete : 0.90
For T and Box Sections which cover the full girder spacing with thin concrete topping (usually about 50 mm), we may not need any shear reinforcement (need only surface roughening) g g) – need to check For I-Sections, we generally require some shear reinforcement at the interface We generally design the web shear reinforcement first (not taught), and extend that shear reinforcement through the interface. Then we check if that area is enough for horizontal shear transfer at the interface.
Final Notes on Composite Behavior
Composite C it section ti iis used d nott only l ffor prestressed t d concrete t sections, but also for steel sections. Benefits is that the slab helps resists compression and helps prevent lateral torsional buckling of the steel section, as well as local bucklingg at the compression p flange. g
If not, we need additional reinforcement If enough, then we do nothing
Final Notes on Composite Behavior
The analysis concept is similar to that of prestressed concrete. There are also:
Effective width and transformed section Shored and Unshored Construction Sh Shear Transfer T f att IInterface t f
b
Final Notes on Composite Behavior
There are various ways to transfer shear at steel-concrete interface
Spirals
Studs
Final Notes on Composite Behavior Shear Stud is one of the most common shear h connectors – it is welded to the top flange of steel girder
Channels
Final Notes on Composite Behavior
Part IV: Things I did not teach but you should be aware of !!! Shear Strength – MCFT Unbonded and External Prestressing Anchorage Reinforcement Camber and Deflection Prediction Detailed Calculation of Prestress Losses Steel Girder with Shear Stud
Shear
Shear - MCFT
Traditionally, the shear design in AASHTO Standard Specification is similar to that of ACI, ACI which is empiricalbased Th axial The i l force f from f prestressing i reduces d the h principal i i l tensile stress and helps close the cracks; thus, increase shear h resistance.
Shear
The shear resisting mechanism in concrete is very complex and we do not clearly understand how to predict it AASHTO LRFD (5.8.3) (5 8 3) uses new theory, theory called “modified compression field theory (MCFT)” Th actuall theory The h is very complicated l d but b somewhat h simplified procedure is used in the code This theory is for both PC and RC
Minimum Transverse Reinforcement
The nominal shear resistance is the sum of shear strength of concrete, concrete steel (stirrups) (stirrups), and shear force due to prestressing (vertical component)
Vn = Vc + Vs + Vp ≤ 0.25f 'c bv dv + Vp Vs =
Av fy dv cot θ s
Vc = 0.083 β f 'c bv dv
We need some transverse reinforcement when the ultimate shear force is greater than ½ of shear strength from concrete and prestressing force
Vu > φ0.5(Vc + Vp )
If we need it, it the minimum amount shall be
Av ≥ 0.083 0 083 f 'c
bv s fy
Minimum Transverse Reinforcement
Maximum Spacing
Unbonded or External Prestressing
For vu