Seoul Nat’l UniversitySeoul Nat l University

Concrete in Plasticity

Sung-Gul HongSung Gul HongSNU

March 2, 2006

From analysis to Design and SynthesisFrom analysis to Design and Synthesis

1900 ElasticityPlasticity

Analysis

1970

PlasticityMoment distribution

MatrixFEM

Design

1970

1980 Strength DesignLimit Designg

Synthesis of Designd A l i

1990

2000

Limit Design

Performance-basedITand Analysis Social Response

History of Theory of PlasticityHistory of Theory of Plasticity

• Gvosdev• Drucker and Prager• W.F Chen• Johansen: Yield Line Theory• Hillerborg: Strip Method• The Technical University of Denmark:

Ni l d B• Nielsen and Braestrup• Swiss Federal Institute of Technology in Zurich:

M eller and Marti• Mueller and Marti• Vecchio and Collinse in Canada• T T C Hsu in U S A• T.T.C.Hsu in U.S.A

Limit Theory for Design of Reinforced Concrete

• Lower Bound- EquilibriumEquilibrium- Yield Condition

Forces Displacement

• Upper Bound- Geometry

Forces

- Geometry- Work equation

Normality Rules

ContentsContents

h f l i i• Theory of Plasticity• Yield Conditions• Theory of Plain Concrete• Disks• Disks• Beams• Slabs• Punching ShearPunching Shear• Bond Strength

Theory of PlasticityTheory of Plasticity

• 1.1 Constitutive EquationsVon Mises’s Flow Rule

Q q ifqQ

λ ∂=

∂2Q 2qiQ∂

Q1Q 1q

Extreme Principles for Rigid-Plastic Materialsε

σσ

-Lower Bound Theorem-Upper Bound Theorem-Uniqueness Theorem

Yield ConditionsYield Conditions

• Concrete- Frictional hypothesisFrictional hypothesis

• Reinforcing bar• Disks• Disks• Slabs

Mohr Coulomb Material σ ε

τ

2 2σ ,ε

τ1 1σ ,ε

Sliding Failure

σφ Separation failure

2 2σ ,εσ

1 1σ ,ε

Determination of Parameters for More-Coulomb material

σ σ

P T i

ττPure Compression Pure Tension

1 3 ck fσ σ− =σ

Pure Shearτ

Failure Modes in Pure ShearFailure Modes in Pure Shear

σSliding failure

σ

τ

σSeparation failure

τ

Failure Modes in Pure TensionFailure Modes in Pure Tension

σSeparation failure

σ

τ

σσ

Sliding failureτ

g

Yield Condition in Plane StressYield Condition in Plane Stress

IσIIσIIIσ IσIIσIIIσ

IIIσ IIσIσ

Yield condition for DiskYield condition for Disk

2 2σ ,εf

1 1σ ,εc-f

c-f

1 1σ , εyfy-f

Disks in Plane StressesDisks in Plane Stressesyσ

τ

σ

xyτ

= +xσxσ

xyτ

σ εyσ 2 2σ ,ε

c-f1 1σ , ε

yfy-f

1 1σ ,ε

-f

1 1σ , ε

cf

Yield Condition for DiskYield Condition for Diskxy

fτ

cf

σ

s YA ft

Φ =

y

cfσ

1Φ +

x

cfσ

x

cfσ

Free Body Diagram for Yield ConditionCondition yσ

xyτ

τ xσxyτxyτ

xσ

xyτ xyyσ

cσsy YA f

cσ

θ sx YA f xσθxyτ

θ sx Yf

yσ

Shear strengthShear strengthσ A f

xytτA f

cσsy YA f

ytσ

( )( ) 2 0c x c y xyf fσ σ τ− Φ − Φ − + =xy

sinctσ θ

sy YA ftθ θ

( )( )c x c y xyf f

yσ2sinsy Y

y c

A ft tσ σ θ= −

cσ

A f σθxyτ

θcosctσ θ

y ct

i θ θsx YA f xσθ θ

sx YA ft

sin cosxy ct tτ σ θ θ=

t2cossx Y

x cA ft t

tσ σ θ= −

Shear Strength of DiskShear Strength of Disk

Relationship between Shear Strength and Reinforcement Ratio in Disk

Reinforcement in Disk based on Plasticity Solution

sx Yx xy

A ft

σ τ= +

sy Yy xy

tA f

tσ τ= +

x yσ σ≤

x xyσ τ≥ −

2x y xyσ σ τ≤y y

Failure Mechanisms in DisksFailure Mechanisms in Disks

Yield conditions for SlabsYield conditions for Slabs• Pure bending -> yield line theory• Pure bending -> yield line theory

and strip methodP t i > S d i h d l• Pure torsion -> Sandwich model

n

ptxx

pt t

Torsion in slabsTorsion in slabs

Top panelp p

( )1 2p o s Yt A f h= − Φp

bottom panelbottom panel

The Theory of Plain ConcreteThe Theory of Plain Concrete

• Constitutive Equations• Dissipation WorksDissipation Works• Lines of Discontinuity• Stress Fields• Strain Fields• Strain Fields• Applications• Concentrated Loadings

Constitutive EquationsConstitutive Equations • Coulomb Material• Yield Condition• Strain Vectors

6 Surfaces• Strain Vectors

1 3 1 2 3,ck fσ σ σ σ σ− = ≥ ≥

3 1 3 2 1,ck fσ σ σ σ σ− = ≥ ≥k fσ σ σ σ σ≥ ≥ f∂

1 2 1 3 2,ck fσ σ σ σ σ− = ≥ ≥

2 1 2 3 1,ck fσ σ σ σ σ− = ≥ ≥ ii

fqQ

λ ∂=

∂2 3 2 1 3,ck fσ σ σ σ σ− = ≥ ≥

k fσ σ σ σ σ= ≥ ≥

iQ∂

3 2 3 1 2,ck fσ σ σ σ σ− = ≥ ≥

Yield Surface in 3 DYield Surface in 3-D3σ

(2)1σ

2σ(2)(6)

(4)

(1) (5)

(3) (4)

2σ(1) (5)

1σ3σ

Plastic Strain VectorPlastic Strain Vector

• Along planes

1

2ε1ε 3εkλ 0 λ−

k f ≥ ≥2

3 λkλ

kλ0

0

λ−1 3 1 2 3,ck fσ σ σ σ σ− = ≥ ≥

f∂ 3

4

λ−kλkλ

00λ

λ−ifqQ

λ ∂=

∂ 5

6

kλkλ0

0 λ−

λ−iQ∂

1

2ε1ε 3εkλ 0 λ−1

2

3

kλ

λkλ

kλ00

0

λ−λ

3

4

λ−kλkλ

00λ− • Along edges

2ε1ε 3ε5

6

kλkλ0

0 λ−

λ−

g g

1/52ε1ε 3ε

1kλ 2kλ

( )

6 0 λ

λλ( )1 2λ λ− +

4/54/2

( )1 2 kλ λ+ 2λ−

( )1 2λ λ− +1λ−

2kλ1kλk

ε +

−=∑

∑ 2/63/6

1λ− 2λ− ( )1 2 kλ λ+

kλ ( )λ λ− + kλ

ε −∑3/61/3

1kλ ( )1 2λ λ+ 2kλ

( )1 2 kλ λ+ 2λ− 1λ−

Plastic Strain Vector at ApexPlastic Strain Vector at Apex1 2 3 cot

1cf c

kσ σ σ ϕ= = = = 3σ

1 2 3 1kϕ

−

(2)( ) ( )kλ λ λ λ (2)(6)(4)

( ) ( )1 1 3 2 4kε λ λ λ λ= + − +

( ) ( )

(1) (5)

(3) (4)

2σ( ) ( )2 4 5 3 6kε λ λ λ λ= + − +

( ) ( ) (1) (5)( ) ( )3 2 6 1 5kε λ λ λ λ= + − +

1σ

Dissipation WorkDissipation WorkAlong plane 1Along plane 1

1 1 1 1 1 1 1 3 cW k fσ ε σ ε σ ε σ λ σ λ λ= + + = − =

Along the edge 1/5

1 1 1 1 1 1 1 3 cf

( )1 2 cW fλ λ= +

g g

At the apex

( )1 2 3 4 5 6 cW fλ λ λ λ λ λ= + + + + +

Yield Condition and Flow Rule for Coulomb Material in Plane Stress

2 2,σ ε2 2,σ ε

( )kλ λ 2 2

( )0, ,kλ λ−

( )0kλ λ1

cfk

( )0, ,kλ λ−

( ), ,0kλ λ−

1 f

( ), ,0kλ λ−1 1,σ εk

( ), ,0kλ λ−( ),0, kλ λ− cf

1 1,σ εcfk

( )0, , kλ λ−

( ), ,0kλ λ−( ),0, kλ λ− cf

( )0 kλ λ

( )

( )0, , kλ λ−

Plane StrainPlane Strain2 2,σ ε

1 1,σ ε( ), ,0kλ λ−

1

( )

1

1 1,σ ε1

cfk

11 cfk −

cf

( ), ,0kλ λ−( )

Modified Coulomb MaterialModified Coulomb Material3σ

6

2

8

9

6

87

3

4

2σ

51

1σ

Modified Coulomb MaterialModified Coulomb Material3σ Surfaces: 7 8 9

62 Edges: 1/7, 3/7, 6/9, 2/9, 4/8, 5/8, 8/7, 7/9, 8/9

Surfaces: 7, 8, 9

897

6

34

51σ2σ

Apex: 4/5/8, 7/8/9/1, 1/3/7, 3/7/6/9, 2/6/9, 8/9/2/4, 7/8/9

51

1

+∑ kεε

+

−>∑

∑ ε∑

Dissipation WorksDissipation Worksε +∑When kεε −

=∑∑

∑∑

cW f ε −= ∑

When kε +

−>∑

∑ ε∑

( )+∑ ∑ ∑( )c tW f f kε ε ε− + −= + −∑ ∑ ∑

Planes and Lines of DiscontinuityPlanes and Lines of Discontinuity

I II

IIII II

Cracking ICrackingIIt II

δ

t

1 u

δThin layer

α ( )max1 sin 12

uε ε αδ

+= = +u

nutu

u

( )min1 sin 12

uε ε αδ

−= = −

n

( )min 2 δ

Dissipation work : Plane StrainDissipation work : Plane StrainI II

u

Along surfaces2 2,σ ε

1 1,σ ε( ), ,0kλ λ−I II

I IIϕ

( )1 1 sin2c

uW f ϕδ

= −1 1,σ ε

1cfk

11 cfk −

cf

I ϕ IIu

1 1,σ ε

( ), ,0kλ λ−( )1 1 sin

2l cW ub f ϕ= −

ϕ uAt apex ϕ α π α≤ ≤ −

I IIϕ

( ) ( ){ }1 1 sin 1 1 sintfW ub f k kα α⎡ ⎤

= − + − − + +⎢ ⎥( ) ( ){ }1 sin 1 1 sin2l c

c

W ub f k kf

α α= + + +⎢ ⎥⎣ ⎦

Dissipation work : Plane StressDissipation work : Plane StressI II

uAt apex at A ϕ α π α≤ ≤

( )1 1 sincfW ub α+

I ϕ IIu

I IIϕ

At apex at A ϕ α π α≤ ≤ −2 2,σ ε

( )0, ,kλ λ−

( )0kλ λ( )1 sin2

cl

fW ubk

α= +

1 1,σ ε

1cfk

( ), ,0kλ λ−

Along other apexes and surfaces ( ), ,0kλ λ−( ),0, kλ λ− cf

d≤ ≥

I IIϕ u( )0, , kλ λ−and α ϕ α π ϕ≤ ≥ −

1 I II( )1 1 sin

2l cW f ub α= −

Coulomb Material

Stress FieldStress Field

c

σ3σ 1σmσ

ϕϕ

στ

( ),σ τc

σ

ττσ

τ

90 ϕ−

Stress on Failure sectionStress on Failure sectiontt στ

τσ

τ( )21 2 tan 2 tannt cσ ϕ σ ϕ= + −

τσ

n nϕ ϕσ

nσ σ=tant cτ τ σ ϕ= = − tannt cτ τ σ ϕ

Stresses from Failure SectionStresses from Failure Section( ) ( )cot cot 1 sin sin 2c cσ ϕ σ ϕ ϕ θ ϕ= + − − +⎡ ⎤⎣ ⎦( ) ( )( ) ( )

cot cot 1 sin sin 2

cot cot 1 sin sin 2x m

y m

c c

c c

σ ϕ σ ϕ ϕ θ ϕ

σ ϕ σ ϕ ϕ θ ϕ

= + +⎡ ⎤⎣ ⎦= + − + +⎡ ⎤⎣ ⎦

( ) ( )cot sin cos 2xy m cτ σ ϕ ϕ θ ϕ= − − +

σx

θτxσ

τσ

xyτ

y

Failure section from a given stressesgFor given and x xyσ τ

tancot

xy

x cτ

βσ ϕ

= −−

i β

Angle θ is determined.

( ) sincos 2sin

βθ ϕ βϕ

+ − =

( )cotcot

1 sin sin 2x

mcc σ ϕσ ϕ

ϕ θ ϕ−

= +− +

Stress σ and τ are calculated.

( )ϕ ϕ

( ) 2cot coscσ σ ϕ ϕ= +( )cot cosm cσ σ ϕ ϕ= +

Equilibrium EquationsEquilibrium EquationsRankine Zone

( )tan0

c σ ϕσ ∂ −∂± σ σ∂ ∂( )

( ) ( )2

0

1 2 tan 2 tan tan0

x y

c c

ϕ

ϕ σ ϕ σ ϕ

± =∂ ∂

⎡ ⎤∂ + − ∂ −⎣ ⎦ ± =

0x yσ σ∂ ∂

= =∂ ∂( ) 0

y x± =

∂ ∂y

Prandtl Zoneσ∂

2 tan cotCe cθ ϕσ ϕ±= − +0

rσ∂

=∂σ∂

2 tantanr C e θ ϕθτ ϕ ±= ±

( )2 tan 0cσ σ ϕθ

∂± − =

∂

Rankine ZoneRankine Zoney

cσ σϕ

3σ c3σ1σ

σ c

σ

τ90 φ−

mσϕ

cτ

( ),σ τσ

45 / 2φ− ( )x

3σ

Strut and Rankine ZoneStrut and Rankine Zone

( )σ τσ

y

( ),σ τ3σ

y

σ σ3σ 1σ

τ45 / 2ϕ45 / 2ϕ−

x τ3σ

Prandtl ZonePrandtl Zone

σ3σ

1σ

( )σ τ3σ ( ),σ τ

3στ

3σ

3σ

Prandtle ZonePrandtle Zone( )σ τ( ),σ τ

σ3σ

1σ

τ

Strain Fields1

2ε1ε 3εkλ 0 λ Strain Fields1

2

kλkλ

00λ−

λ−

1 1⎛ ⎞

3

4

λ−kλkλ

00λ−

11 112 k

ε⎛ ⎞+⎜ ⎟⎝ ⎠

4

5

6

kλkλ

0

0

0 λ−

λ

λ

ε1ε6 kλ0 λ−

ε1ε12 k

εε = −ϕ

112

γ

Homogeneous Strain FieldHomogeneous Strain Fieldn / cos tan

cos cosh u uh

hϕ γ

ϕ ϕ× = =

tanu ϕϕ

cos coshϕ ϕ

uϕ

γh

ϕ

γ90 ϕ+h

t/ cosh ϕ

tan tanu ϕε γ ϕ= =t

11 112 k

ε⎛ ⎞+⎜ ⎟⎝ ⎠tann h

ε γ ϕ

uε1ε1

2 kεε = −

ϕ

0tε = ntuh

γ γ= = 12

γ

Triangular ZoneTriangular Zonen

uγ

h

γ

90 ϕ+ht

uu2u

2u

/ 2γ/ 2γ

90 ϕ+ hh

ApplicationsApplications

uu1uϕ

ββϕu

2uβ

ApplicationsApplications2u

u1u2β β ϕ

1u

2u2u 22

ApplicationsApplications

1u1u

2u2u 22

Design for concentrated loadingg gP

σ

2a

σ

b

1 1 bσ ⎛ ⎞= +⎜ ⎟h

12cf a

+⎜ ⎟⎝ ⎠

fh⎛ ⎞

fb⎛ ⎞

b 2 21.0 1 , 2.482

t

c c

fhK Kf a fσ ⎛ ⎞= + − ≅⎜ ⎟

⎝ ⎠

1 11.5 1 , 3.6t

c c

fbK Kf a fσ ⎛ ⎞= + − ≅⎜ ⎟

⎝ ⎠

Semi empirical formulaSemi-empirical formula2 2A e f•2 2oA e f= •

2 2A c d= •

1 1 tfAK⎛ ⎞⎜ ⎟

2 2A c d= •

2ec 4

0

1 1 tlocal

c

fc KA f

= + −⎜ ⎟⎜ ⎟⎝ ⎠

2e

2 fc

d d

01 1,2 2

a A b A= =

DisksDisks

• Constitutive Equations• Dissipation WorksDissipation Works• Yield Zones• Applications• Strength Reduction due to Initial Cracks• Strength Reduction due to Initial Cracks• Strut and Tie Models• Shear Walls

Disk ElementDisk Element2 2,σ ε2 2

1 1,σ εcfΦ

1 1,σ ε

cf

DissipationDissipationη

( ) ( )2 2

1 1 1u u u

W f ξ η ξ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥Φ Φ⎜ ⎟ ⎜ ⎟

δuξ uη

( ) ( )1 12 cW f ξ η ξ

δ δ δ⎢ ⎥= + Φ + − − Φ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

ξξ

Known SolutionsKnown Solutionsf

pcf

0y

0h y−

fΦ

0h y

cfΦ

L

212 1p cM th fΦ

=Φ ( )

2

2

41

ch fpL

Φ=

+ Φ2 1p cf+ Φ ( )1 L+ Φ

Discontinuous Stress FieldsDiscontinuous Stress Fields

pcf

cfΦ

p

cf cfΦcf cf

Strength Reduction due to Sliding in Initial Cracks

14 1

scsf ν

= ≤

α β

2 1sin cos 0.75sino cfν γ γ γ

= ≤−

δα β

δ

Strut and Tie ModelsStrut and Tie Models

• Strut• Strut and Tie SystemsStrut and Tie Systems• Fans- Non-concentric fans- Concentric fans- Concentric fans- Fans with Bond

Single StrutSingle Strutx h y−0x 0 0

0 0

tan x h yy a x

θ = =+

f

0x

0y

20 1 1

2x a ah h h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

hcf

2h h h⎢ ⎥⎝ ⎠⎣ ⎦

21 a aτ ⎡ ⎤⎛ ⎞⎢ ⎥θ 1 1

2c

a af h hτ ⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦a

20 01 4 1

2y yx a a

h h h h h

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= − + −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠2h h h h h⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

Single StrutSingle Strut

0s Y cA f y tf=

cf

0 12

s YA f ythf h

Φ = = ≤h

cf

2cthf h

θ0y

a0x

( )2

0 1 4 1xP a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= = = Φ − Φ + −⎜ ⎟( )4 12c cf thf h h h

⎢ ⎥= = = Φ Φ + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Fan shaped stress fieldsFan-shaped stress fields

• Non-concentric fan- Uniform Normal stress type

D

Uniform Normal stress type- Uniform Shear stress type

A

• Concentric FanB

• Concentric Fan

C

T

Boundary Stress onBoundary Stress on Non-concentric fan-shaped stress field

y Py

Sw(x)

tcσ ,top

h

x

Z(x) cσ ,botr

q

Uniform normal stress on boundary ofUniform normal stress on boundary of Non-concentric fan-shaped stress field

d =dx+(dw-dz)k+(w-z)dk

-dw

w+dw-[z+dz]

x+(w-x)k

k 1

1

z

k+dk

ddx

dx

-dz

Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (Uniformof Non concentric Fan shaped stress field (UniformNormal stress type)

y σconstτ =

H

P

x

LLdx-dz k+( -z)dk2

⋅

-dzk

k+dk1

1

k+dk

Uniform shear stress on boundary of Non-concentric fan-shaped stress field

yl/2 y

yσ

0

2σ =0xyτ

yσy

yxτ+y-σ t

+

x

T

1σ =0

pl 2-σ

x

+- tτ1σ =0

0

el

dx

-y-σ t

+y-σ t

+- tτtrT-q

2-σ

-- tτ

y tτ

2σ =0

1σ =0

or

xσ

y- tτtr

-pt

os

p

r

s yx- tτ1σ =0

2σ 0

T+dTx

l/2l/2

yy

ξξ

el

kk+dk

11

kdξ

θ

( ) k+dk1

-r

r

dr

Q(x,y)

b b(x ,y )

xx

-ss

-ds+y-σ t

+x

1σ =0

x

dξ

+y-σ t

+y- tτ

+y- tτ

dr

2-σ1σ 0

-qtx' 2-σw

+

dξ R

R'

θ

-ds

-ptx' 2

+-σ tR

Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (UniformShear stress type)

θe

p

m

q

r

qq

Concentric Fan-shaped stress field

dx

2,bottomσσ =

dx hdK+2,

1top dK

dx

ση

=+dx hdK+

2 tσdx

2,topσ

11K K dK+

11

dx

K dK+2,bottomσ

dxStress State in Diagonal Compression Stress FieldsStress State in Diagonal Compression Stress Fields

Single Strut actiong0x

cf0y

hh

′

N

θa

0y′

0Ca

PN τN

sτ

sσ

P 0T

Strut with Diagonal Compression FieldStrut with Diagonal Compression Fieldtτ

cσcott tσ τ θ=

f

N Ncf

θcf

PP P21 1

2s ca aP thfh h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠2s cf h h⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sP Pτ −=

( )0t t h y

τ =′−

Diagonal Compression FieldDiagonal Compression Field

cσ

cffσ =

θc cfσ =

2sin 2fτθ = tanYyrf τ θ=

cf21 1 a a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= + + Φ⎜ ⎟ ( )1τ

= Φ − Φ12 y

cf h h h⎢ ⎥= + − + Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )1y ycf

= Φ Φ

Maximum Shear StrengthMaximum Shear Strengthτ

cf

( )1y ycf

τ= Φ − Φ0.5

cf21 1

2 ya a a

f h h hτ ⎡ ⎤⎛ ⎞⎢ ⎥= + − + Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Φ0.5

2cf h h h⎢ ⎥⎝ ⎠⎣ ⎦

Strut with Fan-Shaped Stress Fieldp

Chap 5 BeamsChap. 5 Beams

• Beams in Bending• Beams in ShearBeams in Shear- Transverse reinforcement

Without transverse reinforcement- Without transverse reinforcement- Effective Concrete Compressive Strength

Arch Actions- Arch Actions- Design

- With Normal Forces

• Beams in Torsion

Beams in BendingBeams in Bendingbcf

dpM 0yh dp

sA s YA fd

A f y bf0s Y cA f y bf=

0y= Φ

2p cM bd f=

dΦ

211M bd f⎛ ⎞= Φ Φ⎜ ⎟12p cM bd f= − Φ Φ⎜ ⎟

⎝ ⎠

Failure Mechanism in BendingFailure Mechanism in Bending

0y 452ϕ

+

Beams in ShearBeams in ShearP P

P Py

CVV

MV

T xθcσ

cotV θT xθ

Lower Bound SolutionLower Bound Solutionfτ

cf( )1ψ ψτ

⎧ −⎪⎨ 1

2cf

= ⎨⎪⎩

0.5

ψ0.50.5

Shift in Tension Force in StringerShift in Tension Force in Stringer

/M h1 cot2

V θ

/M h T

Tmax max/M h T=

TT

Upper Bound SolutionUpper Bound Solution

2β θ= tan ψθ =

⎡ ⎤

2β θ= tan1

θψ

=−

21 12c

a a af h h hτ ψ

⎡ ⎤⎛ ⎞⎢ ⎥= + − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦c ⎣ ⎦

Upper Bound SolutionsUpper Bound Solutionsτ

cf

0.5

( )1cf

τ ψ ψ= −21 a a aτ ⎡ ⎤⎛ ⎞⎢ ⎥

ψ

cf

0 5

1 12c

a a af h h hτ ψ⎛ ⎞⎢ ⎥= + − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ψ

0.5

Shear Capacity without Shear Reinforcement

f 1

b

cfcf

012

y h=

hhT

aP

Influence of Longitudinal Reinforcement on Shear Capacity

sl YlA fΦ =

1 cot2Y

PaT Ph

θ= +

cbhfΦ =

2Y h

cotYwP rfbh

τ θ= =

22 a af h hτ ψ

ψ

⎡ ⎤Φ ⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦cf h hψ⎢ ⎥⎝ ⎠⎣ ⎦

Beams in TorsionBeams in Torsion

• Space truss models• Closed thin walled sectionsClosed thin walled sections• Extended application of beam models in shear

Torsional Strength ModelTorsional Strength Model

Tq2 o

qA

=

AcpA

0 0.85 ohA A=

Cracking Torsional LimitCracking Torsional Limitτ

102

TA t

σ τ= =τ

τ

02A t ττ3 2cpA

2στ02,

4 3cp

cpcp

t A Ap

= =

1σcpTp

σ τ1σ

σ

1 2cpA

σ τ= =

2σ

Cracking TorsionCracking Torsion

113 ckfσ = 압축-인장 파괴 강도: 균열 비틀림3

1 0.7 kfσ = 휨인장 파괴 강도: 균열 휨강도 산정1 0.7 ckfσ

1 f1 6 ckfσ = 압축-인장 파괴 강도: 전단 강도산정

2⎛ ⎞216

cpcr ck

AT f

p⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠6 cpp⎝ ⎠

비틀림 효과 무시비틀림 효과 무시1T T=4 crT T

2 2

1T V⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟전단 비틀림 상관 1

cr crT V+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠전단-비틀림 상관

0.97 crV V= cr

2Af ⎛ ⎞

12cpck

thcp

AfT

pφ

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠cpp⎝ ⎠

비틀림 강도비틀림 강도

Tq =2 o

qA

=

V d 1V qd=1 / sinD V θ=

1V qd=

V qd=

1 q

cott tN V θ=1V qd= t t

비틀림 강도비틀림 강도

1 2o oT V y V x= +

t yv oA f y2 cott yv of y

Vs

θ=s

2 o t yvA A fθcoto t yv

n

fT

sθ=

s

길이방향 철근길이방향 철근

( )2 cot2

no o

TN x y

Aθ= +( )

2 oA

l ylA f N=

T pcot

2n h

lo fl

T pA

A fθ=

o flf

전단-비틀림 조합전단 비틀림 조합

Direct sum

Root-square sum

압축장의 압축파괴 한계압축장의 압축파괴 한계비틀림으로 유발된 압축응력

2 hT pV

비틀림으로 유발된 압축응력

22cos sin 1.7 cos sin

u hcd

o oh

T pVf

ty Aθ θ θ θ= =

전단력으로 유발된 압축응력

cos sinu

cdV

fb d θ θ

=cos sinwb d θ θ

최소철근최소철근

12 w wt k

b s b sA A f+ = ≥2

16 3v t ckyv yv

A A ff f

+ ≥

5 yvck t ff AA A p

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟,min 12l cp hyl yl

A A pf s f

= − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

SlabsSlabs

• Statical Conditions• Geometrical ConditionsGeometrical Conditions• Constitutive Equations• Yield Zones• Upper Bound Solutions• Upper Bound Solutions• Lower Bound Solutions

Upper bound SolutionsUpper bound Solutions

• Assume yield lines • Work equationsWork equations• Minimize the upper bound solutions

Axes of rotationsLines of plastic hingesLines of plastic hingesFan mechanismsCorner leversC

Lower Bound SolutionsLower Bound Solutions2 22

2 22 xy yx m mm px x y y

∂ ∂∂− + = −

∂ ∂ ∂ ∂x x y y∂ ∂ ∂ ∂

2m a bx cx= + +xm a bx cx+ +

2m d ey fy= + +ym d ey fy= + +

hxym g hx my ixy= + + +

Punching ShearPunching Shear

• Upper Bound Solutions• Practical ApplicationsPractical Applications• Eccentric Loading• Effect if Counter-pressure• Edge and Corner Loads• Edge and Corner Loads

Upper Bound SolutionUpper Bound Solution

u

r( )

0,

h

cP f F r r dxπ ′= ∫α

( )r r x=( )

0∫

x 1 / 2d

Shape of failure surfaceShape of failure surfacerr

( )r r x= ( )0

,h

cP f F r r dxπ ′= ∫u α1 / 2d

( )r r x ( )0c ∫

x 1

Truncated cone with half-angle α0 dTruncated cone with half angle α00tan

2dr x α= +

( )( )cos sin sind h l mh α α α+ −( )( )0 0 02

0

cos sin sin2 cosc

d h l mhP fα α α

πα

+=

Use of Euler EquationUse of Euler Equation• Minimization of functional

( ),h

cP f F r r dxπ ′= ∫0∫

h i hx xb

• Complete solution

cosh sinhx xr a bc c

= +

r d⎧r

( )r r x=ϕ

0 0

tan2d x

rx h x h

ϕ⎧ +⎪⎪= ⎨ − −⎪u α/ 2d

( )r r x= 0 0cosh sinhx h x ha bc c

⎪ +⎪⎩

x 1 / 2d

Shear in JointsShear in Joints

• Monolithic Concrete• Strength of Different Types of JointsStrength of Different Types of Joints- Crack as a Joint- Construction Joint- Butt Joints- Butt Joints- Keyed Joints

Monolithic JointMonolithic Joint P

α

u

PcosEW Pu α=

sinR s YD A f u α=1 ( )1 1 sin2c cD f u α= −

Shear CapacityShear Capacityfτ

cf

0.5 ( )1τ ψ ψ=

tanc c

cf fτ ψ ϕ= +

( )1cf

ψ ψ= −

ψc cf f

ψsin1 2t t tf f fτ ϕψ ψ

⎡ ⎤⎛ ⎞ ⎛ ⎞= + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟

0.51 2

1 sinc c c cf f f fψ ψ

ϕ= + +⎢ ⎥⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎣ ⎦

Crack as a JointCrack as a Joint

uϕ

Average yield line

yield line in cement paste

Load carrying capacity by frictionLoad carrying capacity by friction( )tanp α μ+( )tan

1 tanp α μ

τμ α

+=

−pp

σσ

p p( )tan tan tanβ μ β β α+ − pβ

p( )( )tan

pβ μ β β

σβ α μ

=− −

Bond StrengthBond Strength

• Local Bond Strength• Failure Mechanisms in SectionsFailure Mechanisms in Sections• Effect of Transverse Reinforcement

Local Bond Strength ModelsLocal Bond Strength Modelsu i

1csu =cu sincu α=

cossu α=s

cu

P L S Bτ+ +

cos cos cosc c c c cf dlf dlf ndlf ndlfπ π α π α π α= = + +

Failure MechanismsFailure Mechanisms

Closing Remarks and Future TopicsClosing Remarks and Future Topics

• Compatibility based Upper bound Solutions• Ultimate Deformation Estimation Models• Application to Composite structures• How to handle Brittle Failures : Fracture mechanics• How to handle Brittle Failures : Fracture mechanics• Bond Strength applications