Download - RC concrete member behavior

Transcript
Page 1: RC concrete member behavior

MEMBER BEHAVIOR

Ductility, Plastic Hinge, Redistribution

Page 2: RC concrete member behavior

DUCTILITY

Ductility is an essential property of structures responding inelastically during severe earthquakesDefinition: Ductility defines the ability of a structure and selected structural components to deform beyond the elastic limits without excessive strength and stiffness degradation.

Page 3: RC concrete member behavior

GENERAL DUCTILITY

yΔΔ

=μ m

Page 4: RC concrete member behavior

Strain Ductility• Fundamental source of ductility.• Related with the ability of the constituent

materials to sustain plastic strains without significant reduction in stress.

yεε

=με

• ε: imposed total strain• εy: strain at which plastic action starts

Page 5: RC concrete member behavior

Strain Ductility

0≅με 20≅με

Page 6: RC concrete member behavior

Strain Ductility

20>>με

Page 7: RC concrete member behavior

Strain Ductility

• It should be noted that significant structural ductility can be achieved only if inelastic strains can be developed over a reasonable length of that member. If inelastic action is limited to a very small length (local ductility) very large strain ductility demands may arise.

Page 8: RC concrete member behavior

Local Ductility Demand

Δ

PP

Each has area A and length l

Py

ΔyΔ1 = Δu

Pu

Δ2 ~ Δ1

Page 9: RC concrete member behavior

Local Ductility Demand•Note that Δ1 = Δy ≈ Δ2 (a simplifying assumption)•When the applied load reaches Py, the total deflection at the onset of yielding is:

Δy = nΔ1 + Δy•When the load is increased to Pu, the corresponding deflection is:

Δu = nΔ2 + ΔuNow introducing Δ1 = Δy ≈ Δ2 into the equation we have:

Page 10: RC concrete member behavior

Local Ductility Demand

Dividing the nominator by Δy, and simplifying the resulting equation ; we have

( )Linksy

uy

Linksy1

u2

Chainy

uChainD )1n(

nnn

⎟⎟⎠

⎞⎜⎜⎝

Δ+

Δ+Δ=⎟

⎟⎠

⎞⎜⎜⎝

Δ+ΔΔ+Δ

=⎟⎟⎠

⎞⎜⎜⎝

ΔΔ

( ) ( ))1n(

n LinkDChainD +

μ+=μ or,

( ) ( ) n)1n( ChainDLinkD −μ×+=μ

Page 11: RC concrete member behavior

Local Ductility Demand

Now, if n=8 and (μD)Chain= 2 then required ductility demand on the ductile link will be:

( ) 1082)18(LinkD =−×+=μ

For the same assembly, (μD)Chain= 3 is obtained when;

( ) !1983)18(LinkD =−×+=μ

Page 12: RC concrete member behavior

Local Ductility DemandIn the construction of the chain, however, if we use “n”brittle links along with “m” ductile links, than:

( )Linksy

uy

Chainy

uChainD )mn(

mn⎟⎟⎠

⎞⎜⎜⎝

Δ+

Δ+Δ=⎟

⎟⎠

⎞⎜⎜⎝

ΔΔ

( ) ( ))mn(

mn LinkDChainD +

μ+=μ or,

( ) ( )m

n)mn( ChainDLinkD

−μ×+=μ

Page 13: RC concrete member behavior

Local Ductility DemandNow, getting back to the same example; take n=6 and m=3 (so that the total length of the chain does not change) for (μD)Chain= 3, the ductility demand on the ductile links will be:

( ) 73/)693(LinkD =×−×=μ

Remarks: The 1st rehearsal illustrates very large local ductilities are required to provide rather small increases in the overall systemductility.

The 2nd rehearsal shows the importance of the length over which theinelastic action develops. As the length over which inelastic action develops decreases, the required local ductility demand increases very rapidly, which further requires materials having very large strain ductility.

Page 14: RC concrete member behavior

Curvature Ductility

The most common and desirable sources of inelastic structural deformations are rotations in the critical regions. Therefore, it is useful to relate section rotations per unit length to causative bending moments.

This can easily be achieved by performing section analysis to obtain moment-curvature relationship of R/C sections.

Page 15: RC concrete member behavior

Curvature Ductility

In general for beams M’y=0.75My and corresponding curvature is

φy=1.33φ’y

For heavily reinforced beams and for column sections a sectionanalysis have to be performed.

y

m

φφ

=μφ

φ

M

Mmax

0.85MmaxShifting of neutral axisleads to moment increase

φyφ’y

M’y

My

φm φu

Idealized (Plastic hinge behavior)

Observed

Page 16: RC concrete member behavior

PLASTIC HINGEIdealized M-φ response is actually a typical “PLASTIC

HINGE” response. This definition implies that a plastic hinge is a region where concentrated rotations occur.

Page 17: RC concrete member behavior

PLASTIC HINGE

Page 18: RC concrete member behavior

Displacement DuctilityThe most convenient quantity to evaluate the ductility is

displacement. Hence, for the figure below we have:

yΔΔ

=μWhere Δ=Δy+Δp. Δy: the yield and Δp:fullyplastic components of the total lateral tip deflection.

Page 19: RC concrete member behavior

Displacement DuctilityFor frames the total deflection used is commonly that at the level of roof.

Of particular interest in design is the ductility associated with the maximum anticipated displacement Δ=Δm. Equally, if not more important are displacement ductility factors that relates interstory deflections to each other.

Page 20: RC concrete member behavior

Displacement Ductility

While displacement ductilities, in terms of the roof deflection Δ, of the two frames shown above are the same, dramatically different results are obtained when the displacements relevant to the first story are compared.

This figure illustrates that displacement ductility capacity of such frames, μΔ, will largely be governed by the ability of “plastic hinges” to be sufficiently ductile (as measured by individual member ductility).

Page 21: RC concrete member behavior

Relationship b/w μΔ and μφ

Given the curvature distributions at the yield and at the maximum response, the required ductility can be obtained by integration

φμφφ

φ

φμ K

KK

dxx

dxx

y

m

y

m

y

m ===ΔΔ

=∫∫

Δ2

1

)(

)(

Page 22: RC concrete member behavior

Relationship b/w μΔ and μφ• Yield Displacement:Actual curvature diagram is nolinear. However, by adopting a linear moment diagram as shown, φe, the yield displacement can be calculated as (by applying moment-area theorem):

3

2Lyy

φ=Δ

(b) Maximum Displacement:The curvature distribution at the maximum displacement corresponds to a maximum curvature at the base of the cantilever.

Page 23: RC concrete member behavior

Relationship b/w μΔ and μφ(b) Maximum Displacement (con’t)The extent over which plasticity spreads marks the length of the“plastic hinge” developing at the column end. For convenience in calculation, an equivalent plastic hinge length, lp, is defined and over this length a plastic curvature φp=φm-φe is assumed (to be equal to φm-φy)

lp is chosen such that the plastic displacement at the top of the cantilever, Δp, predicted by the simplified approach is the same as that derived from the actual curvature distribution.

Page 24: RC concrete member behavior

Relationship b/w μΔ and μφThus, the plastic rotations occurring within the plastic hinge is:

θp=φplp = (φm-φy) lp

Assuming that θp is concentrated at the mid-height of

the plastic hinge, we have:

Δp=θp (L-0.5lp)= (φm-φy)lp(L-0.5lp)

Page 25: RC concrete member behavior

Relationship b/w μΔ and μφ• Thus the ductility factor is:

( ) ( )−−+=

Δ

Δ+=

Δ

Δ+=

Δ

Δ+Δ=

ΔΔ

=

Δ

Δ

LL

y

ppym

y

p

y

p

y

py

y

ll

3

5.011

as written bemay which

1

2φφφ

μ

μ

Page 26: RC concrete member behavior

Relationship b/w μΔ and μφ• Therefore:

( )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+=

Δ

Δ

LL

LL

L

pp

pp

ll

ll

5.013

11

converselyor

5.0131

μμ

μμ

φ

φ

Page 27: RC concrete member behavior

Example 1• Consider that the cantilever has a plastic hinge length,

approximately equal to 1/10 of its total length, for a displacement ductility factor of 6 what is the required curvature ductility?

( ) ( ) !5.17

1015.01

1013

1615.03

11 =

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

−+==

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+= Δ

LL

Lpp ll

μμφ

As will usually be the case, the curvature ductility factor is much larger than the displacement ductility factor.

Page 28: RC concrete member behavior

PLASTIC HINGE LENGTH• Theoretical values for the equivalent plastic hinge

length lp, based on integration of the curvature distribution for typical members, would make lpproportional to L. Such values however do not match well with the experimentally measured lengths.

Reasons:1) Theoretical M-φ diagram ends abruptly at the base of the

cantilever, while steel tensile stains continue, due to anchorage slip, for some depth into the footing. The elongation of bars beyond the theoretical base leads to additional rotations and tip deflection. This phenomenon is referred to as “tensile strain penetration” or “bond slip’.

Page 29: RC concrete member behavior

PLASTIC HINGE LENGTHReasons:

2) The spread of plasticity resulting from

inclined “flexure-shear cracking”:

Remember that inclined cracks result

in steel stresses, therefore strains,

some distance above the base

(usually proportional to “d”) being

higher than predicted by the bending

moment at that level.

Page 30: RC concrete member behavior

PLASTIC HINGE LENGTH

Good estimates of the plastic hinge length are defined by:Paulay: lp = 0.08L + 0.022×db×fyk ≥ 0.044×db×fyk

Here db is the bar diameter and fyk is the characteristic yield strength of the longitudinal steel, measured in MPa.

Page 31: RC concrete member behavior

PLASTIC HINGE LENGTH

Mattock and Corley: lp = 0.5d + 0.05z

Here “d” is the effective depth of the section and z is the length of the shear span.

2yk

140fρ

zb0.020.003 ⎟⎟

⎞⎜⎜⎝

⎛ ′′++=cuε

Here “b” is the section width ρ’’ is the volumetric ratio of transverse and compression steel and fyk is steel yield strength in MPa. “z” being the shear span is the distance between the critical section and the nearest inflection point.

Page 32: RC concrete member behavior

Example 2Given: 400 mm x 400 mm square section cantilever columnS420 8φ20 longitudinal barsL = 3.0 mPaulay: lp = 0.08 x 3,000 + 0.0022 x 20 x 420

= 425 mmFor S220, we have lp = 406 mm

Page 33: RC concrete member behavior

Example 2Given: 400 mm x 400 mm square section cantilever columnS420 8φ20 longitudinal bars , d = 360 mmL = 3.0 mMattock lp = 0.5d + 0.02z

= 0.5 x 360 + 0.02 x 3000 = 240 mmExperimental results show scattered values of lpranging fom “d/2” to “h” for building columns.

Page 34: RC concrete member behavior

Example 3

• Let us retry Example 1 with revised value of the plastic hinge length.

L = 4 m, h = 0.8 m, 8φ28, fyk = 300 MPaDisplacement ductility factor = 6lp = 0.08 x 4,000 + 0.0022 x 28 x 300 = 505 mm

( ) ( ) 1.15

000,4505.05.01

000,4505.03

1615.013

11 =

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛−

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+= Δ

LLpp ll

μμφ