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Design of concrete structures based on ACI 318 –Lecture note No4

Presented by Dr. Nguyen Dai MinhHanoi, June 2010

Hawaii, Hilton Hotel

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Lecture note 4

Bond, anchorage, development length and serviceability

(analysis based on ACI 318)

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Contents of the lecture1. Fundamentals of flexural bond 2. Bond strength and development length 3. ACI provisions for development length and splices4. Serviceability – introduction5. Cracking in flexural members6. Control of deflections7. Immediate deflections8. Deflections due to long-term loads9. ACI code provisions for control of deflections10. Deflections due to shrinkage and temperature changes11. Moment vs. curvature for reinforced concrete sections 12. Summary

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1. Fundamentals of flexural bond

• If the loads are applied on the beam, the bars would slip longitudinally with respect to the adjacent concrete. The main assumption in analysis and design of RC structures is that the strain in the steel bars and the surrounding concrete should be the same. Therefore it is essential that the bond forces must be developed on the interface between concrete and rebars in order to prevent significant slip from occurring at that interface.

• In practice, it is important to avoid the beam fail by bond failure.

• Hooks or special anchorage devices can be used to increase the bond forces between concrete and rebars in case the development length is not enough.

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Fig 5.1: Bond stress due to flexure: (a) beam before loading, (b) unrestrained slip between concrete and steel, (c) bond forces acting on concrete, (d) bond forces acting on steel

Fig 5.2: Tied-arch action in abeam with little or no bond

anchorage

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a. Bond stress based on simple cracked section analysis

=>

=>

Where U is the unit bond force on the bar

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b. Actual distribution of flexural bond stressThe actual distribution of bond stress along deformed rebars is much more complex than the earlier mentioned equations.

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Fig 5.5: Effect of the bending moment changing and flexural cracks on bond stresses in beams: (a) beam with flexural cracks and M changing, (b) variation of tensile force T in steel along span, (c) variation of bond stress u along span

Notes: the actual bond is highest in the region of high shear force. However, we are interested in the total bond force (the integration or the area of the bond-diagram in Fig 5.5 c) in design for development length.

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2. Bond strength and development length

a. Bond strength

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b. Development length

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As*fs or (As*fy)

fb*π*d*lFig 5.7a: Bond and development length

Note: to prevent splitting using the concrete cover and stirrups or transverse rebars

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c. Factor influencing development length

Factors are concrete tensile strength, concrete cover, spacing of the rebars, the transverse reinforcement (stirrups, transverse bars). Epoxy-coated reinforced bars used in corrosive environmental conditions have a reduced bond with concrete due to the less friction between surrounding concrete and coated bars. The required development length must be increased substantially.

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3. ACI code provision for development and splicesof reinforcement

For detail, refer to Chapter 12 of ACI 318-02

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Shear links need to be added in the rebars cut-off zone

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That are all for bond and development length. These are important for detailing of RC structures and members. However, in every project, a detail specifications for development lengths, overlaps, splices including mechanical splices and typical detailing of the column-beam joints etc. are shown as guidance for engineers to follow.

Now, let’s go to the serviceability limit states analysis (for cracking and deflections).

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4. Serviceability – introduction

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5. Cracking in flexural members

(Note: 4000 psi = 27.58 MPa)

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a. Factors affecting width of cracks

- Bond between concrete and rebars (splitting etc.)

- Stress in reinforcement fs at the service load, can be calculated based on elastic cracked-section analysis. Alternatively, ACI code 10.6.4 says fs may be taken by 0.6fy.

- Concrete cover measured from center of the bar to the surface ofconcrete.

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b. The Gergely-Lutz equation for cracking width (w)

3076.0 Adfw csβ= (6.1)

β, = h2/h1

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d. ACI code provisions for crack control

• For simplifications, β = 1.2. Then, cracking width w is determined by

)2.6(091.0 3 Adfw cs=

•Normally in dry air or protective members: w <= 0.016 in (0.41 mm)

•Humidity, moist air, soil: w <= 0.012 in (0.30 mm)

•Water retaining structures etc,: w <= 0.004 in (0.10 mm)

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6. Control of deflections

The second it is to calculate deflections and to compare those predicted values with the limits specified by the codes or by special requirements.

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7. Immediate deflectionsElastic deflections can be expressed in the general form:

( )EI

portsspanloadsf sup,,=Δ

where f(loads, span, supports) is a function of the particular load, span and support arrangement, eg.: f = 5ql4/374 (for simply supported beams).

When the beam is not cracked, the uncracked section formulation can be used to determine the deflection iuΔ :

utc IE

f (...)=Δ (a)

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After the tension stress in concrete is greater than modulus of rupture fr, or bending moment M > Mcr where Mcr is determined as below:

t

utrcr

ut

tcrrc y

IfMorI

yMff ====σ (6.4)

where yt – the distance from the neutral axis to the tension face

When M > Mcr , the exact variation of I depends on the shape of M diagram. Exact calculation of I is difficult and impossible. Extensive research (Branson 1977) showed that deflection icΔ (subscript c – denotes to cracking) can be determined using the effective moment of inertia Ie:

ec

ic IElPqf ,...),,(

=Δ (b)

where utcra

crut

a

cre IandI

MMI

MMI ≤

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

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1 (6.5)

where Icr is the moment of inertia of the cracked transformed section.

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Due to the nonlinear behavior, the increment between Δ2-Δ1 must be calculated 2 times by reduction from deflection of M2 to that of M1.

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8. Deflections due to long-term loads

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Due to this complexities, in practice using more simplified approach:Δt = λ*Δi (6.6)

where Δt is the additional long-term deflection due to the combined effects of creeps and shrinkages, and Δi is the initial elastic deflection, λ is the coefficient depends on the duration of the sustained load.

The maximum total deflection of the beam under the dead load (W), live load (P) considering the long-term effects can be obtained as follows:1. Calculate the instantaneous deflection Δiw caused by sustained load W by method presented in Sec 6.5 (of the book) or in point 8 of this presentation for immediate deflection.2. Calculate the additional long-term deflection caused by W:

Δtw = λ*Δiw3. Then, the total deflection caused by the sustained part of the load is

Δw = Δiw + Δtw4. Because of nonlinear effect of the reinforced concrete (Fig 6.5), the additional deflection caused by short term live load P is:

Δip = Δi(w+p) - Δip

5. Finally, the total deflection under the sustained load + short-term load isΔ = Δw + Δip

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9. ACI provisions for control of deflections

Beams l/16 l/18.5 l/21 l/8

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• For design purposes, the moment of uncracked section Iut can be replaced by the gross concrete section Ig neglecting reinforcement without serious error.

t

grcr

g

tcrrc y

IfMor

IyMff ====σ (6.7)

and

gcra

crg

a

cre IandI

MMI

MMI ≤

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

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1 (6.8)

The modulus of rupture '5.7 cr ff =

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• Long-term deflection multiplier λ ACI code specifies that additional long-term deflection Δtw due to the

combined effects of creep and shrinkage shall be calculate by

Δt = λ*Δi and '501 μρ

ξμλ+

= (6.11)

Fig 6.8: Time variation of ξ for long-term deflection

bdAs

'' =ρwhere

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• Permissible deflections

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• Concrete shrinkage causes the compression in the rebars and tension in the concrete.

• Curvature due to shrinkage of concrete in an unsymmetrically RC member is shown in Fig 6.10:

10. Deflections due to shrinkage and temperature

Fig 6.10: Shrinkage curvature of a RC beam or slab: (a) cross section; (b) free shrinkage, (c) shrinkage curvature

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Fictitious compressive force: Tsh = (As + A’s) εsh Es

and EI

eTshsh =Φ ,

E = 0.5 Ec due to creep

then gc

gsh

IEeT2

=Φ (6.14)

Branson suggests that a) for steel percentage (ρ-ρ’) ?3%:

( )2/1'

3/1'7.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=Φ

ρρρρρε

hsh

sh (6.15a)

b) for steel percentage (ρ-ρ’) >3%:

hsh

shε

=Φ (6.15b)

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If steel percentages are constant along the span, the deflection due to shrinkage can be determined by 2lK shshsh Φ=Δ (6.16) • Deformations due to the differential temperaturesfrom top to bottom member are also based on equation (6.16) but the curvature Φsh is replaced by Φtemp. calculated as follows:

hT

tempΔ

=Φ α.

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11. Moment vs. curvature for reinforced concrete sections

(It is needed to investigate the inelastic behavior of the structure in seismic design such as q for EN and R for UBC)

unit curvature r1

=ψ

r – radius of curvature

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2cIfM utr

cr = (6.19)For M <

Uncracked beam

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Cracked beam but elastic

The corresponding moment is

2

21 dbjkfM elel = (6.21)

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Cracked beam but inelastic before failure

Process to determine the relationship between M and ψ is based on: -Strain compatibility ψ = ε1 / c1-Force equilibrium: C = T- M = Cz = Tz

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The procedure for inelastic cracked sectional beam is as follows:

1. Select any top face concrete strain ε1 in inelastic range (point 3 in Fig 6.16), between εel and εu (ultimate strain = 0.003) 2. Assume the neutral axis depth, a distance c1 below the top face 3. From the strain diagram, determine εs = εcs (strain compatibility) 4. Compute fs = εsEs but <= fy , and T = As fs 5. Determine C by intergrating numerically under the concrete stress distribution curve 6. Check to see if C = T. If not, the neutral axis must be adjusted upward or downward, for the particular concrete strain that was selected in Step 1, until equilibrium is satisfied. This determines the correct value of c1.

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Curvature, then, can be found from:

1

1

cinelε

=Φ (6.22)

The corresponding M is: Minel = C z = T z (6.23) The procedure can be repeated for newly ε1 . The end results will be a series of points such as 4, 5, 6, 7 etc. Finally, the M-ψ curve can be established. This can represent the yield point and failure point (at εu).

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Fig 6.16: Moment-curvature relation for tensile-reinforced beam

hinged

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This is the summary of the lecture:• Fundamentals of flexural bond • Bond strength and development length • ACI provisions for development length and splices• Serviceability – introduction• Cracking in flexural members• Control of deflections• Immediate deflections• Deflections due to long-term loads• ACI code provisions for control of deflections• Deflections due to shrinkage and temperature changes• Moment vs. curvature for reinforced concrete sections.The next lecture will be about the torsion and compression of concrete

members.

12. Summary

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• Questions and answers• Thank you very much !