REINFORCED CONCRETE sem V P-3b Flexure - ULS (2).pdf

WARSAW UNIVERSITY OF TECHNOLOGY

CONCRETE STRUCTURES

PACKAGE 3b

Ultimate Limit State (ULS) BENDING

2

INTRODUCTION

Bending = Flexure Bending. According to Eurocode 2 - Design of concrete structures – Part 1-1: General rules and rules for buildings – our basic source of principles and rules. Chapter 6.1. „Bending with or without axial force” Flexure. According to American Concrete Institute (ACI) Building Code Requirements. „Flexural members, flexural reinforcement – the reinforcement of flexural members (elements)”, „Strength design of members for flexure (American)”, „Flexure and axial loads (ACI) „. In British Standard (BS) we can find: „Design resistance moment of beams, the ultimate moment of resistance” and, if the axial loads are substantial „Design of column section for ULS”.

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INTRODUCTION GENERAL PRINCIPLES AND REQUIREMENTS

The concrete structures and the structural concrete elements (plain, reinforced (RC), prestressed) should meet the requirements concerning:

• the mechanical resistance and stability • serviceability • durability • fire resistance

Other requirements, such as thermal or sound insulation, are not considered in these lectures.

4

p

MEd = 0,125 pl2

MEd - design bending moment depends on type of the structure and loading.

MRd - design ultimate bending moment depends on the cross-section properties: shape, strength (and other properties) of concrete, yield strength (and other properties) of steel

INTRODUCTION GENERAL PRINCIPLES AND REQUIREMENTS

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BENDING. BASIC DESIGN ASSUMPTIONS

The assumptions made when determining the ultimate moment resistance. They apply to RC and prestressed sections, with or without axial force (only RC and pure uniaxial bending). They concern the critical, cracked cross section.

Two fundamental conditions have to be satisfied:

• Static equilibrium (resulting from the basic laws of nature)

• Compatibility of strains (resulting from many experiments)

BERNOULLI’S HYPOTHESIS

.

6

Before loading

After loading

This assumption applies to regions of beams, slabs and similar types of members for which sections plane before loading remain approximately plane after loading. Usually this theory is applied to the whole length of all beams and slabs.

Plane sections remain plane

7

STRAINS IN STEEL

The strains in bonded reinforcement: - for compression - are the same as that in the surrounding concrete

- for tension - are derived with the use of strain distribution diagram

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NEGLECTING OF CONCRETE TENSILE STRENGTH

The tensile strength of the concrete is negligable

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When using the partial factors method for ULS verification we apply: • the design values of actions (forces/loads applied to the structure and imposed deformations e.g. caused by temperature, uneven settlement) greater than mean values • the design values of the material properties smaller than mean values

PARTIAL FACTORS METHOD

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The stresses in the concrete in compression are derived from the design curves called “stress-strain relations for the design of cross-section” (Eurocode 2).

STRESS-STRAIN RELATIONS FOR CONCRETE UNDER COMPRESSION

AND ULTIMATE STRAINS

The shape of the strain-stress relation depends on the strength of concrete. This relation may be shown for the characteristic values of compressive strength fck or for the design values of this strength fcd . For ULS analysis the relations based on design values are used.

fcd

εc2

0,0035

c

c

0,0020

εcu2

for fck ≤ 50 MPa

11

fc – compressive strength of concrete (in general meaning or the actual value obtained in the particular case of the element or specimen just considered) fcm – mean value of concrete cylinder compressive strength fck – characteristic compressive strength of concrete – the strength value having a prescribed probability of not being attained in a hypothetical unlimited test series – generally corresponds to 95% fractile of the statistical distribution of the compressive strength – approximately about 80% fcm – is used as the basic characteristics of concrete fcd – design value of concrete compressive strength - the value used in ULS analysis γC - partial factor for concrete, the recommended values in persistent and transient situations: γC = 1,5 acc. Eurocode 2, γC = 1,4 acc. to Polish National Annex. Example. For concrete with average cylinder strength 30 MPa,

fck ≈ 25 MPa and fcd ≈ 17 MPa.

COMPRESSIVE STRENGTH OF CONCRETE SYMBOLS

12

For fck ≤ 50 MPa

if 0 < εc < 2‰, then

if 2‰ ≤ εc ≤ 3,5‰, then σc = fcd

250011 ccdc f

n

c

ccdc f

2

11

fcd

εc2

0,0035

c

c

0,0020

εcu2

Parabola-rectangle diagram for concrete under compression for fck ≤ 50 MPa


AND ULTIMATE STRAINS

if 0 < εc < εc2, then

if εc2 ≤ εc ≤ εcu2, then σc = fcd

13

Stress-strain relations (characteristic values) for fck >50 MPa in comparison with the relation for fck = 30 MPa

fck= 60

fck= 90

σc [MPa]

1,0 2,0 3,0 3,5‰

fck= 30 MPa

fck= 50

fck= 55

fck= 80

fck= 70

εc


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The shape of the stress-strain diagram depends on three parameters:

εc2 – the strain at reaching the maximum stress

εcu2 – the ultimate strain

n – the exponent in the formula describing the curved part of the stress-strain graph

The depth of the rectangular part of the stress block in concrete x1 equals χx, where

Values of εc2 εcu2 and χ are given in the table below:

2

21cu

c

4286,07

3

fck

[MPa]

≤ 50 55 60 70 80 90

εc2 [‰] 2,0 2,2 2,3 2,4 2,5 2,6

εcu2 [‰] 3,5 3,1 2,9 2,7 2,6 2,6

χ 0,2908 0,2069 0,1111 0,0385 0

n 2 1,75 1,6 1,45 1,4 1,4

Deformation characteristics for concrete


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There are other diagrams allowed according to Eurocode, fo example bi-linear stress-strain relation:

c

fcd

σc


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The stress in concrete is assumed as equal ηfcd. The effective height of the compression zone λx (concerns stresses) is taken smaller than the depth of the compression zone determined by strains. The factor λ and the factor η, defining the effective strength, come from:

Fc

z = d – 0,5λx

ηfcd εcu3 = εcu2

b

d

x

As1

λx

Fs

Fc – compression stress resultant

Fs – tension stress resultant

MPa 50for 8,0 ckf

MPa 9050for 200

508,0

ck

ck ff

MPa 50for 0,1 ckf

MPa 9050for 200

500,1

ck

ck ff

RECTANGULAR STRESS DISTRIBUTION

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STRESS-STRAIN RELATIONS FOR STEEL

The stresses in the reinforcing steel are derived from the design curves according to figure below (in tension and in compression).

s

fyd

σs

sss

s

yd

s EE

f then , If

yds

s

yd

s fE

f then , If

The quotation from Eurocode: “For normal design, either of the following assumptions may be made: an inclined top branch with a strain limit (see the dotted line shown on the figure above) a horizontal top branch without the need to check the strain limit.”

The second assumption, a horizontal top branch without the strain limit, will be used in this class. The analysis with and without strain limit gives very similar results (in the wide range of reinforcement ratio values these results are just equal). For analysis based on the first assumption, a top branch with a strain limit, the strain limit is 10‰ in Polish Code PN-B-03264.

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FAILURE TYPES IN THE ULTIMATE LIMIT STATE

If the ultimate moment resistance is reached, then in the failing section:

– the greatest strain in concrete equals εcu2,

– the stresses in reinforcement are equal fyd - the failure is initiated by steel – it occurs in normally reinforced (correctly reinforced) sections,

or

– the stresses in reinforcement are smaller than fyd (the strain is smaller than εsy = fyd /Es) - the failure is initiated by concrete – it occurs in strongly reinforced (over-reinforced) cross sections.

correctly reinforced section

x xlim

εcu2

a2

As1 balanced condition

d

As2

a1 over-reinforced section εsy

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• The position of neutral axis in ULS depends on the amount of the tension reinforcement. The depth of neutral axis in balanced conditions is called xlim. For under-reinforced cross-sections x < xlim, for over-reinforced sections x ≥ xlim

• In order to use the yield strength fyd the heights of the compression zone should not be greater than xlim.

• In most designs the over-reinforced sections are not recommended because in such sections the yield strength of reinforcement is not reached. The sections should be correctly reinforced both from the economic and structural point of view.

• In case of x > xlim it is economical to arrange a compression reinforcement

• The concrete cross-section of beams should be chosen so large, that no compression reinforcement is necessary, unless exceptionally (e.g. in local weaker parts, openings) this reinforcement cannot be avoided.

FAILURE TYPES IN THE ULTIMATE LIMIT STATE

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correctly reinforced section

lim

2

lim xxdE

fcu

s

yd

ydcus

cus

fE

dEx

2

2lim

For x = xlim

Limit of the relative (non- dimensional) neutral axis depth

The values of ξlim depend on the strength of concrete and on the yield strength of reinforcement. The values are given in the table on the next page.

ydcus

cus

fE

E

d

x

2

2limlim

εcu2

x xlim

a2

As1 balanced condition

d

As2

a1 over-reinforced section εsy

εsy = fyd/Es

FAILURE TYPES AND THE LIMIT OF THE COMPRESSION ZONE DEPTH ξlim

21

fyd [MPa] 190 210 310 350 420 435

fck ≤ 50 MPa 0,7865 0,7692 0,6931 0,6667 0,6250 0,6167

fck = 55 MPa 0,6667 0.6392 0,5962 0,5877

fck = 60 MPa 0,6517 0,6237 0,5800 0,5714

fck = 70 MPa 0,6353 0,6067 0,5625 0,5538

fck = 80 MPa 0,6265 0,5977 0,5532 0,5445

fck = 90 MPa 0,6265 0,5977 0,5532 0,5445

COMPRESSION ZONE DEPTH ξlim

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EQUILIBRIUM AND FUNDAMENTAL RELATIONS IN ULS THE ARBITRARY CROSS-SECTION

We consider an arbitrary cross-section. We assume, that reinforcement is concentrated in two groups of rebars (rebars – reinforcement bars) As1and As2.

a1

εs1

a2

d b

As1

As2 fcd

x MRd

= z

As2 σs2

As1 σs1

Fc

εcu2

εs2

Symbols (Notation):

b - width of a cross-section d – effective depth of a cross-section x – depth to the neutral axis z – lever arm (of the inner forces), the distance between the force Fc and the reinforcement As1

As1 – cross sectional area of tension reinforcement

σs1, εs1 – stress, strain in As1 (positive for tension) As2 – cross sectional area of reinforcement in compression zone σs2, εs2 – stress, strain in As2 (positive for compression) Fc – compression stress (concrete only) resultant MRd – design ultimate bending moment

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a1

εs1

a2

d b

As1

As2 fcd

x MRd

= z

As2 σs2

As1 σs1

Fc

εcu2

εs2

From conditions of equilibrium and compatibility of strains we obtain the basic relations denoted by (F) - equilibrium of the forces, (M) - equilibrium of the bending moments, and (k) - efficiency of the reinforcement. Always (i.e. for an arbitrary stress distribution) the formula (F), (M) and (k) can be presented as:

The coefficients k1 and k2 are given by the relations:

Usually σs1 = fyd , k1 = 1,0.

where

ydsydsc fkAfkAF 1122

Rdydsc MadfkAM 222

zFM cc

yd

s

fk 1

1

yd

s

fk 2

2

(F)

(k)

(M)


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The stresses in As2 depend on the relation a2/x. Very often for small values of a2/x, σs2 = fyd , k2 = 1,0.

x

xdcus

21

x

axcus

222

if stresses in the reinforcement are in the elastic range, from Hooke’s law follows:

x

xd

f

E

fk

yd

cus

yd

s 21

1

x

ax

f

E

fk

yd

cus

yd

s 2222

(k)

Equations (F), (M) and (k) are the basic expressions for bending resistance of RC members. On the basis of these relations for given material properties and given reinforcement the ultimate bending moment of arbitrary cross-sections may be determined (using computer).

The simple theory concerning rectangular and T-shaped cross-sections is sufficient for most designs. This theory is presented in the next part of the lecture.

a2

εs1

x

εcu2

εs2

d

The strains in As1 and As2:


25

a2

a1 b

d

As1

As2 σs2

As1 σs1

x

fcd εcu2

z

Fc As2

Acc

y

Strains and stresses induced by parabola-rectangle stress diagram

From the principle “plain sections remain plain” appears that the diagram of strains is linear.

RECTANGULAR CROSS-SECTION PARABOLA–RECTANGLE STRESS DIAGRAM

26

Equilibrium of forces: ydsyds

x

c fAkfAkdyb 1122

0

Rdyds

x

c MadfAkdyyb 222

0

In the two preceding equations „y” denotes the coordinate, and:

c

x

c Fdyb 0

c

y

c Mdyyb 0

yd

s

fk 1

1

yd

s

fk 2

2

a2

a1 b

d

As1

As2 σs2

As1 σs1

x

fcd εcu2

z

Fc As2

Acc

y

RECTANGULAR CROSS-SECTION PARABOLA–RECTANGLE STRESS DIAGRAM AND EQUILIBRIUM CONDITIONS

Equilibrium of bending moments:

27

The concrete compressive force Pc = 0,8095bxfcd is situated at the distance 0,4159x from the upper edge of the cross-section (see the next slide).

The equilibrium conditions:

ydscd fAfxb 18095,0

Rdccd MMxdfxb 4159,08095,0

ydscd fAfxb 18,0

Rdccd MMxdfxb 4,08,0

(F)

(M)

THE RECTANGULAR CROSS-SECTION

εcu2

x

Bending. ULS. The strain and stress diagrams. a) the parabola-rectangle stress diagram, b) the rectangular stress distribution

(concrete characteristic strength fck ≤ 50 MPa, ξ ≤ ξlim )

z

As1 σs1

fcd Pc

a1 b

d

As1

0,8 x

fcd

As1 σs1

a) b)

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σc = fcd

2‰

3,5‰ xx 4286,0

7

3

x7

4

cdcdcdc fbxfbxfbxP 8095,07

4

3

2

7

3

The centroid of the red figure is situated at the distance 0,4159 x from the upper edge:

Justification of the formula from the preceding slide:

Area of the rectangular part of the red figure

The parabolic part

THE RECTANGULAR CROSS-SECTION PARABOLA–RECTANGLE STRESS DIAGRAM

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Substituting x in the Eq. (M) by the value taken from the Eq. (F) we obtain:

cd

yds

ydsRdfdb

fAdfAM

1

1 5138,0

cd

yds

ydsRdfdb

fAdfAM

1

1 5,0

a)

cd

yds

fb

fAx

8095,0

1

b) cd

yds

fb

fAx

8,0

1

THE RECTANGULAR CROSS-SECTION

Thank you

REINFORCED CONCRETE sem V P-3b Flexure - ULS (2).pdf

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