11 continuous beams

Continuous Beams

Sagging Moment Region` Hogging Moment Region

Continuous Composite Beam System

Continuous Composite Beam Advantages greater load capacity due to redistribution

of moments greater stiffness and therefore reduce

deflection and vibration.

Disadvantages increase complexity in design susceptible to buckling in the negative

moment region over internal supports. Two forms of buckling may occur: (i) local

buckling of the web and/or bottom flange (ii) lateral torsional buckling.

3

Optimum Span/Depth Ratio of Composite Beam

Simply Supported Beam:

L/D = 18 to 22

Continuous Beam

L/D = 25 to 28

L = Span Length

D= Overall depth, including the concrete or

composite slab

4

Introduction

Continuous beams may be more economical than simply supported beams.

However, special phenomena may occur which must be taken into account in design, such as: – local buckling of compressed plate elements – lateral-torsional buckling – cracking of concrete due to tensile stresses

These all occur in the hogging moment regions. In the sagging moment regions, design checks are similar

to those of simply supported beams.

ft f

b f

hs

tension

compression

P.N.A.

Hogging Moment Resistance depends on the reinforcements within the effective width

Effective Width of the Concrete Flange (1)

The effective width of the concrete flange depends on the distance between the zero-moment points, approximated by Le.

L1 L2 L3

1 10.85eL L=

( )2 1 20.25eL L L= +

3 20.7eL L=4 32eL L=

be1 or be2 = Le/8 < actual width Le = length between points of zero moment

2e1eoeff bbbb ++=

7

Effective Width of the Concrete Flange (2)

L1 L2 L3

1 10.85eL L=

( )2 1 20.25eL L L= +

3 20.7eL L=4 32eL L=

At mid-span or an internal support: 2

01

eff eii

b b b=

= + ∑

/ 8ei e ib L b= ≤At an end support:

2

01

eff i eii

b b bβ=

= + ∑( )0.55 0.025 / 1.0i e eiL bβ = + ≤

Defining

b0: distance between the centers of the outstand shear connectors 8

Hogging Moment Resistance effective width For internal beam, the effective width of concrete slab in the negative (hogging) bending region is given by Le = 0.5L and corresponds to an effective width of L/8, where L is the clear span (rather than L/4 for the positive (sagging) moment in a simply supported beam). This means that the bar reinforcement is concentrated in a relatively narrow width over the internal supports.

beff = L/8

For hogging moment

beff = 0.7L/4

For sagging moment

L1 L2 L3

1 10.85eL L=

( )2 1 20.25eL L L= +

3 20.7eL L=4 32eL L=

if bo = 0, and L=L1= L2

Example : Effective width of edge beam

A continuous beam of uniform section consists of two spans and a cantilever, as shown in the figure below. Calculate the effective width for the mid-span regions AB and CD, for the support regions BC and DE.

A B C E

N

N’ 0.3 2.0

Section N-N’

6 m 8 m 2 m

D

0.15

Units: m

10

For segment AB,

0.85 6 5.1 meL = × =

1 / 8 0.638 m > 0.3 me eb L= =

0.3 2.0

Section N-N’

1 0.3 meb =

2 / 8 0.638 m < 2 me eb L= =

0.15 0.638 0.3 1.088 meffb = + + =

0.15

Example : Effective width of edge beam

11

2

01

eff eii

b b b=

= + ∑

Example 6.1

Perform similar calculation for the rest of the locations,

AB BC CD DE Le 5.1 3.5 5.6 4

Le/8 0.638 0.438 0.7 0.5 b0 0.15 0.15 0.15 0.15 be1 0.3 0.3 0.3 0.3 β1 - - - - be2 0.638 0.438 0.7 0.5

β2 - - - - beff 1.088 0.888 1.15 0.95

All units: m

12

Classification of Composite Cross-Section

EN 1993-1-1 clause 5.5.2: Classification should be according to the less favorable class of elements in compression. A steel component restrained by attaching it to a reinforced concrete element may be placed in a more favorable class. Simply supported composite beams (sagging moment) are almost always in Class 1 or 2, because:

1. the depth of web in compression (if any) is small 2. the connection to the concrete slab prevents local buckling of the steel

flange.

13

1) Composite sections without concrete encasement Compression outstand flange unrestrained from buckling follow EN 1993-1-1 Table 5.2 Web follow EN 1993-1-1 Table 5.2

c t

c

t

Classification of Cross-Section under hogging moment (1)

14

Cross-section classification (5)

Classification boundaries for webs in pure bending and uniform compression (EC3)

Class Pure bending Uniform compression 1 72ε 33ε 2 83ε 38ε 3 124ε 42ε

yf

2N/mm 235=ε

rolled

welded

d / t t trolled

welded

d / t

Class Web subject to bending Web subject to compression

Web subject to bending and compression

Stress distribution

(compression positive)

1 d/t ≤ 72 ε d/t ≤ 33 ε when α>0,5: d/t ≤ 396ε/(13α−1)

when α<0,5: d/t ≤ 36ε/α 2 d/t ≤ 83 ε d/t ≤ 38 ε when α>0,5: d/t ≤

456ε/(13α−1)

when α<0,5: d/t ≤ 41,5ε/α

Stress distribution

(compression positive)

3 d/t ≤ 124 ε d/t ≤ 42 ε when ψ>-1: d/t≤42ε/(0,67+0,33ψ)

when ψ ≤-1:

d/t≤ 62ε.(1−ψ).

Classification for web (negative bending)

+

+

2) Composite sections with concrete encasement

bc

b bc

b

0.8 1.0cbb

≤ ≤

Web encased in concrete can be assumed to be Class 1 or Class 2 provided: the concrete that encases the steel section should be reinforced, mechanically connected to the steel section, and capable of preventing buckling of the web and of any part of the compression flange towards the web (clause 5.5.3(2) of BS EN 1994-1-1).

Classification of encased beam under hogging moment (2)

17

Classification of outstand flanges in uniform compression

Class Type Web uncased Web encased (EC3) (EC4)

1 Rolled 9ε 10ε Welded 9ε 9ε 2 Rolled 10ε 15ε Welded 10ε 14ε 3 Rolled 14ε 21ε Welded 14ε 20ε

yf

2N/mm 235=ε

c t

c

t

rolled welded

c / t


Cross-section classification

Class of the section is defined as class of the element with the less favourable behaviour (e.g.: class 1 web and class 2 flange = class 2 section)

beff

hchp

tw

d 20t εw

20t εw

Exception: if compression flange is at least class 2 and web is class 3, then the section can be considered class 2: – with the same cross-section, if the

web is encased – with an effective web, if the web is

not encased

2) The minimum area of reinforcement for Class 1 and 2 sections:

Additional requirements on Class 1 and 2 sections if the reinforcement is in tension:

1) Ductility requirement on steel reinforcement ductility class B and C steels

s s cA Aρ≥235

y ctms c

sk

f f kf

ρ δ=

Ac= effective area of the concrete flange fy = nominal yield strength of the structural steel, MPa fctm = mean tensile strength of the concrete (EN 1992-1-1, Table 3.1 for normal

weight concrete or 11.3.1 for lightweight concrete, see next slide) fsk = characteristic yield strength of the reinforcement kc = coefficient accounting for the stress distribution prior to concrete cracking δ: = 1 for Class 2 and 1.1 for Class 1 (more reinforcing steels for Class 1)


20

Mechanical properties of concrete

fck (MPa) 25 30 35 40 45 50 55 60

fctm (MPa) 2.6 2.9 3.2 3.5 3.8 4.1 5.2 4.4

Ecm (GPa) 31 33 34 35 36 37 38 39

Extract from Table 3.1 in EN 1992-1-1 (for normal weight concrete)

For light weight concrete, the fctm and Ecm values for a given grade are reduced by the following coefficient:

( )0.4 0.6 / 2200lctm ctmf f ρ= +

ρ refers to the density of lightweight concrete in kg/m3

( )2/ 2200lctm ctmE E ρ=

21

Classification of Cross-Section (8)

( )0

1 0.31 / 2c

c

kh z

= ++

hc

N.A. z0

hc: thickness of the concrete flange, excluding ribs and haunches.

z0: distance between the centroids of the uncracked concrete flange and the uncracked composite section, calculated using n0 for short term loading

0 /a cmn E E=

Ea: modulus of elasticity for structural steel 22

Plastic cross-section resistance

Basic assumptions: – full connection between steel and concrete – steel & reinforcement are full yielded – resistance of concrete in tension is zero

Moment Resistance beff

Maximum Tensile force in reinforcement

sskss fAF γ= /

/a a y aF A f γ= γs = 1.15 γa = 1.0

Rw = dtfy

d

t

Fs

Fa

Maximum Compressive force on steel section

24

Moment Resistance at Hogging Moment region

Two main cases for which formulae are developed – case 1: plastic neutral axis is in the flange of the

steel section – case 2: plastic neutral axis is in the web of the

steel section

25

Plastic cross-section resistance Case 1: Fa > Fs ≥ Rw plastic neutral axis is in the steel flange

sskss fAF γ= /

/a a y aF A f γ=

and

Mpl,Rd = 0.5Faha + Fshs - (Fa - Fs)2/(4bffy)

ft f

b f

hs

tension

compression

P.N.A. hs

PNA from top of steel beam

Rw = Fa – 2bftf fy

Fa

Moment about top of the steel flange

2a s

fy f

F Fzf b−

=

(Fa-Fs)0.5Zf

hs = position of rebars from the beam flange

Simplified Moment Mpl,Rd ≈ 0.5Faha + Fshs

Plastic cross-section resistance Case 2: Fa > Fs < Rw plastic neutral axis is in the web of the steel section

fsk / γs

Fs

P.N.A.

f / γy af / γy a

beff

tension

hchp

ha

Fa

Fatw

zw

h /2a

. . (0.5 ) 0.5pl Rd apl Rd s a s s wM M F h h F z− = + + −

2s

ww y

Fzt f

=

sskss /fAF γ=

and

PNA from centriod of steel beam

/a a y aF A f γ=

Moment about the center of the steel beam

2. . (0.5 ) 0.25 / ( )pl Rd apl Rd s a s s w yM M F h h F t f− = + + −

Analysis of Continuous Beam

In BS EN 1994-1-1, the design of continuous composite beams may be based on two approaches to determine the design bending moments in negative (hogging) and positive (sagging) bending: • Clause 5.4.4 states that linear elastic analysis may be

used for composite beams with all section classifications using maximum permitted moment redistributions

• Clause 5.4.5 states that rigid plastic analysis may be used for Class 1 sections

28

Methods of Analysis

•Simplified table of moment coefficients (BS5950:Part 1: 3-1) •Elastic analysis – uncracked section •Elastic analysis – cracked section •Plastic hinge analysis (class 1 section only)

29

Simplified table of moment coefficients (BS5950 Part 1: 3-1 Table 2.)

* *

* *

* *

This method accounts for pattern loading, cracking of concrete and yielding of steel. 30

Examples (BS5950 Part 1: 3-1 Table 2)

0.79wl2/8

0.50wl2/8 2 spans Non reinforced plastic section

0.52wl2/8

0.67wl2/8

0.80wl2/8

3 spans Reinforced compact section

31

Some restrictions are placed on this method: uniform section with equal flanges and without

haunches. Steel beam should be the same in each span. The loading should be uniformly distributed. Unfactored imposed load should not exceed 2.5

times the unfactored dead load. No span should be less than 75% of the longest. End span should not exceed 115% of the length of

the adjacent span. There should not be any cantilevers.

Restrictions on the use of Simplified moment coefficients

32

Elastic Analysis of Continuous Beam

The elastic analysis method depends on whether the composite cross-section is considered to be uncracked, or cracked in negative bending. For uncracked analysis, the stiffness of the beam is treated as being constant along its length. For cracked analysis, the stiffness of the beam is reduced in the negative (hogging) moment region and hence lower percentage redistributions of moment are permitted in comparison to the uncracked case.

33

Cracked and un-cracked analysis

(b) Crack section

EIg

EIn EIg EIg

Uncracked analysis: use EIg through out Cracked analysis: Use EIn near the support and EIg outside the 15% length of the support

34

Analysis of Continuous Composite Beam Distribution of bending moment

35

Elastic Analysis

Uncracked Section (positive moment) n is the ratio of the elastic moduli of steel to concrete taking

into account the creep of the concrete (may assume n = 13) r is the ratio of the cross-sectional area of the steel section

relative to the concrete section , Aa/(beff hc). Iay is the second moment of area of the steel section

ay

3ceff

2apca

g In12hb

)nr1(4)hh2h(A

I +++

++=

36

Ar = area of reinforcement hr = distance of reinforcement from the top of concrete slab

Cracked Section, Negative Moment

2( 2( ))4( )

a r a p c rn ay

a r

A A h h h hI I

A A+ + −

= ++

37

hr

ha

hc hp

Elastic analysis – moment redistributions

Analysis Method for Composite

Section

Section Class to SS EN 1993-1-1 1 2 3 4

Un-cracked Section 40% 30% 20% 10%

Cracked Section 25% 15% 10% 0%

Maximum moment redistributions for elastic global analysis of continuous composite beams (per cent of the initial value of the bending moment to be reduced)

Resistance against combined bending and shear

Interaction diagram

V Sd

C B

A

V pl.Rd

V pl.Rd0,5

Mf.Rd

_M

Rd

_M

V.Rd

_

Low shear – moment capacity not reduced

Low bending – shear capacity not reduced

High bending and shear – interaction formula

−−⋅−+= −−−−

2

.... 121)(

Rdpl

SdRdfRdRdfRdv V

VMMMM

Moment capacity of flanges only

Problems for continuous beam design

High shear

Lateral-torsional buckling at hogging moment region Shear connection design

Loss of serviceability due to concrete cracking

Lateral-torsional buckling

The theory of lateral-torsional buckling of continuous beams over supports is rather complex.

In reality, lateral-torsional buckling is affected by: – beam distortion / lateral deflection of compressed flange – torsional rigidity of section

In design, two types of simplified approach may be

followed: – simplified calculation of lateral-torsional buckling resistance

according to analogy to steel beams (EC3 approach) – application of certain detailing rules that prevent lateral-torsional

buckling

Lateral-torsional buckling EC3 approach

−− χ= RdLTRdb MM . −

−

=λcr

plLT M

MEC3 LT buckling

curves

In this approach, the elastic critical moment is determined using the so-called “inverted U-frame model”. The use of this model is subject to certain conditions. This model is not discussed here in detail

No lateral-torsional buckling if the lateral-torsional buckling slenderness ratio < 0.4,

0.4LTλ ≤

Lateral Torsional Buckling at Hogging Moment Region

4.0LT <λ

≤ 0,4.

No reduction in capacity due to lateral torsional buckling

43

Prevention of lateral-torsional buckling by bracing

Member Steel grade S235

Steel grade S275

Steel grade S355

Steel grade S420 or S460

IPE/UB 600 550 400 270

HE/UC 800 700 650 500

Maximum depth h (mm) of uncased steel member to avoid lateral-torsional buckling checks (EC 4 Table 6.1 )

Lateral torsional buckling can also be prevented by limiting the depth of the steel

member at the hogging moment region

Shear connection design

Basic rules • Connectors should be ductile • Plastic design of shear connection is possible even if global analysis

is elastic, provided that the end cross-sections of the critical length to be designed are at least Class 2

• In hogging moment regions, use of full shear connection is recommended

• In sagging moment regions, partial shear connection may be applied

Shear Connections in Negative Moment Regions

PRd = design capacity of a shear connector in negative moment regions considering concrete cracking

Total no. of shear connectors = Np + Nn needed between the point of maximum moment and each adjacent support

Nn = Fs/PRd

= tension resistance of reinforcement

s sks

s

A fFγ

=

46

0.851 min ; 1.5 1.15

eff c ck s skp n a y

Rd

b h f A fN N A fP

+ = ⋅ +

0.851 min ; 1.5

eff c ckp a y

Rd

b h fN A f

P

= ⋅

11.15

s skn

Rd

A fNP

= ⋅

Fa Fc

Fs

47

Deflection

Deflection is affected by Pattern loading Cracking of concrete Yielding of rebars

But yielding of rebars and cracking of concrete have less influence on deflections in services than they do on analyses for ultimate limit states. Simplified method is developed for uniform beam in which deflection is estimated based on uniformly distributed load that the hogging end moments M1 and M2 reduce the md-span moment deflection from δo to δc 48

Serviceability deflection

δc = δo{1-0.6(M1+M2)/Mo} δo = Deflection of a simply supported beam Mo = maximum sagging moment in the beam when it is simply supported

Deflection of a continuous beam (simplified)

M1 and M2 are moments after redistribution for pattern loads, etc.

49

Influence of Pattern loading

To account for pattern loading as shown in the figure, reduce the uncracked moments at the internal supports by 40%

M1 and M2 are moments after redistribution for pattern loads, etc.

50

Serviceability – Cracking of concrete

In continuous beams, concrete cracking is mainly due to

tensile stresses in the hogging moment regions

This cracking is prevented by limiting bar spacing or bar diameters in the reinforcement

Serviceability – Cracking of concrete

Limiting bar spacing (for high bond bars only) to avoid cracking over supports

stress in reinforcementσs, N/mm2

maximum bar spacingfor wk = 0,4 mm



160200240280320360

300300250250150100

30025020015010050

20015010050––

this stress is calculated considering tension stiffening

sss σ∆+σ=σ 0sst

ctms

fρα

=σ∆4.0

…with...

unless using a more precise method:

sss σ∆+σ=σ 0

σso = tensile stress in the reinforcement

0.4 ctms

st s

fσα ρ

∆ =

fctm is the mean tensile strength of concrete; ρs is the “reinforcement ratio” expressed as αst = As / Act Act is the area of concrete flange in tension within the effective width As is the total area of reinforcement within the area Act αst is the ratio

aa IAAI

where A and I are the area and second moment of area of the composite section neglecting concrete in tension and any sheeting, and Aa and Ia are the same properties for the bare steel section.

Conclusions

Continuous beams offer advantages over simply supported beams, but special phenomena need particular attention during design in the hogging moment regions

In the case of both elastic and plastic design, cross-section classification and resistance calculation are key issues

Lateral-torsional buckling at the hogging moment regions must be prevented by appropriate detailing or by direct check

In shear connection design, hogging moment regions require full shear connection

In the hogging moment regions, the serviceability limit state of cracking of concrete may be relevant

Example

Homework 7: Continuous Composite Beams

11 continuous beams

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