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3.1 3.1 Frame idealisation, Frame idealisation, classification and classification and

analysisanalysis

Frame modelling for Frame modelling for analysisanalysis

•• Frame componentsFrame components–– BeamsBeams–– BeamBeam--columnscolumns–– JointsJoints

Beam

Beam-column

Joint

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Frame IdealisationFrame Idealisation

Reduction of 3-D framework to plane frames

Frame modelling for Frame modelling for analysisanalysis

•• Framing and jointsFraming and joints

–– Continuous framing: Continuous framing: rigid jointrigid joint–– Simple framing: Simple framing: pinned jointpinned joint–– SemiSemi--continuous framing: continuous framing: semisemi--rigid jointrigid joint

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Joint modelling for frame Joint modelling for frame analysisanalysis

•• The main approaches are:The main approaches are:–– the the traditional approachtraditional approach in which the in which the

joints are considered as (nominally) pinned joints are considered as (nominally) pinned or rigid or rigid

–– the the semisemi--rigid approachrigid approach in which a in which a more realistic model representing the joint more realistic model representing the joint behaviour is used. It is usually introduced behaviour is used. It is usually introduced as a spiral spring at the extremity of the as a spiral spring at the extremity of the member it attaches (usually the beam).member it attaches (usually the beam).

Joint modelling for frame analysisJoint modelling for frame analysis

JOINTMODELLING

BEAM-TO-COLUMN JOINTSMAJOR AXIS BENDING

BEAMSPLICES

COLUMNBASES

SIMPLE

SEMI-

CONTINUOUS

CONTINUOUS

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Global frame Global frame analysisanalysis

•• Aims of global frame analysisAims of global frame analysis–– Determine the distribution of the internal Determine the distribution of the internal

forcesforces–– Determine the corresponding deformationsDetermine the corresponding deformations

•• MeansMeans–– Adequate models incorporating assumptions Adequate models incorporating assumptions

about the behaviour of the structure and its about the behaviour of the structure and its component:component:members and jointsmembers and joints

Requirements for analysisRequirements for analysis

•• Basic principles to be satisfied:Basic principles to be satisfied:

–– Equilibrium Equilibrium throughout the structurethroughout the structure–– CompatibilityCompatibility of deformation between the of deformation between the

frame componentsframe components–– Constitutive lawsConstitutive laws for the frame for the frame

componentscomponents

•• Frame model Frame model -- element modelelement model

–– must satisfy the basic principlesmust satisfy the basic principles

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Frame Frame behaviourbehaviour

Frame

DisplacementLoad

Load parameter

Displacement parameter

Elastic limit

Peak load

Full elasticresponse

λ

Frame behaviourFrame behaviour

•• Actual response of the frame is non linearActual response of the frame is non linear

–– Linear behaviour limitedLinear behaviour limited

–– NonNon--linear behaviour due to: linear behaviour due to: •• Geometrical influence of the actual Geometrical influence of the actual

deformed shape (second order effects)deformed shape (second order effects)•• Joint behaviourJoint behaviour•• Material yieldingMaterial yielding

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Second order effectsSecond order effects

PH

δ

Δ

x

M(h) = Hh + PM(x) = Hx + P δ + P Δ x / h

Δ

PH

h

x

M(h) = HhM(x) = Hx

Displacement

Frame

LoadSway

Consideration for second-order moment

h

Fh + PΔ

Δ

P

F

F x h

P

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Second order effectsSecond order effects•• PP--Δ Δ effect : effect :

–– due to floor sway due to floor sway –– 1st order frame stiffness modified1st order frame stiffness modified–– dominant effectdominant effect

•• PP--δ δ effect :effect :–– due to beamdue to beam--column deflection column deflection –– 1st order member stiffness modified1st order member stiffness modified–– significant only for relatively slender significant only for relatively slender

members which is raremembers which is rare

ImperfectionsImperfections

Φ Φ

Frame imperfection

L

N

Member Imperfection

eo,d

PP--ΔΔ effecteffectPP--δδ effecteffect

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ImperfectionsImperfections

•• Frame imperfectionFrame imperfection–– always to be allowed foralways to be allowed for

•• Member imperfection:Member imperfection:–– only for slender members (rare) in sway only for slender members (rare) in sway

frames, otherwise it is covered in the frames, otherwise it is covered in the relevant buckling curverelevant buckling curve

Use of Notional Horizontal Forces

1. NHF + Other loadings: To allow for frame imperfections such as lack of verticality

2. NHF Alone: To test for sway sensitivity

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Notional horizontal loads(To allow for frame imperfections such as lack of frame verticality)

0.5% of (D+I))4

0.5% of (D+I)3

0.5% of (D+I)2

0.5% of (D+I)1`

φ= 0.005 or 1/200(D+I)1

(D+I)2

(D+I)3`

(D+I)4

=

Notional horizontal loadsNotional horizontal forces should NOT:

a) be applied when considering overturningb) be applied when considering pattern loadingc) be combined with applied horizontal loadsd) be combined with temperature effectse) be taken to contribute to the net reactions at

the foundations.

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Minimum horizontal forces

Factored dead load

1% of DL4

1% of DL3

1% of DL2

1% of DL1

Wind load or

Wind load or

Wind load or

Wind load or

Greater of DL4

DL3

DL2

DL1

DL 1-4 are the total dead load at each floor level

Resistance to horizontal forces

Resistance to horizontal forces may be provided in a number of ways as follows:a) triangulated bracing members.b) moment resisting joints and frame action.c) cantilever columns, shear walls, staircase

and lift shaft enclosures.d) or a combination of these.

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(a)

Resistance to horizontal force is provided by concrete core

Can a braced frame be a sway frame?Can a braced frame be a sway frame?

Yes, when lateral deflection is largeEspecially for high-rise building

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Classification of frames Classification of frames

Frames may be Frames may be 1.1. Braced or unbracedBraced or unbraced ––

depends on how horizontal forces are depends on how horizontal forces are transmitted to the ground.transmitted to the ground.

2.2. Sway or nonSway or non--swaysway --depends on significance or otherwise of Pdepends on significance or otherwise of P--ΔΔeffects.effects.

Purpose of classification of braced and Purpose of classification of braced and unbraced frameunbraced frame

Central Core

Is the bracing adequate?

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Braced multi-storey frameIf frame B is braced by frame A

Stabilizing System to resist all horizontal load

Braced frame designed to resist gravity load only

B A

Independently Braced Frames (5.1.4)

a) The stabilizing system has a lateral stiffness at least 4 times larger than the total lateral stiffness of all the frames to which it gives horizontal support (i.e. the supporting system reduces horizontal displacements by at least 80%).

andb) The stabilizing system is designed to resist

all the horizontal loads applied including the notional horizontal forces.

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Classification of Braced and Unbraced Frame

ΔA ΔB

8.01B

A ≥ΔΔ

−

Frame B is braced by Frame A if

If frame B is not braced by frame AFF ααFF (1(1-- αα))FF

The lateral force F should be distributed or resisted in accordance with the relative lateral stiffness of the respective frame. The stiffer frame will resist higher lateral force.

frametheofstiffnesslateraltheisKKK

K

BA

A ;+

=α

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Example of Simple Braced Example of Simple Braced FrameFrame

Bracing Frame(resist all horizontal load)

Simple FrameK = 0

SwaySway--sensitive Vs Nonsensitive Vs Non--sway Framesway Frame

Δ

P

• If second order effects (P- Δ) are significant –sway sensitive frame

• If second order effects can be ignored in the calculation – non-sway frame

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NonNon--sway framesway frame

Sway sensitive frameSway sensitive frame

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Definition of Sway/non sway FrameDefinition of Sway/non sway Frame

A frame can be classified as nonA frame can be classified as non--sway if its sway if its response to inresponse to in--plane horizontal forces is plane horizontal forces is sufficiently stiff for it to be acceptably accurate sufficiently stiff for it to be acceptably accurate to neglect any additional internal forces or to neglect any additional internal forces or moments arising from horizontal displacements moments arising from horizontal displacements of its nodes.of its nodes.

Classification of Sway / NonClassification of Sway / Non--sway Framesway Frame

A frame can be deemed to be nonA frame can be deemed to be non--sway if,sway if,

λλcrcr ≥≥ 1010Otherwise it is a sway frame.Otherwise it is a sway frame.

λλcr cr can be determined using deflection method in can be determined using deflection method in Annex F.2 of BS5950:Part1:2000Annex F.2 of BS5950:Part1:2000

λλcr cr = = 1 / (200 1 / (200 φφmaxmax ) = h / (200 ) = h / (200 δδmaxmax))

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Annex F.2 – How to determine critical load Ratio λcr ?

δ

0.5%(D+I)

0.5%(D+I)

0.5%(D+I)

0.5%(D+I)

1

δ 2

δ 3

δ 4

(3)(3) Calculate Sway Index of each storey

φδ δ

su L

h=

−

(1)(1) Apply notional horizontal loads(2)(2) Determine inter-storey drift

(4) Compute λcr = 1 / 200 φmax

For nonsway frame

λcr = 1 / (200 φmax ) >10

or φmax < 1/2000

Since φi = Δi/hiΔI = inter-storey deflection,hi = storey height

Δi < hi /2000 for every storey

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Use Notional Load for Classification of Sway Frame Use Notional Load for Classification of Sway Frame (2.4.2.6)(2.4.2.6)

Frame is non sway ifFrame is non sway ifΔΔii < h< hii /2000/2000 for cladded framefor cladded frame

Effect of cladding is not considered in Effect of cladding is not considered in calculating calculating ΔΔii

Δ1

Δ2

Δ4

Δ4NHL4

NHL3

NHL2

NHL1

SummarySummary

Notional loadsNotional loads are used to allow for frame are used to allow for frame imperfections such as lack of verticality and to classify imperfections such as lack of verticality and to classify framesframesNonsway frameNonsway frame: : λλcrcr ≥≥ 1010Sway frame:Sway frame: λλcrcr << 10 10 Braced frame: Braced frame: the horizontal supporting system the horizontal supporting system reduces horizontal displacements by at least 80%.reduces horizontal displacements by at least 80%.Braced frame needs to be designed for gravity load Braced frame needs to be designed for gravity load only. The lateral load resisting system will resist all only. The lateral load resisting system will resist all horizontal forces.horizontal forces.

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Example: Frame ClassificationExample: Frame Classification

Unfactored roof and floor loadsRoof:Dead load Wdr = 3.5 kN/m2

Imposed load Wir = 1.0 kN/m2

Floor:Dead load Wdf = 3.5 kN/m2

Imposed load Wif = 6.0 kN/m2

Roof beam 305 x 127 x 37 UB in grade S275Floor beam 406 x 178 x 60 UB in grade S275Ground to 2nd floor columns 203 x 203 x 60 UC in grade S2752nd floor to roof columns 203 x 203 x 46 UC in grade S275Bracing 168.3 x 6.3 CHS in grade S275

Factored roof and floor loadsConsider the following three load combinations:(1) 1.4 dead + 1.6 imposed(2) 1.0 dead + 1.4 wind (dead load resisting overturning due to

wind)(3) 1.2 dead + 1.2 imposed + 1.2 windGravity loads for load combination 1Roof: wr' = (3.5 X 1.4) + (1.0 X 1.6) = 6.5 kN/m2Floor: wf' = (3.5 x 1.4) + (6.0 x 1.6) = 14.5 kN/m2Gravity loads for load combination 2Roof: wr2 = 3.5 X 1.0 = 3.5 kN/m2Floor: wf2 = 3.5 x 1.0 = 3.5 kN/m2Gravity loads for load combination 3Roof: wr3 = (3.5 X 1.2) + (1.0 X 1.2) = 5.4 kN/m2Floor: wf3 = (3.5 x 1.2) + (6.0 x 1.2) = 11.4 kN/m2

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1 Load combination 1 (Dead + Imposed)Roof level NHF = 0.005 x 0.5 x 28 x 49 x 6.5 = 22.3 kNFloor level NHF = 0.005 x 0.5 x 28 x 49 x 14.5 = 49.7 kNThe result of an elastic analysis on one braced bay (bare frame only) under the action of the notional horizontal forces is shown in Figure 2.Ground – 1st floor

cr3500 5.15

200 3.4λ = =

×

2nd floor – 3rd floor

cr3000 5.36

200 2.8λ = =

×

Therefore, cr 5.15λ =

Since cr 10λ <

The frame is classed as "sway"

λλcrcr = 1 / (200 = 1 / (200 φφmaxmax ) ) = h / (200 = h / (200 δδmaxmax))

λ

2 Load combination 2 (Dead +2 Load combination 2 (Dead + Wind)Wind)Roof level NHF = 0.005 x 0.5 x 28 x 49 x 3.5 = 12 Roof level NHF = 0.005 x 0.5 x 28 x 49 x 3.5 = 12 kNkNFloor level NHF = 0.005 x 0.5 x 28 x 49 x 3.5 = 12 Floor level NHF = 0.005 x 0.5 x 28 x 49 x 3.5 = 12 kNkN

Ground Ground –– 1st floor1st floor

2nd floor 2nd floor –– 3rd floor 3rd floor

Therefore, Therefore,

Since , Since ,

the frame is classed as "nonthe frame is classed as "non--swaysway

cr3500 17.5

200 1.0λ = =

×

7.169.0200

3000cr =

×=λ 7.16

9.02003000

cr =×

=λ

cr3000 16.7

200 0.9λ = =

×

cr 16.7λ =

cr 10λ >

,

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3 Load combination 3 (Dead + Wind + Imposed)Roof level NHF = 0.005 x 0.5 x 28 x 49 x 5.4 = 18.5 kNFloor level NHF = 0.005 x 0.5 x 28 x 49 x 11.4 = 39.1 kNGround – 1st floor

2nd floor – 3rd floor

Therefore,

Since ,

The frame is classified as "sway sensitive" for load combination 3.

cr3500 17.5

200 1.0λ = =

×

cr3000 6.82

200 2.2λ = =

×

cr 6.82λ =

cr 10λ <

SummarySummary

Consider the following three load combinations:(1) 1.4 dead + 1.6 imposed – Sway frame(2) 1.0 dead + 1.4 wind (dead load resisting

overturning due to wind) - nonsway(3) 1.2 dead + 1.2 imposed + 1.2 wind – sway

frame

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Methods of structural Methods of structural analysisanalysis

1st1st--order elastic analysisorder elastic analysis

M

φ

M

Elastic

M

φ

Moment rotation characteristic of the sectionMoment rotation characteristic of the jφ

φ

Mj

ElasticMj

Indefinite linear elastic response of member sections Indefinite linear elastic response of member sections and of jointsand of jointsEquilibrium established for the Equilibrium established for the undeformedundeformed structural structural configurationconfiguration

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2nd order elastic analysis2nd order elastic analysis

Indefinite linearIndefinite linear-- elastic elastic response of member response of member sections and jointssections and joints

Equilibrium established Equilibrium established for the for the deformeddeformedstructurestructure

Allows for PAllows for P--Δ Δ effect effect and, if necessary, for Pand, if necessary, for P--δ δ effect effect

Load parameter

2nd order elastic analysis

Displacement parameter

λcr

M

M

Plastic hinge

Moment rotation characteristics of the member

pl.Rd

Moment rotation characteristics of the jo

Rigid plasticMpl.Rd

φp

Mpl.Rd

φ p

M

Mj,Rdφ p

Rigid plastic

M j,Rd

φp

j

Plastic hinge

RigidRigid--plastic global analysisplastic global analysis

RigidRigid--plastic member section behaviourplastic member section behaviourRigidRigid--plastic joint behaviour when plastic hinges are plastic joint behaviour when plastic hinges are allowed there allowed there

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RigidRigid--plastic global analysisplastic global analysis

Usually a first order Usually a first order analysisanalysisFind critical mechanismFind critical mechanismEasy application for simple Easy application for simple frames e.g. industrial frames e.g. industrial portal framesportal framesServiceability deflection Serviceability deflection checkcheck

Load parameter

Displacement parameter

Plastic mechanism

1

3

2

Critical collapse loadλ

LRP3

W

Beam mechanism

ΦΦ

1

Sway mechanism

Φ Φ

H

2

Δ

A

B

C

D

E A

B D

E

h

H

W

Δ w

W

H

Δ

Φ Φ

Combined mechanism

3plastic hinge location

A

B

C

D

E

h

Δ w

ElasticElastic--perfectly plastic global perfectly plastic global frame analysisframe analysis

ElasticElastic--perfectly plastic response of member perfectly plastic response of member sections and jointssections and joints

M

φ

Plastic hingeMj.

M

φj

M

Plastic hingeMp

Mp

φ

φ

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ElasticElastic--perfectly plastic perfectly plastic analysisanalysis

2nd2nd--order order analysis usually analysis usually usedused

Load appliedLoad appliedin incrementsin increments

““DeteriorationDeterioration””of frame stability of frame stability as plastic hinges as plastic hinges formform

1st hinge

2nd hinge maximum load

elastic buckling load of frame

elastic buckling load

Load parameter

Displacement parameter

branch 1

branch 2branch 3

branch 4

λL2EPP

of deteriorated frame

Frame classification and type of frame analysis

1st order

2nd order2nd order

1st order

BRACINGBRACED UNBRACED

SWAYLATERAL

DISPLACEMENTNON SWAY

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Summary• The frame has first to be idealised

• Then a frame classification is carried out⇒ sway-non sway / braced-unbraced

• On the basis of the frame class, the type of frame analysis is finally selected.– Sway frame – second order analysis– Nonsway frame – first order analysis

READING ASSIGNMENT

• Chapter 3 Section 3.1

• BS5950:Part1 Clause 2.3.2

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