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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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