Höglund_2008

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TALAT 2301 1

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TALAT Lecture 2301

Design of Members

prepared by T. Höglund, Royal Institute of Technology, Stockholm

Objectives:

− To give background to calculation methods for aluminium members in order tounderstand the specific behaviour of statically loaded aluminium alloy structures.

Prerequisites:

− Basic structural mechanics and design philosophy− Structural aluminium alloys and product forms

Note:

This lecture material has been updated during the Leonardo da Vinci projectTraining in Aluminium Structural Design, TAS WP1, June 1998and further updated in the aluMATTER project 2007

Date of Issue: 2008© EAA – European Aluminium Association

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2301 Design of Members

Contents

2301 Design of Members ................................................................................................. 21 General ..................................................................................................................... 5

1.01 Scope ................................................................................................................ 51.02 Symbols ........................................................................................................ 51.03 Safety and serviceability............................................................................... 61.04 Design with regards to instability................................................................. 81.05 Geometrical imperfections ........................................................................... 9

1.051 Extruded profiles ...................................................................................... 9 1.052 Welded profiles....................................................................................... 10

1.06 Residual stresses and variability in material properties ............................. 11

1.061 Residual stresses..................................................................................... 11 1.062 Inhomogeneous distribution of mechanical properties .......................... 14 1.063 Bauschinger effect .................................................................................. 14

1.07 Heat affected zones..................................................................................... 141.08 Stress-strain relationship ............................................................................ 17

2 Design basis ....................................................................................................... 19 2.01 Basic values of strength.............................................................................. 192.02 Design values of strength ........................................................................... 192.03 Design values for reduced strength in the heat-affected zone.................... 212.04 Partial factors (Resistance factors) ............................................................. 212.05 Gross section / net section .......................................................................... 22

3 Local buckling.................................................................................................. 23 3.01 Cross section classes................................................................................... 233.02 Behaviour of slender plates ........................................................................ 243.03 Effective cross section ................................................................................ 263.04 Calculation technique for class 4 cross sections ........................................ 283.05 Calculation of deflections of beams with class 4 cross section.................. 283.06 Breathing .................................................................................................... 29

4 Bending moment ............................................................................................... 31 4.01 Yielding and local buckling........................................................................ 314.02 Classification of cross sections................................................................... 33

4.03 Slenderness parameter ................................................................................ 354.04 Classification of cross section parts ........................................................... 364.05 Effective thickness...................................................................................... 384.06 Effective cross section ................................................................................ 384.07 Welded section ........................................................................................... 404.08 Section with holes....................................................................................... 404.09 Lateral torsional buckling........................................................................... 414.09 Simple method to allow for local buckling ................................................ 44

5 Axial force ......................................................................................................... 46 5.01 General ....................................................................................................... 465.02 Tensile force ............................................................................................... 47

5.03 Compressive force ...................................................................................... 48

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5.031 Euler load, squash load and resistance.................................................. 49 5.032 Reduction factor for flexural buckling ................................................... 50 5.033 Cross section class 4 .............................................................................. 52 5.034 Slenderness parameters.......................................................................... 54 5.035 Buckling length....................................................................................... 55

5.036 Torsional buckling and torsional-flexural buckling............................... 56 5.037 Design of splices and end connections................................................... 57 5.038 Derivation of expression for second order bending moment ................. 58 5.039 Derivation of interaction formula and expression xω ........................... 59

5.04 Welded columns and columns with bolt holes or cut-outs......................... 605.041 Longitudinal welds ................................................................................. 60 5.042 Transverse welds .................................................................................... 60 5.043 Columns with unfilled bolt-holes or cut-outs ......................................... 61

5.05 Uniform built-up members ......................................................................... 626 Shear force ........................................................................................................ 63

6.01 Shear buckling of plate girder webs ........................................................... 636.02 Shear resistance of webs with stiffeners at supports only .......................... 666.03 Plate girders with intermediate stiffeners ................................................... 676.04 Corrugated or closely stiffened webs ......................................................... 69

7 Concentrated loads and support reactions..................................................... 72 7.01 Beam webs without stiffeners..................................................................... 727.02 Beam webs with stiffeners.......................................................................... 75

8 Torsion............................................................................................................... 77 8.01 Shear centre ................................................................................................ 778.02 Closed and open sections............................................................................ 788.03 Torsion without warping ............................................................................ 80

8.04 Torsion with warping ................................................................................. 819 Axial force and bending moment .................................................................... 82

9.01 General ....................................................................................................... 829.02 Bending and axial tension .......................................................................... 829.03 Bending and axial compression.................................................................. 839.04 Strength of beam-column segments. .......................................................... 84

9.041 Rectangular section - plastic theory....................................................... 84 9.042 Rectangular section - strain hardening material ................................... 85 9.043 I-, H- and T-section - strain hardening material.................................... 86 9.044 Biaxial bending of rectangular section .................................................. 88 9.045 Biaxial bending of I- and H-section ....................................................... 89

9.05 Flexural buckling........................................................................................ 919.06 Lateral-torsional buckling .......................................................................... 949.07 Thin walled cross sections.......................................................................... 969.08 Transverse welds ........................................................................................ 969.09 Columns with unfilled bolt-holes or cut-outs............................................. 999.10 Varying applied bending moment .............................................................. 999.11 Derivation of design section for linearly varying bending moment......... 101

10 Deviation of linear stress distribution....................................................... 104 10.01 General ................................................................................................. 10410.02 Shear lag ............................................................................................... 104

10.03 Flange curling of a wide flange ............................................................ 105

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10.04 Lateral deflection of non-symmetrical flanges..................................... 10611 Examples ..................................................................................................... 106

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1 General

The objective of this part of TALAT is to give background to the design methods andrecommendations in EN 1999-1-1 Design of aluminium structures Part 1-1: Generalrules (Eurocode 9) in order to understand the specific behaviour of static loadedaluminium structures. The text is supplemented by a number of design examples.

1.01 Scope

This chapter concern aluminium plate structures and extrusions primarily applicable toaluminium structures in buildings, civil and structural engineering works such as

bridges, hydraulic and offshore structures and temporary structures, e.g. erection and

building scaffolds, masts, cable- and light poles.

1.02 Symbols

Symbols

The symbols used in Eurocode 9 are mostly used in this document as well. In order tofacilitate the reading the most common symbols are given here.

Aeff Effective area Agr Gross area Anet Net areab Widthbhaz Width of heat affected zonec Distance; Outstandd Diameter; Depth

E Modulus of elasticity f o Characteristic strength for

yielding f 0,2 0,2 proof strength f u Characteristic ultimate strength f o,haz Characteristic 0,2 proof strength

in heat-affected zone f u,haz Characteristic ultimate strength

in heat-affected zoneG Shear modulus

I eff Second moment of area of effective cross section ( I

ewith

further subscript) I gr Second moment of area of gross

cross section I net Second moment of area of net

cross section I ser Effective second moment of area

for calculation of deflections L length, span, system lengthl cr Effective (buckling) length

M el Elastic bending momentM pl Plastic bending momentM Ed Bending moment (action),

design valueM Rd Bending moment resistance,

design value N Ed Axial force (action), design

value N Rd Axial force resistance, design

valueq Distributed load

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r Radius t Thicknesst eff Effective thickness ( t e with

further subscript)W el Elastic section modulus

W pl Plastic section modulusW eff Section modulus of effective

cross sectionα Shape factor

β Slenderness ratio of a crosssection part

γ M1 Partial safety factor for theresistance due to overall yielding

γ M2 Partial safety factor for theresistance of net section

ε Strainλ c Slenderness parameter ( λ in

Eurocode 9)l cr /i Slenderness ratio ( λ in Eurocode

9)

ρ c Reduction factor for local buckling

ρ o,haz HAZ softening factor for the0,2 % proof strength

ρ u,haz HAZ softening factor for theultimate strength

σ Stress χ Reduction factor for flexural

buckling χ LT Reduction factor for lateral-

torsional bucklingψ Stress ratioψ Factor defining representative

values of variable actionsψ c Exponent in interaction formulaξ yc Exponent in interaction formulaξ zc Exponent in interaction formulaη c Exponent in interaction formulaω x HAZ softening factor for local

softening along a member

S.I. units

The following S.I. units are used

- Forces and loads kN, kN/m, kN/m 2 - unit mass kg/m 3 - unit weight kN/m 3 - stresses and strength N/mm 2 (=MN/m 2 or MPa)- moments kNm

1.03 Safety and serviceability

The design philosophy and the design procedure are discussed in Aluminium Design, part 4.1. In this chapter a short presentation of the partial factor method is given.Design values of strength of aluminium alloys are given in sub clause 2, Design basis.

In most modern codes of practice structural safety is established by the application of the partial safety factors to the loads (or 'actions') and to the strength (or 'resistance') of components of the structure. The Eurocodes for the design and execution of buildingsand civil engineering structures use a limit state design philosophy defined in Eurocode0, Basis of structural design.

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The partial safety factors for actions ( γf ) depend on an accepted degree of reliability,which is recognised as a national responsibility within the European Community. The

probability of severe loading actions occurring simultaneously can be foundanalytically, if enough statistical information exists, and this is taken into account by theintroduction of a second factor ψ . The design value of the action effects (when the

effects are unfavourable) is then found by taking values of γf dependent on the type of loading and values for ψ that take account of the chances of simultaneous loading. Avalue of γf of 1,35 is suggested for permanent loads, such as the dead load of bridgegirders, and 1,5 for variable loads such as traffic loads or wind loading. The loadingactions on members are usually found by an elastic analysis of the structure, using thefull cross-sectional properties of the members.

The partial safety factors for actions takes account of the possibility of unforeseendeviations of the actions from their representative values, of uncertainty in thecalculation model for describing physical phenomena, and uncertainty in the stochastic

model for deriving characteristic codes.

The partial safety coefficient for material properties ( γM) reflects a commonunderstanding of the characteristic values of material properties, the provision of recognised standards of workmanship and control, and resistance formulae based onminimal accepted values. The value given to γM accounts for the possibility of unfavourable deviations of material properties from their characteristic values,uncertainties in the relation between material properties in the structure and in testspecimens, and uncertainties associated with the mechanical model for the assessmentof the resistance.

A further coefficient, γd, is specified in some codes, and this can be introduced to takeaccount of the consequences of failure in the equation linking factored actions withfactored resistance. It is incorporated in γf . It recognises that there is a choice of reliability for classes of structures and events that take account of the risk to human life,the economic loss in the event of failure, and the cost and effort required to reduce therisk. Typical values are γd = 0,83 for consequence class 1, 0,91 for consequence class 2and 1,0 for consequence class3. The consequence classes 1, 2 and 3 are then defined aslow, medium and high consequence for loss of human life or small, considerable andvery great economic, social or environmental consequences

The ultimate limit states defined by the use of the above factors refer to failure of members or connections by rupture or excessive deformation, transformation of thestructure into a mechanism, failure under repeated loading (fatigue) (STR) and the lossof equilibrium of the structure as a rigid body (EQU).

Serviceability limit states, according to most definitions, correspond to a loss of utility beyond which service conditions are no longer met. They may correspond tounacceptable deformations or deflections, unacceptable vibrations, the loss of the abilityto support load-retaining structures, and unacceptable cracking or corrosion. Becausecertain aluminium alloys in the non-heat-treated condition, or in the work-hardenedcondition, do not have a sharply defined 'knee' to the stress/strain curve, it is sometimes

possible for unacceptable permanent deformation to occur under nominal or working

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loads. The same may be true for alloys that have a substantial amount of welding duringfabrication.

1.04 Design with regards to instability

Extruded and welded members are never totally perfect. They possess a number of imperfections. Residual stresses, heat-affected zones and other variation of material

properties and the Baushinger effect are dealt with in section 1.06 and 1.07. Other typesof imperfections, for example initial curvature and deviation of cross sectionaldimensions, are dealt with in section 1.05. The stress-strain relationship of aluminium isdealt with in section 1.08.

It is of great importance to take the influence of imperfections into consideration,especially for different types of instability phenomena, e.g. flexural buckling, lateral-torsional buckling and plate buckling. To illustrate this, consider the case of flexural

buckling for a bar subjected to an axial load, cf. Figure 1.01. In the past, thecompressive force capacity was calculated with Euler's buckling formula. This formulais valid for a perfectly straight, elastic bar without imperfections. However, in reality,such a bar contains a number of imperfections that reduce the strength. In figure 1.01the behaviour of an idealised Euler column is compared to that of a real column.

It is possible, in the age of computers, to create calculation models that can, with greatdetail, simulate the actual behaviour, but under one condition. Every imperfection of the

beam must be known and correctly modelled and taken into consideration. Residualstresses and variation in material properties have little influence on the behaviour of extruded members. On the other hand these imperfections can have great effect onwelded members.

Welding effects the member by creating residual stresses and reduction in strength of

the material in the heat affected zones.

Initiallystraight bar

Bar with initial deflection

δ

N

N

δ

N cr

N

Figure 1.01 Comparison between buckling behavior of an idealized

Euler bar and of a real bar with imperfections

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