DESIGN OF AXIALLY LOADED COLUMNS - Steel-insdag
DESIGN OF AXIALLY LOADED COLUMNSDESIGN OF AXIALLY LOADED COLUMNS101.0 INTRODUCTIONIn an earlier chapter, the behaviour of practical columns subjected to axial compressiveloading was discussed and the following conclusions were drawn. Very short columns subjected to axial compression fail by yielding. Very longcolumns fail by buckling in the Euler mode.Practical columns generally fail by inelastic buckling and do not conform to theassumptions made in Euler theory. They do not normally remain linearly elastic uptofailure unless they are very slenderSlenderness ratio (λ/r) and material yield stress (fy) are dominant factors affecting theultimate strengths of axially loaded columns.The compressive strengths of practical columns are significantly affected by (i) theinitial imperfection (ii) eccentricity of loading (iii) residual stresses and (iv) lack ofdefined yield point and strain hardening. Ultimate load tests on practical columnsreveal a scatter band of results shown in Fig. 1. A lower bound curve of the typeshown therein can be employed for design purposes.σcTest data (x) from collapse testson practical columns(Mpa)fyxx200xxxxEuler curvexxxx xxxx100xDesign curvexxx x50100150Slenderness λ (λ/r)Fig. 1 Typical column design curve Copyright reservedVersion II10 - 1
DESIGN OF AXIALLY LOADED COLUMNS2.0 HISTORICAL REVIEWBased on the studies of Ayrton & Perry (1886), the British Codes had traditionally basedthe column strength curve on the following equation.( f y σ c ) ( σ e σ c ) η σ e σ c(1)wherefy yield stressσc compressive strength of the column obtained from the positive root of theabove equationπ 2EEuler buckling stress given by 2λσe η a parameter allowing for the effect of lack of straightness and eccentricityof loading.λ Slenderness ratio given by (λ/r)(1a)In the deviation of the above formula, the imperfection factor η was based onη y r2where(2)y the distance of centroid of the cross section to the extreme fibre of thesection. initial bow or lack of straightnessr radius of gyration.Based on about 200 column tests, Robertson (1925) concluded that the initial bow ( )could be taken as length of the column/1000 consequently η is given by y λ r r η 0.001 . λ α α λ r (3)where α is a parameter dependent on the shape of the cross section.Version II10 - 2
DESIGN OF AXIALLY LOADED COLUMNSσc(Mpa)fy200Euler curveDesign curve with α 0.00310050100150λFig.2 Robertson’s Design CurveRobertson evaluated the mean values of α for many sections as given in Table 1:Table1: α values Calculated by RobertsonColumn typeα ValuesBeams & Columns about the major axis0.0012Rectangular Hollow sections0.0013Beams & Universal columns about the minor axis0.0020Tees in the plane of the stem0.0028He concluded that the lower bound value of α 0.003 was appropriate for columndesigns. This served as the basis for column designs in Great Britain until recently. Thedesign curve using this approach is shown in Fig. 2. The Design method is termed "PerryRobertson approach".3.0 MODIFICATION TO THE PERRY ROBERTSON APPROACH3.1Stocky ColumnsIt has been shown previously that very stocky columns (e.g. stub columns) resisted loadsin excess of their squash load of fy.A (i.e. theoretical yield stress multiplied by the areaof the column). This is because the effect of strain hardening is predominant in lowVersion II10 - 3
DESIGN OF AXIALLY LOADED COLUMNSvalues of slenderness (λ). Equation (1) will result in column strength values lower than fyeven in very low slenderness cases. To allow empirically for this discrepancy, recentBritish and European Codes have made the following modification to equation 3 given byη α λIn the unmodified form this will cause a drop in the calculated value of column strengtheven for very low values of slenderness. Such columns actually fail by squashing andthere is no drop in observed strengths in such very short columns. By modifying theslenderness, λ to (λ - λ0) we can introduce a plateau to the design curve at lowslenderness values. In generating the British Design (BS: 5950 Part-1) curves(λ0 0.2 πE / fy) was used as an appropriate fit to the observed test data, so that weobtain the failure load (equal to squash load) for very low slenderness values. Thus incalculating the elastic critical stress, we modify the formula used previously as follows:σe π 2Efor all values of λ λ0(λ λ0 )2( 4)Note that no calculations for σe is needed when λ λe as the column would fail bysquashing at fy.σcTest data (x) from collapse testson practical columns(Mpa)xxxxfy200xxx xxEuler curvexxx100xxx x xxxRobertson’s curvexxλ050[0.2π (E/fy ) ]1/2100150λFig.3 Strut curve with a plateau for low slenderness valuesVersion II10 - 4
DESIGN OF AXIALLY LOADED COLUMNS3.2 Influence of Residual StressesReference was made earlier to the adverse effect of locked-in residual stresses on columnstrengths (see Fig. 4). Studies on columns of various types carried out by the EuropeanCommunity have resulted in the recommendation for adopting a family of design curvesrather than a single “Typical Design Curve” shown in Fig. 3. Typically four columncurves are suggested in British and European codes for the different types of sectionscommonly used as compression members [See Fig. 5(a)]. In these curves, η α (λ -λ0)where λ0 0.2πEfyand the α values are varied corresponding to various sections.Thus all column designs are to be carried out using the strut curves given in [Fig. 5(a)].TTCTTTCTCCCTCCCTCTTCCTTRolled columnWelded boxRolled beamCCFig. 4 Distribution of residual stressesa)b)c)d)300250α 0.002α 0.0035α 0.0055α 0.008(Curve A)(Curve B)(Curve C)(Curve D)200fy150Compressivestrength σc(Mpa)100500050100150200250Slenderness, λFig. 5(a) Compressive strength curves for struts for different values of αVersion II10 - 5
DESIGN OF AXIALLY LOADED COLUMNSThe selection of an appropriate curve is based on cross section and suggested curves arelisted in Table 2.σc20N/mm2(Mpa)fy1505050100150λFig. 5(b) Compressive strength of welded 511.522.5λFig. 5(c) Column Buckling Curves as per IS: 800Version II10 - 63
DESIGN OF AXIALLY LOADED COLUMNSTable 2: Choice of appropriate values of αSectionsHot rolled structural hollow sectionsHot rolled I sectionWelded plateI section (up to 40 mm thick)I section (above 40 mm thick)Welded Box Section(Up to 40 mm thick)(Over 40 mm thick)Rolled I section with Welded coverplates (Up to 40mm thick)(Over 40mm thick)Rolled angle, Channel, T sectionCompound sections - Two rolledsections back to back, Battened orlaced sectionsAxis of bucklingX -Xα 0.002 (Curve A)α 0.002 (Curve A)Y-Yα 0.002 (Curve A)α 0.0035 (Curve B)α 0.0035 (Curve B) α 0.0055 (Curve C)α 0.0035 (Curve B) α 0.008 (Curve D)α 0.0035 (Curve B) α 0.0035 (Curve B)α 0.0055 (Curve C) α 0.0055 (Curve C)α 0.0035 (Curve B) α 0.002 (Curve A)α 0.0055 (Curve C) α 0.0035(Curve B)Check buckling about ANY axis Withα 0.0055 (Curve C)Note: For sections fabricated by plates by welding the Value of fy should be reduced by20 N/mm2 [See Fig. 5(b)].For computational convenience, formulae linking σc and λ are required. The lower root ofequation (1) [based on Perry - Robertson approach] represents the strut curves given Fig.3 and BS: 5950 Part - 1.σ c φ φ 2 f yσ e f ywhere, φ f y (η 1)σ e2and η (λ λ0 )α(5)(6 a )(6b)3.3 Types of Column SectionsSteel suppliers manufacture several types of sections, each type being most suitable forspecific uses. Some of these are described below. It is important to note that columnsmay buckle about Z, Y, V or U axis. It is necessary to check the safety of the columnabout several axes, so that the lowest load that triggers the onset of collapse may beidentified.Version II10 - 7
DESIGN OF AXIALLY LOADED COLUMNSUniversal Column (UC) sections have been designed to be most suitable for compressionmembers. They have broad and relatively thick flanges, which avoid the problems oflocal buckling. The open shape is ideal for economic rolling and facilitates easy beam-tocolumn connections. The most optimum theoretical shape is in fact a circular hollowsection (CHS), which has no weak bending axis. Although these have been employed inlarge offshore structures like oil platforms, their use is somewhat limited because of highconnection costs and comparatively weaker in combined bending and axiallycompressive loads. Rectangular Hollow Sections (RHS) have been widely used inmulti-storey buildings satisfactorily. For relatively light loads, (e.g. Roof trusses) anglesections are convenient as they can be connected through one leg. Columns, which aresubjected to bending in addition to axial loads, are designed using Universal beams (UB).3.4 Heavily Welded SectionsAlthough both hot rolled sections and welded sections have lock-in residual stresses, thedistribution and magnitude differ significantly. Residual stresses due to welding are veryhigh and can be of greater consequence in reducing the ultimate capacity of compressionmembers.3.5 Stipulations of IS: 800The stipulations of IS: 800 follow the same methodology as detailed above. For varioustypes of column cross sections including Indian Standard rolled steel sections (as againstUniversal Column sections), CHS, SHS, RHS and Heavily Welded sections, IS: 800recommends the classifications following the column buckling curves a, b, c and d asdetailed in fig. 5(c) above. Apart from types of column cross-sections, the respectivegeometric dimensions of individual structural elements and their corresponding limitsalso guide Buckling Class. For example, if the ratio of overall height is to the overallwidth of the flange of rolled I section i.e. h/b is greater than 1.2 and the thickness offlange is less than 40 mm, the buckling class corresponding to axis z-z will be guided byCurve a of Fig. 5 (c). This shows for same slenderness ratio, more reserve strength isavailable for the case described above in comparison to built-up welded sections. Forvarious types of column cross-sections, Table 3 (Table 7.2 of Revised IS: 800) definesthe buckling classes with respect to their respective limits of height to width ratio,thickness of flange and the axis about which the buckling takes place (to be determinedbased on the column buckling curves as indicated in fig. 5 (c) corresponding to bothmajor and minor axis):Version II10 - 8
DESIGN OF AXIALLY LOADED COLUMNSTABLE 3:BUCKLING CLASS OF CROSS SECTIONSCross SectionRolled I-Sectionshz-zy-yab40 mm tf 100 mmz-zy-ybcz-zy-ybcz-zy-yddz-zy-ybcz-zy-ycdHot rolledAnyaCold formedAnybGenerally(Except as below)Anybb/tf 30z-zch/tw 30y-ycAnycAnyctfdtwzztf 100 mmh/bf 1.2 :bfWelded I-Sectiontf 100 mmyyytwh ztf 40 mmtftfz htwztf 40 mmzyybbHollow SectionWelded Box SectionyhBucklingClasstf 40 mmh/bf 1.2 :yBucklingabout axisLimitstftwThick welds andzzb yChannel, Angle, T and Solid SectionsyzzyyBuilt-up MemberzzyVersion II10 - 9
DESIGN OF AXIALLY LOADED COLUMNS4.0 EFFECTIVE LENGTH OF COLUMNSThe effective length, KL, is calculated from the actual length, L, of the member,considering the rotational and relative translational boundary conditions at the ends. Theactual length shall be taken as the length from center to center of its intersections with thesupporting members in the plane of the buckling deformation, or in the case of a memberwith a free end, the free standing length from the center of the intersecting member at thesupported end.Effective Length – Where the boundary conditions in the plane of buckling can beassessed, the effective length, KL, can be calculated on the basis of Table 7.5. Whereframe analysis does not consider the equilibrium of a framed structure in the deformedshape (Second-order analysis or Advanced analysis), the effective length of compressionmembers in such cases can be calculated using the procedure given in Appendix E.1. Theeffective length of stepped column in individual buildings can be calculated using theprocedure given in Appendix E.2.Eccentric Beam Connection – In cases where the beam connections are eccentric in planwith respect to the axes of the column, the same conditions of restraint as in concentricconnection, shall be deemed to apply, provided the connections are carried across theflange or web of the columns as the case may be, and the web of the beam lies within, orin direct contact with the column section. Where practical difficulties prevent this, theeffective length shall be taken as equal to the distance between points of restraint, in nonsway frames.Stipulations of IS: 800 – Method for determining Effective Length for Stepped ColumnsSingle Stepped Columns – Effective length in the plane of stepping (bending about axis zz) for bottom and top parts for single stepped column shall be taken as given in Table E.2Note: The provisions of E.2.1 are applicable to intermediate columns as well with stepping on either side,provided appropriate values of I1and I2 are takenDouble Stepped Columns – Effective lengths in the plane of stepping (bending about axisz-z) for bottom, middle and top parts for a double stepped column shall be taken as perthe stipulations of Appendix E3Coefficient K1 for effective length of bottom part of double stepped column shall be takenfrom the formula:I2222t1 K 1 t 2 K 2 K 3 (1 n2 ) 1I ' avK1 1 t1 t 2()whereK1, K2, and K3 are taken from Table E.6,Version II10 - 10
DESIGN OF AXIALLY LOADED COLUMNSTABLE 4: EFFECTIVE LENGTH OF PRISMATIC COMPRESSION MEMBERSBoundary ConditionsSchematicEffectiveAt one endAt the other estrainedRestrained0.65 LNote – L is the unsupported length of the compression member (7.2.1).Version II10 - 11
DESIGN OF AXIALLY LOADED COLUMNSDesigns of columns have to be checked using the appropriate effective length forbuckling in both strong and weak axes. A worked example illustrating this concept isappended to this chapter.5.0 STEPS IN DESIGN OF AXIALLY LOADED COLUMNS AS PER IS: 800The procedure for the design of an axially compressed column as stipulated in IS: 800 isas follows:(i)Assume a suitable trial section and classify the section in accordance with theclassification as detailed in Table 3.1 (Limiting Width to Thickness Ratios) of theChapter 3 of IS: 800. (If, the section is slender then apply appropriate correctionfactor).(ii)Calculate effective sectional area, Ae as defined in Clause 7.3.2 of IS: 800(iii)Calculate effective slenderness ration, KL/r, ratio of effective length KL, toappropriate radius of gyration, r(iv)Calculate λ from the equation, λ non-dimensional effective slenderness ratio f y f cc (v)2π 2ECalculate φ from the equation, φ 0.5[1 α (λ - 0.2) λ2]Where,α Imperfection factors for various Column Buckling Curves a, b, c and d aregiven in the following Table: (Table 7.1 of IS: 800)TABLE 5:(vi)( r)f y KLIMPERFECTION FACTOR, αBuckling Classabcdα0.210.340.490.76 Calculate χ from equation, χ 1( φ φ 2 λ2 0 .5 )(vii)Choose appropriate value of Partial safety factor for material strength, γm0 fromTable 5.2 of Chapter 5 of IS: 800(viii)Calculate design stress in compression, fcd , as per the following equation (Clause7.1.2.1 of IS: 800):f y / γ m0f cd χ f y / γ m0 f y / γ m022 0.5φ φ λCompute the load Pd, that the compression member can resist Pd Ae fcd[(ix)Version II]10 - 12
DESIGN OF AXIALLY LOADED COLUMNS(x)Calculate the factored applied load and check whether the column is safe againstthe given loading. The most economical section can be arrived at by trial anderror, i.e. repeating the above process.6.0CROSS SECTIONAL SHAPES FOR COMPRESSION MEMBERS ANDBUILT- UP COLUMNSAlthough theoretically we can employ any cross sectional shape to resist a compressiveload we encounter practical limitations in our choice of sections as only a limited numberof sections are rolled by steel makers and there are sometimes problems in connectingthem to the other components of the structure. Another limitation is due to the adverseimpact of increasing slenderness ratio on compressive strengths; this virtually excludesthe use of wide plates, rods and bars, as they are far too slender. It must be speciallynoted that all values of slenderness ratio referred to herein are based on the leastfavourable value of radius of gyration, so that (λ/r) is the highest value about any axis.6.1 Rolled Steel SectionsSome of the sections employed as compression members are shown in Fig. 6. Singleangles [Fig 6(a)] are satisfactory for bracings and for light trusses. Top chord membersof roof trusses are usually made up of twin angles back to back [Fig 6(b)]Double angle sections shown in Fig. 6(b) are probably the most commonly used membersin light trusses. The pair of angles used has to be connected together, so they will act asone unit. Welds may be used at intervals – with a spacer bar between the connectinglegs. Alternately “ stitch bolts”, washers and “ring fills” are placed between the angles tokeep them at the proper distance apart (e.g. to enable a gusset to be connected).When welded roof trusses are required, there is no need for gusset plates and T sections[Fig 6(c)] can be employed as compression members.Single channels or C-sections [Fig. 6(d)] are generally not satisfactory for use incompression, because of the low value of radius of gyration. They can be used if theycould be supported in a suitable way in the weak direction.Circular hollow sections [Fig. 6(e)] are perhaps the most efficient as they have equalvalues of radius of gyration about every axis. But connecting them is difficult butsatisfactory methods have been evolved in recent years for their use in tall buildings.The next best in terms of structural efficiency will be the square hollow sections (SHS)and rectangular hollow sections, [Fig. 6(f)] both of which are increasingly becomingpopular in tall buildings, as they are easily fabricated and erected. Welded tubes ofcircular, rectangular or square sections are very satisfactory for use as columns in a longseries of windows and as short columns in walkways and covered warehouses. For manyVersion II10 - 13
DESIGN OF AXIALLY LOADED COLUMNSstructural applications the weight of hollow sections required would be only 50% of thatrequired for open profiles like I or C sections.(a) Single Angle(d) Channel(b) Double Angle(e) Circular HollowSection (CHS)(c) Tee(f) Rectangular HollowSection (RHS)Fig 6: Cross Section Shapes for Rolled Steel Compression MembersThe following general guidance is given regarding connection requirements:When compression members consist of different components, which are in contact witheach other and are bearing on base plates or milled surfaces, they should be connected attheir ends with welds or bolts. When welds are used, the weld length must be not lessthan the maximum width of the member. If bolts are used they should be spacedlongitudinally at less than 4 times the bolt diameter and the connection should extend toat least 1 ½ times the width of the member.Single angle discontinuous struts connected by a single bolt are rarely employed. Whensuch a strut is required, it may be designed for 1.25 times the factored axial load and theeffective length taken as centre to centre of the intersection at each end. Single anglediscontinuous struts connected by two or more bolts in line along the member at each endmay be designed for the factored axial load, assuming the effective length to be 0.85times the centre to centre distance of the intersection at each end.For double angle discontinuous struts connected back to back to both sides of a gusset orsection by not less than two bolts or by welding, the factored axial load is used
DESIGN OF AXIALLY LOADED COLUMNS σc (Mpa) fy λ Euler curve Design curve with α 0.003 200 00 1 50 100 150 Fig.2 Robertson’s Design Curve Robertson evaluated the mean values of α for many sections as given in Table 1: Table1: α values Calculated by Robertson Column
Axially-loaded Members Stiffness and flexibility Factor of safety, allowable stresses and loads Changes in length under non-uniform conditions (intermediate axial loads, prismatic segments, continuously varying loads or dimensions) Elasto-plastic analysis 2 Axially-loaded Members Structural components subjected only to tension or compression:
Chapter 2 Axially Loaded Members . Structural components subjected only to . tension. or . compression: solid bars with straight longitudinal axes, cables and coil springs – can be seen in truss members, connecting rods, spokes in bicycle wheels, columns in buildings, and struts in aircraft engine mounts. 2.2 Changes in Lengths of Axially .
wrapping an existing damaged masonry column with carbon fibre reinforced polymer (CFRP) sheet. The study focused onthe rehabilitation ofdamaged columns but the results obtained may also be applicable to strengthening of undamaged columns. 18 columns were tested. Each column was initially loaded axially until cracking was observed in the masonry.
Civil Engineering Design (1) 10 Dr. C. Caprani 2. Short Braced Axially Loaded Columns 2.1 Development The design of such columns is straightforward. The ultimate force is the sum of the stress areas of the steel and concrete: cu0.67 y uz c sc mm f f NAA γγ
A maximum rotation of the pile head of 0.5 is usually demanded. Regarding axially loaded piles an important question is how the axial ultimate pile capacity can be predicted with sufficient accuracy. The ß-method commonly used in offshore design (e.g. API, 2000) is known to either over-or underestimate pile capacities, dependent on the boundary
CIVL 4135 Chapter 3. AXIALLY LOADED MEMBERS 45 3.3.1. Example 1 Given: 4 # 8 bars Assume: fc′ 4000 psi fy 40 ksi area of steel 4(area of # 8 bars) 4(0.79) 3.16in2 (see ACI 318 - ba
4.2. Elastic Deformation of an Axially Loaded Member 4.4. Statically Indeterminate Axially Loaded Member 4.3. Principle of Superposition 4.5. The Force Method of Analysis 4.6. Thermal Stress CHAPTER 5. TORSION 5.1. Torsional Deformation of a Circular Shaft 5.2. The Torsion Formula 5.3. Power Transmission 5.4. Angle of Twist 5.5.
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