nfatec – l13 – columns (27/09/2004)library.tee.gr/digital/m2247/m2247_l13_eng.pdf• structural...

NFATEC – L13 – Columns (27/09/2004)

{LASTEDIT}Roger 27/09/2004{/LASTEDIT}

{LECTURE}

{LTITLE}Columns{/LTITLE}

{AUTHOR}John Ermopoulos{/AUTHOR}

{EMAIL}[email protected]{/EMAIL}

{OVERVIEW}

• Structural members subjected to axial compression are known as columns or struts

• Stocky columns may be unaffected by overall buckling

• Stocky columns may fail by local buckling or squashing

• Columns with high slenderness fail by elastic buckling

• Columns of medium slenderness are sensitive to the effects of imperfections and fail by inelastic buckling

• Imperfections in real columns reduce the capacity below that predicted by theory

• For design, a probabilistic approach using column curves is adopted

• The design buckling resistance is found by reducing the compression resistance of the cross section by a reduction factor for the relevant buckling mode.

{/OVERVIEW}

{PREREQUISITES}

• Euler buckling

• Cross-section classification

• Concept of effective lengths

{/PREREQUISITES}

• None

{OBJECTIVES}

On successful completion of this lecture you should:

• be able to describe the difference in behaviour of stocky and slender columns

• be able to recognise the sources of imperfection in real columns and the need for a probabilistic approach to design

• be able to compare the ECCS column curves

• be able to calculate the non-dimensional slenderness of a column

• be able to calculate the reduction factor for the relevant buckling modes for columns of different cross-sectional shapes

{/OBJECTIVES}

{REFERENCES}

• Structural Stability Research Council, Galambos, T.V., (ed) Guide to Stability Design Criteria for Metal Structure, 4th edition, John Wiley, New York, 1988

• prEN1993-1-1:2003, Eurocode 3: Design of steel structures Part 1.1 General rules and rules for buildings, 20/01/2003

• Timoshenko, S.P. and Gere, J.M., Theory of Elastic Stability, McGraw -Hill, New York, 1961

• Trahair, N.S. and Bradford, M.A., The Behaviour and Design of Steel Structures, E&F Spon, 1994

{/REFERENCES}

{SECTION}

{STITLE}Introduction{/STITLE}

{SUMMARY}

{SUMTITLE}Compression members{/SUMTITLE}

{PPT}Lecture13Intro.pps{/PPT}

Although most columns in practice are subject to bending as well as compression, the case on compression only needs to be considered as a basic case.

{DETAIL}

The term compression member is generally used to describe structural components subjected only to axial compression loads; this can describe columns (under special loading conditions) but generally refers to compressed pin-ended struts found in trusses, lattice girders or bracing members. If they are subjected to significant bending moments in addition to the axial loads, they are called beam-columns.

This lecture deals with compression members and, therefore, concerns very few real columns because eccentricities of the axial loads and transverse forces are generally not negligible. Nevertheless, compression members represent an elementary case which leads to the understanding of the compression effects in the study of beam-columns. Because most steel compression members are rather slender, buckling can occur. The lecture briefly describes the different kinds of compression members and explains the behaviour of both stocky and slender columns; the buckling curves used to design slender columns are also given.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Stocky columns{/SUMTITLE}

Stocky columns are not affected by overall buckling

{PPT}Lecture13StockyColumns.pps{/PPT}

{DETAIL}

Stocky columns have a very low slenderness such that they are unaffected by overall buckling of the member. In such cases the compressive capacity of the member is dictated by the compressive resistance of the cross-section, which is a function of the section classification. Class 1, 2, 3 cross-sections are all unaffected by local buckling and hence the design compression resistance is taken as the design plastic resistance, {ECLINK}EC3 Part 1.1 6.2.4(2){/ECLINK}

{PPT}Lecture13Notprone.pps{/PPT}

{EQN}EQ1.gif{/EQN} (1)

For Class 4 cross-sections, local buckling in one or more elements of the cross-section prevents the attainment of the squash load and thus the design compression resistance is limited to the local buckling resistance, {ECLINK}EC3 Part 1.1 6.2.4(2){/ECLINK}

{PPT}Lecture13Prone.pps{/PPT}

{EQN}EQ2.gif{/EQN}

where {EQN}Aeff.gif{/EQN} is the area of the effective cross-section as determined in accordance with clause 6.2.2.5.

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Axially compressed members.{/TTITLE}

{QUESTION}

{QTITLE}Columns{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Is it common in real structures to have columns subjected only to axial compression?{/QTEXT}

{ANSWER}Yes

{CHECKMARK}0{/CHECKMARK}

{UNCHECKMARK}1{/UNCHECKMARK}

{CHECK}No, it is not common. {/CHECK}

{/ANSWER}

{ANSWER}

No



{UNCHECK}It is not common. {/UNCHECK}

{/ANSWER}

{FEEDBACK}Generally, due to the vertical and horizontal loading, bending moments and shear forces also appear in the members. Only in the case of pin-ended members (e.g. in trusses, in lattice girders or in bracing members) axial compression only can be developed.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Behaviour of stocky columns{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Stocky columns are unaffected by overall buckling and thus the design compression resistance is always taken as the design plastic resistance{/QTEXT}

{ANSWER}Yes


{CHECK}No- It depends on the section classification.{/CHECK}


{UNCHECK}It depends on the section classification.{/UNCHECK}

{/ANSWER}

{ANSWER}No


{CHECK}Correct - it depends on the section classification.{/CHECK}


{UNCHECK}It depends on the section classification.{/UNCHECK}

{/ANSWER}

{FEEDBACK}It depends on the section classification. For Class 4 cross-sections, local buckling prevents the attainment of the squash load and thus the design compression resistance is limited to the local buckling resistance.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Resistance of a stocky column{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Match the formula, which gives the design compression resistance of a stocky column, to the Class of the cross section. {/QTEXT}

{ANSWER}

Class 1, 2 and 3 cross-sections

{MARK}1{/MARK}

{MATCH}

{EQN}EQ1.gif{/EQN}

{/MATCH}

{REASON}Class 1, 2 and 3 sections are able to develop yield stress over the entire section{/REASON}

{/ANSWER}

{ANSWER}

Class 4 cross-sections

{MARK}1{/MARK}

{MATCH}

{EQN}EQ2.gif{/EQN}

{/MATCH}

{REASON}Class 4 sections are unable to develop yield stress due to the possibility of local buckling, and this is accounted for by using a reduced cross-sectional area.{/REASON}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Slender steel columns{/STITLE}

{SUMMARY}

{SUMTITLE}Modes of behaviour{/SUMTITLE}

Slender steel columns exhibit two different types of behaviour. Columns with very high slenderness are dominated by elastic buckling, whilst columns of a medium slenderness are more adversely affected by imperfections.

{PPT}Lecture13Slender.pps{/PPT}

{DETAIL}

Depending on their slenderness, columns exhibit two different types of behaviour: those with high slenderness present a quasi elastic buckling behaviour whereas those of medium slenderness are very sensitive to the effects of imperfections.

The Euler critical load {EQN}Ncr.gif{/EQN} is equal to:


and it is possible to define the Euler critical stress as:


By introducing the radius of gyration, {EQN}iIA.gif{/EQN} and the slenderness, {EQN}EQ6.gif{/EQN} for the relevant buckling mode, Equation (4) becomes:


Plotting the curve of critical stress as a function of the slenderness on a graph (Figure 1), with the horizontal line representing perfect plasticity, {EQN}sigma.gif{/EQN}={EQN}fy.gif{/EQN}, shown, it is interesting to note the idealised zones representing failure by buckling, failure by yielding and safety.

{PPT}Lecture13Bucklingcurve.pps{/PPT}

{IMAGE}Fig1.gif{/IMAGE}

{TIMAGE}Figure 1 Euler buckling curve and modes of failure

A=Failure by yielding, B=Failure by buckling, E=Euler buckling curve{/TIMAGE}

The intersection point P, of the two curves represents the maximum theoretical value of slenderness of a column compressed to the yield strength. This limiting slenderness when the critical stress is equal to the yield strength of the steel is given by: {ECLINK}EC3 Part 1.1 6.3.1.3(1){/ECLINK}

{PPT}Lecture13Limitingslend.pps{/PPT}


where:


Therefore the limiting slenderness is equal to 93,9 for steel grade S235, to 86,8 for steel grade

S275 and to 76,4 for steel grade S355.

Figure 1 may be redrawn in a non-dimensional form, see Figure 2, by dividing the Euler critical stress by the yield strength and the slenderness by the limiting slenderness. This is helpful as the same plot may then be applied to struts of different slenderness and material strengths.

{PPT}Lecture13Nondimenscurve.pps{/PPT}


{TIMAGE}Figure 2 Non-dimensional buckling curve{/TIMAGE}

{PPT}Lecture13Realsteelcol.pps{/PPT}

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Real columns{/SUMTITLE}

Real columns fail at stresses below the theoretical limits due to a range of imperfections.

{DETAIL}The real behaviour of steel columns is rather different from the idealised behaviour described above. Columns generally fail by inelastic buckling before reaching the Euler buckling load due to various imperfections in the "real" element: initial out-of-straightness, residual stresses, eccentricity of axial applied loads and strain-hardening. The imperfections all affect buckling and, therefore, the ultimate strength of the column.

{PPT}Lecture13Effectsofimperf.pps{/PPT}

Compared to the theoretical curves, the real behaviour shows greater differences in the range of medium slenderness than in the range of large slenderness. In the zone of the medium values of slenderness (representing most practical columns), the effect of structural imperfections is significant and must be carefully considered. The greatest reduction in the theoretical value is in the region of limiting slenderness. The lower bound curve is obtained from a statistical analysis of test results and represents the safe limit for loading.

{PPT}Lecture13Imperf-slenderness.pps{/PPT}

A column can be considered slender if its slenderness is larger than that corresponding to the point of inflexion of the lower bound curve, shown in Figure 3. The ultimate failure load for such slender columns is close to the Euler critical load {EQN}Ncr.gif{/EQN} and is thus independent of the yield stress.


{TIMAGE}Figure 3 Real column test results and buckling curves

A=Medium slenderness, B=Large slenderness, I=Point of inflexion{/TIMAGE}

Columns of medium slenderness are those whose behaviour deviates most from Euler's theory. When buckling occurs, some fibres have already reached the yield strength and the ultimate load is not simply a function of slenderness; the more numerous the imperfections, the larger the difference between the actual and theoretical behaviour. Out-of-straightness and residual stresses are the imperfections which have the most significant effect on the behaviour of this kind of column. Residual stresses can be distributed in various ways across the section as shown in Figures 4 and 5. Residual stresses combined with axial stresses cause yielding to occur in the cross-section and the effective area able to resist the axial load is, therefore, reduced.

{PPT}Lecture13Residstresses.pps{/PPT}


{TIMAGE}Figure 4 Residual stress pattern

(a)=Example of residual stresses due to hot rolling (b)=Example of residual stresses due to welding

{/TIMAGE}


{TIMAGE}Figure 5 Combination with axial stresses{/TIMAGE}

An initial out-of-straightness {EQN}eo.gif{/EQN}, produces a bending moment giving a maximum bending stress (see Figure 6a), which when added to the residual stress, gives the stress distribution shown in Figure 6b. If the maximum stress is greater than the yield stress the final distribution will be partly plastic and sections of the member will have yielded in compression, as shown in Figure 6c.

{PPT}Lecture13Initialoutofstraight1.pps{/PPT}


{TIMAGE}Figure 6 Partially yielded compression member

A=Yielded zones{/TIMAGE}

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Axially compressed steel columns{/TTITLE}

{QUESTION}

{QTITLE}The difference between stocky and slender columns.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Is the following statement true or false: Stocky columns have a very low slenderness such that they are not affected by overall buckling of the member. On the other hand slender columns, depending on their slenderness, present either a quasi elastic

buckling behaviour (high slenderness) or they are very sensitive to the effects of imperfections (medium slenderness).{/QTEXT}

{ANSWER}True


{CHECK}Yes, it is true.{/CHECK}


{UNCHECK}No, it is true.{/UNCHECK}

{/ANSWER}

{ANSWER}False


{CHECK}No, it is true.{/CHECK}


{UNCHECK}It is true.{/UNCHECK}

{/ANSWER}

{FEEDBACK}This is clear from Figure 1.




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Regarding the failure in a slender column by buckling.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Can a slender column fail by yielding prior to buckling?{/QTEXT}

{ANSWER}Yes


{CHECK}No, it is not possible.{/CHECK}


{UNCHECK}It is not possible.{/UNCHECK}

{/ANSWER}

{ANSWER}No


{CHECK}It is not possible.{/CHECK}


{UNCHECK}No, it is not possible.{/UNCHECK}

{/ANSWER}

{FEEDBACK}See Figure 1 and check the safety zone.




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}The three zones corresponding to safety, failure by yielding and failure by buckling.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}In Figure 1 does the white area correspond to the safe zone.




{/QTEXT}

{ANSWER}Yes


{CHECK}See Fig. 1{/CHECK}


{UNCHECK} See Fig. 1{/UNCHECK}

{/ANSWER}

{ANSWER}No




{UNCHECK} See Fig. 1{/UNCHECK}

{/ANSWER}

{FEEDBACK}See Figure 1 and Section for slender columns.




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}The physical meaning of limiting slenderness.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}The limiting slenderness corresponds to the intersection between the horizontal line representing perfect plasticity with the Euler's buckling curve.{/QTEXT}

{ANSWER}True


{CHECK}Yes, see Fig. 1.{/CHECK}


{UNCHECK}No, it is true, see Fig. 1.{/UNCHECK}

{/ANSWER}

{ANSWER}False


{CHECK}No, it is true, see Fig. 1.{/CHECK}


{UNCHECK}It is true.{/UNCHECK}

{/ANSWER}

{FEEDBACK}See Figure 1 and Section for slender columns.




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Verification of limiting slenderness values.{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Using {EQN}E.gif{/EQN} = 210 kN/mm2, match the values of the limiting slenderness for steel of design grades S235, S275 and S355. {/QTEXT}

{ANSWER} Steel grade S235

{MARK}1{/MARK}

{MATCH}93,9{/MATCH}

{REASON}See equations 6 and 7{/REASON}

{/ANSWER}

{ANSWER}

Steel grade S275

{MARK}1{/MARK}

{MATCH}86,8 {/MATCH}


{/ANSWER}

{ANSWER}

Steel grade S335

{MARK}1{/MARK}

{MATCH}76,4{/MATCH}


{/ANSWER}

{FEEDBACK}Use equations 6 and 7 with {EQN}fy.gif{/EQN} =235 N/mm2 for S235, {EQN}fy.gif{/EQN} =275 N/mm2 for S275 and {EQN}fy.gif{/EQN} =355 N/mm2 for S355.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Regarding the ideal columns in the praxis.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Do the ideal columns meet in the praxis?{/QTEXT}

{ANSWER}Yes


{CHECK}No, they do not meet in the praxis.{/CHECK}


{UNCHECK}They do not meet in the praxis.{/UNCHECK}

{/ANSWER}

{ANSWER}No


{CHECK}No, they do not meet in the praxis.{/CHECK}


{UNCHECK}No, they do not meet in the praxis.{/UNCHECK}

{/ANSWER}

{FEEDBACK}No, because of the various imperfections that exist in real structures (i.e. initial out-of-straightness, residual stresses, eccentricity of axial applied loads and strain-hardening){/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}The greatest difference between the critical Euler stress and the existing stress at failure due to buckling in a "real" member.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}The greatest difference between the critical Euler stress and the existing stress at failure due to buckling in a "real" member corresponds to the limiting slenderness.{/QTEXT}

{ANSWER}True


{CHECK} Yes, See Fig. 3.{/CHECK}


{UNCHECK}The answer is true.{/UNCHECK}

{/ANSWER}

{ANSWER}False


{CHECK}The answer is true. See Fig. 3.{/CHECK}


{UNCHECK}Yes.{/UNCHECK}

{/ANSWER}

{FEEDBACK}See Fig. 3.




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Magnitude of slenderness.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}The slenderness is considered to be large if it is greater than the limiting slenderness.{/QTEXT}

{ANSWER}Yes


{CHECK}No, see Fig. 3.{/CHECK}


{UNCHECK}The answer is no.{/UNCHECK}

{/ANSWER}

{ANSWER}No


{CHECK}Yes.{/CHECK}


{UNCHECK}The answer is no.{/UNCHECK}

{/ANSWER}

{FEEDBACK}The areas of medium and large slenderness are determined by the point of inflexion at the lower bound curve (see Fig. 3).




{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Initial out-of-straightness in a member.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}If an initial out-of-straightness {EQN}eo.gif{/EQN} exists in a member, then bending moments are produced.{/QTEXT}

{ANSWER}Yes




{UNCHECK}See Fig. 6{/UNCHECK}

{/ANSWER}

{ANSWER}No




{UNCHECK}See Fig. 6{/UNCHECK}

{/ANSWER}

{FEEDBACK}An initial out-of-straightness {EQN}eo.gif{/EQN}, produces a bending moment giving a maximum bending stress (see Figure 6a), which when added to the residual stress, gives the stress distribution shown in Figure 6b. If maximum stress is greater than the yield stress the final distribution will be partly plastic and sections of the member will have yielded in compression, as shown in Figure 6c.


{TIMAGE}Figure 6 Partially yielded compression member A=Yielded zones{/TIMAGE}

{/FEEDBACK}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Non-dimensional slenderness{/STITLE}

{SUMMARY}

{SUMTITLE}Definition{/SUMTITLE}

EC3 defines the non-dimensional slenderness in Part 1.1, 6.3.1.3(1)

{DETAIL}

EC3 defines the non-dimensional slenderness as the following, {ECLINK}EC3 Part 1.1 6.3.1.3(1) {/ECLINK}

{PPT}Lecture13Nondimslender.pps{/PPT}

{EQN}EQ11.gif{/EQN} for class 1, 2 and 3 cross-sections (8)

and {EQN}EQ12.gif{/EQN} for class 4 cross-sections (9)

where {EQN}Ncr.gif{/EQN} is the elastic critical force for the relevant buckling mode based on the gross cross sectional properties

{EQN}lcr.gif{/EQN} is the buckling length in the buckling plane considered

{EQN}i.gif{/EQN} is the radius of gyration about the relevant axis, determined using the properties of the gross cross-section.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}ECCS buckling curves{/STITLE}

{SUMMARY}

{SUMTITLE}Allowance for imperfections{/SUMTITLE}

The ECCS buckling curves are based on the results of more than 1000 tests and are represented in terms of an imperfection factor to account for geometrical out-of-straightness, and residual stresses. These are related to the process by which a column is manufactured – hot rolled, welded, or cold rolled.

{PPT}Lecture13Eurobuclingcurves1.pps{/PPT}

{DETAIL}

The ECCS buckling curves are based on the results of more than 1000 tests on various types of members (I, H, T etc), with different values of slenderness (between 55 and

160). A probabilistic approach, using the experimental strength, associated with a theoretical analysis, allows curves to be drawn describing column strength as a function of the reference slenderness. A half sine-wave geometric imperfection of magnitude equal to 1/1000 of the length of the column and the effect of residual stresses relative to each kind of cross-section are accounted for.

{PPT}Lecture13Assumptions.pps{/PPT}

The ECCS buckling curves (a, b, c or d) are shown in Figure 7. These give the value for the reduction factor of the resistance of the column as a function of the reference slenderness for different kinds of cross-sections (referred to different values of the imperfection factor ).

{PPT}Lecture13ECCS.pps{/PPT}


{TIMAGE}Figure 7 European buckling curves{/TIMAGE}

EC3 expresses the ECCS curves by the mathematical expression for the reduction factor:{ECLINK}EC3 Part 1.1 6.3.1.2(1) {/ECLINK}


where: {EQN}EQ14.gif{/EQN} (11)

The imperfection factor {EQN}alpha.gif{/EQN}depends on the shape of the column cross-section considered, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed); values for the imperfection factor, which increase with the imperfections, are given in Table 1.

{PPT}Lecture13Imperfactor.pps{/PPT}

Buckling curve Imperfection factor a0 0,13 a 0,21 b 0,34 c 0,49 d 0,76

{FIGURE}Table 1 Imperfection factors {ECLINK}EC3 Part 1.1, Table 6.1{/ECLINK}{/FIGURE}

{PPT}Lecture13Selectbucklingcurve.pps{/PPT}

Table 2 helps with the selection of the appropriate buckling curve as a function of the type of cross-section, of its dimensional limits and of the axis about which buckling can occur.

Cross section type

Limits

Rolled sections

{IMAGE}Table 2 Rolled Sections.gif{/IMAGE}

{EQN}hb.gif{/EQN}>1,2 {EQN}tf.gif{/EQN} {EQ40mm

40mm <

{EQN}tf.gif{/EQN} {EQ100mm

{EQN}hb.gif{/EQN}{EQN}LTEQ.gif{/EQN}1,2 {EQN}tf.gif{/EQN} {EQ

100mm {EQN}tf.gif{/EQN} > 1 Welded I-sections

{IMAGE}Table 2 Welded Sections.gif{/IMAGE}

{EQN}tf.gif{/EQN} {EQN}LTEQ.gif{/EQN

{EQN}tf.gif{/EQN} > 4 Hollow sections

{IMAGE}Table 2 Hollow Sections.gif{/IMAGE}

hot finished

cold formed Welded box sections

{IMAGE}Table 2 Welded Box Sections.gif{/IMAGE}

generally (except as belo

thick welds: {EQN}a.gif{/EQN}>0,5{EQN}b.gif{/EQN}/{EQ{EQN}h.gif{/EQN}/ {E<30

U-, T-, and solid sections

{IMAGE}Table 2 U Sections.gif{/IMAGE}

L-sections

{IMAGE}Table 2 L Sections.gif{/IMAGE}

{FIGURE}Table 2 Selection of appropriate buckling curve for a cross section {ECLINK}EC3 Part 1.1, Table 6.2{/ECLINK}{/FIGURE}

{/DETAIL}

{TEST}

{TTITLE}ECCS buckling curves{/TTITLE}

{QUESTION}

{QTITLE}Imperfections of ECCS buckling curves.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}What kinds of imperfections do the ECCS buckling curves account for? (Check only one of the following options){/QTEXT}

{ANSWER}A half sine-wave geometric imperfection of magnitude equal to 1/1000 of the length of the column


{CHECK}ECCS buckling curves do not account only this kind of imperfection {/CHECK}


{UNCHECK}ECCS buckling curves do not account only this kind of imperfection {/UNCHECK}

{/ANSWER}

{ANSWER} The effect of residual stresses.


{CHECK}ECCS buckling curves do not account only this kind of imperfection {/CHECK}


{UNCHECK}ECCS buckling curves do not account only this kind of imperfection {/UNCHECK}

{/ANSWER}

{ANSWER} Both the above


{CHECK} Yes - ECCS buckling curves take into account both A and B imperfections {/CHECK}


{UNCHECK}ECCS buckling curves take into account both A and B imperfections {/UNCHECK}

{/ANSWER}

{FEEDBACK}A half sine-wave geometric imperfection of magnitude equal to 1/1000 of the length of the column and the effect of residual stresses relative to each kind of cross-section are accounted for in ECCS buckling curves.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}The imperfection factor{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Which of the following parameters are accounted for by the imperfection factor? {/QTEXT}

{ANSWER}The shape of the column cross-section


{CHECK}Yes, the imperfection factor accounts for the shape of the column cross-section{/CHECK}


{UNCHECK}The imperfection factor accounts for the shape of the column cross-section, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed).{/UNCHECK}

{/ANSWER}

{ANSWER}The direction in which buckling can occur (y axis or z axis)


{CHECK}Yes, the imperfection factor accounts for the direction in which buckling can occur (y axis or z axis) {/CHECK}



{/ANSWER}

{ANSWER}The fabrication process used on the compression member (hot-rolled, welded or cold-formed).


{CHECK}Yes, the imperfection factor accounts for the fabrication process used on the compression member (hot-rolled, welded or cold-formed).{/CHECK}



{/ANSWER}

{ANSWER}The length of the column


{CHECK} The imperfection factor accounts only for the shape of the column cross-section, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed).{/CHECK}


{UNCHECK} The imperfection factor does not account for the length of the column{/UNCHECK}

{/ANSWER}

{ANSWER}The steel grade


{CHECK} The imperfection factor accounts only for the shape of the column cross-section, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed).{/CHECK}


{UNCHECK} The imperfection factor does not account for the steel grade{/UNCHECK}

{/ANSWER}

{ANSWER}The slenderness of the column


{CHECK}The imperfection factor accounts only for the shape of the column cross-section, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed).{/CHECK}


{UNCHECK} The imperfection factor does not account for the slenderness of the column{/UNCHECK}

{/ANSWER}

{FEEDBACK} The imperfection factor accounts only for the shape of the column cross-section, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed). Although other parameters influence column behaviour they are accounted for by other means.{/FEEDBACK}

{/QUESTION}

{/TEST}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Design of compression members{/STITLE}

{SUMMARY}

{SUMTITLE}Design calculation procedure{/SUMTITLE}

The design steps for compression members involve calculating the effective length (about both cross-sectional axes if they are different). From this the non-dimensional slenderness can be determined for an assumed cross-section. The appropriate imperfection factor is related to the type of member and the thicknesses. This enables the reduction factor to be calculated which leads directly to compression strength.

{PPT}Lecture13Designsteps1.pps{/PPT}

{DETAIL}

To design a simple compression member it is first necessary to evaluate its two effective lengths, in relation to the two principal axes, bearing in mind the expected connections at its end. The verification procedure should then proceed as follows:

• geometric characteristics of the shape, and its yield strength give the reference non-dimensional slenderness.

• the reduction factor is calculated, taking into account the forming process and the shape thickness, using one of the buckling curves and the non-dimensional slenderness.

The design buckling resistance of a compression member is then taken as:

{EQN}EQ15.gif{/EQN} for class 1, 2, 3 cross-sections (12)

and {EQN}EQ16.gif{/EQN} for class 4 cross-sections (13)

If this is higher than the design axial load, the column is acceptable; if not, another larger cross-section must be chosen and checked.

{/DETAIL}

{TEST}

{TTITLE}Design of compression members{/TTITLE}

{QUESTION}

{QTITLE}Meaning of variables in the equation of the design buckling resistance of a compression member.{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Match the variables {EQN}xsi.gif{/EQN} and {EQN}Aeff.gif{/EQN} in the equation of the design buckling resistance of a compression member:

{EQN}EQ15.gif{/EQN} to the descriptions.{/QTEXT}

{ANSWER}

{EQN}xsi.gif{/EQN}

{MARK}1{/MARK}

{MATCH} the reduction factor {/MATCH}

{REASON}{EQN}xsi.gif{/EQN} is the reduction factor{/REASON}

{/ANSWER}

{ANSWER}

{EQN}Aeff.gif{/EQN}

{MARK}1{/MARK}

{MATCH}the effective cross-sectional area{/MATCH}

{REASON}{EQN}Aeff.gif{/EQN} is the effective cross-sectional area{/REASON}

{/ANSWER}

{FEEDBACK}See Section for non-dimensional slenderness{/FEEDBACK}

{/QUESTION}

{PPT}Lecture13Summativetest.pps{/PPT}

{QUESTION}

{QTITLE}Summative test.{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}For a rolled H section (not class 4, {EQN}t.gif{/EQN} < 100, S275) with a slenderness of 130 about the minor axis calculate the reduction factor from EC3, Part 1.1,

equation 6.49 (here equations 10 and 11). Verify the result with that taken from EC3, Part 1.1, Fig 6.4 (here Fig. 7).



{/QTEXT}

{ANSWER} reduction factor equals to 0,315


{CHECK}

{/CHECK}

{/ANSWER}

{FEEDBACK} a. For a rolled H section with {EQN}t.gif{/EQN} < 100, S275 and buckling about the minor axis (z-z), Table 2 leads to buckling curve "c".

b. From Table 1 and for buckling curve "c" the imperfection factor is 0,49.

c. For S275 - {EQN}fy.gif{/EQN} =275 N/mm2 - equation. 7 gives {EQN}epsilon.gif{/EQN} =0,924 and equation 6 gives

{EQN}lamda1.gif{/EQN}=86,76

d. 130/86,76=1,498 (equation 8 for class 1, 2 and 3 cross-sections).

e. From equations 11 and 10, {EQN}phi.gif{/EQN} =1,94 and {EQN}xsi.gif{/EQN} =0,315.

f. From Fig. 7 (or from EC3, Part 1.1, Fig. 6.4), {EQN}xsi.gif{/EQN} =0,315.



{/FEEDBACK}

{/QUESTION}

{/TEST}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Concluding summary{/STITLE}

{SUMMARY}

• A stocky column (with non-dimensional slenderness not greater than 0,2) can achieve the full plastic resistance of the cross-section and buckling does not need to be checked, although the local buckling may reduce the capacity of class 4 sections.

• If the non-dimensional slenderness is greater than 0,2, reduction of the load resistance must be considered because of buckling. Columns with medium slenderness fail by inelastic buckling and slender columns by elastic buckling.

•

European βυχκλινγ χυρϖεσ γιϖε τηε ρεδυχτιον φαχτορ φορ τηε ρελεϖαντ βυχκλινγ µοδε δεπενδινγ ον τηε σηαπε οφ τηε χροσσ−σεχτιον, τηε φορµινγ−προχεσσ, τηε ρεφερενχε σλενδερνεσσ ανδ τηε αξισ αβουτ ωηιχη βυχκλινγ χαν οχχυρ. Τηεψ τακε ιντο αχχουντ εξπεριµενταλ ανδ τηεορετιχαλ αππροαχηεσ ανδ γιϖε ρελιαβλε ρεσυλτσ.

• The design buckling resistance is calculated as a "knocked down" design compression resistance of the cross-section, using the reduction factor for buckling.

{/SUMMARY}

{/SECTION}

{/LECTURE}

nfatec – l13 – columns (27/09/2004)library.tee.gr/digital/m2247/m2247_l13_eng.pdf• structural...

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