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Chapter 03 A Chapter 03 A

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Dr Zahid Ahmad Siddiqi 1 Design of Compression Members Dr Zahid Ahmad Siddiqi 2 INTRODUCTION When a load tends to squeeze or shorten a member the stresses produced are said to be compressive in nature and the member is called a ID: 529556

ahmad zahid buckling siddiqi zahid ahmad siddiqi buckling column length load columns members ratio compression member stresses figure loads

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Slide1

Chapter 03 A

Dr. Zahid Ahmad Siddiqi

1

Design of Compression MembersSlide2

Dr. Zahid Ahmad Siddiqi

2

INTRODUCTION

When a load tends to squeeze or shorten a member, the stresses produced are said to be compressive in nature and the member is called a

compression member

.

P

P

Figure 3.1. A Simple Compression MemberSlide3

Dr. Zahid Ahmad Siddiqi

3

Examples are struts (short compression members without chances of buckling), eccentrically loaded columns, top chords of trusses, bracing members, compression flanges of beams and members that are subjected simultaneously to bending and compressive loads.

The term

column

is usually used for straight vertical member whose length is considerably greater than the cross-sectional dimensions. Slide4

Dr. Zahid Ahmad Siddiqi

4

Short vertical members subjected to compressive loads are often called

struts

or simply compression members.

There are two significant differences between the

behaviour of tension and compression members, explained as under:

There are no chances of buckling in tension members, whereas the strength of a compression member most dominantly depends on buckling phenomenon.Slide5

Dr. Zahid Ahmad Siddiqi

5

The tensile loads tend to hold a member straight even if the member is not initially in one line and is subjected to simultaneous bending moments.

In contrast, the compressive loads tend to bend the member out of the plane of the loads due to imperfections, simultaneous bending moment or even without all these.Slide6

Dr. Zahid Ahmad Siddiqi

6

Tests on majority of practical columns show that they will fail at axial stresses well below the elastic limit of the column material because of their tendency to buckle.

For these reasons, the strength of compression members is reduced in relation to the danger of buckling depending on length of column, end conditions and cross-sectional dimensions.Slide7

Dr. Zahid Ahmad Siddiqi

7

The longer a column becomes for the same cross-section the greater is its tendency to buckle and the smaller is the load it will support.

When the length of a compression member increases relative to its cross-section, it may buckle at a lower load.

After buckling the load cannot be sustained and the load capacity nearly approaches zero. Slide8

Dr. Zahid Ahmad Siddiqi

8

The condition of a column at its critical buckling load is that of an unstable equilibrium as shown in Figure 3.2.Slide9

Dr. Zahid Ahmad Siddiqi

9

The three possible states of equilibrium are shown in the same figure.

Referring to part (a) of Figure 3.2, if the ball is given movement and released, it comes back to the original position showing a

Stable Equilibrium

.

If ball is displaced and released in part (b), it retains its new position but do not return to its original position. This condition is called

Neutral Equilibrium. Slide10

Dr. Zahid Ahmad Siddiqi

10

The ball in part (c) is Unstable because if the ball is displaced and released it do not return back to its original position and do not retain its new position.

In the first case, the restoring forces are greater than the forces tending to upset the system.

Due to an infinitesimal small displacement consistent with the boundary conditions or due to small imperfection of a column, a moment is produced in a column trying to bend it. Slide11

Dr. Zahid Ahmad Siddiqi

11

At the same time, due to stress in the material, restoring forces are also developed to bring the column back to its original shape.

If restoring force is greater than the upsetting moment, the system is stable but if restoring force is lesser than the upsetting moment, the system is unstable.

Right at the transition point when restoring force is exactly equal to the upsetting moment, we get neutral equilibrium. Slide12

Dr. Zahid Ahmad Siddiqi

12

The force associated with this condition is the critical or buckling load.

Returning back to the behaviour of a compression member, relatively rigid end conditions of the member, not allowing the member to rotate freely at these points, reduce the effect of length up to certain extent making the load carrying capacity a little improved. Slide13

Dr. Zahid Ahmad Siddiqi

13

Other factors, such as the eccentricity of load application, imperfection of column material, initial crookedness of columns, erection stresses and residual stresses from manufacture, help to buckle the column at a lesser load.Slide14

Dr. Zahid Ahmad Siddiqi

14

The presence of rivet or bolt holes in tension members reduces the area available for resisting loads; but in compression members the rivets or bolts are assumed to fill the holes and the entire gross area is available for resisting load.

The ideal type of load on a column is a concentric load and the member subjected to this type of load is called

concentrically loaded column

. Slide15

Dr. Zahid Ahmad Siddiqi

15

The load is distributed uniformly over the entire cross-section with the centre of gravity of the loads coinciding with the centre of gravity of the columns.

Due to load patterns, the live load on slabs and beams may not be concentrically transferred to interior columns. Slide16

Dr. Zahid Ahmad Siddiqi

16

Similarly, the dead and live loads transferred to the exterior columns are, generally, having large eccentricities, as the centre of gravity of the loads will usually fall well on the inner side of the column.

In practice, majority of the columns are

eccentrically loaded compression members

.Slide17

Dr. Zahid Ahmad Siddiqi

17

Slight initial crookedness, eccentricity of loads, and application of simultaneous transverse loads produce significant bending moments as the product of high axial loads (P) multiplied with the eccentricity,

e

.

This moment,

P x e, facilitates buckling and reduces the load carrying capacity.

Eccentricity, e, may be relatively smaller, but the product

(P x e) may be significantly larger. This effect is shown in the Figure 3.3.Slide18

Dr. Zahid Ahmad Siddiqi

18Slide19

Dr. Zahid Ahmad Siddiqi

19

The AISC Code of Standard Practice specifies an acceptable upper limit on the out-of-plumbness

and

initial crookedness

equal to the length of the member divided by 500.

Stub column is defined as a short compression test specimen that is long enough to allow strain measurements but short enough to avoid elastic and plastic buckling.Slide20

Dr. Zahid Ahmad Siddiqi

20

RESIDUAL STRESSES

Residual stresses are stresses that remain in a member after it has been formed into a finished product.

These are always present in a member even without the application of loads.

The magnitudes of these stresses are considerably high and, in some cases, are comparable to the yield stresses (refer to Figure 3.4). Slide21

Dr. Zahid Ahmad Siddiqi

21

The causes of presence of residual stresses are as under:

Uneven cooling

which occurs after hot rolling of structural shapes produces thermal stresses, which are permanently stored in members.

The thicker parts cool at the end, and try to shorten in length. Slide22

Dr. Zahid Ahmad Siddiqi

22

While doing so they produce compressive stresses in the other parts of the section and tension in them.

Overall magnitude of this tension and compression remain equal for equilibrium.

In I-shape sections, after hot rolling, the thick junction of flange to web cools more slowly than the web and flange tips. Slide23

Dr. Zahid Ahmad Siddiqi

23

Consequently, compressive residual stress exists at flange tips and at mid-depth of the web (the regions that cool fastest), while

tensile residual stress

exists in the flange and the web at the regions where they join.

Slide24

Dr. Zahid Ahmad Siddiqi

24

2.

Cold

bending of members

beyond their elastic limit produce residual stresses and strains within the members.

Similarly, during fabrication, if some member having extra length is forced to fit between other members, stresses are produced in the associated members.

3.

Punching of holes and cutting operations during fabrication also produce residual stresses.Slide25

Dr. Zahid Ahmad Siddiqi

25

Welding

also produces the stresses due to uneven cooling after welding.

Welded part will cool at the end inviting other parts to contract with it.

This produces compressive stresses in parts away from welds and tensile stresses in parts closer to welds.Slide26

Dr. Zahid Ahmad Siddiqi

26Slide27

Dr. Zahid Ahmad Siddiqi

27Slide28

Dr. Zahid Ahmad Siddiqi

28

SECTIONS USED FOR COLUMNS

Single angle, double angle, tee, channel, W-section, pipe, square tubing, and rectangular tubing may be used as columns.

Different combinations of these structural shapes may also be employed for compression members to get built-up sections as shown in Figure 3.5. Slide29

Dr. Zahid Ahmad Siddiqi

29Slide30

Dr. Zahid Ahmad Siddiqi

30

Built-up sections are better for columns because the slenderness ratios in various directions can be controlled to get equal values in all the directions.

This makes the column economical as far as the material cost is concerned .

However the joining and

labour

cost is generally higher for built-up sections. Slide31

Dr. Zahid Ahmad Siddiqi

31

The total cost of these sections may become less for greater lengths.

The joining of various elements of a built-up section is usually performed by using

lacing

.

LIMITING SLENDERNESS RATIO

The slenderness ratio of compression members should preferably not exceed 200

.Slide32

Dr. Zahid Ahmad Siddiqi

32

INSTABILITY OF COLUMNS

When buckling occurs in columns, we say that columns have become unstable. The instability may be due to

local

or

overall buckling

.Slide33

Dr. Zahid Ahmad Siddiqi

33

Local Instability

During local instability, the individual parts or plate elements of cross-section buckle without overall buckling of the column.

Width/thickness ratio

of each part gives the slenderness ratio

(

λ

=b/t), which controls the local buckling.

Slide34

Dr. Zahid Ahmad Siddiqi

34

Local buckling should never be allowed to occur before the overall buckling of the member except in few cases like web of a plate girder.

An

Un-stiffened Element

is a projecting piece with one free edge parallel to the direction of the compressive force.

The example is half flange

AB in figure 3.6.Slide35

Dr. Zahid Ahmad Siddiqi

35Slide36

Dr. Zahid Ahmad Siddiqi

36

A Stiffened Element is supported along the two edges parallel to the direction of the force.

The example is web

AC

in the same figure.

For un-stiffened flange of figure,

b is equal to half width of flange (b

f/2) and

t is equal to tf

.

Hence, bf/2t

f ratio is used to find λ.

Slide37

Dr. Zahid Ahmad Siddiqi

37

For stiffened web, h is the width of web and

t

w

is the thickness of web and the corresponding value of

λ

or b/t ratio is

h/tw

, which controls web local buckling. Slide38

Dr. Zahid Ahmad Siddiqi

38Slide39

Dr. Zahid Ahmad Siddiqi

39Slide40

Dr. Zahid Ahmad Siddiqi

40Slide41

Dr. Zahid Ahmad Siddiqi

41

Overall Instability

In case of overall instability, the column

buckles as a whole

between the supports or the braces about an axis whose corresponding slenderness ratio is bigger as shown in Figures 3.7 to 3.9.Slide42

Dr. Zahid Ahmad Siddiqi

42Slide43

Dr. Zahid Ahmad Siddiqi

43Slide44

Dr. Zahid Ahmad Siddiqi

44Slide45

Dr. Zahid Ahmad Siddiqi

45

Note:

Single

angle sections may buckle about their weak axis (

z-axis

shown in Design Aids and Figure 3.10).

Calculate

Le / r

z to check the slenderness ratio.

In general, all un-symmetric sections having non-zero product moment of inertia (

Ixy) have

a weak axis different from the y-axis.Slide46

Dr. Zahid Ahmad Siddiqi

46Slide47

Dr. Zahid Ahmad Siddiqi

47

Unsupported Length

It is the length of column between two consecutive supports or braces denoted by

L

ux

or

L

uy in the x & y directions, respectively.

A different value of unsupported length may exist in different directions and must be used to calculate the corresponding slenderness ratios. Slide48

Dr. Zahid Ahmad Siddiqi

48

To calculate unsupported length of a column in a particular direction, only the corresponding supports and braces are to be considered neglecting the bracing preventing buckling in the other direction.Slide49

Dr. Zahid Ahmad Siddiqi

49

Effective Length Of Column

The length of the column corresponding to

one-half sine wave of the buckled shape

or

the length between two consecutive inflection points or supports

after buckling is called the effective length.Slide50

Dr. Zahid Ahmad Siddiqi

50

BUCKLING OF STEEL COLUMNS

Buckling is the sudden lateral bending produced by axial loads due to initial imperfection, out-of-straightness, initial curvature, or bending produced by simultaneous bending moments.

Chances of buckling are directly related with the slenderness ratio

KL/r

and hence there are three parameters affecting buckling.

Effective length factor (K), which depends on the end conditions of the column.Slide51

Dr. Zahid Ahmad Siddiqi

51

Unbraced length of column (

Lu), which may be the unbraced length in strong direction or unbraced length in weak direction, whichever gives more answer for

KL

u

/r

.

Radius of gyration (

r), which may be r

x or ry

(strong and weak direction) for uniaxially or biaxially symmetrical cross-sections and least radius of gyration (rz) for un-symmetrical cross-sections like angle sections.

Following points should be remembered to find the critical slenderness ratio:Slide52

Dr. Zahid Ahmad Siddiqi

52

Buckling will take place about a direction for which the corresponding slenderness ratio is maximum.

For unbraced compression members consisting of angle section, the total length and

r

z

are used in the calculation of

KL/r

ratio.

For steel braces, bracing is considered the most effective if tension is produced in them, due to buckling.Slide53

Dr. Zahid Ahmad Siddiqi

53

Braces that provide resistance by bending are less effective and braces having compression are almost ineffective because of their small x-sections and longer lengths.

The brace is considered effective if its other end is connected to a stable structure, which is not undergoing buckling simultaneously with the braced member.Slide54

Dr. Zahid Ahmad Siddiqi

54

The braces are usually provided inclined to main members of steel structures starting from mid-spans to ends of the adjacent columns.

Because bracing is most effective in tension, it is usually provided on both sides to prevent buckling on either side.

Bracing can be provided to prevent buckling along weak axis.

KL/r

should be calculated by using

K

y, unbraced length along weak axis and r

y. Slide55

Dr. Zahid Ahmad Siddiqi

55Slide56

Dr. Zahid Ahmad Siddiqi

56Slide57

Dr. Zahid Ahmad Siddiqi

57Slide58

Dr. Zahid Ahmad Siddiqi

58

Bracing can also be provided to prevent buckling along the strong axis.

KL/r in this case should be calculated by using

K

x

, the unbraced length along strong axis and

r

x.

The end condition of a particular unsupported length of a column at an intermediate brace is considered a hinge. The reason is that the rotation becomes free at this point and only the lateral movement is prevented.Slide59

Dr. Zahid Ahmad Siddiqi

59

EFFECTIVE LENGTH FACTOR (K)

This factor gives the

ratio of length of half sine wave of deflected shape after buckling to full-unsupported length

of column.

In other words, it is the

ratio of effective length to the unsupported length.

This depends upon the end conditions of the column and the fact that whether

sidesway is permitted or not.Slide60

Dr. Zahid Ahmad Siddiqi

60

Greater the K-value, greater is the effective length and slenderness ratio and hence smaller is the buckling load.

K

-

value in case of

no sidesway is between

0.5 and 1.0, whereas, in case of appreciable

sidesway, it is always greater than or equal to 1.0Slide61

Dr. Zahid Ahmad Siddiqi

61

Sidesway

Any appreciable lateral or sideward movement of top of a vertical column relative to its bottom is called

sidesway

,

sway

or lateral drift.

If sidesway is possible, K-value increases by a greater degree and column buckles at a lesser load.

Sidesway in a frame takes place due to:

Lengths of different columns are unequal.Slide62

Dr. Zahid Ahmad Siddiqi

62

Sections of columns have different cross-sectional properties.

Loads are un-symmetrical.

Lateral loads are acting.

I

I

2I

Figure 3.11. Causes of Sidesway in a Building FrameSlide63

Dr. Zahid Ahmad Siddiqi

63

Sidesway may be prevented in a frame by:

Providing shear or partition walls.

Fixing the top of frame with adjoining rigid structures.

Provision of properly designed lift well or shear walls in a building, which may act like backbone of the structure reducing the lateral deflections.

Shear wall is a structural wall that resist shear forces resulting from the applied transverse loads in its own plane and it produces frame stability.Slide64

Dr. Zahid Ahmad Siddiqi

64

Provision of lateral bracing, which may be of following two types:

Diagonal bracing, and

Longitudinal bracing

Unbraced Frame:

It is defined as the one in which the resistance to lateral load is provided by the bending resistance of frame members and their connections without any additional bracing.Slide65

Dr. Zahid Ahmad Siddiqi

65

K-Factor for Columns having Well Defined End Conditions

Effective length factor and the buckled shape of some example cases are given in Figure 3.12.

Design Aids may be used for other end conditions. Slide66

Dr. Zahid Ahmad Siddiqi

66Slide67

Dr. Zahid Ahmad Siddiqi

67Slide68

Dr. Zahid Ahmad Siddiqi

68

Make correctionSlide69

Dr. Zahid Ahmad Siddiqi

69Slide70

Dr. Zahid Ahmad Siddiqi

70Slide71

Dr. Zahid Ahmad Siddiqi

71Slide72

Dr. Zahid Ahmad Siddiqi

72

K-Factor for Frame or Partially Restrained Columns

Consider the example of column

AB

shown in Figure 3.13. The ends are not free to rotate and are also not perfectly fixed.

Instead these ends are

partially fixed with the fixity determined by the ratio of relative flexural stiffness of columns meeting at a joint to the flexural stiffness of beams meeting at that joint.

Slide73

Dr. Zahid Ahmad Siddiqi

73

This ratio is denoted by

G or and is determined for each end of the column by using the expression given below:

Slide74

Dr. Zahid Ahmad Siddiqi

74Slide75

Dr. Zahid Ahmad Siddiqi

75

Alignment charts, given in Design Aids, are then used to find the effective length factors.

The method to use these charts is explained in Figure 3.14. (This Figure does not give the actual values).

First step is to select the alignment chart depending upon the presence or absence of the sidesway.

Next, points are marked on two outer lines for values of

G

or at end A and

B of the column, according to the provided scale. Slide76

Dr. Zahid Ahmad Siddiqi

76Slide77

Dr. Zahid Ahmad Siddiqi

77

These points are then joined by a straight edge and the K-value is read from the central line according to its graduations.

K-Value for Truss & Braced Frames Members

The effective length factor,

K

, is considered equal to 1.0 for members of the truss & braced frames columns. In case the value is to be used less than one for frame columns, detailed buckling analysis is required to be carried out and bracing is to be designed accordingly.Slide78

Dr. Zahid Ahmad Siddiqi

78

ELASTIC BUCKLING LOAD FOR LONG COLUMNS

A column with pin connections on both ends is considered for the basic derivation, as shown in the Figure 3.15.

The column has a length equal to

l

and is subjected to an axial compressive load, P.

Buckling of the column occurs at a critical compressive load, P

cr.Slide79

Dr.

Zahid Ahmad Siddiqi

79Slide80

Dr. Zahid Ahmad Siddiqi

80

The lateral displacement for the buckled position at a height

y from the base is u.

The bending moment at this point

D

is

M = Pcr

x u (1)

This bending moment is function of the deflection unlike the double integration method of structural analysis where it is independent of deflection.

The equation of the elastic curve is given by the Euler-Bernoulli Equation, which is the same as that for a beam.Slide81

Dr. Zahid Ahmad Siddiqi

81Slide82

Dr. Zahid Ahmad Siddiqi

82

The solution of this differential equation is:

u = A cos (C x y) + B sin (C x y) (VI)

where, A and B are the constants of integration.

Boundary Condition No. 1:

At

y = 0, u = 0

0 = A cos (0°) + B sin (0°)

A = 0 u = B sin (C x y) (VII)Slide83

Dr. Zahid Ahmad Siddiqi

83

Boundary Condition No. 2: At

y =

l

, u = 0

From Eq. (VII): 0 = B sin (Cl)

Either B = 0 or sin (Cl) = 0 (VIII)

If B = 0,

the equation becomes u = 0, giving un-deflected condition. Only the second alternate is left for the buckled shape. Slide84

Dr. Zahid Ahmad Siddiqi

84

Hence from Eq. IX:

The smallest value of

P

cr

is for

n = 1,

and is given below: Slide85

Dr. Zahid Ahmad Siddiqi

85

For other columns with different end conditions, we have to replace

l

by the effective length,

l

e =

Kl.

The same expression may be converted in terms of area of cross-section and radius of gyration using the expression I=Ar2.Slide86

Dr. Zahid Ahmad Siddiqi

86

Equations

XII

and

XIV

give the Euler elastic critical buckling load for long columns. It is important to note that the buckling load determined from Euler equation is independent of the strength of steel used.Slide87

Dr. Zahid Ahmad Siddiqi

87

The most important factor on which this load depends is the

Kl/r

term called the

slenderness ratio

.

Euler critical buckling load is inversely proportional to the square of the slenderness ratio.

With the increase in slenderness ratio, the buckling strength of a column drastically reduces.Slide88

Dr. Zahid Ahmad Siddiqi

88

In the above Equations:

K

l

/r

=

slenderness ratio P

cr = Euler’s critical elastic buckling load

Fe

= Euler’s elastic critical buckling stress

Long compression members fail by elastic buckling and short compression members may be loaded until the material yield or perhaps even goes into the strain-hardening range.Slide89

Dr. Zahid Ahmad Siddiqi

89

However, in the vast majority of usual situations

failure occurs by buckling after a portion of cross-section has yielded.

This is known as inelastic buckling.

This variation in column behavior with change of slenderness ratio is shown in Figure 3.16.Slide90

Dr. Zahid Ahmad Siddiqi

90

F

cr

KL/r (R)

F

y

C

D

B

A

R

c

Compression Yielding

Inelastic Buckling (straight line or a parabolic line

Is assumed

Euler’s Buckling (Elastic Buckling)

Elastic Buckling

0.4

F

y

approximately

Short

Columns

Intermediate

Columns

Long

Columns

(KL/r)

max

Slide91

Dr. Zahid Ahmad Siddiqi

91

Thank You for giving Attention