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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

Figure 3.1. A Simple Compression Member

P

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

The ball in part (c) is Unstablebecause 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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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).

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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 (bf/2) and t is equal to tf.

Hence, bf/2tf ratio is used to find λ.

Dr. Zahid Ahmad Siddiqi

For stiffened web, h is the width of web and tw is the thickness of web and the corresponding value of λ or b/t ratio is h/tw, which controls web local buckling.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

Single angle sections may buckle about their weak axis (z-axis shown in Design Aids and Figure 3.10).

Calculate Le / rzto 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.

Dr. Zahid Ahmad Siddiqi

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

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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/rand hence there are three parameters affecting buckling.

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

Dr. Zahid Ahmad Siddiqi

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 KLu/r.

- Radius of gyration (r), which may be rx 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:

Dr. Zahid Ahmad Siddiqi

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 rz 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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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/rshould be calculated by using Ky, unbraced length along weak axis and ry.

Dr. Zahid Ahmad Siddiqi

Bracing can also be provided to prevent buckling along the strong axis. KL/rin this case should be calculated by using Kx, the unbraced length along strong axis and rx.

- 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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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 sideswayis between 0.5 and 1.0, whereas, in case of appreciable sidesway, it is always greater than or equal to 1.0

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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 Frame

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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 land is subjected to an axial compressive load, P.

Buckling of the column occurs at a critical compressive load, Pcr.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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)

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

The smallest value of Pcris for n = 1, and is given below:

Dr. Zahid Ahmad Siddiqi

For other columns with different end conditions, we have to replace l by the effective length, le = Kl.

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

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

Kl/r = slenderness ratio

Pcr = 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.

Dr. Zahid Ahmad Siddiqi

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.

Dr. Zahid Ahmad Siddiqi

D

Fy

Inelastic Buckling (straight line or a parabolic line

Is assumed

C

Euler’s Buckling (Elastic Buckling)

Fcr

B

Elastic Buckling

0.4 Fy

approximately

A

Rc

KL/r (R)

Intermediate

Columns

Long

Columns

Short

Columns

(KL/r)max

Dr. Zahid Ahmad Siddiqi

Thank You for giving Attention

Dr. Zahid Ahmad Siddiqi

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