Video 1
This new module which will deal with compression member. A structure member when subjected to axial force only then such type of structure member is called compression member. Now the different type of compression members is termed as different way like in case of RCC building such type of compression member is called column, which is basically a vertical member which carries the load from beam or from the floor and transferring from upper floor to lower floor. Similarly, in case of steel building this type of compression member is called stanchion, again the compressive member in a roof truss or bracing is called strut. Similarly, the principal compression in a crane is called boom. So what we could see that the same compression member is termed in different names for different cases. Now when we will go for calculation of strength of the compression member or when we will go to design a compression member then we have to see that what are the types of failure may come for a compression member, depending on the type of failure we have to find out what is the strength can be carried by that particular member. So in case of compression member, there are different type of failures occurs out of them one is called squashing. Squashing basically happens when the length of the compression member is quite less compared to its transverse direction cross section area. Say for example if a member is short and under compression load then in such case, the crushing will come intothe picture and full strength will attain at its yield strength and therefore the failure load can be calculated in strength into the cross sectional area. Failure load will be calculated simply the yield strength into its cross sectional area. So in such cases, we could see that the member fails due to crushing of the material that is what one type of failure we could observe. Another failure is the local buckling and it happens due to its cross sectional configuration incase of a steel building. Say, for example, we oftenly used the channel section, so in such case what will happen that due to compression this web may buckle individually, this flange may buckle individually or some other part of the member may buckle which is called local buckling. So it has to be taken care while calculating the compressive strength of the member. Then another type of failure will be that overall flexural buckling. When the member lengthalong its axis is quite high compared to its cross sectional dimension then such type of buckling occurs which is called flexural buckling. Say for example we have a long column means compared to its lateral dimension, so in such cases, it may buckle in this way. Sobefore going to fail due to crushing it may fail due to buckling.So such type of buckling phenomena has to be taken care while considering the strength of the compression member. Again if we see the cross section say for example if the cross section is something like this then we can see it may buckle about this axis, it may buckle about this axis. So we have it will buckle in which direction, it will buckle about the weaker section. So, in this case, it will buckle about this then another failure may come due to torsionalbuckling. Torsional buckling failure occurs due to torsional moment, the member gets twisted about the shear centre in the longitudinal axis. So torsional buckling may occur may be in case of angle section or channel section depending on the type of load means acting in aparticular place means load will be compressive but where it is acting whether it is acting in the member axis or it is Cg depending on that the torsion will come into the picture. Another scope of buckling is called flexural-torsional buckling. This is nothing but the buckling which occurs when the member bends and twists simultaneously, that means member will bend again it will twist means it may twist like this may be like this. Such type of failure happens generally in case of unsymmetrical cross section. So unsymmetrical cross section means say for example channel section it is symmetrical in one direction but it is unsymmetrical in another direction. So in which direction it is unsymmetrical depending on that we have to consider whether it is an unsymmetrical cross section or symmetrical crosssection and accordingly the torsional buckling will come into the picture. Now while calculating the compressive strength of a compression member we have to find out means what type of effect is coming on a particular compressive member. This effect depends in length of the compression member like one is called short compression member, short compression member means what I told earlier that suppose a member is length is quite short compared to its width and thickness means if its width and thickness is substantial with respect to moment that means the L by R ratio is quite low in that case the failure stress will be equal to the yield stress and there will be no buckling, buckling will not happen in this case. So here it will fail due to yielding of the material so this happens when short compression members are means when the members have short compression. And another type of member which is called long compression member. In this case, stress will occur due to buckling due to the long length of the member and quite less cross sectionarea. So in this case buckling may happen before yielding of the stress that is why we need to consider whether it is long compression member or not and accordingly we have to find out what type of means stress is developing due to buckling or due to due to yielding accordingly the compressive strength of the member will be consider. Another case is an intermediate compression member. In case of intermediate compression, member failure occur due to the combined effect of crushing and buckling. Intermediate compression means in practise most of the members are considered as intermediate compression member because in this case, the member will undergo both the stress, one isdue to crushing, due to compression of the member it will shorten its length and it will crushand another is due to its length it will buckle something like this.So buckling stress will come into picture as well as crushing stress will come into picture. So both the effect we have to consider and we have to find out the failure strength of the member and most of the cases the compression member acts as intermediate compression member where both the effects will be will have to be taken care. Now for finding out the compressive strength of a member as we see that one is the crushing value we have to find out the yield strength of a particular material and accordingly the strength of the compression member can be calculated. Another case is it may buckle due to buckling then what will be the buckling force and whatwill be the stress that we have to find out and for that Euler has considered an ideal column and has found out a critical load for buckling that critical load which has been obtained due tobuckling are derived on the basis of this few assumptions. That is one is the material is homogeneous and isotropic, that means the material along the throughout its length will be homogeneous and isotropic, there will be no change of material properties. Then another assumption is material is perfectly elastic that means upto elastic limit this buckling theory will be will be considered and this will be true for upto elastic limit. Then no imperfection that means member will be perfectly straight as its initial state and there will beno flaw of the geometry and material across the member. With these assumptions, Euler has suggested a buckling theory which is given here that is if a compressive load, P is acting along this member and if buckle happens like this then at a distance of x the displacement will be y and the governing differential equation will be That Pcr is the critical load which can be found from this governing differential equation and from the governing differential equation the lowest value can be found as Where l is the effective length and EI is the modulus of rigidity. Now effective length will nothing but the length where the two moment contra flexures are occurring distance between that two like in case of if it is fixed it will buckle like this, soeffective length will be l. Therefore the critical stress can be found as Where A is the cross sectional area of the column Further, Where r is the radius of gyration and λ is nothing but the slenderness ratio.The radius of gyration means the minimum radius of gyration. In two direction radius of gyration will occur and about minimum radius of gyration, it will fail first that is why the minimum radius of gyration will be considered. Thus, the critical stress is inversely proportional to the slenderness ratio. So critical stress will be increasing if the value of λ is less or reversely I can say the critical stress will be less if the slenderness ratio will be more. So the critical stress using Euler buckling theory can be found from this, which will be used for deriving the compressive streng th of the column.
 
Video 2
Now for an ideal strut, the strength curve of a column can be derived if the strut is axially loaded and initially straight with pin-ended then this can be derived in this way where x axis will be the slenderness ratio that is l/r and y axis will be the compressive strength of the of the material. So here we see the path is varying from A to C and then C to B, right. So column fails when the compressive strength is greater than or equal to the values defined by ACB thatmeans this is the path defined and if column stress is coming somewhere here or here thatn means it is failed. So if the column stress is going to be greater than the stress defined by this path ACB then I can say that column is going to fail and this AC is basically failure by yielding and if we consider low slenderness ratio then failure may happen due to yielding and failure may happen due to buckling for high slenderness ratio and you see failure will happen due to buckling if λ is greater than λc. Now plastic yield defined by f c=f y and this is defined by the elastic buckling stress. f c=σcr=f y=250 MPa This constant is for a particular value of E and fy.So what we could see from this curve that if the slenderness ratio value becomes more thanm88.85 then it will fail by elastic buckling and if it is less than that it will fail by plastic yield. Now the same can be written in a non-dimensional form as well, which is shown here where in y axis it is fc/fy and it will be 1 because fc and fy will be equal here and λc will be 1 here means ´ λ will be 1 and in the x direction ´ λ=(σf cr y )1/2 have been plotted. So the non-linear curve will be elastic buckling and the linear curve will be plastic yield. Sothis is how the strength curve for an ideal strut can be developed by Euler’s theory. But this cannot be applied for a practical case because there are certain parameters which willeffect on the compressive strength of the member. Therefore we will consider a differentformula however this formula is based on the Euler buckling theory as well as some otherfactor also has been included so that we will discuss later.Now if we see the factors which are going to effect the strength of compressive member we can see that first is the material property of the member. So the design compressive strength fcd depends on fy.Another factor is the length of the member because we have seen the Euler critical load is inversely proportional to the radius of gyration and there by the length. So if the length is more than definitely compressive load carrying capacity will be less. Then another factor is cross sectional configuration means in case of RCC member there is no problem because there will be no local buckling because generally in case of RCC member either rectangular section, square section, or circular section we use in general. But in case of steel member we use different type of built up section say for example built up section or rolled section, this is a built up section we are using channel facing each other channel face to face or we can make some I section also say I section also we can use. Sohere what we can see that due to cross sectional configuration local buckling of the flange or web may happen so that has to be taken into consideration. Another factor is the support condition because in case of hinge support the effective length (l) will be simply the overall length (L), but if both support is fixed support then its buckling curve will be like this and we know that l will be basically L/2, right. So effective length is going to reduce. So as the length of the member effects on the compressive strength in the member, therefore, supports conditions also effects on that. Next factor is the imperfection. Now imperfection means that material may not be isotropic truly and homogeneous then geometric variation of the column may be there, that means cross sectional cross section throughout the length of the column may not be same exactlythen eccentricity may not have exactly eccentricity. So these imperfections also effect on the strength of the member therefore that has to be also taken care in our design. Another is the residual stress, if residual stresses are there in the member then the compressive strength is going to be different so that aspects also have to keep in mind. Now coming to the cross section of the member if we see that in case of column or compression member different type of steel rolled sections which are available in the market can be used for compression member, like we can use single angle section however while using single angle section if the load act on a leg of a member then the eccentricity will develop and therefore the torsional buckling will come into picture. So we have to take carethe strength of the member accordingly. Similarly for double angle also we can use in this way or T sections can be used for compressive member, most popularly used compressive member is channel section which we oftenly used for compressive member. The hollow circular section also we use, rectangular hollow sections also are used. So these are few steel rolled section which is commonly usedfor compressive membe. Then some built-up sections are also used, like channel face to face this is one type of builtup section we use, then channel back to back this also use. So now suppose channel face to face or back to back if we use say for example this, now we cannot use simple like this because if we see in the elevation it will be something like this, right. So unless we tie then unless we provide some lacing then it will not act as a monolithic. Therefore we have toprovide some joint means in terms of batten plate or some lacing has to be provided, so that throughout the length it acts as a monolithic member. So this has to be taken care. And another is built-up box section means with four plates one can make built-up box section. Then plated I section, then built-up I section like this some of the commonly used built-up compression members are there. Then we will come to the effective length factor, effective length factor means here we see the le as we have written the effective length is equal to K×L, where K is the effective lengthfactor. Now this K depends on the restrained condition of the member, as I have told that in case of say suppose fixed-fixed column it will buckle like as shown in the above figure, so two points of contra-flexure will be developed as shown in the figure, the point of contraflexure means where the moment is becoming 0. Now the distance between two points of contra-flexure becoming as effective length le, right where capital L is the total length of the member, right. So what will be the value of K, so here theoretically we got K value as half, that means le is equal to L/2. However in IS code, this is considered as 0.65 because it will not be perfectly fixed and itwill not be perfectly 0.5, theoretically, though we are getting 0.5 we are going to consider as 0.65 with a certain conservative factor of safety, right. Similarly when columns with both ends are pinned we are considering K value as 1 that means le is equal to L. Again columns with one end fixed and other end pinned in this case theoretical value though it is coming 0.7, in codal provision, it is 0.8.Again column with one end fixed and other end free means like cantilever column theoretical value is 2 also we are considering 2, means in IS code also it is considering 2. Columns partially restrained at each end will be 1, however, it is considered as 1.2. Similarly columns with one end unrestrained and other end rotation partially restrained, it is 2. So when we are going to consider the effective length of a member we have to go to table 11 of IS 800:2007 and as per table 11 of IS 2007, we have to find out the effective length of the member. This snapshot of the code of table 11 of IS 800:2007 has been shown here, the effective length factor has been given here. Now another thing we will discuss here that is the effective length of column in the frame because in practical cases we have to calculate the effective length in a frame and it is not a separate member column member is not a separate or compressive member is not separate, it is inside the frame. So what should be the effective length of the column in a frame that we have to know and that is given in annexure D of the IS 800:2007 in clause 7.2.