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Module 1: Compression Members

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Video 1
Hello, now I am going to discuss about the design strength of compression member, in lastclass I have discussed different factors effecting compressive strength in a compressionmember. So from the earlier discussions we have seen the main three parameters which areeffecting on the on the compressive strength of a member, one is the material strength of themember that means what is the yield strength of the member depending on that thecompressive strength will vary.So while calculating the compressive strength or while developing the expression forcompressive strength we need to consider the material strength that is yield strength.Next factor is the slenderness ratio, we have seen from the Euler’s critical load and Euler’sbuckling formula that the compressive strength varies inversely with the slenderness ratio. Sowhile developing the compressive strength formula means the expression for compressivestrength we need to incorporate the slenderness ratio in that also, so this has to be taken care.Another aspects which are of the importance we have to give that is the local bucklingbecause one is crushing, another is overall buckling, and another is local buckling. Sobecause of the configuration of the member cross section of the member the local buckling ofthe flange or web may happen. So that aspects has to also be taken care in the derivation ofthe expression for compressive strength.So these three aspects has to be considered and we have to develop a reasonable accurateformula and that formula should be capable of reflecting the actual behaviour.
So let us see how people have started developing formula and what we are adapting. So fourdifferent approaches have been considered for finding the design column formula. So one isthe formula based on the maximum strength, this is one approach in which people have tried.Another is formula based on the yield limit, which is called Perry-Robertson formula andbasically this approach is considered by our Indian Code the IS 800:2007 has also adapted themultiple column curves based on the Perry Robertson formula and this is basically similar tothe British code BS 5950 (part-1) 2000.This the formula which have been derived is similar to the British code and the formula wasprescribed by Perry-Robertson who has proposed. So this has been adapted but we willdiscuss about this in details and another two formula are also adapted to establish columndesign formula that is formula based on tangent modulus theory and Empirical formula suchas Merchant-Rankine formula. So these four basic approach are observed to establish columndesign formula and we may recall the earlier code that is IS 800:1984, which was establishedas per Merchant-Rankine formula.So before going to this new version we will shortly discuss about the Merchant-Rankineformula, how the earlier code was adapted and then we will come to this formula.
So in IS 800:1984 Merchant-Rankine formula was adapted where we know the basic formulais as follows:
Where, fe is the elastic stress and fy is the yield stress. So considering two stress, one is elasticstress and yield stress the equivalent stress f can be related as this formula.
Here we can see that two things has been taken care, one is the material property which is theyield strength as fy and another is the fcc that is elastic critical stress which comes frombuckling.So global buckling and the squashing effect has been taken into consideration but anotherimportant parameter which has not been considered is the geometrical configuration of thecross section of the member.So before crushing or global buckling due to local buckling the member may fail from whichit is not possible to capture this formula.
Therefore this formula we are no longer using, we are using the new developed formulawhich is based on Perry-Robertson formula.
According to Perry-Robertson formula the multiple design curve has been adapted by the IScode, in IS code figure 8 these column buckling curves are given and this is based on thePerry-Robertson theory. According to Perry-Robertson theory three things has beenconsidered as I told one is the material strength fy, another is the elastic critical buckling fccand another is the local buckling and this local buckling has been considered in terms of thisclass a, b, c, d. We can see here four type of graphs have been proposed depending on thebuckling class and in table 7 this buckling class have been defined. We can see here thataccording to buckling class a, b, c, d that imperfection factor α has been introduced and this αvalue is 0.21 in case of class a, 0.34 in class of b, 0.49 in case of class c and 0.76 in case ofclass d.So according to buckling class the imperfection factor has been proposed and according tothat imperfection factor the column buckling curve for class a, b, c, d have been derived. Hereyou can see that along X direction it is ´ λ=√ffcr y , non-dimensional and along y directionfcd/fy where fcd is the design compressive stress and fy is the yield stress and here fy is the yieldstress and fcr is the elastic critical stress, right.So we could see here that based on Perry-Robertson theory we have considered three meanswe have taken into consideration three effects one is due to material fy, another is due toglobal buckling and other one is the new component which has been adapted in this code is the class of the structure means buckling class a, b, c, d so according to buckling class thishas been changed, right.
Similarly for weld section weld I section, if tf is less than equal to 40, then buckling class willbe b about z-z axis, c about y-y axis and if tf is greater than equal to 40, this will be c andabout y-y will be d. Similarly for Hollow section for hot rolled about any axis it will be a andfor cold rolled cold formed section it will be b.
Video 2
Now coming to weld box section, here also buckling class has been defined. Similarly I amnot going into details, we can find out in the code, in table 10. The only thing I want tomention that when we are using channel section, angle section, T section or some solidsection, solid section means rectangular section or circular section then the buckling classwill be c about any direction about any direction, say for angle about y-y axis, about z-z axisit will be c for both the direction.Similarly for channel section also it will be c, for T section also it will be c and for built-upmember any direction the buckling class has also been considered as c.
So these are the thing we need to find out from table 10 of IS 800, for calculating theimperfection factor α. Now this imperfection factor will be used in the formula given in theclause 7.1.2 of IS 800:2007, where it is mentioned that P should be less than Pd where P is theexternal compressive load and Pd is the design compressive load, so design compressive loadshould be greater than the external load. The design compressive load Pd can be found fromthe following formula:
Pd=Ae f cdWhere Ae is the effective sectional area effective sectional area which is defined in clause7.3.2.Unlike tension member in case of compression member effective sectional area will be thegross area in general. If generally we do not deduct the whole area because considering thebolt or rivet whatever we provide are intact with the member. So the gross area will becomethe effective sectional area. And fcd is the design compressive stress of axially loadedcompressive member. So we have to find out fcd the design compressive stress.Now design compressive stress has been given in the code which is written as
χ is basically stress reduction factor which depends on the radius of gyration, length andimperfection factor that means basically the slenderness ratio and imperfection factor, right.So fcd value we can find out where phi can be calculated from the following formula,
Here K is the length factor to find out basically KL means the effective length and r is theradius of gyration, fy is the yield strength of the member and E is the modulus of elasticity ofthe material.So from this I can find out the value of λ, right and once I find out the value of λ, I can findout the value of ϕ with the help of imperfection factor α. So once we can find out ϕ ,we can find out the value of fcd, right. So this is how one can find out the design compressivestrength of a member.
The strength of the member will be considered with respect to minimum radius of gyration.Then α is the imperfection factor as given in table 7 of IS 800:2007 and this χ is the stressfactor for different buckling class, slenderness ratio and yield stress, right. So three thingshave been taken into consider buckling class, slenderness ratio and yield stress.
Now this is a TDS process to find out the value of fcd, therefore in the IS code some valuesare given in a tabular form so that we do not have to calculate the entire things again andagain.
So if we see the table 8a to table 8d of IS 800:2007 this is given for different yield stress f ywith different λ, say 10, 20, 30 like this, the value say this is 250, so different type of valuesare given then the reduction factor value are given for different value of λ and fy, right. Soreduction factors are given. So say for example if we have λ in between 20 and 30 and fy as250 then we can interpolate between these two to find out a particular value of reductionfactor with respect to particular value of slenderness ratio.
Say for example if λ is equal to 22 we can interpolate between these two to find out the χvalue with respect to 22, right. So this tables 8a to 8d one can find out the value of reductionfactor according to the buckling class a, b, c or d we can choose table 8a or 8b or 8c or 8d andwe can find out the value.
Similarly we can find out the design compressive stress in table 9 (a to d) for variousbuckling class.
So in table 9 (a to d) we can see that for different value of λ and for different value of fy yieldstress, we can find out the value of fcd. Say for example say 10, 20, 30 like these λs arevarying and fy is say 200, 250, 300 like this it is varying. So for a particular value of λ we canfind out the value of fy and then from the table 9 a, b, c or d according to the buckling classwe can find out the value of fcd, say for example this is x1, this is x2 for 10 and 20.Then for λ is equal to say 12, we can find out the value of fcd in between x1 and x2 means inbetween x1 and x2 we can find out the value of fcd by interpolating the values, right. So veryquickly one can find out the fcd value by using table 9, table 9 have four tables a, b, c, d withrespect to buckling class a, b, c, d.
Now another thing we have to remember while designing the compressive member that is theallowable slenderness ratio. So that has been defined in table 3 of IS 800:2007. The allowableslenderness ratio for compressive member that is like carrying loads resulting from dead loadand superimposed load the limiting value is 180 that means the effective slenderness ratio KLby r should not become more than 180 in case of dead load and live load or superimposedload.When carrying loads resulting from wind and seismic loads provided the deformation of sucha member does not adversely affect the stress in any part of the structure we can increase upto250, so allowable slenderness ratio is 250 for such type of cases.Now normally acting as a tie in a roof truss for a bracing member which is not consideredeffective when subjected to reversal of stress resulting from acting wind or earthquake forcesthat can be made as 350, so allowable slenderness ratio will become 350.
Similarly lacing bars in columns will be 145, when the built-up sections will be used then thelacing bars has to be designed, so in such cases the slenderness ratio will should not be morethan 145.And different elements or components in built-up section there it should be less than 50, sodifferent components in built-up section the allowable slenderness ratio should not exceeds50. So this is how we have to follow this codal provision while designing the member, right.So in today’s lecture what we could see that design column curve can be obtained from thePerry-Robertson formula which takes care the material properties, material strength whichtakes the slenderness ratio that means buckling and also the buckling class of the member thatmeans the configuration of the structure cross sections which effects as a local buckling.So to take care these three effects the yielding, global buckling, and local buckling theformula has been derived, right and it is observed from the literature that using such type offormula the behaviour of the compression members are similar to this formula whatever weare getting that means after numerous experimental verifications this formula has beenadapted by our code and we can now use this formula for finding out the design strength ofthe compressive member and we can design accordingly.So in next class we will see how to calculate the design strength of a compression member,we will see how to calculate the design strength of a built-up compression member and thenwe will go for design of compression members. So for today’s lecture we will we will like toconclude here, thanks.