Unit4 B Failure of Columns Eulers Theory and Rankines Theory PDF

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Failure of Columns Eulers Theory and Rankines Theory...

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TOS2:UNIT4B FAILUREOF COLUMNS:EULER’S ANDRANKINE’S THEORY 1. To Understand the Failure of Axially Loaded Columns 2. To Understand Euler’s Theory and Short and Long Columns 3. To Understand the Limitations of Euler’s Theory 4. To Understand Effective Lengths for various End Conditions 5. To Understand Slenderness Ratio (λ) and its value for Buckling for various Materials 6. To Understand Rankine’s Theory.

1. Euler’s Theory 2.Rankines’s Theory

54

TOS 2: Unit4B: Failure of Columns; Euler’s Theory and Rankine’s Theory 4.B.1 Introduction: A column is a vertical member subjected to either axial loading or eccentric loading.

An axially loaded column may undergo two different kinds of deformations depending upon the height of the column; it’s cross sectional area and the load to which it is subjected. These deformations are as follows 1. The column may get compressed, eventually leading to crushing of the material as the load increases beyond a certain point. 2. The column may bend outwardly leading to a condition called buckling and finally may fail due to this buckling. To get a clear picture of this take a full chalk and then break a small piece of the same. Apply a pressure on this small piece of chalk with your closed fist. Upon application of a certain pressure, the material of the chalk just crumbles or gets crushed under the pressure. Apply the same pressure on a full piece of chalk. It is very likely that the piece of chalk breaks simply by snapping in the middle. The chalk material being brittle snaps before it bends. Imagine a load being applied to a walking stick. The stick starts bending outside and upon application of a certain load is likely to snap across the middle. This load at which it starts bending outward is called the Buckling or Crippling Load. However now imagine that the cross section of the walking stick is much more i.e. it has a greater diameter. Would such a stick bend outwardly?

Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta

55 Hence we can safely conclude that the failure of an axial compression member is dependent of three variables 1. Axial Compressive load applied 2. Cross section of the member 3. Height of the member/ (Effective length – end conditions, to be discussed later) Accordingly a column is classified as follows 1. Short column – one which fails by crushing. 2. Long column – one which fails by buckling.

4.B.2 Euler’s Theory 4B.2.A.Assumptions in Euler’s Theory: 1. 2. 3. 4. 5.

The material of the column is Isotropic and Homogenous The cross section is uniform throughout the length of the column. The load is placed axially and the column is straight when the load is placed. The column is long and is going to fail by buckling only The stresses are within the elastic limit.

4B.2.B. PE = Euler’s Load The load calculated by Euler’s theory is called as Crippling Load or Critical Load or  

 = () Where  =  Or Euler’s Load or Buckling Load E = Young’s Modulus of the Material of the Column I = Moment of Inertia Minimum = bd³/12 for rectangular sections and = πD⁴/64 for circular sections Le = Effective Length of the Column Member

4.B.3 Limitations Of Euler’s Theory

 =

  () 

If K is radius of Gyration (the distance at which the area of a plane lamina can be safely assumed to be concentrated so that the Moment of Inertia of the area about its own c.g. is the same as the double moment of area about that point. So =  

 =

    ()

Same as

 



= () 

  

The term Le/K is known as the slenderness ratio and is denoted by λ. λ = Le/K The term PE/A is the stress in the body i.e. Force/Area. As the value of stress in steel is limited to 250N/mm², the value of slenderness ratio for steel from the above equation limits to 88.84 or 90 Substitute PE/A = 250N/mm², E = 2 x 10⁵N/mm² λ = 88.84=90.

Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta

56 Thus we can see that the value of λ for steel is limited to 90, below which the column will fail due to crushing and above which it will fail due to buckling. Thus Euler’s theory is limited to long columns. This Limitation of Euler’s Theory is overcome by another theory known as Rankine’s Theory and is discussed later on in this chapter. Greater the Slenderness Ratio Lesser the Buckling Load and Lesser the Slenderness Ratio Greater the Buckling Load We shall discuss the relationship between Buckling Load, Slenderness Ratio and Material after the next discussion

4.B.4 Effective Length of a Column Member It would be good for us to study the term Effective length and its relation to the actual length with the help of the following different end conditions for Columns. 1. Both ends Fixed  = /2 2. One end Fixed and one end Hinged

 =

 √

3. Both ends Hinged  = 4. One end Fixed and One end Free  = 2 These conditions and their implications can be seen in the diagrams given below.

4.B.5 Slenderness Ratio and Material: As explained earlier the ratio of the effective length of a column to the least radius of gyration of its cross section is called the slenderness ratio (sometimes expressed with the Greek letter lambda, λ). This ratio affords a means of classifying columns. Slenderness ratio is important for design considerations. All the following are approximate values used for convenience. 1. A short steel column is one whose slenderness ratio does not exceed 50; an intermediate length steel column has a slenderness ratio ranging from about 50 to 200, and are dominated by the strength limit of the material, while a long steel column may be assumed to have a slenderness ratio greater than 200 and its behavior is dominated by the modulus of elasticity of the material. 2. A short concrete column is one having a ratio of unsupported length to least dimension of the cross section equal to or less than 12. If the ratio is greater than 12, it is considered a long column (sometimes referred to as a slender column). 3. Timber columns may be classified as short columns if the ratio of the length to least dimension of the cross section is equal to or less than 10. The dividing line between intermediate and long timber columns cannot be readily evaluated Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta

57 In practice the ideal conditions are never [i.e. the strut is initially straight and the end load being applied axially through centroid] reached. There is always some eccentricity and initial curvature present. It is realized that, due to the above mentioned imperfections the strut will suffer a deflection which increases with load and consequently a bending moment is introduced which causes failure before the Euler's load is reached. In fact failure is by stress rather than by buckling and the deviation from the Euler value is more marked as the slenderness-ratio l/k is reduced. For values of l/k < 120 approximately, the error in applying the Euler theory is too great to allow of its use. The following is a graphical representation of the above discussion. Allowing for the imperfections of loading of strut, actual values at failure must lie within and below line CBD. Other formulae have therefore been derived to attempt to obtain closer agreement between the actual failing load and the predicted value in this particular range of slenderness ratio I.e. l/k=40 to l/k=100.

Problem 1. A steel rod 5.5m long and having a diameter of 4.5cm is used as a column with one end fixed and the other free. Determine crippling load if E=2 X 10⁵ N/mm².  

Solution:  = () π= 3.14 E= 2 X 10⁵ N/mm² Imin = πD⁴/64 = 3.14 X (45)⁴/64 = 201186.91mm⁴(D = 4.5 cm = 45mm) Le = 2L = 2 X 5500 = 11000mm as L= 5.5 m = 5500mm (One end Fixed and the other free) Hence Substituting

PE =

  

=

$.&'²))&* + )*&&,-..&

= 3278N = 3.278kN. () &&***² Answer: The Buckling Load as per Euler’s Theory is 3.278kN In all these problems it is a good idea to write down all the necessary terms in Newtons and mm and then substitute the same.

Problem 2. A steel column fixed at one end and free at the other end has Ixx = 39210.8cm⁴ and Iyy = 2985.2cm⁴. It is required to carry a safe load of 240kN. , with factor of safety of 2.5. Calculate maximum height using Euler’s Formula. E=2 X 10⁵ N/mm². Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta

58

Solution: 1. When we have Ixx not equal to Iyy we choose the lower I as with this I the PE will be lower and will be the limiting buckling load in both directions. Hence I = Imin = Iyy = 2985.2 cm⁴ = 2985.2 x 10⁴mm⁴ Safe Load =

/0123425 62/047489280:

Hence Safe Load = Actual Load x F.O.S = 240 X 2.5 = 600 X 10³N

Hence PE = 600000N and E = 2 X 10⁵N/mm² PE =

  () 

600000 =

$.&'²;;&*+;.,$,- ''&C>$,-

= 4171431.52 N = 4171kN. Answer: The failure Load is 4171kN

As Load coming on Column = 4000kN < 4171kN which it can bear the column is safe.

{Note: The Column is a Short column having an effective height of 1.5m. Here the Buckling Load of 73862kN is about 16.7 times the Crushing load of 4421kN. The Rankine’s Load of 4171kN is a number closer to the Crushing Load and smaller than the Crushing Load }

Problem 7. A Hollow Square Column has a cross-section of 400mm x 400mm with 10mm wall thickness. Length of the Column is 4m, with one fixed and the other hinged. Crushing Stress = 320N/mm². E = 2 X 10⁵N/mm².

Solution: Cross-sectional Area = 400 x 400 – 380 x 380 = 15600mm² Ixx = Iyy = Imin = 400 x 400³/12 – 380 x 380³/12= 39572 x 10⁴mm⁴ Le = 4000/√2 = 2828mm (One End Fixed One End Hinged) Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta

60

1. PC = σc. A = 320 X 15600 = 4992000N = 4992kN  

D.EFGHGHEIJHDOJLGPEI⁴ = 97570.49kN GKGK² '..Q.>*.'. = 4749.03kN. Answer: The failure '..C.>*.'.

2. PE = () = 3. P =

Load is 4749.03kN

{Note: The Column is a Short column having an effective height of 2.828m. Hence the Buckling Load of 97570.49kN is about 19 times the Crushing load of 4992kN. The Rankine’s Load of 4749.49kN is a number closer to the Crushing Load and smaller than the Crushing Load. }

Problem 8. A Concrete Column of size 230 x 400 is used as a column with both ends hinged of height 10.0m. If E = 0.20 X 10⁵N/mm². Find Failure Load by Rankine's Theory. Crushing Stress in Concrete is 25N/mm² Solution: E = 0.20 X 10⁵N/mm², σc = 25N/mm² Cross- Sectional Area = 230 x 400 = 92000mm²

Imin = 400 x 230³/12 = 405.56 x 10⁶mm⁴ Le = 10.0m = 10000mm (Le=L as both ends are hinged.) 1. PC = σc. A = 25 X 92000 = 2300000N = 2300kN.  

D.EFG HI.GIHEIJ HFIJ.JRMEI⁶ EIIII² $**Q>...>$

2. PE = () = 3. P =

$**C>...>$

= 799731N = 799.73kN

= 593.41kN

{Note: The Column is a Long column having an effective height of 10.0m. Hence the Crushing Load of 2300kN is about 2.87 times the Buckling load of 799.73kN. The Rankine’s Load is a number closer to the Buckling Load and smaller than the Buckling Load. } My Way of simplifying calculations: If the Column is a Short Column PE will be large compared to PC. Let PE = y x PC So P =

@A.@B

or P =

@AC@B T@AQ@A

Hence P =

@A(UC&)

=

@AQTQ@A @ACTQ@A V VC&

x PC where y = PE/PC (ratio of larger load to smaller load)

Hence in the last problem after finding PC = 2300kN and PE = 799.3kN y = PC/PE = 2300/799.73 = 2.875 Hence y/(y+1) = 2.875/3.875 = 0.742 P = 0.74 x PE = 0.742 x 799.73 = 593.54kN. Hence the P by Rankine’s Theory is a number smaller than (to the ratio of y/y+1) than the smaller of the two loads between PE and PC.

Suggested Theory Questions: Unit 4B I. II. III. IV. V. VI. VII.

What are Short and Long Columns? What are the assumptions of Euler’s Theory of Buckling Write down Euler’s Formula and explain each term Explain the Limitation of Euler’s Theory. Draw Various End Conditions of Column and show their effective lengths Explain Slenderness Ratio and its relationship to Buckling Write down Rankine’s Formula for Failure Load of Columns and Explain how it over comes the Limitations of Euler’s Theory

Unit 4B Failure of Columns; Euler's Theory and Rankine's Theory

Compiled by Ar. Arthur Cutinho @ Er. Sujata Mehta...