DANotes: Buckling: Tutorial problems


Buckling problems



In the following problems, buckling mode interaction is not considered and the material is steel unless otherwise stated.


1 ex 01 Show that the post-buckling paths for the perfect mechanism are :   p = 2P/kL = cosθ.
Plot the   p-θ paths, and ascertain whether equilibrium is stable, neutral or unstable.
If the mechanism is imperfect, with an initial out- of- straightness θo when the springs are free, show that the maximum load capacity is :   p* = [ 1 - ( sinθo )2/3 ]3/2
Plot   p* versus   θo , and also the equilibrium paths for some representative   θo .
 
2 ex 02 The links of the mechanism are connected by a torsion spring of stiffness   k (Nm/rad) whose strain energy is   1 /2 2   where   Δ is the spring's angular deflection. The mechanism may embody either one of two imperfections :
( i) The load is not coaxial with the mechanism's axis, but is offset by the moment arm :   e = ε L   measured perpendicular to the upper link.
( ii) The mechanism is not straight initially, the links being inclined at   θo - in which configuration the spring is undeformed.
Plot the equilibrium paths for the perfect mechanism and
-for the imperfect mechanism ( i) with   θo = 0 and   ε = 0, 0.05, 0.1, 0.2, 0.5
-for the imperfect mechanism (ii) with   ε   = 0 and   θo = 0, 2.5, 5, 10, 20o
Comment on the paths' shapes. Show that the mechanism is always stable.
 
3 ex 03 The mechanism is inclined initially at angle   α, when the spring is free.
Application of the load   P causes displacement through angle   θ.
Show that the equilibrium path is
      p   =   P/2kL   =   sinφ ( 1 - cosα / cosφ )     where       φ = α - θ
            and plot this for   α = 22.5o   in the range   -10o ≤ θ ≤ 50o.
Prove that the local maximum/minimum loads are   p* = [ 1 - ( cosα )2/3 ]3/2   and state clearly what happens when   P increases from zero.
4 A sub-sea pipeline, of nominal size   100 mm and   400 MPa yield strength, lies at   200 m.
Using a safety factor of   2, select a suitable pipe wall thickness from AS 1835 to withstand buckling.
A corrosion allowance of   1 mm is applicable both inside and outside the pipe. [ 6.3 mm ]
 
5
( a) A straight steel rod,   20 mm diameter and   200 mm long, is fixed at one end. The free end is subjected to a compressive load offset from the rod's centreline by   1 mm.
What load is permissible if the maximum stress is limited to   200MPa ?   [ 37.1 kN]
( b) Repeat ( a) but with rod diameter halved.   [ 4.2 kN]
( c) Repeat ( a) but with rod length doubled.   [ 19.6 kN]
( d) Repeat ( a) but with load eccentricity doubled.   [ 28.6 kN]

6 A column is characterised by   S = 200 MPa and   σc = 150 MPa.
Determine the maximum permissible compressive stress predicted by each of the three imperfect column models - equations ( h) - for   η = 0.1 and for   η = 1.0.
Comment on the significance of the form and magnitude of the imperfections demonstrated by these values.   [ 121.7, 124.6, 125.0;   66.7, 69.3, 69.7 MPa]
 
7 A   1 m long pin-ended column is made from a   250 MPa yield,   10 mm thick, 80x80 mm equal angle section. Assuming an equivalent eccentricity of   L/1000, determine the maximum allowable load with a safety factor of   2.   [ 153 kN]
Repeat this if the column is nominally perfect with   ση = 60 MPa.   [ 156 kN]
 
8
( a) The critical Euler stress of a column is   σc and the eccentricity ratio is   η. It is made from a brittle material with compressive and tensile strengths   Sc and St respectively.
Prove the following algorithm from which the direct compressive stress   σd may be calculated :
      If         1/( Sc - St ) - η/( Sc + St )   <   1/( 2 σc )
      then     σ2d + σd [ St - ( 1 - η ) σc ] - St σc   =   0
      else       σ2d - σd [ Sc + ( 1 + η ) σc ] + Sc σc   =   0
      Hint : Apply equation ( g) to the convex side where bending stresses are tensile
( b) A   400 mm long end-fixed column is made from a cast iron whose compressive strength is   765 MPa. The cross-section is square,   20x20 mm, and due to fears of casting inaccuracies, an eccentricity ratio of   1 is assumed for estimating purposes.
What maximum load can the column withstand on this basis ?   [ 85kN]

9 Trial-and-error methods must be employed when applying equation ( 3) to the design of ductile columns. Input constraints necessary for the design process include the column equivalent length,   L, the actual load,   F, and desired safety factor,   n.
A range of sections is chosen - of one material whose known properties include the elastic modulus,   E, and yield strength, S.   A value for the imperfection stress,   ση , also results from this choice.
The trial-and-error process commences by selecting one particular section from the range, of area   A, and minimum second area moment   I. Equation ( 3) enables calculation of the corresponding safety factor. This process is repeated until the safety factor is just larger than the desired value.
( a)   Prepare a calculation sequence using consistent units to assist in the above task.
( b)   A 2.4 m long, nominally straight pin-ended column has to support a   300 kN load with a safety factor of   2. A range of   250 MPa yield steel sections is available for which the imperfection stress,   ση = 60 MPa.
What size of section should be used if the sections are :-
  1. geometrically similar, with I/A2 = 0.15 ?   [ 4200 mm2]
  2. tubes with an outside diameter-to-thickness ratio of 15 ?   [ 120 mm diameter]
  3. standard channels whose properties are listed in the references ?   [ 229x89 mm]
     
10 ex 10 The sketch illustrates the right half of an ideal pin-ended column which is subjected to a transverse load   Q at its centre, in addition to the compressive load   P.
Using the nomenclature of the preceding notes, show that the elastic curve is :
v/L   =   ( Q/P ) [ sin( π√ψ.φ )/( π√ψ .cos( π/2 √ψ )) - φ ]
Hence derive the design equation for this loading, analogous to equation ( 2) above :
      σd   =   S - σQ.tan( π/2 √ψ )/( π/2 √ψ )
            where   σd = nF/A   and   σQ = 1/2Q L ymax /I   is the maximum bending stress due to the transverse load   Q acting alone.
Plot   σd versus slenderness ratio for some representative values of the parameter   σQ - say 0, 5, 20, 50, 100, 200 MPa - and compare the behaviour with that of imperfect columns with eccentricity ratio as the parameter.



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