22. Equivalent strut length
Having derived the result for the buckling load of a strut with pinned ends the Euler loads for other end conditions may all be written in the same form, n2El i.e. Pe = — (2.7)
where I is the equivalent length of the strut and can be related to the actual length of the strut depending on the end conditions. The equivalent length is found to be the length of a simple bow (half sine-wave) in each of the strut deflection curves shown in Fig. 2.6. The buckling load for each end condition shown is then readily obtained.
The use of the equivalent length is not restricted to the Euler theory and it will be used in other derivations later.
Pinned - pinned
M"
Fixed-pinned P
Fixed-pinned P
Ends fixed in direction but not in position P
n'El
Fig. 2.6. "Equivalent length" of struts with different end conditions. In each case I is the length of a single bow.
23. Comparison of Euier theory with experimental results (see Fig. 2.7)
Between L/k = 40 and L/k = 100 neither the Euler results nor the yield stress are close to the experimental values, each suggesting a critical load which is in excess of that which is actually required for failure-a very unsafe situation! 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.
(a) Straight-line formula
P = ayA[l - n(L/k)] the value of n depending on the material used and the end condition.
(b) Johnson parabolic formula
the value of b depending also on the end condition.
Neither of the above formulae proved to be very successful, and they were replaced by:
(c) Rankine-Gordon formula
PR Pe Pc where Pe is the Euler buckling load and Pc is the crushing (compressive yield) load = ayA. This formula has been widely used and is discussed fully in §2.5.
2.4. Euler "validity limit"
From the graph of Fig. 2.7 and the comments above, it is evident that the Euler theory is unsafe for small L/k ratios. It is useful, therefore, to determine the limiting value of L/k below which the Euler theory should not be applied; this is termed the validity limit.
Euler curve
Euler curve
Stenderness ratio L/k
Fig. 2.7. Comparison of experimental results with Euler curve.
Stenderness ratio L/k
Fig. 2.7. Comparison of experimental results with Euler curve.
The validity limit is taken to be the point where the Euler oe equals the yield or crushing stress Uy, i.e. the point where the strut load
Now the Euler load can be written in the form n 2 EI ix2EAk2
where C is a constant depending on the end condition of the strut. Therefore in the limiting condition
7T2EAk2
The value of this expression will vary with the type of end condition; as an example, low carbon steel struts with pinned ends give L/k — 80.
2.5. Rankine or Rankine-Gordon formula
As stated above, the Rankine formula is a combination of the Euler and crushing loads for a strut
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