Process$-$Oriented Problem $10\mathrm{.}002-$ Euler buckling various ends $-$ DEPENDENT MULTIPART PROBLEM $-$ ASSIGN ALL PARTS

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Euler$$s buckling formula can be expressed as

Pcr $=\backslash$pi $2$EI$($KL$)2$

where Pcr is the critical buckling load, E is the column$$s Young$$s modulus, I is the column$$s moment of inertia, and L is the column$$s length. Derived using a quantity called effective length, the constant K depends upon the column$$s end conditions.

This problem will compare various end conditions of a slender column under compression. The studied column has a length of L $=1$ meters, and its square cross$-$section has a side length of b $=2$ centimeters. The material is a grade of steel with E $=200$ GPa and $\backslash$sigma y $=500$ MPa.

Process$-$Oriented Problem $10\mathrm{.}002.$c$.$one $-$ Fixed$-$Free Equation

As illustrated, the easiest deflected buckling shape shows the column bending laterally: there is no deflection at the fixed end, and the maximum displacement at the free end. This deflected shape is no longer a half$-$sine shape. The effective length is the multiple of the column$$s length that generates a half$-$sine shape. What is the effective length of the fixed$-$free column? $($You must provide an answer before moving to the next part.$)$

Multiple Choice

half of the length, L$/2$

one$-$fourth of the length, L$/4$

the full length, L

twice the length, $2$L