The observation cage C has a weight of 250 kip and through a...

The observation cage $C$ has a weight of 250 kip and through a system of gears, travels upward at constant velocity along the $A-36$ steel column, which has a height of 200 ft. The column has an outer diameter of $3 \mathrm{ft}$ and is made from steel plate having a thickness of 0.25 in. Neglect the weight of the column, and determine the average normal stress in the column at its base, $B,$ as a function of the cage's position $y$ Also, determine the displacement of end $A$ as a function of $y$

Key Concepts

Axial Normal Stress

Axial normal stress is the compressive (or tensile) force per unit cross?sectional area in a structural member. In situations where a load is applied at one end (or partially along the member), the stress at any cross?section is found by dividing the total force transmitted by the area through which it acts. Here, this concept is used to relate the weight of the cage (i.e. the applied compressive force) to the normal stress at the column’s base as the cage changes position.

Axial Deformation

Axial deformation refers to the change in length of a member when it is subjected to an axial load. This deformation is given by the relationship ? = (F·L)/(E·A), where F is the load, L the length over which the load is applied, E the modulus of elasticity of the material, and A the cross?sectional area. In problems where only a portion of the member is loaded (such as when a moving load is applied at a variable position), the effective length L in the formula changes accordingly, affecting the displacement of the unloaded or load?exit point.

Thin?Walled Cylinder Cross?Section

For a cylinder made of thin-walled material, the cross-sectional area can be approximated by the product of the circumference and the wall thickness, rather than using the full area of a solid circle. This concept is used to determine the area over which the load is distributed in a column with a large outer diameter and comparatively small thickness, and is essential for calculating both the normal stress and the axial deformation.

Load Position Dependency

In structures with moving loads or loads that are applied only to a part of the member, the response (both stress and displacement) depends on the load’s position. When the cage ascends the column, only the portion of the column below its current position carries the entire load, while the portion above remains unloaded. This variable loading scenario means that the normal stress at the base and the overall axial deformation of the column must be expressed as functions of the cage’s position along the column.

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