Initial Conditions

This problem involves both temperature and mechanical loads. Furthermore, the ceiling will restrict the overall deflection of the pole as the temperature increases. The ceiling restriction makes this an indeterminate problem to the first degree (one redundant support or restraint). As such, in addition to the standard equilibrium equations, one compatibility relationship is needed to solve this problem.

Compatibility Relationship

When all loads, including thermal, are applied, the structure will have a set deflection, which can be used to define the compatibility condition.

Using the concept of superposition, each type of deflection can be isolated. The first deflection, is due to the two shelf loads, F₁ and F₂, labeled δ_F. These will compress the pole.

Next, the temperature change will cause the pole to increase by δ_T. This deflection assumes there is no ceiling. But there is a ceiling that will restrict the deflection and cause a compression force and deflection, δ_R. The total deflection cannot exceed the initial gap of 1 mm. Thus, the compatibility relationship is

     -δ_F + δ_T - δ_R = 1 mm

Before finding the pole deflections, the cross-sectional area of the pipe needs to be calculated.

     A = π (2.5² - 2.25²) cm²
        = 3.731 cm² = 0.0003731 m²

Load Deflection Using Superposition

Load Deflection
The two loads, F₁ and F₂, each cause the pole to decrease in length. The deflection from each load can be added together using the principle of superposition, giving the total deflection from the applied loads as

      δ_F = δ_F1 + δ_F2

     
       δ_F = 0.0004842 m = 0.4842 mm

Thermal Deflection
The expansion of the pole due to the temperature change (assuming no ceiling) is

      δ_F = L α ΔT
       = (2.4 m - 0.001 m) (23 × 10^-6/^oC) (50^oC - 20^oC)
       = 0.0016553 m = 1.6553 mm

This total thermal deflection may not be possible due to the ceiling. If it is too great, the ceiling will induce a reaction force into the pole that must be considered. This is calculated in the next paragraph.

Ceiling Reaction Deflection
It is assumed that the pole will expand enough for the pole to press against the ceiling. This will cause a reaction force on the pole that will compress the pole. The total compression load is not known but is related to the deflection,

        = 8.8081 × 10^-8 R₂ m = 0.000088081 R₂ mm

Now that all the deflections have been established, they can be substituted into the compatibility relationship,

     -δ_F + δ_T - δ_R = 1 mm

     -0.4842 + 1.6553 - 0.000088081 R₂ = 1.0

     R₂ = 1,943 N

The reaction is positive, thus the initial assumption that the ceiling will cause compression was correct.

The floor reaction can be found by using the static equilibrium equation,

      ΣF_y = 0
     R₁ - F₁ - F₂ - R₂ = 0
     R₁ = (-5.494 - 5.494 - 1.943) kN

     R₁ = 12.93 kN