91Ó°ÊÓ

{A} bar \(A B C\) of length \(L\) consists of two parts of equal lengths but different diameters. Scgment \(A B\) has diameter \(d_{1}=100 \mathrm{mm},\) and segment \(B C\) has diameter \(d_{2}=60 \mathrm{mm} .\) Both segments have length \(L / 2=0.6 \mathrm{m} . \mathrm{A}\) Iongitudinal hole of diameter \(d\) is drilled through segment \(A B\) for one- half of its length (distance \(L / 4=0.3 \mathrm{m}\) ). The bar is made of plastic having modulus of elasticity \(E=4.0\) GPa. Compressive loads \(P=110 \mathrm{kN}\) act at the ends of the bar. (a) If the shortening of the bar is limited to \(8.0 \mathrm{mm}\) what is the maximum allowable diameter \(d_{\max }\) of the hole? (See figure part a.) (b) Now, if \(d_{\max }\) is instead set at \(d_{2} / 2,\) at what distance \(b\) from end \(C\) should load \(P\) be applied to limit the bar shortening to \(8.0 \mathrm{mm} ?\) (See figure part b.) (c) Finally, if loads \(P\) are applied at the ends and \(d_{\max }=d_{2} / 2,\) what is the permissible length \(x\) of the hole if shortening is to be limited to \(8.0 \mathrm{mm}\) ? (See figure part c.)

Step by step solution

01

Identify Parameters and Equations

Given the bar's properties, identify its lengths, diameters, load, and allowable shortening. Calculate each segment's cross-sectional area: \(A_{AB} = \frac{\pi}{4}(d_1^2-d^2)\) and \(A_{BC} = \frac{\pi}{4}d_2^2\). Use the formula for shortening \(\Delta L = \frac{PL}{EA}\) to inform subsequent calculations.

02

Calculating Shortening in Segment AB with Drilled Hole

Calculate the allowable shortening for segment \(AB\) using \(\Delta L_{AB} = \frac{PL_{AB}}{E A_{AB}}\). Substitute \(L_{AB} = 0.3\, \text{m}\), \(A_{AB} = \frac{\pi}{4}(0.1^2-d_{\max}^2)\), and solve for \(d_{\max}\) such that \(\Delta L_{AB} + \Delta L_{BC} = 8 \times 10^{-3} \text{ m}\).

03

Calculating Stress in Segment BC

Using \(A_{BC} = \frac{\pi}{4}(0.06^2)\), calculate shortening in \(BC\) with \(\Delta L_{BC} = \frac{P L_{BC}}{E A_{BC}}\). Confirm calculations balance with deductions from actual shortening.

04

Determine Maximum Allowable Hole Diameter (a)

For the allowable shortening and given equation, solve for \(d_{\max}\) when \(\frac{110 \times 10^3 \times 0.3}{4 \times 10^9 \times \frac{\pi}{4}(0.1^2-d_{\max}^2)}\) solves to balance with segment facilitations, helping affirm \(d_{\max}\).

05

Load Distance from End C for Limited Shortening (b)

With defined \(d_{\max} = 0.03\, \text{m}\), reapportion balance using new load location formula from remaining bar settings using \(L = 1.2 / 2\) and balance based computations from known conditions.

06

Maximum Hole Length with Set End Loads (c)

Relating \(x\) of the hole by combining proportions of initial shortening using \(L_{AB} - x\) and \(x\) settings; resolving by iterations corresponds the hole capacities in terms of the confirmed allocations bounded by a difference of \(8.0 \text{ mm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compressive Load

A compressive load is a force that tends to squeeze or crush an object, trying to reduce its size. In structural mechanics, understanding compressive loads is crucial, as they impact how materials and structures are designed.
Key points about compressive loads:

• Compressive load acts along the axis of the object, usually opposite in direction, pushing towards the center.
• This type of load can lead to material deformation, such as shortening or buckling, if excessive.
• In the context of structural mechanics, it is essential to calculate how much compressive load a material can handle before failure.

In our exercise with the bar of length, the compressive load applied is 110 kN. This is the force acting at the ends of the bar, causing it to potentially shorten by a specific amount, limited here to 8 mm.

Modulus of Elasticity

The modulus of elasticity, also known as Young's modulus, is a measure of a material's ability to resist deformation under stress. It is a fundamental property in structural mechanics that indicates how much a material will stretch or compress when a load is applied.
Points to consider about modulus of elasticity:

• The modulus of elasticity is expressed in pascals (Pa) or gigapascals (GPa), which shows the stiffness of the material.
• Materials with a high modulus of elasticity, like metals, are stiffer and exhibit less deformation.
• It is used in the equation \[ \Delta L = \frac{PL}{EA} \] to calculate the deformation (\( \Delta L \)), where \( P \) is the load, \( L \) is the original length, \( E \) is the modulus of elasticity, and \( A \) is the cross-sectional area.

For the bar in this exercise, the plastic material has a modulus of elasticity of 4.0 GPa, which informs us how the material will react to the applied load causing potential shortening.

Cross-sectional Area

The cross-sectional area of an object is the area of a specific section of that object sliced perpendicular to a given axis. In structural mechanics, it is crucial in determining how a structure will respond to loads.
Considerations for cross-sectional area:

• A larger cross-sectional area can distribute stress more effectively, reducing the risk of structural failure.
• When the compressive load is applied, the change in length is inversely proportional to the cross-sectional area; more area typically means less deformation.
• The calculation for cross-sectional area of a cylindrical shape is derived from the formula \[ A = \frac{\pi}{4}d^2 \] for the full solid part and \[ A = \frac{\pi}{4}(d_1^2 - d_2^2) \] if hollow, reflecting the need to subtract inner areas when holes are present.

In the problem, different segments of the bar have unique cross-sectional areas due to varying diameters and a longitudinal hole, affecting how the compressive load impacts each segment.

Longitudinal Hole

A longitudinal hole in a structure is a hollow section running along the length, created to reduce weight or allow passage, but it also affects structural integrity.
Factors regarding longitudinal holes:

• Introducing a hole changes the cross-sectional area, reducing the material's ability to absorb compressive loads without deforming.
• Engineering must carefully calculate the balance between functionality and structural stability when determining the size of such holes.
• In our exercise, a longitudinal hole is drilled through a segment of the bar, significantly impacting calculations for permissible stress and shortening. Finding the optimal diameter for this hole ensures the bar doesn't shorten beyond 8 mm.

By understanding how a longitudinal hole interacts with compressive stress, one can optimize design while maintaining the structural integrity of a component.