91Ó°ÊÓ

A rectangular cross-section refers to the shape of the bar's surface when it is sliced perpendicular to its length. This shape is crucial in determining how the material will handle stress and strain.
The dimensions of a rectangular cross-section are its width and thickness.

• In this exercise, the bar maintains a constant thickness (\(t\)) throughout its length.
• However, its width varies due to the linear taper, ranging from \(b_1 = 100 \, \mathrm{mm} \) to \(b_2 = 150 \, \mathrm{mm}\).

Managing these dimensions is essential because they directly influence the stress distribution across the material under load. A wider section can withstand more force without yielding compared to a narrower part.
Understanding rectangular cross-sections is fundamental to evaluating the mechanical properties of materials and ensuring their structural integrity.

Young's modulus is a material property that measures its stiffness. It is defined as the ratio of stress to strain in the elastic region of a stress-strain curve.
Mathematically, it is expressed as:
\[E = \frac{\sigma}{\varepsilon} \]
Where:

• \(\sigma\) is the stress (force per unit area) applied to the material.
• \(\varepsilon\) is the strain (deformation per unit length) experienced by the material.

In our problem, Young's modulus for the material is given as \(200 \, \text{GPa}\) (GigaPascals). This value indicates how much the material will deform under a given load.Understanding Young's modulus is essential for predicting how materials will behave under different stresses.
A higher Young's modulus means the material is less likely to deform significantly, maintaining its shape better under tension.

Linear taper refers to how the width of the bar gradually changes from one end to the other. In this exercise, the taper is described as linear, implying that the change in width is gradual and occurs evenly along the length of the bar.
This is expressed in the equation:
\[ b(x) = b_1 + \frac{b_2 - b_1}{L} x \]
Here:

• \(b(x) \) is the width at a distance \(x\) from the smaller end.
• \(L\) is the total length of the bar.

The assumption of a small angle of taper ensures that the variation is slight, simplifying the calculations.
This gradual change in width affects how the stress is distributed throughout the material, where sections with varying widths will experience different stress levels.
Understanding linear taper is important for this problem because it directly affects how elongation is calculated and how evenly the stress is distributed along the bar.

The stress-strain relationship is a fundamental concept in material science, crucial for understanding how materials deform under various loads.

When a material is subjected to tension, it experiences stress, which in turn causes strain:

• **Stress** \(\sigma\): Force applied per unit area.
• **Strain** \(\varepsilon\): The deformation or elongation per unit length.

The relationship between stress and strain is typically linear in the elastic region, represented by the equation:
\[ \varepsilon = \frac{\sigma}{E} \]
Where \(E\) is Young's modulus.
In this exercise, we calculated the elongation of the bar, understanding that different widths across the bar mean varying stress and hence strain at each point.
Integrating the expression for strain along the length of the bar gives the total elongation:
\[ \delta = \int_0^L \varepsilon \, dx \]
This integration helps in determining how much the bar will extend when subjected to the tensile force \(P\).
Understanding this relationship is key to predicting material behavior and solving problems involving changes in shape due to applied forces.