Problem 35 - An Introduction to the Mechanics of Solids in SI Units

Question

A bookshelf is made out of \(6 \mathrm{~mm}\) plate glass. For long-time service, ordinary plate glass cannot safely be stressed to more than about \(7 \mathrm{MN} / \mathrm{m}^{2}\) in tension. If the supports are located in the optimum position, estimate the average weight of books per unit length which can be placed along the shelf.

Step-by-Step Solution

Answer

The average weight of books per unit length which can be placed along the shelf is approximately \(4280 \mathrm{kg/m}\).

Step 1: Understand and Note Down Given Information

The thickness of the glass shelf is \(6 \mathrm{mm}\) or \(0.006 \mathrm{m}\) (as we need values in SI units for calculation). The shelf can handle a maximum stress of \(7 \mathrm{MN/m^2}\) or \(7 \times 10^6 \mathrm{N/m^2}\). We have to find the weight per unit length.

Step 2: Relate Force, Pressure, and Area

The formula to calculate force with pressure and area is \(Force = Pressure \times Area\). The resulting force will give us the total weight the shelf can bear in newtons because weight is essentially a force.

Step 3: Apply the Dimensions of the Shelf

Since we need to find the weight per unit length, the area would be the cross-sectional area of the shelf, which is its thickness multiplied by a unit length: \(Area = Thickness \times 1 \mathrm{m} = 0.006 \mathrm{m^2}\). So, \(Force = 7 \times 10^6 \mathrm{N/m^2} \times 0.006 \mathrm{m^2} = 42000 \mathrm{N}\).

Step 4: Convert the Result to Weight

As weight is the force due to gravity, it can be calculated using the formula \(Weight = Force ÷ Gravity\), where gravity is approximately \(9.81 \mathrm{m/s^2}\). So, \(Weight = 42000 \mathrm{N} ÷ 9.81 \mathrm{m/s^2} = 4280 \mathrm{kg}\). This is the weight the shelf can bear per unit length.

Key Concepts

Plate Glass Strength

Plate glass is a versatile material, often used in construction and design due to its clarity and strength. However, understanding its strength is essential when using it in load-bearing roles, like a shelf. The strength of plate glass is primarily determined by its ability to resist stress without breaking. For ordinary plate glass, the threshold is about 7 MN/m² in tension. This means that if the tensile stress acting on the glass exceeds this value, the glass will likely fracture.

When working with glass, it's also crucial to consider the thickness, as thicker glass can generally sustain more stress. A 6 mm thick plate, as used in our example, strikes a balance between weight, cost, and load-bearing capacity for normal use cases.

Ensuring supports are placed optimally is vital to evenly distribute loads and prevent stress concentration, which could exceed the glass's tensile strength even if the average stress is below the threshold.

Stress Calculations

Stress calculations are foundational in ensuring that a structure can tolerate the forces applied to it without failing. In the case of our plate glass shelf, stress is defined by the force per unit area. This is calculated using the formula: \[\text{Stress} = \frac{\text{Force}}{\text{Area}}\]

From our solution, we understand that the permissible stress for the glass is 7 MN/m². To ensure that we do not exceed this during use, we calculate how much force this glass can withstand without breaking. The force is essentially the load the shelf will bear, which directly relates to how much weight can be placed on it.

The area over which this force is distributed is the cross-sectional area of the glass. For a shelf, this is the product of its thickness (in this case, 6 mm or 0.006 m) and a unit length of the shelf. The same principle that guarantees the glass's durability can be used to ensure the safety of a glass-made structure.

Load Bearing Capacity

The load-bearing capacity is a critical factor for any structural element, detailing how much load it can handle before failure. For a plate glass shelf, this capacity is influenced by the stress and the strength of the material, as discussed. The specific calculation shows how engineering principles are applied to ascertain this practical limit.

Calculating the load bearing capacity involes multiplying the maximum permissible stress by the area over which this stress acts. This can be quantified as: \[\text{Force} = \text{Stress} \times \text{Area}\]

For a practical application, as solved, the plate glass can bear up to 42000 N, translating into 4280 kg weight per meter length after considering the gravitational effect.

• First, note the maximum stress level the material can handle (7 MN/m²).
• Calculate the force based on the cross-sectional area.
• Finally, convert this force into weight using gravitational constants.

These steps ensure that the materials, such as glass, are used safely, considering their nature and strengths, thus enhancing the reliability of glass as a structural component.