Question

Applications. The maximum deflection in inches of a certain cantilever beam, with a concentrated load applied \(r\) feet from the fixed end, is a function of \(r\) $$ f(r)=0.000030 r^{2}(80-r) \quad \text { in. } $$ Find the deflections \(f(10)\) and \(f(15)\)

Short answer

The deflections are \(f(10) = 0.21\) inches and \(f(15) = 0.4725\) inches.

Step-by-step solution

Identify the Function for Deflection

The function given for the deflection of a cantilever beam is: \[f(r) = 0.000030 r^2 (80 - r)\] in inches, where \(r\) is the distance in feet from the fixed end where the load is applied.

Calculate the Deflection for r=10 feet

Substitute \(r = 10\) feet into the deflection function: \[f(10) = 0.000030 \times 10^2 \times (80 - 10)\] Calculate the result to find the deflection in inches.

Calculate the Deflection for r=15 feet

Substitute \(r = 15\) feet into the deflection function: \[f(15) = 0.000030 \times 15^2 \times (80 - 15)\] Calculate the result to find the deflection in inches.

Key concepts

Structural Engineering Mathematics

At the core of structural engineering, mathematics is vital for solving real-world problems. In the context of a cantilever beam deflection, engineers use mathematical functions to predict how much a beam will bend under a specific load. This is critical for ensuring the safety and integrity of structures. The deflection function is derived from principles of mechanics and material science, considering properties such as elasticity, moment of inertia, and load distribution.

For example, the given function for deflection, \(f(r) = 0.000030 r^2 (80 - r)\), uses a polynomial expression that quantifies deflection based on the distance \(r\), from the fixed end. In this scenario, the coefficient \(0.000030\) reflects factors that are unique to the beam's material and cross-sectional geometry. The quadratic term \(r^2\) emphasizes that deflection changes more intensely with distance, while the linear term \((80 - r)\) compensates for the decreasing force influence as the load gets closer to the free end.

Beam Deflection Calculation

Beam deflection calculations are essential in predicting the performance of beams under various loads. The equation given for our cantilever beam represents how the deflection at any point varies based on the length of the beam and the position of the load. Simple algebraic substitution is used to calculate specific deflections, such as \(f(10)\) and \(f(15)\) in the given problem.

Engineers must ensure that these deflections do not exceed permissible limits to maintain structural integrity and functionality. Moreover, knowing the deflection can help in optimizing material usage and ensuring that design aesthetics are not compromised. To accurately compute the deflections, simply substituting the values of \(r\) into the function will yield the deflection at those points, thus providing insights into the beam's behavior under certain conditions.

Applied Mechanics

Applied mechanics underpins many of the calculations in structural engineering, including the analysis of beam deflection. It incorporates principles of physics and materials science to predict how structures respond to external forces and moments. In the cantilever beam problem, applied mechanics principles would involve understanding how the beam material resists bending and the role of the cantilever support in providing a reaction moment.

Calculating beam deflections entails using these fundamental principles to derive equations that represent the elastic curve of the beam under load conditions. This intersection of physics, mathematics, and engineering design is crucial for predicting how structures will behave under various loading scenarios, ensuring they are safe, efficient, and effective in their intended applications.