Das Widerstandsmoment \(W\) eines Balkens mit rechteckigem Querschnitt wird nach der Formel $$ W=W(b ; h)=\frac{1}{6} b h^{2} $$ berechnet ( \(b:\) Breite des Balkens; \(h\); Höhe des Balkens). Welche prozentuale Anderung erfährt das Widerstandsmoment eines Balkens der Breite \(b=18 \mathrm{~cm}\) und der Dicke \(h=10 \mathrm{~cm}\), wenn man die Balkenbreite um \(5 \%\) vergr?ßert und gleichzeitig die Balkendicke um \(10 \%\) verkleinert?

Beam mechanics is a branch of engineering mechanics that focuses on the behavior of structural elements known as beams. Beams are long, slender members subjected to different kinds of loads, like bending, shear, and axial forces. Understanding beam mechanics is crucial in designing structures such as buildings, bridges, and aircraft. The main goal is to ensure that a beam can withstand applied loads without failing or deforming excessively. One key concept in beam mechanics is the resistance moment, which determines a beam's ability to resist bending. It is essential for the stability and durability of structures.

The cross-sectional area of a beam greatly affects its structural integrity. It is the shape and size of the section cut at right angles to the beam’s length. In our exercise, the beam has a rectangular cross-section. Four dimensions primarily define it: width (\b\text \b) and height (\b\text \b). These measurements are essential because they influence the resistance moment of a beam. The formula used to calculate the resistance moment is \b \b is: \[ W = \frac{1}{6} b h^2 \] This shows how width and height contribute to the beam's ability to resist bending. If we change the cross-section dimensions, the resistance moment changes. This affects how much load the beam can safely carry.

Calculating percentage changes is a common task in engineering to understand how variables influence performance. In the exercise, we calculate the percentage change in the resistance moment of a beam when its width and height are altered. Here's a brief overview of the steps:Understand the original dimensions and their effect on resistance.Calculate the original resistance moment.Determine the new dimensions after the specified changes.Calculate the new resistance moment.Compare the new resistance moment to the original and compute the percentage change. The formula for percentage change is: \[\text{Percentage change} = \frac{W' - W} {W} * 100 \] Substituting our values gives us a percentage change of -14.95%, indicating a reduction.

Engineering mathematics is a critical tool for solving practical problems in engineering. It involves mathematical techniques and equations that are applied to engineering concepts. In our exercise, we encounter formulas and calculations essential for solving real-world engineering problems. These include:Using formulas like the one for resistance moments.Applying percentage calculations to understand variations.Both these steps require a solid grasp of algebra and arithmetic. Understanding these mathematical principles allows engineers to design safe, efficient, and cost-effective structures. Engineering mathematics bridges the gap between theoretical concepts and practical applications, making it an invaluable skill for aspiring engineers.