(untitled) (مجلة الفاروق للعلوم, 2026)
This paper presents a professional general-mechanics study of a slender cantilever beam subjected to a combined transverse tip load and uniformly distributed load. The work integrates closed-form Euler-Bernoulli beam theory, one-dimensional Hermite finite-element modeling, stress evaluation, modal analysis, and a parametric design study. The objective is to provide a complete and reproducible mechanical analysis workflow that connects fundamental mechanics equations with engineering decision metrics such as maximum deflection, extreme-fiber stress, natural frequency, mesh convergence, and factor of safety. A steel beam with a length of 1.2 m, rectangular cross-section of 30 mm by 12 mm, tip load of 50 N, and distributed load of 20 N/m is used as the reference case. The analytical solution predicts a tip deflection of 37.46 mm, maximum fixed-end stress of 103.33 MPa, and first bending natural frequency of 6.96 Hz. The finite-element model reproduces the static displacement response and converges rapidly for stress and modal predictions; with 32 beam elements, the first four natural frequencies agree with the analytical solution with errors below 0.001%. A thickness sensitivity study shows that increasing the beam thickness from 8 mm to 20 mm reduces the tip deflection from 126.4 mm to 8.1 mm and increases the first natural frequency from 4.64 Hz to 11.60 Hz. The study is suitable as a professional research paper template in general mechanics, mechanical design, finite-element verification, and structural vibration.