COMBINED EFFORTS 😉 Combination of LOADS

Causes of Normal and Shear Stresses in Beams
📌 Normal stresses ($\sigma$) are perpendicular to the cut face, while tangential (shear) stresses ($\tau$) lie within the plane.
📏 Normal stresses are caused by axial stress (constant across the section, $\sigma_x = P/A$) or bending moment ($M$, resulting in a stress gradient $\sigma_x = \frac{My}{I_z}$).
⚙️ Shear stresses are primarily generated by shear force ($V$, maximum at the center of the section) or torsional moment ($T$, maximum at the outer fiber).

Calculation of Normal Stresses
📐 Bending moment creates a stress profile that varies linearly from zero at the neutral axis (passing through the centroid) to maximum tension and compression at the outer fibers.
➕ Total normal stress in a section is the superposition of the axial stress and the stresses induced by all applied moments ($\sigma_{total} = \sigma_{axial} + \sigma_{M_z} + \sigma_{M_y}$).
⬆️ The most critical position experiences the summation of tensile stresses (e.g., the upper-left corner when both moments and axial load cause tension there).

Calculation of Shear Stresses and Neutral Axis Shift
〰️ Shear stress ($\tau$) due to shear force ($V$) is zero at the outer edges and maximum at the center, calculated using $\tau = \frac{VQ}{It}$ (where $Q$ is the first moment of area).
🌀 Torsional shear stress is proportional to the distance from the center and is maximum at the outer boundary.
⬇️ When both axial force and bending moment ($M_z$) are present, the neutral axis shifts away from the side experiencing net tension. The new neutral axis position ($y$) can be found by setting the total normal stress to zero: $P + \frac{M_z y}{I_z}A = 0$.

Key Points & Insights
➡️ Superposition Principle is crucial: Total stress at any point is the algebraic sum of stresses generated by axial load, bending moments (about both axes), and torsion.
➡️ For a rectangular beam under shear, the maximum shear stress occurs at the center and is equal to $1.5$ times the average shear stress ($V/A$).
➡️ The neutral axis under combined loading is located where the normal stress ($\sigma$) is zero; its position shifts based on the relative contributions of the axial load and the bending moments.
➡️ In the provided example, Point 1 (upper corner) was critical due to the summation of tensile stresses from the axial load ($P$) and the moment $M_z$.

📸 Video summarized with SummaryTube.com on Mar 01, 2026, 04:27 UTC