Stresses in Beams

Cheatsheet Content

### Bending Stress - **Definition:** Stress caused by bending moment in a beam. - **Formula (Flexure Formula):** $$\sigma_x = -\frac{My}{I}$$ Where: - $\sigma_x$ = normal stress in the x-direction (along the beam's length) - $M$ = bending moment at the cross-section - $y$ = perpendicular distance from the neutral axis to the point where stress is calculated - $I$ = moment of inertia of the cross-section about the neutral axis - **Maximum Bending Stress:** Occurs at the points farthest from the neutral axis (top or bottom surfaces). $$\sigma_{max} = \frac{Mc}{I} = \frac{M}{S}$$ Where: - $c$ = distance from the neutral axis to the outermost fiber - $S = I/c$ = section modulus ### Shear Stress - **Definition:** Stress caused by shear force in a beam. - **Formula (Jourawski's Formula):** $$\tau_{xy} = \frac{VQ}{Ib}$$ Where: - $\tau_{xy}$ = shear stress at the point - $V$ = shear force at the cross-section - $Q$ = first moment of area (static moment) about the neutral axis for the area above (or below) the point where shear stress is calculated. $Q = A' \bar{y}'$ - $A'$ = area above (or below) the point - $\bar{y}'$ = distance from the neutral axis to the centroid of $A'$ - $I$ = moment of inertia of the entire cross-section about the neutral axis - $b$ = width of the cross-section at the point where shear stress is calculated #### Example: Shear Stress in a Rectangular Beam A rectangular beam with width $b=100$ mm and height $h=200$ mm is subjected to a shear force $V=50$ kN. Calculate the maximum shear stress. **Solution:** 1. **Calculate the Moment of Inertia ($I$):** $I = \frac{bh^3}{12} = \frac{(100 \text{ mm})(200 \text{ mm})^3}{12} = \frac{800 \times 10^6}{12} \text{ mm}^4 = 66.67 \times 10^6 \text{ mm}^4$ 2. **Determine the maximum shear stress formula for a rectangular beam:** The maximum shear stress in a rectangular beam occurs at the neutral axis and is given by $\tau_{max} = \frac{3}{2} \frac{V}{A}$. 3. **Calculate the cross-sectional area ($A$):** $A = b \times h = 100 \text{ mm} \times 200 \text{ mm} = 20000 \text{ mm}^2$ 4. **Calculate the maximum shear stress:** $\tau_{max} = \frac{3}{2} \frac{50 \times 10^3 \text{ N}}{20000 \text{ mm}^2} = \frac{3}{2} \times 2.5 \text{ N/mm}^2 = 3.75 \text{ N/mm}^2 = 3.75 \text{ MPa}$ ### Neutral Axis - **Definition:** The axis in the cross-section of a beam where there is no normal stress ($\sigma_x = 0$) due to bending. - **Location:** Passes through the centroid of the cross-section for homogeneous materials. - **Calculation for Composite Sections:** $$ \bar{y} = \frac{\sum A_i y_i}{\sum A_i} $$ Where: - $A_i$ = area of the i-th component - $y_i$ = distance from a reference axis to the centroid of the i-th component ### Moment of Inertia ($I$) - **Definition:** A measure of a beam's resistance to bending. - **Parallel Axis Theorem:** Used to find $I$ about an axis parallel to the centroidal axis. $$ I = I_c + A d^2 $$ Where: - $I_c$ = moment of inertia about the centroidal axis - $A$ = area of the section - $d$ = distance between the centroidal axis and the parallel axis - **Common Shapes:** - **Rectangle:** $I = \frac{bh^3}{12}$ (about centroidal axis) - **Circle:** $I = \frac{\pi r^4}{4}$ (about centroidal axis) ### Stress Distribution - **Bending Stress:** - Linear distribution across the depth of the beam. - Zero at the neutral axis. - Maximum at the top and bottom fibers. - Compressive on one side of NA, tensile on the other. - **Shear Stress:** - Typically parabolic distribution for rectangular sections. - Zero at the top and bottom free surfaces. - Maximum at the neutral axis for rectangular sections. - For a rectangular section: $\tau_{max} = \frac{3}{2} \frac{V}{A}$ ### Example: Rectangular Beam - **Cross-section:** Width $b$, Height $h$. - **Neutral Axis:** At $h/2$ from top/bottom. - **Moment of Inertia ($I$):** $\frac{bh^3}{12}$ - **Section Modulus ($S$):** $\frac{bh^2}{6}$ - **Max Bending Stress ($\sigma_{max}$):** $\frac{6M}{bh^2}$ - **Max Shear Stress ($\tau_{max}$):** $\frac{3}{2} \frac{V}{A} = \frac{3V}{2bh}$ (at NA)