Flexural rigidity; T-beam bending stresses (60 kN loads)

Flexural rigidity

Flexural rigidity is the product $EI$ of Young's modulus and the moment of inertia of the section about the neutral axis. It measures the resistance of a beam to bending: from $\dfrac{M}{EI} = \dfrac{1}{R} = \dfrac{d^2y}{dx^2}$, the curvature and deflection of a beam are inversely proportional to $EI$.

Importance: a larger $EI$ gives smaller deflection and curvature for the same loading. It is used to check deflection limits, to compare sections (deep sections have large $I$), and it appears in the deflection and column formulas ($\delta \propto \dfrac{WL^3}{EI}$, $P_{cr} = \dfrac{\pi^2EI}{L_e^2}$).

Bending stresses in the beam

Reading of the figure: simply supported beam with span $AD = 1 + 1.5 + 1 = 3.5$ m and two 60 kN loads at B (1 m from A) and C (1 m from D). The T-section has a flange $7.5 + 5 + 7.5 = 20$ cm wide and 5 cm thick, and a web 5 cm wide and 15 cm deep (total depth 20 cm). The flange is taken at the top (if the flange is at the bottom, the tensile and compressive values below are interchanged).

Reactions and maximum moment: by symmetry $R_A = R_D = 60$ kN. The maximum moment is constant between B and C:

$$M_{max} = 60\times1 = 60\ \text{kN·m} = 60\times10^6\ \text{N·mm}$$

Centroid (from the bottom of the web):

$$\bar y = \frac{(20\times5)(17.5) + (5\times15)(7.5)}{100 + 75} = 13.214\ \text{cm}$$

Moment of inertia about the neutral axis:

$$I = \left[\frac{20(5)^3}{12} + 100(17.5-13.214)^2\right] + \left[\frac{5(15)^3}{12} + 75(13.214-7.5)^2\right] = 5900.30\ \text{cm}^4$$

Stresses ($\sigma = My/I$, $M = 60\times10^5$ N·cm):

$$\sigma_{tension}\ (\text{bottom},\ y = 13.214\ \text{cm}) = \frac{60\times10^5 \times 13.214}{5900.30} = 13437.6\ \text{N/cm}^2 = 134.38\ \text{N/mm}^2$$ $$\sigma_{compression}\ (\text{top},\ y = 6.786\ \text{cm}) = \frac{60\times10^5 \times 6.786}{5900.30} = 6900.4\ \text{N/cm}^2 = 69.00\ \text{N/mm}^2$$

Answer: maximum tensile stress $= 134.38$ N/mm² (bottom of the web); maximum compressive stress $= 69.00$ N/mm² (top of the flange).