Load Stress Relation - EOC

Load Stress Relation \(M=Fy\) (Fd) \[\sigma=\frac{F}{A}\] Compression duz x day Exyday fension \[M_{2}=E_{k}\iint_{A}y^{2}dA\] note \(I_{Z}=\iint_{A}y^{Z}\cdot dA\) \[M_{2}=-E\kappa I_{2}\Rightarrow\sigma_{x}=-\frac{M_{x}}{I_{2}}y\] assumptions to these bending relations maximum deflection is around the same scale as the cross sectional dimension of the beam (ie small deformations) Linear elastic benavious Long, slender beam- Length of the beam should be at least 8 times the cross sectional dimension of the beam bending moment & shearing force are the only internal forces present bending moment acts along the principal axis of the cross-sechon Calculating maximum stress (mas I lymer) e.gill cantilever beam Cross-section IMI BMD \[M_{male}=PL\] \[y_{mar}=\frac{a}{2}\] \[\sigma_{max}=\frac{p_{L}}{9(a)^{3}}{a}}\cdot\frac{a}{2}=\frac{6p_{L}}{a^{3}}\] 2.9% simply supported beam \[|M_{amax}|=\frac{P}{3}\cdot\frac{4}{2}\] BMD JP woss-section \[=\frac{P}{4}\] \[\frac{p}{2}\] end reactions \[I=\frac{\pi c^{2}}{4}\] \[y_{max}=r\] \[\sigma_{max}=\frac{\frac{P_{L}}{\frac{\pi r^{2}}{4}}\cdot r=\frac{P_{L}}{\pi r^{3}}\] PO