[ \sigma = \fracPA ]
[ \fracd^2 vdx^2 = \fracM(x)EI ]
[ \sum F_x = 0 \quad \sum F_y = 0 \quad \sum M_z = 0 ]
[ P_cr = \frac\pi^2 EI(KL)^2 ]
[ \tau_\textavg = \fracVQI b ]
In 3D:
(( b \times h )) maximum shear (at neutral axis): structural analysis formulas pdf
| Case | Max Deflection (( \delta_\textmax )) | Location | |------|-------------------------------------------|----------| | Cantilever, end load (P) | (\fracPL^33EI) | free end | | Cantilever, uniform load (w) | (\fracwL^48EI) | free end | | Simply supported, center load (P) | (\fracPL^348EI) | center | | Simply supported, uniform load (w) | (\frac5wL^4384EI) | center | | Fixed-fixed, center load (P) | (\fracPL^3192EI) | center | | Fixed-fixed, uniform load (w) | (\fracwL^4384EI) | center | For a prismatic beam (rectangular cross-section approximation):
[ V(x) = -\int w(x) , dx + C_1 ] [ M(x) = \int V(x) , dx + C_2 ] For pure bending of a linear-elastic, homogeneous beam:
[ \fracdVdx = -w(x) \quad \textand \quad \fracdMdx = V(x) ] [ \sigma = \fracPA ] [ \fracd^2 vdx^2
Integral forms:
| End condition | (K) | |---------------|-------| | Pinned-pinned | 1.0 | | Fixed-free | 2.0 | | Fixed-pinned | 0.7 | | Fixed-fixed | 0.5 |
Where: ( M ) = internal bending moment, ( y ) = distance from neutral axis, ( I ) = moment of inertia of cross-section. The differential equation: structural analysis formulas pdf
Member force (axial): [ F = \sigma A = \frac\delta AEL ] Carry-over factor (for prismatic member): 1/2 Member stiffness: [ k = \frac4EIL \quad (\textfixed far end) \quad \textor \quad k = \frac3EIL \quad (\textpinned far end) ]