In 3D:
Where ( v(x) ) = vertical deflection. Common solutions:
(( b \times h )) maximum shear (at neutral axis):
Where: ( P ) = axial load, ( A ) = cross-sectional area, ( L ) = original length, ( E ) = modulus of elasticity. For a beam with distributed load ( w(x) ) (upward positive): structural analysis formulas pdf
Slenderness ratio:
[ 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:
[ \sum F_x = \sum F_y = \sum F_z = 0 ] [ \sum M_x = \sum M_y = \sum M_z = 0 ] Normal stress: In 3D: Where ( v(x) ) = vertical deflection
[ \fracd^2 vdx^2 = \fracM(x)EI ]
[ P_cr = \frac\pi^2 EI(KL)^2 ]
[ \fracdVdx = -w(x) \quad \textand \quad \fracdMdx = V(x) ] ( A ) = cross-sectional area
| 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):
Distribution factor at joint: [ DF = \frack_i\sum k ] Rectangle (width (b), height (h)): [ I = \fracb h^312, \quad A = bh ]
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) ]
(radius (r)): [ I = \frac\pi r^44, \quad A = \pi r^2 ]
| End condition | (K) | |---------------|-------| | Pinned-pinned | 1.0 | | Fixed-free | 2.0 | | Fixed-pinned | 0.7 | | Fixed-fixed | 0.5 |