Where ( v(x) ) = vertical deflection. Common solutions:
[ \sum F_x = 0, \quad \sum F_y = 0 ]
[ \tau_\textmax = \frac3V2A ] Critical load for a slender, pin-ended column:
Where: ( V ) = shear force, ( Q ) = first moment of area about neutral axis, ( I ) = moment of inertia, ( b ) = width at the point of interest. structural analysis formulas pdf
Effective length factors (K):
In 3D:
[ 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: Where ( v(x) ) = vertical deflection
| End condition | (K) | |---------------|-------| | Pinned-pinned | 1.0 | | Fixed-free | 2.0 | | Fixed-pinned | 0.7 | | Fixed-fixed | 0.5 |
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):
Where: ( M ) = internal bending moment, ( y ) = distance from neutral axis, ( I ) = moment of inertia of cross-section. The differential equation: For a beam with distributed load ( w(x)
[ \fracKLr, \quad r = \sqrt\fracIA ] For a pin-jointed truss in equilibrium at each joint:
[ \delta = \fracPLAE ]
(( b \times h )) maximum shear (at neutral axis):
Integral forms: