Applied Mechanics and Strength of Materials
Applied Mechanics and Strength of Materials supplies the fundamental equations used everywhere else in civil engineering: equilibrium of force systems, centroid and moment of inertia, friction, and the elastic behaviour of bars and shafts under load.
Statics
For coplanar concurrent forces in equilibrium, the sum of forces in x and in y is zero. Lami theorem applies to exactly three concurrent forces: each force is proportional to the sine of the angle between the other two. For parallel forces, add a moment equation to the force equations.
Stress and strain
Stress = P/A and strain = change in length / original length, with E = stress/strain. Hooke law holds up to the proportional limit. Poisson ratio is lateral strain divided by longitudinal strain, typically 0.25 to 0.33 for metals. A bar of length L, area A and modulus E under axial load P extends by PL/AE.
Worked example
A 20 mm diameter steel rod, 2 m long, carries 40 kN; E = 2 x 10 to the fifth N/mm2. Area = pi d squared/4 = 314.16 mm2. Stress = 40000/314.16 = 127.3 N/mm2. Extension = PL/AE = 40000 x 2000/(314.16 x 2 x 10 to the fifth) = 1.27 mm. Strain = 1.27/2000 = 6.37 x 10 to the minus 4.
Torsion and columns
For a circular shaft, shear stress/r = T/J = G theta/L, with J = pi d to the fourth/32 for a solid shaft. Power transmitted = 2 pi N T / 60. Euler buckling load = pi squared EI / L squared, valid for slender columns; below a slenderness ratio of about 80 use the Rankine-Gordon formula instead.
Traps
- Using diameter where radius belongs in J and in the torsion formula.
- Mixing newtons with kilonewtons, or millimetres with metres.
- Applying Lami theorem to four forces.
- Ignoring the effective length factor for fixed, hinged and free-standing columns.