### Lecture 26: Stresses in Beams - I

##### Course Description :

Analysis of stresses :

Body forces, Surface forces, Internal Force, Stress at a point, Components of stress in Rectangular coordinates, Stress tensor, Principal stresses, Transformation, Equations, Stress invariants, Plane stress, Mohrs circle for plane stress, Octahedral stresses, Differential equations of equilibrium, Components of stresses in cylindrical and Polar coordinates,

Analysis of Strain :

Deformable bodies, Concepts of normal strain and shear strain, Strain components at a point, Transformation equations, Principal strains, Mohrs circle of strains, Compatibility conditions, Displacement equation of equililibirum, Plane strain.

Stress-Strain relations :

Uniaxial tensile test, Elasticity, Anelasticity , Work-hardening, anisotropy, homogeneity and continuity, generalized Hookes law, Lames constants, Modulus of rigidity, Bulk modulus, relation between the elastic constants, Principle of superposition, Uniqueness theorem, Thermal effects.

Uniaxial Loading :

Bars of variable cross-section, Statically indeterminate problems in a tension and compression, Thin cylindrical and spherical vessels.

Torsion :

Geometry of deformation of a twisted circular shaft, Stress and deformation in twisted circular solid and hollow shafts, Strain energy due to torsion , Power transmitted by circular shafts.

Bending of Beams :

Bending moment and shear force diagrams, Stresses due to bending, bending equation, shear stresses in symmetrical elastic beams transmitting both shear and bending moment.

Deflection of Beams :

The moment curvature relation, Macaulays and moment-area method, Castiglianos theorem.

Combined Stresses :

Beam subjected to bending and shear, shaft subjected to bending and torsion, short columns.

Stability of Columns :

Stable and unstable equilibrium, Eulers formula for long columns, Rankines formula.

Springs :

Types of Springs. Close coiled and open coiled springs