Italian engineer Alberto Castigliano (1847 – 1884) developed a method of determining deflection of structures by strain energy method. Weighted Residual Methods. 1. The strain energy is composed of two distinct forms – volume changes and distortion (angular change). For the linearly elastic structure it may be seen that OABCDO OABEO Area OBC = Area OBE i.e. Where, Strain energy of prismatic bar with varying sections: Strain energy of non-prismatic bar with varying axial force: A x = Cross-section of differential … Eccentric Axial Loading, Analysis of transverse shear load ,SFD & BMD & Numerical problem, Analysis of Eccentric Transverse Shear load & SFD ,BMD & TMD, Torsion equation & Analysis of pure torsion & twisting moment diagram (TMD), Eq. ot . Description of the Method. One Dimensional Examples. Strain Energy: The work done by the load in straining the body is stored within the strained material in the form of strain energy. Overview of Strength of Materials (in Hindi) 6:47 mins. In the absence of energy losses, such as from friction, damping or yielding, the strain energy is equal to the work done on the solid by external loads. The strain energy methods are most commonly used for solving the unknown factors in complicated structural problems. Strain Energy Equations General Case - Constant Area with Varying . of E & G, Bulk modulus (K) , & Expression for volumetric strain under tri-axial loading, Numerical problems on Volumetric strain & Elastic constant, Analysis of Pure axial loading on Stepped bar ( Bar in series), Case 2 - Analysis of Axial loading on Stepped bar ( Bar in series), Numerical problems on stepped bar & Analysis of Bar fixed at both end (Statically Indeterminate Bar), Numerical problems on Bar fixed at both end ( Statically Indeterminate Bar) & Shortcut for reactions, Important Numerical problems on Statically Indeterminate Bar & Shortcut for reactions, More Important Numerical problems on Statically Indeterminate Bar, Analysis of Axial load on Tapered bar & Calculate maximum stress induced & elongation of tapered bar, Analysis of Tapered bar fixed at both ends (Statically Indeterminate Tapered bar), Elongation of Prismatic bar due to its self weight & Numerical problems, Elongation of Conical bar due to its self weight & Question- when both self weight & Axial load, Comparison among Prismatic bar under Axial loading , Prismatic & Conical bar due to its self weight, Strain Energy, Resilience,& Toughness, Proof Resilience, Modulus of Resilience, Modulus of Toughness, Calculate Strain Energy under Axial loading,under bending,under twisting, Numerical problems for the calculation of Strain Energy, Strain Energy of a rectangular block under a shear load, Strain Energy of a beam under pure Bending, Numerical problems of Strain Energy of a cantilever beam & simply Supported beam & draw SFD,BMD, Strain energy of Prismatic shaft & Stepped shaft under Pure Torsion, Bars in Parallel / Composite bar (Statically Indeterminate bar) under axial loading & Numerical prob, Understanding of Thermal Stress,Thermal stress during free expansion & completely restricted, Thermal stress during completely or Partially restricted expansion/ Compression & Numerical problem, Analysis of Thermal Stresses in Compound bar (Bars in series), Numerical problems on Thermal Stresses in Compound bar (bars in series) & Important points, Analysis of Thermal Stresses in Composite bar (bars in Parallel), Numerical problems for the calculation of Thermal Stresses in Composite bar (bars in Parallel), Important numerical problems for the calculation of Thermal Stress, Beam,Beam classification, Types of Rigid supports, Types of Loads acting on Beams, Calculation of Loads, Determine Support reactions acting on beam, Numerical problems for the calculation of Support reactions acting on beam, Important points to draw SFD & BMD and Sign convention, Numerical problems on SFD(Shear force diagram) and BMD (Bending moment diagram), Numerical problems on SFD and BMD & location of point of contraflexture, Numerical problems on SFD & BMD, location of contraflexture point, radius of curvature, Draw SFD & BMD for uniform distributed load, uniformly varying load, Draw SFD & BMD for uniformly varying load, Draw SFD and BMD for double uniformly varying load & Numerical problems, Important Numerical problems on SFD and BMD, Numerical problems on SFD and BMD, Relationship b/w load intensity,Shear force and bending moment, GATE previous year problems on SFD and BMD, GATE Previous year problems on SFD AND BMD, GATE problems on SFD and BMD & Important points related to bending stress, Deflection of beams-Objective, Basic understanding, Location of maximum slope & maximum deflection & Double Integration method’s procedure, Double Integration method – Calculate maximum slope and maximum deflection, Remember important values of Max slope & max Deflection for various beams under different loading, Important problems for the calculation of maximum slope, maximum deflection , location and support, Numerical problems on deflection of beams, More Problems for the calculation of maximum slope and maximum deflection, Important problems of Deflection of beams, Moment Area Method for the calculation of deflection and slope, Problems -Use Moment area method for the calculation of deflection and slope, Calculate slope and deflection by using moment area method, Important problems for the calculation of slope, deflection and reactions using moment area method, Castigliano's Theorems- Strain energy method for the calculation of slope and deflection, Problems for the calculation of deflection and slope using Castigliano's Theorems, More Problems for the calculation of slope and deflection using Castigliano's Theorem, Solve these important problems using Castigliano's Theorems, Maxwell's Reciprocal Theorem for the calculation of deflection and solve Some Problems, Most important problems of Deflection of beams, Analysis of Thin Cylinder (Thin Pressure Vessels), Analysis of Thin Cylinder & thin spheres -part 2, Exp. The equations are written below for convenience. Develop understanding of Stress-strain diagram, Calculate yield stress by offset method, Ductility,Stree-strain diagram for brittle Material, Strength, Design criterion- Strength criterion & Rigidity Criterion, Numerical problems on Design criterion & ductile & brittle fracture under tension test, Analysis of Impact load, Gradually applied load,& dead load & Axial impact load, Important numerical problems on Impact loading & also Gate Questions, Comparison b/w Impact Axial load & Static Axial load & Numerical problems & Fatigue load & its type, Normal and Shear stress developed on a oblique plane under axial loading, Problems for calculation of Normal and Shear stress on oblique plane and some important points, Graphical & matrix representation of Tri-axial & Bi-axial state of stress & for strain (3D & 2D), Normal & Shear stress developed on a oblique plane passing through a point under Bi-axial stress, Problems for calculation of Normal and Shear stress on oblique plane under Biaxial state of stress, Types of strain-Longitudinal Strain, Lateral strain, Poisson Ratio, Volumetric strain, Shear Strain, Deformation in vector form, Numerical problem, Relationship between Elastic constants –E,G,K, Upper limit and lower limit of G & K in terms of E as variation of Poisson’s ratio & Def. His Theorem of the Derivatives of Internal Work of Deformation extended its application to the calculation of relative rotations and displacements between points in the structure and to the study of beams in flexure. Strain energy, Put Or Proof Resilience: The maximum strain energy that can be stored in a material is known as proof resilience. ij corresponding to the increment of strain is obtained from the elasticity law ˙ ij= C ijkl kl (8.14a) ˙ ij= C ijkl kl (8.14b) Therefore, by eliminating C ijkl ˙ ij ij= ij ˙ ij (8.15) The total strain energy of the elastic system is the sum of the elastic strain energy stored and the work …

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