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Strain Energy or Resilence  It is defined as, " energy absorbed in a body when deformation is done on material " . Depend on Material . Strain Energy Density is defined as, " strain energy per unit volume " . Independent of material . Used to tell which deformation component is greater with direction. If greater in specific direction   ↔  crack propagates in that direction. Stress (in Dynamic effect)  >  Stress (in Static effect) Types of Loading Energy Conservation It states that, " Energy can neither be created nor be destroyed but can transfer in other forms " . We only deal with mechanical energy. External work = internal energy or strain energy . If this external force is in elastic limit, strain energy restores body back to its original undeformed position. Note: For Castigliano's Theorem  ↔  See Energy Method . With the application of force on Car chassis , How it behaves? References: Material from Class Lectures + Book named Engineering Mec...

Material Property Relationships There are three principles which inter-related (applies to Uniaxial direction): Hooke's Law Poisson's Ratio Principle of Superposition The relationship between E, ν, G is: Dilatation Rate When an elastic material is subjected to normal stress, its volume will change. The change in volume is given by: Theory of Elastic Failure or Yielding When dealing with design of structures/components   ➞  physical properties are found using simple stress conditions (most usual tensile test). Important for determining allowable working stresses to avoid failure. These theories are for predicting the behavior of materials adequate correlation b/w estimated component life and under service load condition for ductile/brittle material applications. Max. Principal Stress Theory (Rankine) Max. Shear Stress Theory (Guest-Tresca) Max. Principal Strain (Saint-Venant) Total Strain Energy per unit volume (Haigh) Shear Strain Energy per unit volume (Maxwell-Von-Miss...

Strain Transformation Principal Strain and stresses can occur in the same directions. Material Properties Relation (Young, bulk Rigidity Modulus)  ⇼  Hooke's Law General State of Strain   ⇼  Є X , Є Y , Є Z and ૪ X , ૪ Y , ૪ Z . Stress (normal or shear)/ Strain (normal or shear)  ⇼  vary with element orientation. Transformation equations for Plane strain derived from: Interpretation of Experimental measurements  Represent in graphical form for plane strain (Mohr's Circle). Geometry and independent of material properties. Mohr's Circle It is defined as.,  " A graphical method for determining normal and shear Shear stresses without using the stress transformation equations " . While considering the circle  CCW   ⇼  Shear strain positive upward & Normal strain positive towards right. The  construction of Mohr's circle  (with normal and shear stresses are known) is quite easy which include following steps: Draw a set o...

Stress Transformation It is defined as, " the state of plane stress at the point is uniquely represented by two normal stress components and one shear component acting on an element (face). To be equivalent, these three components will be different for each specific orientation Ө of the element at that point " . If no load applied  ⇔  normal and shear stresses are zero. Stress Tensor ( has magnitude & multiple directions )   ⇔  convenient way of expressing the stress state inside a material. Plane-Stress Transformation Equations A method of transforming the normal and shear stress components from x and y to x' and y'. Max. In-Plane Principal Stresses Corresponding planes on which stresses act are called the Principal Planes.  2Ө P1   - 2Ө P2   =  180°  Max. In-Plane Shear Stress Mohr's Stress Circle It is defined as., " A graphical method for determining normal and shear Shear stresses without using the stress transformation equations " ....

Statically Indeterminate Beams It is defined as, " a beam subjected to transverse loads with more than two reaction components because the equilibrium equations are not sufficient to determine all the reactions " . For the determinacy of coplanar 2D beam (n = no. of parts, r = no. of reactions)  r = 3n  →  Statically Determinate r > 3n   →  Statically Indeterminate Deflection of Beams Beams are designed on evaluation of Shear Force and Bending Moment distributions to calculate stresses and deformation.  Most of the beams in real world  →  Statically Indeterminate . Methods to analyze SIB is: Double Integration Method Superposition Graphical Method (Moment Area) Energy Method (versatile and apply on every condition) Beams depends on the: Stiffness of the material Cross-sectional dimensions of the beam Configuration of the applied loads  Supports Double Integration Method For a hand solution of a problem   →  Click Here . Ener...

Introduction To Structural OR Concrete Design Beams must have adequate strength against different types of failure which are: Shear  more dangerous than Flexural (or Bending) failure because it creates additional tensile stresses. E.g.: Airplane wing (act as cantilever beam ) and made of Nanocomposites, composites, aluminum. Following are the types of failures in Beam: Flexural (or bending) failure Diagonal Tension failure Shear-Tension failure   Shear-Compression failure Following are the types of Shear : Longitudinal Shear Transverse Shear Shear Failure Diagonal Tension Failure   ↠  Shear failure of reinforced concrete beam (difficult to predict). Only valid for Homogenous beams . When we apply load on beam  ↠  Bending as well as Shear stresses are produced. Shear stress have maximum value at Neutral axis N.A. Bending stress have maximum value at Extreme fibers. At maximum bending stress   ↠  shear stress = 0 . Assumptions for shear stresses i...

Structural loads Structural loads are the forces and moments applied to the structure. They are further divided into following types: Surface loads   ↔  externally applied load on the surface (like point load, line load, pressure, bending, torsion). Body loads   ↔  developed internally within the structure (gravity, weight, thermal load). There are other types of loads with respect to time: Static load   ↔  if load is not a function of time (have constant magnitude, directions and location). Dynamic load   ↔  if load is a function of time . Quasi-Static load   ↔  if load varies slowly with time (taken as static load). Dynamic loads are further divided as follows: Steady-State loads   ↔  which maintain same character (like frequency, amplitude, etc) over the long term. Transient loads   ↔  which change their character with time. Taxonomy of Structure Classification of structures (complex systems) by decomposing them...

Solids A body is and remains continuous under the action of external force. That means: Consist of continuous material points. Can sustain shear. Neighboring points remains neighbors. Solid Mechanics We will discuss what steps are taken out to understand the dynamic or static response of structure. Steps are: Computation or Modelling  Finite element method  Boundary element method Molecular dynamics simulation Theory or Analytical Modelling Continuum mechanics Micromechanics  Constitutive model Experiments  Material characterization Strain measurement  High strain rate Fatigue Defining a Problem Describe the goal and desired information. Model the geometry of solid. Determine the loading applied to the solid. Describe what modelling must be included in the model. Choose a constitutive law which describe the behavior of material. Choose a method of analysis. Solve the problem. While defining a problem, we need t consider failure mechanisms as under: Critical loa...