Principles of Engineering

Strength of Materials

Moment of Inertia is an indication of the stiffness of a particular shape. The higher the moment of inertia, the less the deflection

Moment of Inertia = 1/12 (b)(h³)

Buckling - Failure caused by bending, giving way, or crumpling due to excessive force

Centroid - A point whose coordinates are the averages of the corresponding coordinates of a given set of points and which for a given plane or three-dimensional figure (as a triangle or sphere) corresponds to the center of mass of a thin plate of uniform thickness and consistency or a body of uniform consistency having the same boundary.

• center of gravity or center of mass
• Symbol =
• Centroid = Center of Gravity
• Triangle:
• Centroid = h/3;  b/3
   
• Irregular Shapes:
• Section off shape for easy calculations
• Label each section i.e. A₁, A₂, A₃, etc.
• Find centroid of each section ( X & Y)
• Calculate Area of each section
• Use table below:

 

 

Section	Area	X	Y	A x X	A x Y
A1
A2
A3
	add above			add above	add above
Total Area
				divide by total area	divide by total area
			Centroid	X =	Y =

• Stress - A material’s internal resistance to force; symbol: σ, “sigma”; calculated by dividing the force in the material by the area of the material that is subjected to the force:

stress is a physical quantity and is defined as the force per unit area. F is force, A is surface area.

σ=F/A

Stress can be of different types: tensile, compressive, shear . Tensile stress pulls, compressive stress pushes and shear stress acts parallel to the surface.

• Axial - Force aligned along the central axis of a material
   

  Formula:  s = P / A 

  P = Pressure

  A = Cross Sectional Area

• Shear - Force that acts parallel to the surface of the material.
   

  Formula:  t = P / A

  P = Pressure

  A = Cross Sectional Area

Bending Stress:

   or  

 

s                 = bending stress

M = moment

C = maximum fiber distance

I = moment of inertia of cross section

S= section modulus= I / C

 

• Strain - Strain is a measurement of deformation in a structure due to applied forces.
  If a rod  of length L and constant cross section area A has a force P applied at the end, the deformation will be d

   

Strain is calculated from:

                              Strain =  Deformation

                                             Original Length

or

ε = d / L

Stress and Strain are related through the stress strain curve and Hooke’s Law in the elastic region

• Hooke's Law: In most solid objects, the relation between load and deformation is linear. This is known as Hooke's Law. 

E is the Elastic Modulus.

E is the slope of the line in the elastic region

• Deflection - A measure of deformation of a structure due to applied loads.