STRENGTH OF MATERIALS

BEAM= A structural element which has one dimension considerably larger than the other two dimensions and is supported at few points to carry lateral or transverse loads in axial plane is called a beam.

TYPES OF BEAMS-

1. Simply supported beam= It rests on a knife edge. The reaction and support for vertical load are perpendicular to beam.
2. Hinged beam= If one end of beam is hinged and other end rests on roller, the beam is said to be hinged.
3. Cantilever beam= If the beam is fixed at one end and free at the other end , it is called a cantilever beam.
4. Overhanging beam= If the beam is projecting beyond the support, it is called overhanging beam.
5. Fixed beam= In a fixed beam, there are moments and forces due to reactions at fixed ends.
6. Continuous beam= If a beam is supported on more than two supports,it is called continuous beam.

TYPES OF LOADING ON BEAM-

1. Point load
2. Uniformly Distributed loading-

                Force (F)= (W × Length of beam)

3. Uniformly Varying loading

            Force (F)= (½ W × Length of beam)

SHEAR FORCE (V)= Shear force at a section in a beam is a force that tries to shear off  the section and is defined as algebraic sum of all forces acting normal to the axis of beam either to the left or right of the section.

BENDING MOMENT (M)= Bending Moment at a section in a beam is a moment that tries to bend the beam and is defined as the algebraic sum of moment of all forces about the section acting either left or right of the section.

RELATION BETWEEN  ‘V’ , ‘M’  &  LOAD  ‘W’ –

POINT OF CONTRAFLEXTURE OR INFLEXION- It is defined as a point which represents the section on the beam where bending moment is zero or bending moment changes its sign.

NOTE- The bending moment will be maximum or minimum         where shear force is zero

SHEAR FORCE AND BENDING MOMENT DIAGRAM FOR SOME CASES-

DEGREE OF DETERMINANCY (Di)= (Total number of external reactions at support) ­ (Total available equilibrium conditions)

Di= R – 3  (for plane)                  

Di= R – 6  (for space)

IF    Di > 0 , stable and indeterminate

       Di< 0 , unstable and determinate

       Di= 0 , stable and determinat

PURE BENDING= When the state of loading on a beam segment is absolutely free from shear stress and is subjected to a constant bending moment, then this state is called pure or simple bending.

                          

SIMPLE BENDING MOMENT EQUATION-

                             

Where symbols have their usual meanings

Assumptions made for deriving above equation-

• Material of beam is homogenous and isotropic
• Loading is within elastic limit
• Beam is initially straight and is of uniform section throughout
• Transverse section of beam remains plane and perpendicular to neutral surface after bending
• Radius of curvature of beam during bending is large compared to its transverse dimensions

SECTION MODULUS (Z)= It is defined as the ratio of moment of inertia of a section about neutral axis to the distance of the outermost layer from neutral axis.

                                                Z=
                                  
Section modulus tells about the strength of the section of beam.

Section  Modulus  & Moment of inertia of some sections-

   NOTE -  bending  stress = σ 

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