Shear Force and Bending Moment Theory: Analyzing Internal Forces for Structural Analysis a, Slides of Mechanics of Materials

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Shear Force Diagram &

Bending Moment Diagram

Types of Beam

1.Simply supported beam:- A beam supported freely on the walls or columns at its both ends is known as simply supported beam 2.Over hanging beam:- A beam is freely supported on two supports. But its one end or both the ends are projected beyond the support.

Types of Beam…

5.Continuous Beam:-

• A beam supported on more then two support is known as continuous beam.

Types of loads

1.Point load or Concentrated load:- When a load is acting on a relatively small area it is considered as point load or concentrated load. W = point load, its is given in N or kN. W

Types of loads…

3.Uniformly Varying load:-

A load which is spread over a beam in such a manner that its intensity varies uniformly on each unit is called uniformly varying load. w = U.V.L, It is given in N/m or kN/m. w

Determinate & Indeterminate Beams

1.Determinate Beam

A beam is said to be statically determinate if the total no. of unknown

reactions are equal to the no. of conditions of static equilibrium.

Conditions of Static equilibrium = 3

No. of unknown reactions = 3, so

The beam is statically determinate

A B

VB

HA

VA

Determinate & Indeterminate Beams…

We will limit our study to shear force and bending moment diagrams of statically determinate beams. Commonly encountered statically determinate beams are, a. Cantilever Beam b. Simply Supported Beam c. Over-hanging Beam

Shear Force & Bending Moment…

Consider a section x-x at a distance 6m from left hand support A Imagine the beam is cut into two pieces at section x-x and is separated, as shown in figure. 5kN 10kN^ 8kN 4m (^) 5m 5m^ 1m A C (^) D B RA = 8.2 kN (^) RB=14.8kN E x x 6 m

Shear Force & Bending Moment...

Thus the section x-x considered is subjected to forces 3. 2 kN and moment

39. 2 kN-m as shown in figure. The force is trying to shear off the section and hence is called shear force. The moment bends the section and hence, called bending moment. 5kN A 8.2 kN 10kN 8kN B 14.8 kN 3.2 kN 3.2 kN 39.2 kN-m 39.2 kN-m

Shear Force & Bending Moment...

 Shear force at a section : The algebraic sum of the vertical forces acting on the beam either to the left or right of the section is known as the shear force at a section.  Bending moment (BM) at section : The algebraic sum of the moments of all forces acting on the beam either to the left or right of the section is known as the bending moment at a section 3.2 kN 3.2 kN F

F

Shear force at x-x M Bending moment at x-x 39.2 kN

Sign Convention for shear force

F

F

F

F

+ ve shear force (^) - ve shear force

Sign Convention for bending moment

The bending moment is considered as Sagging Bending Moment if it tends to bend the beam to a curvature having convexity at the bottom as shown in the Fig. given below. Sagging Bending Moment is considered as positive bending moment. Fig. Sagging bending moment [Positive bending moment ] Convexity

Shear Force and Bending Moment Diagrams

(SFD & BMD)

 Shear Force Diagram (SFD):

 The diagram which shows the variation of shear force along the length of the beam is called Shear Force Diagram (SFD).

 Bending Moment Diagram (BMD):

 The diagram which shows the variation of bending moment along the length of the beam is called Bending Moment Diagram (BMD).

Point of Contra flexure [Inflection Point]:

• It is the point on the bending moment diagram where bending

moment changes the sign from positive to negative or vice versa.

• It is also called “Inflection point”. At the point of inflection point

or contra flexure the bending moment is zero.