Shear force-bending moment and shear stresses-bending stresses

Types of beams.

There are several types of beams commonly used in construction and engineering, including:

1. Simply Supported Beam. A beam supported at both ends, which can only experience bending and not axial forces.

2. Cantilever Beam: A beam that is fixed at one end and free at the other, often used in bridges and overhangs.

3. Fixed Beam: A beam that is fixed at both ends, preventing any movement or rotation.

4. Continuous Beam: A beam that is supported along its length by multiple supports, often usedin large structures such as bridges and buildings.

5. Propped Cantilever Beam: A cantilever beam that is supported by an intermediate prop, reducing the bending moment at the fixed end.

6. Overhanging Beam: A beam that extends beyond one or both of its supports, often used in building design to create a projecting roof or balcony

Types of Load.

Three main types of loads in strength of materials are

1. Uniformly distributed load (UDL)

2. Concentrated or point load.

3. uniformly varying load

UDL refers to a load that is evenly distributed over a structural member, causing an even stress distribution.

Concentrated or point load is a load that acts at a single point on a structural member, causing a high stress concentration at that point.

A uniformly varying load, also known as a gradually increasing or decreasing load, is a type of load in strength of materials where the magnitude of the load changes linearly over the length of a structural member. This type of load results in a parabolic distribution of stress along the member.

Define shear force

Define shear force and bending moment. Ans. Shear force: Shear force at any cross section of the beam is the algebraic sum of vertical forces on the beam acting on the right side or left side of the section is called shear force.

A shear force is the resultant vertical force acting on the either side of a section of a beam.

Bending Moment:

Bending moment at any section of a beam is the algebraic sum of moment of all forces acting on the right or left side of a section is called as bending moment.

State relation between rate of loading, shear force and bending moment.

Ans. a) Relation between rate of loading and shear force dF/dx = W The rate of change of shear force with respect to the distance is equal to the intensity of loading.

b) Relation between shear force and bending moment. dm/dx = F The rate of change of bending moment at any section is equal to the shear force at that section with respect to the distance

Define the point of contra flexure.

How is the point of contra flexure located for a beam? Point of contra-flexure: -The point at which bending moment diagram changes the sign from positive to negative or vice versa or the point at which BM is zero is 02 called as point of contra-flexure Location of point of contra-flexure

i)At the point of contra-flexure B.M is zero.

ii)Take B.M at the point of contra-flexure and equate with zero.

iii)The distance (location) of point of contra-flexure will be find from either end of beam.

Point D, B.M. = 0 , Point D is Point of Contra flexure

Flexural formula

Write the flexural formula. State the meaning of each term

M=Maximum Bending moment which is equal to moment of resistance of a beam

I=Moment of inertia of the beam section about the neutral axis Bending stress in layer at a distance

'y’ fromN.A. y =Distance of the layer from N.A. of the beam cross section

E =Modulus of elasticity of the beam material

R= Radius of curvature of a bent of beam

Assumptions in the theory of simple bending

State any four assumptions in the theory of simple bending.

1.The material of the beam is homogeneous and isotropic i.e. the beam made of the same material throughout and it has the elastic properties in all the directions.

2.The beam is straight before loading and is of uniform cross section throughout.

3.The beam material is stressed within its elastic limit and this obeys Hooke’s law

4.The transverse sections which were plane before bending remain plane after bending.

5.The beam is subjected to pure bending i.e. the effect of shear stress is totally neglected.

6.Each layer of the beam is free to expand or contract independently of the layer above or below it.

7.Young’s modulus E for the material has the same value in tension and compression

Neutral axis and moment of resistance.

With reference to theory of simple bending, explain neutral axis and moment of resistance.

Ans. Neutral Axis: The fibers in the lower part of the beam undergo elongation while those in the upper part are shortened. These changes in the lengths of the fibers set up tensile and compressive stresses in the fibers. The fibers in the centroidal layer are neither shortened nor elongated. These centroidal layers which do not undergo any extension or compression is called neutral layer or neutral surface.

When the beam is subjected to pure bending there will always be one layer which will not be subjected to either compression or tension. This layer is called as neutral layer

Moment of resistance:

Moment of resistance of the beam is the moment of couple formed by the total compressive force acting at the Centre of gravity of the compressive stress diagram and the total tensile force acting at the Centre of gravity of the tensile stress diagram. Moment of couple = C x Z or T x Z. This moment is called the moment of resistance of the beam and is denoted by Mr.

Shear stress distribution

shear stress distribution diagram for circular beam section