September 07, 2024

June 13, 2022

- Building Construction (76)
- Building Materials (74)
- Columns (2)
- Concrete Beam (3)
- Concrete Construction Techniques (4)
- Concrete Mix Design (9)
- Concrete Repair (11)
- Concrete Slab (10)
- Construction Equipment (15)
- Construction News (7)
- Design of Structures (15)
- Engineering Drawing (1)
- Estimation (2)
- Geotechnical engineering (25)
- Highway Engineering (11)
- Innovations (27)
- Material Testing (8)
- Matrix Analysis of Structures (2)
- Mechanical Engineering (3)
- Transportation Engineering (9)

- October 20239
- September 20234
- August 20231
- July 202312
- June 202310
- May 202311
- April 202310
- March 202318
- February 20234
- January 20238
- December 20229
- November 20229
- October 202212
- August 202217
- July 202211
- June 202215
- May 20221
- April 202213
- March 202215
- February 202214
- January 202223
- December 20216
- November 20212
- October 20217
- September 20215
- August 202118
- July 202122
- June 202115
- May 20216
- April 202110
- March 202111
- February 202132
- January 202123
- December 202011
- November 20203
- September 202019
- August 202012
- July 20204
- June 202010
- May 202012
- April 202010
- March 20206
- February 20203
- November 20192
- October 20194
- September 20193
- August 201915
- July 20199
- June 20191
- May 20193
- April 20192
- March 20193
- February 20195
- October 20182
- August 20182
- June 20183
- April 20187
- March 201813
- February 20189
- January 20184
- December 20171
- October 20172
- September 201711
- August 20176
- July 201713
- June 20177
- May 20176
- April 20171
- March 20172
- January 20171

Neenu
August 21, 2019

Simple bending or pure bending is defined as the phenomenon of the development of stresses throughout the length of the beam due to the action of bending moment exclusively. The stress development throughout the length of the beam is called **bending stress.**

Beam Construction; Image Courtesy: Molin Precast Products |

Simple bending can be explained by the below beam and load arrangement. As shown in the figure, the BM diagram of the corresponding beam arrangement has a constant moment along the beam length AB and the SF has nothing to contribute to that length.

Fig.1. Simple Bending Theory Explanation |

M/I = E/R = 𝛔/y

- The material of the beam is homogeneous and isotropic
- The transverse section of the beam remains plane before and after bending.
- The value of young's modulus is the same in tension and compression
- The beam is initially straight and all the longitudinal filaments bend into circular arcs with a common center of curvature
- The radius of curvature is large compared with the dimensions of the cross-section.
- Every layer of the beam is free to expand or contract independently of the layer below it.

As shown in the figure above, consider a beam small beam section ABCD with a length of dx. The N-N forms the neutral axis of the beam element. Section AB and CD are perpendicular to the neutral axis N-N. The beam element under the action of bending gets deformed as shown in figure-2(b).

The layers of the beam before bending do not remain the same after bending. The layer AC and BD have deformed to A'C' and B'D' respectively. The layer gets shortened at top and expands at the bottom layer. The beam layer does not undergo any change at the neutral layer. This means,

As we move from the bottom layer to the neutral layer, the length of the layers decreases. Hence, the increase or decrease of the length of the layer is dependent on its distance from the neutral axis. This theory of bending is called the theory of simple bending.

Given:

The radius of the Neutral layer = R

The angle subtended by A'B' and C'D' at O = θ

The radius of the Neutral layer = R

The angle subtended by A'B' and C'D' at O = θ

Then,

Strain Variation along the Depth of Beam

Original length of layer = EF = dx;

Original length of Neutral layer = NN = N'N' = dx

Hence, the strain in layer EF is directly proportional to the distance of the layer from the neutral axis. This relation shows the variation of the strain along the depth of the beam. The variation of strain is linear.

Stress Variation along Depth of Beam Young's modulus E = stress in layer EF / Strain in layer

The moments caused due to these internal forces must be equal and opposite the BM caused at section f of the beam.

The moment of resistance is defined as the algebraic sum of moments about the neutral axis of the internal forces developed in the beam.

If the cross-sectional area of the element is dA, then

Total Moment of resistance

**M/I = E/R = 𝛔/y ;**

The above equation is called the Bending Equation/ Flexural Formula.

Strain Variation along the Depth of Beam

Original length of layer = EF = dx;

Original length of Neutral layer = NN = N'N' = dx

From Figure,

N'N' = R x θ = dx

E'F' = (R + y ) θ

Strain in the layer EF = Increase in the length of layer EF / (Original Length)

= (E'F' -EF)/EF

= ( (R +y) θ - (Rx θ))/dx

= (R θ + y θ - R θ)/ dx

= y θ/dx

= y θ/R θ

=y/R

N'N' = R x θ = dx

E'F' = (R + y ) θ

Strain in the layer EF = Increase in the length of layer EF / (Original Length)

= (E'F' -EF)/EF

= ( (R +y) θ - (Rx θ))/dx

= (R θ + y θ - R θ)/ dx

= y θ/dx

= y θ/R θ

=y/R

Hence, the strain in layer EF is directly proportional to the distance of the layer from the neutral axis. This relation shows the variation of the strain along the depth of the beam. The variation of strain is linear.

Stress Variation along Depth of Beam Young's modulus E = stress in layer EF / Strain in layer

E = 𝛔 /(y/R)

=> E = 𝜎R/y

=> E/R = 𝛔/y

Moment of Resistance The layer above the neutral axis experience compressive force Fc and the layers below the neutral axis experience tensile force Ft.

=> E = 𝜎R/y

=> E/R = 𝛔/y

Moment of Resistance The layer above the neutral axis experience compressive force Fc and the layers below the neutral axis experience tensile force Ft.

For equilibrium:

Fc= Ft

Fc= Ft

The moments caused due to these internal forces must be equal and opposite the BM caused at section f of the beam.

The moment of resistance is defined as the algebraic sum of moments about the neutral axis of the internal forces developed in the beam.

If the cross-sectional area of the element is dA, then

Total Moment of resistance

M = ∫ (𝛔 dA . y) = ∫(Ey. y. dA)/R

=> M = (E/R)∫y^2 . dA

Moment of Inertia I = ∫y^2. dA

Hence, we get

M = EI/R

Hence, we get

The above equation is called the Bending Equation/ Flexural Formula.

March 02, 2022

January 19, 2022

March 22, 2024

July 02, 2020

July 17, 2021

April 15, 2024

August 11, 2021

September 10, 2017

March 02, 2022

Created By SoraTemplates | Distributed By Gooyaabi Templates

Close Menu

## 0 Comments

Commenting Spam Links Are Against Policies