Theory

Bending & Beam Theory

Bending: The behaviour of a slender structural element subjected to an external load applied perpendicular to the longitudinal axis of the element.

Transverse load applied to a pinned element:

M = Bending moment, this depends on the load (F) and the dimensions (x)

u = displacement which is a function of x

R = radius of curvature and can be assumed to be constant

Euler-Bernoulli beam theory assumptions:

1. The beam is initially straight, unstressed and symmetric.

2. The material of the beam is linearly elastic, homogeneous and isotropic.

3. The proportional limit is not exceeded so there is no plastic deformation.

4. Young’s modulus for the material is the same in tension and compression.

5. All deflections are small, so that planar cross-sections remain planar before and after bending. (e.g. it goes back to where it was)

6. Applied load is pure bending moment (e.g. external forces cause no twisting)

When bending an element, one surface of the material stretches in tension while the opposite surface compresses. Unlike axial/linear loads, bending creates non-uniform stress through the element. Therefore, there is a line or region of zero stress between the two surfaces, called the neutral axis.

Tensile stress is parallel to the neutral axis and constant across the cross section of the beam.

If a bending moment causes compression at the top surface, it is considered to be a positive moment.

If a bending moment causes tension at the top surface, it is considered to be a negative moment.

Beam theory equation:

σ = Stress y = Distance from neutral axis

M = Bending Moment

I = Second moment of area

E = Young’s Modulus

ĸ = Curvature

The second moment of area for a material is a geometrical constant that relates to the cross-sectional area of the beam.

Flexural rigidity:

There are three arrangements of load F, each creating a distribution of the moment M(x).

Bending Stiffness: The resistance offered by a structure while undergoing bending.

Integrate the Beam theory equation twice to get to find the bending stiffness for a rectangular beam:

S = Bending StiffnessF = Force

δ = Maximum Deflection

C₁= Geometry Constant (typically taken as 1)

E = Young’s Modulus

I = Second Moment of Area

L = Length of Beam

Second Moment of Area:

I = Second Moment of Area A = Cross-sectional Area

y = Maximum Distance from Neutral Axis

This means that the stresses across the width of the beam section can be determined if the following values are known:

· The applied bending moment (M)

· The location of the centroid (i.e. the Neutral Axis)

· The second moment of area for the section (I)

Examples of I:

Rectangular Cross-section:

This can be evaluated to find a standard solution for a rectangular beam with uniform stress across the width:

This gives:

Circular Cross-section:

Semi-circular cross-section:

I-beam cross-section:

This is the best component for construction as it can withstand both shear and tension forces.

Circular Tube cross-section:

Rectangular Tube cross-section: