Embedded Files
Bending and beam theory
Bending : characterises the behaviours of a slender structural element subjected to an external load applied perpendicular to the longitudinal axis of that element.
Curvature:
The beam must also have constant curvature.
This is denoted by the kappa symbol and defined as 1/R, with R being the radius the beam would make if extended at its current angle.
Curvature: K = 1/R
We can model how a beam bends under axial forces and bending according to the Euler-Bernoulli theory.
Euler-Bernoulli Theory assumptions:
1.Beam is initially straight, not under stress and symmetric.
2.The material of the beam is linearly elastic, homogenous and isotropic.
3.The proportional limit is not exceeded - only in the elastic region and no plastic deformation.
4.Young's modulus for the material is the same in compression and tension.
5. All deflections are small, so that planner cross sections remain planar before and after bending.
6.Applied load in pure bending moment. (no twisting)
Note: When bending an element, one surface of the material stretches in tension while the opposite side compresses, therefore there is a line or region between the surfaces known as the 'neutral axis'. Stress increases with distance from this axis.
Beam theory equation :
Bending stiffness: otherwise known as flexural rigidity is defined as the resistance of a beam against bending deformation.
To find the bending stiffness for a rectangular beam you integrate the beam theory equation twice giving:
Second moment of inertia: The sum of all elementary products of area elements and the square of their respective distances from the centroidal axis.
Different examples of I (shape dependent):
Rectangular cross section:
Circular:
Semi-circular:
Circular hollow tube:
Rectangular tube:
I beam cross section:
Key Words
Torsion - twisting due to an applied force
Shaft - a structural member which is long and slender and subject to a torque (moment) about its long axis
Torque - Twisting action that occurs when you apply a twisting force
Shear Stress - stress (force per unit area) parallel to the surface
Torsion
As demonstrated in the drawings, you can see the differences between the torsion of a circular shaft and a square shaft. The surface distortions are caused by the stresses during contortion.
Torsional Stress:
As stated, the cross sections of circular shafts remain undistorted. When a torque is applied to the ends of of a circular shaft the shaft with twist with an angle . The twist angle starts at zero and increases linearly as a function of x. You can see the line of the plane as the dotted black line, due to the twisting angle this has now moved to the solid black line.
The angle will be a function of shaft length L, and the stiffness G which is the shear modulus-measure of stiffness of a material. There is an inner cone angle,θ, which is constant along the length.
Stresses
The distribution of shearing stresses due to torsional loads is assumed to be uniform. Thus the distribution of stress around a circle will be centered at its radius. This however also means that any imperfections such as surface roughness can concentrate in these areas which will increase the magnitude of stress at that spot which could potentially lead to breakage.
We can use this equation to find the shear stress a
Among other things
Where is shear stress
T is torque
G is shear modulus (this is specific to the material and can usually just be googled)
θ is twisting angle
L is length
J is polar second moment of area
r is radius
Maximum Shear Stress
As the maximum stresses produced during torsion are orientated 45 degrees around the shaft we are able to rearrange the shear stress equation to find a maximum shear stress, where the material will fail.
A Side Note - Polar Second Moment of Area:
The Polar Second Moment of Area depends on the shape of the shaft. Here are some example equations:
For a Solid Cylinder For a Hollow Cylinder
Rmax is the maximum radius of a cylinder while Rint is the internal radius
Quiz
Now that you've learned all about beam theory, try this quiz below!
Page updated
Report abuse