Beam Stress and Superposition

Normal Stress

The normal stress in a beam is a resultant from axial loading.  This type of loading is a compressive or tensile force that acts through the centroid of the section.  The normal stress can be calculated with the following equation.

Although the normal stress can be uneven through the cross section of the beam near the points of loading it is usually taken as an average in the stress calculation.

Bending Normal Stress

Bending stress is a resultant of vertical or horizontal loading on a beam.  This type of loading creates a compressive and tensile stress.

Bending Shear Stress

Bending shear stress is the resultant of vertical or horizontal loading.  When the beam is loaded the fibers throughout the beam are elongating and contracting through each layer of the beam from the normal stresses, this difference in elongation creates a shear stress that transfers through each layer of the beam.

Torsional Shear Stress

Torsional shear stress is the resultant of a torque on the beam.  When torque is applied to the beam each cross section is experiencing this shear stress in the plane of the cross section. 

The generalized equation for pure torque shear on non-circular beams is given as

Bearing Stress

Bearing stress acts in a direction normal to the axis of the beam and does not add to the normal stresses from compression and bending.  These stresses are a resultant of the beam bearing on a support and are checked locally on the beam because the stresses do not radiate significantly throughout the beam.  

Warping Stress

One stress that is not commonly accounted for in general beam design is the warping stress.  This stress occurs when a torque is applied to a non-circular beam where the cross section of the beam will warp out of plane and cause normal and shear stresses throughout the affected section. 

Shear stress from warping

Normal stress from warping

Adding Stresses – Superposition

Each of these stresses represent either a normal or shear stress that can be combined to find the total stress acting on a finite point in the beam in each orthogonal direction.  An example of this superposition is shown below with each of the stress types.

Care must be taken when adding these stresses together as they do not always align in the same direction.  For instance, the shear stress vector from pure torque at the top of a shaft acts in a direction orthogonal to the bending shear, but in the middle of the shaft both shears act in the same direction and can be added directly.  Another example is bearing stress, this stress acts orthogonally to the axis of the beam and does not add to the normal stress in the calculations (this is an assumed behavior, in reality a bearing stress will act radially on the beam).

Nate Nissen

Vitruvius Engineer