### 3.4.1. Measures of Stress

Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019

3.4.1.1. Uniaxial Stress

Uniaxial Stress is a measure of the average axial load over the cross-sectional area of a structural member. Axial stress is achieved by an axial load applied along the axis of a straight member.

Figure 3.4.1‑1: Axial Stress

Note that in this text we are using the symbol f for axial stress. The Greek letter σ (sigma) is also often used for axial stress.

For the purposes of most analyses, the change in area of the cross-section caused by Poisson’s effect is not accounted for in the calculation of the value of stress.

This is in part due to the fact that most material strength data is calculated from test failure loads using the original cross section of the test article used to develop the strength data.

Ref  (MIL-HNDBK-5H, 1998) Section 1.4.4.5 “It should be noted that all stresses are based on the original cross-sectional area of the test specimen, without regard to the lateral contraction of the specimen, which actually occurs during the test.”

3.4.1.2. Bending Stress

Bending stress occurs whenever a member is loaded ‘off axis’. Bending stress occurs around the neutral axis of the section. The neutral axis of the section experiences no bending stress effects and lies on the centroid of the cross section. The neutral axis, or plane, stays at a constant length under bending effects as it experiences zero strain.

The outer fiber of the cross section experiences the highest bending stress. The maximum bending stress is usually calculated at the outer fiber, at a distance ‘y’ from the neutral axis.

Figure 3.4.1‑2: Bending Stress – Cantilever Example with Point Load

In the figure above, the example of a cantilevers is used. In a cantilever beam, the moment is developed over the length of the beam. The moment at any point along the cantilever beam is calculated by multiplying the load by the distance from the applied load to the point along the beam of interest.

It follows that the maximum moment occurs at the furthest point (towards the support) from the applied load.

You can download a spreadsheet that calculates the shear force and bending moment for a cantilever beam with a point load applied at the free end here:

A comprehensive set of beam analysis methods is defined in Section 8 of this book.

Figure 3.4.1‑3: Bending Stress – Cantilever Example with Applied Moment

Note that the triangular distribution of bending stress through the thickness of the cantilever beam depends on material elasticity. In the case where the stress exceeds the proportional limit of the material plastic bending, see section 14.1.1.

For an initial assessment, it is conservative to assume the material remains perfectly elastic for predicted stress values.

3.4.1.3. Shear Stress

Shear Stress is the component of stress coplanar with the material cross section. In the case of a cantilever beam, the shear stress is constant along the length of the beam. This type of shear is called transverse shear.

Note that in this text we are using the symbol f s for shear stress. The Greek letter τ (tau) is also often used to denote shear stress.

The distribution of shear stress in a beam does not affect the bending or axial stress distribution.

It is common to assume that the shear stress is constant across a cross section.

Figure 3.4.1‑4: Average Shear Stress Distribution

Note that the use of the letter ‘V’ is used to denote a load that results in a shear reaction from the structure it is applied to. The nature of the applied load is the same whether ‘P’ or ‘V’ is used. The type of load induced in the structure from the applied load is different.

In reality, the shear stress varies across the cross section according to the following relationship:

The shear stress distribution for a rectangular cross section beam is shown below:

Figure 3.4.1‑5: Actual Shear Stress Distribution

We have created several spreadsheets that calculate the parabolic shear distribution for common cross sections. These spreadsheets also calculate the average shear stress and compares the average shear stress with the peak parabolic shear stress:

### 3.4.1. Measures of Stress

Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019

3.4.1.1. Uniaxial Stress

Uniaxial Stress is a measure of the average axial load over the cross-sectional area of a structural member. Axial stress is achieved by an axial load applied along the axis of a straight member.

Figure 3.4.1‑1: Axial Stress

Note that in this text we are using the symbol f for axial stress. The Greek letter σ (sigma) is also often used for axial stress.

For the purposes of most analyses, the change in area of the cross-section caused by Poisson’s effect is not accounted for in the calculation of the value of stress.

This is in part due to the fact that most material strength data is calculated from test failure loads using the original cross section of the test article used to develop the strength data.

Ref  (MIL-HNDBK-5H, 1998) Section 1.4.4.5 “It should be noted that all stresses are based on the original cross-sectional area of the test specimen, without regard to the lateral contraction of the specimen, which actually occurs during the test.”

3.4.1.2. Bending Stress

Bending stress occurs whenever a member is loaded ‘off axis’. Bending stress occurs around the neutral axis of the section. The neutral axis of the section experiences no bending stress effects and lies on the centroid of the cross section. The neutral axis, or plane, stays at a constant length under bending effects as it experiences zero strain.

The outer fiber of the cross section experiences the highest bending stress. The maximum bending stress is usually calculated at the outer fiber, at a distance ‘y’ from the neutral axis.

Figure 3.4.1‑2: Bending Stress – Cantilever Example with Point Load

In the figure above, the example of a cantilevers is used. In a cantilever beam, the moment is developed over the length of the beam. The moment at any point along the cantilever beam is calculated by multiplying the load by the distance from the applied load to the point along the beam of interest.

It follows that the maximum moment occurs at the furthest point (towards the support) from the applied load.

You can download a spreadsheet that calculates the shear force and bending moment for a cantilever beam with a point load applied at the free end here:

A comprehensive set of beam analysis methods is defined in Section 8 of this book.

Figure 3.4.1‑3: Bending Stress – Cantilever Example with Applied Moment

Note that the triangular distribution of bending stress through the thickness of the cantilever beam depends on material elasticity. In the case where the stress exceeds the proportional limit of the material plastic bending, see section 14.1.1.

For an initial assessment, it is conservative to assume the material remains perfectly elastic for predicted stress values.

3.4.1.3. Shear Stress

Shear Stress is the component of stress coplanar with the material cross section. In the case of a cantilever beam, the shear stress is constant along the length of the beam. This type of shear is called transverse shear.

Note that in this text we are using the symbol f s for shear stress. The Greek letter τ (tau) is also often used to denote shear stress.

The distribution of shear stress in a beam does not affect the bending or axial stress distribution.

It is common to assume that the shear stress is constant across a cross section.

Figure 3.4.1‑4: Average Shear Stress Distribution

Note that the use of the letter ‘V’ is used to denote a load that results in a shear reaction from the structure it is applied to. The nature of the applied load is the same whether ‘P’ or ‘V’ is used. The type of load induced in the structure from the applied load is different.

In reality, the shear stress varies across the cross section according to the following relationship:

The shear stress distribution for a rectangular cross section beam is shown below:

Figure 3.4.1‑5: Actual Shear Stress Distribution

We have created several spreadsheets that calculate the parabolic shear distribution for common cross sections. These spreadsheets also calculate the average shear stress and compares the average shear stress with the peak parabolic shear stress: