Basic stress analysis calculations

Mechanical engineers need to be able to calculate things! One of the most important things a mechanical engineer must know are basic stress analysis calculations. This guide covers all the fundamental aspects of this area.

But first let’s get right down to basics. What is stress? Stress is the amount of internal force that is sustained and exerted on the molecular level between the particles of a material. Stress is the result of external forces applied on something, so it is present in all things found on our planet at all times since gravity is generating a weight force for everything that has mass. All kinds and types of forces applied on a material create stress in it, and this stress is typically invisible to our eyes as it occurs on the molecular level. This is why stress isn’t only generated by the application of an external load or force, but also due to temperature or chemical changes that may increase the molecular activity on a material, or thanks to specialized manufacturing methods that achieve a kind of stress storing into something like concrete and glass.

The only way that we can notice the existence of stress is by observing a kind of a deformation that takes place. For example, when a weightlifter lifts the metallic barbell we can observe that there is a noticeable bend on the sides near the weight plates. This temporary deformation is called “elastic stress” or strain, and has a certain limit up to which it remains temporary. Should this limit be surpassed, the deformation becomes permanent and the stress is called viscous or plastic stress. Simply put, stress is an internal resistance of a body against its deformation, so it has a limit and that limit is defined by the molecular structure of the material that constitutes the body.

Types of Stress

There are three types of basic stresses that are categorised based on how exactly they affect the body that sustains them, namely the compressive stress, shearing stress, and tensile stress.

• Tensile stress is the material’s resistance to tearing, so it is generated when forces of opposite direction are pulling it apart. A classic example of tensile stress is the game of “tug of war” where two teams pull a rope apart.
• Compressive stress is the opposite to tensile stress, meaning that the forces are compressing the material. An example of this is you sitting on your chair with your weight pressing the chair rod down and the ground resistance force pressing it upwards. This results in the generation of compressive stress on the center of the rod.
• Shear stress is the resistance generated by the material on a specific cross sectional point, and against deforming opposite forces applied on itself or objects/materials that are connected to it. An example of this is the act of cutting a piece of paper with a scissor, applying opposite forces on its sides that cut the paper material on the point of a cross section where the shear stress is generated.

Basic Stress Analysis Calculations

Stress is symbolized with “σ” and is measured in N/m² or Pascal (Pa) which is actually an SI unit of pressure. Shear stress is symbolized with “τ” for differentiation. As expected by the units, stress is given by dividing the force by the area of its generation, and since this area (“A”) is either sectional or axial, the basic stress formula is “σ = F/A”.

By experiment or through software simulation, we can figure out when a material is elongating or compressing with the strain formula which is “ε = ΔL/L”. This is the division of the change in the material’s length to its original length. As the stress value increases, the strain increases proportionally up to the point of the elastic limit which is where the stress becomes viscous/plastic from elastic.

After having calculated the stress and the strain, we can calculate the modulus of elasticity which is given by the formula: “Ε = σ/ε”. This is also called the “Young’s modulus” and is a measure of the stiffness of a material.

Another important element that we can calculate in the context of the basic stress analysis is the “Poisson’s ration” (μ) or the ratio of the lateral strain to the longitudinal strain. This ratio is especially interesting for the analysis of structural elements such as beams, slabs and columns.

Additionally, if we have elements that are subjected to both tension and compression at the same time, we use the bending stress formula which is “σb = 3 FL/2wt²” where F is the force, L is the length of the structural element, w is the width, and t is its thickness. Similarly, for the calculation of the bending modulus, we use the formula “Eb = FL³/4wt³y” with y being the deflection at the load point.

Finally, no “basic stress analysis calculations” guide would be complete without explaining how to calculate the max stress based on a selected safety factor. The safety factor is given by the formula “fs = Ys / Ds”, with Ys being the yield strength of the material and Ds the design stress, both defined during the experimental phase. Then we conclude by calculating the Maximum allowable stress as = ultimate tensile strength / factor of safety.

Summary table of basic stress analysis formulae

Basic stress formula	σ = F/A	σ = Stress, measured in N/m^2 or Pascals (Pa). Instead of σ use τ for shear stress. A = Area (this can be either sectional or axial)
Basic strain formula	ε = ΔL/L	ε = Strain ΔL = Change in length L = Initial length
Modulus of elasticity (Youngs modulus)	Ε = σ/ε	E = Modulus of elasticity σ = Stress ε = Strain
Poisson’s Ratio	υ = – εt / εl	υ = Poisson’s ratio εt = Transverse strain εl= Longitudinal or axial strain
Bending stress	σb = 3 FL/2wt2	F = Force L = Length of the structural element w = Width t = Thickness
Bending modulus	Eb = FL3/4wt3y	F = Force L = Length of the structural element w = Width t = Thickness y = Deflection at the load point
Factor of Safety (FoS)	fs = Ys / Ds	fs = Factor of Safety (FoS) Ys = Yield strength of the material Ds = Design stress
Maximum allowable stress	UTS/fs	UTS = Ultimate Tensile Strength fs = Factor of Safety (FoS)