Engineering components may be subjected to direct tension or compression, shear, bending, torsion or combinations of these loading modes. The resulting stress, strain and deformation depend not only on the magnitude of the load, but also on its distribution, direction, duration and rate of application.
This page provides an overview of component loading and the relationship between applied loads, stress, strain, stiffness and possible failure. Detailed analysis methods are covered by the related RoyMech references linked below.
The principal mechanical loading modes encountered in machine components and structures include:
Many engineering components are subjected to more than one loading mode at the same time. A shaft, for example, may carry torsion from transmitted power while simultaneously being subjected to bending from gears, pulleys or its own weight.
For linear-elastic behaviour and sufficiently small deformation, stresses and deformations produced by individual loads may often be combined using the principle of superposition.
An axial load acting through the centroidal axis of a uniform member produces direct tensile or compressive stress. Under the ideal uniform-stress assumption:
where F is the axial force and A is the loaded cross-sectional area.
Shear loading acts tangentially to a section or interface. A simple average shear stress may be written:
where V is the shear force. The actual shear-stress distribution depends upon geometry and loading and is not necessarily uniform.
A bending moment produces longitudinal tensile and compressive stresses across a section. For linear-elastic beam bending under the assumptions of simple beam theory:
where M is the bending moment, y is the distance from the neutral axis and I is the second moment of area about that axis.
Torque applied to a member produces shear stress. For a circular shaft in linear-elastic torsion:
where T is torque, J is the polar second moment of area, r is radial position, G is shear modulus, θ is angle of twist and L is the shaft length.
See the RoyMech Torsion reference for detailed torsional analysis.
Component response is affected by how a load is applied as well as by its magnitude. Important loading conditions include:
The distinction is important because a component that safely supports a particular static load may not safely support the same nominal load when it is applied suddenly, repeatedly or dynamically.
External loads produce internal forces and moments within a component. Stress describes the intensity of these internal actions over an area.
For uniform direct loading:
The SI unit of stress is the pascal:
Engineering stresses are commonly expressed in MPa, where:
The simple expression F/A gives an average direct stress. Real stress distributions can be non-uniform because of bending, holes, shoulders, threads, contacts, load introduction and other geometric effects.
Strain describes deformation relative to the original dimensions of a material element. Strain is dimensionless.
For an original length L undergoing a small change in length ΔL:
Volumetric strain is the change in volume divided by the original volume:
Engineering shear strain is conventionally represented by γ. For a small shear deformation:
where Δx is the relative tangential displacement of two initially parallel planes separated by normal distance h. For small deformation, γ is also the change in angle between the corresponding material lines, expressed in radians.
Stress and strain need not result solely from directly applied external mechanical forces. Important additional causes include:
An engineering component is normally designed to behave in a predictable way throughout its intended loading range. The required behaviour depends upon its function.
Component design therefore involves more than simply ensuring that the calculated stress is below a material strength value. Stiffness, stability, fatigue resistance, wear, temperature, environment, dimensional tolerances and functional requirements may also govern the design.
A component has failed when it can no longer perform its required function. This does not necessarily require complete fracture.
Possible failure or unacceptable conditions include:
Ductile materials commonly exhibit appreciable plastic deformation before fracture under suitable loading conditions. Brittle materials generally undergo much less plastic deformation and can fracture rapidly once a critical stress or fracture condition is reached.
Actual failure behaviour depends on the material, stress state, temperature, loading rate, geometry, defects and environment. It should therefore not be assumed that a particular material will always fail in one predetermined mode.
A component that was adequate when new may later fail under a load below its original design capability because its condition has deteriorated. Causes include:
Some engineering failure modes do not follow the simple progressive relationship between static load and deformation. A component may appear satisfactory until a critical condition is reached or damage accumulates over many loading cycles.
Repeated or fluctuating stresses can initiate and propagate cracks even when the maximum nominal stress is below the material's static tensile strength. Fatigue behaviour depends on stress range, mean stress, stress concentrations, surface condition, environment, material and number of cycles.
Long slender members subjected to axial compression can lose stability by buckling. The critical load depends strongly on member stiffness, length and end restraint and may be much lower than the load required to cause material compressive yielding.
See the RoyMech Buckling and Strut reference.
A rotating shaft can experience large lateral vibration when its rotational speed approaches a critical speed associated with the shaft-system natural frequency. Mass distribution, stiffness, support conditions and damping all influence the critical speeds and response.
When the frequency of a periodic excitation approaches a natural frequency of a mechanical system, the vibration response can become large. The actual amplitude depends strongly on damping as well as the magnitude and frequency of the excitation.
Engineering material properties are commonly established using standardised laboratory tests performed under controlled conditions.
The uniaxial tensile test provides important mechanical properties including, as applicable:
See the RoyMech Stress-Strain Curve reference for further information.
Common indentation hardness tests include Brinell, Vickers and Rockwell methods. Hardness provides information about resistance to local indentation and can, for particular material classes and conditions, be correlated empirically with properties such as tensile strength.
Electrical resistance strain gauges are widely used to measure local surface strain in loaded components. With suitable material properties and stress-state assumptions, measured strains can be used to determine corresponding stresses.
Within the linear-elastic range, stress and strain are related by elastic material constants.
For uniaxial normal stress:
where E is Young's modulus.
For simple shear in the linear-elastic range:
where G is the shear modulus.
Bulk modulus describes resistance to uniform volumetric compression. Using pressure p as positive in compression:
where volumetric strain is defined positive for an increase in volume. Under positive hydrostatic pressure the volume decreases, so εv is negative and K remains positive.
If compressive volumetric strain is instead defined as a positive magnitude, the same relationship is commonly written simply as K = p / εv,comp.
For a homogeneous, isotropic, linear-elastic material, Young's modulus E, shear modulus G, bulk modulus K and Poisson's ratio ν are related. Two useful relationships are:
These relationships do not generally apply unchanged to anisotropic, non-linear or inelastic material behaviour.