To calculate the stresses and deformations in structures, we must know the material behaviour, which can be obtained only by experiments.
Through standardised tests, the material properties are established. One such test is the tensile test, resulting in the stress-strain diagram.
This section addressesd mainly materials with linear-elastic behaviour, which obey Hooke's Law.
Strength, stiffness and ductility are important material properties and can be described as follows:
- strength - the resistance that has to be overcome to break the cohesion of the material
- stiffness - the resistance against deformation
- ductility - the capacity to undergo large strains before fracture occurs
The tensile test is often used to determine the strength, stiffness and ductility of a material. In tensile tests, a specimen of the material in the form of a bar is slowly stretched until fracture occurs. For each applied elongation, ΔL, the required strength, F, is measured and both values are plotted on a F- ΔL diagram; force-elongation diagram:
The figure above shows the force-elongation diagram (not to scale) for hot rolled steel (mild steel) in tension.
There are four stages to this F- ΔL diagram:
- Linear-elastic stage - path OA
This part of the diagram is practically straight. Up to point A there is a proportionality (linear relationship) between the force, F, and elongation, ΔL. If the force in OA is removed, the same path is followed in the opposite direction until reaching point O again; the bar returns to its original length. This type of behaviour is elastic,
- Yield stage (or platic stage) - path AB
Path AB of the diagram generally includes a number of 'bumps' but is otherwise generally horizontal. This means the elongation of the bar increases with a nearly constant foce. This phenomenon is known as yielding or plastic flow of the material.
- Strain hardening stage - path BC
When deformation becomes larger, the material may offer additional resistance. The required force to obtain the elongation increases. This is called strain hardening.
- Necking stage - path CD
Beyond point C, the force decreases with increasing elongation. Locally, the bar produces a pronounced necking that increasees until fracture occus at D. At fracture the force falls away and both parts of the bare spring back a little elastically.
If somewhere between point A (the limit of proportionality) and point D (at which fracture occurs) the force is removed, the test bar will revert a little elastically. The return path (unloading path) is a nearly straight line parallel to OA. In the F-ΔL diagram above this is shown by means of the dashed line. Once the force has been removed the bar demonstrates a permanent set or plastic elongation ΔLp; the elastic elongation was ΔLe.
The F-ΔL diagram depends not only on the material, but also on the dimensions of the test bar, namely the length, L, between the prismatic part of the bar, and the area, A, of the cross-section.
If the bar is twice as long and subjected to the same force, then the elongation is twice as large. The elongation ΔL is therefore proportional to the length of the bar.
To eliminate the influence of the length of the test bar, we plot ε = ΔL/L on the horizontal axis instead of ΔL. The dimensionless deformation quantity
is referred to as the strain of the bar.
If the cross-section, A, of the bar is twice as large, a doubled force, F, is required to achieve the same elongation, ΔL. To get an equal elongation, ΔL, each bar has to be subjected to a normal force, F, and the total load on the system of two bars is 2·F. Therefore to force, F, is proportional to the area, A, of the cross-section of the bar.
To eliminate the influence of the area of the cross section, we plot the quantity
along the vertical axis instead of F; σ is the normal stress in the cross-section.
In general, the normal stress varies across the cross-section and σ = F/A should be seen as the "average" normal stress in the cross-section. If the cross-section is homogeneous (the cross-section consists of the same material everywhere) and the cross-section in question is far enough away from the ends of the bar where the forces are applied (these are disruption zones), then the normal stress due to tensile force is roughly constant over the cross-section.
By converting the force-elongation diagram (F-ΔL diagram) into a stress-strain diagram (σ-ε diagram) we eliminate the influence of the bar geometry on the result of the tension test. So test bars of various dimensions lead to approximately the same σ-ε diagrams.
The values found by experiments are of course subject to dispersion. In addition, they depend on the way the experiments are performed, such as the speed at which the force is increased. For all materials the test results are influenced by temperature, and for wood and concrete, humidity also plays a role.
The figure below shows a σ-ε diagram with a distinct yield range.
The specific quantities by which the shape of the stress-strain diagram is more or less determined are:
- fy - the yield point: also referred to as yield stress or yield strength. Strength quantities in the σ-ε diagram are indicated by the symbol f instead of σ.
- ft - the tensile strength: also referred to as ultimate (tensile) stress. To calculate the stress σ, the force may be divided by the original area, A, of the cross-section, or by the actual cross-section, A', which will have decreased from A through transverse contraction and necking. Since A' is less than A, the 'true' stress F/A' is larger than the nominal stress F/A. In practice, attention is restricted to the nominal stress.
- εy - the yield strain, that is the strain at the start of the yield stage.
- εpl - the strain at the end of the yield stage.
- εt - the strain associated with the tensile strength, ft.
- εu - the strain at fracture.
In the elastic range, there is a linear relationship between the stress and strain:
The proportionality factor, E, is a material constant and is known as the modulus of elasticity or Young's modulus. The modulus of elasticity characterises the resistance (stiffness) of the material with respect to deformations due to change in length. In the σ-ε diagram, the modulus of elasticity is equal to the slope E = σ/ε of the linear-elastic stage.
Since the strain is dimensionless, the modulus of elasticity has the dimension of stress (force/area).
In the figure above, the concepts of stiffness, strength, etc., are shown in the σ-ε diagram.
- a stiff material has a larger modulus of elasticity, E, than a compliant material.
- a hard material has a larger yield point, fy, than a soft material.
- a strong material has a higher tensile strength, ft, than a weak material.
- a ductile material has a larger strain, εu, at fracture than a brittle material.
Ductile materials include most metals, such as steel and aluminium. For metals, the σ-ε diagrams for tension and compression are generally equal, so the compressive strength, f'c (*), is equal to the tensile strength, ft.
Materials in which fracture occurs with minor strain are known as brittle materials. Examples include concrete, stone, cast iron, glass. With stone-like materials, the diagrams for tension and compression generally differ and the compressive strength is generally larger than the tensile strength.
* Note: in mechanics, it is convention to call normal stresses positive if they are tensile. If one is primarily dealing with comrpessive stresses, it may be convenient to call compressive stresses positive. In such cases, the prime (eg. f'c) is used for the change in sign.
For materials with a sufficiently long yield stage (ductile materials such as steel), the σ-ε diagram is often simplified to that of the figure below for an elastic-plastic material.
In engineering practice, we are mainly interested in the situation in which the structure or part of the structure reaches a so-called limit state. Here, we distinguish between ultimate limit states and serviceability limit states:
- Ultimate limit states are states at which the structure or part of it collapses. This may be due to a loss of equilibrium (eg. through turning over, sliding, floating or instability) or to a loss of carrying capacity (because the structure is not strong enough in one or more of its parts to transfer the forces to which they are subjected.
- Serviceability limit states are states in which the structure or part of the structure no longer functions appropriately (eg. due to excessive deformations, vibrations, cracking, etc.), often long before the structure collapses.
When in an ultimate limit state the structure collapses because one or more structural parts are no longer strong enough to transfer the forces, the material will be loaded to its ultimate in these parts, and ductile materials will be loaded far into the plastic region. The associated ultimate force (yield force) for ductile materials is determined bu the theory of plasticity.
In a serviceability limit state the deformations are generally so small that they are on the linear-elastic path of the σ-ε diagram, sufficiently far from the yield point. Calculations relating to the serviceability limit states are therefore performed according the the linear theory of elasticity, based on the proportionality between stress and strain.
The proportionality between stress and strain was found by Robert Hooke (1635-1703) and is known as Hooke's Law. Hooke formulated the law 'as is the tension, so is the force' in 1678.
is Hooke's law in its simplest form
Note that the use of the word "law" can be somewhat misleading. The character of this law is somewhat different to those of other generally applicable laws such as those of Newton. Hooke's law is no more than a good representation of certain results found by experiments. The approach is very good for the elastic stage in metals.
For wooden beams subject to moderate forces the approach is reasonable; time-dependent influences are corrected by a creep factor.
For concrete, the approximation is not so good. Under compression, the relationship between stress and strain is barely linear. Time-dependent influences (shrinkage and creep) are other complicating factors. However, in serviceability limit states, a linear-elastic material behaviour is also assumed for concrete. The time-dependent effects are taken into account in modulus of elasticity.
The figure below shows the first part of the linear-elastic stage for different materials in one σ-ε diagram. The slope of each path represents the modulus of elasticity, E = σ/ε, the material property that characterises the stiffness of the material against deformation through a change in length. The figure provides an idea of the differences in stiffness between the various materials in the elastic stage.