The stress-strain curves from tensile tests are well-known. They are performed according to EN ISO 6892 and provide us with valuable information about material behavior under uniaxial (tensile) loading. A specimen and measured length L is defined typically as \(L=5.65 \cdot \sqrt{A_0}\), and deformation measured on this length \(\Delta L\) is observed in relation to axial force. The strain \(\varepsilon\) is calculated as:
\[\varepsilon=\frac{\Delta L}{L}\]
The stress is calculated using constant, initial area \(A_0\):
\[ \sigma = \frac{N}{A_0} \]
where N is applied normal force. That is how we obtain the engineering stress-strain curve. An example of tensile tests is shown below for S235.
There are criteria for the structural steel at the beginning of codes regarding \(f_y\) and \(f_u\) ratio and elongation at failure. For example in EN 1993-1-1 – Cl. 3.2.2:
- \(f_u/f_y \ge 1.10\)
- elongation at failure not less than 15%
- \(\varepsilon_u \ge 15 \varepsilon_y\), where \(\varepsilon_u\) is elongation at ultimate strength and \(\varepsilon_y=f_y/E\)
The strain at fracture must be therefore higher than 15%, and from the experiments we often get values like 30%. Why do we use only 5% as a plastic strain limit?
There are three main reasons:
- We do not want to reach necking, i.e. go beyond \(\varepsilon_u\), which is strain at \(f_u\)
- Numerical error is growing with increasing plastic strain
- Uniaxial loading in tensile is not the most dangerous
Numerical error of shell finite elements
Engineers like to create models that are a simplification of reality. The engineering stress-strain curve is a simplification. In reality, due to Poisson coefficient (typically assumed for steel \(\nu=0.3\)) the loaded area shrinks from \(A_0\), and the stress is actually higher. This effect cannot be captured by shells, because their thickness remains constant. The real strain is also not \(\varepsilon=\Delta L/L\) but \(\varepsilon=\ln (\Delta L/L+1)\). We can plot the error as a ratio of linear strain to logarithmic strain. The error grows almost linearly with increasing strain. At \(\varepsilon= 5\%\) we get 2.5% numerical error. At \(\varepsilon= 10\%\), we get 5% numerical error. This is the threshold we generally do not want to overstep. So we should always stay below \(\varepsilon= 10\%\).
Effect of stress triaxiality
Structural engineers know stress triaxiality from soil mechanics. Also, in concrete structures, e.g. design of base plates, we utilize stress triaxiality by increasing the strength of partially loaded areas confined by larger bulk of concrete (EN 1992-1-1 – 6.7 allows for maximum increase by 3 times, while research shows even much higher values, e.g. 15 times for reinforced concrete). Stress triaxiality affects steel structures as well.
Stress triaxiality of steel elements is a phenomenon studied by mechanical engineers. There is a limited number of research studies into structural steel used in civil engineering. The stress triaxiality appears:
- At necking for uniaxial loading. We never want to get so far in design.
- In notches - e.g. bolt holes.
- In confined elements, e.g. a beam flange butt welded to a stiff column flange.
Multiaxial compressive stress is positive and prevents any fracture.
On the other hand, multiaxial tensile stress leads to:
- increase strength,
- decreased strain at failure.
The triaxiality is quantified by triaxiality parameter:
\[\eta=\frac{\sigma_H}{\bar \sigma}\]
where \(\sigma_H\) is hydrostatic stress and \(\bar \sigma\) is von Mises stress. \(\eta = -1/3\) means triaxial compression, for which no rupture is possible. The strain may theoretically be increased infinitely without any damage. For tensile test before necking, \(\eta=1/3\). It is an open question, what is the highest positive triaxial parameter in structural steel connections. There are limitations on the number and radii of notches by detailing rules, e.g. bolt hole sizes, bolt spacings, and bolt edge distances. This requires further study, but expectations are that the triaxiality parameter stays below 1.
The figure below is taken from the very influential paper by Yingbin Bao, Tomasz Wierzbicki: On fracture locus in the equivalent strain and stress triaxiality space, doi:10.1016/j.ijmecsci.2004.02.006:
Again, further research is necessary, but a number of studies of notched specimens can be found in the literature. For the specimens of mild steel, the plastic strain at initial damage does not drop below 5%. This may not hold true for high-strength steel (\(f_y \ge 460 \textrm{ MPa}\)).
Shell elements are simplified and through-thickness stress is always 0. For shell elements, the triaxiality is at most 2/3. The effect of triaxiality may therefore be taken into account only indirectly. It is possible that a model with variable strain limit will be implemented in the future to IDEA StatiCa models.
Validation of plastic strain limit
Plastic strain limit is needed for the failure mode of tensile rupture. This failure mode (or limit state in American terminology) uses typically a combination of yield strength and ultimate strength. Strain hardening is very limited in IDEA StatiCa models. In total 529 specimens where tensile rupture of plates governed were collected from the literature, and IDEA StatiCa Connection models were created in this study. The results depend on plastic strain limit and meshing. For the default setting of meshing and plastic strain limit \(\varepsilon=5%\), the summary is as follows:
"Using measured material and geometric properties without resistance factors applied, the strength from IDEA StatiCa was less than or equal to the experimentally observed strength for all but 12 specimens out of 529 (9 of which were fabricated with high strength steel, \(f_y = 847 \textrm{ MPa}\)."
It can be concluded that \(\varepsilon=5\%\) with default meshing is suitable for the failure mode of tensile rupture for mild steel.
References
Yingbin Bao, Tomasz Wierzbicki: On fracture locus in the equivalent strain and stress triaxiality space, doi:10.1016/j.ijmecsci.2004.02.006
Junhe Lian: A generalised hybrid damage mechanics model for steel sheets and heavy plates, https://publications.rwth-aachen.de/record/564339/files/564339.PDF?version=1
https://doi.org/10.1016/j.msea.2020.140332
DOI: 10.2495/MC190211
DOI: 10.1115/1.3078390