# True stress and engineering stress pdf

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- True Stress-True Strain Models for Structural Steel Elements
- From engineering to true strain, true stress
- True Stress-True Strain Models for Structural Steel Elements

*Think about pulling a bar in tension. Load divided by cross-sectional area is force, or stress. But what cross section are you considering?*

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## True Stress-True Strain Models for Structural Steel Elements

Arasaratnam, K. Sivakumaran, M. A standard uniaxial tensile test, which establishes the engineering stress-strain relationship, in general, provides the basic mechanical properties of steel required by a structural designer. Modern numerical analysis techniques used for analysis of large strain problems such as failure analysis of steel structures and elements metal forming, metal cutting, and so forth, will require implementation and use of true stress-true strain material characterization.

This paper establishes a five stage true stress-strain model for A and W steel grades, which can capture the behavior of structural steel, including the postultimate behavior of steel, until fracture. The proposed model uses a power law in strain hardening range and a weighted power law in the postultimate range. The true stress-true strain model parameters were established through matching of numerical analysis results with the corresponding standard uniaxial tensile test experimental results.

The material constitutive relationship so derived was then applied to predict the load-deformation behavior of coupons with a hole in the middle region subjected to direct tension loading. The predicted load-deformation behavior of perforated tension coupons agreed well with the corresponding test results validating the proposed characterization of the true stress-true strain relationship for structural steel.

The finite-element- FE- method-based numerical analysis and other numerical analysis techniques are widely used in research involving structural steel and in the analysis and design of steel structures and elements. In research, numerical modeling techniques are often used to effectively expand the limited experimental results and used to investigate the influence of relevant parameters associated with a problem.

Such simulations models for structural steel, however, require the use of realistic material stress-strain relationships, often extending up to fracture. Mechanical behavior of metallic type material, such as that of steel, is generally established by means of uniaxial tension test. Such tension test protocol [ 1 ], which was primarily created only for use in comparison of different steels, establishes the engineering stress and the engineering strain.

Figure 1 shows a typical engineering stress-strain relationship for steel solid line , where the stress was calculated as load divided by the original cross-section area of the tension coupon, and the engineering strain was calculated as change in length divided by the original gauge length.

Such calculations, which do not recognize the area changes during increasing loads, are used for convenient of measurements of dimensions and will always show an elastic range Region-I , strain hardening range Region-IV , and a strain softening range Region-V. The stress-strain relationship established on the basis of instantaneous deformed dimensions of the test coupon is known as the true stress-true strain relationship dash line in Figure 1.

For all practical purposes, the engineering relations and the true relations would coincide up to yield point; however, the two relations would diverge beyond this point. Figure 1 shows the qualitative differences between the engineering stress-strain relation and the true stress-strain relation. Accurate numerical modeling of large strain problems such as failure analysis of steel structures and elements, metal forming, metal cutting, and so forth, will require implementation and use of true stress-true strain material characterization.

The objective of this investigation is to develop true stress-true strain relationships for structural steels in general, and for A and W steel grades in particular. This paper establishes five-stage true stress-true strain models for structural steels, based on numerical simulations calibrated against experimental uniaxial tension test results.

Subsequently, the accuracy of these proposed models was established through comparisons with the experimental uniaxial tension test results associated with tension coupons having a small size central hole.

The stress parameters are established using the original cross-section area of the specimen, and the average strain within the gauge length is established using the original gauge length. Because of the use of original dimensions in engineering stress-strain calculations, such relations will always show an elastic range, strain hardening range, and a strain softening range. In general, the strain softening is associated with the necking range of the test.

Once the specimen begins to neck, the distribution of stresses and strains become complex and the magnitude of such quantities become difficult to establish [ 2 ]. Owing to the nonuniform stress-strain distributions existing at the neck for high levels of axial deformation, it has long been recognized that the changes in the geometric dimensions of the specimen need to be considered in order to properly describe the material response during the whole deformation process up to the fracture [ 3 , 4 ].

The true stress-true strain relationship is based on the instantaneous geometric dimensions of the test specimen. Figure 1 illustrates the engineering stress-strain relationship and the true stress-true strain relationships for structural steels. These relationships can be divided into five different regions as follows. Region-I Linear Elastic Range During the initial stages of loading, stress varies linearly proportional to strain up to a proportional limit.

The corresponding true stress and the true strain, which recognize the deformed geometrics of the section during tests, can be established directly from the engineering stress and the engineering strain based on the concept of uniform stress, small dimensional change, and incompressible material, which is valid for steel. The difference between true stress and engineering stress at proportional limit stress may be about 0.

The value for m must be determined from the uniaxial tension test. Region-IV Strain Hardening At the end of yield plateau, strain hardening begins with a subsequent increase in stress. Region-IV includes the strain hardening range up to ultimate strength when the test specimen may begin to exhibit necking. However, a power law is often used to relate the true stress to the true strain in this strain hardening region [ 6 , 7 ]. Region-V Strain Softening This region represents the behavior of the material in the apparent strain softening region.

As explained earlier, the apparent strain softening is due to the use of the original cross-sectional area, and should the actual cross-sectional area be used, the stress and strain would continue to increase. The true stress-strain relations cannot be established in this region from engineering stress-strain values; thus, an experimental-numerical iterative approach was used in this study to derive the true stress-strain material characterization for this region.

Zhano and Li [ 8 ] proposed that the parameters for a true stress-true strain relation be determined by using iterative FE method with an experimental tensile load-extension curve as a target. Although this method establishes the true stress-true strain relations from standard tensile test results without measurements of the deformed dimensions of the test specimens, the main shortcoming is that the entire stress-strain relation during necking is treated as an unknown and a trial and error procedure is used for a series of strain intervals until good correlation with the experimental results is attained.

By nature, Zhano and Li [ 8 ] proposed method is computationally intensive and time consuming. Ling [ 9 ] proposed a weighted-average method for determining the uniaxial true stress versus true strain relation during necking.

This method requires identification of a lower and an upper bound for the true stress-strain function during necking and expresses the true stress-strain relation as the weighted average of these two bounds.

The weighting constant w has to be established in an iterative manner by numerical simulation of a tensile test until a good correlation is achieved between the calculated and the experimental load extension curve. In summary, this paper proposes a five stage characterization for the true stress-true strain relations for structural steel. The next section describes an experimental program conducted to establish the above parameters for ASTM A steel and the W steel grades and to validate the proposed model.

The A is a relatively new steel grade for building construction in North America. The true stress-true strain model parameters were established through amalgamation of experimental and numerical modeling techniques. The test program considered twenty eight tensile coupons, fourteen each from two different steel grades, namely, ASTM A steel and the W steel. For each steel grade, eight coupons were taken from the flanges and six coupons were from the web of the section.

The fabrication dimensions of the tensile coupons were in accordance with ASTM A [ 1 ] specifications and recommendations. For each specimen, three thickness measurements and three width measurements were taken at different locations within the reduced cross-section of the tensile coupons, and the average thickness and the average width of the test coupons were established.

The thickness of the flange coupons was about 9. The initial gross cross-sectional area of each specimen was calculated based on these average dimensions.

Some test specimens, which were used for the validation of the proposed model, had a central hole. The net area at the hole location was established based on measured hole diameter. Three identical flange and web coupons with no holes shown as F1, W1, etc.

Five remaining flange coupons and the three remaining web coupons were used as perforated tension coupons having different diameter holes at the centre of the specimens. The photographic image of the test specimens solid sample with no holes, and perforated samples is shown in Figures 2 a and 2 b , respectively.

Figure 3 shows the engineering stress-engineering strain relationships obtained during these tests. As evident from this figure, consistent results were obtained for three identical specimens. Furthermore, the specimens from the web exhibited yield plateau, whereas no such behavior was observed in the specimens taken from the flange. Table 1 summarizes the mechanical properties established from the solid coupon tensile tests.

These coupons reached the ultimate strength at the strain of Figure 4 shows a representative calculation corresponding to W web element. Complete power law relationships for A, W flange and web elements are given in Table 2. The task is to match the finite element numerical analysis results with the corresponding experimental results in this region. The model used the 4-node shell elements with six degrees of freedom per node.

This element can be employed to model thick and thin general shell structures, and it accounts for finite strains by allowing for changes in the element thickness [ 12 ]. The model also incorporated a geometric imperfection maximum amplitude of 0.

The analysis incorporated both geometric and material nonlinearities von Mises yield criterion and isotropic strain hardening rule. One edge of the model was fully restrained while the other end was subjected to a uniform displacement. The true stress and strain relationship for Regions-I, II, III, and IV used in the analysis model was derived from the engineering stress-strain curve obtained from tension coupon tests as described above and as given in Table 2.

Figure 5 shows a representative FE model used to reproduce the standard coupon test and the associated failure of the model due to necking followed by fracture. This figure also shows the boundary conditions used in the FE model. Figure 6 shows the resulting FE predicted responses along with the experimental responses of three identical tension coupons A flange. Table 2 shows the values of the weighting constants for A, W flange and web elements. Table 3 summarizes the experimental and FE predicted values for the engineering stresses and strains at fracture.

The predicted stresses and strains were in good agreement with the corresponding experimental values considering the three identical specimens. Figure 7 shows the resulting true stress-true strain model for A flange element. The proposed true stress-true strain constitutive relations were further validated by incorporating them in a finite element model for tension coupons having a central hole and through comparison of the FE numerical results with the corresponding experimental results.

This part of the investigation considered sixteen test cases consisting of eight A steel grade and eight W steel grade. Each steel grade considered five flange specimens and three web specimens containing a central hole. Figure 2 shows these test specimens. Figure 8 shows a representative test specimen with a hole and the corresponding FE model. This figure also shows the experimental failure mode and the fracture during FE analysis. Overall, visually similar failure patterns were observed.

Figure 9 establishes the comparison between the FE results and the corresponding test results for the perforated specimens obtained from the flanges and webs of the A steel section. As can be seen in this figure, the stress-strain responses obtained through FE model incorporating the proposed true stress-true strain constitutive relations showed a reasonably good agreement with the test responses of similar samples.

Similar comparisons were also made on the W steel perforated tension members. Again, the numerical simulations agreed well with the experimental results, particularly in predicting the ultimate strengths of perforated samples. Table 4 presents the experimentally and numerically obtained ultimate strength values for the perforated coupons. Steel structures construction often necessitates fabrication of holes in the flanges of steel beams [ 14 ]. If one has to build finite element models for such studies or other similar studies on steel structures and elements, then such FE models require realistic material stress-strain relationships, which can capture the fracture of steel as well.

Traditional uniaxial tension tests provide engineering stress-engineering strain results which are not accurate particularly in the strain hardening range and in the postultimate strength range. This investigation developed true stress-true strain relationships for structural steels in general, and for A and W steel grades in particular.

## From engineering to true strain, true stress

Every component in a linear motion system experiences some form of loading due to applied forces or motion. Strain is the deformation or displacement of material that results from an applied stress. The most common way to analyze the relationship between stress and strain for a particular material is with a stress-strain diagram. For many materials, the proportional limit and the elastic limit are the same or nearly equal. In the stress-strain curve shown here, the proportional limit and the elastic limit are assumed to be the same.

First of all, you may check that your experimental data from a uniaxial tension test is expressed in terms of true stress vs. Be aware that experimental data always includes some degree of error and thus tends to be somewhat noisy or erratic. Input of noisy experimental data may cause spurious behavior, particularly in the case of the default, 3-iteration plane stress plasticity algorithm for shells. The effective plastic strain values input in defining a stress vs. True stress is input directly for the stress values. For metals, E is very large compared to the yield stress so it's fairly common practice in the case of metals to just subtract off a constant value equal to the strain at initial yield from all subsequent strain values. In any case, the first plastic strain value should be input as zero and the first stress value should be the initial yield stress.

## True Stress-True Strain Models for Structural Steel Elements

Arasaratnam, K. Sivakumaran, M. A standard uniaxial tensile test, which establishes the engineering stress-strain relationship, in general, provides the basic mechanical properties of steel required by a structural designer.

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