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Effect of strain ageing on the mechanical properties of partially damaged structural mild steel
Construction and Building Materials 77 (2015) 83–93
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Effect of strain ageing on the mechanical properties of partially damaged structural mild steel Sajjad Hosseini a, Amin Heidarpour a,⇑, Frank Collins a, Christopher R. Hutchinson b a b
Department of Civil Engineering, Monash University, Melbourne VIC3800, Australia Department of Materials Engineering, Monash University, Melbourne VIC3800, Australia
h i g h l i g h t s Strain ageing effect on mechanical properties of damaged steel is investigated. Two-stage experimental program are conducted to incorporate strain ageing effect. Changes in microstructure of steel due to strain ageing effect are examined. Parameters of Ramberg–Osgood model are calibrated by incorporating strain ageing. It is shown strain ageing may have significant effect on behaviour of structures.
a r t i c l e
i n f o
Article history: Received 16 October 2014 Received in revised form 8 December 2014 Accepted 10 December 2014
Keywords: Strain ageing Damaged structures Steel material Ramberg–Osgood Micro-structure
a b s t r a c t This paper addresses the strain ageing effects on the mechanical properties of the partially damaged structural mild steel. Since repairing partly damaged structures may not occur immediately, the strain ageing effect can significantly influence the structural behaviour. The changes due to this effect have not so far been considered in the civil engineering design guidelines. In order to investigate strain ageing effects, two-stage experimental tests are carried out on the mild-steel specimens. In the first stage, partial damage is made using quasi-static loading. During the second stage, the strength and ductility of the specimens are examined after 2 and 7 days ‘ageing’ at room temperature and the results are compared with the corresponding no-age samples. The microstructure of the specimens is examined using scanning electron microscopy (SEM). To illustrate the effect of strain ageing on the global behaviour of steel structures, a numerical example is provided in which strain ageing impacts on loading capacity and deflection of a steel beam. Finally, the stress–strain relation of partially damaged mild-steel material incorporating strain ageing effects is expressed by calibrating the parameters of Ramberg–Osgood model. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In various situations, repair and rehabilitation of partially damaged steel structures are implemented in place of demolishing the structure and constructing a new one. As a result, a better understanding of the mechanical properties of the partially damaged materials under different loading scenarios is required. Once a structure is partly damaged (ie. strained), it may take several days until the repair process of the structural component(s) commences. This lapsed time (named as ageing in this paper) may significantly influence the structural response in terms of strength, ductility, energy absorption, etc.
⇑ Corresponding author. Tel.: +61 3 99024435; fax: +61 3 99054944. E-mail address: [email protected] (A. Heidarpour). http://dx.doi.org/10.1016/j.conbuildmat.2014.12.021 0950-0618/Ó 2014 Elsevier Ltd. All rights reserved.
Although various loading scenarios such as static, thermal, high-strain or seismic loadings may cause partial damage in structural members, only damages made by static loading are considered in this paper. Foundation settlement is a common source of damage in structures in which structural components may experience large relative displacements beyond the material yield point. Other examples in which unexpected large static loading may cause the damage include lateral ground water pressure applied to bridge piers, temporary loads during repairing process, and large hydrostatic pressures applied to the structures under tsunami. The strain ageing phenomenon, which occurs because of the time gap between the occurrence of damage caused by static loading and the repair process commencement, can substantially change the mechanical properties of the steel material such that utilizing the conventional constitutive material models without incorporating strain aging effects may lead to a completely incorrect assessment.
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Strain aging effects have been investigated by researchers worldwide in the field of materials engineering, e.g. [1,2], and it is well known that they must be taken into account during the production of some metal components, such as steel sheet panels for automotive applications. The physical origin of the strain aging effect in steels can be explained in the following way. The deformation of metals, including steels, occurs through the generation and movement of linear defects known as dislocations [2]. The strength of a metal is controlled by the stress required to move dislocations over substantial distances in the microstructure. Dislocations exhibit a long range strain field, and this strain field can interact with the strain field of solute atoms that are present in alloys. In the case if steels, the most important solutes from the point of view of strain ageing, are Carbon and Nitrogen. Even if present in small amounts, these solute species can diffuse within the material, even at room temperature, and segregate to dislocations, and hence partially relax their long range strain field. The consequence on the mechanical properties of the steels, is that subsequent loading can require a much higher stress level to then ‘un-pin’ the dislocations from the Carbon or Nitrogen solute atmospheres. This is the physical basis of the strain aging effect, e.g. [2]. This effect is especially important in steels because Carbon and Nitrogen can diffuse within the material at room temperature whereas other elements typically found in steels, such as Mn, are essentially immobile at room temperature. In the materials science and engineering literature, both experimental and theoretical/modelling aspects of the strain ageing effects have been investigated. The effect has most commonly been observed in typical mild steels but also occurs in dual phase steels, pearlitic steel wires [3] and even ultra low Carbon bake hardening (BH) steels [4]. Although most well known in the steel community, strain ageing effects are also observed in Ni-base alloys such as Inconel [5], structural intermetallics such as TiAl [6] and Al3Ti [7], Ti alloys [8], and Cu based alloys [9]. As well as experimental studies there have also been a large number of modelling studies of different aspects of the solute-dislocation interaction that is at the core of the strain aging effect. These include considerations of the strain [10], temperature [2] and solute content dependence [2]. Although these different aspects of strain aging have been investigated in the materials science context, strain aging can also play a significant role in designing and rehabilitating of structural elements. It is worth noting that there are two common types of ‘aging’ in civil engineering applications: short and long term aging. Long term aging refers to the time effect due to some phenomena such as creep and shrinkage in concrete, and creep in steel which occurs over a very long period of time. Although long term aging has been widely discussed in the literature, short term aging has not been considered in detail. This type of ageing (strain-ageing) is considered in this work. Currently, there is no appropriate proposed formulation in the prescriptive codes of practice in civil engineering to include the effect of strain ageing in design and evaluation of partly damaged steel structures [11–14], and therefore the outcome of this research can be used as a platform in rational analysis and design of steel structures. In this paper, changes in mechanical properties of partially damaged steel material due to strain ageing are addressed. The strain ageing effect on the stress and strain values is explored through two-stage experimental tests. Using scanning electron microscopy (SEM) images taken from the fracture surfaces of the broken specimens, the strain ageing effect on the microstructure of the material is also investigated. Moreover, the significant effect of the strain ageing on the global behaviour of steel structures is explored through a numerical example and the parameters of the
Ramberg–Osgood model are also calibrated so that the strain ageing effect is incorporated into the stress–strain relations. 2. Experimental program 2.1. Test material and specimen The coupon specimens are made of Grade 350 mild steel plates, approximately equivalent to ASTM A633A. The chemical composition of Grade 350 mild steel considered in this study is listed in Table 1 and the engineering stress–strain curve of the material under quasi-static loading is depicted in Fig. 1. The tests are conducted based on the requirements of standard ASTM E 8M-04 [15]. The tensile test coupons, with enlarged shoulders for gripping, are accurately machined from the primary steel plates. The measurements are made within the gauge length in reduced section and the specimens mounted in the machine from the shoulder part during the test. The specimen’s typical geometry is demonstrated in Fig. 2. Both faces of each specimen are ground to give uniform thickness and a smooth finish across the entire surface of the specimens. 2.2. Testing equipment An electromechanical testing machine, Instron 5982 dual column testing system with a 100 kN load cell, is employed in order to perform tensile testing. The axial strains are measured using a MTX LX 500 non-contact laser extensometer in which the value of strain is recorded by measuring the relative displacements between the reflective stickers glued on the specimen within the gauge length. In order to check the accuracy of the laser extensometer, the strain measurement in the first test was conducted by using both strain gauges and non-contact laser extensometers and a good agreement was achieved. Fig. 3 depicts 100 kN Instron testing machine, laser extensometer and specimen used in this study. In order to perform microstructural examinations, optical microscopy (OM) and a JEOL 7001F FEG scanning electron microscope (SEM) are employed. The images taken by both microscopes from the fracture surfaces are used to determine the fracture strain as well as to better interpret the effect of strain ageing on the fracture characteristics of the material.
Table 1 Chemical composition of Grade 350 mild steel. C
Si
Mn
S
P
0.22%
0.55%
1.70%
0.030%
0.040%
Fig. 1. Engineering stress–strain curve of Grade 350 steel material under quasistatic loading.
Fig. 2. Geometry of the specimen.
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Specimen
Instron 5982 100kN testing machine MTS Laser extensometer (LX1500)
Fig. 3. Testing machines used to conduct uniaxial tensile tests on specimens. 2.3. Testing procedure In this study, statically damaged specimens at room temperature are considered. In order to investigate the strain ageing effect, a two-stage loading case is applied. During stage I, the specimen is subjected to uniaxial tensile loading until partial damage occurs in the specimen. Since the damage level may affect the mechanical properties of the material, three different damage levels (D) of 1.5%, 13% and 19% strain shown in Fig. 1 are considered. D = 1.5% represents the strain close to the onset of strain-hardening whilst D = 19% represents the ultimate strain (eu). Once the magnitude of the strain reaches the pre-defined damage level the test is terminated. After a lapsed time T, the specimen then undergoes stage II testing in which uniaxial tensile loading is applied to the specimen to failure. Three levels of elapsed time including T = 0 (no-age), T = 2 days and T = 7 days, are examined so that the effect of the strain ageing on the partially damaged materials can be studied. All tests are performed under strain controlled conditions. It is worth noting that the different test cases are denoted in the form of D13 T7, where the blank box (h) in front of D and T are respectively filled in by the damage level applied in the first stage of the experimental tests and the age of the specimen. For example, D13T7 represents the test specimen under damage level of 13% strain with 7 days age. A minimum of two samples are tested for each condition and their results are averaged. However, a third specimen is tested if there is a variation of more than 5% from the average value. 2.4. Stage I: partial-damage quasi-static tests During stage I testing, uniaxial tensile tests are carried out using an Instron 5982 with loading capacity of 100 kN. Tests are performed under strain controlled condition with the displacement being appropriately adjusted for the strain rate at
yield to be equal to 104 s1, which is within the 0.005 0.002 min1 range specified by ASTM E21-92 [16]. After reaching the target damage level, the test is terminated and the specimen is unloaded using the manual control of Instron 5982 dual column testing system so that no load remains on the specimen. The specimen is removed and stored in an appropriate place out of environmental conditions including humidity, corrosion, rust, etc. The specimen is ready for the second stage of testing once it reaches the target ageing time defined in the previous section.
2.5. Stage II: uniaxial quai-static tests to failure In the second stage of testing, the partially damaged specimen is loaded into the Instron 5982 machine and the uniaxial strain-controlled test is conducted to failure. The setup of the experiments in the second stage is similar to that of the first stage whilst the reflective stickers are checked before running the test to make sure they are in a good condition.
3. Results and discussion The effect of strain ageing on the ultimate strength (fu), ultimate strain (eu) and fracture strain (ef) has been investigated. Whilst the magnitudes of ultimate strength and ultimate strain are obtained from the stress–strain curves as shown in Fig. 4, the magnitude of the fracture strain is obtained using two different methods: (i) using the data recorded by non-contact laser extensometer during
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Fig. 4. Definition of mechanical properties on the stress–strain curve.
the test; and (ii) using the reduction in the fracture surface area calculated through the formulation available in literature [17]. 3.1. Stress-strain curves The stress–strain curves for steel specimens with different damage levels and strain ageing parameters are shown in Fig. 5a–c. From Fig. 5a–c, it can be seen that strain aging can significantly influence the material behaviour. This is especially obvious in samples damaged to 13% (Fig. 5b) and 19% (Fig. 5c). As a result, considering such a significant modification to the overall mechanical behaviour, the primary design based on the original stress–strain curve may lead to inaccurate evaluation of the structural components. It is seen that the overall behaviour of the steel material is profoundly dependent on the damage level and the strain ageing. For a low damage level of 1.5% (Fig. 5a), the strain ageing effect is insignificant (except for an effect on the fracture strain), even after holding for 7 days. At damage levels of 13% (Fig. 5b) and 19% (Fig. 5c), there is a large effect of strain aging on the ultimate stress and whilst this is larger after 7 days ageing compared to 2 days ageing, the great majority of the increase occurs within 2 days. The ultimate strain (eu) also noticeably decreases in the 13% and 19% damage samples after strain aging such that a transition from ductile behaviour to non-ductile behaviour is clearly observed. This non-ductile behaviour may lead to a catastrophic collapse of the structure and that is why a proper stress–strain curve reflecting this phenomenon needs to be considered in the building evaluation process. Moreover, the plateau part in the plastic region of the mild-steel stress–strain curve vanishes so that distinct values for yield strength and ultimate strength will no longer be available. Therefore, due to strain ageing the strain hardening in the partially damaged steel material may completely vanish when the value of the partial damage is equal to or greater than damage level (DT). In order to find DT, a series of two-stage uniaxial tensile tests were conducted on steel specimens with different values of partial damage of 1.5%, 5% and 7% and the stress–strain curves to failure after 7 days are shown in Fig. 6. It can be seen from this figure that whilst a minor strain hardening occurs at 5% damage level, no strain hardening occurs at the corresponding specimen after 7% damage. Consequently, it can be seen that for the mild steel studied in this research work the damage level DT takes a value between 5% and 7%. It is also worth noting that plastic analysis should not be applied on the partially damaged steel structures (with damage level of equal to or greater than 7%) affected by strain ageing since due to significant change in the values of ultimate strength, the steel material does not meet the requirements
Fig. 5. Stress–strain curves at (a) 1.5% damage level (b) 13% damage level (c) 19% damage level.
Fig. 6. Stress–strain curves of steel material at different damage levels after 7 days.
of ultimate stress to yield stress ratios recommended by Eurocode 3 [18], AS4100 [19], AS/NZS 4600 [20] and AISC [21]. Similarly, as shown in Fig. 7, a series of two-stage uniaxial tests are conducted on the steel specimens after 13% damage and different values of strain ageing. These experiments indicate that the strain ageing effect becomes significant within 6–12 h and the peak point after strain hardening has vanished within 12–24 h after occurrence of the partial damage.
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Fig. 8 it is seen that the stress increase in 19% damage level specimens is greater in the first 48 h compared to 13% damage level specimens. 3.3. Ductility Ductility plays an important role in the structures behaviour under extreme actions. A low-ductile structure under large static loading may experience a catastrophic collapse and strain aging can influence the ductility and energy absorption of the structural members which depends on the values of ultimate strain and fracture strain. Fig. 7. Stress–strain curves of steel material under 13% damaged specimens at different ages.
3.2. Strength The effect of strain ageing on the magnitudes of the ultimate stress is summarised in Table 2. It is worth noting that the values shown in Table 2 are normalised to the ultimate stress of the virgin material. It can be seen from this table that at damage level (D) equal to 1.5%, strain aging has an insignificant effect on the values of ultimate strength. However, at 13% damage level, a substantial increase in the magnitudes of ultimate stress after 2 (+22.5%) and 7 days ageing (+26.2%) is observed. The 7 days ageing specimens show only slightly greater strength than the 2 days ageing specimens which indicates that 2 days ageing is adequate for the main part of strain aging effect on material strength. At 19% damage level, the variation of the material strength under strain ageing is similar to the 13% damage level; however, the increase in ultimate strength of the aged specimens compared to the original specimen is greater (33.1% and 34.3% respectively for 2 and 7 days aging). The increase in the values of ultimate stress (Dr) with respect to ageing time for damage levels of 13% and 19% is shown in Fig. 8. Dr is defined as the difference between ultimate strength of the strain aged specimen and that of the virgin specimen. From
Table 2 Variation of the normalised ultimate stress and strain values with damage level under different ages.
No age 2 days 7 days
Variation of the ultimate stress
Variation of ultimate strain
1.50%
13%
19%
1.50%
13%
19%
1.000 0.998 1.021
1.000 1.225 1.262
1.000 1.331 1.343
0.923 0.919 0.897
0.332 0.028 0.029
0.051 0.035 0.035
3.3.1. Ultimate strain Table 2 shows the effect of the strain ageing on the magnitudes of the ultimate strain for the partly damaged steel specimens. It can be seen that under 1.5% damage level, which is close to onset of strain hardening, strain ageing has only a moderate effect on the values of the ultimate strain. The magnitude of eu for the 2 days aged specimen is not affected by strain ageing whilst 7 days aged specimens experience slightly lower ultimate strain compared to the 2 days age specimens. When the damage level increases, the magnitude of ultimate strain decreases such that those specimens which are subjected to 19% damage level experience less ultimate strain compared to those with 13% damage level. However the reduction of the ultimate strain depends significantly on the strain ageing for which at D = 13%, the value of eu reduces by 30% after 7 days as shown in Table 2. Also, the ultimate strain values of 2 and 7 days age specimens under D = 13% or 19% damage levels are close which indicates that strain aging has had most of its effect on the material ductility within 2 days. It is worth noting that 19% damage level almost corresponds to the ultimate strain of the virgin material and therefore the value of eu is small. However, the ultimate strain of the mild-steel material reduces by 50% after 2 and 7 days. 3.3.2. Fracture strain Generally, determination of the fracture strain is a complicated issue which is due to non-uniform deformation within the gage length of the specimens once necking starts. There are several methods reported in the literature to measure the value of the fracture strain. Two methods are employed in this study. In the first method the fracture strain ef is defined as
ef ¼ ln
A0 Af
ð1Þ
where A0 is the area of the original cross-section and Af is the fracture surface area. Whilst the magnitude of A0 is obtained from the geometry of the unbroken specimen, the magnitude of Af is obtained through image processing by using an optical microscope. Kim et al. [17] presented an alternative method where the fracture strain is expressed by
ef ¼ ðet þ ew Þ
ð2Þ
and
et ¼ ln
Fig. 8. Stress increase due to strain ageing versus time.
t 1 þ 2t3 þ t2 ; 4t 0
ew ¼ ln
w w0
ð3Þ
where et , and ew are strains in the thickness and width directions, respectively and w and w0 are the specimen width before and after fracture, respectively. The magnitudes of t1, t2 and t3 are obtained from the fracture surface shown in Fig. 9. Fracture strain values measured using both methods can be found in Table. 3.
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Fig. 9. Fracture strain parameters used in Eq. (3).
Table 3 Comparison of fracture strain values measured by using the measured area through image processing (Eq. (1)) and using reduced area method (Eq. (2)). Eq. (1)
1.50% 13% 19%
Eq. (2)
No-age
2 days
7 days
No-age
2 days
7 days
1 0.99 0.99
0.97 0.82 0.73
0.95 0.8 0.72
1.00 0.99 0.99
0.97 0.92 0.90
0.91 0.89 0.89
4. Microstructural examination Microstructural examination was also performed in this study using SEM. Fig 10a shows low and higher magnifications images of the bulk microstructure of the mild steel. The white regions are the pearlitic aggregates in the microstructure (consisting of fine lamellae of ferrite (Fe) and cementite (Fe3C), and the dark regions are the ferrite grains. These are images typical of mild steel. SEM images of the fracture surfaces of the samples are shown in Fig. 10b–d. It can be seen from Fig. 10b–d that a ductile failure is observed in the steel specimens undergone different damage levels at various ages. 4.1. Rationalization of the ageing time and damage level dependence of the strain ageing effect The strain ageing effect is ultimately due to the pinning of dislocations introduced into the steel during the partial damage, by segregated Carbon and/or Nitrogen atoms [2]. This effect depends on both the damage level and the time between the partial damage and subsequent deformation. The level of partial damage is important because this determines the number of dislocations that are introduced into the steel (the dislocation density, q (m/m3)), and consequently their average separation (k). The ageing time is important because it is during this time that the Carbon and/or Nitrogen atoms can diffuse to the dislocations introduced during partial damage and it is this solute segregation to dislocations that causes the strain ageing effect. Both solute diffusion and the evolution of the dislocation density during deformation are sufficiently well understood in the field of materials science and engineering, that estimates can be made to check the reasonableness of the strain ageing effects observed in this study. Consider, the sample partially damaged to 13%. From Fig. 1, we can see that the proof stress of the mild steel is 340 MPa and after 13% strain, the flow stress has risen to 490 MPa. This provide a strain hardening increment of 150 MPa. The strengthening that occurs during straining of alloys such as mild steel has two
components – a kinematic component which has its origins on the plastic mismatch between the pearlite regions and the ferrite regions of the microstructure [22], and an isotropic contribution which is due to the formation of dislocations. The kinematic contribution may represent up to 30% of the strengthening increment [22]. As a result, we can estimate an upper and lower bound for the strengthening from dislocations: 105–150 MPa. The actual dislocation density responsible for this strengthening can be calculated using Taylor’s equation [2] (Eq. (4))
q¼
Dr aMGb
2 ð4Þ
where a is a constant with value 0.3, M is the Taylor factor (3) G is the shear modulus of Fe and b is the Burgers vector of a dislocation in Fe. From the dislocation density, the average spacing of dislocations can be calculated, p1ffiffiqffi, an the upper and lower estimate of the mean dislocation separation is plotted in Fig. 11 for the sample with 13% partial damage. The strain ageing effect depends on the diffusion of Carbon and/ or Nitrogen atoms to these dislocations. The diffusion pffiffiffiffiffiffi distance (x) at room temperature can be calculated using x Dt where D is the Carbon diffusivity [23] at room temperature and t is time, and this is also plotted as a function of time in Fig. 11. Obviously, with longer times, the Carbon atoms can diffuse longer distances. Since the strain ageing effect depends on the pinning of the dislocations, the shorter the distances the Carbon needs to diffuse to pin the dislocations, the shorter the ‘ageing’ required to give rise to the strain ageing effect. As can be seen in Fig. 11, the average dislocation spacing and the diffusion distance of Carbon atoms at room temperature are of the same magnitude for times roughly 6–12 h. This indicates that for ageing times much shorter that 6 h, for the 13% damage sample, we will probably not see much strain ageing effect, and for times greater than 12 h we should expect to see a significant effect. In this study, a range of delay times after partial damage were examined, but Fig 7 in particular for the 13% damage situation, illustrates the systematic effect of times of 6 h, 12 h, 24 h 2 days and 7 days. The strain ageing effect becomes significant at times between 6 and 12 h which is consistent with the reasoning used to generate Fig. 11. It is also clear from Table 2 that the increment in strength resulting from the strain ageing effect saturates with time after damage and before subsequent testing. The saturation appears after around 2 days and this is also illustrated in the stress strain curves themselves showing only a slightly greater strain ageing effect for samples held for 7 days compared with samples held for 2 days. The reason for this saturation effect of the strain aging is a saturation in the segregation of Carbon to the dislocations after 2 days. Longer hold-
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Fig. 10. SEM images of fracture surface at 500 and 2000 magnifications (a) virgin material (b) specimen at 1.5% damage after 7 days (c) specimen at 13% damage after 7 days (d) specimen at 19% damage after 7 days.
ing does not lead to any further segregation and as a result no further change to the stress required to ‘un-pin’ the dislocations upon subsequent deformation.
Overall, the observations of the effect of holding time at room temperature after damage and before subsequent testing, as well as the effects of different levels of damage are perfectly under-
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Fig. 11. Comparison of the average diffusion distance for Carbon in mild steel at room temperatures as a function of time, and comparison with the lower and upper estimates of the average dislocation spacing after 13% damage.
standable within the framework of what is currently understood about the physics of the strain ageing effect. 5. Numerical study 5.1. Illustration In order to investigate the effect of the mechanical properties of the material affected by strain aging on the behaviour of the damaged structural elements, a numerical example is presented in this section. In this example, the behaviour of a typical steel beam shown in Fig. 12 is examined with and without incorporating the strain aging effect. A simply supported steel beam with length of L = 6 m is considered. The cross-section is taken as 460UB 67.1 [24]. It is assumed that lateral restraints are provided such that no local or member flexural buckling occurs [25–28]. The steel beam is subjected to two concentrated loads at distance of 2 and 4 m from the left support. It is worth noting that the beam is loaded under strain controlled conditions whilst the material and geometric non-linearities are taken into account. The steel beam is statically loaded until the maximum strain at the extreme fibres is equal to 13. Thus, the beam is unloaded for which residual deformation equal to d takes a place in the steel beam. At this stage in order to elaborate further on the strain aging effect, two different scenarios are considered. In the first scenario, the deformed structure is loaded until the maximum strain is equal to the ultimate strain of the virgin material without incorporating the aging effect. In other words, it is assumed that the beam is loaded immediately after partial damage occurrence for which the virgin material stress–strain curve is followed. In this case, the beam experiences an additional displacement of 0:29d whilst the maximum load that the steel beam could sustain is 586 kN. In the second scenario, it is assumed that the deformed steel beam is loaded after 7 days since the partial damage occurrence until the maximum strain at the extreme fibre is equal to the ultimate strain of the material affected by the strain aging effect. The corresponding stress–strain curve depicted in Fig. 12 is used. It is seen that the beam may experience only an additional displacement equal to 0:07d whilst the maximum load that the structure could sustain is 421 kN. This simple example illustrate the fact that the ductility of the material reduces significantly due to strain
Fig. 12. Ageing effect on the global behaviour of a typical steel beam (a) beam cross section (b) geometry of deformed beam (c) stress–strain behaviour of two scenarios.
ageing effects and this phenomenon needs to be considered for the situation of partially damaged steel structures. 5.2. Ramberg–Osgood model The Ramberg–Osgood model is employed in this section to develop the stress–strain relations. It is a simplified model to describe the stress–strain curve in terms of few parameters including Young’s modulus and yield strength. The Ramberg–Osgood model can be expressed in a generic form by
r e¼ þk E
r rty
n ð5Þ
in which e is strain, r is stress, rty is yield stress and k, n are constants. The value of k is assumed to be 0.002 [29] whilst n can be determined from
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Table 4 Variation of parameter n in Ramberg–Osgood model for partially damaged mild-steel material at different ages.
1.5% damage level n 13% damage level n 19% damage level n
No-age
2 days
7 days
11.45
11.53
11.52
245.59
17.55
5.46
54.5
50.63
44.63
Fig. 14. Comparing Ramberg–Osgood curves with experimental data at 13% damage level (a) no-age (b) 2 days (c) 7 days.
where rtu is ultimate strength and eus is plastic strain at the end of uniform elongation and can be represented in the form of
Fig. 13. Comparing Ramberg–Osgood curves with experimental data at 1.5% damage level (a) no-age (b) 2 days (c) 7 days.
n¼
Lnðeus =0:2Þ Lnðrtu =rty Þ
ð6Þ
eus ¼ 100 er
rtu E
ð7Þ
Using the stress–strain curves for the partially damaged mildsteel material developed in the previous sections, the values of n can be tabulated with respect to the damage level and strain ageing time, as shown in Table 4. At 1.5% damage level, the changes of parameter n due to aging is insignificant. At 13% damage level, the
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those obtained from the experimental tests. As shown in Figs. 13– 15, the calibrated model can describe well the stress–strain curve for partially damages steel material at different ages.
6. Conclusion The effect of strain aging on the mechanical properties of partially damaged Grade 350 structural mild steel was investigated in this paper. It is shown that the strain aging can substantially affect the strength and ductility of the partly damaged steel structures. Two-stage experimental tests were conducted to explore the strain ageing effects in which time and damage level were taken into account as two effective variables. Moreover, microstructure examination was conducted using SEM. The following conclusions can be drawn from this study: - The strain ageing effect depends strongly on the damage level. For higher damage levels, the effect of strain ageing is greater. The ultimate stress may increase up to 40% whilst the reduction in ultimate strain might be up to 80%. Moreover, the fracture strain may reduce around 10% whilst the reduction rate depends on the measurement method; - For Grade 350 steel, the strain ageing effect becomes significant at a time between 6 and 12 h after the initial occurrence of damage. The strain hardening capacity of the steel after high partial damage vanishes at a time between 12 and 24 h after damage; - Using the obtained stress–strain curves for the partly damaged steel material, the effect of strain ageing on the global behaviour of a steel beam was studied. It was shown that due to strain ageing the loading capacity of the steel beam may decrease up to 30%. - Due to significant effects of strain ageing on the ductility and strength of damaged steel structures, the relevant codes of practice need to take this effect into account.
Acknowledgement The research work presented in this paper was supported by Australian Research Council through a Discovery Project (DP1096454) awarded to the second author. References
Fig. 15. Comparing Ramberg–Osgood curves with experimental data at 19% damage level (a) no-age (b) 2 days (c) 7 days.
value of n decreases from 245.59 for no-ageing specimen to 17.55 and 5.46 for 2 days and 7 days age specimens, respectively. At 19% damage level, the reduction of n with strain ageing is lower compared to that of 13% damage level as it decreases from 54.5 for no-age specimen to 50.63 and 44.63 for 2 and 7 days age specimens, respectively. Using the parameters shown in Table 4 the stress–strain curves obtained from the Ramberg–Osgood model are compared with
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Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Effect of strain ageing on the mechanical properties of partially damaged structural mild steel Sajjad Hosseini a, Amin Heidarpour a,⇑, Frank Collins a, Christopher R. Hutchinson b a b
Department of Civil Engineering, Monash University, Melbourne VIC3800, Australia Department of Materials Engineering, Monash University, Melbourne VIC3800, Australia
h i g h l i g h t s Strain ageing effect on mechanical properties of damaged steel is investigated. Two-stage experimental program are conducted to incorporate strain ageing effect. Changes in microstructure of steel due to strain ageing effect are examined. Parameters of Ramberg–Osgood model are calibrated by incorporating strain ageing. It is shown strain ageing may have significant effect on behaviour of structures.
a r t i c l e
i n f o
Article history: Received 16 October 2014 Received in revised form 8 December 2014 Accepted 10 December 2014
Keywords: Strain ageing Damaged structures Steel material Ramberg–Osgood Micro-structure
a b s t r a c t This paper addresses the strain ageing effects on the mechanical properties of the partially damaged structural mild steel. Since repairing partly damaged structures may not occur immediately, the strain ageing effect can significantly influence the structural behaviour. The changes due to this effect have not so far been considered in the civil engineering design guidelines. In order to investigate strain ageing effects, two-stage experimental tests are carried out on the mild-steel specimens. In the first stage, partial damage is made using quasi-static loading. During the second stage, the strength and ductility of the specimens are examined after 2 and 7 days ‘ageing’ at room temperature and the results are compared with the corresponding no-age samples. The microstructure of the specimens is examined using scanning electron microscopy (SEM). To illustrate the effect of strain ageing on the global behaviour of steel structures, a numerical example is provided in which strain ageing impacts on loading capacity and deflection of a steel beam. Finally, the stress–strain relation of partially damaged mild-steel material incorporating strain ageing effects is expressed by calibrating the parameters of Ramberg–Osgood model. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In various situations, repair and rehabilitation of partially damaged steel structures are implemented in place of demolishing the structure and constructing a new one. As a result, a better understanding of the mechanical properties of the partially damaged materials under different loading scenarios is required. Once a structure is partly damaged (ie. strained), it may take several days until the repair process of the structural component(s) commences. This lapsed time (named as ageing in this paper) may significantly influence the structural response in terms of strength, ductility, energy absorption, etc.
⇑ Corresponding author. Tel.: +61 3 99024435; fax: +61 3 99054944. E-mail address: [email protected] (A. Heidarpour). http://dx.doi.org/10.1016/j.conbuildmat.2014.12.021 0950-0618/Ó 2014 Elsevier Ltd. All rights reserved.
Although various loading scenarios such as static, thermal, high-strain or seismic loadings may cause partial damage in structural members, only damages made by static loading are considered in this paper. Foundation settlement is a common source of damage in structures in which structural components may experience large relative displacements beyond the material yield point. Other examples in which unexpected large static loading may cause the damage include lateral ground water pressure applied to bridge piers, temporary loads during repairing process, and large hydrostatic pressures applied to the structures under tsunami. The strain ageing phenomenon, which occurs because of the time gap between the occurrence of damage caused by static loading and the repair process commencement, can substantially change the mechanical properties of the steel material such that utilizing the conventional constitutive material models without incorporating strain aging effects may lead to a completely incorrect assessment.
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Strain aging effects have been investigated by researchers worldwide in the field of materials engineering, e.g. [1,2], and it is well known that they must be taken into account during the production of some metal components, such as steel sheet panels for automotive applications. The physical origin of the strain aging effect in steels can be explained in the following way. The deformation of metals, including steels, occurs through the generation and movement of linear defects known as dislocations [2]. The strength of a metal is controlled by the stress required to move dislocations over substantial distances in the microstructure. Dislocations exhibit a long range strain field, and this strain field can interact with the strain field of solute atoms that are present in alloys. In the case if steels, the most important solutes from the point of view of strain ageing, are Carbon and Nitrogen. Even if present in small amounts, these solute species can diffuse within the material, even at room temperature, and segregate to dislocations, and hence partially relax their long range strain field. The consequence on the mechanical properties of the steels, is that subsequent loading can require a much higher stress level to then ‘un-pin’ the dislocations from the Carbon or Nitrogen solute atmospheres. This is the physical basis of the strain aging effect, e.g. [2]. This effect is especially important in steels because Carbon and Nitrogen can diffuse within the material at room temperature whereas other elements typically found in steels, such as Mn, are essentially immobile at room temperature. In the materials science and engineering literature, both experimental and theoretical/modelling aspects of the strain ageing effects have been investigated. The effect has most commonly been observed in typical mild steels but also occurs in dual phase steels, pearlitic steel wires [3] and even ultra low Carbon bake hardening (BH) steels [4]. Although most well known in the steel community, strain ageing effects are also observed in Ni-base alloys such as Inconel [5], structural intermetallics such as TiAl [6] and Al3Ti [7], Ti alloys [8], and Cu based alloys [9]. As well as experimental studies there have also been a large number of modelling studies of different aspects of the solute-dislocation interaction that is at the core of the strain aging effect. These include considerations of the strain [10], temperature [2] and solute content dependence [2]. Although these different aspects of strain aging have been investigated in the materials science context, strain aging can also play a significant role in designing and rehabilitating of structural elements. It is worth noting that there are two common types of ‘aging’ in civil engineering applications: short and long term aging. Long term aging refers to the time effect due to some phenomena such as creep and shrinkage in concrete, and creep in steel which occurs over a very long period of time. Although long term aging has been widely discussed in the literature, short term aging has not been considered in detail. This type of ageing (strain-ageing) is considered in this work. Currently, there is no appropriate proposed formulation in the prescriptive codes of practice in civil engineering to include the effect of strain ageing in design and evaluation of partly damaged steel structures [11–14], and therefore the outcome of this research can be used as a platform in rational analysis and design of steel structures. In this paper, changes in mechanical properties of partially damaged steel material due to strain ageing are addressed. The strain ageing effect on the stress and strain values is explored through two-stage experimental tests. Using scanning electron microscopy (SEM) images taken from the fracture surfaces of the broken specimens, the strain ageing effect on the microstructure of the material is also investigated. Moreover, the significant effect of the strain ageing on the global behaviour of steel structures is explored through a numerical example and the parameters of the
Ramberg–Osgood model are also calibrated so that the strain ageing effect is incorporated into the stress–strain relations. 2. Experimental program 2.1. Test material and specimen The coupon specimens are made of Grade 350 mild steel plates, approximately equivalent to ASTM A633A. The chemical composition of Grade 350 mild steel considered in this study is listed in Table 1 and the engineering stress–strain curve of the material under quasi-static loading is depicted in Fig. 1. The tests are conducted based on the requirements of standard ASTM E 8M-04 [15]. The tensile test coupons, with enlarged shoulders for gripping, are accurately machined from the primary steel plates. The measurements are made within the gauge length in reduced section and the specimens mounted in the machine from the shoulder part during the test. The specimen’s typical geometry is demonstrated in Fig. 2. Both faces of each specimen are ground to give uniform thickness and a smooth finish across the entire surface of the specimens. 2.2. Testing equipment An electromechanical testing machine, Instron 5982 dual column testing system with a 100 kN load cell, is employed in order to perform tensile testing. The axial strains are measured using a MTX LX 500 non-contact laser extensometer in which the value of strain is recorded by measuring the relative displacements between the reflective stickers glued on the specimen within the gauge length. In order to check the accuracy of the laser extensometer, the strain measurement in the first test was conducted by using both strain gauges and non-contact laser extensometers and a good agreement was achieved. Fig. 3 depicts 100 kN Instron testing machine, laser extensometer and specimen used in this study. In order to perform microstructural examinations, optical microscopy (OM) and a JEOL 7001F FEG scanning electron microscope (SEM) are employed. The images taken by both microscopes from the fracture surfaces are used to determine the fracture strain as well as to better interpret the effect of strain ageing on the fracture characteristics of the material.
Table 1 Chemical composition of Grade 350 mild steel. C
Si
Mn
S
P
0.22%
0.55%
1.70%
0.030%
0.040%
Fig. 1. Engineering stress–strain curve of Grade 350 steel material under quasistatic loading.
Fig. 2. Geometry of the specimen.
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Specimen
Instron 5982 100kN testing machine MTS Laser extensometer (LX1500)
Fig. 3. Testing machines used to conduct uniaxial tensile tests on specimens. 2.3. Testing procedure In this study, statically damaged specimens at room temperature are considered. In order to investigate the strain ageing effect, a two-stage loading case is applied. During stage I, the specimen is subjected to uniaxial tensile loading until partial damage occurs in the specimen. Since the damage level may affect the mechanical properties of the material, three different damage levels (D) of 1.5%, 13% and 19% strain shown in Fig. 1 are considered. D = 1.5% represents the strain close to the onset of strain-hardening whilst D = 19% represents the ultimate strain (eu). Once the magnitude of the strain reaches the pre-defined damage level the test is terminated. After a lapsed time T, the specimen then undergoes stage II testing in which uniaxial tensile loading is applied to the specimen to failure. Three levels of elapsed time including T = 0 (no-age), T = 2 days and T = 7 days, are examined so that the effect of the strain ageing on the partially damaged materials can be studied. All tests are performed under strain controlled conditions. It is worth noting that the different test cases are denoted in the form of D13 T7, where the blank box (h) in front of D and T are respectively filled in by the damage level applied in the first stage of the experimental tests and the age of the specimen. For example, D13T7 represents the test specimen under damage level of 13% strain with 7 days age. A minimum of two samples are tested for each condition and their results are averaged. However, a third specimen is tested if there is a variation of more than 5% from the average value. 2.4. Stage I: partial-damage quasi-static tests During stage I testing, uniaxial tensile tests are carried out using an Instron 5982 with loading capacity of 100 kN. Tests are performed under strain controlled condition with the displacement being appropriately adjusted for the strain rate at
yield to be equal to 104 s1, which is within the 0.005 0.002 min1 range specified by ASTM E21-92 [16]. After reaching the target damage level, the test is terminated and the specimen is unloaded using the manual control of Instron 5982 dual column testing system so that no load remains on the specimen. The specimen is removed and stored in an appropriate place out of environmental conditions including humidity, corrosion, rust, etc. The specimen is ready for the second stage of testing once it reaches the target ageing time defined in the previous section.
2.5. Stage II: uniaxial quai-static tests to failure In the second stage of testing, the partially damaged specimen is loaded into the Instron 5982 machine and the uniaxial strain-controlled test is conducted to failure. The setup of the experiments in the second stage is similar to that of the first stage whilst the reflective stickers are checked before running the test to make sure they are in a good condition.
3. Results and discussion The effect of strain ageing on the ultimate strength (fu), ultimate strain (eu) and fracture strain (ef) has been investigated. Whilst the magnitudes of ultimate strength and ultimate strain are obtained from the stress–strain curves as shown in Fig. 4, the magnitude of the fracture strain is obtained using two different methods: (i) using the data recorded by non-contact laser extensometer during
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Fig. 4. Definition of mechanical properties on the stress–strain curve.
the test; and (ii) using the reduction in the fracture surface area calculated through the formulation available in literature [17]. 3.1. Stress-strain curves The stress–strain curves for steel specimens with different damage levels and strain ageing parameters are shown in Fig. 5a–c. From Fig. 5a–c, it can be seen that strain aging can significantly influence the material behaviour. This is especially obvious in samples damaged to 13% (Fig. 5b) and 19% (Fig. 5c). As a result, considering such a significant modification to the overall mechanical behaviour, the primary design based on the original stress–strain curve may lead to inaccurate evaluation of the structural components. It is seen that the overall behaviour of the steel material is profoundly dependent on the damage level and the strain ageing. For a low damage level of 1.5% (Fig. 5a), the strain ageing effect is insignificant (except for an effect on the fracture strain), even after holding for 7 days. At damage levels of 13% (Fig. 5b) and 19% (Fig. 5c), there is a large effect of strain aging on the ultimate stress and whilst this is larger after 7 days ageing compared to 2 days ageing, the great majority of the increase occurs within 2 days. The ultimate strain (eu) also noticeably decreases in the 13% and 19% damage samples after strain aging such that a transition from ductile behaviour to non-ductile behaviour is clearly observed. This non-ductile behaviour may lead to a catastrophic collapse of the structure and that is why a proper stress–strain curve reflecting this phenomenon needs to be considered in the building evaluation process. Moreover, the plateau part in the plastic region of the mild-steel stress–strain curve vanishes so that distinct values for yield strength and ultimate strength will no longer be available. Therefore, due to strain ageing the strain hardening in the partially damaged steel material may completely vanish when the value of the partial damage is equal to or greater than damage level (DT). In order to find DT, a series of two-stage uniaxial tensile tests were conducted on steel specimens with different values of partial damage of 1.5%, 5% and 7% and the stress–strain curves to failure after 7 days are shown in Fig. 6. It can be seen from this figure that whilst a minor strain hardening occurs at 5% damage level, no strain hardening occurs at the corresponding specimen after 7% damage. Consequently, it can be seen that for the mild steel studied in this research work the damage level DT takes a value between 5% and 7%. It is also worth noting that plastic analysis should not be applied on the partially damaged steel structures (with damage level of equal to or greater than 7%) affected by strain ageing since due to significant change in the values of ultimate strength, the steel material does not meet the requirements
Fig. 5. Stress–strain curves at (a) 1.5% damage level (b) 13% damage level (c) 19% damage level.
Fig. 6. Stress–strain curves of steel material at different damage levels after 7 days.
of ultimate stress to yield stress ratios recommended by Eurocode 3 [18], AS4100 [19], AS/NZS 4600 [20] and AISC [21]. Similarly, as shown in Fig. 7, a series of two-stage uniaxial tests are conducted on the steel specimens after 13% damage and different values of strain ageing. These experiments indicate that the strain ageing effect becomes significant within 6–12 h and the peak point after strain hardening has vanished within 12–24 h after occurrence of the partial damage.
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Fig. 8 it is seen that the stress increase in 19% damage level specimens is greater in the first 48 h compared to 13% damage level specimens. 3.3. Ductility Ductility plays an important role in the structures behaviour under extreme actions. A low-ductile structure under large static loading may experience a catastrophic collapse and strain aging can influence the ductility and energy absorption of the structural members which depends on the values of ultimate strain and fracture strain. Fig. 7. Stress–strain curves of steel material under 13% damaged specimens at different ages.
3.2. Strength The effect of strain ageing on the magnitudes of the ultimate stress is summarised in Table 2. It is worth noting that the values shown in Table 2 are normalised to the ultimate stress of the virgin material. It can be seen from this table that at damage level (D) equal to 1.5%, strain aging has an insignificant effect on the values of ultimate strength. However, at 13% damage level, a substantial increase in the magnitudes of ultimate stress after 2 (+22.5%) and 7 days ageing (+26.2%) is observed. The 7 days ageing specimens show only slightly greater strength than the 2 days ageing specimens which indicates that 2 days ageing is adequate for the main part of strain aging effect on material strength. At 19% damage level, the variation of the material strength under strain ageing is similar to the 13% damage level; however, the increase in ultimate strength of the aged specimens compared to the original specimen is greater (33.1% and 34.3% respectively for 2 and 7 days aging). The increase in the values of ultimate stress (Dr) with respect to ageing time for damage levels of 13% and 19% is shown in Fig. 8. Dr is defined as the difference between ultimate strength of the strain aged specimen and that of the virgin specimen. From
Table 2 Variation of the normalised ultimate stress and strain values with damage level under different ages.
No age 2 days 7 days
Variation of the ultimate stress
Variation of ultimate strain
1.50%
13%
19%
1.50%
13%
19%
1.000 0.998 1.021
1.000 1.225 1.262
1.000 1.331 1.343
0.923 0.919 0.897
0.332 0.028 0.029
0.051 0.035 0.035
3.3.1. Ultimate strain Table 2 shows the effect of the strain ageing on the magnitudes of the ultimate strain for the partly damaged steel specimens. It can be seen that under 1.5% damage level, which is close to onset of strain hardening, strain ageing has only a moderate effect on the values of the ultimate strain. The magnitude of eu for the 2 days aged specimen is not affected by strain ageing whilst 7 days aged specimens experience slightly lower ultimate strain compared to the 2 days age specimens. When the damage level increases, the magnitude of ultimate strain decreases such that those specimens which are subjected to 19% damage level experience less ultimate strain compared to those with 13% damage level. However the reduction of the ultimate strain depends significantly on the strain ageing for which at D = 13%, the value of eu reduces by 30% after 7 days as shown in Table 2. Also, the ultimate strain values of 2 and 7 days age specimens under D = 13% or 19% damage levels are close which indicates that strain aging has had most of its effect on the material ductility within 2 days. It is worth noting that 19% damage level almost corresponds to the ultimate strain of the virgin material and therefore the value of eu is small. However, the ultimate strain of the mild-steel material reduces by 50% after 2 and 7 days. 3.3.2. Fracture strain Generally, determination of the fracture strain is a complicated issue which is due to non-uniform deformation within the gage length of the specimens once necking starts. There are several methods reported in the literature to measure the value of the fracture strain. Two methods are employed in this study. In the first method the fracture strain ef is defined as
ef ¼ ln
A0 Af
ð1Þ
where A0 is the area of the original cross-section and Af is the fracture surface area. Whilst the magnitude of A0 is obtained from the geometry of the unbroken specimen, the magnitude of Af is obtained through image processing by using an optical microscope. Kim et al. [17] presented an alternative method where the fracture strain is expressed by
ef ¼ ðet þ ew Þ
ð2Þ
and
et ¼ ln
Fig. 8. Stress increase due to strain ageing versus time.
t 1 þ 2t3 þ t2 ; 4t 0
ew ¼ ln
w w0
ð3Þ
where et , and ew are strains in the thickness and width directions, respectively and w and w0 are the specimen width before and after fracture, respectively. The magnitudes of t1, t2 and t3 are obtained from the fracture surface shown in Fig. 9. Fracture strain values measured using both methods can be found in Table. 3.
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Fig. 9. Fracture strain parameters used in Eq. (3).
Table 3 Comparison of fracture strain values measured by using the measured area through image processing (Eq. (1)) and using reduced area method (Eq. (2)). Eq. (1)
1.50% 13% 19%
Eq. (2)
No-age
2 days
7 days
No-age
2 days
7 days
1 0.99 0.99
0.97 0.82 0.73
0.95 0.8 0.72
1.00 0.99 0.99
0.97 0.92 0.90
0.91 0.89 0.89
4. Microstructural examination Microstructural examination was also performed in this study using SEM. Fig 10a shows low and higher magnifications images of the bulk microstructure of the mild steel. The white regions are the pearlitic aggregates in the microstructure (consisting of fine lamellae of ferrite (Fe) and cementite (Fe3C), and the dark regions are the ferrite grains. These are images typical of mild steel. SEM images of the fracture surfaces of the samples are shown in Fig. 10b–d. It can be seen from Fig. 10b–d that a ductile failure is observed in the steel specimens undergone different damage levels at various ages. 4.1. Rationalization of the ageing time and damage level dependence of the strain ageing effect The strain ageing effect is ultimately due to the pinning of dislocations introduced into the steel during the partial damage, by segregated Carbon and/or Nitrogen atoms [2]. This effect depends on both the damage level and the time between the partial damage and subsequent deformation. The level of partial damage is important because this determines the number of dislocations that are introduced into the steel (the dislocation density, q (m/m3)), and consequently their average separation (k). The ageing time is important because it is during this time that the Carbon and/or Nitrogen atoms can diffuse to the dislocations introduced during partial damage and it is this solute segregation to dislocations that causes the strain ageing effect. Both solute diffusion and the evolution of the dislocation density during deformation are sufficiently well understood in the field of materials science and engineering, that estimates can be made to check the reasonableness of the strain ageing effects observed in this study. Consider, the sample partially damaged to 13%. From Fig. 1, we can see that the proof stress of the mild steel is 340 MPa and after 13% strain, the flow stress has risen to 490 MPa. This provide a strain hardening increment of 150 MPa. The strengthening that occurs during straining of alloys such as mild steel has two
components – a kinematic component which has its origins on the plastic mismatch between the pearlite regions and the ferrite regions of the microstructure [22], and an isotropic contribution which is due to the formation of dislocations. The kinematic contribution may represent up to 30% of the strengthening increment [22]. As a result, we can estimate an upper and lower bound for the strengthening from dislocations: 105–150 MPa. The actual dislocation density responsible for this strengthening can be calculated using Taylor’s equation [2] (Eq. (4))
q¼
Dr aMGb
2 ð4Þ
where a is a constant with value 0.3, M is the Taylor factor (3) G is the shear modulus of Fe and b is the Burgers vector of a dislocation in Fe. From the dislocation density, the average spacing of dislocations can be calculated, p1ffiffiqffi, an the upper and lower estimate of the mean dislocation separation is plotted in Fig. 11 for the sample with 13% partial damage. The strain ageing effect depends on the diffusion of Carbon and/ or Nitrogen atoms to these dislocations. The diffusion pffiffiffiffiffiffi distance (x) at room temperature can be calculated using x Dt where D is the Carbon diffusivity [23] at room temperature and t is time, and this is also plotted as a function of time in Fig. 11. Obviously, with longer times, the Carbon atoms can diffuse longer distances. Since the strain ageing effect depends on the pinning of the dislocations, the shorter the distances the Carbon needs to diffuse to pin the dislocations, the shorter the ‘ageing’ required to give rise to the strain ageing effect. As can be seen in Fig. 11, the average dislocation spacing and the diffusion distance of Carbon atoms at room temperature are of the same magnitude for times roughly 6–12 h. This indicates that for ageing times much shorter that 6 h, for the 13% damage sample, we will probably not see much strain ageing effect, and for times greater than 12 h we should expect to see a significant effect. In this study, a range of delay times after partial damage were examined, but Fig 7 in particular for the 13% damage situation, illustrates the systematic effect of times of 6 h, 12 h, 24 h 2 days and 7 days. The strain ageing effect becomes significant at times between 6 and 12 h which is consistent with the reasoning used to generate Fig. 11. It is also clear from Table 2 that the increment in strength resulting from the strain ageing effect saturates with time after damage and before subsequent testing. The saturation appears after around 2 days and this is also illustrated in the stress strain curves themselves showing only a slightly greater strain ageing effect for samples held for 7 days compared with samples held for 2 days. The reason for this saturation effect of the strain aging is a saturation in the segregation of Carbon to the dislocations after 2 days. Longer hold-
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Fig. 10. SEM images of fracture surface at 500 and 2000 magnifications (a) virgin material (b) specimen at 1.5% damage after 7 days (c) specimen at 13% damage after 7 days (d) specimen at 19% damage after 7 days.
ing does not lead to any further segregation and as a result no further change to the stress required to ‘un-pin’ the dislocations upon subsequent deformation.
Overall, the observations of the effect of holding time at room temperature after damage and before subsequent testing, as well as the effects of different levels of damage are perfectly under-
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Fig. 11. Comparison of the average diffusion distance for Carbon in mild steel at room temperatures as a function of time, and comparison with the lower and upper estimates of the average dislocation spacing after 13% damage.
standable within the framework of what is currently understood about the physics of the strain ageing effect. 5. Numerical study 5.1. Illustration In order to investigate the effect of the mechanical properties of the material affected by strain aging on the behaviour of the damaged structural elements, a numerical example is presented in this section. In this example, the behaviour of a typical steel beam shown in Fig. 12 is examined with and without incorporating the strain aging effect. A simply supported steel beam with length of L = 6 m is considered. The cross-section is taken as 460UB 67.1 [24]. It is assumed that lateral restraints are provided such that no local or member flexural buckling occurs [25–28]. The steel beam is subjected to two concentrated loads at distance of 2 and 4 m from the left support. It is worth noting that the beam is loaded under strain controlled conditions whilst the material and geometric non-linearities are taken into account. The steel beam is statically loaded until the maximum strain at the extreme fibres is equal to 13. Thus, the beam is unloaded for which residual deformation equal to d takes a place in the steel beam. At this stage in order to elaborate further on the strain aging effect, two different scenarios are considered. In the first scenario, the deformed structure is loaded until the maximum strain is equal to the ultimate strain of the virgin material without incorporating the aging effect. In other words, it is assumed that the beam is loaded immediately after partial damage occurrence for which the virgin material stress–strain curve is followed. In this case, the beam experiences an additional displacement of 0:29d whilst the maximum load that the steel beam could sustain is 586 kN. In the second scenario, it is assumed that the deformed steel beam is loaded after 7 days since the partial damage occurrence until the maximum strain at the extreme fibre is equal to the ultimate strain of the material affected by the strain aging effect. The corresponding stress–strain curve depicted in Fig. 12 is used. It is seen that the beam may experience only an additional displacement equal to 0:07d whilst the maximum load that the structure could sustain is 421 kN. This simple example illustrate the fact that the ductility of the material reduces significantly due to strain
Fig. 12. Ageing effect on the global behaviour of a typical steel beam (a) beam cross section (b) geometry of deformed beam (c) stress–strain behaviour of two scenarios.
ageing effects and this phenomenon needs to be considered for the situation of partially damaged steel structures. 5.2. Ramberg–Osgood model The Ramberg–Osgood model is employed in this section to develop the stress–strain relations. It is a simplified model to describe the stress–strain curve in terms of few parameters including Young’s modulus and yield strength. The Ramberg–Osgood model can be expressed in a generic form by
r e¼ þk E
r rty
n ð5Þ
in which e is strain, r is stress, rty is yield stress and k, n are constants. The value of k is assumed to be 0.002 [29] whilst n can be determined from
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Table 4 Variation of parameter n in Ramberg–Osgood model for partially damaged mild-steel material at different ages.
1.5% damage level n 13% damage level n 19% damage level n
No-age
2 days
7 days
11.45
11.53
11.52
245.59
17.55
5.46
54.5
50.63
44.63
Fig. 14. Comparing Ramberg–Osgood curves with experimental data at 13% damage level (a) no-age (b) 2 days (c) 7 days.
where rtu is ultimate strength and eus is plastic strain at the end of uniform elongation and can be represented in the form of
Fig. 13. Comparing Ramberg–Osgood curves with experimental data at 1.5% damage level (a) no-age (b) 2 days (c) 7 days.
n¼
Lnðeus =0:2Þ Lnðrtu =rty Þ
ð6Þ
eus ¼ 100 er
rtu E
ð7Þ
Using the stress–strain curves for the partially damaged mildsteel material developed in the previous sections, the values of n can be tabulated with respect to the damage level and strain ageing time, as shown in Table 4. At 1.5% damage level, the changes of parameter n due to aging is insignificant. At 13% damage level, the
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those obtained from the experimental tests. As shown in Figs. 13– 15, the calibrated model can describe well the stress–strain curve for partially damages steel material at different ages.
6. Conclusion The effect of strain aging on the mechanical properties of partially damaged Grade 350 structural mild steel was investigated in this paper. It is shown that the strain aging can substantially affect the strength and ductility of the partly damaged steel structures. Two-stage experimental tests were conducted to explore the strain ageing effects in which time and damage level were taken into account as two effective variables. Moreover, microstructure examination was conducted using SEM. The following conclusions can be drawn from this study: - The strain ageing effect depends strongly on the damage level. For higher damage levels, the effect of strain ageing is greater. The ultimate stress may increase up to 40% whilst the reduction in ultimate strain might be up to 80%. Moreover, the fracture strain may reduce around 10% whilst the reduction rate depends on the measurement method; - For Grade 350 steel, the strain ageing effect becomes significant at a time between 6 and 12 h after the initial occurrence of damage. The strain hardening capacity of the steel after high partial damage vanishes at a time between 12 and 24 h after damage; - Using the obtained stress–strain curves for the partly damaged steel material, the effect of strain ageing on the global behaviour of a steel beam was studied. It was shown that due to strain ageing the loading capacity of the steel beam may decrease up to 30%. - Due to significant effects of strain ageing on the ductility and strength of damaged steel structures, the relevant codes of practice need to take this effect into account.
Acknowledgement The research work presented in this paper was supported by Australian Research Council through a Discovery Project (DP1096454) awarded to the second author. References
Fig. 15. Comparing Ramberg–Osgood curves with experimental data at 19% damage level (a) no-age (b) 2 days (c) 7 days.
value of n decreases from 245.59 for no-ageing specimen to 17.55 and 5.46 for 2 days and 7 days age specimens, respectively. At 19% damage level, the reduction of n with strain ageing is lower compared to that of 13% damage level as it decreases from 54.5 for no-age specimen to 50.63 and 44.63 for 2 and 7 days age specimens, respectively. Using the parameters shown in Table 4 the stress–strain curves obtained from the Ramberg–Osgood model are compared with
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