Introduction to Philosophy of design of steel structures
In this lecture the Philosophy of design of steel structures has been briefly discussed.
1.1 Historical Development of Structural Steel in the World
Ancient times culture were the first users of iron some 3 to 4 millenniums ago. Their language was altered to Indo – European and they were native of Asia Minor. There is archaeological evidence of usage of iron dating back to 1000 BC, when Indus valley, Egyptians and probably the Greeks used iron for structures. Thus, iron industry has a long ancestry.
Wrought iron had been produced from the time of middle ages, if not before, through the firing of iron ore and charcoal in “bloomery”. This method was replaced by blast furnaces from 1490 onwards. With the aid of water-powered bellows, blast furnaces were used for increased output and continuous production. A century later, rolling mill was introduced for enhanced output. The traditional use of wrought iron was principally as dowels and ties to strengthen masonry structures. As early as 6th century, iron tie-bars had been incorporated in arches of Haghia Sophia in Istanbul. Renaissance domes often relied on linked bars to reinforce their bases. A new degree of sophistication was reached in the 1770 in the design of Pantheon in Paris.
Till the 18th century the output of charcoal fired blast furnaces was almost fully converted to wrought iron production, with about 5% being used for casting. Galleries for the House of Commons in England were built of slender cast iron columns in 1706 and cast iron railings were erected around St. Paul’s Cathedral in London in 1710. Abraham Darby discovered smelting of iron with coke in 1709. This led to further improvements by 1780s when workable wrought iron was developed. The iron master Henry Cort took out two patents in 1783-84, one for a coal-fired refractory furnace and the other for a method of rolling iron into standard shapes. Without the ability to roll wrought iron (into standard shapes), structural advances, which we see today, would never have taken place.
Technological revolution, industrial revolution and growth of mills continued in the West and this increased the use of iron in structures. Large-scale use of iron for structural purposes started in the Europe in the later part of the 18th Century. The first of its kind was the 100 feet Coalbrookadale arch bridge in England constructed in 1779. This was a large size cast iron bridge. The use of cast iron as a primary construction material continued up to about 1840 and then onwards, there was a preference towards wrought iron, which is more ductile and malleable. The evolution of making better steel continued with elements like manganese being added during the manufacturing process. In 1855, Sir Henry Bessemer of England invented and patented the process of making steel. It is also worth mentioning that William Kelly of USA had also developed the technique of making steel at about the same time. Until the earlier part of the 19th century, the ‘Bessemer process’ was very popular. Along with Bessemer process, Siemens Martin process of open-hearth technique made commercial steel popular in the 19th century. In the later part of the 19th century and early 20th century, there had been a revolution in making better and newer grades of steel with the advent of newer technologies. This trend has continued until now and today we have very many variety of steels produced by adding appropriate quantities of alloying elements such as carbon, manganese, silicon, chromium, nickel and molybdenum etc to suit the needs of broad and diverse range of applications.
There are numerous examples of usage of iron in India in the great epics Ramayana and Mahabharatha. However the archaeological evidence of usage of iron in our country, is from the Indus valley civilisation. There are evidences of iron being used in some instruments. The iron pillar made in the 5th century (standing till today in Mehrauli Village, Delhi, within a few yards from Kutub Minar) evokes the interest and excitement of all the enlightened visitors. Scientists describe this as a “Rustless Wonder”. Another example in India is the Iron post in Kodachadri Village in Karnataka, which has 14 metres tall “Dwaja Stamba” reported to have remained without rusting for nearly 1½ millennia. The exciting aspects of these structures is not merely the obvious fact of technological advances in India at that time, but in the developments of techniques for handling, lifting, erecting and securing such obviously heavy artefacts. These two are merely examples besides several others. We can see several steel
structures in public buildings, railway stations and bridges, which testifies the growth of steel in the past. The “Rabindra Sethu” Howrah Bridge in Calcutta stands testimony to a marvel in steel. Even after its service life, Howrah Bridge today stands as a monument. The recent example is the Second Hooghly cable stayed bridge at Calcutta, which involves 13,200 tonnes of steel. Similarly the Jogighopa rail-cum-road bridge across the river Brahmaputra is an example of steel intensive construction, which used 20,000 tonnes of steel. There are numerous bridges, especially for railways built, exclusively using steel.
As far as production of steel in India is concerned, as early as in 1907, Jamsetji Nusserwanji Tata set up the first steel manufacturing plant at Jamshedpur. Also at the same time in 1905, Tata Institute for research and development works was established in Bangalore, Karnataka which was later renamed as Indian Institute of Science, Bangalore. Later Pandit Jawaharlal Nehru realised the potential for the usage of steel in India and authorised the setting up of major steel plants at Bhilai, Rourkela and Durgapur in the first two five year plans. In Karnataka Sir Mokshakundam Visweswarayya established the Bhadravati Steel Plant. The annual production of steel in 1999-2000 has touched about 25 million tonnes and this is slated to grow at a faster rate.
1.2 Stress-strain Behaviour of Steel
The primary characteristics of structural steel include mechanical and chemical properties, metallurgical structures and weldability. In the past structural engineers have tended to focus only on the tensile properties (longitudinal yield stress and ultimate tensile strength), with some attention paid to the deformability as measured by the elongation at fracture of a tension specimen. Since the modulus of elasticity, E, is constant for all practical purposes for all grades of steel, it has rarely been a consideration other than for serviceability issues. Weldability was assumed to be adequate for all such steels. Deformability or ductility was similarly assumed to be satisfactory, in part because the design specification has offered only very limited, specific requirements.
The stress-strain curve for steel is generally obtained from tensile test on standard specimens as shown in Fig.1.1. The details of the specimen and the method of testing is elaborated in IS: 1608 (1995). The important parameters are the gauge length ‘Lc’ and the initial cross section area So. The loads are applied through the threaded or shouldered ends. The initial gauge length is taken as 5.65 (So)1/2 in the case of rectangular specimen and it is five times the diameter in the case of circular specimen. A typical stress-strain curve of the tensile test coupon is shown in Fig.1.2 in which a sharp change in yield point followed by plastic strain is observed. When the specimen undergoes deformation after yielding, Luder’s lines or Luder’s bands are observed on the surface of the specimen as shown in Fig.1.3.
These bands represent the region, which has deformed plastically and as the load is increased, they extend to the full gauge length. This occurs over the Luder’s strain of 1 to 2% for structural mild steel. After a certain amount of the plastic deformation of the material, due to reorientation of the crystal structure an increase in load is observed with increase in strain. This range is called the strain hardening range. After a little increase in load, the specimen eventually fractures. After the failure it is seen that the fractured surface of the two pieces form a cup and cone arrangement. This cup and cone fracture is considered to be an indication of ductile fracture. It is seen from Fig.1.2 that the elastic strain is up to εy followed by a yield plateau between strains εy and εsh and a strain hardening range start at εsh and the specimen fail at εult where εy, εsh and εult are the strains at onset of yielding, strain
hardening and failure respectively. Depending on the steel used, εsh generally varies between 5 to 15 εy, with an average value of 10 εy typically used in many applications. For all structural steels, the modulus of elasticity can be taken as 205,000 MPa and the tangent modus at the onset of strain hardening is roughly 1/30th of that value or approximately 6700 MPa.
Certain steels, due to their specific microstructure, do not show a sharp yield point but rather they yield continuously as shown in Fig. 1.4. For such steels the yield stress is always taken as the stress at which a line at 0.2% strain, parallel to the elastic portion, intercepts the stress strain curve. This is shown in Fig. 1.4. A schematic diagram of the tensile coupon at failure is shown in Fig.1.5. It is seen that approximately at the mid section the area is ‘S’ compared to original area S0. Since S is the actual area experiencing the strain, the true stress is given by ft = P/S, where P is the load.
However S is very difficult to evaluate compared to S0 and the nominal stress or the engineering stress is given by fn = P/ S0 . Similarly, the engineering strain is taken as the ratio of the change in length to original length. However the true strain is obtained when instantaneous strain is integrated over the whole of the elongation, given by
where ft and fn are the true and nominal stresses respectively and εt and εn are the true and nominal strain respectively.
1.3 Experimental Investigation of a True Stress-True Strain Model
In this section, experimental investigation of a true stress-true strain model is described. A standard uniaxial tensile test, in general, provides the basic mechanical properties of steel required by a structural designer; thus, the mill certificates provide properties such as yield strength Fy , ultimate strength Fu, and strain at fracture εf . 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. As the load increases and when the specimen begins to fail, the cross-section area at the failure location reduces drastically, which is known as the “necking” of the section. 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. Owing to the non-uniform 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. The true stress-true strain relationship is based on the instantaneous geometric dimensions of the test specimen. Figure 1.6 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 proportional limit stress Fpl is typically established by means of 0.01% strain offset method. Thus, the engineering stress can be related to engineering strain as follows: Fe = Eεe in the range Fe < Fpl and εe < εpl, where E is the initial elastic modulus of steel, which is often taken as 200,000MPa. 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 In compressible material, which is valid for steel. Resulting relations are Ft = Fe (1 + εe) and εt = ln(1 + εe), where Ft and Fe are the true stress and engineering stress and εt and εe are the true strain and the engineering strain, respectively. The difference between true stress and engineering stress at
Figure 1.6 The engineering stress-strain relations and the proposed true stress-true strain material model
Region-II (Nonlinear Elastic Range). This range represents a region between the proportional limit and the yield point. The yield point Fy may be conveniently established as 0.2% strain offset method. In this region, the variation of stress-strain relationship can be idealized as Fe
Fpl+Et (εe−εpl), which is valid in the range Fpl < Fe < Fy. Here, Et is the tangent modulus given as Et = (Fy −Fpl)/(εy −εpl). The true stress and true strain can be obtained as in the linear elastic range as follows: Ft = Fe(1 + εe) and εt = ln(1 + εe), where εpl < εe < εy .
Region-III (Yield Plateau). Some steels may exhibit yield plateau. The engineering stress in
this region can be assumed as a constant value of Fy , which is valid in the range εy < εe <
εsh, where εsh is the strain at the onset of strain hardening. The ratio between εsh and εy is
defined here as m = εsh/εy . The value for m must be determined from the uniaxial tension
test. The true stress and true strain can be obtained as in the linear elastic range as; Ft =
Fy(1+εe) and εt = ln(1+εe), where εy < εe < εsh.
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. Though this region involves a nonlinear stress-strain relation, it is postulated that the true stress and the true strain can be obtained using the relations Ft = Fe(1 + εe) and εt = ln(1 + εe). However, a power law is often used to relate the true stress to the true strain in this strain hardening region. A power law of the form Ft = Fut · (εt/εut)n is proposed herein, where Fut and εut are the true stress and true strain associated with the ultimate tensile strength Fu. The value for n must be established for different steel grades which may be achieved using a least square analysis of the corresponding experimental results. This range is valid for εsh < εe < εu.
Region-V (Strain Softening). This region represents the behaviour 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. It was 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. It was also proposed a weighted-average method for determining the uni-axial 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. A power-law fit method, which represents strain hardening region of the flow curve, can be used as the lower bound whereas a linear strain hardening model can be used as the upper bound. Accordingly, the lower bound power law is Ft = Fut · (εt/εut)n, which was established in Region-IV and the upper bound linear hardening model could be Ft = (a0 + a1εt), where constants are a0 = Fut · (1 − εut) and a1 = Fut. Based on the weighted-average method, the true stress-strain relation in the postultimate strength region (Region-V) may be derived as Ft = Fut[w · (εt/εut)n + (1 − w) · (1 + εt − εut)], where w is the unknown weighting constant. 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.
The A992 is a relatively new steel grade for building construction in North America. The 350W steel is the Canadian standard CSA G40.21 steel, which is somewhat equivalent to ASTM A572 Grade 50 steel.
The true stress-true strain model parameters were established through amalgamation of experimental and numerical modeling techniques. The test program described here considered twenty eight tensile coupons, fourteen each from two different steel grades, namely, ASTM A992 steel and the 350W steel. The tensile coupons for this investigation were cut along the rolling direction (length direction) of standard W310 × 39 (W12 × 26) wide flange beam sections. 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 A370-10 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 9.1mmand thickness of the web coupons was 5.8 mm. The width of the specimens was about 40 mm. 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. The specimen ID (identification) used in this investigation is based on net area/gross area ratio of the test specimen. In the specimen ID related to the experimental investigation, A992/350W indicates the steel grade followed by F/W, which indicates the flange/web, followed by the value of net area/gross area ratio. For example, Specimen ID-A992-F-0.8 refers to a coupon cut from the flange of the A992 steel with net area/gross area ratio of 0.8. Three identical flange and web coupons with no holes (shown as F1, W1, etc., in Figure 1.7 and Table 1.1) were used to establish the mechanical characteristics of the steel grades under consideration. 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. Holes with net area/gross area ratios varying from 0.5 to 0.9 in increments of 0.1 were prepared for the flange coupons, whereas holes with net area/gross area ratios varying from 0.5 to 0.9 in increments of 0.2 were considered for the web coupons. The photographic image of the test specimens (solid sample with no holes, and perforated samples) is shown in Figures 1.7(a) and 1.7(b), respectively. The coupons were tension tested in a Tinius Olsen machine with an axial load capacity of 600 kN. Each test specimen was first aligned vertically and centered with respect to the grips of the machine’s loading platforms. Two extensometers having gauge lengths of 200mm and 50mm were attached on either face of the test coupon. The larger extensometer was used to establish the overall engineering stress-strain curve of the coupons, whereas the smaller extensometer, which had a greater sensitivity, allowed a more accurate estimation of the initial modulus (E) and the proportional limit stress (Fpl). Figure 1.8 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 1.1 summarizes the mechanical properties established from the solid coupon tensile tests. The average yield strength Fy and ultimate strength Fu of the A992-flange coupons were calculated to be 444MPa and 577MPa, respectively, resulting in the Fy/Fu ratio of 0.77. The average strains corresponding to the ultimate strength εu and at fracture ε f were measured to be 13.8% and 20.8%, respectively. Note that the above strains were based on 200mm gauge length. The average Fy and Fu values for the A992-web coupons were 409MPa and 573MPa, respectively, resulting in the Fy/Fu ratio of 0.71. These coupons reached the ultimate strength at the strain of 15.6% and fractured at the strain of 21.4%. The 350Wflange coupons had the Fy and Fu values of 427MPa and 578MPa, respectively, resulting in the Fy/Fu ratio of 0.74. The average εu and ε f values associated with these coupons were 13.9% and 22.0%, respectively. The average Fy and Fu values of 350W-web coupons were measured to be 416MPa and 582MPa, respectively, resulting in the Fy/Fu ratio of 0.71. These coupons had average εu and ε f of 15.3% and 19.5%, respectively. The Fy/Fu ratio value for the A992-flange coupon was 4%higher than that of the 350W-flange coupon.
The true stress-true strain model parameters for Regions-I, II, and III were extracted from these results and are shown in Table 1.2. The Region-IV requires the power law parameter n, which was established through linear regression of the test results corresponding to that region. The test results considered for this region is between points C and D in Figure 1.6 and is valid for true stress-true strain region between points C1 and D1 shown in Figure 1.6. Figure 1.9 shows a representative calculation corresponding to 350W web element. The experimental engineering stress and strains were first converted to true stress and strains, and then the strain hardening portion of the relationship was used to obtain a power law fit, which resulted in n = 0.1628 for 350W web element. Complete power law relationships for A992, 350W flange and web elements are given in Table 1.2. The Region-V requires establishment of a weighting constant w, which is found here by trial and error. The task is to match the finite element numerical analysis results with the corresponding experimental results in this region. Here, the tensile test coupon was modeled using the finite element analysis package ADINA. 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. Also, this shell element can be efficiently used with plastic multilinear material models for large-displacement/large-strain analyses. Each shell element employed 2 × 2 integration points in the mid surface (in r-s plane) and 3 Gauss numerical integration points through thickness (in t-direction). The model also incorporated a geometric imperfection (maximum amplitude of 0.1% of the width—40mm) of a half sine wave along the gauge length in order to cause diffuse necking. 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. For analysis of members with mid-hole, which is presented in the next section, a finer mesh was used for a 50mm length of the middle region, where the strain gradient is expected to be large. 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 1.2. The material model in Region-V first requires a true fracture strain εft (point E1 shown in Figure 1.6). A Study undertaken previously indicated that the localized fracture strains for structural steel under uni-axial tensile load could vary between 80% and 120%. Therefore, this study considered a true fracture strain of 100% (i.e., εft = 100%.) corresponding to point E1. Figure 1.10 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. The weighting constant w for Region-V has to be established in an iterative manner by numerical simulation of tensile tests until a good correlation is achieved between the calculated and the experimental load extension curves. In order to illustrate the influence of the weighting constant, three different values for w = 1.0, 0.6, and 0.4 were considered in the numerical simulations. Figure 1.11 shows the resulting FE predicted responses along with the experimental responses of three identical tension coupons (A992 flange). The weighting factor w = 1.0, which represents the Region-V by a power-law hardening model, results in a numerical response well below the experimental curves. However, for w = 0.4, the numerical curve was slightly above the experimental curve and sustains larger fracture strain.
The weighting value w = 0.6 gives the best fit for this set of experimental results. Although a suitable weight constant w to reproduce the experimental stress-strain curve needs to be established by trial and error approach, only a few trials were required in this study. Table 1.2 shows the values of the weighting constants for A992, 350W 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. The stresses at fracture varied as high as a maximum 3%, whereas the fracture strain differed by a maximum 5% when compared to the corresponding experimental values. Figure 1.12 shows the resulting true stress-true strain model for A992 flange element.
Steel structures construction often necessitates fabrication of holes in the flanges of steel beams. 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 uni-axial tension tests provide engineering stress-engineering strain results which are not accurate particularly in the strain hardening range and in the post ultimate strength range. This investigation developed true stress-true strain relationships for structural steels in general, and for A992 and 350W steel grades in particular. This paper established five-stage true stress-true strain constitutive models for structural steels, based on numerical simulations calibrated against experimental uni-axial tension test results. The proposed model uses a power law in strain hardening range and a weighted power-law in the post ultimate range. The true stress-true strain model parameters were established through a combination of experimental and numerical modeling techniques. The stresses and strains at fracture for the standard coupons based on numerical analysis differed by less than 5% when compared to the corresponding
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results from the experiment. The proposed material constitutive relation was further verified through comparison of finite element analysis load deformation behavior with the corresponding experimental results for perforated tension coupons.
1.4 Structural Steel from IS 800-2007
1.4.1 The provisions in this section are applicable to the steels commonly used in steel construction, namely, structural mild steel and high tensile structural steel.
1.4.2 All the structural steel used in general construction, coming under the purview of this standard shall before fabrication conform to IS 2062.
1.4.3 Structural steel other than those specified in 1.4.2 may also be used provided that the permissible stresses and other design provisions are suitably modified and the steel is also suitable for the type of fabrication adopted.
126.96.36.199 Steel that is not supported by mill test result may be used only in unimportant members and details, where their properties such as ductility and weldability would not affect the performance requirements of the members and the structure as a whole. However, such steels may be used in structural system after confirming their quality by carrying out appropriate tests in accordance with the method specified in IS 1608.
The properties of structural steel for use in design, may be taken as given in 188.8.131.52 and 184.108.40.206.
220.127.116.11 Physical properties of structural steel irrespective of its grade may be taken as:
Unit mass of steel, p = 7850 kg/m~
Modulus of elasticity, E = 2.0 x 10s N/mm2 (MPa)
Poisson ratio, p = 0.3
Modulus of rigidity, G = 0.769 x 10s N/mm2 (MPa)
Co-efficient of thermal expansion cx.= 12 x 10’ /“c
18.104.22.168 Mechanical properties of structural steel
The principal mechanical properties of the structural steel important in design are the yield stress, fy; the tensile or ultimate stress, fu; the maximum percent elongation on a standard gauge length and notch toughness. Except for notch toughness, the other properties are determined by conducting tensile tests on samples cut from the plates, sections, etc, in accordance with IS 1608. Commonly used properties for the common steel products of different specifications are summarized in Table 1.4
Table 1.4 Tensile Properties of Structural Steel Products