CHAPTER 2 LITERATURE REVIEW

Transcription

1 8 CHAPTER 2 LITERATURE REVIEW 2.1 GENERAL A brief review of the research carried out during the past years related to the behaviour of bolted steel angle tension members is presented herewith. Literature pertaining to both hot-rolled steel tension members and cold-formed steel tension members is presented. The review includes experimental, analytical and numerical investigations carried out in the past years. 2.2 SHEAR LAG Tension members are used in a variety of structures such as trusses, transmission towers etc. The most widely used structural shapes are the angle section and the channel sections. Angles may be used as single angles or double angles and the connection may be bolted or welded. For practical reasons, it is unusual to connect the entire cross-section to the gusset plate. As a result, highly non uniform stresses will be generated near the connection and this can cause localized yielding in parts of the cross-section. Thus, the whole cross-section may not be fully utilized which causes a reduction in the net section efficiency. This loss of efficiency of the section is due to shear lag effect. An accurate estimation of this non uniform stress distribution is necessary for determination of load carrying capacity of angles under tension. This non-uniform stress distribution across cross-section of the angle

2 9 connected by only one leg to the gusset is shown in Figure 2.1. The concept of effective net area has been traditionally used to account for this nonuniform stress distribution. Less stressed material Outstanding leg (A 2 ) More stressed material Connected leg ( A 1 ) Figure 2.1 Non uniform stress distribution across the cross-section of the angle Effect of Shear Lag Shear lag reduces the effectiveness of tension member components that are not connected directly to a gusset plate or other anchorage. The efficiency of a member can be increased by reducing the areas of such components relative to the area of the member as a whole. The distance from a fastener plane (gusset plate) to the C.G of the area tributary to it is a convenient measure of the distribution of the cross-sectional area of a member. For example, the tributary area of an angle member of Figure 2.2 is the entire area of the angle and the co-ordinate x from the fastener plane to the centroid of the area is a measure of the efficiency of the cross section.

3 10 Figure 2.2 Tributary areas of angle section Shear Lag Factor Shear lag is influenced by the length and eccentricity of the connection. The effect of these two parameters can be expressed as an efficiency coefficient given by where K 4 = 1- ( x / L) (2.1) K 4 = Shear lag factor x refers to the distance from the face of the connection to the centre of gravity of the member, last one). L is the length of connection (distance from the first fastener to the 2.3 STUDIES ON HOT ROLLED MEMBERS Chesson and Munse (1963) investigated a wide range of truss type tension members using both test results obtained from their own experiments and from others. The parameters studied included different cross sectional configurations, connections, materials, and fabrication methods. An empirical equation to calculate the net section efficiency was

4 11 proposed. It was based on the test results of 218 specimens among which there were 56 single angles and 33 double angles. Both riveted and bolted connections were examined. An empirical equation to calculate the net section efficiency was proposed. Chesson and Munse found that the net section efficiency of tension members with bolted or riveted end connections was a function of a number of factors and it could be expressed as follows: where A ne = K 1 K 2 K 3 K 4 A n (2.2) A ne = effective net area of cross-section A n = net area of cross-section K 1 = Q < 1 K 2 = 0.85 for members with punched holes = 1 for members with drilled holes K 3 = A n /A g A g = gross area of cross-section K 4 = 1 x /L. K 1 is the factor that accounts for the ductility of material, in which, the term Q is the percent reduction in the area at rupture of a standard tensile test coupon (51mm gauge length). K 2 is the fabrication factor that accounts for the reduction in efficiency due to the effect of punching the holes. K 3 is a geometry factor that accounts for the effect of hole spacing on the connection. K 4 is the shear lag factor. This factor takes into account for both the eccentricity in the connected part and the connection length. In the expression for K 4, x refers to the distance from the face of the connection to the centre of gravity of the member, and L represents the connection length and is taken as the distance between extreme fasteners. Kennedy and Sinclair (1969) investigated the influence of the edge distance and the end distance on net section efficiency. In this investigation,

5 single angle, single bolted connections were tested. In order to simulate the fabrication of members in field conditions, all the specimens were cut to length by shearing and all holes were punched. The test results showed that minimum edge and end distances were required to develop the yield strength of the cross section. March (1969) conducted a series of tests on single angle members in tension and compression. The effects of plastic behaviour were studied during ultimate loading of the sections. March stated that as the extreme fibres of the section yield, the line of action of the load would move, as well as the eccentricity. Based on these observations, he proposed that the net effective area (A ne ) could be calculated as follows: A ne 2 (Lc L0t)t dt L 0.04 L c (2.3) where L c = width of the connected leg L 0 = width of the unconnected leg. t = thickness of the section L = distance from the point of loading to the innermost bolt. d = diameter of the bolt hole. For unequal leg angles, this formula gave a good prediction if the long leg was connected. However, the prediction was rather optimistic if the short leg was connected. Hardash and Bjorhovde (1985) tested 28 specimens to develop an improved design method for gusset plates. Gage between lines of bolts, edge distance, bolt spacing and number of bolts were considered as the strength parameters. Gusset plates fastened with two lines of bolts were tested. Test specimens had a gage length of 51, 76 and 101 mm, edge distance of 25, 38

6 13 mm, and pitch distance of 38 and 51mm. Connections had two to five bolts in a bolt line and diameter of bolt holes were 14 and 17 mm. The average material properties of 27 specimens had a yield strength of 229 MPa and an ultimate strength of 323 MPa. One specimen had a yield strength value of 341 MPa and ultimate strength of 444 MPa. Test plates had a basic failure mode consisting of tensile failure across the last row of bolts, along with an elongation of the bolt holes. Load deformation curves of each test specimen was obtained and it was observed that the drop in strength from the ultimate load to second strength plateau corresponded approximately to the ultimate strength of the net area at the last row of bolts. Ultimate shear resistance was more difficult to define, because, the shear stress behaviour varied among the test specimens. Shear stress was found to be dependent on the connection length and a new block shear capacity equation, which includes the connection length factor, was developed. Murty et al. (1988) summarized the design approaches for computing the ultimate strength of bolted single angles in accordance with the following five specifications: the American Association of State Highway and Transportation Officials, the American Institute of Steel Construction, American society of Civil Engineers, the Canadian Standards Association, and the British Standards Institute. These specifications were compared with the test results of Nelson (1953), and an experimental program carried out. They observed that certain combinations of end distance, and pitch may cause the block shear mode of failure instead of net section failure. Although there was no specific design equation proposed, they concluded that a distinction should be made when predicting the ultimate strengths of angles connected by the long leg or short leg.

7 14 Epstein (1992) performed an experimental study on double-row, staggered, and unstaggered bolted connections of structural steel angles. The basic connections to be tested were pairs of angles, 8 mm thick, connected by two rows of 8 mm diameter bolts in two rows on a 150 mm leg. Outstanding legs of the angles vary between 90, 210 and 150 mm. An end and edge distances of 38 mm, a bolt diameter of 19 mm were used in the connections. The effect of several parameters in the connection geometry was investigated. Test results were compared with the current code provisions and a revised treatment was suggested by inclusion of a shear lag factor to the equation. Gaylord (1992) presented a similar equation as Munse and Chesson (1963). He suggested that the effective net area of tension member was a function of four factors: steel ductility, fabrication methods, connection efficiency, and shear lag effects. Their expression was as follows: A eff = K 1 K 2 K 3 K 4 A n (2.4) where K 1 = ductility factor = R < = 1.0 K 2 = fabrication factor = 0.85 for punching effects = 1.0 for drilling effects. K 3 = efficiency coefficient K 4 = shear lag factor. R = percent reduction in the cross sectional area of a tensile coupon at failure. The efficiency coefficient and the shear lag factors were similar to that presented by Munse and Chesson (1963). Gaylord suggested that the effect of punched bolt holes reduced net section capacity by 15%. Also, they suggest that the inclusion of ductility factor had an effect on connection efficiency. The R value was determined experimentally from tensile coupon

8 15 tests. More ductile steels would allow for a better distribution of stress concentrations along a cross sectional area than lower ductile steels. Wu and Kulak (1993) conducted an experimental program to investigate the shear lag effect on single and double angle tension members. The parameters studied include length of members, length of the connection, size and disposition of the cross-section, including angle thickness and whether the long leg or short leg was connected, out of plane stiffness of the gusset plate for the single angle cases. Based on the test results, the following design formula was proposed: T r = 0.85 Φ (F u A cn + βf y A o ) (2.5) where T r = factored resistance of the member Φ = resistance factor =0.90 F u = ultimate tensile strength of the material F y = yield strength of the material A cn = net area of the connected leg at the critical cross-section, computed by taking the diameter of holes 2mm larger than the nominal size if the holes were punched. A o = gross area of the outstanding leg β = 1.0 for members with four or more fasteners per line in the connection. = 0.5 for members with fewer than four fasteners per line in the connection. Gross et al (1995) tested ten A588 Grade 50 and three A36 steel single angle tension members with various leg sizes that failed in block shear. A588 Grade 50 steel had a yield and ultimate strength of 427 and 545 MPa and A36 steel had a yield and ultimate strength value of 310 and 469 MPa, respectively. Bolt holes having a diameter of 21 mm and a bolt hole spacing

9 16 of 64 mm and an end distance of 38 mm were used in all specimens. The edge distance was varied between 32, 38, 44 and 50mm. Test results were compared with the AISC-ASD and AISC-LRFD equation predictions and it was observed that code treatments accurately predict failure loads for A36 and A588 specimens. Cunnigham et al (1995) performed a statistical study to assess the American block shear load capacity predictions. Even though, both ASD and LRFD equations predicts the failure loads with a reasonable level of accuracy on average, it was observed that both the ASD and LRFD block shear predictions have drawbacks in terms of anticipated failure modes. It is evident from the test results that tension and shear planes do not rupture simultaneously as assumed in ASD specification. Thus, Cunnigham set the geometric and material parameters that had been investigated, and studied several other parameters such as in-plane shear eccentricity and tension eccentricity. Some equations, which include different types of failure modes and variables, were presented to predict block shear load capacity. Kulak and Grondin (2001) performed a statistical study on evaluation of LRFD rules for block shear capacities in bolted connections with test results. It was stated that there were two equations to predict the block shear capacity but the one including the shear ultimate strength in combination with the tensile yield strength seemed unlikely. Examination of the test results on gusset plates reveals that there is not sufficient tensile ductility to permit shear fracture to occur. Mohan Gupta and Gupta (2002) presented simple equations for predicting the load carrying capacity of single and double angles in tension, for net section failure based on previously published experimental results. These experimental results were divided into two parts: first which follow all

10 17 the following four conditions, and the second which violate one or more of the following conditions. The conditions are: a) The pitch is not less than the minimum and not more than the maximum, b) minimum edge distance is provided, c) minimum end distance is provided and d) thickness of angle sections is not less than that normally used in structural works. The following conclusions are obtained. 1. There is no significant difference between the net section efficiencies of single and double angle tension members, because of the shear lag effect. As such, there is no need to make any distinction between single and double angles in working out the load carrying capacities. 2. The following equations, based on net section efficiency, may be used to predict the load carrying capacity of single and double angles with relatively longer connection lengths. For unequal angles connected by their long length, R n = 0.85A n f u (2.6) For equal angles R n = 0.80A n f u (2.7) For unequal angles connected by their short leg, R n = 0.75A n f u (2.8) where R n = Net section strength A n = Net cross-sectional area f u = Ultimate tensile strength of the material

11 18 3. There is an acute shortage of test results with relatively shorter connection lengths. Gupta Mohan and Gupta (2005) analysed the Indian standard for design of steel structures (IS ) which follows working stress method, and found that the provisions for design of angle tension members were conservative for single angles when the number of bolts were relatively more and less conservative, for single angles and double angles with lesser number of bolts. For block shear failures, these predictions were least conservative. They adopted these provisions in the revision of the code in the limit state format and in these, provisions were adequate for single angles with three or more bolts. The net section strength of single and double angle members was adequately represented by the equation where R n = fu (A 1 +A 2 k) (2.9) k = 3A 1 /(3A 1 +A 2 ) for three or more bolts per row. k = A 1 /(A 1 +A 2 ) for two bolts per row. The equation to predict the block shear strength is R b = f u A nt +f ys A gv (2.10) where The strength based on yielding of gross section is R g = f y.a g (2.11) A 1 = Net area of the connected leg A 2 = Gross area of the unconnected leg. A g = Gross cross-sectional area A gv = Gross area subjected to shear A nt = Net area subjected to tension

12 19 R n = Net section strength R g = Gross section strength R b = Block shear rupture strength f u = Ultimate tensile strength of steel f y = Yield strength of steel f ys = Shear yield strength of steel ( = 0.6fy) k = Reduction factor, to account for the effects of connection eccentricity and shear lag. They also reported that design strengths were obtained by using specified minimum values of yield stress and ultimate stress in association with partial safety factors. The lower of these gave the maximum factored load carrying capacity of angles in tension. The factor of safety obtained as a result indicated adequate representation of the design strengths. The above research studies reveals that the effect of shear lag on hot-rolled steel structural connections was well understood. Hence most of the codal provisions had already incorporated necessary design guidelines. 2.4 STUDIES ON COLD-FORMED STEEL TENSION MEMBERS Bryan (1993) presented a paper on the design of bolted joints in cold-formed steel sections. Design expressions using joint flexibility for the bearing strength of bolted joints and for the joint moment under load were given. It was shown how the design expressions may be used to estimate the moment capacity and moment/rotation relationship of bolt groups, and how this information may be used to give an economical design of structural assemblies.

13 20 La Boube and Yu (1995) conducted an experimental and analytical study at the University of Missouri Rolla, to expand the knowledge and understanding of the behaviour of cold-formed steel bolted connections. This research consisted of two parts. The first part concentrated on the tensile capacity, bearing capacity and the interaction of tension and bearing capacities of flat sheet cold-formed steel bolted connections. For the specimens that failed in bearing, the results showed that the AISI specification was a good predictor of the ultimate strength while the AISC specification was not. For the specimens that failed in net section, both the AISI and AISC specifications were deemed to be good predictors. In the second part, the tensile capacity and bearing capacity of bolted connections of flat sheet, angle and channel cold-formed steel members were addressed. For the angle and channel sections that failed by net section fracture, the studies had shown that the current AISC specification formulation for addressing the influence of shear lag was unacceptable for cold-formed steel connections. Based on the tests results, the equations that could estimate the influence of shear lag on the tensile capacity of bolted connections were derived for cold-formed angle and channel sections and were stated as follows: where For angle sections, U = x /L 0.9 (2.12) For channel sections, U = x /L 0.9 (2.13) U = net section efficiency, x = distance from the face of the connection to the center of gravity of the member L = connection length Seleim and LaBoube (1996) studied the behaviour of low ductility steel in cold-formed steel connections. Single lap bolted connections were

14 21 studied to assess the influence of steels not meeting the specified ductility requirements. The conclusions obtained from forty-seven tests results were: AISI equations provide conservative strength prediction for the limit state edge shearing parallel to the direction of loading, specimens that failed in a bearing type mode were subjected to small deformation prior, out-of-plane shearing and fracture in the net section did not occur for specimens governed by net-section failure as per AISI provisions. Test results indicated that failure modes in low ductility steels were inconsistent with observed failure modes in adequate ductility steels. Chung and Lau (1999) conducted an experimental investigation on cold-formed steel members with bolted moment connections. Two lipped C sections back to back with four bolts per member were used as beam and column members. Four modes of failure such as bearing failure in section web around bolt hole, lateral torsional buckling of gusset plate, flexural failure of connected member and combined compression and bending failure of column member were identified. Among sixteen tests, the moment resistance of bolted moment connections with four bolts per member was found to lie between 42% and 84% of the moment capacities of the connected members. It was observed that moment connections among cold-formed steel members were structurally feasible and economical through rational design. Yip and Cheng (2000) performed an experimental program consisting of 23 angle and channel specimens to study the shear lag effect. The connection length and cross sectional geometry are two major parameters studied. With the test results, the net section efficiency(u) and the behaviour of the specimen were discussed. Finite element method was used to model and analyze the test specimens. A parametric study was also set up using the developed finite element models to investigate the factors affecting the net section efficiency of angle and channel sections. With the results obtained

15 22 from the parametric study, it was concluded that the current design equations give inconsistent predictions on the net section efficiency of cold-formed tension members. It was found that the net section efficiency does not only depend on the connection length(l) and eccentricity( x ), but also the flat width to thickness(w/t) and flat width to bolt diameter (w/d) ratios. Based on this observation, new net section efficiency equations were developed using non-linear regression analysis for both angle and channel sections as a) For equal leg angle members connected by one leg or unequal leg angles with long leg connected i) with one bolt in the line of force U = (w/t) 0.3 (w/d) (2.14) ii) with two or more bolts in the line of force U = ( x /L ) 0.41 (w/t) 0.36 (w/d) (2.15) b) For channel members connected by the web i) with one bolt in the line of force U = (w/t) 0.4 (w/d) (2.16) ii) with two or more bolts in the line of force U = ( x /L ) 0.85 (w/t) 0.54 (w/d) (2.17) Rogers and Hancock (2000) investigated the failure modes of bolted sheet steel connections loaded in shear. The load capacity formulations presented in the American Iron and Steel Institute specification could not accurately predict the failure modes of these connections when loaded in shear. A modification to the bearing coefficient provisions to account for the reduced bearing resistance of the materials was necessary and was suggested. A revision of the net section fracture design method was also required.

16 23 Recommendations concerning the procedure used to identify the net section fracture and bearing failure modes were also made. Chi Ling pan (2004) investigated the shear lag effect on bolted cold formed steel tension members. Fifty four Channel sections with different dimensions tested by using bolted connections were discussed. The comparisons were made between test results and predictions computed based on several specifications such as AISC, AISI, AS/NZS etc. The predictions according to AISI and AS/NZS seemed to be overestimating as compared to the test results. The computed values based on the AISC specification provided good correlation with the test results. To study the stress distribution at the various locations of the cross-section of specimen, finite element software ANSYS was used. It was observed that stress distribution over the entire section of the specimen was not uniform. The stresses in the connected element (web) were larger than the stresses in the unconnected elements (flanges). He also observed that the ratio of connection eccentricity to connection length x /L, and the ratio of unconnected element width to connected element width W u /W c were the two factors which had mainly influenced the tensile strength of channel sections. The tensile strength may be estimated by applying the following empirical equation for the channel section, P = UA n F u (2.18) where U = ( x /L) 0.14 (W u /W c ) (2.19) x = Connection eccentricity L = Connection Length W u = width of unconnected elements W c = width of connected elements

17 24 Valdier Francisco de paula et al (2008), presented experimental results of 66 specimens carried out on cold-formed steel angles fastened with bolts under tension. Out of the 66 specimens, four angles have one bolt and remaining 62 specimens have more than one bolt per line. These 66 specimens showed net section failure with two or more bolts in the crosssection of cold-formed angles. The specimens tested had equal or different legs, different cross-sections, various thicknesses and a varied number of bolts and bolt lines. He conducted multiple linear regression analysis and suggested the expression for net section efficiency (U) which depended on the geometrical factors such as connection eccentricity ( x ), connection length (L), width of connected leg of the angle (b c ), net width of the angle with connected leg (b cn ), width of unconnected leg (b d ), nominal bolt diameter (d) and angle thickness (t). The proposed equation is U = ( x /L) (0.63b cn b d 0.47d-1.70t)/b c (2.20) The effect of shear lag on cold-formed steel sections were much limited when compared to studies on hot-rolled steel sections. The American Iron and Steel Institute, Australian/ New Zealand and British Standard codes were recently revised and incorporated the provisions. Hence there is a need to investigate the behaviour of cold-formed steel angles under tension. 2.5 STUDIES ON NUMERICAL INVESTIGATION Epstein and Chamarajanagar (1996) developed analytical model for a series of single angle tests with staggered bolted connections. A 20 node brick element was used in the finite element modeling of the angle sections to capture the stress concentration effect in the vicinity of bolt holes. The material nonlinear effects were modeled using the von Mises yield criterion

18 25 and the material stress-strain curve was assumed to be elastic perfectly plastic. In this study, a strain based failure criterion in which failure was assumed to have occurred once the maximum strain reached five times the initial yield strain was employed to capture the failure load. The bolts were assumed to be rigid and the load was transferred from the gusset plate to the angle fully by the bearing of bolts. The longitudinal and the in-plane transverse displacements of the nodes attached to the bearing surfaces were coupled to one another. This finite element study included only the material non-linear effects and the geometric non-linear effects were considered to be negligible. Kulak and Wu (1997) conducted a finite element analysis to evaluate the stress distribution of the critical cross section at ultimate load. A large strain four-node quadrilateral shell element with six degrees of freedom per node was used in the finite element modeling of the double angle members. The gusset plate was modelled using elastic four node quadrilateral shell element as yielding of the gusset plate was not observed in the experimental tests. An elasto plastic von Mises yield criterion was adopted to represent the material non-linear effects. The material stress-strain curve was described by a multilinear isotropic hardening behaviour. Based on the symmetry consideration of the specimen, only half the length of the specimen was modelled. Similarly, due to the symmetry of the double angle members about gusset plate, only one of the pair angles was modelled. In the finite element model, the effect of bolts was modelled by coupling the longitudinal and in-plane transverse degrees of freedom of the nodes attached to the hole surfaces on which the bolts bear against during deformation. The finite element model included both geometric as well as material non-linear effects. In the analysis, the failure load of the angle section was taken as the load corresponding to the last converged step. At failure, significant necking of the net area between the leg edge and lead bolt hole was observed.

19 26 Epstein McGinnis (2000) conducted a second study aimed at refining the tools developed in Epstein s 1996 work. The boundary conditions and the solution procedure were identical to the 1996 Epstein study. Although this finite element study included only the material nonlinearity as represented by a simple elastic-perfectly-plastic yield criterion, the finite element results indicated a reasonably good correlation with the experimental results. Chung and Ip (2000) investigated the finite element modeling of bolted connections between cold-formed steel strips and hot-rolled steel plates under shear. The modeling was done with three-dimensional solid elements using the results of the coupon tests. Twelve lap shear tests with two steel grades, one bolt diameter and two washer sizes were carried out to caliber the finite element models. The load-extension behaviour of the bolted connections agreed well with the test data for extensions upto 3mm in terms of both initial and the final slopes and also the maximum load carrying capacities. The patterns of yielding, strength degradation and the strain distribution the connections were established in detail using finite element modeling. Typical strain levels in cold-formed steel strips in the vicinity of bolt holes were found to be 40%. Therefore it is important to incorporate reduced strength at larger strains for accurate prediction of the load-carrying capacities of bolted connections. Cem Topkaya (2004) aimed to develop simple block shear capacity equations that are based on principles of mechanics in this study. A parametric study was conducted to identify important parameters that influence the block shear response. Specimens tested by three independent research teams were modeled and analyzed. Analysis was performed with a finite element program ANSYS. Gusset plates were modeled with six node triangular plane stress elements, whereas angles and tee sections were modeled with ten node tetrahedral elements. These element types were

20 27 capable of showing high material and geometric nonlinearities. The nonlinear stress strain behavior of steel was modeled using von Mises yield criterion with isotropic hardening. A generic true stress- true strain response was used in all analysis. Throughout the analysis the Newton-Raphson method is used to trace the entire nonlinear load-deflection response and failure load was assumed to be the maximum load reached during the loading history. He presented three equations based on the analysis performed to predict block shear load capacity Gupta Mohan and Gupta (2004) conducted finite element analysis to evaluate the stress distribution in the angle at design loads predicted by equations developed earlier on the basis of experimental results. Detailed finite element analysis was conducted on three bolted angle specimens. These three angle specimens had two, three and four bolts at each end respectively. Since the thickness of the angles and the gusset plates used were all less than one inch, shell elements were used to represent all of the connection and member components. A plastic quadrilateral shell element was used to model the angles and the gusset plates. This element had six degrees of freedom at each of the four nodes. Yielding was determined using the von Mises yield criteria. From the analysis it was observed that in three bolts and four bolts connections, the zones of high stresses lay along the critical section in the connected and the un-connected leg. In a two bolt connection, the stresses were mainly concentrated on the connected leg only, and in the un-connected leg, the stresses were relatively low. The magnitude and the distribution of stresses at critical section for three bolts and four bolts connection was almost the same. The resulting stress distribution justified the use of area along the gross shear plane in block shear strength prediction equation. The distribution and concentration of von Mises stresses indicated that block shear failure

21 28 might occur in a two bolt connection, and net section failure might occur in three and four bolts connection. Only limited studies are reported on the numerical investigation of steel angle sections connected to gusset plate. It is very difficult to model the specimens because behaviour depends on a number of parameters like geometry of section, material properties etc. But numerical analysis using finite element method will be useful to investigate the behaviour and to predict the modes of failure.