Intermediate HSS bracing members during seismic excitations: modeling, design, and behavior

Abstract

Concentric hollow structural section (HSS) bracing members are used frequently in steel framed structural systems to resist seismic excitations. Finite element modeling of the HSS braces that utilizes the true stress-strain curves produces hysteresis responses that are reasonable matches to the experimental response. True stress-strain curves are obtained from coupon tests or stub-column tests while utilizing an exponential function or strain hardening rule with a trial and error procedure to obtain the hysteresis behavior. In the current study, the true stress-strain curves are directly obtained from tests on stub-columns extracted from the full scale HSS bracing members away from the mid-length plastic hinge after cyclic testing. Two experimental tests (Shaback 2001 and Haddad 2004) were used to validate the model. Results indicate that the stress-strain curves for these braces are not unique. A refined damage accumulation model for ultra-low-cycle fatigue is implemented to predict fracture of the brace tests. The refined damage model is then used in the finite element modeling to predict fracture of braces in a chevron braced frame of an eight-storey building subjected to selected ground motions analyzed using OpenSees program. Results indicate that all braces could sustain the selected earthquake records without fracture.

Keywords

HSS / FEM / stress-strain curves / damage model

Introduction

During earthquakes, braces in concentrically braced steel frames undergo several cycles of tensile and compressive excursions that cause the brace to yield and buckle. Local buckling occurs before fracture takes place at the mid-length of the brace if the connections are designed and detailed properly.

The most important parameters of the brace that affect the hysteresis behavior are its slenderness ratio and width-to-thickness ratio. The width-to-thickness ratio of the HSS braces tested by Shaback [¹] and Haddad [²] were between 7 and 17, covering the lower limits for width-to-thickness ratio presented by the AISC and CSA seismic provisions from 2005 to 2015 [³–⁷]. The range of slenderness ratio was between 64.6 and 96.3, placing the test specimens in the intermediate brace range: 50 – 110, as reported by Wijesundara et al. [⁸]. The test braces were fabricated from CSA G40.21-350W [⁹] steel which is similar in material content to the United States ASTM A500 Grade B or C steels [¹⁰]. The braces were designed according to the weak–brace strong–gusset concept with a free length around two times the gusset thickness as suggested by Astaneh-Asl et al. [¹¹] to allow for the free formation of plastic hinges at the far ends of the specimens in the gusset plates. The gusset plates were made of hot rolled steel with minimum yield strength of 300 MPa. The gusset plates were inserted in slots at the ends of the braces. The asymmetric loading protocol began with compression loading as presented by Shaback [¹] and Haddad [²].

In the present study, 18 HSS bracing members (Shaback [¹] and Haddad [²]) were modeled using the Abaqus finite element code [¹²]. The results of the experimental tests were used to validate the axial and lateral hysteresis results of the corresponding finite element analyses. A refined damage accumulation model based on the significant cumulative plastic strain was able to predict the fracture life of the 18 HSS braces. The refined model was then used to estimate the fracture of braces in a chevron braced frame of an eight-storey building subject to different earthquake ground records.

Finite elements models have the advantage over fiber models of being able to model the local buckling phenomenon. However, the finite element models are very sensitive to the material model, especially the stress-strain curves of the brace materials. Therefore, the objective here was to develop a finite element model that utilizes a material model based on behaviors obtained from the stub-column tests, and a refined damage model to predict fracture of HSS braces. Recent seismic provisions in terms of width-to-thickness ratio were examined. Lastly, the fracture life of braces in an eight-storey chevron frame in a building subjected to different ground motions was predicted. The results lead to recommendations for future concentric braced frame research.

Stub-column tests

Stub-column tests were introduced to provide information about the material behavior of HSS sections: test specimens should be designed according to the procedure described in the Structural Stability Research Council (Galambos [¹³], (Technical Memorandum No. 3)). In general, the stub-column member should be short enough to exclude member buckling when compressed, but long enough to contain the initial residual stress pattern as the brace member cut from the same stock. However, the stub-column member for cold-formed sections should be sufficiently long to include the effects of local buckling and cold forming on the column behavior. The sub-column results presented here have not been published previously.

For the brace tests of Shaback [¹] and Haddad [²], 500 mm long stub-column specimens were tested. The ends were cold-sawn and machined plane and perpendicular to the longitudinal axes of the columns. Four transducers (of 10 mm stroke) were installed at the middle of the outer faces of the stub-column member (one at each face) with some clearance. These transducers cover a central length of 250 mm.

Thirty three compressive tests were conducted: three for each heat mill and section size used for the tested specimens (Table 1) before and after cyclic testing of the large scale braces. Twenty four stub-column tests were conducted from the same stock of the HSS material before cyclic testing, and nine specimens from the HSS material were tested after cyclic testing of the braces. In the latter, the stub-columns were extracted from locations away from the mid-length plastic hinges. The engineering properties – the Young’s modulus of the materials and the strength at 0.2% strain offset were computed: average properties are reported in Table 2. Typical sub-column stress-strain curves are shown in Fig. 1. As may be seen, neither the engineering stress-strain curves (produced from stub-columns before cyclic testing) nor the cyclic (true) stress-strain curves (produced from stub-columns extracted from the large scale braces after cyclic testing) are identical. The true stress-strain curves are obtained by subtracting the elastic part from the cyclic stress-strain curves shown in Fig. 4. The true stress-strain curves were used in the finite element analyses presented here. Use of these material characteristics in the finite element modeling was expected to produce hysteresis behavior equivalent to that seen in the large scale brace tests of Shaback [¹] and Haddad [²]. The nine stub-columns that were subjected to the cyclic loading protocol during the testing of the large scale braces, were compressed to produce the actual (true) stress-strain curves in the range of strains and stresses measured (Fig. 4) in the brace tests.

Finite element modeling

Nonlinear finite element analyses were performed using the Abaqus code [¹²]. Four node quadrilateral shell elements with nine integration points (Simpson integration) through their thickness were used. The combined isotropic-kinematic hardening model with data type equal to half a cycle was used in the analysis of all 18 HSS braces listed in Table 1. Patran preprocessor [¹⁴] was used to sketch the geometry and to perform the meshing of the braces. High mesh density was applied at the expected plastic hinge locations. The equivalence option was used to connect the parts of the model. The input file was adjusted manually for the material model and was introduced to the Abaqus environment where first modal shape was obtained through linear buckling analysis while multiplying the mid-length out-of-plane displacement by the value of the initial imperfection as measured experimentally (Table 1). The gusset plates were assumed to be perfectly connected to the HSS. The weld was not modeled in the finite element analysis. The far ends of the HSS braces were constrained with the fixed boundary conditions and the applied displacement history that were assigned through the steps of the nonlinear analysis. Specimen 4B with end connection is shown in Fig. 2. The Young’s moduli are listed in Table 2 and the Poisson’s ratio was taken as 0.3. The full Newton Raphson method was implemented in a static based approach to satisfy equilibrium. The number of nodes, elements used in modeling each of Shaback [¹] and Haddad [²] tests are (16730, 13580), respectively.

Refined material and damage models

The combined isotropic-kinematic hardening material model is capable of simulating the expansion, contraction and the shift of the yield surface in stress space to model the brace behavior under cyclic loading. The translation in the yield surface is a function of the plastic strains and the plastic hardening modulus. The translation of the yield surface is subtracted from the corresponding stresses according to the Ziegler’s [¹⁵] kinematic hardening rule (Haddad [²]. The stress-strain curves obtained from the stub-column tests on specimens cut from sections away from the mid-length plastic hinge of the cyclically tested HSS braces (Table 1) were used in the finite element analysis. Compressive cyclic envelopes that matched the experimental results were expected to be produced automatically during the triple actions of global buckling, local buckling and yielding.

A damage model refined from that presented by Haddad [²] is implemented herein to predict fracture. A strain-based approach using incremental deformation plasticity theory, based on the significant cumulative plastic strain, was implemented in the refined damage model to predict the behavior and fracture life of the braces. The significant cumulative plastic strain is defined as:

(1){\epsilon }_P^{*}\hspace{0.17em}=\hspace{0.17em}{\displaystyle {\sum }_{tensile\hspace{0.17em}-\hspace{0.17em}cycles}{\displaystyle \underset{{\epsilon }_{\mathrm{1}}}{\overset{{\epsilon }_{\mathrm{2}}}{\int }}\exp \hspace{0.17em}(|\mathrm{1.5}T|)\hspace{0.17em}\mathrm{d}{\epsilon }_t\hspace{0.17em}-\hspace{0.17em}{\displaystyle {\sum }_{compressive\hspace{0.17em}-\hspace{0.17em}cycles}{\displaystyle \underset{{\epsilon }_{\mathrm{1}}}{\overset{{\epsilon }_{\mathrm{2}}}{\int }}\exp \hspace{0.17em}(|\mathrm{1.5}T|)\hspace{0.17em}\mathrm{d}{\epsilon }_c}}}},

where T is the triaxiality ratio which is defined as the mean stress over the effective or the von Mises stress. Cycles are considered tensile when triaxiality is greater than 1/3 and vice versa. In the current study, damage evolution occurs when triaxiality is greater than 1/3. The 1/3 is a revised simplification for maximum principle stress [¹⁶].

The stiffness in the direction of the maximum principal stress, when the significant cumulative plastic strain reached a threshold value, was reduced to zero at the appropriate integration points. Cracks develop on the compressive side at the mid-length plastic hinge of the specimen. Hence, the stiffness at this location will gradually decrease causing the specimen to fail due to numerical error.

The effect of mesh density, on the significant cumulative plastic strains, along the mid-length plastic hinge, is shown in Fig. 3. The number of elements (5, 10, 15, 20, 25, 30, 35, 40, 45, 50, and 55) corresponds to element length of (10, 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0.9) times the thickness of the HSS, respectively. The size of elements across the depth and width of the HSS was kept constant. In the mesh density analysis, the specimens were subjected to the same applied displacement histories until the point where the first cracking was noted experimentally. The specimens investigated here had different cross-sectional size, thickness, length, and yield strength. Therefore, from one specimen to another, the effective slenderness ratio of the brace, the width-to-thickness thickness ratio and the yield strength of the HSS were different. The effect of test repeatability on the significant cumulative plastic strain was also investigated. The significant cumulative plastic strain at which the first crack occurred during the experimental tests converged to a single value (0.8) at an element length that is equal to the thickness of the HSS. Reducing the element length substantially below the thickness of the HSS along the length of the mid-length plastic hinge, results in non-convergence in the finite element analysis. The element size was kept constant for the analysis of all specimens to obtain the hysteresis loops. The significant cumulative plastic strain of an average value equal to 0.8 is set in the current damage accumulation model to predict fracture.

Geometric measurements were made at both ends of the HSS specimens before making the end slots for the gusset plates. As expected, the total width or depth (b) and thickness (t) measurements are above the minimum specified nominal width or depth and thickness as specified by CSA-S16-09 [⁶] standards. Variations in experimental measurements of the flat width or depth are compared to the flat width or depth as calculated by the CSA-S16-09 [⁶] code (b-4t) as seen in Table 3. The comparison suggests that the folding and welding of the specimens will leave some differences in measurements of the flat width and depth of the specimens. It should be mentioned that the AISC 360-10 [¹⁷] defines the flat width as (b-3t) while Euro code 3 [¹⁸] defines the flat width as (b-2t).

Model verification

The experimental axial-hysteresis stress-strain loops for specimens 3A, 4A, 1B, 3B (Shaback [¹] and specimens 2, 4, 8 (Haddad [²]) are shown in Fig. 4(a). The equivalent finite element axial-hysteresis stress-strain loops for specimens 4A, 1B, 3A are shown in Fig. 4(b). For all specimens except specimen 3A, the tensile capacity of the axial-hysteresis stress-strain loops followed the engineering stress-strain curves for the corresponding stub-column monotonic tests. The corresponding stress-strain curves of stub-column cyclic tests for specimens extracted from locations away from the mid-length of the braces, represent the true stress-strain curves that are utilized in the finite element analysis to obtain the axial-hysteresis stress-strain loops. The elastic part was subtracted from stress-strain curves so that initial yielding starts at zero plastic strain. The initial yield stress at zero equivalent plastic strain (defined in this study at 0.025% proof stress) varied between 315 and 350 MPa for the true stress-strain curves as shown in Fig. 1. The previous initial stress range is below the minimum yield strength specified by the AISC code. The Young’s modulus and Poisson’s ratio are substituted in the elastic part of the finite element analysis. Excellent agreement is seen between the experimental and finite element results. The axial-hysteresis strains are between 2 and 1 in compression and in tension, respectively. The previous range is covered by the stress-strain curves obtained from the stub-column cyclic and monotonic tests. It should be mentioned that local buckling occurred at greater strain values in all stub-column tests than in the large scale brace tests. In the latter, the local buckling at the mid-length plastic hinge is shown in Fig. 6(a) for specimen 4B, as an example.

The axial and lateral hysteresis loops with their equivalent finite element analysis loops for specimens 4B, 3C, 4, 6, and 8 are shown in Fig. 5 in terms of load versus displacement. The hysteresis behavior for specimen 4B is described during the finite element analysis in the following paragraphs.

Specimen 4B had the largest section size and was nearly the longest among the specimens tested. Its slenderness ratio was 84.8 and the width-to-thickness ratio, (b-4t)/t, was 12.9. In both the finite element analysis and the actual experiment, specimen 4B was subjected to four elastic cycles of compression-tension loading. The specimen buckled into a bow shape during the compressive side of the fifth cycle. Two plastic hinges formed progressively in the free length at the two end gusset plates. The full tension side of the same cycle straightened the specimen. The specimen buckled at lower axial displacements in the sixth compressive cycle than in the fifth. This is mainly attributed to the plastic hinges which formed at both ends of the specimen. During the same cycle, the bow shape buckling became larger. The two free length plastic hinges formed completely during the sixth compressive cycle. Fillet weld cracking was noticed in the experiment in both the compression and tension sides of the gusset plate adjacent to free length. The weld was not modeled in the finite element model. The brace showed a marked decrease in compressive load capacity in the cycles following the fifth cycle. The tension side of the sixth cycle fully yielded the specimen. Stable plastic behavior of the gusset plates was evident at all times during the finite element analysis and experiment.

The specimen kinked at the mid-length during the compressive side of cycles seven and eight indicating the beginning of the development of the third plastic hinge there. At the mid-length plastic hinge, local buckling began during the compressive side of the ninth cycle. Local buckling is characterized by inward and outward bulging of the specimens’ compressive web and flanges, respectively. The outward bulging is closer to the compressive web rather than the tension web. A small inward bow of the tension web was also noticed. The local buckling partially disappeared during the tension side of the same cycle. Local buckling continued to increase in severity during the following cycles as shown in Fig. 6(a).

In the experiment, the continued working of the mid-length region resulted in the tearing of the flanges on the corners at the compressive side of the locally buckled region during the tension excursion of the 11th and final cycle of loading. A small size bent was present at the compressive corners and web at the mid-length plastic hinge upon compressive buckling in both finite element analysis and the experiment. This bent is usually severe at the compressive corners as shown in Fig. 6(a) due to the geometric nature of local buckling and bending. The bent opens during tensile loading and becomes concave during compressive buckling leading to crack formation due to the cyclic rotational demand and reduction in the cross-sectional area at those locations. The axial strain hysteresis loops are shown in Fig. 7(a) for the strain gauge shown in Fig. 6. The reduction in strain gauge measurements due to the formation of the small size bent on the compressive side of cycles 11 and 12 may be seen in Fig. 7(b). Tearing and subsequent propagation of the tear resulted in the failure of the specimen upon increasing levels of axial displacement as shown in Fig. 6(b).

All specimens followed the same trend in behavior until the initiation of cracking at the compressive face of the mid-length plastic hinge. It should be mentioned that the small size bent developed in neither the finite element modeling nor the experiment for specimens 8, 9 and 10. The width-to-thickness ratio for specimens 8, 9, and 10 is very small as listed in Table 1. Therefore, from the compressive loading point of view, local buckling was not severe for these three specimens, thus the initiation of cracking was delayed. Hence, the number of cycles to crack initiation is greater for the previous three specimens than that for all other specimens as listed in Table 4. From the tensile loading point of view, the previous three specimens were subjected to limited tension yielding since their width-to-thickness ratio is small. Thus, braces with small width-to-thickness ratio can survive 15–20 cycles before crack initiation if subjected to limited tension yielding. For specimen 3C, crack initiation and fracture occurred in cycles 16 and 17, respectively as will be reasoned and explained in the following paragraph. Specimen 6 with the thicker gusset plates fractured at the inner connection of the gusset to the HSS.

Discussion of results

Code limits in relation to brace parameters (test repeat, slenderness ratio and width–to–thickness ratio)

Specimen 2 is a repeat test to specimen 4A. Similarly, specimen 4 is a repeat test to 3B. Specimens 2 or 4A and specimens 4 or 3B had different section size. For repeat tests, the experimental axial-hysteresis stress-strain behavior was similar in terms of tensile and compressive capacities as shown in Fig. 4. Specimen 1B had the same section size and lower slenderness ratio compared to specimen 3B. The axial hysteresis loops for specimen 3B encompassed the corresponding axial-hysteresis loops for specimen 1B. However, the tensile capacities of specimens 1B and 3B were similar.

Specimens 8 and 3A had the same length with width-to-thickness ratios of 7 and 17, respectively. While the tensile stress-strain capacity of specimen 8 followed the engineering stress-strain curve for the stub-column monotonic tests as shown in Fig. 4, 3A had lower tensile capacity; suggesting that the width-to-thickness ratio for specimen 3A is not acceptable during seismic excitations. Specimen 3A had a slenderness ratio of 90.2. The AISC 2010 [⁴] and current AISC 2015 [⁵] seismic provisions have a lower limit on the width-to-thickness ratio of 13.25 with no lower limit on slenderness ratio, while the CSA-S16-09 [⁶] seismic provision has a lower limit on the width-to-thickness ratio of 17.64 with a lower limit on slenderness of 70. The latter limit on the width-to-thickness ratio is applicable in the CSA-S16-09 [⁶] seismic provisions when the effective slenderness ratio is less or equal 100. Specimens 4 or 3B had similar slenderness ratios to specimen 3A with a width-to-thickness ratio of 13. Specimens 4 or 3B followed the engineering stress-stress curve. Specimens 2 or 4A had lower slenderness ratio than specimen 3A with a width-to-thickness ratio of 16.1 and a slenderness ratio of 83.9. Specimens 2 or 3B followed the engineering stress-strain curve. Specimens 7 and 2A with a slenderness ratio of (64.6, 68.4) and a width-to-thickness ratio of (13, 16.1), had the first crack occurring in cycles 8 and 9, respectively. Please note that the applied displacement history used in the quasi-static cyclic tests of Shaback [¹] and Haddad [²] was very demanding in compression but not that demanding in tension. The specimens in the tests of Shaback [¹] and Haddad [²] were subjected to increasing displacement reversals with amplitudes greater in compression than in tension. Thereafter, the tensile displacement amplitude was kept constant to avoid exceeding the tensile capacity of the actuators (2.7 MN). Meanwhile, the compression displacement amplitudes were increased and then kept constant to avoid exceeding the stroke capacity of the actuators. From the above analogy and to satisfy the near-filed and far-field loading protocols, both the AISC and CSA seismic provisions should follow the AISC 2005 [³] seismic provisions in terms of a lower limit on the width-to-thickness ratio of 15.41 in addition to the CSA-S16-09 [⁶] seismic provisions in term of a lower limit on the slenderness ratio of 70. Braces with the previous suggested limits on width-to-thickness ratio and slenderness ratio may sustain 10 high amplitude cycles before crack initiation at the mid-length of the brace. The braces must be designed with a minimum thickness of gusset plates while taking the frame action into account [¹⁹,²⁰]) to avoid the fracture in the gusset plates. The capacity of the gusset plate will be reduce due to frame action. However, the gusset should not be very thick as explains below from the strain mechanism point of view.

Effect of initial imperfection on brace behavior

For numerical simulation, Hassan et al. [²¹] recommended an initial imperfection range: 0.05%–0.2% of the brace length. In the previous range, the upper limit is recommended by Ziemian [²²] while an initial imperfection value equal to 0.1% of the brace length is recommended by Chajes [²³]. Hu [²⁴], Wijesundara et al. [²⁵], Nascimbene et al. [²⁶] used initial imperfection values equal to 0.1%, 0.0714%, 0.143% of the brace length, respectively. The arbitrary selection of the initial imperfection value as a fixed percentage of the brace length can lead to non-negligible differences of the brace buckling and prediction of structural response D’ Aniello et al. [²⁷,²⁸]. In the current study, the initial imperfection values (Table 1) that were implemented in the finite element analysis are in the range: 0.0004%–0.06% of the brace length, as measured experimentally. It should be mentioned that specimen 3, 7, and 10 buckled in the opposite direction of the maximum initial imperfection during the experiments. Specimen 5 initially buckled in the direction of the maximum initial imperfection during the compressive side of the fourth cycle and reversed the direction of buckling during the following cycle and continued with the reversed buckling shape during cyclic loading until fracture occurred. The previous incidents are attributed to the small initial imperfection values and the variation of stresses along the length of the specimens in addition to the thicker gusset plates for specimen 5. For the latter case, greater stress values are present at the free length of the gusset plates and at the mid-length of specimen 5 compared to the stresses at the regions between the previous expected plastic hinge locations. Due to the high stiffness of thicker gussets that are not easy to be straightened when pulled in tension, the brace segments between previous expected plastic hinge locations may concave in an opposite direction to the initial buckling shape (cycle no. 4) forcing the brace to reverse the direction of buckling (cycle no 5).

Interestingly, Li and Wu [²⁹] showed that the governing imperfection distribution that should be adopted within finite element analysis to capture the actual load carrying capacity does not always follow the lowest buckling mode, which distinguishes it from the existed specification. The same behavior was noticed in the finite element analysis with the small initial imperfection values even with the first buckling mode that was implemented in the finite element analysis.

Effect of gusset thickness on brace behavior

It is worth mentioning that high accumulation of plastic strains was present in the gusset plates for specimen 3C at the earlier stages of cyclic loading. For specimen 3C, the tensile capacity of the HSS (2284kN) was slightly below the tensile capacity of the gusset plates (2318kN). Specimen 3C has the highest number of cycles prior to fracture among all specimens of Shaback [¹]. This suggests that yielding of the gusset plates is preferable since it allows the brace to develop large inelastic deformation. The gusset plate should be designed to develop the full tensile capacity of the HSS in tension without buckling in compression. Increasing the gusset plate tensile resistance extensively reduces the fracture life of the brace [²]. However, the frame action should be considered in the design of gusset plates. In tension, the frame deformation effects have caused an increase in the von Mises and the first principal stresses located at the Whitmore section of the gusset, as well as the increase in the combined effects of the factored tension and shear forces at the gusset edges [²⁰]. In compression, the frame effects have caused a reduction in the capacity of the gusset plates [¹⁹]. The effective width of gusset plates is controlled by the Whitmore width at an angle of 30 degrees, a slight increase in the thickness of gusset plates may be justified to avoid fracture of gusset plates especially when frame action is considered. A resistance factor of 0.9 for yielding and ultimate capacities should be used in all design expressions in the design procedure described by Haddad and Tremblay [³⁰] when checking failure modes of gusset buckling capacity, tear-out failure in the brace and in the gusset plate, tension yielding on the Whitmore section of the gusset plate, and failure of welds.

Validation of critical value for plastic strain

The significant cumulative plastic strain obtained from the refined damage model is of an average value of 0.8 for mesh size that is equal to the thickness of the HSS. The significant cumulative plastic is dependent on the cumulative plastic strain that has additional components (normal and shear strain) other than axial strain. The additional strain components are not measured by strain gauges. However, the main component is the axial strain. For specimen 5, strain gauge 8 is located very close to the crack that initiates at the compressive face of the mid-length plastic hinge. The axial strain hysteresis loops for strain gauge 8 is shown in Fig. 8(a). The measurement of the critical absolute sum of tensile minus compressive increments of the axial plastic strains is equal to 0.67 as shown in Fig. 8(b). The hysteresis behavior for strain gauge 8 is very similar to the axial hysteresis behavior of the whole brace. Strain gauge measurements at mid-length plastic hinge are at least ten times greater than global strain measurements for the whole brace. The strain gauge measurement for specimen 6 at the same location did not catch up with specimen 5 since fracture of specimen 6 with the 2 inch thick plate occurred at the net-section.

Case study

Chevron braces are a common configuration for providing lateral-load resistance in steel-framed buildings where braces are installed in an inverted V-shape there allowing free space for openings. Bracing types could be HSS, W-shapes, WT-shapes and double angles. The high compressive resistance of HSS bracing members is preferable. The brace refined damage model is used to predict the fracture life of all braces in an 8-storey chevron concentrically braced frame in a typical building located on a site class C in Victoria, B.C., Canada, subjected to four ground motions. The floor height at each floor level is equal to 3.8 m. The brace inclination angle is equal to 45 degrees. The same building plan with different number of floor levels was analyzed for different purposes by Robert and Tremblay [³¹], Chen and Tirca [³²], Tirca et al. [³³]. In the current study, the moderately ductile braced frame in the typical building shown in Fig. 9, is re-designed according to the provisions of NBCC (NRCC 2010) [³⁴] and CSA S16-09 [⁶]. The following gravity loads are considered in the calculations: a roof dead load of 3.4 kPa, a roof live load of 1.48 kPa, a floor dead load of 3.5 kPa, a floor live load of 2.4 kPa. The weight of floor partitions is equal to 1.0 kPa and the weight of the exterior walls is equal to 1.2 kPa.

The dynamic response-spectrum analysis method was used to determine seismic design forces in the braces. All braces are square HSS tubes conforming to CSA G40.21-350W [⁹]. The braces at the 7th floor level are selected as representatives to all brace. The size of the braces at the 7th floor is HSS 152 mm × 152 mm × 9.5 mm, which is the same as specimen 4B. In the building frame, the distance between the ductile hinges forming in the gusset plates upon brace buckling is 4836 mm. For specimen 4B, the length between the hinges was 4882 mm. The latter is very close to the length of the brace in the prototype building. The same finite element model for specimen 4B is used to examine the fracture life of the brace in the braced frame with a slightly modified version concerning the gusset plate thickness. The gusset plate thickness was equal to 19.1 mm when designed according to the Uniform Force Method and satisfied the failure modes of gusset buckling capacity, tear-out failure in the brace and in the gusset plate, tension yielding on the Whitmore section of the gusset plate, and failure of welds.

Nonlinear seismic response history analyses of the 8-storey building were re-carried out using the OpenSees program [³⁵]. The bracing members were re-modeled using force based nonlinear beam-column elements with fiber discretization of the cross-section [³⁶,³⁷]. The Giuffré-Menegotto-Pinto (Steel02) material model was selected to account for Bauschinger effect and simulate kinematic-isotropic strain hardening response. The nominal yield strength of 350 MPa was assigned to the steel material. Initial sinusoidal camber with maximum amplitude of 0.2% of the unsupported member length was specified for the bracing members [²²]. A co-rotational formulation was chosen to consider geometric nonlinearities for the braces. The analyses were performed for four different earthquake records from the PEER [³⁸] ground motion database: Nos. 057, 767, 986, and 1006, as listed in Table 5. Hence, a total of eight axial deformation histories were obtained for the two representative braces, which is of interest because some include several cycles in compression while others have more demand in tension.

The time histories axial deformation responses for the two braces (Left and Right) at the 7th floor, as monitored between the two gusset plate plastic hinges in the response history analyses were applied to the finite element model to verify crack initiation limit state. In the finite element model of the brace, the material properties used for specimen 4B were used after scaling the strength values so that yield strength becomes equal to 350 MPa, same as in the response history analyses. The brace axial deformation at yield is 8.3 mm.

Excellent agreement is found between the brace hysteretic response predicted by the OpenSees and Abaqus models for both braces and all ground motions as shown in Fig. 10. In all cases, the braces sustained larger axial deformations in compression as a result of the downward resultant brace load that was imposed at the beam mid-span after buckling of the braces. Yielding in tension only occurred in the Right brace under records Nos. 57, 767, and 1006. The use of heavier or stiffer profiles for braced-intercepted beams can be beneficial in reducing the brace ductility demand in compression and in activating the brace yielding in tension [³⁹,⁴⁰]. Local buckling did not develop in the finite element analysis of the 7th floor braces. Local buckling makes braces vulnerable to fracture [²]. These braces did not fracture under the four earthquake records according to the damage models. In addition, no crack initiation was detected in the Left and Right braces located at all other floor levels for the previous four records of ground motions.

The incremental dynamic analysis was carried in OpenSees program [³⁵] by applying the following scaling factors, SF: 1.5 and 2.0 to the records of ground motion, Nos. 057, 767, 986, and 1006. For the ground motion record No. 767, the results for SF = 1.0, 1.5 and 2.0 are plotted in Fig. 11 for the 4.0–16.0 s time interval. As shown in the figure, the Left brace is subjected to a series of tension excursions before it experiences a series of sequence of significant compression half-cycles. The opposite is observed for the Right brace. No crack initiation was detected in the Left and Right braces at all floor levels for the previous magnifications of all ground motions as could be seen from Fig. 12 for the second and seventh floors for ground motion record No. 767. From Fig. 12, the significant cumulative plastic strains are equal to 0.65 and 0.43 for the Left and Right braces of the second floor, respectively. Meanwhile, significant cumulative plastic strains are equal to 0.23 and 0.22 for the Left and Right braces of the seventh floor, respectively. Therefore, is more critical to experience damage when the brace is subjected to a series of tension excursions before compression excursions for the displacement histories used in the case study presented here. It should be also mentioned that no crack initiation was detected when using gusset plate thickness equal to 22.1 mm, suggesting that a slight increase in the gusset thickness would not affect the crack initiation limit state for the previous records of ground motions.

Conclusions and recommendations

1) The finite element model incorporating the refined damage model was able to simulate the axial and lateral hysteresis behavior of concentric braces up to and including fracture.

2) The engineering stress-strain curves that results from stub-column tests are not unique. Thus, the corresponding true stress-strain curves that produce the hysteresis response of HSS braces are not unique.

3) The hysteresis response of the HSS braces could fairly be obtained utilizing the engineering stress-strain curves obtained from stub-columns extracted from tested HSS braces away from the mid-length plastic hinge.

4) The significant cumulative plastic strain at which cracking occurs is sensitive to mesh size.

5) The mesh density analysis of finite element model revealed that the significant cumulative plastic strain at which the first cracking occurred is of average value equal to 0.8 at an element size equal to the thickness of the HSS.

6) Brace fracture was found to be more critical in braces that are subjected to a series of tension excursions before compression excursions are applied. In addition, a slight increase in the gusset plate thickness would not affect the fracture life of the brace.

7) Coupon and notch bar tests on small scale specimens (e.g.: sections of the webs, corners and welds) are recommended for HSS braces after testing away from the mid-length plastic hinges to check the significant cumulative plastic strain value at which first cracking occurs in the HSS.

8) The effect of corner radius of the HSS bracing members on the hysteresis behavior and fracture life is recommended for future research.

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