An analytical formulation to extract the capacity curve of steel structures

Among the methods for evaluating the nonlinear performance of structures, pushover analysis is an appropriate alternative instead of direct time history analysis. To accurately extract the capacity curve of a structure, according to the loading regulations/protocols such as FEMA-356 and ATC-40, lateral loads are incrementally applied to the structure in experimental tests until the structural failure occurs. Because of the cost and time-consuming nature of experimental tests, proposing mathematical/analytical methods could be the appropriate tools to predict the capacity curves of a system. The present study proposes a new method to find the capacity curves of cantilever steel beams based on mathematical formulations, structural analysis, and material properties. The reason to select a simple beam in this study is to shed more light on the unknown aspects of the system’s behavior. Therefore, in this research, the effect of axial load is ignored to clarify the nonlinear behavior of complicated systems such as frames. The strains, stresses, and other responses corresponding to large geometric deformations have been extracted in two cases with and without strain hardening by considering changes in the behavior of materials. The proposed method has been verified using the finite-element method with Abaqus software. The results indicate that the proposed method has acceptable accuracy and could be applied in the pushover analysis of steel structures.


List of symbols L u
The length of the beam between the point with M y and the support The load that causes plastic moment at the support P y The load that causes yielding moment at the support Δ u Displacement of point B due to plastic deformation Δ y−B Displacement of the free end of the beam due to the deformation of the elastic length of the beam at the point of beam failure Δ Displacement of the free end of the beam under P P Displacement of the free end of the beam due to only the plastification of the beam at the support A g Area of the beam cross-section M y = S * F y Yielding moment θ Rotation of the whole beam length d Beam cross-section depth

Introduction
Researchers in the field of structural and earthquake engineering use pushover analysis, which is a nonlinear static analysis method, to study the behavior of structures, such as large deformations and ductility. Pushover analysis is also applied to investigate the nonlinear performance of structures and is an alternative to time history analysis. To obtain the pushover curve, the structure is constructed in real size or on a smaller scale in the laboratory, and under predetermined instruction according to the loading regulations such as FEMA-356 and ATC-40, the lateral load is applied to the structure so that the structure eventually collapses.
Obviously, for steel structures, the first step of the pushover curve is describing their elastic behavior. In the first step, the behavior of the structure is linear and has the highest lateral stiffness, and the deformations are relatively small. When the material experiences inelastic behavior and forming of the first plastic hinge is started, the structure enters the nonlinear region. Gradually, the lateral stiffness of the structure is reduced until the structure became a mechanism. The last point of this curve indicates the maximum displacement and collapse load of the structure. In addition, the area under this curve shows the ductility capacity of the structure. Push-analysis was applied first in 1975 for systems of one degree of freedom (Freeman, 1975). This method was then gradually developed by other researchers, e.g., Shibaca and Sozen (1976), Saiidi and Sozen (1981), Fajfar and Fischinger (1988), Moghadam (2002), and Chopra and Goel (2002).
The seismic evaluation of concentrically braced frames is investigated experimentally by Zhao et al. (2017). In their research, the effects of axial loading and column shape are considered in the quasi-test of braced frames. Progressive collapse analysis of steel structure using a semi-rigid beam model is studied by Zhao et al. (2019). They studied the effect of bending and axial loading on the collapse mechanism of steel frames and confirmed that the simultaneous effects of these forces have a significant influence on the mechanical performance of the system.
There is some research for collapse analysis and pushover analysis of RC systems. Assessing seismic collapse of low ductility RC frames studied by Xian et al. (2020). They utilized incrementally dynamic analysis (IDA) for collapse assessment of RC frames. A numerical approach to pushover analysis of slender cantilever bridge piers was proposed by Di Re et al., (2022). Their study is based on an iterative approach to enforce the element equilibrium under large displacements, efficiently accounting for P-Delta effects induced by vertical loads.
Today, despite many advances in nonlinear analysis methods, researchers do not use analytical methods to obtain the pushover curve. Since in this case, the structure must tolerate large geometric nonlinear deformations to extract the pushover curve. High accuracy, low cost, and time savings are the advantages of the analytical method in comparison with experimental or numerical methods. Therefore, in this study, the authors used concepts such as the first and second moment-area theories, a combination of static equilibrium equations, materials' properties, and structural geometry in the case of large deformations, to propose an analytical method for extracting the pushover curve for a cantilever beam. The reason to select a simple beam for this study is to discover all aspects of the system's nonlinear behavior. The presented formulation in this research can obtain the pushover curve point by point and with high accuracy in the inelastic case for nonlinear geometric deformations. In addition, by solving the numerical example, it is shown that the strain hardening phenomenon cannot be ignored at large deformations, and if the strain hardening phenomenon is ignored, a large error in calculations will occur. In this research, the following assumptions in analytical and numerical modelings are considered.
• The beam has been restrained along the plastic hinge so that damages such as localized web and flange local buckling, and diagonal web buckling will not occur. • Lateral buckling will not occur during the beam and the failure of the beam is occurred only due to the rupture of the material.

Beam geometry and material properties
According to Fig. 1, consider the cantilever beam of L length and moment of inertia of I, and the modulus of elasticity of E. To prevent lateral buckling, the beam laterally restraint along its length and is fully fixed in support A, with a concentrated load P applied to its free end. If the load uniformly and gradually increases to collapse load ( P U ) , the free end deflection and the beam length rotation will increase accordingly. During loading, the flexural moment at support A gradually changes from the elastic flexural moment to the flexural yielding moment M y and then to the plastic flexural moment M P . If the strain hardening is considered, the flexural moment increases to reach the beam collapse moment ( M U ) , and if the strain hardening phenomenon is ignored, at the plastic flexural moment, the rotation continues to the collapse point.
In order to correctly apply the material properties, especially the tangential modulus in the inelastic and strain hardening regions of steel, according to Fig. 2, the modified strain stress curve has been used according to the suggestion of Boeraeve et al. (1993) and Gioncu and Mazzolani, (2003). The stress-strain curve continues until the final steel stress F U with a slope of E h = 0.03E after reaching the strain h , which corresponds to the end of the steel-plastic step, and after the stress reaches F U , it continues with a zero slope until it reaches the final strain U .

Small deformations' analysis P ≤ P y
All types of classic analytical methods in structural analysis references, such as virtual work, and first and second moment-area theories could be used to calculate the forces corresponding to elastic small deformation (Kassimali, 2018). At this point, the behavior of the entire length of the beam from the support to the free end is elastic.

Large deformations' analysis P > P y
The large deformation involves deformations after the yield point (formation of M y at the support) which consists of four distinct stages: a-formation of M P at the support; b-reaching the behavior of materials to the end of the steel-plastic step means reaching the maximum strain to h ; c-reaching the maximum stress to F U or in other words, reaching the maximum strain on the beam cross-section to hs ; d-reaching flexural moment at the support to M U (maximum flexural moment tolerable by the beam section) or in other words, reaching the maximum strain on the beam section to U . If the beam is designed such that local and general buckling failures, crippling under the load, etc. do not occur in the beam and the failure is limited to the failure of the beam caused by the flexural moment reaching the nominal flexural strength of the beam, the curvature diagram ( ∅ = M∕EI ) could be shown in large deformations as shown in Fig. 3 (right). The relation of force and moment in elastic rage according to Fig. 3 is expressed as the following equation: According to Fig. 4, at large deformations, the beam consists of two parts: plastic part (length AB ′ ) and elastic part (from point B ′ to the free end of the beam). In Fig. 4a, the parameter θ is the rotation of the whole beam, which consists of two parts: a-the rotation of the beam due to the (1) deformation of the plastic region u , b-the rotation of the elastic region y−U . In this figure, r is a tangent line at the point B ′ . Based on the geometry relations, in Fig. 4b the arc AB ′ is approximated by an arc of a circle, the angle of the chord AB ′ along the horizon is equal to u ∕2 and the length of the chord itself is equal to AB � = 2L u ∕ u sin u ∕2 . Point B ′ , which corresponds to point B before loading, is the position where the stress is reached to the yielding stress in the farthest section of fiber. According to Fig. 4a, the flexural moment at B ′ is obtained from the following equation: Given that the maximum stress in the farthest section of fiber at the point B ′ (Fig. 4a) has reached F y , based on the superposition principle, the following equation at point B ′ is written: (2)  (2) and (3) and simplifying them, the following equation is obtained: According to Fig. 4c and writing the support-based moment equilibrium equation for part AB ′ , the following equation is obtained: By placing the value of x B ′ in Fig. 4b and the value of M B ′ from Eq. (2) in Eq. (5) and simplifying it, the following equation is obtained: Combining Eqs. (4) and (6), a system of two nonlinear unknown parameters (P and L u ) is obtained according to the following equation: From the system of Eq. (7) not only the collapse load P U could be found but also the pushover curve can be obtained in the range of large deformations ( P > P y ) point by point. By placing M y ≤ M A ≤ M U , the load P corresponding to M A can be found. With the values of L u and P, the value of Δ H can be obtained from Eq. (1). It should be noted that in Eq. (7), the parameter u is not unknown because according to the first moment-area theory, the area of the hatched area ( Fig. 3 on the right) is equal to tan u and according to the material properties, according to the following equation, the value u is known in terms of L u : Based on the curvature-strain relationship in the section subjected to flexural moment (for small deformations: ε = y * ∅ and large deformations:ε = y * tan∅ , y: distance from the neutral fiber), assuming that for the inelastic deformations the relation ε = n y is established (for ST37 steel, ε U = 167 y means that the value of n at the moment of failure is 167 (Popov, 1990)), it can be concluded that at any curvature such as ∅ A , the curvature equation can be written in y (Eq. (12)). If the strain hardening phenomenon in steel is ignored, M U = M P , i.e., the cross-section of the beam at a fixed moment M P = Z * F y will increase the curvature until it is finally collapsed at the curvature ∅ U . However, if the strain hardening phenomenon in steel is considered, the flexural moment of the section collapse is equal to M U = Z * F U ( F U represents the maximum tolerable tensile stress of steel) which will occur at the curvature ∅ U (Fig. 5).
According to Fig. 5 and the assumption that the crosssection of the beam is symmetrical for the flexure axis, the curvature at the moment of failure and the yielding of the section are obtained from Eqs. (9) and (10), respectively, and by defining, the curvature ductility is equal to the ratio of the collapse ductility to the section yielding ductility (Eq. 11): As well, in the general case of inelastic deformation ∅ A > ∅ y , the curvature is obtained according to the following equation: By substituting ∅ A from Eq. (12) in Eq. (8), u can be presented in terms of yield strain y and L u in terms of the next following equation: The free end displacement and the rotation of the elastic part, due to the deformation of the elastic part of the beam, are obtained from the following equations, respectively: When the system of Eq. (7) for P U and M U is solved, the obtained parameter u is correspond to the moment of failure of the beam and can be calculated from Eq. (17). By definition, rotational ductility is the total rotation at the moment of failure divided by the rotation corresponding to the formation of M y at the support (Eq. (18)): According to Figs. 3 and 4, the total deflection of the free end of the beam consists of three parts: a-Δ u , which is the deflection of point B, which according to the second moment-area is equal to the moment of the hatched region in Fig. 3 around point B, b-deflection due to rotation of the elastic region due to deformation of the plastic region, i.e., L − L u * sin u , c-Δ y−B , which is the deflection due (12) ∅ A = tan −1 2 * n y d , to deformation of the elastic region B ′ C . The sum of these three parts has been shown in the following equation: Displacement ductility is the total deflection at the moment of collapse divided by the deflection corresponding to the formation of M y at the support Δ = Δ U Δ y .

Plastic hinge length calculation
The plastic flexural moment M P will occur in the ∅ P curvature (Fig. 5). Steel sections reach ∅ P at the strain much less than h (Fig. 2) (this ratio is about 1.1-1.25 for I-shaped sections), i.e., � P = (1.1−1.25)� y (Chen & Sohal, 2013) and for ST37 steel, ∅ h = 10∅ y (Popov, 1990). Therefore, in the interval ∅ P -∅ h , although the strain at the section height will increase significantly, the stress along with the height of the cross-section is constant and equal to F y . Therefore, at this distance, the flexural moment at a constant cross-section is equal to M P = Z * F y . If a plastic moment M P is placed in Eq. (7) instead of M A (Fig. 6), a nonlinear Eq. (20) is obtained instead of a nonlinear Eq. (7): By solving the system of nonlinear Eq. (20), the length L P and the load P P and using them the displacement Δ P , the rotation of the whole beam y−B + P , which all correspond to the formation of a plastic hinge at the support A are obtained. Note that due to the strain hardening phenomenon for steel, by forming a plastic hinge at the support, the end of the resistance of the cantilever beam is not reached. As well, by calculating the value of L u corresponding to M U by solving the system of Eq. (7), the length L u − L P , where the strain of all the fibers at the height of the section has exceeded the yield strain y , can be calculated. Considering the very small P , the nonlinear equation system (20) can be replaced by the linear equation system (21) cos P ≅ 1, sin P ≅ P : According to Fig. 6 (right) if M A = M P , i.e., for the case where the beam has reached the plastic flexural moment in the support, the area of the hatched region is equal to P , also, by calculating L P and P P from the system of Eq. (21),   other parameters such as deflection and rotation can be obtained:

Results
A steel (ST37) beam with a length of 3 m and a given crosssection according to Fig. 7 is considered. The pushover curve is plotted for the two modes a and b by the method described in the present study. The stress-strain curve used for steel has been presented in Fig. 8, where F y is the yielding stress and F U is the ultimate stress of the steel under tension: According to the section geometry, mechanical properties such as the moment of inertia and modulus of elastic and plastic sections are presented in Table 1.
(22) P = ∅ y + ∅ P L P 2 → P = (n + 1) 2 ∅ y L P ∅ P = n∅ y , for I section ∶ n = 1.1 ∼ 1.25 . moment-area theories, the results can be calculated in the case of small deformations according to Table 2.

Large deformations P > P y
To obtain the coordinates of the points in the pushover curve for P > P y , according to the stress-strain diagram (Fig. 8), different points must be investigated and calculated, which are examined as follows.
Formation of the plastic hinge at the support P = P P According to studies (Chen & Sohal, 2013) for I-shaped sections, there is ∅ P = 1.2∅ y . According to Table 2 and Fig. 5, ∅ P = 2 P d = 0.0144 , thus P = 0.00144 < h = 0.012 , so it can be seen that P is smaller than the strain hardening h = 0.012 . Therefore, the section of the beam reaches the plastic moment before the strains reach the strain hardness, so the calculations of both modes with and without strain hardening are the same for M A = M P . Using the system of Eq. (21), the results are obtained and summarized in Table 3: Table 3 Results for large deformations ( P = P P )  Table 4 Results for large deformations (P = P h ) 0.0007(m) 0.032(rad) 0.066L uh 0.01rad 0.091(m)

Small deformations P = P y
If beam analysis is performed for P = P y with classic analytical methods such as virtual work or the first and second Results at the end of the plastic step P = P h The end of the plastic step is a fixed point in the steel stress-strain diagram for both cases with and without the strain hardening phenomenon. Therefore, the calculations are the same for both cases. Using the system of Eqs. (7), the following equation is obtained, that by solving this nonlinear system, the values of L uh and P h are obtained: Solutions obtained from MATLAB software using fsolve command: It can be seen that there is no significant difference between P P (Table 3) and P h and the reason is that first, the distance between P and h in the steel strain stress diagram is very small and second, for both cases, the stress in the total cross-section depth is constant and equal to F y , so M P = M h . But the reason for the very small difference between them is that by increasing the free end displacement of the beam from Δ P = 0.0405m to Δ h = 0.091m , the length in flexural moment decreases, and to keep the flexural moment in the support constant, the value of force should be increased from P P -P h . Using the results of the system of Eq. (23), the parameters have been calculated and summarized in Table 4.

Results at the end of strain hardening zone P = P hs
For both cases of with and without strain hardening phenomenon, the strain in the farthest section of the fiber is equal to hs , except that in the case of strain hardening phenomenon, the stress in the farthest section of the fiber is F U , but in the case of ignoring the strain hardening phenomenon, the stress is still constant along with the cross-section and is equal to F y (Fig. 8). The values ∅ hs and hs are obtained from the following equations:

Results with considering the strain hardening phenomenon
According to the stress distribution on the cross-section, the flexural moment will be equal to M hs = 21.27tm . After substituting Eq. (7) in the system of equations and solving it using MATLAB, the following solutions are obtained for L uhs and P hs : Using the above results, the rest of the necessary parameters such as explained in "Results at the end of the plastic step P = P h " are obtained and presented in Table 5.

Results without considering the strain hardening phenomenon
In the case of ignoring the strain hardening phenomenon, when the maximum strain on the cross-section reaches hs , the stress is still constant at the cross-section depth and equal to F y (Fig. 8), so M hs = 14.87tm . After substituting the parameters in the system of Eqs. (7) and solving the device using MATLAB using the fsolve command, the following solutions are obtained for L uhs and P hs : hs = tan −1 0.168L uhs . Using the above results, the rest of the necessary parameters such as "Results at the end of the plastic step P = P h " are obtained, presented in Table 6.

Results at the end point of the pushover curve P = P U
At the endpoint, due to the maximum strain on the crosssection reaching the strain in which the necking phenomenon occurs in steel, the rupture will occur immediately and it is the end of strength and the beam is collapsed. For both cases of with and without strain hardening phenomenon, the strain in the farthest sectional fiber is equal to U , except that in the case of the strain hardening phenomenon, the stress in the total cross-section depth is constant and equal to F U , but in the case of ignoring the strain hardening phenomenon, it is equal to F y (Fig. 8). The values of ∅ U and u are obtained from the following equations:

Results considering the strain hardening phenomenon
Considering the constant stress on the cross-section, the flexural moment will be equal to M hs = z * F U = 22.93tm . After substituting in the system of Eqs. (7) and solving the device using MATLAB, the following solutions are obtained for L u and P U : Using the above results, the rest of the necessary parameters such as "Results at the end of the plastic step P = P h " are obtained, presented in Table 7.
Results without considering the strain hardening phenomenon In the case of ignoring the strain hardening phenomenon, as in "Results without considering the strain hardening phenomenon", the stress is still constant at the cross-section depth and is equal to F y (Fig. 8), so M U = z * F y = 14.87t.m . After substituting the parameters in the set of Eqs. (7) and  solving the system using MATLAB, the following answers are obtained for L u and P U : Using the above results, the rest of the necessary parameters such as "Results at the end of the plastic step P = P h " are obtained, which are summarized in Table 8.
According to the calculated results, the curvature, rotational and displacement ductility in the case of with and without considering the strain hardening have been calculated and the results have been presented in Table 9.
To conveniently compare the responses in different cases, all results obtained in "Results" are summarized in Table 10.

Numerical modeling
The finite-element analysis software ABAQUS/CAE 6.14-3 is utilized for simulation and numerical analyses of the proposed beam. For mesh generation in the ABAQUS software, the two-dimensional deformable wire element is used for linear and nonlinear analyses. For finite-element mesh, the length of the beam (3 m) is divided into 15 beam elements (B21). Then the results of the proposed analytical and numerical methods are compared to illustrate the accuracy of the analytical formulations. Based on the obtained results in all cases the pushover carves (P-Delta, M-Teta) are illustrated in Figs. 9 and 10, respectively. The results show that there is a good agreement between the responses of the proposed analytical method and the numerical modeling.

Discussion and conclusion
Based on the studied model, the following results could be drawn: • In this research, the pushover curve for a steel beam using structural analysis theory, material properties, and geometry of the deformed structure are obtained analytically. The advantage of the proposed method is reaching acceptable accuracy instead of using experimental tests or numerical modelings. • In deformations corresponding to elastic strain and plastic strain, the strain hardening phenomenon does not have a significant effect on the behavior of the beam and in nonlinear analysis, its effect can be ignored and the behavior of the materials could be considered elastic and perfectly plastic. • At large deformations, the effect of the strain hardening phenomenon is significant and its effect should not be ignored. One of the important effects of strain hardening is that the strength of the material and cross-sectional ductility is increased at the collapse moment. Therefore, using the elastic perfectly plastic for the strain-stress diagram, significant errors will appear in the calculation of the capacity curve and the results are not reliable. • According to the results of this study, the plastic flexural moment will increase by about 54% if the effects of strain hardening are taken into account. As well, when strain hardening is considered, the rotational ductility has increased by about 203% and the displacement ductility has increased by about 173%. • When the stain hardening is included in the formulation, reaching flexural moment (M) to M P is not the end of the strength and by increasing force, the flexural moment can increase up to M U . In this case, along with the member a space/distance appears between the M P and M U locations. This distance is the length at which the strain of all the fibers at the cross-section depth is greater than y . The results show that this distance has a significant effect on seismic performance, and energy dissipation, and increases the ductility of the system.