Mechanism analysis of steel frames considering moment

Abstract

Considering the interaction of flexural moment and shear force in the steel frames with haunch or intermediate beam length and eccentrically braced frames with intermediate link length is a major concern of the structural analysis and design. This article contains two stages. In the first stage, to investigate the moment–shear interaction for the highly ductile steel I-sections, a study is carried out using finite element analysis and a simple and practical relationship is developed. In the second stage, a simple approach based on virtual work method for assemblage of interconnected rigid bodies is employed to consider collapse mechanisms with mixed hinges. Using this approach, the applicability of the proposed relationship is demonstrated for a one-bay portal frame by considering all possible collapse mechanisms including those containing mixed hinges. Some numerical examples are presented using the proposed approach. Results indicate that, in general, the effect of moment–shear interaction on the load capacity of the structures cannot be ignored, and the capacity could be estimated, without using step-by-step analysis. Finally, by satisfying kinematic compatibility requirements and normality condition, simplified relations are derived to estimate post-mechanism deformations of plastic hinges for a prescribed lateral drift.

Keywords

moment–shear interaction steel I-sections eccentrically braced frames collapse mechanism steel frames mixed hinges

Introduction

Steel frames and eccentrically braced frames (EBFs) are some of the widely used systems in building industry. Inelastic behavior of EBFs depends on the link length. Based on AISC 341-16 (2016a), for links of length $1.6 M_{p} / V_{p}$ or less, where $M_{p}$ and $V_{p}$ are plastic moment and plastic shear capacities of the link, respectively, shear yielding dominates the links’ inelastic response, while for links of length $2.6 M_{p} / V_{p}$ or greater, flexural yielding is dominant. If the link length falls in between the magnitudes of $1.6 M_{p} / V_{p}$ and $2.6 M_{p} / V_{p}$ , the inelastic response will be controlled by the combination of both shear and flexural yielding. Similar to EBFs, shear yielding, flexural yielding, or the combination of both may govern the frame inelastic response depending on the frame beam length. In the frames with haunches, the length of the uniform parts could be short or intermediate, too, requiring consideration of flexure and shear interaction.

Several researchers have investigated the effect of moment–shear, M-V, interaction on beam capacity. By means of upper and lower bound theorems of plasticity, Drucker (1956) examined a cantilever and a simple beam with rectangular cross sections and developed a moment–shear interaction relationship. Neal (1961a) extended Drucker’s work to consider the axial force effect and obtained a moment–shear–axial forces interaction relationship. He also presented a lower bound solution for I-section beams (Neal, 1961b). Using Neal’s equation, Montuori et al. (2014) presented a procedure for calculating the ultimate end moments and shear forces of intermediate links of EBFs with inverted Y-scheme. By ignoring the effect of vertical load, they also estimated plastic deformation of intermediate links. Basler (1961) derived an interaction relationship for I-section by assuming a simple stress distribution, which had been used in AISC specifications for consecutive years until AISC 360-05 (2005). Liu et al. (2009) obtained a failure criterion for steel member cross sections considering the combination of bending moment shear force and axial force. Kazemi and Erfani (2007a, 2007b, 2009) presented an element with zero or non-zero length that it considers moment–shear interaction. Their model includes multiple and dissimilar yield surfaces with a kinematic hardening rule and a developed non-associated flow rule. Kazemi and Asl (2011) considered moment–shear–axial interaction yield surface in deformation space and included strength and stiffness degradation. They also presented a mixed element based on damage-plasticity. For slender plate girders, Shahabian and Roberts (2008) presented a moment–shear interaction formula which covers a range of web panel width to height ratio from 1 to 2 and slender ratio from 150 to 300. For design of beams with reduced web section connections, an interaction relationship has been presented (Kazemi et al., 2012; Momenzadeh et al., 2017).

Azizinamini et al. (2007) tested eight steel I-girders with simple supports and reported that all the specimens sustained loads more than the predicted values. They state that the moment–shear interaction can be ignored, in practice. Based on the experimental results, White et al. (2008) indicate that the shear and flexural strengths can be calculated without the need for the consideration of their interaction, using ANSI/AISC 360-05 (2005) provisions for the flexural resistance and Basler’s model (White and Barker, 2008) or Cardiff model (White and Barker, 2008) for the shear capacity. Daley et al. (2016) examined moment–shear interaction for welded I-girders with thin unstiffened webs. They concluded that it does not need to consider the moment effect for evaluating of shear strength. Based on their observations, the commentary of ANSI/AISC 360-16 (2016b) states, “Consideration of shear and bending interaction is not required because the shear and flexural resistances can be calculated with a sufficient margin of safety without considering this effect.” On the other hand, Lee et al. (2013) investigated the interaction behaviors of I-girders and developed a new interaction relationship. They concluded that interaction effect cannot be neglected when failure occurred by yielding due to combined bending moment and shear force. By developing simplified approaches, it will not be needed to ignore the flexural shear interaction.

For moment frames, Grigorian and Grigorian (2012a, 2012b) introduced a simplified method to calculate the sequences of the plastic hinges formation and corresponding lateral deformation up to mechanism load. Using theory of plastic mechanism control, Montuori et al. (2015) presented a closed-form design procedure to prevent occurrence of any partial or story mechanisms and to achieve the global-type collapse mechanism. This concept specialized for steel moment frames (Dell’Aglio et al., 2017) and reinforces concrete frames (Montuori and Muscati, 2017).

The contribution of the concrete slab to the combined shear and bending behavior of steel–concrete composite beams have been investigated by several researchers (Liang et al., 2004, 2005; Nie et al., 2004; Vasdravellis and Uy, 2014). They showed that concrete slab significantly improves the combined shear and bending behavior of composite beams.

This study is limited to the highly ductile steel I-section members without considering concrete slab, where their webs could yield, and the effect of moment–shear, M-V, interaction is challenging. This article includes two parts. In the first part, M-V interaction of steel I-section beam is investigated. A new relationship is constructed for M-V interaction using refined inelastic finite element (FE) analysis, based on the modeled samples. The proposed simplified relationship is comparable with the other available ones and is suitable for the plastic analysis of the steel frames. In the second part, the proposed relationship has been used, directly, to predict the collapse load of single-bay portal frame considering all its possible collapse mechanisms, without requiring step-by-step or stage-by-stage nonlinear analysis. A simplified approach is developed to consider collapse mechanisms which include mixed hinges. Also, the possibility of internal plastic hinge development for the beams carrying vertical load is considered, using a simple concept. To show the applicability of the suggested approach, several numerical examples are presented. In general, pre-mechanism plastic deformations in the inelastic hinges depend on the load path and require incremental analysis, but to estimate the post-mechanism plastic deformation of plastic hinges for a prescribed lateral drift, simplified relations are presented.

Numerical study for development of moment–shear yield surface

In order to develop a yield surface relationship for I-section beams in the space of bending moment and shear force, EBFs, with different link lengths were modeled using the nonlinear FE analysis software ABAQUS (2014).

ABAQUS S4R shell element, which is a four nodes shell element using reduced integration method, is utilized for meshing all model geometries. The hourglass phenomenon is controlled by checking the ratio of the total artificial energy to the total internal energy be less than 0.01. Also, mesh convergence study is performed to find the suitable element size for achieving both accuracy of the results and optimal computational time. The nonlinear static-modified Riks method (arc-length method) is applied for the analysis, in which the geometric nonlinearity option is activated. Initial geometric imperfections are introduced into the model based on the first two buckling modes of the models which are calculated by a linear eigenvalue buckling analysis. Accordingly, the effects of large deformation, local buckling, and post-buckling behavior are simulated. The maximum value of initial imperfections of $h_{w} / 200$ is considered, where $h_{w}$ is the beam web height. It would be mentioned that because all the sections selected for this study are highly ductile doubly symmetric I-section, the initial imperfections just affect the post-yielding behavior of the member. Increasing the initial imperfections causes the local buckling to occur sooner but not earlier than the section yielding.

To verify the modeling techniques discussed above, in a preliminary study, several existing experimental results (Cooper et al., 1964; D’Apice et al., 1966; Nishino and Okumura, 1968; Vasdravellis and Uy, 2014) were used. The first verification study is a simply supported I-section beam, specimen BS-1 of (Vasdravellis and Uy, 2014), which was loaded by a mid-span point load. The beam cross section is IPE330 and the span length is 0.8 m.

Figure 1 presents the curves of shear force versus deflection at the mid-span for the present finite element method (FEM) results and the experimental and the FEM results reported by Vasdravellis and Uy (2014). The comparison shows that the present FE model is able to accurately predict the initial stiffness, ultimate strength, and post-buckling behavior of the specimen (Figure 1). The material stress–strain curve of the steel assumed to be similar to one reported by Kirkland et al. (2015) with yield stress of 358 MPa.

Figure 1.

Comparison between the present FEM result and the experimental and FEM results reported by Vasdravellis and Uy (2014).

The second verification study contains three simply supported I-section girder, where each beam has two symmetric panels. The present numerical results are compared with some available experimental results in Table 1. The dimensions and material properties of the specimens are similar to the one proposed by Graciano and Ayestarán (2013) and are reported in Table 1, where a is the distance between transverse stiffeners, $h_{w}$ and $t_{w}$ are height and thickness of web, $b_{f}$ and $t_{f}$ are width and thickness of flange, and $f_{yw}$ and $f_{yf}$ are web and flange yield stress, respectively. According to the experiments, only the right panel is restricted for out-of-plane buckling, and the load is applied vertically on the top flange at the girder center between two stiffeners. Mesh convergence study resulted in approximate mesh size of $h_{w} / 40$ . The shear strengths of the experiments, $V^{\exp}$ , FEM results conducted by Graciano and Ayestarán (2013), and present FEM results, $V^{FE}$ , are also listed and compared in Table 1. The difference in shear strengths, $Δ V / V$ , is from 1.7% to 7.4%. Regarding the lack of the detailed information about the imperfections, the accuracy of the modeling is acceptable. Although the first two specimens have slender webs and the third one has a noncompact web, the FEM results could predict the capacity of the beams.

Table 1.

Comparison of the experimental with the present FEM results and material properties and dimensions of the girders.

Girder	a (mm)	$h_{w}$ (mm)	$t_{w}$ (mm)	$b_{f}$ (mm)	$t_{f}$ (mm)	$f_{yw}$ (MPa)	$f_{yf}$ (MPa)	$V^{\exp}$ (kN)	$V^{FE}$ (kN)	$Δ V / V$ (%)	$V^{FEA}$ (kN)Reported by Graciano and Ayestarán (2013)
LS1 D’Apice et al. (1966)	1270	1270	4.76	355	38.1	323	234	808	794	1.7	811
H1T1 Cooper et al. (1964)	3810	1270	9.98	459	24.8	745	703	2542	2602	2.4	2413
G8 Nishino and Okumura (1968)	2851	1080	8.9	221	22.2	372	431	996	1070	7.4	973

FEM: finite element method.

EBFs are suitable structures for studying M-V interaction. Therefore, eight EBFs, designed based on ANSI/AISC 341-16 (2016a), were modeled which have different link lengths and stiffeners. In all models, the beam length and column height are 8.0 and 3.6 m, respectively. In each model, the connections of beam to column and the brace to both beam and column are rigid, while the column support is pinned (Figure 2).

Figure 2.

Schematic drawing of EBFs used in FE study.

An ideal elastic-perfectly plastic von Mises material, with $F_{y} = 345 MPa$ , and elasticity modulus, E, of $200 GPa$ is used in the analyses. The Poisson ratio, $ν$ , is also assumed to be 0.3. All sections are highly ductile doubly symmetric I-section (Figure 3).

Figure 3.

Dimensions of (a) beam, (b) column, and (c) brace sections.

The intermediate and end transverse stiffeners of the links are plates of 380 mm × 124 mm × 12 mm, and 380 mm × 124 mm × 10 mm, respectively. Based on ANSI/AISC 341-16 (2016a), equally spaced intermediate stiffeners are considered on one side of the link web, while end stiffeners are considered on both sides. Table 2 represents the frame link lengths, their corresponding number of intermediate stiffeners, and dominant inelastic response of links based on ANSI/AISC 341-16 (2016a). All frames were pushed, using arc-length method, in which lateral load pattern was applied to both left and right of the frame, while displacement of the link middle point was controlled.

Table 2.

Categorization of the sample EBFs based on ANSI/AISC 341-16 (2016a).

Model no.	Link length (m)	Number of intermediate stiffeners	Link dominant inelastic response	Inelastic link rotation angle $(γ_{p})$	Plastic disp. $(Δ_{p})$ (mm)
1	5.0	0	Flexural	0.02	45
2	2.4	4	Mixed (upper bound length)	0.02	21.6
3	2.0	4	Mixed	0.047	42
4	1.8	4	Mixed	0.06	48.6
5	1.7	5	Mixed	0.067	51
6	1.6	5	Mixed	0.073	52.8
7	1.5	3	Mixed (lower bound length)	0.08	54
8	0.5	1	Shear	0.08	18

EBFs: eccentrically braced frames.

The target displacement was selected as 1.5 times of the design story drift. The design story drift $(Δ_{d})$ is sum of the elastic and plastic story drift $(Δ_{d} = Δ_{e} + Δ_{p})$ . Similar to ANSI/AISC 341-16 (2016a), the plastic story drift of each frame is calculated based on inelastic link rotation angle as follows

Δ_{p} = \frac{eh}{L} γ_{p}

(1)

where h, L, and e are column height, beam length, and link length, respectively. $γ_{p}$ , the inelastic link rotation angle, is limited by ANSI/AISC 341-16 (2016a) based on links’ length. Table 2 also lists the inelastic links rotation angles and their corresponding plastic displacement for all models.

Based on the above assumptions, the eight described EBFs were modeled and analyzed. As an example, for the FE model of the EBF with the link of length 1.7 m, mesh convergence study was resulted in 25,823 elements. Figure 4 represents von Mises stress distribution in the deformed shape of the model. Also, the effect of moment–shear interaction on the deformations can be seen in Figure 4, where the link beam not only rotates but also vertically slips, with respect to the beam outside of the link.

Figure 4.

Von Mises stress distribution in a deformed EBF with link of length 1.7 m. The deformations are exaggerated five times.

Proposed moment–shear interaction relationship

Based on the numerical investigation explained in the previous section, a general relationship considering M-V interaction is developed. The ultimate bending moments and shear forces, obtained from the numerical analysis, were normalized by plastic bending moment $(M_{p}^{FE})$ and plastic shear force $(V_{p}^{FE})$ , respectively. $M_{p}^{FE}$ and $V_{p}^{FE}$ were obtained from FE analyses, which were just 1.004 and 1.003 times of the theoretical values of $M_{p} = Z F_{y} = 867.05 kN m$ and $V_{p} = 0.6 A_{w} F_{y} = 943.9 kN$ , where Z and $A_{w}$ are plastic section modulus and web cross-sectional area of the beam, respectively.

By examining the numerical results and other models, equation (2) is proposed

f = {(\frac{| M | - M_{y}}{M_{p} - M_{y}})}^{2} + {(\frac{| V | - V_{y}}{V_{p} - V_{y}})}^{2} - 1 = 0 \begin{matrix} M_{y} < | M | < M_{p} \\ V_{y} < | V | < V_{p} \end{matrix}

(2)

where $M_{y}$ and $V_{y}$ are yield moment and yield shear, respectively, and M and V are the reduced moment and shear strength, respectively. Equation (2), as a yield surface, assumes that there is no any interaction between the shear and flexural mechanisms for $| M | \leq M_{y}$ and $| V | \leq V_{y}$ . Figure 5 shows that the proposed relationship can predict the FE results well. In this figure, the values of $M_{p}$ , $M_{y}$ , $V_{p}$ , and $V_{y}$ , used for plotting, were calculated based on section properties, $M_{p} = Z F_{y}$ , $M_{y} = S F_{y}$ , $V_{p} = 0.6 A_{w} F_{y}$ , $V_{y} = 0.4 A_{w} F_{y} = 2 / 3 V_{p}$ , where S is the elastic section modulus.

Figure 5.

Comparison of proposed M-V interaction relationship with FE results.

The effect of stiffeners on M-V interaction relationship was also investigated with analyzing all EBF samples with and without stiffeners. Comparing the results shows that the effect of stiffeners is about 3% for shorter links and it tends to zero for the links of length greater than 1.7 m. Therefore, the effect of stiffeners on the capacity of highly ductile members can be neglected.

Comparison with existing relationships

Table 3 lists three M-V interaction relationships available in the literature. Neal (1961b) presented a lower bound solution for considering M-V interaction in I-sections represented in Table 3. Basler’s relationship assumes that when the web is fully plastic under shear, the section still can carry moment equal to the flanges moment capacity, $M_{f}$ , but when the moment reaches the section moment capacity, $M_{p}$ , then the section does not have any shear capacity (Basler, 1961). Lee et al. (2013) used two different stress distributions for I-sections for $M_{f} < | M | \leq M_{y}$ and for $M_{y} < | M | \leq M_{p}$ . Their relationships are presented in Table 3, where $S_{r}$ is defined by equation (3), where $A_{f}$ and $A_{w}$ are flange and web cross-sectional area, respectively

S_{r} = \frac{6 A_{f}}{A_{w}} [(1 + \frac{A_{w}}{4 A_{f}}) \frac{M}{M_{p}} - 1]

(3)

Table 3.

Some of the existing M-V interaction relationships.

Neal (1961b)	${(\frac{V}{V_{p}})}^{2} + {(\frac{\| M \| - M_{f}}{M_{p} - M_{f}})}^{2} = 1$ for $M_{f} < \| M \| \leq M_{p}$
Basler (1961)	${(\frac{V}{V_{p}})}^{2} + \frac{\| M \| - M_{f}}{M_{p} - M_{f}} = 1$ for $M_{f} < \| M \| \leq M_{p}$
Lee et al. (2013)	$\frac{V}{V_{p}} = - 0.176 S_{r}^{3} - 0.004 S_{r}^{2} - 0.043 S_{r} + 1.0$ for $M_{f} < \| M \| \leq M_{y}$
Lee et al. (2013)	$\frac{V}{V_{p}} = 0.5 + 0.28 \sqrt{3 - 2 S_{r}}$ for $M_{y} < \| M \| \leq M_{p}$

Figures 6 and 7 compare the proposed relationship with others’ for IPB300 and IPB1000 sections, respectively. As can be seen in these figures, the difference in predictions of the relationships for IPB1000 (Figure 7) is more significant than IPB300 (Figure 6). In the both figures, the proposed relationship lies between case of without interaction (Azizinamini et al., 2007; Daley et al., 2016; White et al., 2008) and others’ interaction relationships (Basler, 1961; Lee et al., 2013; Neal, 1961b). It means that the proposed relationship prediction for the shear capacity is more than those of existing interaction relationships. Those relationships are derived from the lower bound plasticity analysis by assuming a stress distribution on a beam cross section, while the proposed relationship was obtained by investigating FE results considering more realistic stress distribution based on theory of plasticity.

Figure 6.

Comparison of the proposed relationship with others’ for IPB300.

Figure 7.

Comparison of the proposed relationship with others’ for IPB1000.

Also, examination of the numerical results showed that the proposed relationship is more suitable for an associated plasticity model in the force space and could be used for estimating plastic rotational and plastic slip at the developed mixed plastic hinges in the frame. Although the proposed relationship is obtained based on EBFs, it can be used for any highly ductile member with I-section in steel frames and other suitable cases.

Mechanism analysis of steel frames using the proposed M-V interaction relationship

In many structural design policies, one needs to estimate the collapse load as the primary goal of design. Plastic mechanism analysis is a suitable approach, which could be used, at least, for the basic and preliminary design. For moment frames, plastic design has been developed and used since 1948 (Horne and Morris, 1981). Due to complexity of mixed hinge analysis, limited work has been introduced for practical application in design. Any M-V relationship can be used in the nonlinear step-by-step static or dynamic analysis of frames, provided to accompany a flow rule for plastic deformation.

In order to show the applicability of the proposed interaction equation for plastic analysis of structures, a single-bay portal frame with pinned supports subjected to a uniform gravity load, w, and a lateral load, H, is considered. The height and span length of the frame are denoted by h and L, respectively. It is assumed that the columns are stronger than the beams and the axial load will not affect the results. In the mechanism method, the collapse load is the minimum load obtained by comparing all possible mechanisms’ loads.

Virtual work method is employed to obtain the mechanism loads. The principle of virtual work is an alternative way of expressing the equilibrium condition. In the virtual work method, a fictitious work is computed with a set of statically admissible forces assumed to remain constant while they do work on a set of infinitesimal, kinematically admissible displacements. The force and displacement distribution may be independently prescribed. The statically admissible force system is defined as one satisfying the equilibrium and force boundary conditions. A kinematically admissible displacement is one satisfying any prescribed displacement boundary conditions and to be continuous in the interior of the body (Malvern, 1969). In the mechanism analysis of a frame, the structure consists of rigid members interconnected by the plastic joints, which can transfer internal forces. The virtual work of such joint forces must be added to the work of external forces (Malvern, 1969).

In the next section, the frame is analyzed without considering the M-V interaction, and in the following section, the frame is analyzed with considering the M-V interaction.

Frame collapse loads obtained without considering M-V interaction

Ignoring the M-V interaction, five mechanisms are possible to occur. Two of them are beam mechanisms (Figure 8(a) and (b)), one is a story mechanism (Figure 9(a)), and two other ones are mixed mechanism (Figure 9(b) and (c)). For the mixed mechanisms, for any given uniform gravity load, w, the corresponding lateral collapse load, H, is calculated.

Figure 8.

Possible beam mechanisms of the frame without M-V interaction consideration. (a) Mechanism 1: beam flexural (BF) mechanism and (b) mechanism 2: beam shear (BS) mechanism.

Figure 9.

Possible story and mixed mechanisms of the portal frame, without considering M-V interaction effect. (a) Mechanism 3: story mechanism by forming two flexural plastic hinges at the beam ends (F-F); (b) Mechanism 4: mixed mechanism by forming one flexural plastic hinge at the beam’s right end, and another one somewhere along the beam span (IF-F); (c) Mechanism 5: mixed mechanism by forming just one shear plastic hinge at the beam’s right end (-S).

Figure 8(a) is a beam flexural (BF) mechanism, mechanism 1, formed by two flexural plastic hinges at beam ends and one at the middle of the beam, with the collapse load of $w_{1} = 16 M_{p} / L^{2}$ . The second beam mechanism under uniform gravity load is a beam shear (BS) mechanism (mechanism 2) formed by two shear plastic hinges shown in Figure 8(b), with the collapse load of $w_{2} = 2 V_{p} / L$ .

Figure 9(a) presents the story mechanism (mechanism 3), which is formed by two flexural plastic hinges at the beam ends with the collapse load of $H_{3} = 2 M_{p} / h$ . Figure 9(b) and (c) demonstrates the mixed mechanisms. The fourth mechanism (mechanism 4) includes one flexural plastic hinge at the beam end and another one somewhere along the beam span (Figure 9(b)). Virtual work equation for mechanism 4 can be expressed by equation (4)

Hh θ + \frac{1}{2} wLx θ - M_{p} θ - 2 M_{p} \frac{x}{L - x} θ - M_{p} θ = 0

(4)

By minimizing the lateral load, H, the location of the interior hinge, x, could be determined. Also, by noting the fact that at the interior plastic hinge the shear force should be zero, the interior hinge location could be determined

x = L - \sqrt{\frac{4 M_{p}}{w}}

(5)

Substituting equation (5) into equation (4) results in the lateral collapse load given as a function of w

H_{4} = \frac{2 L}{h} \sqrt{w M_{p}} - \frac{w L^{2}}{2 h}

(6)

The last mechanism (mechanism 5) shown in Figure 9(c) is formed by just one shear plastic hinge at the beam end. Virtual work relationships for mechanism 5 is expressed by equation (7), which results in the lateral collapse load as a function of w (equation (8))

Hh θ + w (L θ) \frac{1}{2} L - V_{p} L θ = 0

(7)

H_{5} = \frac{V_{p} L}{h} - \frac{w L^{2}}{2 h}

(8)

Hence, if the moment–shear interaction can be ignored, for the conditions that w is less than both $w_{1}$ and $w_{2}$ , the lateral collapse load of the frame, $H_{c}$ , is the minimum obtained from the last three mechanisms as follows

H_{c} = min (H_{3}, H_{4}, H_{5})

(9)

Frame collapse loads obtained by considering M-V interaction

To calculate the collapse load of the portal frame incorporating M-V interaction effect, five mechanisms, mechanism 6 to mechanism 10, shall be appended to the previous investigated mechanisms. Mechanism 6 is the third beam mechanism, formed by two mixed hinges (M) at the beam ends and one flexural plastic hinge (F) at the beam center, and it is named beam mixed (BM) mechanism (Figure 10(a)).

Figure 10.

(a) Mechanism 6: beam mixed (BM) mechanism of the frame by forming two M-V mixed hinges at the beam ends and one flexural plastic hinge at the beam center. F and M stand for flexural plastic hinge and mixed hinge, respectively. (b) and (c) are frame virtual deformations used in writing virtual work relationships for mechanism 6.

To solve mechanism 6, two equilibrium equations and the M-V interaction equation are required. As a substitute for the direct equilibrium equations, two independent virtual deformations are considered (Figure 10(b) and (c)).

For the virtual deformation of Figure 10(b), which considers only virtual rotation at the beam ends and the center, one could obtain

\frac{w l^{2}}{4} - 2 M^{*} = 2 M_{p}

(10)

For the virtual deformation of Figure 10(c), which considers only vertical slip at the beam ends, it could be seen that

2 V^{*} - wL = 0

(11)

The third equation relates $M^{*}$ and $V^{*}$ , using the yield surface of equation (2) as follows

{(\frac{| M^{*} | - M_{y}}{M_{p} - M_{y}})}^{2} + {(\frac{| V^{*} | - V_{y}}{V_{p} - V_{y}})}^{2} - 1 = 0

(12)

Accordingly, the collapse gravity load, $w^{*}$ , can be found by solving above three equations. Considering mechanisms 1 and 2, therefore, the collapse uniform gravity load, $w_{c}$ , that causes the frame to collapse can be obtained according to equation (13)

w_{c} = min (\frac{16 M_{p}}{L^{2}}, \frac{2 V_{p}}{L}, w^{*})

(13)

By means of virtual work method or static equilibrium, the ultimate collapse load can be determined. However, the corresponding plastic rotation and shear plastic displacement of plastic hinges remain unknown and their determination requires incremental or stage-by-stage analysis based on assumed load path, which are out of the scope of this article.

By forming just two hinges at the beam ends, there are three possible mixed mechanisms (mechanisms 7 to 9), which include interaction of gravity and lateral loads. In mechanism 7, Figure 11(a), both hinges are mixed hinges, M, while in mechanisms 8 and 9, Figure 11(b) and (c), there is one flexural plastic hinge, F, and one mixed hinge.

Figure 11.

Possible mixed mechanisms of the portal frame, considering M-V interaction effect. F and M stand for flexural plastic hinge and mixed hinge, respectively. (a) Mechanism 7 (M-M), (b) mechanism 8 (F-M), and (c) mechanism 9 (M-F).

Regarding the mechanism 7, there are five unknowns, $H, V_{1}, M_{1}, V_{2}, M_{2}$ , for a specified value of w. To obtain the lateral collapse load, $H_{c}$ , of mechanism 7, virtual deformations and two interaction equations could be used. Figure 12 presents three independent chosen virtual deformations.

Figure 12.

(a)–(c) Portal frame virtual deformations used in writing virtual work relationships.

For the virtual deformation of Figure 12(a), which only contains virtual rotations at the beam ends, one could obtain

M_{1} + M_{2} = Hh

(14)

In Figure 12(b), to simplify the virtual work equation, only anti-symmetric vertical slips are considered at the beam ends. In this way, the gravity load does not do any work, and $θ = 2 Δ / L$ . Therefore, the virtual work relationship yields to

2 Hh - V_{1} L - V_{2} L = 0

(15)

In the third virtual deformation, Figure 12(c), the same vertical slips are considered at the beam ends, to prevent the appearance of lateral load in the virtual work equation, so results in

- V_{1} + V_{2} = wL

(16)

Two equations can be obtained from yield surface of equation (2) as follows

{(\frac{| M_{1} | - M_{y}}{M_{p} - M_{y}})}^{2} + {(\frac{| V_{1} | - V_{y}}{V_{p} - V_{y}})}^{2} - 1 = 0

(17)

{(\frac{| M_{2} | - M_{y}}{M_{p} - M_{y}})}^{2} + {(\frac{| V_{2} | - V_{y}}{V_{p} - V_{y}})}^{2} - 1 = 0

(18)

The lateral collapse load of mechanism 7 can be obtained by solving simultaneously equations (14) to (18), which contains two nonlinear equations of (15) and (16). These set of equations were solved using MATLAB software (MathWorks, Inc., 2014).

Similar to mechanism 7, the lateral collapse load of mechanism 8 is calculated by setting $M_{1} = M_{p}$ and solving equations (14) to (16) and (18) and mechanism 9 can be solved by setting $M_{2} = M_{p}$ and solving equations (14) to (17).

Finally, the last possible mechanism, mechanism 10, consists of one mixed hinge at the beam right end and one flexural plastic hinge along the beam (Figure 13(a)). To obtain lateral collapse load of mechanism 10, the $V^{*}$ , $M^{*}$ , and x are obtained according to equations (19) to (21), where equations (19) and (20) are derived from static equilibrium relationships based on the free body diagram shown in Figure 13(b), and equation (21) is obtained using yield surface of equation (2). It would be mentioned that the interior hinge is formed at the location of the maximum moment in the beam, which is equivalent to setting the shear force at interior hinge to zero

V^{*} = w (L - x)

(19)

M^{*} + M_{p} = \frac{w {(L - x)}^{2}}{2}

(20)

{(\frac{| M^{*} | - M_{y}}{M_{p} - M_{y}})}^{2} + {(\frac{| V^{*} | - V_{y}}{V_{p} - V_{y}})}^{2} - 1 = 0

(21)

Figure 13.

Mechanism 10 (IF-M): forming by two hinges consists of one mixed hinge at the beam right end and one flexural plastic hinge along the beam. (b) Free body diagram of the part of the beam between flexural and mixed hinges. (c) Frame virtual deformations used in writing virtual work relationship.

The only remaining unknown, which is the lateral collapse load could be calculated using the virtual work equation based on the chosen virtual deformation shown in Figure 13(c), where $θ = Δ / L$ is the virtual drift. In this figure, the internal flexural (IF) plastic hinge does not have any rotation, and just a vertical slip, $Δ$ , is considered at the beam right end. The virtual work relationship results in

H = V^{*} \frac{L}{h} - \frac{w L^{2}}{2 h}

(22)

Numerical examples

To study the dependency of the portal frame capacity on the moment–shear interaction, some numerical examples are presented. For this purpose, three portal frames with height 3 m and span lengths 6, 9, and 11 m are considered. The beam section of all frames is IPB1000, so the beam span to depth ratios, $L / d$ , 6, are 9 and 11, respectively. Similar to plastic design charts for moment frames (Horne and Morris, 1981), for any assumed gravity load, corresponding lateral collapse load is calculated by looking at the results of the possible mechanisms. Therefore, for a specified gravity load, w, the frame lateral capacity load, $H_{c}$ , can be obtained by investigating all possible mechanisms from the set of 10 mechanisms. In order to make a comparative study, dimensionless frame lateral load $(Hh / M_{p})$ is plotted versus dimensionless frame gravity load $(w h^{2} / M_{p})$ for all frames (Figures 14 to 16). In these figures, “F,”“IF,”“M,” and “S” represent flexural plastic hinge, IF plastic hinge, mixed hinge, and shear plastic hinge, respectively. As an example, “IF-F” demonstrates the mechanism formed by one IF plastic hinge and one flexural (F) plastic hinge at the beam end (mechanism 4). As can be seen in the figures, depending on the amount of gravity load, w, the collapse mechanisms could vary.

Figure 14.

Frame capacity for a portal frame with h = 3 m, L = 6 m, and beam section of IPB1000 (L/d = 6).

Figure 15.

Frame capacity for a portal frame with h = 3 m, L = 9 m, and beam section of IPB1000 (L/d = 9).

Figure 16.

Frame capacity for a portal frame with h = 3 m, L = 11 m, and beam section of IPB1000 (L/d = 11).

Each figure contains two curves with and without M-V interaction effect. As depicted in Figures 14 to 16, the capacity curve including M-V interaction effect (mixed hinge) lies below the one in which the interaction effect is neglected. The maximum differences are about 5% in Figure 14, 13% in Figure 15, and 63% in Figure 16. Each figure contains additional diagram showing the relative location of interior plastic hinge along the beam $(X_{0} / L)$ . According to these diagrams, with increasing gravity load (w), interior plastic hinge moves from the left end of the beam toward the beam center.

For the frame with shorter span, as can be seen in Figure 14, the curve “independent hinge” contains three regions. When w is less than $1.1 M_{p} / h^{2}$ , mechanism 3 (F-F) will occur, while mechanism 4 (IF-F) will happen for w between $1.1 M_{p} / h^{2}$ and $1.2 M_{p} / h^{2}$ , and if w is larger than $1.2 M_{p} / h^{2}$ and less than $2.2 M_{p} / h^{2}$ , mechanism 5 (-S) will occur. BS mechanism, mechanism 2 (BS), will occur for the gravity load of $w = 2.2 M_{p} / h^{2}$ . Similarly, the curve “mixed hinge” includes four regions of “F-F,”“F-M,”“IF-M,” and“-S.” It can be observed that the M-V interaction affects the frame lateral load capacity for a range of gravity load from $0.45 M_{p} / h^{2}$ up to $1.3 M_{p} / h^{2}$ .

Figures 15 and 16 show that despite the larger value of the beam span to depth ratio, $L / d = 9 and 11$ , mixed hinge would occur for a range of gravity load.

For the frame with the larger span, Figure 16, beam mechanism occurs by forming mixed hinges (mechanism 6), while using independent hinge assumption, flexural plastic hinges form the beam mechanism (mechanism 1). As can be seen in this figure, considering M-V interaction results in 3.5% decrease in the frame gravity load capacity.

To investigate the effect of span to depth ratio of the beam $(L / d)$ on the frame capacity, Figures 17 and 18 present the lateral load capacity as a function of $L / d$ and w, for the case of ignoring and considering the M-V interaction effect, respectively. It can be observed from those figures that by reducing $L / d$ from 13 to 5, the lateral load capacity increases. For the shorter spans, the capacity depends on the vertical load, too. In these cases by decreasing $L / d$ from 5 to 3, the frame capacity decreases for the intermediate values of w. For the lower values of w, by ignoring M-V interaction, it is constant and independent of w and L, but considering M-V interaction for $L / d = 3$ , mechanism 8 (F-M) will govern, which will reduce the lateral load capacity of the frame (Figure 18).

Figure 17.

Frame capacity ignoring M-V interaction effect for a portal frame with h = 3 m, beam section of IPB1000, and different span to depth ratio of the beam.

Figure 18.

Frame capacity considering M-V interaction effect for a portal frame with h = 3 m, beam section of IPB1000, and different span to depth ratio of the beam.

Post-mechanism deformation at plastic hinges

To ensure satisfactory behavior of a structure in a severe earthquake, the inelastic deformation in the plastic hinges should not exceed their inelastic deformation capacity. Hence, it would be useful to relate the plastic hinge deformation to the plastic story drift angle. For the strong ground motion, the design or expected drift angle depends on the intensity of earthquake, and it can be estimated using seismic design codes (ASCE 41-17; ASCE 7-16). The inelastic capacity of plastic hinges could be obtained from test results and also estimated using existing codes (ASCE 41-17). Pre-mechanism deformations in the inelastic hinges depend on the load path and require incremental analysis. However, the post-mechanism deformation of plastic hinges could be estimated by satisfying kinematic compatibility requirements and normality condition as a flow rule. In the case of just flexural hinges or just shear hinges, the flow rule will be one-dimensional and practically post-mechanism deformation of plastic hinges could be estimated just by satisfying kinematic compatibility requirements. However, in case of existence of mixed hinges, it is necessary to consider the relation of the plastic hinge rotation and shear deflection of any mixed hinges using a flow rule. In this section, for the single-bay portal frame defined previously, simplified relations are derived to estimate the post-mechanism deformation of plastic hinges for a prescribed lateral drift. For sake of brevity, just two mechanisms are presented (mechanisms 4 and 10).

Since mechanism 4 forms by two flexural hinges with equal rotations, and there is not any mixed hinges, so the post-mechanism plastic rotation of both hinges, $ϕ_{pm}$ , could be obtained just by satisfying kinematic compatibility requirement as follows

ϕ_{pm} = (\frac{L}{L - x}) θ_{pm}

(23)

where $θ_{pm}$ is the story plastic drift ratio as shown in Figure 19(a) and x could be obtained using equation (5).

Figure 19.

Plastic hinges deformations in (a) mechanism 4 (IF-F) and (b) mechanism 10 (IF-M).

Regarding mechanism 10, kinematic compatibility requirements result in

θ_{pm} = \frac{L - x}{L} ϕ_{pm} + \frac{δ_{pm}}{L}

(24)

where $δ_{pm}$ is the plastic shear deflection of the mixed hinge as depicted in Figure 19(b). The plastic hinge location, x, for this mechanism could be obtained by solving equations (19) to (21). Using normality condition, the mixed hinge plastic shear deflection can be related to its plastic rotation as follows

δ_{pm} = \frac{\partial f / \partial V}{\partial f / \partial M} ϕ_{pm} = (\frac{V - V_{y}}{M - M_{y}}) {(\frac{M_{p} - M_{y}}{V_{p} - V_{y}})}^{2} ϕ_{pm}

(25)

Now, substituting equation (25) in equation (24) results in

ϕ_{pm} = \frac{L θ_{pm}}{L - x + (\frac{V_{2} - V_{y}}{M_{2} - M_{y}}) {(\frac{M_{p} - M_{y}}{V_{p} - V_{y}})}^{2}}

(26)

where $V_{2}$ and $M_{2}$ are shear force and bending moment at the mixed hinge at incipient collapse, respectively. Finally, $δ_{pm}$ could be obtained by substituting equation (26) in equation (25).

As a numerical representation, the portal frame with beam length of 9 m, which its capacity curve presented in Figure 15, is considered. The normalized plastic rotation, $ϕ_{pm} / θ_{pm}$ , and shear deflection, $δ_{pm} / L θ_{pm}$ , of the hinge at the right end of the beam are shown in Figure 20.

Figure 20.

Normalized plastic deformations at the hinge at the right end of the beam for the frame with beam of length 9 m.

Conclusion

Plastic analysis of frames and EBFs with intermediate link length is governed by the combination of flexural and shear yielding. In the current work, a simple and design-oriented relationship was developed and proposed to account the moment–shear (M-V) interaction of highly ductile steel I-sections within the framework of plastic design. Furthermore, using the proposed relationship, a mechanism analysis was carried out on a one-bay portal frame considering all possible collapse mechanisms. A simplified approach was developed to solve a set of nonlinear equations for the collapse mechanisms, considering M-V interaction. Considering mixed hinges adds five more mechanisms to the primary mechanisms based on the independent flexural and shear plastic hinges. The effect of gravity load and possibility of forming plastic hinges along the beams are considered and their exact locations are determined. To investigate the dependency of the portal frame capacity on the moment–shear interaction, numerical examples were solved. The obtained results indicate that, in general, the effect of M-V interaction on the capacity of frames cannot be ignored. Finally, by satisfying kinematic compatibility requirements and normality condition, post-mechanism deformations at plastic hinges were obtained for a prescribed lateral drift ratio.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

ABAQUS (2014) ABACUS Version 6.14 Documentation. Providence, RI: Dassault Systèmes Simulia Corporation.

AISC (2005) Specification for Structural Steel Buildings (ANSI/AISC 360). Chicago, IL: American Institute of Steel Construction.

AISC (2016a) Seismic Provisions for Structural Steel Buildings (ANSI/AISC 341). Chicago, IL: American Institute of Steel Construction.

AISC (2016b) Specification for Structural Steel Buildings (ANSI/AISC 360). Chicago, IL: American Institute of Steel Construction.

ASCE-41-17 (2017) ASCE Standard, ASCE/SEI, 41–17: Seismic Evaluation and Retrofit of Existing Buildings. Reston, VA: American Society of Civil Engineers.

ASCE-7-16 (2017) ASCE Standard, ASCE/SEI, 7–16: Minimum Design Loads and Associated Criteria for Buildings and Other Structures. Reston, VA: American Society of Civil Engineers.

Azizinamini

Hash

Yakel

et al . (2007) Shear capacity of hybrid plate girders. Journal of Bridge Engineering 12: 535–543.

Basler

(1961) Strength of plate girders under combined bending and shear Proc. ASCE, 87, ST7, Reprint No. 187 (210) (61-14). Available at: https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=1070&context=engr-civil-environmental-fritz-lab-reports

Cooper

Lew

Yen

(1964) Welded constructional alloy steel plate girders, Proc. ASCE, 90 ST1, Reprint No. 239 (64–3). Available at: https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=1072&context=engr-civil-environmental-fritz-lab-reports

10.

Daley

Brad Davis

White

(2016) Shear strength of unstiffened steel I-section members. Journal of Structural Engineering 143: 11.

11.

D’Apice

Fielding

Cooper

(1966) Static tests on longitudinally stiffened plate girders, June 1966, Publication No. 310(66-16). Fritz Laboratory Reports. Paper 1874.

12.

Dell’Aglio

Montuori

Nastri

et al . (2017) A critical review of plastic design approaches for failure mode control of steel moment resisting frames. Ingegneria Sismica 34: 82–103.

13.

Drucker

(1956) The effect of shear on the plastic bending of beams. Journal of Applied Mechanics 23: 509–523.

14.

Graciano

Ayestarán

(2013) Steel plate girder webs under combined patch loading, bending and shear. Journal of Constructional Steel Research 80: 202–212.

15.

Grigorian

(2012a) Lateral displacements of moment frames at incipient collapse. Engineering Structures 44: 174–185.

16.

Grigorian

(2012b) Recent developments in plastic design analysis of steel moment frames. Journal of Constructional Steel Research 76: 83–92.

17.

Horne

Morris

(1981) Plastic Design of Low-Rise Frames. London: HarperCollins.

18.

Kazemi

Erfani

(2007a) Analytical study of special girder moment frames using a mixed shear–flexural link element. Canadian Journal of Civil Engineering 34: 1119–1130.

19.

Kazemi

Erfani

(2007b) Mixed shear-flexural (VM) hinge element and its applications. Scientia Iranica 14: 193–204.

20.

Kazemi

Erfani

(2009) Special VM link element for modeling of shear–flexural interaction in frames. Structural Design of Tall and Special Buildings 18: 119–135.

21.

Kazemi

Asl

(2011) Modeling of inelastic mixed hinge and its application in analysis of the frames with reduced beam section. International Journal of Steel Structures 11: 51.

22.

Kazemi

Momenzadeh

Asl

(2012) Study of reduced beam section connections with web opening. In: Proceedings of the 15th world conference on earthquake engineering 15WCEE, Lisbon, 24–28 September.

23.

Kirkland

Kim

et al . (2015) Moment–shear–axial force interaction in composite beams. Journal of Constructional Steel Research 114: 66–76.

24.

Lee

Yoo

(2013) Flexure and shear interaction in steel I-girders. Journal of Structural Engineering 139: 1882–1894.

25.

Liang

Bradford

et al . (2004) Ultimate strength of continuous composite beams in combined bending and shear. Journal of Constructional Steel Research 60: 1109–1128.

26.

Liang

Bradford

et al . (2005) Strength analysis of steel–concrete composite beams in combined bending and shear. Journal of Structural Engineering 131: 1593–1600.

27.

Liu

Grierson

(2009) Combined MVP failure criterion for steel cross-sections. Journal of Constructional Steel Research 65: 116–124.

28.

Malvern

(1969) Introduction to Continuum Mechanics. Engle Cliffs, NJ: Prentice Hall.

29.

MathWorks, Inc. (2014) MATLAB: R2014a. Natick, MA: MathWorks, Inc.

30.

Momenzadeh

Kazemi

Hoseinzadeh Asl

(2017) Seismic performance of reduced web section moment connections. International Journal of Steel Structures 17: 1–13.

31.

Montuori

Muscati

(2017) Smart and simple design of seismic resistant reinforced concrete frame. Composites Part B: Engineering 115: 360–368.

32.

Montuori

Nastri

Piluso

(2014) Rigid-plastic analysis and moment–shear interaction for hierarchy criteria of inverted Y EB-Frames. Journal of Constructional Steel Research 95: 71–80.

33.

Montuori

Nastri

Piluso

(2015) Advances in theory of plastic mechanism control: closed form solution for MR-Frames. Earthquake Engineering & Structural Dynamics 44: 1035–1054.

34.

Neal

(1961a) The effect of shear and normal forces on the fully plastic moment of a beam of rectangular cross section. Journal of Applied Mechanics 28: 269–274.

35.

Neal

(1961b) Effect of shear force on the fully plastic moment of an I-beam. Journal of Mechanical Engineering Science 3: 258–266.

36.

Nie

Xiao

Chen

(2004) Experimental studies on shear strength of steel–concrete composite beams. Journal of Structural Engineering 130: 1206–1213.

37.

Nishino

Okumura

(1968) Experimental investigation of strength of plate girders in shear. In: 8th congress of the international association for bridges and structural engineering (IABSE Theme IIC), New York, September.

38.

Shahabian

Roberts

(2008) Behaviour of plate girders subjected to combined bending and shear loading. Scientia Iranica 15: 16–20.

39.

Vasdravellis

(2014) Shear strength and moment-shear interaction in steel-concrete composite beams. Journal of Structural Engineering 140: 04014084.

40.

White

Barker

(2008) Shear resistance of transversely stiffened steel I-girders. Journal of Structural Engineering 134: 1425–1436.

41.

White

Barker

Azizinamini

(2008) Shear strength and moment-shear interaction in transversely stiffened steel I-girders. Journal of Structural Engineering 134: 1437–1449.

Mechanism analysis of steel frames considering moment–shear interaction

Abstract

Keywords

Introduction

Numerical study for development of moment–shear yield surface

Proposed moment–shear interaction relationship

Comparison with existing relationships

Mechanism analysis of steel frames using the proposed M-V interaction relationship

Frame collapse loads obtained without considering M-V interaction

Frame collapse loads obtained by considering M-V interaction

Numerical examples

Post-mechanism deformation at plastic hinges

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References