An overview of the plastichinge analysis of 3D steel frames
 VanLong Hoang^{2},
 Hung Nguyen Dang^{1}Email authorView ORCID ID profile,
 JeanPierre Jaspart^{2} and
 JeanFrançois Demonceau^{2}
https://doi.org/10.1186/s4054001500169
© Hoang et al. 2015
Received: 12 August 2015
Accepted: 25 November 2015
Published: 23 December 2015
Abstract
An overview of plastichinge model for steel frames under static loads is carried out in this paper. Both rigidplastic and elasticplastic methods for framed structures are reviewed, including advantages and disadvantages of each method. It concerns both analysis and optimization methodologies. The modeling of 3D plastic hinges by using the normality rule of the plasticity is described. The paper also touches on the consideration of strain hardening in the plastichinge modeling. Related to take into account different phenomena (distributed plasticity, imperfections, stiffness degradation, etc.), the practical modeling of members is summarized. How to consider behaviors and cost of beamtocolumn connections is discussed. The existing methods to capture large displacements are briefly presented, as well as global formulations for different types of analysis and optimization procedures. For the illustration, several numerical examples are carried out, including a “loss a column” scenario in the robustness analysis.
Keywords
Background
Steel frames show a high nonlinear behavior due to the plasticity of the material and the slenderness of members. How to approach the “actual” behavior of steel frames has been a large subject in the research field of constructional computation. In general, either the plasticzone or the plastichinge approach is adopted to capture the inelasticity of material and geometric nonlinearity of a framed structure.
In the plasticzone method, according to the requirement of refinement degree, a structure member should be discretized into a mesh of finite elements where the nonlinearities are involved. Thus, this approach may describe the “actual” behavior of structures, and it is known as a “quasiexact” solution. However, although tremendous advances in computer hardware and numerical techniques were achieved, plasticzone method is still considered as an “expensive” one, requiring considerable computing burden. Moreover, software based on the plasticzone approach requires the expertise of users.
On the other hand, the plastichinge approach demands only one beamcolumn element per physical member to assess approximately the nonlinear properties of the structures; so the computation time is considerably reduced. In addition, computer programs using the plastichinge model are familiar to the habit of engineers. Thanks to these advantages, it appears that the plastichinge method is more widely used in practice by engineers than the plasticzone method. Wherefore, the improvement in the accurateness of the plastichinge approach has been an attractive topic since over the past 60 years.
The present paper consists in an overview of the plastichinge approach for 3D steel frames. Both the rigidplastic and elasticplastic methods for framed structures are reviewed, including the advantages and disadvantages of each method. It concerns both analysis and optimization methodologies. Furthermore, a description of the modeling of 3D plastic hinges by using the normality rule of the plasticity is done. The consideration of strain hardening in the plastic modeling is also touched on. By taking into account different phenomena (distributed plasticity, imperfections, stiffness degradation, etc.), we summarize the practical modeling of members. How to consider behaviors and cost of beamtocolumn connections is discussed. The existing methods to capture large displacements are briefly presented, as well as global formulations for different types of analysis and optimization procedures. For the illustration, several numerical examples are carried out, including a “loss a column” scenario in the robustness analysis.
Generality on the behavior of frames
Plastic behavior

Monotonous loading where all applied loads are monotonically increased with a unique loading factor.

Fixed repeated loading where the loads are repeated (loading, unloading and reloading and so on), but the protocol is defined (defined history).

Arbitrary repeated loading where each load varies independently with arbitrary history, but within their limits (maximum and minimum values, Fig. 1).
The monotonous loading and the fixed repeated loading are two particular cases of the arbitrary repeated loading. Therefore, in this paper, the terms ‘complex loads’ and ‘simple loads’ may be used to indicate the arbitrary repeated loading and the monotonous/fixed repeated loadings, respectively.
In the practice of construction, a structure may be subjected to various kinds of load, for example: dead load, live load, wind load, effects of earthquake, etc. The dead load consists of the weight of the structure itself and its cladding. The dead load remains constant, but other loads vary continually. Those variations are normally independent and repeated with arbitrary histories. It is clear that the structure is normally subjected to the loads with arbitrary histories; so the simple loads are used as a simplification in calculations.
 1.
The structure returns to the elastic range after having some plastic deformations (Fig. 3a); the structure is referred to as shakedown (plastic stability/plastic adaptation).
 2.
Plastics deformation constitutes a closed cycle (Fig. 3b), the structure is presumed to be failed by alternating plasticity (lowcycle fatigue);
 3.
Plastic deformation implies an infinitely progress (Fig. 3c), the structure is considered to be failed by incremental plasticity.
The alternating plasticity or the incremental plasticity behaviors may be accepted in some exceptional load cases (as a seismic event), while these behaviors should be avoided in normal state where the shakedown behavior should be planed. The shakedown analysis aims to determine the load domain for the complex load cases (Fig. 2) such that the shakedown occurs in the structure. In other words, the shakedown analysis is a straight method (“one step”) to avoid the alternating plasticity or the incremental plasticity in structures without knowing the loading histories.
Geometric nonlinearities
Plastic methods for framed structures
During the past 60 years, the theories of plasticity, stability and computing technology have recorded great achievements that constitutes the basis allowing scientists to develop successfully plastic methods for structures. The framed structures are often regarded as benchmark to build up computation methods for other kinds of structure. Up to now, plastic methods for framed structures can be classed into two groups: direct methods and stepbystep method.
Direct methods
The term “direct methods” consists in the rigidplastic methods that the load multiplier can be directly identified without any intermediate states of structures. The direct methods are based on the static and kinematic theorems—two fundamental theorems of the limit analysis, which lead to static approach and kinematic approach, respectively.
In the 1950s, the first plastic methods (e.g., trial and error method, a combination of mechanism method and plastic moment distribution method) were proposed by Baker, Neal, Symonds and Horne (see Neal [62]). Since the 1970s, the direct methods have been largely developed thanks to the application of mathematical programming; in particular, the linear programming problem can be generally solved through the simplex method (see Dantzig [20]). An overall picture on the application of the mathematical programming to structural analysis can be found from: the stateoftheart report of Grierson [26]; the book edited by Cohn [13]; the stateoftheart papers and the key note of Maier [56, 57]; the book edited by Smith [71]; and other papers by Cocchetti [12], NguyenDang [66]. Additionally, some interesting computer programs were built up, e.g. DAPS [68], STRUPLANALYSIS [25], CEPAO [29, 32, 66] where the linear programming technique is combined with the finite element method that enables automatic procedures.

Limit analysis

Shakedown analysis

Limit optimization

Shakedown optimization
Even if the shakedown problem is classed in the rigidplastic analysis, the elastic behavior of structures is needed for the shakedown analysis/optimization.
Advantages of the direct methods

capable of taking full advantage of mathematical programming achievements.

suitable to solve structures subjected to arbitrary repeated loading (shakedown problem).

possible to unify into unique computer program because the algorithms of the direct methods for different procedures are similar, such as: limit or shakedown, analysis or optimization, frames or plate/shell, etc.

not influenced by local behaviors of structures, namely the elastic return (a phenomenon often occurs in the stepbystep methods). There exists sometimes degenerate phenomenon in the simplex method but it is treated by the lexicographical rule (see Dentzig [20]).
Limitations of the direct methods

when the geometric nonlinearity conditions are taken into account, so it poses a great challenge.

when solving largescale frames, because the direct methods belong to “one step” approaches.
Stepbystep methods
Stepbystep methods or elasticplastic incremental methods are based on the standard methods of the elastic analysis. The loading process is divided into various steps. After each loading step, the stiffness matrix is updated to take into account nonlinear effects. In comparison with the elastic solution, only the physical matrix is varied to consider the plastic behavior. The stepbystep methods take advantage of large experiences of the linear elastic analysis by the finite element method. One may find many useful computational algorithms and techniques in many text books (e.g., Bathe [2, 3], among others). Commercial software for structural analysis has been almost developed by adapting the stepbestep methods.
Compared to the direct methods, the stepbystep methods have the following features:
Advantages of the stepbystep method

The geometric nonlinearity is appropriately taken into account.

The stepbystep methods furnish a complete redistribution progress prior to the collapse of structures.

With the progress in both computing hardware and numerical technology, the modeling of complex structures, even very large scales, may be dealt with.
Limitations of the stepbystep methods

For the case of arbitrary loading histories (shakedown problem), the stepbystep methods are cumbersome and embed many difficulties.

With the elasticplastic analysis of frames, this method is influenced by the local behavior of structures, such as the elastic return, it may lead to an erroneous solution.
Plastichinge modeling
Plastichinge modeling is an important issue of the plastic analysis for framed structures; it influences not only the accurateness but also the formulation procedure. To model plastic hinges, the yield surface is firstly needed to be defined and then the relationship between forces and plastic deformations at the plastic hinges is necessary to be established.
Yield surfaces

Yield surface of AISC [1] (Fig. 5c):$$\begin{aligned} \left n \right + (8/9)\left {m_{y} } \right + (8/9)\left {m_{z} } \right = 1\,{\text{for}}\,\left n \right \ge 0.2; \hfill \\ (1/2)\left n \right + \left {m_{y} } \right + \left {m_{z} } \right = 1\,{\text{for}}\,\left n \right < 0.2; \hfill \\ \end{aligned}$$(3)
Orbison’s yield surface is very suitable to the elasticplastic analysis by stepbystep method for 3D steel frames, it has been widely applied (see Orbison [67], Liew [51, 52], Kim [42–45], Chiorean [11], among others). On the other hand, the polyhedrons [e.g. the sixteenfacet polyhedron, Eq. (3)] obviously are the unique way allowing the use of the linear programming technique in the rigidplastic analysis.
Other definitions of yield surface have been also used in some researches (e.g., Izzuddin [37]). In particular, adapted yield surfaces for various shape section of steel profiles can be found in Meas [60], where the coefficients in Eq. (2) are varied for each type of cross section.
Plastic deformation
The application of the normality rule for the 3D plastic hinge was detailed in Hoang [32] for the stepbystep method where Orbison’ yield surface was adopted; or in Hoang [30, 33] for the rigidplastic analysis using the polyhedron yield surface.
Force: plastic deformation relationship
Strain hardening consideration
In Eqs. (8), (9), (10), \(\varphi\) is the yield surface of the cross section [e.g., Orbison’s yield surface in Eq. (2)]; H is the strain hardening modulus (or plastic modulus), it is assumed constant (linear hardening low); \(\bar{\varepsilon }^{p}\) is the effective strain that is defined below; \(\bar{\varepsilon }_{l}^{p}\) is the limit effective strain. How to determine these parameters, and also how to involve the yield surface given by Eq. (8) into the global formulation procedure can be found in Hoang [31].
Equations (8), (9), (10) describe, respectively, the elastic range, the hardening range, and the flowed range (Fig. 8b). It shows that a nonlinear hardening rule is approximated through bilinear procedures [Eqs. (9) and (10)]. In the space of internal forces, Φ and ϕ have the same shape, i.e., Φ is an expansion of ϕ.
Member modeling
Concerning the rigidplastic analysis, a member is considered to be rigid body, no deformation is allowed. Therefore, this section mainly devotes to the member modeling in the elastic–plastic analysis by the stepbystep methods, only the compatibility and equilibrium relations (in “Beamcolumn element formulation”) can be used for the both rigidplastic and elasticplastic analysis.
Even if the assumption of small deformation is adopted but different effects (namely Pδ, distributed plasticity, local and lateraltorsional buckling, etc.) should be taken into account for the member behavior. There exist several ways to involve the mentioned effects; the present paper summarizes a practical technique comprising two separate procedures: (1) establish the fundamental relations (compatibility, equilibrium and constitutions) using the elastic linear beam theory (Bernoulli beam) and (2) practically include the different effects to the member formulation. These two procedures will be presented in “Beamcolumn element formulation” and “Taking into account different effects” below, respectively.
Beamcolumn element formulation

e _{k} be the vector of total deformations (generalized strains) of the element extremities in the local axes xyz and \({\mathbf{e}}_{k}^{p}\) be the plastic part of e _{k}.

d _{k} be the vector of nodal displacements in the global axes XYZ;

s _{k} be the vector of internal forces at the element extremities in the local axes xyz;

f _{k} be the vector of external loads applied at the two nodes in the global axes XYZ.
As the small strain is assumed for the member formulation and no special particularity is assigned for the geometric matrix (B _{k}), the explicit form is classical and it is not presented here. The input data to build up the matrix B _{k} are: the total length of the member chord (Fig. 10) and the matrix of direction cosines of the element. On the other hand, as the matrix D _{k} describes the elastic relationship of the member, so the plastic deformation must be eliminated, i.e., the term (ee ^{P}) in Eq. (13)]. Moreover, the elastic length (Fig. 10), which is the difference between the total length and the axial plastic deformations, needs to be used in the matrix D _{k}. The matrix D _{k} includes obviously the Young modulus and mechanical properties of the cross section.
Taking into account different effects
In this section, several techniques existing in the literature to take into account different effects that influence to the member behavior are summarized. Only the principles are presented, the detailed formulation is referred to the corresponding literature.
Pδ effect
The stability functions have been largely used to include the influence of the axial force to the member stiffness (Pδ effect). Normally, one finite element can model one physical member when the stability functions are applied. To introduce the stability functions into the formulation of the element, only the physical relation (matrix \({\mathbf{D}}_{k}^{ep}\) in Eq. (14)] is modified, the explicit forms may be found in many texts (e.g. Chen [8]).
Spread of plasticity at plastic hinge
Imperfections (residual stress and geometric imperfection) and distributed plasticity
It can be found in the literature some other ways to take into account the plasticity along the member, for example: in Izzuddin [37] an adaptive mesh is used, allowing to detect whether plastic hinges occur along the member length; or in Chiorean [11] the Ramberg–Osgood forcestrain relationship was adopted to model the gradual yielding of cross sections; in Liu [54] an element including a plastic hinge was built up, allowing the occurrence of plastic hinge within the element length at an arbitrary location.
Lateraltorsional buckling effect
A procedure to consider the lateraltorsional buckling effect in the plastichinge analysis of 3D steel frames was proposed in Kim [44]. The procedure consists in two steps: (1) using the Standards (e.g. European/American ones) to compute the lateraltorsional buckling strength of the members; (2) replacing the plastic strengths of the cross section in the yield surface [Eq. (1)] by the obtained lateraltorsional buckling strengths. By this way, the lateraltorsional buckling is practically taken into account. In some works (e.g., [27, 47]), the lateraltorsional effect has been included during the element formulations, so this effect is sophisticatedly considered in avoiding fiber/plate/shell/solid elements. In Jiang [40], a mixed element formulation has been proposed to take into account both geometry and material nonlinearities including the lateraltorsional buckling effect.
Local buckling effect
With a similar idea for including the lateraltorsional buckling, a process to consider the local buckling effect was also proposed in Kim [45], where the plastic strength of the section in the yield surfaces involves the local buckling phenomenon.
In another context, based on the classification of cross sections given in Eurocode 3, part 11 [23], a procedure to check the local buckling phenomenon at critical sections was proposed in Hoang [30]. At each calculation step, the positions of the neutral axes for the critical sections are determined from the internal forces (axial force and two bending moments). When the neutral axes of the cross sections are defined, compression parts of the sections can be checked against the local buckling.
Consideration of connections
The connections, as beamtocolumn joints and column bases, make up an important portion of framed structures. The connection behavior shows strong influence on the frame behavior and the cost of connections occupies a considerable part of the frame cost. Therefore, the consideration of the connection behavior and the cost in the frame analysis and the optimisation has been an intensive topic during the past 30 years. In the following, how to practically introduce the connection characteristics and cost to the plastichinge analysis and the optimisation of steel frames is summarized.
Modeling of connections
Effect of initial stiffness of connections
Effect of partial strength connections
Effect of connection cost
f(R) is function of the connection rigidity (R). A detailed expression was provided in Simöes [66], accordingly the conventional length is increased by 20 % if it consists in pinner connections and 100 % if the extremities of the connections are fully rigid.
The conventional lengths of the members replace the actual lengths in the objective function of the optimization procedure (“Weight function”) that means the connection cost is considered in the optimal procedure.
Relation between s and ν
S  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0 

ν  25.0000  10.0000  6.2667  4.4000  3.2800  2.5333  2.0000  1.1667  0.5185  0.0000 
Weight function
Formulation for the whole frame
The previous sections have presented the modeling of plastic hinges, members and connections. This section aims to summarize the formulation procedures for the whole frames with various types of analysis.
Step 1: Preparation of input

s _{0}: vector of plastic capacities (axial force and two bending moments) of the cross sections (plastic hinges)

n _{p}: vector of axial plastic capacities of the cross sections (it is a subvector of s _{0}).

l: vector of member lengths (or conventional lengths if the connection behaviors are considered)

f: vector of applied loads (in the global axes)

s _{E}: envelop vector of elastic responses according to the domain of considered loading (the structure is considered purely elastic), it involves two extreme values: the positive s _{Emax} and the negative s _{Emin}

B: the compatibility matrix (B ^{T} is the equilibrium matrix), see Eq. (11)

D ^{ep}: the physical matrix, see Eq. (14)

N: vector containing gradients of the yield surfaces, see Eq. (5)

d: vector of displacements (in the global system axes)

ρ: vector of residual internal forces (in the local axes)

λ: vector of plastic magnitudes

s: vector of internal forces (in the local axis)
Input and variables according to different types of problem
Type of problem  Input  Unknowns 

Elastic–plastic analysis  B, D ^{ep} , f  d 
Limit analysis  B, N, f, s _{0}  d, λ 
Shakedown analysis  B, N, s _{E} , s _{0}  d, λ 
Limit optimization  B, N, f  n _{p} , s 
Shakedown optimization  B, N, s _{E}  n _{p} , ρ 
Step 2: Global formulation
Global formulation
Type of problem  Global formulation 

Elastic–plastic analysis [29]  \({\mathbf{d}} = {\mathbf{K}}^{  1} {\mathbf{f}}\quad {\text{with}}\quad {\mathbf{K}} = {\mathbf{B}}^{\text{T}} {\mathbf{D}}^{ep} {\mathbf{B}}\) 
Limit analysis [27]  \({\text{Min}}\quad \varOmega ({\dot{\varvec{\lambda }}},{\dot{{{\bf d}}}}) = {\mathbf{s}}_{0}^{\text{T}} {\dot{\varvec{\lambda }}}\quad {\text{with}}\quad \left\{ {\begin{array}{*{20}c} {{{{\bf N}\dot{\varvec{\lambda} }}}  {{{\bf B}\dot{\bf {d}}}} = {\mathbf{0}}} \\ {{\bar{\mathbf{f}}}^{\text{T}} {\dot{{{\bf d}}}} > 0} \\ {{\dot{\varvec{\lambda }}} \ge {\mathbf{0}}} \\ \end{array} } \right.\) 
Shakedown analysis [27]  \({\text{Min}}\quad \varOmega ({\dot{\varvec{\lambda }}},{\dot{\mathbf{d}}}) = {\mathbf{s}}_{0}^{\text{T}} {\dot{\varvec{\lambda }}}\quad {\text{with}}\quad \left\{ {\begin{array}{*{20}c} {{{{\bf N}\dot{\varvec{\lambda }}}}  {{{\bf B}\dot{{\bf d}}}} = {\mathbf{0}}} \\ {{\mathbf{s}}_{\text{E}}^{\text{T}} {{{\bf N}\dot{\varvec{\lambda} }}} > 0} \\ {{\dot{\varvec{\lambda }}} \ge {\mathbf{0}}} \\ \end{array} } \right.\) 
Limit optimization [30]  \({\text{Min}}\;{\text{Z}}({\mathbf{n}}_{\text{p}} ,{\mathbf{s}}) = {\mathbf{n}}_{\text{p}}^{\text{T}} {\mathbf{l}}\quad {\text{with}}\quad \left\{ \begin{aligned} {\mathbf{B}}^{\text{T}} {\mathbf{s}} = {\mathbf{f}} \hfill \\ {\mathbf{N}}^{\text{T}} {\mathbf{s}} \le {\mathbf{s}}_{ 0} \hfill \\ \end{aligned} \right.\) 
Shakedown optimization [30]  \({\text{Min}}\;{\text{Z}}({\mathbf{n}}_{\text{p}} ,{\varvec{\uprho}}) = {\mathbf{n}}_{\text{p}}^{\text{T}} {\mathbf{l}}\quad {\text{with}}\quad \left\{ \begin{array}{l} {\mathbf{B}}^{\text{T}} {\varvec{\uprho}} = {\mathbf{0}} \hfill \\ {\mathbf{N}}^{\text{T}} {\mathbf{(s}}_{\text{e}} + {\varvec{\uprho )}} \le {\mathbf{s}}_{ 0} \hfill \\ \end{array} \right.\) 
Step 3: Solving procedure
Solving procedure
Type of problem  Solving method 

Elastic–plastic analysis  Incrementaliterative strategies 
Limit analysis  Simplex algorithm 
Shakedown analysis  
Limit optimization  
Shakedown optimization 
Large displacements
The plasticity of material and different effects of geometry are dealt with in “Plastichinge modeling” (plastichinge modeling) and “Member modeling” (member formulation). This section concerns the methods to capture large displacement (Fig. 9) and aims to update the deformed configuration of the chord. Generally, either the conventional secondorder approach or corotational approach is adopted in the literature.
On the other hand, in the corotational approach, the deformed configuration of the chord is used to update the fundamental relationships; so this approach can be appropriate for the structure as far as very large displacements with a high accuracy. The corotational approach has been abundantly interpreted in the literature (e.g., Battini [4], Crisfield [14, 15], Izzuddin [35, 36], Mattiasson [59], Souza [72] and Teh [73]), maybe with various terminologies (namely Convected/Eulerian/etc. formulations). The main objective of researches is to treat the finite rotation in the space and there exist actually several techniques. In the following, a quite simple technique to calculate the rotation of the member around its axis, socalled “mean rotation” formulation, is chosen to present.
Numerical examples
Features of the CEPAO program
Features  Problem types  

Elastic–plastic analysis  Limit analysis  Shakedown analysis  Limit optimization  Shakedown optimization  
Orbison yield surface (“Plastichinge modeling”)^{a}  ×  
AISC yield surface (“Plastichinge modeling”)  ×  ×  ×  ×  
Pδ effect (“Member modeling”)  ×  
Spread of plasticity in the plastic hinge (“Member modeling”)  ×  
Initial imperfections (“Member modeling”)  ×  
Lateraltorsional buckling (“Member modeling”)  ×^{(c)}  
Local buckling (“Member modeling”)  ×  
Member stability check^{b}  ×  ×  
Conventional secondorder approach (“Large displacements”)  ×  
Corotational approach (“Large displacements”)  ×  
Semirigid connection (“Consideration of connections”)  ×  
Partial connection (“Consideration of connections”)  ×  ×  ×  ×  × 
Cost of connection (“Consideration of connections”)  ×  × 
Robustness analysis
In recent years, the robustness analysis has become a relevant topic in the research field. At University of Liege, the “loss a column” scenario is under developement by analytical, numerical and experimental approaches [see Demonceau [19] and Huvelle [34]). The main idea is to model the behavior of structures after loss of a column. In this state, the frame geometry is considerably modified and the internal forces in the frame members are strongly varied (even from purely in bending to purely in tension). This situation is out of the classical concept of the plastichinge analysis, simple models of plastic hinge (for example: neglecting of plastic axial deformations) and the conventional secondorder approach may be not adequate. In the following, some typical examples concerning the “loss a column” scenario are analyzed by the CEPAO program. The CEPAO results are compared with the results provided by FINELG—a nonlinear finite element software developed at the University of Liege [24]. The FINELG model can be considered a plasticzone analysis where both material and geometric aspects are considered.
Example a1
Example a2
Example a3
Example b
Example bthe data of the frame
Frame layout (Fig. 25)  

Used profiles (European profiles)  
Story  Columns  Beams in the X direction*  Beams in the Y direction 
1st to 3rd  HL 1000 × 883  IPE 400  IPE O 600 
4th to 6th  HL 1000 × 591  IPE 400  IPE 600 
7th to 9th  HEM 1000  IPE 400  IPE 550 
10th to 12th  HEM 400  IPE 400  IPE 500 
13th to 15th  HEM 300  IPE 400  IPE 400 
Limit and shakedown analysis for 3D steel frames
Example c1: sixstory space frame
Example c2: twentystory space frame
These two 3D steel frames have been used as benchmarks in the literature concerning the advanced nonlinear analysis of steel frames (e.g. Orbison [67], Liew [51, 52], Kim [46], Chiorean [11] and Cuong [16]). The frames were also analyzed by CEPAO and the results were validated by previous works Hoang [29, 31]. In the present paper, these examples are represented to highlight the incremental plasticity and the phenomenon of alternating plasticity in the frames.
Concerning the loading domain in the two examples, two cases are considered for the shakedown analysis: a) 0 ≤ μ_{1} ≤ 1, 0 ≤ μ_{2} ≤ 1 and b) 0 ≤ μ_{1} ≤ 1, −1 ≤ μ_{2} ≤ 1. For the fixed or proportional loading, obviously we must have: μ_{1} = μ_{2} = 1. The uniformly distributed loads are lumped at the joints of frames.
Examples c—load factors of the frames given by CEPAO
Type of analysis  Load factors  Limit state  

Example c1  Example c2  
Limit analysis  2.412  1.698  Formation of a mechanism 
Shakedown analysis, domain load a  2.311  1.614  Incremental plasticity 
Shakedown analysis, domain load b  1.670  0.987  Alternating plasticity^{a} 
Secondorder analysis [31]  2.033  1.024 
It appears that in the case of symmetric horizontal loading (wind load for example), the alternating plasticity occurs and corresponding load factors are very small, less than the load factors given by the secondorder analysis, even the secondorder effect is not yet considered in the shakedown analysis.
In the case where the alternating plasticity occurs, one may verify the results as the following. For example, with the sixstory frame and the load domain b, the alternating plasticity occurs at section B (Fig. 27); in this point one has:
The elastic envelop: \(M_{y}^{ + } = M_{y}^{  } = 186.42\) (kNm); \(N^{ + } = N^{  } = 13.46\) (kN); \(M_{z}^{ + } = M_{z}^{  } = 1.22\) (kNm);
The plastic capacity (W12x53): \(M_{py} = 318.70\) (kNm); \(N_{p} = 2525.00\) (kN); \(M_{pz} = 119.50\) (kNm).
Limit optimization of 2D semirigid frame
As mentioned, a strategy of stability check according to Eurocode 3 is adopted for individual members. However, the secondorder effect has not yet been considered in the optimization problem.
Summary
A quite complete picture on the plastichinge analysis and the optimization of 3D steel frames under static loads is made out in the present paper. From the modeling of plastichinges, members as well as connections to the global formulation, a whole frame is dealt with. Both the rigidplastic and elasticplastic methods are addressed; both the analysis and optimization procedures are concerned.
It points out that the elasticplastic analysis by the stepbystep method is an efficient tool to globally analyze steel frames. Using the standard codes for beam columns to practically take into account different effects within the member length allows modeling the local behavior of the frame; furthermore complex formulation can be avoided. By applying the normality rule for plastic hinges and the corotational approach for geometrical nonlinearities, the plastichinge approach can describe the structure behavior with a high accuracy as far as with very large displacement. In comparison with the plasticzone model, the plastichinge approach shows very good agreement results while the computation cost is strongly reduced. Accordingly, exceptional states of structures, as defined in robustness or progressivecollapse analysis, can be resolved by the plastichinge model instead of the plasticzone method. However, for identifying alternating plasticity/incremental plasticity in structures, the stepbystep method is still powerless when arbitrary history is considered for the loads. Moreover, algorithm for optimization design by using the stepbystep method has not yet been straightforwardly deduced from the analysis problem.
On the other hand, the rigidplastic analysis is quite attractive within the case of arbitrary loading histories that are the nature of almost loads applying on structures. With the shakedown analysis, the alternating plasticity and incremental plasticity phenomena can be straightforwardly analyzed without knowing the loading histories. Moreover, the rigidplastic method takes full advantages of mathematic programming achievements in both the analysis and optimization algorithms. However, there remain many difficulties for the rigidplastic method to take into account geometrically nonlinearities that are very explicit within steel frames.
The plastichinge approach can involve the connection behaviors without difficulty. The consideration of connection cost in the optimization problem provides more possibilities to obtain economical designs. The burden may arise from the mechanical modeling and the modeling of connection cost, because the connection configurations are largely varied.
It may be an interesting direction of future research to combine the two approaches: the rigidplastic and elasticplastic methods, for the sake of taking full advantage of both methods.
Declarations
Authors’ contributions
VLH, DHN, JPJ and JFD carried out the study. VLH drafted and revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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