## 1. Introduction

This chapter presents advanced analysis methods for space steel frames which consider both geometric and material nonlinearities. The geometric nonlinearities come from second-order

Geometric imperfections result from unavoidable errors during the fabrication or erection. There are three methods to model the geometric imperfections: (1) the explicit imperfection modeling, (2) the equivalent notional load, and (3) the further reduced tangent modulus. The explicit imperfection modeling for braced and unbraced members is illustrated in Fig. 2(a). For braced members, out-of-straightness is used instead of out-of-plumbness. This is due to the fact that the

Residual stresses are created in the hot-rolled sections due to uneven cooling of the cross-section. Typical residual stress pattern for a hot-rolled wide flange section is illustrated in Fig. 3. When a member is subjected to a compressive force, the fibers which have the highest values of compressive residual stress will yield first, and the fibers with the tensile stress will yield last. It means that the yielding over the cross-section is a gradual process. Hence, the stress-strain curve for a stub column is smooth instead of linear elastic-perfectly plastic in the case of coupon as shown in Fig. 4(a). The gradual yielding over the cross-section is caused not only by residual stress but also by flexure as shown in Fig. 4(b). Although the stress-strain relationship of steel is assumed to be linear elastic-perfectly plastic, the moment-curvature relationship has a smooth transition from elastic to fully plastic. This is because the section starts to yield gradually from extreme fibers which have the highest stresses. Material nonlinearities can be taken into account using various methods based on the degree of refinement used to represent yielding. The elastic plastic hinge method allows a drastic simplification, while the plastic zone method uses the greatest refinement.

In the current design approach, the strength and stability of a structural system and its members are treated separately, and hence, the information about the failure modes of a structural system is not provided. This disadvantage is overcome by using a second-order inelastic analysis called “advanced analysis”. Advanced analysis indicates any methods that efficiently and accurately capture the behavior and the strength of a structural system and its component members. This chapter will present two advanced analysis methods: (1) the refined plastic hinge method and (2) the fiber method. In these methods, the geometric nonlinearities are captured using the stability functions, while the material nonlinearities are considered using the refined plastic hinge model and fiber model. The benefit of employing the stability functions is that it can accurately capture geometrical nonlinear effects by using only one element per member, and hence, this leads to a high computational efficiency as demonstrated by the works of Thai and Kim (2008; 2009; 2011b; 2011c; 2011d; 2012).

## 2. Advanced analysis

### 2.1. Stability functions accounting for second-order effects

Considering a beam-column element subjected to end moments and axial force as shown in Fig. 5. Using the free-body diagram of a segment of a beam-column element of length x, the external moment acting on the cut section is

where

Using

The general solution of Eq. (2) is

The constants

Substituting Eq. (4) into Eq. (3), the deflection

and rotation

The end rotation

(7) |

Eq. (7) can be written in matrix from as

where

Eqs. (9) and (10) are indeterminate when the axial force is zero (i.e.

(11) |

where

(12) |

where

### 2.2. Refined plastic hinge model accounting for inelastic effects

The refined plastic hinge model is an improvement of the elastic plastic hinge one. Two modifications are made to account for a smooth degradation of plastic hinge stiffness: (1) the tangent modulus concept is used to capture the residual stress effect along the length of the member, and (2) the parabolic function is adopted to represent the gradual yielding effect in forming plastic hinges. The inelastic behavior of the member is modeled in terms of member force instead of the detailed level of stresses and strains as used in the plastic zone method. As a result, the refined plastic hinge method retains the simplicity of the elastic plastic hinge method, but it is sufficiently accurate for predicting the strength and stability of a structural system and its component members.

#### 2.2.1. Gradual yielding due to residual stresses

The Column Research Council (CRC) tangent modulus concept is employed to account for the gradual yielding along the member length due to residual stresses. The elastic modulus

Equation (13) is plotted in Fig. 6. The tangent modulus

#### 2.2.2. Gradual yielding due to flexure

The tangent modulus concept is suitable for the member subjected to axial force, but not adequate for cases of both axial force and bending moment. A gradual stiffness degradation model for a plastic hinge is required to represent the partial plastification effects associated with flexure. The parabolic function is used to represent the smooth transition from elastic stiffness at the onset of yielding to the stiffness associated with a full plastic hinge. The parabolic function

The value of parabolic function

where

For AISC-LRFD yield surface (AISC, 2005)

For modified Orbison yield surface (McGuire et al., 2000)

where_{y}, M_{yp}, M_{zp} are axial load, and plastic moment capacity of the cross-section about

When the force point moves inside or along the initial yield surface

When the parabolic function for a gradual yielding is active at both ends of an element, the incremental member force and deformation relationship in Eq. (12) is modified as

(17) |

where

(18) |

where

### 2.3. Fiber model accounting for inelastic effects

The concept of fiber model is presented in Fig. 9. In this model, the element is divided into a number of monitored sections represented by the integration points. Each section is further divided into

The incremental force and deformation relationship, Eq. (12), which accounts for the

where

in which the axial stiffness

(23) |

in which

Section deformations are represented by three strain resultants: the axial strain

Section force vector

Section deformation vector

The incremental section force vector at each integration points is determined based on the incremental element force vector

where

The section deformation vector is determined based on the section force vector as

where

(27) |

Following the hypothesis that plane sections remain plane and normal to the longitudinal axis, the incremental uniaxial fiber strain vector is computed based on the incremental section deformation vector as

where

Once the incremental fiber strain is evaluated, the incremental fiber stress is computed based on the stress-strain relationship of material model. The tangent modulus of each fiber is updated from the incremental fiber stress and incremental fiber strain as

Eq. (32) leads to updating of the element stiffness matrix

### 2.4. Shear deformation effect

To account for transverse shear deformation effect in a beam-column element, the member force and deformation relationship of beam-column element in Eq. (12) should be modified. The flexibility matrix can be obtained by inversing the flexural stiffness matrix as

where

where

The basic force and deformation relationship including shear deformation is derived by inverting the flexibility matrix as

(35) |

The member force and deformation relationship can be extended for three-dimensional beam-column element as

(36) |

in which

(37) |

where

### 2.5. Element stiffness matrix

The incremental end forces and displacements used in Eq. (38) are shown in Fig. 10(a). The sign convention for the positive directions of element end forces and displacements of a frame member is shown in Fig. 10(b). By comparing the two figures, the equilibrium and kinematic relationships can be expressed in symbolic form as

where

(40) |

Using the transformation matrix, the nodal force and nodal displacement relationship of element may be written as

where

It should be noted that Eq. (43) is used for the beam-column member in which side-sway is restricted. If the beam-column member is permitted to sway, additional axial and shear forces will be induced in the member. These additional axial and shear forces due to member sway to the member end displacements can be related as

where

in which

By combining Eqs. (43) and (47), the general force-displacement relationship of beam-column element obtained as

where

### 2.6. Solution algorithm

The generalized displacement control method proposed by Yang and Shieh (1990) appears to be one of the most robust and effective method because of its general numerical stability and efficiency. This method is adopted herein to solve the nonlinear equilibrium equations. The incremental form of the equilibrium equation can be rewritten for the th iteration of the th incremental step aswhere

Once the displacement increment vector

The total applied load vector

where the load factor

The load increment parameter

where

For the iterative step

where

## 3. Numerical examples

In this section, three numerical examples are presented to verify the accuracy and efficiency of two proposed analysis methods: (1) the refined plastic hinge method and (2) the fiber method. The predictions of strength and load-displacement relationship are compared with those generated by commercial finite element packages and other existing solutions. The first example is to show how the stability functions capture the

### 3.1. Elastic buckling of columns

The aim of this example is to show the accuracy and efficiency of the stability functions in capturing the elastic buckling loads of columns with different boundary conditions. Fig. 11 shows cantilever and simply supported columns. The section of columns is W8×31. The Young’s modulus and Poisson ratio of the material are

Fig. 12 shows the load-displacement curves of the columns predicted by the present element and the cubic frame element of SAP2000. Since the present element is based on the stability functions which are derived from the closed-form solution of a beam-column subjected to end forces, it can accurately predict the buckling load of columns with different boundary conditions by using only one element per member. Whereas the cubic frame element of SAP2000, which is based on the cubic interpolation functions, overpredicts the buckling loads by 18% and 16% for the cantilever column and simply supported column, respectively, when the columns are modeled by one element per member. The load-displacement curves shown in Fig. 12 indicate that SAP2000 requires more than five cubic elements per member in modeling to match the results predicted by the present element. This is due to the fact that when the member is divided into many elements, the

### 3.2. Two story space frame

A two-story space subjected to combined action of gravity load and lateral load is depicted in Fig. 13 with its geometric dimension. The Young modulus, Poisson ratio, and yield stress of material are

The ultimate loads of the frame obtained by different methods are presented in Table 1. The load-displacement responses of the frame are also plotted in Fig. 14. It can be seen that the results of the present element are well compared with those of De Souza (2000) using the force-based method. It should be noted that only one element per member is used in present study and De Souza (2000). The B23 element of ABAQUS overestimates ultimate strength of this frame if each framed member is modeled by less than fifty B23 elements. The difference between B23 element and present element is negligible when more than fifty B32 elements are used, and the ultimate strength and load-displacement curve obtained by ABAQUS and present study are then close each other.

### 3.3. Twenty-story space frame

The last example is a large scale twenty-story space steel frame as shown in Fig. 15. The aim of this example is to demonstrate the capability of two proposed methods in predicting the strength and behavior of large-scale structures. A50 steel with yield stress of 344.8 Mpa, Young’s modulus of 200 Gpa, and Poisson’s ratio of 0.3 is used for all sections. The load applied to the structure consists of gravity loads of 4.8 kN/m^{2} and wind loads of 0.96 kN/m^{2} acting in the Y-direction. These loads are converted into concentrated loads applied at the beam-column joints. The obtained results are also compared with those generated by Jiang et al. (2002) using the mixed element method.

Jiang et al. (2002) used both the plastic hinge and spread-of-plasticity elements to model this structure to shorten the computational time because the use of a full spread-of-plasticity analysis is very computationally intensive. When a member modeling by one plastic hinge element detected yielding to occur between the two ends, it was divided into eight spread-of-plasticity elements to accurately capture the inelastic behavior. In this study, each framed member is modeled by only one proposed element. The load-displacement curves of node A at the roof of the frame obtained by the present elements and mixed element of Jiang et al. (2002) are shown in Fig. 16. The ultimate load factor of the frame is also given in Table 2. A very good agreement between the results is seen.

## 4. Conclusion

This chapter has presented two advanced analysis methods for space steel frames. In these methods, the geometric nonlinearities are captured using the stability functions, while the material nonlinearities are considered using the refined plastic hinge model and fiber model. The benefit of using the stability functions is that they require only one element per member, and hence, minimize the modeling and solution time. The advantage of refined plastic hinge model is its simplicity and efficiency. However, it is limited to steel material. Although the fiber model is a little bit time consuming compared to the refined plastic hinge model, it can be used for both steel and concrete or concrete-filled steel tubular structures as shown in the works of Thai & Kim (2011a).