**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 *P* and *P* effects (see Fig. 1.) as well as geometric imperfections, while the material nonlinearities are due to gradual yielding associated with residual stresses and flexure. The *P* effect results from the axial force acting through the relative displacement of the ends of the member, so it is referred to as a member chord rotation effect. The *P* effect is accounted in the second-order analysis by updating the configuration of the structure during the analysis process. The *P* effect is caused by the axial force acting through the lateral displacement of the member relative to its chord, so it is referred to as a member curvature effect. The *P* effect can be captured by using stability functions. Since the stability functions are derived from the closed-form solution of a beam-column subjected to end forces, they can accurately capture the *P* effect by using only one element per member. Another way to capture the *P* effect without using stability functions is to divide the member into many elements, and consequently, the *P* effect is transformed to the *P* effect.

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 *P* effect due to the out-of-plumbness is vanished by braces. The limitation of this method is that it requires the determination of the direction of geometric imperfections which is often difficult in a large structural system. In the equivalent notional load method, the geometric imperfections are replaced by equivalent notional lateral loads in proportion to the gravity loads acting on the story as described in Fig. 2(b). The drawback of this method is that the gravity loads must be known in advance to determine the notional loads before analysis. Another way to account for the geometric imperfections is to further reduce the tangent modulus. The advantage of this method over the explicit imperfection modeling and

© 2012 Thai, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

equivalent notional load methods is its convenience and simplicity because it eliminates the inconvenience of explicit imperfection modeling and equivalent notional load methods.

Advanced Analysis of Space Steel Frames 67

Residual stresses are created in the hot-rolled sections due to uneven cooling of the crosssection. 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 crosssection 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

**Figure 3.** Typical residual stress pattern for a hot-rolled wide flange section

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;

greatest refinement.

2011b; 2011c; 2011d; 2012).

**Figure 1.** The *P* and *P* effects of a beam-column

**Figure 2.** Geometric imperfection methods

Residual stresses are created in the hot-rolled sections due to uneven cooling of the crosssection. 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 crosssection 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.

66 Advances in Computational Stability Analysis

**Figure 1.** The *P*

P

<sup>L</sup> =L/1000

Braced member

P

0.004P

Braced member

L

**Figure 2.** Geometric imperfection methods

and *P* effects of a beam-column

equivalent notional load methods is its convenience and simplicity because it eliminates the inconvenience of explicit imperfection modeling and equivalent notional load methods.

(a) Explicit imperfection modeling

(b) Equivalent notional load

P

Unbraced member

P

0.002P

Unbraced member

L

=L/500

L

**Figure 3.** Typical residual stress pattern for a hot-rolled wide flange section

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).

(a) Due to residual stress (b) Due to flexure

Advanced Analysis of Space Steel Frames 69

(4)

(7)

(9)

Using <sup>2</sup> *k P EI* / , Eq. (1) is rewritten as

The general solution of Eq. (2) is

and rotation *y* is given as

The end rotation *A*

<sup>2</sup> *MA M M B A y kP <sup>x</sup>*

1 2 2 2 sin cos *MM M <sup>A</sup> B A y C kx C kx <sup>x</sup>*

and <sup>2</sup> <sup>2</sup>

sin sin *<sup>A</sup> <sup>B</sup> kL x x*

sin sin *<sup>A</sup> <sup>B</sup>*

1 cos 1 1 1 1 <sup>0</sup> (a) sin sin

1 2 2 1 *A A B B*

(a) 2 2cos sin

(b) 2 2cos sin

(6)

1 1 1 1 cos 1 (b) sin sin

(8)

(5)

The constants *C*1 and *C*2 are determined using the boundary conditions *y yL* 0 0

sin

1 cos 1 1 sin cos 1 sin

*y kx kx M kx M EIk kL L EIk kL L*

1 cos 1 11 1 cos sin cos

*y kx kx M kx M EIk kL kL EIk kL kL*

1 2

Substituting Eq. (4) into Eq. (3), the deflection *y* can be written as

*kL*

where 1 *S* and 2 *S* are the stability functions defined as

1

*S*

2

*S*

compressive, the stability functions are redefined as

 and *<sup>B</sup>* 

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

cos

2 2

can be obtained as

*A A B*

*kL y MM EIk kL kL EIk kL kL*

*kL y L <sup>M</sup> <sup>M</sup> EIk kL kL EIk kL kL*

> *M SS EI M SS L*

sin cos

*kL kL kL kL*

*kL kL kL*

sin

*kL kL kL*

*kL kL kL*

<sup>1</sup> *S* and 2 *S* account for the coupling effect between axial force and bending moments of the beam-column member. For members subjected to an axial force that is tensile rather than

*B A B*

*MA <sup>B</sup> kL M <sup>C</sup> EIk kL*

*EIL EI*

(2)

*EILk EIk*

(3)

*MA <sup>C</sup> EIk*
