**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

$$M\_{ext} = M\_A + Py - \frac{M\_A + M\_B}{L}x = -EIy'' \tag{1}$$

where *E* , *I* , and *L* are the elastic modulus, moment of inertia, and length of an element, respectively.

**Figure 5.** Beam-column with double-curvature bending

Advanced Analysis of Space Steel Frames 69

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

$$y'' + k^2 P = \frac{M\_A + M\_B}{EIL} x - \frac{M\_A}{EI} \tag{2}$$

The general solution of Eq. (2) is

68 Advances in Computational Stability Analysis

**Figure 4.** Gradual yielding of steel member

external moment acting on the cut section is

P

P

**Figure 5.** Beam-column with double-curvature bending

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

*ext A*

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

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

*M M M M Py x EIy <sup>L</sup>*

where *E* , *I* , and *L* are the elastic modulus, moment of inertia, and length of an element,

MA MB

L EI = Constant <sup>A</sup>

MA M=-EIy''

x

(MA+ MB) L

*A B*

B

<sup>P</sup><sup>y</sup>

(1)

P

x

**2. Advanced analysis** 

respectively.

$$y = C\_1 \sin kx + C\_2 \cos kx + \frac{M\_A + M\_B}{E \text{IL}} x - \frac{M\_A}{E \text{I} k^2} \tag{3}$$

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

$$\mathbf{C}\_1 = -\frac{M\_A \cos kL + M\_B}{Elk^2 \sin kL} \text{ and } \mathbf{C}\_2 = \frac{M\_A}{Elk^2} \tag{4}$$

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

$$y = -\frac{1}{Elk^2} \left[ \frac{\cos kL}{\sin kL} \sin kx - \cos kx - \frac{x}{L} + 1 \right] M\_A - \frac{1}{Elk^2} \left[ \frac{1}{\sin kL} \sin kx - \frac{x}{L} \right] M\_B \tag{5}$$

and rotation *y* is given as

$$y' = -\frac{1}{Elk} \left[ \frac{\cos kL}{\sin kL} \cos kx + \sin kx - \frac{1}{kL} \right] M\_A - \frac{1}{Elk} \left[ \frac{1}{\sin kL} \cos kx - \frac{1}{kL} \right] M\_B \tag{6}$$

The end rotation *A* and *<sup>B</sup>* can be obtained as

$$\begin{aligned} \theta\_A = \mathbf{y}'(0) &= -\frac{1}{Elk} \left[ \frac{\cos kL}{\sin kL} - \frac{1}{kL} \right] M\_A - \frac{1}{Elk} \left[ \frac{1}{\sin kL} - \frac{1}{kL} \right] M\_B & & \text{(a)}\\ \theta\_B = \mathbf{y}'(L) &= -\frac{1}{Elk} \left[ \frac{1}{\sin kL} - \frac{1}{kL} \right] M\_A - \frac{1}{Elk} \left[ \frac{\cos kL}{\sin kL} - \frac{1}{kL} \right] M\_B & & \text{(b)} \end{aligned} \tag{7}$$

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

$$
\begin{Bmatrix} M\_A \\ M\_B \end{Bmatrix} = \frac{EI}{L} \begin{bmatrix} \mathbf{S}\_1 & \mathbf{S}\_2 \\ \mathbf{S}\_2 & \mathbf{S}\_1 \end{bmatrix} \begin{Bmatrix} \theta\_A \\ \theta\_B \end{Bmatrix} \tag{8}
$$

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

 1 sin cos (a) 2 2cos sin sin *kL kL kL kL S kL kL kL kL kL kL* (9)

$$S\_2 = \frac{kL\left(kL - \sin kL\right)}{2 - 2\cos kL - kL\sin kL} \tag{b}$$

<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 compressive, the stability functions are redefined as

$$\begin{aligned} S\_1 &= \frac{kL\left(kL\cosh kL - \sinh kL\right)}{2 - 2\cosh kL - kL\sinh kL} \\ S\_2 &= \frac{kL\left(\sinh kL - kL\right)}{2 - 2\cosh kL - kL\sinh kL} \end{aligned} \tag{10}$$

Advanced Analysis of Space Steel Frames 71

(13)

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

system and its component members.

modulus *<sup>t</sup> E* can be written as

*2.2.2. Gradual yielding due to flexure* 

be noted that only a simple relationship for

when 0.5 *<sup>y</sup> P P* .

parabolic function

*2.2.1. Gradual yielding due to residual stresses* 

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

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 *E* (instead of moment of inertia *I* ) is reduced to account for the reduction of the elastic portion of the cross-section since the reduction of the elastic modulus is easier to implement than a new moment of inertia for every different section. The rate of reduction in stiffness is different in the weak and strong direction, but this is not considered since the dramatic degradation of weak-axis stiffness is compensated for by the substantial weak-axis plastic strength. This simplification makes the present method more practical. The CRC tangent

1.0 for 0.5 (a)

*t y*

*E E PP*

*y y t y*

*P P E PP*

4 1 for 0.5 (b)

*t y y*

*P P <sup>E</sup> E P PP*

0 for (c)

Equation (13) is plotted in Fig. 6. The tangent modulus *<sup>t</sup> E* is reduced from the elastic value

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

calibration with plastic zone solutions of simple portal frames and beam-columns. It should

representing the gradual stiffness degradation is obtained based on a

is required to describe the degradation in

Eqs. (9) and (10) are indeterminate when the axial force is zero (i.e. 0 *kL* ). To overcome this problem, the following simplified equations are used to approximate the stability functions when the axial force in the member falls within the range of -2.0 ≤ ≤ 2.0

$$\begin{aligned} S\_1 &= 4 + \frac{2\pi^2 \rho}{15} - \frac{(0.01\rho + 0.543)\rho^2}{4 + \rho} - \frac{(0.004\rho + 0.285)\rho^2}{8.183 + \rho} \\ S\_2 &= 2 - \frac{\pi^2 \rho}{30} + \frac{(0.01\rho + 0.543)\rho^2}{4 + \rho} - \frac{(0.004\rho + 0.285)\rho^2}{8.183 + \rho} \end{aligned} \tag{11}$$

where <sup>2</sup> 2 2 / ( )/ *<sup>e</sup> P P P EI L kL* . For most practical applications, it gives excellent correlation to the "exact" expressions given by Eqs. (9) and (10). However, for other than the range of -2.0 ≤ ≤ 2.0, the conventional stability functions in Eqs. (9) and (10) should be used. The incremental member force and deformation relationship of a three-dimensional beam-column element under axial force and end moments can be written as

$$\begin{aligned} \left\{ \begin{array}{c} \text{A}P\\ \Delta M\_{yA}\\ \Delta M\_{yB}\\ \Delta M\_{zA}\\ \Delta M\_{zB}\\ \Delta M\_{zB}\\ \Delta T \end{array} \right\} &= \begin{bmatrix} \frac{EA}{L} & 0 & 0 & 0 & 0 & 0\\ 0 & S\_{1y}\frac{EI\_y}{L} & S\_{2y}\frac{EI\_y}{L} & 0 & 0 & 0\\ 0 & S\_{2y}\frac{EI\_y}{L} & S\_{1y}\frac{EI\_y}{L} & 0 & 0 & 0\\ 0 & 0 & 0 & S\_{1z}\frac{EI\_z}{L} & S\_{2z}\frac{EI\_z}{L} & 0\\ 0 & 0 & 0 & S\_{2z}\frac{EI\_z}{L} & S\_{1z}\frac{EI\_z}{L} & 0\\ 0 & 0 & 0 & S\_{2z}\frac{EI\_z}{L} & S\_{1z}\frac{EI\_z}{L} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{GI\_z}{L} \end{bmatrix} \begin{Bmatrix} \text{A}\delta\\ \Delta\theta\\ \Delta\theta\_{yA}\\ \Delta\theta\_{zA}\\ \Delta\theta\_{zB}\\ \Delta\theta\_{zB}\\ \Delta\theta \end{Bmatrix} \tag{12}$$

where *P* , *MyA* , *MyB* , *MzA* , *MzB* , and *T* are the incremental axial force, end moments with respect to *y* and *z* axes, and torsion, respectively; , *yA* , *yB* , *zA* , *zB* , and are the incremental axial displacement, the end rotations, and the angle of twist, respectively; 1*<sup>n</sup> S* and 2*<sup>n</sup> S* are stability functions with respect to *n* axis *n yz* , given in Eqs. (9) and (10); and *EA* , *<sup>n</sup> EI* , and *GJ* denote the axial, bending, and torsional stiffness, respectively.

#### **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*

70 Advances in Computational Stability Analysis

1

 

 

where <sup>2</sup> 2 2 / ( )/ *<sup>e</sup>*

 *P P P EI L kL* 

> *EA L*

*P S S*

*M EI EI S S <sup>M</sup> L L*

*S*

2

other than the range of -2.0 ≤

*S*

written as

*zB* 

, and

stiffness, respectively.

1

*S*

2

*S*

(a) 2 2cosh sinh

(10)

≤ 2.0

 

(12)

(11)

*kL kL kL*

(b) 2 2cosh sinh

*kL kL kL*

Eqs. (9) and (10) are indeterminate when the axial force is zero (i.e. 0 *kL* ). To overcome this problem, the following simplified equations are used to approximate the stability

2 2 2

<sup>2</sup> (0.01 0.543) (0.004 0.285) <sup>4</sup> (a) 15 4 8.183

1 2

*S S*

*z z*

*z z z z*

> *GJ L*

> > , *yA* , *yB* , *zA* ,

00000

0 000

 

 

0 000

*yA yA y y y y yB yB zA z z zA*

000 0

*zB zB*

000 0

where *P* , *MyA* , *MyB* , *MzA* , *MzB* , and *T* are the incremental axial force, end

twist, respectively; 1*<sup>n</sup> S* and 2*<sup>n</sup> S* are stability functions with respect to *n* axis *n yz* , given in Eqs. (9) and (10); and *EA* , *<sup>n</sup> EI* , and *GJ* denote the axial, bending, and torsional

00000

2 1

are the incremental axial displacement, the end rotations, and the angle of

*S S L L*

 

 

. For most practical applications, it gives

≤ 2.0, the conventional stability functions in Eqs. (9) and

(0.01 0.543) (0.004 0.285) <sup>2</sup> (b) 30 4 8.183

2 2 2

excellent correlation to the "exact" expressions given by Eqs. (9) and (10). However, for

(10) should be used. The incremental member force and deformation relationship of a three-dimensional beam-column element under axial force and end moments can be

cosh sinh

*kL kL kL kL*

*kL kL kL*

sinh

functions when the axial force in the member falls within the range of -2.0 ≤

1 2

*y y y y*

*EI EI*

*L L*

2 1

*M EI EI*

*M L L T EI EI*

moments with respect to *y* and *z* axes, and torsion, respectively;

 

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 *E* (instead of moment of inertia *I* ) is reduced to account for the reduction of the elastic portion of the cross-section since the reduction of the elastic modulus is easier to implement than a new moment of inertia for every different section. The rate of reduction in stiffness is different in the weak and strong direction, but this is not considered since the dramatic degradation of weak-axis stiffness is compensated for by the substantial weak-axis plastic strength. This simplification makes the present method more practical. The CRC tangent modulus *<sup>t</sup> E* can be written as

$$E\_t = 1.0E \quad \text{for} \quad P \le 0.5P\_y \tag{a}$$

$$E\_t = 4\frac{P}{P\_y}\left(1 - \frac{P}{P\_y}\right)E \quad \text{for} \quad 0.5P\_y < P \le P\_y \tag{b} \tag{13}$$

$$E\_t = 0 \quad \text{for} \ P > P\_y \tag{c}$$

Equation (13) is plotted in Fig. 6. The tangent modulus *<sup>t</sup> E* is reduced from the elastic value when 0.5 *<sup>y</sup> P P* .

#### *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 representing the gradual stiffness degradation is obtained based on a calibration with plastic zone solutions of simple portal frames and beam-columns. It should be noted that only a simple relationship for is required to describe the degradation in

stiffness associated with flexure. Although more complicated expressions for can be proposed, simple expression for is needed for keeping the analysis model simple and straightforward.

Advanced Analysis of Space Steel Frames 73

, the element stiffness is

1.0 , the member forces will

1.0 ). If the force point moves beyond the

0.5 , the element

(15)

88 22 for (a) 99 99

2 2 for (b) 2 9 <sup>9</sup>

2 2 4 22 62 42 3.5 3.0 4.5 *z y z y zy*

where / *<sup>y</sup> p PP* , / *m MM z z pz* (strong-axis), / *m MM y y py* (weak-axis); Py, Myp, Mzp are

*p m m pm pm mm* (16)

*yz yz*

*y z y z*

*pm m pm m*

*<sup>p</sup> mm p m m*

axial load, and plastic moment capacity of the cross-section about *y* and *z* axes.

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

initial yield surface and inside the full yield surface 0.5 1.0

When member forces violate the plastic strength surface

reduced to account for the effect of plastification at the element end. The reduction of element stiffness is assumed to vary according to the parabolic function in the Eq. (15b).

be scaled down to move the force point return the yield surface based on incremental-

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

**Figure 8.** Plastification surface

iterative scheme.

remains fully elastic (i.e. no stiffness reduction,

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

**Figure 6.** Stiffness reduction due to residual stress

**Figure 7.** Stiffness degradation function

The value of parabolic function is equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed. The parabolic function can be expressed as (see Fig. 7.)

$$\begin{aligned} \eta &= 1.0 \text{ for } \ a \le 0.5\\ \eta &= 4a \begin{pmatrix} 1 - a \end{pmatrix} \text{ for } 0.5 < a \le 1.0 \\ \eta &= 0 \text{ for } a > 1 \end{aligned} \tag{14}$$

where is the force-state parameter which can be expressed by AISC-LRFD or modified Orbison yield surfaces as (seeFig. 8.).

For AISC-LRFD yield surface (AISC, 2005)

#### Advanced Analysis of Space Steel Frames 73

$$\alpha = p + \frac{8}{9}m\_y + \frac{8}{9}m\_z \quad \text{for} \quad p \ge \frac{2}{9}m\_y + \frac{2}{9}m\_z \tag{a}$$

$$\gamma \quad \gamma \quad \gamma \quad \tag{15}$$

$$\alpha = \frac{p}{2} + m\_y + m\_z \quad \text{for} \quad p < \frac{2}{9}m\_y + \frac{2}{9}m\_z \tag{b}$$

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

72 Advances in Computational Stability Analysis

proposed, simple expression for

**Figure 6.** Stiffness reduction due to residual stress

**Figure 7.** Stiffness degradation function

Orbison yield surfaces as (seeFig. 8.).

For AISC-LRFD yield surface (AISC, 2005)

a plastic hinge is formed. The parabolic function

 

The value of parabolic function

where

straightforward.

stiffness associated with flexure. Although more complicated expressions for

is needed for keeping the analysis model simple and

is equal to 1.0 when the element is elastic, and zero when

can be expressed as (see Fig. 7.)

(14)

 

 

 

 

1.0 for 0.5 (a) 4 1 for 0.5 1.0 (b)

0 for 1 (c)

is the force-state parameter which can be expressed by AISC-LRFD or modified

can be

$$\alpha = p^2 + m\_z^2 + m\_y^4 + 3.5p^2m\_z^2 + 3.0p^6m\_y^2 + 4.5m\_z^4m\_y^2 \tag{16}$$

where / *<sup>y</sup> p PP* , / *m MM z z pz* (strong-axis), / *m MM y y py* (weak-axis); Py, Myp, Mzp are axial load, and plastic moment capacity of the cross-section about *y* and *z* axes.

**Figure 8.** Plastification surface

When the force point moves inside or along the initial yield surface 0.5 , the element remains fully elastic (i.e. no stiffness reduction, 1.0 ). If the force point moves beyond the initial yield surface and inside the full yield surface 0.5 1.0 , the element stiffness is reduced to account for the effect of plastification at the element end. The reduction of element stiffness is assumed to vary according to the parabolic function in the Eq. (15b). When member forces violate the plastic strength surface 1.0 , the member forces will be scaled down to move the force point return the yield surface based on incrementaliterative scheme.

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

$$
\begin{bmatrix}
\Delta P\\ \Delta M\_{ya} \\ \Delta M\_{yB} \\ \Delta M\_{zB} \\ \Delta M\_{zB} \\ \Delta T
\end{bmatrix} = \begin{bmatrix}
\underline{E}\_t A & 0 & 0 & 0 & 0 & 0 \\
0 & k\_{ijy} & k\_{ijy} & 0 & 0 & 0 \\
0 & k\_{ijy} & k\_{ijy} & 0 & 0 & 0 \\
0 & 0 & 0 & k\_{iz} & k\_{ijz} & 0 \\
0 & 0 & 0 & k\_{ijz} & k\_{ijz} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{GJ}{L}
\end{bmatrix} \begin{bmatrix}
\Delta \delta \\ \Delta \theta\_{yA} \\ \Delta \theta\_{yB} \\ \Delta \theta\_{zA} \\ \Delta \theta\_{zB} \\ \Delta \theta\_{zB}
\end{bmatrix} \tag{17}
$$

where

$$k\_{\dot{}\dot{}\dot{}\dot{y}} = \eta\_A (\mathcal{S}\_1 - \frac{\mathcal{S}\_2^2}{\mathcal{S}\_1} (1 - \eta\_B)) \frac{E\_t I\_y}{L} \tag{a}$$

$$k\_{ijy} = \eta\_A \eta\_B S\_2 \frac{E\_t I\_y}{L} \tag{b}$$

$$k\_{jjy} = \eta\_B (S\_1 - \frac{S\_2^2}{S\_1}(1 - \eta\_A)) \frac{E\_t I\_y}{L} \tag{c}$$

$$k\_{\rm iiz} = \eta\_A (S\_3 - \frac{S\_4^2}{S\_3}(1 - \eta\_B)) \frac{E\_t I\_z}{L} \tag{d}$$

$$k\_{ijz} = \eta\_A \eta\_B S\_4 \frac{E\_t I\_z}{L} \tag{e}$$

$$k\_{j\natural z} = \eta\_B (S\_3 - \frac{S\_4^2}{S\_3}(1 - \eta\_A)) \frac{E\_t I\_z}{L} \tag{f}$$

where *<sup>A</sup>* and *<sup>B</sup>* are the values of parabolic functions at the ends A and B, respectively.

#### **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 *m* fibers and each fiber is represented by its area *Ai* and coordinate location corresponding to its centroid , *i i y z* . The inelastic effects are captured by tracing the uniaxial stress-strain relationship of each fiber on the cross sections located at the selected integration points along the member length.

The incremental force and deformation relationship, Eq. (12), which accounts for the *P* effect can be rewritten in symbolic form as

$$
\begin{Bmatrix} \Delta F \end{Bmatrix} = \begin{bmatrix} K\_e \end{bmatrix} \begin{Bmatrix} \Delta d \end{Bmatrix} \tag{19}
$$

(18)

Advanced Analysis of Space Steel Frames 75

*<sup>T</sup>*

 *<sup>T</sup> yA yB zA zB <sup>d</sup>*

y

zi

<sup>z</sup> <sup>C</sup>

C

yi

Fiber

**Figure 9.** Fiber hinge model

*e*

*EA L*

A

z x

y

1 2

*y y y y*

*EI EI S S*

*L L EI EI S S*

*y y y y*

2 1

*L L <sup>K</sup>*

1 2

*EI EI S S L L EI EI S S L L*

*z z z z*

*z z z z*

> *GJ L*

B

x

(22)

00000

0 0 0 0

L

0 0 0 0

000 0

000 0

00000

2 1

*yA yB zA zB F PM M M M T* (20)

(21)

where

#### Advanced Analysis of Space Steel Frames 75

$$
\begin{bmatrix} \Delta F \end{bmatrix} = \begin{bmatrix} \Delta P & \Delta M\_{yA} & \Delta M\_{yB} & \Delta M\_{zA} & \Delta M\_{zB} & \Delta T \end{bmatrix}^T \tag{20}
$$

$$
\begin{bmatrix} \Delta d \end{bmatrix} = \begin{bmatrix} \Delta \delta & \Delta \theta\_{yA} & \Delta \theta\_{yB} & \Delta \theta\_{zA} & \Delta \theta\_{zB} & \Delta \phi \end{bmatrix}^T \tag{21}
$$

**Figure 9.** Fiber hinge model

74 Advances in Computational Stability Analysis

where

where *<sup>A</sup>* 

where

 and *<sup>B</sup>* 

00000

*L*

(b)

(e)

*<sup>e</sup>* (19)

(17)

(18)

0 000 0 000 0 00 0 0 00 0

*iiy ijy yA yA ijy jjy yB yB iiz ijz zA zA zB ijz jjz zB*

*t*

*M k k M k k*

*M k k M k k*

*P L*

*E A*

0 0000

*S E I*

*S L*

 

*S E I*

*S L*

 

*S L*

 

*S L*

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 *m* fibers and each fiber is represented by its area *Ai* and coordinate location corresponding to its centroid , *i i y z* . The inelastic effects are captured by tracing the uniaxial stress-strain relationship of each fiber on the cross sections located at the selected

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

*FKd*

 

( (1 )) (a)

*t y*

*t y*

*t z*

*t z*

are the values of parabolic functions at the ends A and B, respectively.

( (1 )) (c)

( (1 )) (d)

( (1 )) (f)

*T GJ*

2 2 1

1

*E I*

*t y*

*L*

*<sup>S</sup> E I k S*

3

*t z*

*L <sup>S</sup> E I k S*

3

*jjz B A*

*iiz A B*

*jjy B A*

*ijy A B*

*k S*

*k S*

 

*k S*

*ijz A B*

**2.3. Fiber model accounting for inelastic effects** 

integration points along the member length.

effect can be rewritten in symbolic form as

 

*E I k S*

*iiy A B*

$$\mathcal{K}\_{\varepsilon} = \begin{bmatrix} \frac{EA}{L} & 0 & 0 & 0 & 0 & 0\\ 0 & S\_{1y}\frac{EI\_y}{L} & S\_{2y}\frac{EI\_y}{L} & 0 & 0 & 0\\ 0 & S\_{2y}\frac{EI\_y}{L} & S\_{1y}\frac{EI\_y}{L} & 0 & 0 & 0\\ 0 & 0 & 0 & S\_{1z}\frac{EI\_z}{L} & S\_{2z}\frac{EI\_z}{L} & 0\\ 0 & 0 & 0 & S\_{2z}\frac{EI\_z}{L} & S\_{1z}\frac{EI\_z}{L} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{GI\_z}{L} \end{bmatrix} \tag{22}$$

in which the axial stiffness *EA* , bending stiffness *<sup>n</sup> EI* , and torsional stiffness *GJ* of the fiber element can be obtained as

$$EA = \sum\_{j=1}^{h} w\_j \left(\sum\_{i=1}^{m} E\_i A\_i\right)\_j \tag{a}$$

$$EI\_y = \sum\_{j=1}^{h} w\_j \left(\sum\_{i=1}^{m} E\_i A\_i z\_i^2\right)\_j \tag{b}$$

(23)

Advanced Analysis of Space Steel Frames 77

(27)

(29)

*xL xL*

sec *qk Q* (28)

*e q* (30)

(31)

(32)

0 0 0 /1 /0

1 0 0 0 00

() 0 / 1 / 0 0 0

<sup>1</sup>

1 11

*i ii m mm*

*k EAyz EAz EAz*

*m mm*

1 11

*i ii m mm*

*i ii*

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

> 1 1 2 2

*y z y z*

*y z*

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

*i*

*E*

is updated from the incremental fiber stress and incremental fiber strain as

... ... ... 1 *m m*

*i*

*i*

*e* 

Eq. (32) leads to updating of the element stiffness matrix *Ke* in Eq. (22) and section stiffness matrix sec *k* in Eq. (29) during the iteration process. Based on the new tangent

1 1

1 11

2

*i ii i iii i i i*

*EAy EAyz EA y*

*i iii i ii i ii*

*i i i i ii i i*

*EA y EAz EA*

where *B x* is the force interpolation function matrix given as

where sec *k* is the section stiffness matrix given as

sec

where is the linear geometric matrix given as follows

section deformation vector as

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

2

*Bx x L x L*

$$EI\_z = \sum\_{j=1}^{h} w\_j \left(\sum\_{i=1}^{m} E\_i A\_i y\_i^2\right)\_j \tag{c}$$

$$\mathcal{G}I = \sum\_{j=1}^{h} \boldsymbol{w}\_{j} \left[ \sum\_{i=1}^{m} \left( y\_{i}^{2} + z\_{i}^{2} \right) \mathbf{G}\_{i} A\_{i} \right]\_{j} \tag{4}$$

in which *h* is the total number of monitored sections along an element; *m* is the total number of fiber divided on the monitored cross-section; *wj* is the weighting factor of the *th <sup>j</sup>* section; *<sup>i</sup> <sup>E</sup>* and *Gi* are the tangent and shear modulus of *th <sup>i</sup>* fiber, respectively; *<sup>i</sup> <sup>y</sup>* and *<sup>i</sup> <sup>z</sup>* are the coordinates of *th <sup>i</sup>* fiber in the cross-section. The element stiffness matrix is evaluated numerically by the Gauss-Lobatto integration scheme since this method allows for two integration points to coincide with the end sections of the elements. Since inelastic behavior in beam elements often concentrates at the end of member, the monitoring of the end sections of the element is advantageous from the standpoint of accuracy and numerical stability. By contrast, the outermost integration points of the classical Gauss integration method only approach the end sections with increasing order of integration, but never coincide with the end sections and, hence, result in overestimation of the member strength (Spacone et al., 1996).

Section deformations are represented by three strain resultants: the axial strain along the longitudinal axis and two curvatures *<sup>z</sup>* and *<sup>y</sup>* with respect to *z* and *y* axes, respectively. The corresponding force resultants are the axial force *N* and two bending moments *Mz* and *My* . The section forces and deformations are grouped in the following vectors:

$$\text{Section force vector } \begin{Bmatrix} Q \end{Bmatrix} = \begin{bmatrix} M\_z & M\_y & N \end{bmatrix}^T \tag{24}$$

$$\text{Section deformation vector } \begin{Bmatrix} q \\ \end{Bmatrix} = \begin{bmatrix} \mathcal{X}\_z & \mathcal{X}\_y & \mathcal{E} \end{bmatrix}^T \tag{25}$$

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

$$\begin{Bmatrix} \Delta Q \end{Bmatrix} = \begin{bmatrix} B(x) \\ \end{bmatrix} \begin{Bmatrix} \Delta F \end{Bmatrix} \tag{26}$$

where *B x* is the force interpolation function matrix given as

76 Advances in Computational Stability Analysis

element can be obtained as

*<sup>i</sup> <sup>z</sup>* are the coordinates of *th*

(Spacone et al., 1996).

vectors:

longitudinal axis and two curvatures *<sup>z</sup>*

incremental element force vector *F* as

*th*

in which the axial stiffness *EA* , bending stiffness *<sup>n</sup> EI* , and torsional stiffness *GJ* of the fiber

2

(a)

(b)

(c)

(23)

*i* fiber, respectively; *<sup>i</sup> y* and

with respect to *z* and *y* axes,

(25)

*Q MMN z y* (24)

along the

(d)

*i* fiber in the cross-section. The element stiffness matrix is

2

*j i i ii j i j*

2 2

in which *h* is the total number of monitored sections along an element; *m* is the total number of fiber divided on the monitored cross-section; *wj* is the weighting factor of the

evaluated numerically by the Gauss-Lobatto integration scheme since this method allows for two integration points to coincide with the end sections of the elements. Since inelastic behavior in beam elements often concentrates at the end of member, the monitoring of the end sections of the element is advantageous from the standpoint of accuracy and numerical stability. By contrast, the outermost integration points of the classical Gauss integration method only approach the end sections with increasing order of integration, but never coincide with the end sections and, hence, result in overestimation of the member strength

1 1

*j ii j i j h m y j i ii j i j*

*h m*

*EA w E A*

1 1

*h m z j i ii j i j*

*EI w E A y*

*EI w E A z*

1 1

*GJ w y z G A*

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

respectively. The corresponding force resultants are the axial force *N* and two bending moments *Mz* and *My* . The section forces and deformations are grouped in the following

Section force vector *<sup>T</sup>*

Section deformation vector *<sup>T</sup>*

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

*z y <sup>q</sup>* 

 

*Q Bx F* (26)

 and *<sup>y</sup>* 

1 1

*<sup>j</sup>* section; *<sup>i</sup> <sup>E</sup>* and *Gi* are the tangent and shear modulus of *th*

*h m*

$$
\begin{bmatrix} B(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & \left(\mathbf{x} / L - 1\right) & \mathbf{x} / L & 0 \\ 0 & \left(\mathbf{x} / L - 1\right) & \mathbf{x} / L & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \tag{27}
$$

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

$$\begin{Bmatrix} \Delta q \end{Bmatrix} = \begin{bmatrix} k\_{\text{sec}} \end{bmatrix}^{-1} \begin{Bmatrix} \Delta Q \end{Bmatrix} \tag{28}$$

where sec *k* is the section stiffness matrix given as

$$\begin{aligned} \begin{bmatrix} \boldsymbol{k}\_{\text{sec}} \end{bmatrix} = \begin{bmatrix} \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{y}\_{i}^{2} & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{y}\_{i} \boldsymbol{z}\_{i} & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \left( -\boldsymbol{y}\_{i} \right) \\\sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{y}\_{i} \boldsymbol{z}\_{i} & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{z}\_{i}^{2} & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{z}\_{i} \\\sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \left( -\boldsymbol{y}\_{i} \right) & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \boldsymbol{z}\_{i} & \sum\_{i=1}^{m} \boldsymbol{E}\_{i} \mathbf{A}\_{i} \end{bmatrix} \end{aligned} \tag{29}$$

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

$$
\begin{Bmatrix}
\Delta e \\
\end{Bmatrix} = \begin{bmatrix}
\Gamma \\
\end{bmatrix} \begin{Bmatrix} \Delta q \\
\end{Bmatrix} \tag{30}
$$

where is the linear geometric matrix given as follows

$$
\begin{bmatrix}
\Gamma \\
\end{bmatrix} = \begin{bmatrix}
\dots & \dots & \dots \\
\end{bmatrix} \tag{31}
$$

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

$$E\_i = \frac{\Delta \sigma\_i}{\Delta e\_i} \tag{32}$$

Eq. (32) leads to updating of the element stiffness matrix *Ke* in Eq. (22) and section stiffness matrix sec *k* in Eq. (29) during the iteration process. Based on the new tangent

modulus of Eq. (32), the location of the section centroid is also updated during the incremental load steps to take into account the distribution of section plasticity. The section resisting forces are computed by summation of the axial force and biaxial bending moment contributions of all fibers as

$$\begin{aligned} \{\mathcal{Q}\_R\} = \begin{Bmatrix} M\_z \\ M\_y \\ M\_y \\ N \end{Bmatrix} = \left\{ \begin{array}{l} \sum\_{i=1}^m \sigma\_i A\_i \left(-y\_i\right) \\ \sum\_{i=1}^m \sigma\_i A\_i z\_i \\ \sum\_{i=1}^m \sigma\_i A\_i \\ \sum\_{i=1}^m \sigma\_i A\_i \end{array} \right\} \tag{33} \end{aligned} \tag{33}$$

Advanced Analysis of Space Steel Frames 79

(37)

(38)

(39)

2 2

*ii jj ij ii s ii jj ij ij s A ii jj ij s ii jj ij s A B ii jj ij ij s ii jj ij jj s B ii jj ij s ii jj ij s*

*k k k k A GL k k k k A GL*

2 2 2 2

*M k k k A GL k k k A GL M k k k k A GL k k k k A GL*

*EA*

*M C C M C C*

*M C C M C C*

*P L*

*iiy*

*C*

*C*

*C*

*C*

*C*

*C*

**2.5. Element stiffness matrix** 

*ijy*

*jjy*

*iiz*

*ijz*

*jjz*

beam-column element as

in which

2 2

The member force and deformation relationship can be extended for three-dimensional

*k k k A GL k k k A GL*

0 0 000

*L*

(d)

0 000 0 000 000 0 000 0

*iiy ijy yA yA ijy jjy yB yB iiz ijz zA zA zB ijz jjz zB*

00000

(a) <sup>2</sup>

(b) <sup>2</sup>

(c) <sup>2</sup>

(e) <sup>2</sup>

(f) <sup>2</sup>

*T GJ*

2

 

*iiy jjy ijy iiy sz*

*k k k k A GL*

*k k k A GL k k k k A GL*

*iiy jjy ijy sz iiy jjy ijy ijy sz*

2

2

*iiy jjy ijy sz iiy jjy ijy jjy sz*

*k k k A GL k k k k A GL*

2

*iiy jjy ijy sz iiz jjz ijz iiz sy*

*k k k A GL k k k k A GL*

2

*ijz sy*

*k A GL*

2

*iiz jjz ijz ijz sy*

*k k k A GL k k k k A GL*

*k k k k A GL*

2

*iiz jjz ijz sy*

where *Asy* and *Asz* are the shear areas with respect to *y* and *z* axes, respectively.

*k k k A GL*

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

*iiz jjz ijz sy iiz jjz ijz jjz sy*

*iiz jjz*

 

*k k*

#### **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

$$
\begin{Bmatrix}
\Delta\theta\_{\text{MA}}\\\Delta\theta\_{\text{MB}}\\\Delta\theta\_{\text{MB}}
\end{Bmatrix} = \begin{bmatrix}
k\_{\text{jj}} & -k\_{\text{ij}}\\k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ij}}^2 & k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ij}}^2\\-k\_{\text{ij}} & k\_{\text{ii}}\\\-k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ij}}^2 & k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ij}}^2 \\\end{bmatrix} \begin{Bmatrix}
\Delta\mathbf{M}\_A\\\Delta\mathbf{M}\_B
\end{Bmatrix} \tag{34}
$$

where *MA* and *MB* are the slope of the neutral axis due to bending moment. The flexibility matrix corresponding to shear deformation can be written as

$$
\begin{Bmatrix} \Delta\theta\_{SA} \\ \Delta\theta\_{SB} \end{Bmatrix} = \begin{bmatrix} 1 & 1 \\ \overline{GA\_S L} & \overline{GA\_S L} \\ 1 & 1 \\ \overline{GA\_S L} & \overline{GA\_S L} \end{bmatrix} \begin{Bmatrix} \Delta\mathcal{M}\_A \\ \Delta\mathcal{M}\_B \end{Bmatrix} \tag{35}
$$

where *GAS* and *L* are shear stiffness and length of the element, respectively. The total rotations at the two ends *A* and *B* are obtained by combining Eqs. (34) and (35) as

$$
\begin{Bmatrix} \Delta\theta\_A\\ \Delta\theta\_B \end{Bmatrix} = \begin{Bmatrix} \Delta\theta\_{MA} \\ \Delta\theta\_{MB} \end{Bmatrix} + \begin{Bmatrix} \Delta\theta\_{SA} \\ \Delta\theta\_{SB} \end{Bmatrix} \tag{36}
$$

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

#### Advanced Analysis of Space Steel Frames 79

$$
\begin{Bmatrix}
\Delta\mathcal{M}\_A\\\Delta\mathcal{M}\_B
\end{Bmatrix} = \begin{Bmatrix}
\frac{k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ii}}^2 + k\_{\text{ii}}A\_s\text{GL}}{k\_{\text{ii}} + k\_{\text{jj}} + 2k\_{\text{ij}} + A\_s\text{GL}} & \frac{-k\_{\text{ii}}k\_{\text{jj}} + k\_{\text{ii}}^2 + k\_{\text{ij}}A\_s\text{GL}}{k\_{\text{ii}} + k\_{\text{jj}} + 2k\_{\text{ij}} + A\_s\text{GL}}\\\frac{-k\_{\text{ii}}k\_{\text{jj}} + k\_{\text{ij}}^2 + k\_{\text{ij}}A\_s\text{GL}}{k\_{\text{ii}} + k\_{\text{jj}} + 2k\_{\text{ij}} + A\_s\text{GL}} & \frac{k\_{\text{ii}}k\_{\text{jj}} - k\_{\text{ij}}^2 + k\_{\text{jj}}A\_s\text{GL}}{k\_{\text{ii}} + k\_{\text{jj}} + 2k\_{\text{ij}} + A\_s\text{GL}}
\end{Bmatrix} \begin{Bmatrix}
\Delta\theta\_A\\\Delta\theta\_B\\\Delta\theta\_B
\end{Bmatrix} \tag{37}
$$

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

$$
\begin{bmatrix}
\Delta P\\ \Delta M\_{yA} \\ \Delta M\_{yB} \\ \Delta M\_{zB} \\ \Delta M\_{zB} \\ \Delta T
\end{bmatrix} = \begin{bmatrix}
\frac{EA}{L} & 0 & 0 & 0 & 0 & 0 \\
0 & C\_{\dot{i}\dot{y}} & C\_{\dot{i}\dot{y}} & 0 & 0 & 0 \\
0 & C\_{\dot{i}\dot{y}} & C\_{\dot{i}\dot{y}} & 0 & 0 & 0 \\
0 & 0 & 0 & C\_{\dot{i}\dot{z}} & C\_{\dot{i}\dot{z}} & 0 \\
0 & 0 & 0 & C\_{\dot{i}\dot{z}} & C\_{\dot{i}\dot{z}} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{GJ}{L}
\end{bmatrix} \tag{38}
$$

in which

78 Advances in Computational Stability Analysis

contributions of all fibers as

**2.4. Shear deformation effect** 

 and *MB* 

inverting the flexibility matrix as

where *MA* 

modulus of Eq. (32), the location of the section centroid is also updated during the incremental load steps to take into account the distribution of section plasticity. The section resisting forces are computed by summation of the axial force and biaxial bending moment

1

*i z m R y i ii i m*

*Q M A z*

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.

*M*

*N*

The flexibility matrix can be obtained by inversing the flexural stiffness matrix as

*m*

1

1

*i*

2 2

*kk k kk k M k k M*

*jj ij MA ii jj ij ii jj ij A MB ij ii B ii jj ij ii jj ij*

*k k*

*kk k kk k*

1 1

*GA L GA L M*

1 1 *SA S S A SB B S S*

*GA L GA L*

where *GAS* and *L* are shear stiffness and length of the element, respectively. The total

*A MA SA B MB SB*

 

 

The basic force and deformation relationship including shear deformation is derived by

rotations at the two ends *A* and *B* are obtained by combining Eqs. (34) and (35) as

2 2

are the slope of the neutral axis due to bending moment. The

*M*

(36)

(33)

(34)

(35)

*ii i*

*A y*

*i i*

*A*

flexibility matrix corresponding to shear deformation can be written as

$$\mathbf{C}\_{\dot{\imath}\dot{\jmath}\mathbf{y}} = \frac{k\_{\dot{\imath}\dot{\jmath}}k\_{\dot{\jmath}\dot{\jmath}\mathbf{y}} - k\_{\dot{\imath}\dot{\jmath}\mathbf{y}}^2 + k\_{\dot{\imath}\dot{\jmath}\mathbf{y}}A\_{sz}GL}{k\_{\dot{\imath}\dot{\jmath}\mathbf{y}} + k\_{\dot{\jmath}\dot{\jmath}\mathbf{y}} + 2k\_{\dot{\imath}\dot{\jmath}\mathbf{y}} + A\_{sz}GL} \tag{a}$$

$$\mathbf{C}\_{\dot{\imath}\dot{\jmath}\mathbf{y}} = \frac{-k\_{\dot{\imath}\dot{\jmath}\mathbf{y}}k\_{\dot{\jmath}\dot{\jmath}\mathbf{y}} + k\_{\dot{\imath}\dot{\jmath}\mathbf{y}}^2 + k\_{\dot{\imath}\dot{\jmath}\mathbf{y}}A\_{sz}GL}{k\_{\dot{\imath}\dot{\jmath}\mathbf{y}} + k\_{\dot{\jmath}\dot{\jmath}\mathbf{y}} + 2k\_{\dot{\imath}\dot{\jmath}\mathbf{y}} + A\_{sz}GL} \tag{b}$$

$$\mathbf{C}\_{jjy} = \frac{k\_{\rm{ijy}}k\_{\rm{jjy}} - k\_{\rm{ijy}}^2 + k\_{\rm{jjy}}A\_{sz}GL}{k\_{\rm{ijy}} + k\_{\rm{jjy}} + 2k\_{\rm{ijy}} + A\_{sz}GL} \tag{39}$$

$$\mathbf{C}\_{\hat{i}\hat{z}} = \frac{k\_{\hat{i}\hat{z}}k\_{\hat{j}\hat{z}} - k\_{\hat{i}\hat{z}}^2 + k\_{\hat{i}\hat{z}}A\_{s\hat{y}}GL}{k\_{\hat{i}\hat{z}} + k\_{\hat{j}\hat{z}} + 2k\_{\hat{i}\hat{z}} + A\_{s\hat{y}}GL} \tag{d}$$

$$\mathbf{C}\_{ijz} = \frac{-k\_{i\bar{i}z}k\_{j\bar{j}z} + k\_{i\bar{j}z}^2 + k\_{i\bar{j}z}A\_{s\bar{y}}GL}{k\_{i\bar{i}z} + k\_{j\bar{j}z} + 2k\_{i\bar{j}z} + A\_{s\bar{y}}GL} \tag{e}$$

$$\mathbf{C}\_{j\dot{\jmath}z} = \frac{k\_{\dot{\imath}\dot{\imath}z}k\_{j\dot{\jmath}z} - k\_{\dot{\jmath}\dot{z}}^2 + k\_{j\dot{\jmath}z}A\_{s\dot{y}}GL}{k\_{i\dot{\imath}z} + k\_{j\dot{\jmath}z} + 2k\_{i\dot{\jmath}z} + A\_{s\dot{y}}GL} \tag{f}$$

where *Asy* and *Asz* are the shear areas with respect to *y* and *z* axes, respectively.

#### **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

$$\begin{array}{cc} \{f\_n\} = \left[ \begin{array}{c} T \end{array} \right]\_{6 \times 12}^{\mathrm{T}} \{F\} & \text{ (a)}\\ \{d\} = \left[ \begin{array}{c} T \end{array} \right]\_{6 \times 12} \{d\_L\} & \text{ (b)} \end{array} \tag{40} $$

Advanced Analysis of Space Steel Frames 81

(47)

*n nL K d* (43)

*s sL K d* (45)

(46)

2 2

*fL L K d* (48)

*fLns f f* (49)

*KK K n s* (50)

12 12 6 12 6 6 6 12 [ ] [] [ ] [] *<sup>T</sup> K TKT n e* (44)

Using the transformation matrix, the nodal force and nodal displacement relationship of

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

*s s*

*G G*

*s s*

0 / / 000

*M M LM M L*

*zA zB yA yB*

0 0 0 000 0 0 0 000 0 0 0 000

/ / 0 000 / 0 / 000

*f*

*f*

where *Ks* is the element stiffness matrix due to member sway expressed as

12 12

*G MML P L*

*K*

2 2

*M M L PL*

 

*yA yB <sup>s</sup>*

column element obtained as

*zA zB*

*s T*

*G G*

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

where *Kn* is the element stiffness matrix expressed as

member sway to the member end displacements can be related as

element may be written as

in which

where

where *f <sup>n</sup>* and *dL* are the nodal force and nodal displacement vectors of the element expressed as

$$\begin{aligned} \{f\_n\}^T &= \{r\_{n1} \quad r\_{n2} \quad r\_3 \quad r\_4 \quad r\_5 \quad r\_6 \quad r\_7 \quad r\_8 \quad r\_9 \quad r\_{10} \quad r\_{11} \quad r\_{12}\} & & \tag{41} \\ \{d\_L\}^T &= \{d\_1 \quad d\_2 \quad d\_3 \quad d\_4 \quad d\_5 \quad d\_6 \quad d\_7 \quad d\_8 \quad d\_9 \quad d\_{10} \quad d\_{11} \quad d\_{12}\} & & \tag{5} \end{aligned} \tag{41}$$

and *F* and *d* are the basic member force and displacement vectors given in Eqs. (20) and (21), respectively. 6 12 *<sup>T</sup>* is a transformation matrix written as

**Figure 10.** Force and displacement notations

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

$$\begin{Bmatrix} f\_n \end{Bmatrix} = \begin{bmatrix} K\_n \end{bmatrix} \begin{Bmatrix} d\_L \end{Bmatrix} \tag{43}$$

where *Kn* is the element stiffness matrix expressed as

$$\begin{bmatrix} \mathbf{K}\_n \end{bmatrix}\_{\mathbf{1} \mathbf{2} \times \mathbf{1} \mathbf{2}} = \mathbf{[}T\mathbf{]}\_{6 \times \mathbf{12}}^{\mathbf{I}} \mathbf{[}K\_e\big|\_{6 \times 6} \mathbf{[}T\big|\_{6 \times \mathbf{12}} \tag{44}$$

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

$$\begin{Bmatrix} f\_s \end{Bmatrix} = \begin{bmatrix} K\_s \end{bmatrix} \begin{Bmatrix} d\_L \end{Bmatrix} \tag{45}$$

where *Ks* is the element stiffness matrix due to member sway expressed as

$$
\begin{bmatrix} \mathbf{K}\_s \end{bmatrix}\_{\text{12}\times\text{12}} = \begin{bmatrix} \begin{bmatrix} \mathbf{G}\_s \end{bmatrix} & -\begin{bmatrix} \mathbf{G}\_s \end{bmatrix} \\\\ -\begin{bmatrix} \mathbf{G}\_s \end{bmatrix}^T & \begin{bmatrix} \mathbf{G}\_s \end{bmatrix} \end{bmatrix} \tag{46}
$$

in which

80 Advances in Computational Stability Analysis

*T n nn T L*

and (21), respectively. 6 12

*T*

6 12

**Figure 10.** Force and displacement notations

*<sup>T</sup>*

where *f*

expressed as

kinematic relationships can be expressed in symbolic form as

*n*

frame member is shown in Fig. 10(b). By comparing the two figures, the equilibrium and

*L*

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

{} { } (a) {} { } (b)

and *F* and *d* are the basic member force and displacement vectors given in Eqs. (20)

is a transformation matrix written as

*L L L L*

*<sup>n</sup>* and *dL* are the nodal force and nodal displacement vectors of the element

1 0 0 0001 0 0 0 00 0 0 1/ 0 1 0 0 0 1/ 0 0 0 0 0 1/ 0 0 0 0 0 1/ 0 1 0 0 1/ 0 0 0 1 0 1/ 0 0 0 0 0 1/ 0 0 0 0 0 1/ 0 0 0 1 0 0 0 1000 0 0 100

*L L L L*

(41)

(a) (b)

(40)

(42)

 6 12 6 12

 

*fT F dT d* 

*f r r rrrrrrrr r r d dddddddddd d d*

*T*

$$\begin{bmatrix} \mathbf{C}\_s \\ \end{bmatrix} = \begin{bmatrix} 0 & \left(M\_{zA} + M\_{zB}\right)/L^2 & \left(M\_{yA} + M\_{yB}\right)/L^2 & 0 & 0 & 0\\ \left(M\_{zA} + M\_{zB}\right)/L^2 & P/L & 0 & 0 & 0 & 0\\ \left(M\_{yA} + M\_{yB}\right)/L^2 & 0 & P/L & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \tag{47}$$

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

$$\begin{Bmatrix} f\_L \end{Bmatrix} = \begin{bmatrix} K \\ \end{bmatrix} \begin{Bmatrix} d\_L \end{Bmatrix} \tag{48}$$

where

$$\left\{ f\_L \right\} = \left\{ f\_n \right\} + \left\{ f\_s \right\} \tag{49}$$

$$
\begin{bmatrix} K \\ \end{bmatrix} = \begin{bmatrix} K\_n \\ \end{bmatrix} + \begin{bmatrix} K\_s \\ \end{bmatrix} \tag{50}
$$

#### **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 *j* th iteration of the *i* th incremental step as

$$
\left[\boldsymbol{\mathcal{K}}\_{j-1}^{i}\right]\left[\boldsymbol{\Delta}\boldsymbol{D}\_{j}^{i}\right] = \boldsymbol{\mathcal{A}}\_{j}^{i}\left\{\hat{\boldsymbol{P}}\right\} + \left\{\boldsymbol{R}\_{j-1}^{i}\right\}\tag{51}
$$

Advanced Analysis of Space Steel Frames 83

(59)

(60)

*<sup>i</sup> Dj* denote the displacement

effect accurately and

,

is calculated as

*GSP*

iteration of the previous incremental step; and <sup>ˆ</sup> *<sup>i</sup> Dj* and

first example is to show how the stability functions capture the *P*

fibers on the cross-section are used in the fiber model.

**3.1. Elastic buckling of columns** 

shown in Fig. 11.

the *j* th iteration of the *i* th incremental step, as defined in Eqs. (52) and (53).

For the iterative step *j* 2 , the load increment parameter *<sup>i</sup>*

where <sup>1</sup> 1

**3. Numerical examples** 

*T*

ˆ ˆ

1 1 1 1 1 1 1

*<sup>T</sup> i i D D*

> *j*

*D D*

*<sup>T</sup> i i*

 

increments generated by the reference load and unbalanced force vectors, respectively, at

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

efficiently. The second one is to show how well the refined plastic hinge model and fiber hinge model predict the strength and behavior of frames. The last one is to demonstrate the capability of two proposed methods in predicting the strength and behavior of a large-scale twenty-story space frame. Five integration points along the length of a member and eighty

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 200,000 *E* MPa and 0.3

respectively. The buckling load of the columns is obtained using the load-deflection analysis. The geometric imperfection is modeled by equivalent notional lateral loads as

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

ˆ

*j i <sup>j</sup> <sup>T</sup> i i*

ˆ ˆ

*D D*

ˆ *<sup>i</sup> D* is the displacement increment generated by the reference load at the first

*j*

ˆ ˆ

*D D*

where 1 *<sup>i</sup> Kj* is the tangent stiffness matrix, *<sup>i</sup> Dj* is the displacement increment vector, *P*ˆ is the reference load vector, <sup>1</sup> *<sup>i</sup> Rj* is the unbalanced force vector, and *<sup>i</sup> j* is the load increment parameter. According to Batoz and Dhatt (1979), Eq. (51) can be decomposed into the following equations:

$$\left\{\mathbf{K}\_{j-1}^{i}\right\}\left\{\boldsymbol{\Delta}\hat{\mathbf{D}}\_{j}^{i}\right\}=\left\{\hat{\mathbf{P}}\right\}\tag{52}$$

$$\left\{ \left[ \boldsymbol{K}\_{j-1}^{i} \right] \left| \left\{ \Delta \overline{\boldsymbol{D}}\_{j}^{i} \right\} \right. \right. \right. = \left\{ \left. \boldsymbol{R}\_{j-1}^{i} \right\} \tag{53}$$

$$\mathcal{L}\left\{\Delta \boldsymbol{D}\_{j}^{i}\right\} = \mathcal{L}\_{j}^{i}\left\{\Delta \hat{\boldsymbol{D}}\_{j}^{i}\right\} + \left\{\Delta \overline{\boldsymbol{D}}\_{j}^{i}\right\} \tag{54}$$

Once the displacement increment vector *<sup>i</sup> Dj* is determined, the total displacement vector *<sup>i</sup> Dj* of the structure at the end of *<sup>j</sup>* th iteration can be accumulated as

$$\left\{D\_{\dot{j}}^{i}\right\} = \left\{D\_{\dot{j}-1}^{i}\right\} + \left\{\Delta D\_{\dot{j}}^{i}\right\} \tag{55}$$

The total applied load vector *i <sup>j</sup> P* at the *j* th iteration of the *i* th incremental step relates to the reference load vector *P*ˆ as

$$
\left\{P\_{\vec{j}}^{i}\right\} = \Lambda\_{\vec{j}}^{i} \left\{\hat{P}\right\} \tag{56}
$$

where the load factor *<sup>i</sup> j* can be related to the load increment parameter *<sup>i</sup> j* by

$$
\Lambda^i\_{\rangle} = \Lambda^i\_{\rangle - 1} + \mathcal{X}^i\_{\rangle} \tag{57}
$$

The load increment parameter *<sup>i</sup> j* is an unknown. It is determined from a constraint condition. For the first iterative step *j* 1 , the load increment parameter *<sup>i</sup> j* is determined based on the generalized stiffness parameter *GSP* as

$$
\mathcal{X}\_1^i = \mathcal{X}\_1^1 \sqrt{|GSP|} \tag{58}
$$

where <sup>1</sup> 1 is an initial value of load increment parameter, and the *GSP* is defined as

$$GSP = \frac{\left\{\Delta\hat{D}\_1^1\right\}^T \left\{\Delta\hat{D}\_1^1\right\}}{\left\{\Delta\hat{D}\_1^{i-1}\right\}^T \left\{\Delta\hat{D}\_1^i\right\}}\tag{59}$$

For the iterative step *j* 2 , the load increment parameter *<sup>i</sup> j* is calculated as

$$\mathcal{A}\_{\hat{j}}^{i} = -\frac{\left\{\Delta \hat{D}\_{1}^{i-1}\right\}^{T} \left\{\Delta \overline{D}\_{\hat{j}}^{i}\right\}}{\left\{\Delta \hat{D}\_{1}^{i-1}\right\}^{T} \left\{\Delta \hat{D}\_{\hat{j}}^{i}\right\}}\tag{60}$$

where <sup>1</sup> 1 ˆ *<sup>i</sup> D* is the displacement increment generated by the reference load at the first iteration of the previous incremental step; and <sup>ˆ</sup> *<sup>i</sup> Dj* and *<sup>i</sup> Dj* denote the displacement increments generated by the reference load and unbalanced force vectors, respectively, at the *j* th iteration of the *i* th incremental step, as defined in Eqs. (52) and (53).

## **3. Numerical examples**

82 Advances in Computational Stability Analysis

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 *j* th iteration of

> 1 1 <sup>ˆ</sup> *i ii i K D PR j jj j*

increment parameter. According to Batoz and Dhatt (1979), Eq. (51) can be decomposed into

 <sup>ˆ</sup> *i ii i D DD <sup>j</sup> jj j* 

<sup>ˆ</sup> *i i*

1 *ii i j j j*

1 1 1 *i* 

is an initial value of load increment parameter, and the *GSP* is defined as

*P*ˆ is the reference load vector, <sup>1</sup> *<sup>i</sup> Rj* is the unbalanced force vector, and *<sup>i</sup>*

*<sup>i</sup> Dj* of the structure at the end of *<sup>j</sup>* th iteration can be accumulated as

where the load factor *<sup>i</sup> j* can be related to the load increment parameter *<sup>i</sup>*

*j* 

based on the generalized stiffness parameter *GSP* as

condition. For the first iterative step *j* 1 , the load increment parameter *<sup>i</sup>*

*i*

is the tangent stiffness matrix,

Once the displacement increment vector

The total applied load vector

The load increment parameter *<sup>i</sup>*

the reference load vector *P*ˆ as

(51)

<sup>1</sup> ˆ ˆ *i i K DP j j* (52)

1 1 *i ii K DR j jj* (53)

<sup>1</sup> *ii i DD D jj j* (55)

*<sup>j</sup> P* at the *j* th iteration of the *i* th incremental step relates to

*<sup>i</sup> Dj* is determined, the total displacement vector

*j j P P* (56)

(57)

*GSP* (58)

is an unknown. It is determined from a constraint

*<sup>i</sup> Dj* is the displacement increment vector,

*j* 

(54)

*j* by

> *j*

is determined

is the load

**2.6. Solution algorithm** 

the *i* th incremental step as

the following equations:

where 1 *<sup>i</sup> Kj*

where <sup>1</sup> 1 

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 *P* effect accurately and efficiently. The second one is to show how well the refined plastic hinge model and fiber hinge model predict the strength and behavior of frames. The last one is to demonstrate the capability of two proposed methods in predicting the strength and behavior of a large-scale twenty-story space frame. Five integration points along the length of a member and eighty fibers on the cross-section are used in the fiber model.

#### **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 200,000 *E* MPa and 0.3 , respectively. The buckling load of the columns is obtained using the load-deflection analysis. The geometric imperfection is modeled by equivalent notional lateral loads as shown in Fig. 11.

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 *P* effect is transformed to the *P* effect, and hence, the results of cubic element are close to the obtained results.

Advanced Analysis of Space Steel Frames 85

P

2 *e* 2 *EI <sup>P</sup> L* 

P

0.004P

0.002P

2 <sup>2</sup> 4 *<sup>e</sup> EI <sup>P</sup> L* 

**Figure 12.** Load-displacement curves of steel columns

0

0

0.2

0.4

0.6

P/Pe

0.8

1

1.2

0.2

0.4

0.6

P/Pe

0.8

1

1.2

(a) Cantilever column

Present, 1 element SAP2000, 1 elements SAP2000, 2 elements SAP2000, 5 elements

0 50 100 150 200 250 300 350 400 Horizontal displacement at the free end (mm)

(b) Simply supported column

Present, 1 element/member SAP2000, 1 element/member SAP2000, 2 element/member SAP2000, 5 element/member

0 50 100 150 200 250 300 Horizontal displacement at the midspan (mm)

**Figure 11.** Steel columns
