WO2003076731A1

WO2003076731A1 - Non-linear analysis method of steel structures

Info

Publication number: WO2003076731A1
Application number: PCT/KR2002/000305
Authority: WO
Inventors: Seung-Eock Kim
Original assignee: Sejong University
Priority date: 2002-02-25
Filing date: 2002-02-25
Publication date: 2003-09-18

Abstract

This invention is regarding the practical software of nonlinear analysis for design of steel structures, in the field of civil and architectural engineering. To achieve the objectives mentioned above, stability functions accounting for nonlinearity, softening plastic hinge, modified stiffness matrix accounting for shear deformation of the member are used in this software. Using this analysis software, three-dimensional steel frames can be analyzed economically.

Description

[THE TITLE OF INVENTION]

NON-LINEAR ANALYSIS METHOD OF STEEL STRUCTURES

[TECHNICAL FIELD] This invention is regarding the practical software of nonlinear analysis for design of steel structures, in the field of civil and architectural engineering.

[BACKGROUND ART] Over the past twenty years, research has developed and validated various methods of performing second-order inelastic analysis on steel frames. Most of these studies can be categorized into two main types: (1) plastic zone and plastic hinge. In the plastic zone method, a frame member is subdivided into several finite elements, and the cross-section of each finite element is further subdivided into many fibers. Although the plastic zone solution is known as the "exact solution", it is not be used in daily engineering design, because it is too intensive in computation. In the plastic hinge method, only one beam-column element per member can capture the second-order effect. Therefore the plastic hinge method has a clear advantage over the plastic zone method. The plastic hinge method, however, generally over predicts the actual strength and stiffness of member.

Chen and Kim extended the modified plastic hinge analysis including gradual yielding and second-order effects. The analysis provides very good agreements with plastic zone method, but it can be used for only the two-dimensional structure.

Several nonlinear analysis methods for the three-dimensional structure have been developed by Orbison, Prakash, Powell, and Liew. Orbison's method is an elastic-plastic hinge analysis without considering shear deformations. The material nonlinearity is considered by the tangent modulus Et and the geometric nonlinearity is by a geometric stiffness matrix. DRAIN- 3DX was developed by Prakash and Powell. The material nonlinearity is considered by the stress-strain relationship of the fibers in a section. The geometric nonlinearity caused by axial force is considered by the use of the geometric stiffness matrix, but the nonlinearity caused by the interaction between the axial force and bending moments is not considered. Liew's analysis is a refined plastic hinge method considering shear deformations. Liew's method overestimates the strength and stiffness of the member subjected to significant axial force.

Thus a new analysis method having the advantage of existing analysis as well as reducing analysis time is required.

[DISCLOSURE OF INVENTION]

To achieve the objectives mentioned above, stability functions accounting for nonlinearity, softening plastic hinge, modified stiffness matrix accounting for shear deformation of the member are used in this software.

Stability Functions Accounting for Second-Order Effect

To capture second-order (large displacement) effects, stability functions are used to minimize modeling and solution time. Generally only one or two elements are needed per a member. The simplified stability functions reported by Chen and Lui (1992) are used here. Considering the prismatic beam-column element in Fig. 1, the incremental force- displacement relationship of this element may be written as M_ S_x S₂ 0 Θ_Λ

El

M_B S₂ S_! 0 θ

0 o

(1) where ι , °z - stability functions

^A , ^B = incremental end moments P = incremental axial force n a

A , ^B = incremental joint rotations ^e = incremental axial displacement

-4 , , L = area, moment of inertia, and length of beam-column element

E = modulus of elasticity.

The stability functions given by Eq. (1) may be written as π-sfp sin(π-sjp) - π²p cos(π-sfp) if P < 0

2-2 cos(π-s[p) - π-sfp s_n(π-sfp)

S = π²p cos .(π-sfp) - π-sfp siτ_h.(π-sfp) if P > 0 2-2 cosb.(π-sfp) + π-sfp smh(π-sfp)

(2)

(3) where P = ^p/(^{π EI}l^L ) _t P is positive in tension.

The force-displacement equation may be extended for three-dimensional beam-column element as

where P , ^yA , ^yB , ^ , -B , and, T are axial force, end moments

with respect to ^y V and ^z axes and torsion respectively. ° s , θ ^yA , θ ^yB , f ^zA ,

*B , and, Φ are the axial displacement, the joint rotations, and the angle

of twist, i , °2 _> 3 _{; anc}j ^ _are the stability functions with respect to ^y and ^z axes, respectively.

CRC Tangent Modulus Model Associated with Residual Stresses

The CRC tangent modulus concept is used to account for gradual yielding(due to residual stresses) along the length of axially loaded members between plastic hinges. 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-directions, but this is not considered since the dramatic degradation of weak-axis stiffness is compensated for by the substantial weak-axis plastic strength(Chen and Kim 1997). This simplification makes the present methods practical. From Chen and Lui(1986). the CRC E_t is written as E = 1.0E P ≤ 0.5P. for (5a)

P P

E_t = 4—E(l-—)

P y P y for P > 0.5P,

(5b)

Parabolic Function for Gradual Yielding due to Flexure

The tangent modulus model 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 bending. We shall introduce the softening plastic hinge model to represented the transition from elastic to zero stiffness associated with a developing hinge. When softening plastic hinges are active at both ends of an element, the slope-deflection equation may be expressed as

where

y = n_Aη_Bs_ ^E , (7b)

k j,jy. = η_B S₁ -^(l-η_A)) ^EtIy s_ L (7c)

El = η_Aη_Bs (7e)

The terms ^ and ^^B is a scalar parameter that allows for gradual inelastic stiffness reduction of the element associated with plastification at end A and B. This term is equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed. The parameter ^ is assumed to vary according to the parabolic function: ^{= 1}-° for a ≤ 0.5 (_8a) η = 4 (l - a) _{for a} > 0.5 (8b) where ^a is a force-state parameter that measures the magnitude of axial force and bending moment at the element end. The term ^a may be expressed by AISC-LRFD and Orbison, respectively:

Shear deformation

To account for transverse shear deformation effects in a beam- column element, the stiffness matrix may be modified as

where, Q _ _. kjj_yk_jjy k_ijy + k^A_szGL

'uy k_iiy +k_jiy +2k_yy +A_s GL (10a)

(10b)

Q _ h_y ^k _jjy -ki_jy +k_jjyA_szGL jy k -u_;;y +k_Αjy, +2k υ_;;y.A A GL

(10c)

(lOd)

(lOe)

Modification of Element Stiffness for the Presence of Plastic Hinges

If the state of forces at any cross-section equals or exceeds the plastic section capacity, a plastic hinge is formed where a slope discontinuity occurs. Therefore, the slope-deflection equation need to be modified to reflect the change in element behavior due to formation of plastic hinges at the element ends. If a plastic hinge forms in an element at end A, the slope-deflection equation from Eq. (14) is written as

where Δ ^ and AM x.pcA are the changes in plastic moment capacity at the end A as P changes. θ_yA and θ^ can be solved from the second and fourth rows of Eq. (11) as

C θ _{θ =} ^^M^ ^ijy^σyB

'yA c uy (12a)

Back-substituting Eq. (12a) and (12b) into the first, third, fifth, and sixth rows of Eq. (18), the modified slope-deflection equation can be expressed as

Eq. (13) represents the modified slope-deflection equation of a frame element with a plastic hinge formed at end A. If a plastic hinge forms at end B, a similar approach can be followed. The corresponding slope-deflection equations

If plastic hinges form at both ends, ^A _and ^UB _can be written in terms of the change in moment at the respective end of the element. The resulting slope-deflection equation is

where AM_ypcA , AM_ypcB AM_zpcΛ , and, AM_zpcB are the changes in the plastic moment capacity at the respective end of the members as P changes. Eq. (13) through (15) account for the presence of plastic hinge(s) at the element end(s). They may be written symbolically as

{/.} = [ -] ⁺ &} ⁺ {/_*} (16) where [K_eh] is the modified basic tangent stiffness matrix due to the presence of plastic hinge(s). [f_eh] is an equilibrium force correction vector that results from the change in moment capacity as P changes. Since the force-point movement remains on the plastic strength surface of a member, the plastic strength surface requirement of a section is not violated by the change of member forces after the full plastic strength of a cross-section is reached. In Fig. 4, after the force- point reaches point Q on the plastic strength surface and the member axial force increases, a corresponding moment decrease results in the member. The force-point then moves from Q to R. The moment at a plastic hinge increases from point R, which decreases the axial forces causing the force-point to relocate to Q.

Element Stiffness Matrix

The end forces and end displacements used in Eq. (9). The sign convention for the positive directions of element end forces and end displacements of a frame member is shown in Fig. 5(b). By comparing the two figures, we can express the equilibrium and kinematic relationships in symbolic form as

R} =P ₁₂K} ₍i₈)

\^{Jn )} _and t A _are g _end force and displacement vectors of a frame member expressed as IJnf ⁼ Vnl ^rn2 ^Vnl ^r A n5 ^Tιι6 il ^Yni ^Vn9 ^r«10 '^"nil ^r«12 ( 19a)

{d_L} = {dχ d₂ d₃ d₄ d₅ d₆ d₇ d_s d₉ d_x0 d_xx d_X2) (19b)

^^e> and ^ ^e> are the end force and displacement vectors in Eq. (13).

Lrl Je^χi2 i_{s a} transformation matrix written as -1 0 0 0 0 1 0 0 0 0 0 1_ 0 0 0 1 0 0 0 0 0 0

L L 1

0 0 0 0 0 0 J_ 0 1 0

L

[ 16x12 ~ 1_

0 0 0 1 0 0 0 0 0 L L 1

0 J_ 0 0 0 0 0 0 0 0 1

L 0 0 0 1 0 0 0 0 -1 0 0 (20)

Using the transformation matrix by equilibrium and kinematic relations, the force-displacement relationship of a frame member may be written as

{f_n} = [K_n]{d_L} (21)

I* "-^■ is the element stiffness matrix expressed as

Eq. (22) can be subgrouped as

where

where

E_tA _b _ C_ih + 2C_yz + C,_z c L L² L

C u_lhy, h " — C ^uz i = ^CUy J =

Eq. (23) is used to enforce no sidesway in the member. If the member is permitted to sway, an additional axial and shear forces will be induced in the member. We can relate this additional axial and shear forces due to a member sway to the member end displacements as

{/,} =[ ;]tø} (26)

where "- _; t ^L) _{^ anc}j L ^s] _are end force vector, end displacement vector, and the element stiffness matrix. They may be written as

U s ) ~ Vsl ^rs2 ^rs3 ^rs ^rs5 s. ^r_l ^r_% ^Ts_- ^r.l<- 'ill 'J12 J (27a)

(27b)

where

and

M_zA +M_{zB b =} M_yA +M_yB a - c = -

L² L² (29)

By combining Eq. (21) and Eq. (26), we obtain the general beam- column element force-displacement relationship as

where

{ } = {/,}+{/.} (31)

(32)

Numerical Implementation

The simple incremental method, as a direct nonlinear solution technique, is used in the analysis. Its numerical procedure is straightforward in concept and implementation. The advantage of this method is its computational efficiency. This is especially true when the structure is loaded into the inelastic region since tracing the hinge-by- hinge formation is required in the element stiffness formulation. For a finite increment size, this approach only approximates the nonlinear structural response, and equilibrium between the external applied loads and the internal element forces is not satisfied. To avoid this, an improved incremental method is used in this program. The applied load increment is automatically reduced to minimize the error when the change in the element stiffness parameter ( Aη ) exceeds a defined tolerance. To prevent plastic hinges from forming within a constant- stiffness load increment, load step sizes less than or equal to the specified increment magnitude are internally computed so plastic hinges form only after the load increment. Subsequent element stiffness formations account for the stiffness reduction due to the presence of the plastic hinges. For elements partially yielded at their ends, a limit is placed on the magnitude of the increment in the element end forces.

The applied load increment in the above solution procedure may be reduced for any of the following reasons: (1) Formation of new plastic hinge(s) prior to the full application of incremental loads; (2) The increment in the element nodal forces at plastic hinges is excessive; (3) Non-positive definiteness of the structural stiffness matrix. As the stability limit point is approached in the analysis, large step increments may overstep a limit point. Therefore, a smaller step size is used near the limit point to obtain accurate collapse displacements and second- order forces.

[BEST MODE FOR CARRING OUT THE INVENTION] Column with Three-Dimensional Degree of Freedom

A simply supported column with three-dimensional degree of freedom is shown in Fig. 6. W8X31 column of A36 steel is used for the analysis. The column strength calculated by the proposed analysis, Euler

solution, and DRAIN-3DX based on the slenderness parameter λ_c are

compared in Fig. 7.

The strength of the proposed analysis compares well with Euler's theoretical solution. The maximum error from the proposed analysis is 1.31% for the practical range of columns ( λ_c ≤ 2.0 ). However, DRAIN-

3DX produces the maximum error of 21.16%. The large error value is a result of not considering the interaction of the axial force and bending moments when considering geometric nonlinear effect.

Where Table 1. L is length of column; λ_c is slenderness parameter about weak axis is written as

Euler buckling strength for weak axis written as π ET, ,„ , .

Orbison's six-story space frame ignoring lateral torsional buckling

Fig. 8 shows Orbison's six-story space frame (Orbison, 1982). The yield strength of all members is 250 MPa (36 ksi) and Young's modulus is 206,850 MPa (30,000 ksi). Uniform floor pressure of 4.8 kN/m² (100 psf) is converted into equivalent concentrated loads on the top of the columns. Wind loads are simulated by point loads of 26.7 kN (6 kips) in the Y- direction at every beam-column joints.

The load-displacement results calculated by the proposed analysis compare well with those of Liew and Tang's (considering shear deformations) and Orbison's (ignoring shear deformations) results (Table 2, Table 3, and Fig. 9). The ultimate load factors calculated from the proposed analysis are 2.057 and 2.066. These values are nearly equivalent to 2.062 and 2.059 calculated by Liew, Tang and Orbison, respectively. Table 2 Analysis result considering shear deformation

Table 3 Analysis result ignoring shear deformation

[BRIEF DISCRETION OF DRAWINGS]

Figure 1 is a simply supported 3 -dimension column.

Figure 2 is a graph of modified CRC tangent modulus.

Figure 3 is a graph which shows parabolic function represents refined plastic hinge Figure 4 is a graph of nonlinear analysis procedure. Figure 5 is a program flow diagram Figure 6 is exampled 3D column for buckling analysis Figure 7 is a column strength curve of Figure 6 Figure 8 is exampled 3D asymmetric 6 -story frame Figure 9 is load-displacement curve of Figure 8

[INDUSTRIAL APPLICABILITY]

(1) Usage in the analysis and design of civil steel structures

(2) Usage in the analysis and design of architectural steel structures

Claims

[CLAIMS]

[CLAIM1]

In nonlinear analysis software for steel structure design, (A) If the axial force is compression, the following stability functions are used to capture the geometric nonlinearity,

If the axial force is tension, the following stability functions are used,

(B) The softening plastic analysis equations below are used for the member subjected to both axial force and moments, η = l :a ≤ 0.5 η = 4a(l-a) : > 0.5 where a is a force-state parameter that measures the magnitude of axial force and bending moments at the element end. The term a may be expressed by

P M_y M_z P 2 M_y 2 _.

(2) a - — < — + - ^2Py ^My_P M_ψ P y 9 M yp 9 M zp (C) To account for transverse shear deformation effects in a beam- column element, the stiffness matrix may be modified as;

where, A is gross area, G is shear elastic coefficient, L is length of member, and k is written as j- _ ~_τr Jy ^~ JJY ⁺ tfy-^Λsz -L

1-r/τyv

C_IIY + CJJY + 2 _IJY + A_SZGL

c c " _/jy + _JJYA_SZGL

C_jjy + C_JJY + 2C _jjy + A_SZGL

ι_ _ ^IIZ^J_JZ ^IJZ ⁺ j_ZA_SγGL ^IIZ ^"■^" ^IJZ "■" 2 _/-/Z + A_SYGL

^IZ "■" jyz + 2C_/JZ + A_sy ⁽_τL

C C " G-_Z "■" CJ_JZA_SY ⁽JL JJZ

(-iiz + C_/z + 2C_/JZ + A_syGL

[CLAIM2]

For the member subjected to additional Y-axis and Z-axis shear force caused by lateral displacement, the following equations in the software are used;

where P is axial force of beam-column member.

[CLAIM3]

The main item of the claim is the practical nonlinear analysis software for design of steel structures.