Nonlinear analysis of three-dimensional steel frames with semi-rigid connections

Computers & S~rucrures Vol. 41, No. 5. pp. 99%1009. 1991 Printed in Great Britain.

0045.7949/91 s3.00 +o.aa Pergamon Press plc

NONLINEAR ANALYSIS OF THREE-DIMENSIONAL STEEL FRAMES WITH SEMI-RIGID CONNECTIONS

S.-H. Hsre~ and G. G. DEIERLHN

School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, U.S.A.

(Received 3 October 1990)

Abstract--With the ultimate aim of improved methods for the realistic limit state design of structures, a method has been developed for incorporating nonlinear connection response in the analysis of three-dimensional steel structures. The method is implemented in an interactive graphics analysis and design program which can model both geometric and material nonhnearities in framed structures. The connection model includes nonlinear moment-rotation response for both major- and minor-axis bending. Standardized models for different types of connections which are calibrated to existing test data and are amenable to design are presented. Application of the method is presented for a realistic case study consisting of a low-rise three-dimensional structure with partially restrained connections.

1. INTRODUCTION

In recent years, the advancement of computer tech- nology together with the growing body of research on connection behavior has increased concern for including connection flexibility in the analysis and design of steel structures. As evidenced by several representative publications [l-5], considerable research has been conducted to study both the behavior and modeling of semi-rigid connections and the effects of connection flexibility on the behavior and design of steel frames. Developments in this area are reflected in recent design specifications which address considerations for including connection effects in analysis and design. For example, the AISC LRFD specification [6] classifies more realistically the types of steel construction into two categories: Type FR (Fully Restrained) construction, and Type PR (Partially Restrained) construction. For Type PR construction, the AISC specification instructs the designer to explicitly take into account the effect of connection flexibility in the design analysis, though specific guidelines are not provided. Another example of a recent specification which addresses the issue of including connection flexibility in design is the latest working draft of Eurocode 3 which includes specific criteria and connection modeling parameters for semi-rigid frames [7].

this paper is to describe a prototype computer-aided analysis and design system for two- and three- dimensional steel structures with semi-rigid connections and to demonstrate its application through a realistic case study.

1.1. Background

Previous research on structures with semi-rigid connections which is utilized in the present study is summarized in detail by Hsieh [8]. Briefly, this work can be categorized into the following two areas:

1.1.1. Behavior and modeling of semi-rigid connec - tions. Many tests are reported in the literature which have focussed on the in-plane moment-rotation (M-6) behavior of connections. In this research, a collection of test data which was originally assembled by Goverdhan [9] and synthesized into a computer- ized database by Kishi and Chen [lo, 111 was used. This data bank includes 303 tests conducted by over 38 researchers. Various techniques have been proposed for representing the M-0 response of connections, some based on simple linear approximations and others on more sophisticated nonlinear func- tions. As described below, in this research a nonlinear equation is used which is based on models previously proposed by Richard and Abbott [12] and Kishi and Chen [ 111.

In spite of the progress made in research, in 1.1.2. Design behavior and analysis of semi-rigid engineering practice connection flexibility is still not connections. The development of frame analysis pro- usually explicitly considered in analysis and design, grams which incorporate connection models, and the and the use of partially restrained connections in use of these to study the design and behavior of unbraced frames is often avoided. Reasons for this framed structures with semi-rigid connections have include the complexities and uncertainties in predict- been reported by several researchers. Various models, ing connection behavior with analytic models, the analysis schemes, and solution methods have been unavailability of commercial structural analysis soft- studied. Previous investigations have primarily dealt ware which effectively handles realistic connection with the following: (a) the effect of connection flexi- response, and lack of consistent methodologies for bility on the force distribution, maximum load-carry- using nonlinear analysis in design. The purpose of ing capacity, deformations (drifts), and stability of

996 S.-H. HSIEH and G. G. DEIERLEIN

frames, (b) methods of implementing semi-rigid connections into frame analysis, (c) validation of different connection models for design, (d) suggestion of design procedures to account for connection flexibility, and (e) evaluation of potential benefits of considering connection flexibility in design. Most previous research focused on two- dimensional analysis and in-plane moment-rotation characteristics of connection; few three-dimensional frame analysis programs which account for nonlinear connection flexibility have been reported in the literature.

1.2. Interactive computer-aided analysis and design

system

With the growing availability of lower cost engineering workstations, interactive computer- aided analysis and design systems are becoming a viable tool for design practice. One prototype system for the analysis and design of three-dimensional steel frames using finite element analysis techniques [13-l 61 has been developed at Cornell University. For statically loaded structures, this system consists of two programs, CU-PREPF and CU-STAND. CU- PREPF is a program for the definition of two- and three-dimensional framed structures; CU- STAND is a program for the elastic and inelastic analysis and design of three-dimensional steel framed structures.

The system described above has been used in the past to study the realistic behavior and limit state design of steel structures. Through interactive computer graphics, CU-PREPF provides a user- friendly and efficient means of defining the structural geometry, member properties, boundary con- ditions, and loadings. The analysis formulation in CU-STAND includes both geometric and material nonlinearities. Geometric nonlinearities are handled through the use of geometric stiffness matrices and an updated Lagrangian method; material nonlinearities are considered through a concentrated plasticity approach assuming elastic-perfectly plastic member response based on a three-parameter yield surface. Various solution and analysis control schemes are provided along with features for graphically monitoring the structural response by viewing the deflected shape of the structure, load- deformation response curves, etc. An example of the menuing system and type of graphical information which can be displayed is shown in Fig. 1. CU- STAND also includes built-in features for semi- automated design checks and redesign based on a subset of the AISC LRFD specification [6]. One of the limitations in the original version of CU-STAND (version 2.0) is that all joints were modeled as fully rigid. In this research, the ability to include nonlinear joint flexibility is incorporated into CU-STAND.

ORtlY

SCALE COHN. SCALE

HINGE5 IJNBALRNCE

INFO PRINT

1~ HELP 1 PHOTO 1 RETURN 1

Fig. 1. Typical menu page. and graphical viewport in CU-STAND.

3-D steel frames with semi-rigid connections 997

I .3. Objectives and scope

The objective of this research is to develop and demonstrate a tool to analyze and design two- and three-dimensional steel structures with semi-rigid connections for static loads using finite element analysis techniques. In this paper, the following aspects of the work are reported:

1.3.1. Development of connection model for use in analysis and design. Based on previous research on the behavior and modeling of semi-rigid connection, an analytic model suitable for the objective of this research is developed to represent connection behavior for design applications.

1.3.2. Modeling of connections in three-dimensional finite element analysis. A nonlinear connection element is presented which takes into account the connection flexibility using a finite element approach. A detailed formulation for the connection element is presented.

1.3.3. Computer implementation of semi-rigid connections. The semi-rigid connection model is implemented in the interactive analysis and design system, CU-STAND. Interactive computer graphics is utilized for defining and editing the connection model and for assigning connection properties to the structural model.

1.3.4. Design example. Capabilities and features of the analysis and design system are demonstrated using a three-dimensional building frame.

2. MOMENT-ROTATION MODEL FOR CONNECTIONS

As reported by previous researchers, the most significant connection effects on building frame behavior are associated with moment-rotation behavior, and therefore, only major- and minor-axis rotational flexibility of connections is considered in the present work. This behavior is modeled by a nonlinear equation which is calibrated to test data and normalized for use in design.

2.1. moment-rotation equation

A four-parameter power equation is adopted to represent the moment-rotation relationship of connections. This nonlinear equation was first presented by Richard and Abbot [ 121, and was later used by Kishi and Chen [I l]. With this equation, the moment-rotation (M-8) relations~p of the connection is represented by the following expression

M= (1)

As shown in Fig. 2, K, = initial stiffness of the connection, KP = strain-hardening stiffness of the connection, MO = a reference moment, and n = a shape parameter. By differentiating eqn (1) with

respect to 8, the tangent stiffness can be expressed as the following

K,=; = (4 - KJ n (*+1)/n + %- (2)

I) This equation was chosen because it can model the

observed experimental data well and the four parameters are related to the physical response of the connection. One advantage of this model is that it easily encompasses more simple models. For example, eqn (1) becomes a linear model if K, = KP, an elastic-perfectly plastic model if KP = 0, and a bilinear model when n is large.

To allow for unloading of the connections associated with nonproportional loading and inelastic force redist~bution, the two alternative unloading curves shown in Fig. 2 are provided. One alternative is based on linear unloading parallel to the initial stiffness, K, , The other is based on a nonlinear unloading curve given by the following equation, where the peak moments and rotations reached during loading are h4, and 6. (see Fig. 2)

M=iv*- We-K#‘,-@ We--K&~,- 0) ’ ‘M

=fo I) -$(% - 0 (3)

2.2. Determination of parameters

The four parameters of the model may be determined by several means according to the specific needs in analysis and design. If experimental data are available, the most precise representation is obtained through curve-fitting the model directly to the data. Where test data are not available, as in typically the case in design, the parameters may be determined using analytic formulations for the connection strength and stiffness if the connection details are known. In design practice, however, it is usually the case that the connection details may not be known until after the structural members have been sized. As described below, a third method proposed for deter- mining the parameters is based on using standardized curves to provide the general shape of the response curve and analytic (design) methods to calculate the nominal connection strength. The standardized curves are obtained from statistical analysis of normalized curves which were curve-fit to experimental data previously collected by Kishi and Chen [lo, 1 l] and Goverdhan [St].

Curve $tting from experimental data. An optimiz- ation approach utilizing the conjugate-gradient method is used to find a set of parameters (MO, Ke, KP, and n) which gives the best curve-fit to experimental response data [8]. An example of the curve- fitting results for top- and seat-angle connections with

998 S.-H. Hsizu and G. G. DEIERL.EIN

double web angles (TSAW) is shown in Fig. 3. In general, the curve-fitting results produce good agreement with the experimental results.

Standardized connection reference curves. In structural design practice, it is ~likely that specific information regarding the connection details will be known during preliminary design, and even during final design, this information may not be available until after the structural members have been sized. Since connection flexibility will affect the structural response and therefore the required member sizes, there is a need to develop some means of amounting for connection behavior in the analysis during the design process before final member sizes are selected. One solution is to use standardized connection reference curves which are based on test data and normalized for use in design.

To generalize eqns (lH3) for design, the moment- rotation expressions are first normalized with respect to a reference value of moment which is defined herein as the nominal connection capacity, M,. The normalized expressions are identical to eqns (l)-(3) except that M, M,, K,, and Kp are replaced by M’ = M~~~~, M; = M~lM~~, K:= KJ M, and K; = KJM,,, respectively. An example is presented below to illustrate how the normalized curves are developed for top- and seat-

Fig. 2. Moment vs rotation model for semi-rigid connections.

angle connections with double web angles (TSAW connections).

Using the curve-fitting technique described previously, eqn (1) was calibrated to experimental data for the TSAW connections, resulting in the curves

o Test dala

- Curve-f i 1

F I 1 4 10.0 20.0 30.0 40.0 SD.0

Relallve rolatlon (111000 red. 1

Fig. 3. Comparison between curve-fitting results and experimental results for TSAW connections.


shown in Fig. 3. The curves were then normalized by a value of M, equal to the moment resisted at an applied rotation of 0.02 radian. This value was chosen after considering several alternate normalization schemes, further details of which are reported by Hsieh [8]. The normalization procedure results in the set of curves shown in Fig. 4. For a given type of connection, this procedure provides a convenient means of condensing the data from a large number of tests by eliminating variations due to scale (strength) effects.

From the normalized curves shown in Fig. 4, the three standard reference curves shown in Fig. 5 were developed. The center (TSAW-Ave) curve was obtained by fitting a curve through the average of the normalized set of curves in Fig. 4. The upper and lower curves in Fig. 5 reflect a variation from the average curve of plus or minus two standard devi- ations. Assuming the variation in connection response is random and normally distributed, the region between the upper and lower curves in Fig. 5 encompasses roughly 95% of the sampled data. Similar curves can be developed for other connection types. Parameters for the three curves shown in Fig. 5 are presented in Table 1.

The aim of this approach is to establish a library of standard reference curves for common connection configurations. Then, for analysis and design of the overall structure, only the connection type and nominal capacity would need to be defined. Standard curves such as those shown in Fig. 5 can be used to investigate the design for the range of expected structural behavior without unnecessary concern over the precise behavior of the final connection details.

3. FINITE ELEMENT ANALYSIS OF SEMI-RIGID FRAMES

As noted previously, the connection model was incorporated into the nonlinear analysis and design program, CU-STAND. An important aspect of the model implementation is that it did not require fundamental changes to the existing geometric and material nonlinear model for the elastic-plastic beam-column elements in CU-STAND. The following is a summary of the formulation and computer implementation.

3.1. Modeling of the beam-column element

The beam-column element in CU-STAND includes both geometric and material nonlinearities. A brief review of the formulation is given in this section; more details are contained in [16].

Beam-columns are modeled as line elements with 12 degrees-of-freedom, consisting of three trans- lations and three rotations, at each end of the element. Common beam theory assumptions, such as homogeneous and isotropic material, plane sections remain plane, doubly symmetric prismatic sections with no cross-section distortion, and small strain theory, are employed in the formulation of the element stiffness. In linear elastic analyses, the element stiffness is the conventional linear elastic stiffness matrix, [k,] (see, for example, Chapter 4 in [18]). For second-order analyses, geometric nonlinearities are handled through the use of element geometric stiffness matrices [k,] and an updated Lagrangian formulation. The nodal coordinates and the terms in [k,] are updated at the end of each incremental/iterative

1.4

1.2

I Honenl rallo

10.0 20.0 30.0 40.0 50.0 Aelallve rolallon ~I/1000 red.)

Fig. 4. Normalized moment-rotation connection curves for TSAW connections.

1000 S.-H. HSIEH and G. G. DEIERLEIN

1.4 + llonenl rallo

1 I I I 4 1 I I 8 I I I 1 .I I I 0 I 1 1

5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

Relative rotallon (111000 rab.)

Fig. 5. Standardized moment-rotation curves for TSAW connections.

load step. For inelastic analyses, material nonlinearities are included through the use of element plastic reduction matrices [k,,] which are based on a three- parameter yield surface for modeling cross-section plastification due to axial load and major- and minor- axis bending. This is a concentrated plasticity model approach where it is assumed that zero-length plastic hinges form at the end of each element. Details of the stiffness matrices [k,], [kg], and [k,] are provided in

1161. The incremental element equilibrium equations can

be written in the following form

IdsI = We1 + [&I + tk,lW~ = [k,lIduI. (4)

In eqn (4), {ds} is the vector of incremental element end forces; {du} is the vector of incremental element

displacements; [k,], [kg], and [k,] are elastic, geometric, and plastic reduction matrices, respectively; and [k,] is the resulting element tangent stiffness matrix. Depending on the type of analysis (e.g., first-order or second-order, elastic or inelastic), [kg] and/or [k,] may not be included in the analysis.

The global incremental equilibrium equations are written as the following

W) = KIWI. (5)

In eqn (5), (dP} is the incremental load vector applied on the entire structure, {dU} is the global incremental displacement vector, and [K,] is the global stiffness matrix obtained by assembling the transformed element tangent stiffness matrices,

kl.

Table 1. Standard reference curve parameters for TSAW connections

Curve MO’ Ke Kp’ n

TSAW-Upper 0.93 436 4.0 1.62

THAW-Ave 0.90 266 7.6 1.40

‘I’SAW-Lower 0.80 132 12.0 2.00


3.2. Modeling of semi-rigid connections

Zero length connection elements are used to permit relative flexural rotations between connected members; the connections do not allow for relative tor- sional rotation or translational displacements. When a semi-rigid connection is specified at one end of a member, the global rotational degrees-of-freedom at the corresponding structural node are associated with the connection element. The corresponding local rotational degrees-of-freedom between the member end and connection are treated as additional global unknowns of the structural system and are included in the global equilibrium equations [i.e., eqn (5)]. Condensation is not used here because it is not as efficient for nonlinear analyses in which stifiness matrices are updated many times. In the formulation, the additional rotational degrees- of-freedom described above are always measured with respect to the local member coordinates even though they are treated as global unknowns in

eqn (5). To introduce the local degrees-of-fr~dom into

the global solution system, the conventional element transformation matrices are modified for the elements with semi-rigid connections. The modified matrices are used to transform the element stiffness matrices from the local to the global coordinate system with some of the unknowns retaining their local coordinate reference axes.

A key advantage of this approach is that the existing nonlinear formulation for the beam-column elements is unaffected. This avoids the difficulty (particularly for three-dimensional problems) of directly formulating the connection flexibility into the nonlinear element stiffness matrices, [Jr,] and [k,]. The use of separate connection elements also facilitates further modifications to the connection model (for example, to account for the finite size of the connec-

tion or for including additional connection degrees- of-freedom). On the other hand, a ~~dv~~ge of this approach is that it increases the total number of degrees-of-freedom in the global system of equations. However, this disadvantage is becoming less significant with the continuing improvement of computer hardware.

Example formulation. An example is presented to demonstrate the formulation and transformation of the beam-column element stiffnesses with nonlinear connection ~exibility. A portion of a three-dimensional structure discretized in a global coordinate system with orthogonal axes X, Y and Z is shown in Fig. 6. The connection elements Aa and Bb connect the beam ab to the nodes A and B. For clarity, the connection elements are shown ‘exploded’ to a finite length but are actually zero-length. The local (or element) coordinate system of the beam ab with orthogonal axes x, y and z is also shown. In general, the local coordinate axis (x, y, z) are arbitrarily ori- ented with respect to the global coordinate axes X Y, Z).

In Fig. 6, the degr~s-of-fr~dom {DOFs) at nodes A and B are represented using the displacement components in the global coordinates, and the DOFs of beam ab are shown in terms of local coordinates at both ends of the beam. In the displacement components, AV and (&, the first subscript i refers to the global or local com- ponent axes and the second subscript j refers to the global or member node designation. These subscripts are also used in developing the expressions for the forces and moments, 41 and M,, respectively.

In the example considered, it is assumed that the connection element Aa includes both major-axis and minor-axis rotational flexibility while the connection Bb is rigid. Aside from major- and minor-axis flexure, no other deformations are considered in the connection Aa.

-& _,Semi-rigid Rigid

Beam ab

Fig. 6. Structural d&ret&ion for a beamxolumn with zero-length connections.

1002 S.-H. HSEEI and G. G. DEIERLEIN

Matrix notations. The following matrix notations are used for the modeling formulation

[y13 X 3 = the conventional rotation matrix of beam ab

[II2 x z = identity matrix

[T,Lxe= the first four rows of [G]

[T2 I2 x 6 = the last two rows of [G].

Member transformation matrix. At end a of beam ab, relative major- and minor-axis rotations between the beam end and the node to which it is connected (0,, and 19,~) are allowed. Transfer of the global DOFs at node A into the local coordinate system is done by the following equation

e YA I J e ZA

Adding 6,, and oz. to eqn (6), results in the following equation which relates the local beam DOFs to the mixed (local and global) DOFs considered in the global system of equations

F“,l,., 0

0 PI 2x2

A XA

A YA -I

A ZA

e XA

e YA

e ZA

e N

9 in

(7)

Similarly, equilibrium gives the following transformation of forces from the local to the mixed global system

‘[T,l,.e 0

0 m2x

F XA F YA

F ZA

MXA

MYA

MzA

Mya

M,,

=[G%.,Wh.,. (8)

Since, in this particular case, no connection flexibility is considered at end b of the beam, the displacement and force transformations are aiven in the following standard forms

=Pl

A XB r 1 A rB

(9)

(10)

Using eqns (7)--(10), the transformation matrix of beam ab can be written as the following

PI [Gll,x, 0

I2 x 14 = 0 1 [G’&., ’

(11) This matrix is then used to transform the element stiffness matrix from the local to the global system, where the equilibrium equations from beam ab in the global system are given as follows:

In eqn (15) Kbh X 14 = Kblf, X 12M12 X drbL2 x l4r [k,] = element tangent stiffness matrix of the beam ab [eqn (4)], and {Fl}, {F2}, {D 1}, and {02} are defined in eqns (7x10). Note that e,, and oz, in {Dl} are local rotations at end a of the beam and Mya and M, in {F 1) are the corresponding member end moments measured in local coordinates.

Stiffness matrix of connection Aa. Given K, = the tangent stiffness of connection Aa for major-axis (z-axis) bending, KY = the tangent stiffness for minor- axis (y-axis) bending, and the relative connection rotations are (6iA-6lo) and (fJ,,-e,,), respectively, the equilibrium equations for connection element Aa are given as the following

e YA

ii

e =k14x4 e:,: . (13)

I J 8 TO

3-D steal framea with semi-rigid connections 1003

In eqn (13), [k,] is the tangent stiffness of connection Aa measured in local coordinates.

Tr~formation of connection stiftiress. From com- patibility at node A, the internal (local) displacements are related to the global displacements by the following equation in which [TJ was defined previously

8

11 uA =tT*l*Yb

e L4 1 A X.4 A YA A e;

e e;

(14)

Note that the first three columns of [T,] are null for the present case of i~tesim~ joint size and non- eccentric member ends.

Adding 13~~ and 6, to eqn (14), and using a similar transformation for forces, the displacement and force tr~fo~ations at the connation are given by eqns (15) and (16)

F212xt5

2 ~.

A XA

A YA

A ZA 0 XA

B YA

8 ZA

8 Ye

e II?

FXA

FYA

F ZA

MXA

MYA

MZA

MY#

Ma,

(15)

! (16)

Using the transformation matrix [fc] from eqns (IS) and (16), the equilibrium equations for connection Aa in the global system are given as the follow-

ing, where Kclr X r = V’CIS’~ #cl, x ,V’cl~ x 8 and [kc] = element stiffness matrix of connection AP measured in local coordinates

(17)

Extension for other situations. The transformation procedure demonstrated above in eqns (6)+7) can be modified to cover connections with only major- or minor-axis rotational flexibility and for beam- column elements with connections at both ends. All of these cases are included in the computer implementation in CU-STAND.

3.3. Computer ~ple~~t~tio~

In addition to implementing the connection model noted above, control menu’s were added in CU- STAND for definition of the connection parameters and to locate connations by graphically attaching them to specific members. One of the connection editor menus is shown in Fig. 7 through which the user can interactively assign the four parameters which define the shape of the connection model and the nominal connection strength, M,. Based on the user input, a plot of the moment rotation curve for the connection is shown in the viewport in the upper left portion of the screen. In the program, the four connection parameters can be either specified directly by the user, chosen from a library of values for standard connection types, or generated from moment-rotation data using the built-in curve-fitting routine. The nominal connection strength, M, can be specified either as an absolute value or as some fraction of the plastic moment of the member, Mti, to which the connection is attached. The latter option is particularly suited to an iterative design process where connection and members properties are un- known at the outset and updated in the course of design.

4. APPLICATION TO DESIGN

A case study is presented to demonstrate application of the compu~r-~ded analysis system for investigating the design and behavior of a low-rise three-dimensional steel frame with partially restrained connections. The frame geometry, member sizes, and loading am shown in Fig. 8. Partially restrained TSAW (top- and seat-angle with double web angles) beam-column connections are used for framing in both directions and the column base plate details are modeled as EEPS (extended end plate with stiffeners) connections. The member sixes shown were designed based on the AISC-LRFD specification provisions with members forces calculated by the nonlinear analysis described below. In the analysis

CA.5 41/s-I

1004 S.-H. Hsrtxand G. G. DEJERLEIN

model, beams were discretized into four elements and gravity loads were applied at the quarter points of beams in the Noah-South (N-S) direction to ap- proximate loads applied through the one-way joist floor system. In this case, the strength limit state for the frame was controlled by the factored gravity load combination, 1.2 DL + 1.6 LL.

In the analysis, all beam-column connections and the column base plate connections were modeled using the procedure described previously. The TSAW beam-column connections were modeled using the set of normalized average parameters presented in Table 1 (i.e., TSAW-Ave). The connection design strength was assigned as a fraction of the plastic moment capacity of the connected members. For framing in the N-S direction which carried most of the gravity load, the nominal connection strength, M,, was taken as 40% of the plastic moment capacity of the connected member (i.e., 0.4 M,,), and for framing in the E-W direction, M, was taken as 0.8 Mph. The column base plate connections were modeled as EEPS connections connections with M En = 1.5 Mph. The normalized parameters for the EEPS connections are equal to the following values based on [17]: k:= 185, Kj = 1.6, MA= 1.05 and n = 1.54.

For strength design purposes, a reduced (design) connection strength, MC,,, is used in the analysis which is equal to 0.65 times the nominal strengths

assumed above (i.e., ML = Mcd = 0.65 M,), The reduction factor of 0.65 was chosen based on a re- liability analysis described in (171. As will be demonstrated, the reduced value of the connection strength provides a conservative estimate of the overall strength and stiffness of the frame. Based on this approach, the following values of Ml,, were used in the analysis: TSAW (N-S), M;,, = 0.26 Mph; TSAW

(E-w), M:, = 0.56 Mph; and EEPS column base, M& = 1 .O Mph. For the subsequent inelastic analyses described below, resistance (reduction) factors were also applied to the member yield surfaces to reflect the design, rather than the nominal, member cross- section strengths (e.g., the design plastic moment was taken as 0.9 M,,). In this way, resistance factors are included in the inelastic analyses based on an approach suggested by Ziemian et al. 1191 which follows the philosophy of the AISC-LRFD specification.

Once the member sizes were selected based on the analysis and design assumptions described above, additional nonlinear (second-order inelastic) analyses were made to investigate the limit strength of the frame and its sensitivity to the assumed connection properties. The connection models used in the five cases are summarized in Table 2. In case A, which is the basic case, the connection properties are the same as those used in the o~~na1 analysis used to deter- mine the member sizes. In cases B and C, the parameters are modified to reflect the upper and

1.11 “’ “lmr h,,

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n . _....... II’. 8.. . re’

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HELP PHOTO RETURN

Fig, 7. Menu page used for defining connection properties.

Case

L A I

IB


b w 81 83

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b s 61 83

a )(I H-t

b 82

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84

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84

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i -t

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DL: Root 25 psf, Floor 75 psf

LL: Roof 27 psf. Fkor 42 psf (‘)

WL: 20 psf in each diiion

(‘) Live beds are reduced values per ANSi A66.1 (1666)

Bl - Roof W16X26 Bl - Flaw w21 x44 82 - Roof W12X14 62-w W 16X26 W-Roof w21 x44 B3-Flow W24X76 W-Roof W12X26 B4-Fbor w21x44 86 WlPXl6

A.l, A.6. D.l, 0.6 W12X26 6.1, Cl, B.6. C.6 W12X35 A.2 - 6 W 12X35 8.2 - 5 w12x63 C.2-6 W 12X45 0.2 - 5 W12X26

All Steel Grade A36 (36 k8i)

Fig. 8. Low-rise building frame with partially restrained connections,

Table 2. Assumed connection properties

North-South F’raming East-We& Framing

curve Men’ curve Men’

TSAW-Ave 0.26 Mpb TSAW-Ave 0.52 Mph

TSAW-Upper 0.26 Mpb TSAW-Upper 0.52 Mpb

Column Reee Rate

Curve Men’

EEPS-Ave 1.0 Mpb

EEPS-Upper 1.0 Mpb

C TSAW-Lower 0.26 Mpb TSAW-Lower 0.52 Mpb EEPS-Lower 1 .OMpb

D TSAW-Ave 0.40 Mpb TSAW-Ave 0.80 Mpb EEPS-Ave 1.5 Mpb

E Rigid Rigid Rigid I

1006 S.-H. Hsran and G. G. DEIERLEIN

0.4

‘0.2 I I : : ; ; ; ; ; ; : (

I.0 2.0 3.0 4.0 5.0 6.0 1.0 9.0 9.0 IO.0

Drilt in N-S Direction (inches)

Fig. 9. Load-deformation response under factored gravity loads.

lower bound connection response curves shown previously in Fig. 5. In case D, the unreduced nominal connection strength (i.e., without the reduction factor of 0.65) is used in the analysis. Note that the differences between cases A, B, and C is mainly with regard to the shape of the response curve, whereas in case D the strength is varied. Finally, in case E, all connections are treated as perfectly rigid. As noted previously, the design strength was governed by the gravity load combination and the limit state analyses were conducted for this loading.

The overall load-deformation behavior for each case is shown in Fig. 9 where the applied load ratio is plotted versus the lateral roof drift in the N-S direction, Note that since the frame is not symmetric in the N-S direction, drift occurred under the gravity

load combination. In Fig. 9, and throughout this example, the applied load is reported in terms of a fraction of the factored gravity load where, for example, an applied load ratio of 1.0 corresponds to application of the full factored load combination (1.2 DL + 1.6 LL). The analyses were stopped when the inelastic limit point was reached as detected by zero diagonal terms in the global stiffness matrix. Numerical values corresponding to the analysis results are reported in Tables 36.

As shown in Fig. 9, all of the frames with partially restrained connections experienced large inelastic drifts prior to reaching their inelastic limit strength, but these drifts were not excessive prior to application of the full factored gravity load combination. For example, for case A the drift at the full factored load (applied load ratio = 1 .O) was approximately 0.6 in. which corresponds to a drift index of H/767 where H is the building height. However, at the limit point the drift for this frame equaled 8.1 inches which corresponds to a drift index of H/57. The large drifts at the limit point demonstrate that even for low rise build- ings, where partially restrained connections are used, second order (geometrically nonlinear) effects should be considered.

The location of plastic hinges at the inelastic limit points for the partially restrained frame (case A) is shown in Fig. 10. In general, all of the partially restrained frames (cases A-D) had a similar pattern of hinges. As indicated in Fig. 9 and Table 3, the first hinges formed at 94-105% of the full factored load; this is expected since the original AISC-LRFD design for case A was based on the most critical members reaching their plastic limit strength at an applied load ratio of approximately 1.0. As shown in the inset of Fig. 9, for the partially restrained frames, the first hinges occurred due to large minor-axis bending moments in the top story columns along column line A (columns A.2-A.5). In the rigid frame, the first

Table 3. Applied load ratios from inelastic analysis for gravity loads

Condition Partially Be&mined Rigid

A B C D E

1st Hinge 1.02 0.95 1.04 1.05 0.94

Limit Point 1.24 1.22 1.26 1.30 1.36

Table 4. Lateral drift in north-south directions under gravity loads (in.)

Load Level Partially Beatmined Rigid

A B C D E

Pull Factored 0.61 1.02 0.45 0.47 0.19

Limit Point 8.13 8.51 7.38 6.88 0.76

3-D steel frames with semi-rigid connections

Table 5. Midspan beam deflections under gravity loads: beam B3-roof level (in.)

1007

--- ~~ Load Level Partially F&trained Rigid

A B C D E

FuIl Factored 2.03 1.98 2.08 1.77 0.49

Limit point 4.65 4.31 4.83 4.13 2.31

Table 6. Connection rotations under gravity loads (radian)

Load Level PartiaIIy Restrained

A B C D

hinges formed at approximately the same applied load ratio but at different locations; in case E the first hinges formed due to negative bending at the interior ends of beams B4 adjacent to column line C.

In general, the difference in the inelastic limit strength due to variations in the normalized connection parameters (K:; K;, Mh and n) was not significant. As reported in Table 3, the applied load ratios at the inelastic limit for cases B and C were within 1.6% of case A. By comparing cases A and D, it was evident, however, that the connection strength used in the analysis had a greater effect on the response where a 50% increase in connection strength (l/0.65) increased the overall system strength by approximately 5%. While this is not a major increase in strength, it is somewhat significant if considering that, for the case with perfectly rigid connections (case E) the total strength increase was only 10% compared to case A.

Differences between the connection response parameters caused a large variation in calculated lateral drifts at full factored loads, but relatively little differences in drifts at the limit points. Also, the differences in parameters had little influence on the calculated vertical beam deflections. As reported in Table 4, under the full factored loads, the calculated roof drift for case B was 67% greater and for case C was 26% less than for case A. The 50% increase in connection strength in case D decreased the drift by 23% compared to case A, and for the rigid frame (case E) the drift was 69% less than case A. At the limit point, however, the drifts for cases A, B, and C were all within lo%, and the drift for case D was 15% less than case A. Overall, the rigid frame experienced much less drift and at the limit point the drift was 90% less than that of the partially restrained frames. As shown in Table 5, the beam deflections for cases B and C were all within 3-8% of those for case A, and

Fig. 10. Hinge locations for partially restrained frame at the inelastic limit point under gravity loads.

1008 S.-H. HSIEH and G. G. DHERLEIN

the deflections for case D were ll-13% less than those for case A. Beam deflections in the rigid frame were 50-75% less than those for case A.

The maximum connection rotations for each case are reported in Table 6. As shown, at the full factored load the maximum rotations (roughly 0.017-0.018 radian for cases A-C and 0.014 for case D) were less than 0.020 radian. This lends validity to the analysis since the nominal connection capacity was defined as the moment resisted at a rotation of 0.020 radian. At the limit point, the maximum rotations were between 0.043 and 0.051 radian. In general, these rotations seem to be in a permissible range based on the deformation capacities reported from tests. It should be noted that in other examples which the authors have studied, connection rotations calculated in the analysis were sometimes in excess of 0.050 radian, particularly where local beam mechanisms form prior to reaching the inelastic limit point.

5. SUMMARY AND CONCLUSIONS

The development and application of an interactive computer-aided analysis and design system for three- dimensional steel framed structures have been presented. The three main components of this work presented herein include (1) development and calibration of a nonlinear connection model amenable to analysis and design, (2) stiffness formulation of a three-dimensional beam-column element to model the effects of nonlinear connections and geometric and material nonlinear member response, and (3) demonstration of the computer implementation using an interactive-graphics user interface.

A unique feature of the computer-aided system is the development and calibration of normalized connection response curves for use in design. The implementation is particularly well suited for practical use during development of a building design when the detailed connection properties are not yet known.

Using the system, a three-dimensional low-rise building frame was designed and analyzed to assess the influence of partially restrained connections on the inelastic limit state response under gravity loading. The results indicate that the overall frame response is relatively insensitive to modest variations in the parameters which define the shape of the normalized connection response curves. Of greater import- ance with regard to overall behavior is the nominal strength of the connections used in the analysis. The results indicate that in general an increase in the connection strength (and indirectly the stiffness) results in a higher inelastic limit state. The observed load-deformation response demonstrates the need to consider nonlinear geometric (second-order) behavior for evaluating the strength limit state of frames with partially restrained connections. Computer- aided analysis and design systems such as the one presented offer an effective and rational means

for evaluating the realistic performance of partially restrained frames.

Acknowledgements-The work reported in this paper was supported by the National Center for Earthquake Engin- eering Research (Project 88-1005A and 87-1008) and the National Science Foundation (Grant MM-8908870). The contributions of faculty, staff, and student colleagues are gratefully acknowledged, in particular the advice of Pro- fessors J. F. Abel and W. McGuire of Cornell University and Professor T. W. Lin of National Taiwan University, and the many individuals who contributed to the development of the computer programs utilized in this work. Finally, the assistance of Professors D. W. White and W. Chen of Purdue University who provided the data bank of connection tests used in this work is appreciated. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

REFERENCES

1. W. F. Chen (Ed.), Connection flexibility and steel frames. Proceedings of a session held in conjunction with the AISC convention, Detroit, MI, U.S.A., October 24, 1985), Structural Division of the ASCE, New York (1985).

2. D. Anderson, F. Biilaard, D. A. Nethercot and R. Zandonini, Analysis and design of steel frames with semi&id connections. IABSE Survevs S-39/87 11987).

3. W. F. ?hen (Guest Ed.), Special is&e on join; flex;- bility in steel frames. 1. Construct. Steel. Res. Vol. 8 (1987).

4. W. F. Chen (Guest Ed.), Special issue on steel beam-to- column building connections. J. Construct. Steel. Res.

5.

6.

7.

8.

9.

10.

11.

12.

Vol. 10 (1988).- R. Bjorhovde, J. Brozzetti and A. Colson (Eds), Con- nections in steel structures-behaviour, strength and design. Proceedings of a state-of-art workshop on Con- nections and the Behaviour, Strength and Design of Steel Structures, France, May 25-27, 1987. Elsevier, London (1988). Maniai of Steel Construction, Load and Resistance Factor Des&. 1st Ed. AISC. Chicago. IL (1986). Eurocode io. 3-Design of Steel &uctu;es, Edited Draft Issue 3. Commission of European Communities, April (1990). S. H. Hsieh, Analysis of three-dimensional steel frames with semi-rigid connections. Structural Engineering Report No. 90-1, School of Civil and Enironmental Engineering, Cornell University, Ithaca, NY (1990). A. V. Goverdhan, A collection of experimental moment-rotation curves and evaluation of predicting eauations for semi-rigid connections. M.S. thesis, De- partment of Civil Engineering, Vanderbilt ‘University, Nashville, TN (1983). N. Kishi and W. F. Chen, Data base on steel beam-to- column connections. CE-STR-86-26, School of Civil Engineering, Purdue University, West Lafayette, IN (1986). W. F. Chen and N. Kishi, Semirigid steel beam-to- column connections: data base and modelling. J. Struct. Engng. 115, 105-119 (1989). R. M. Richard and B. J. Abbot, Versatile elastic-plastic stress-strain formula. J. Engng. Mech. Div. ASCE 101, 511-515 (1975).

13. W. McGuire and J. L. Castaner, Applications ot computer graphics to building design. Proceedings, Third Conference on Steel Developments, Australian Institute of Steel Construction, Monash University (1985).


14. G. G. Deierlein, W. McGuire, J. F. Abel and S. Srivastav, Some interactive graphics and parallel pro- cessing for earthquake engineering. In Computer Utiliz- ation in Structural Engineering (Edited by J. K. Nelson). ASCE Structures Congress, San Francisco (1989).

15. W. McGuire, G. G. Deierlein, T. K. Sooi and Y. T. Zhao, Illustrated Primer to CU-PREPF, CU-STAND and CU-QUAND. Struct. Engrg. Report 89-12, School of Civil and Environmental Engineering, Cornell Uni- versity, Ithaca, NY (1989).

16. S. H. Hsieh, G. G. Deierlein, W. McGuire and J. F. Abel, Technical manual for C&STAND. Struct. Engrg.

Report 89-13, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY (1989).

17. Y. J. Shen, Computer aided design of steel frames with partially restrained connections. M.S. thesis, School of Civil and Environmental Engineering, Cornell Univer- sity, Ithaca, NY (1990).

18. W. McGuire and R. H. Gallagher, Matrix Structural Analysis. John Wiley NY (1979).

19. R. Ziemian, D. W. White, G. G. Deierlein and W. McGuire, One approach to inelastic analysis and design. Proceedings of 1990 National Steel Construction Conference, AISC, Chicago, 19-1 to 19-19 (1990).

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Nonlinear analysis of three-dimensional steel frames with semi-rigid connections