Sample - jrosspub.com · 80 Semi-rigid Connections Handbook where f and g are some mathematical functions.M is the moment at the connection, and θ r is the relative rotation of the

79

4Modeling of Connections

Norimitsu KishiProfessor, Civil Engineering Unit, College of Environmental Engineering, Muroran Institute of Technology, Muroran 050-8585, Japan

Masato KomuroAssociate Professor, Civil Engineering Unit, College of Environmental Engineering, Muroran Institute of Technology, Muroran 050-8585, Japan

4.1 General Remarks .................................................................................................................... 794.2 Behavior Under Monotonic Loading ................................................................................... 80

LinearModel•PolynomialModel•ExponentialModel•PowerModel•BoundingLineModel•Ramberg-OsgoodModel•Richard-AbbottModel

4.3 Behavior Under Cyclic Loading ........................................................................................... 89IndependentHardeningModel•KinematicHardeningModel•BoundingSurfaceModel with Internal Variables

References ................................................................................................................................ 92

4.1 General RemarksIn the conventional analysis and design of steel framework, the behavior of connections is ideal-ized as perfectly rigid or ideally pinned. However, numerous experimental investigations have clearly shown that actual connections behave nonlinearly due to gradual yielding of connection components such as plates and angles, bolts, etc. A beam-to-column connection is generally sub-jected to axial force, shear force, and bending moment. For most connections, however, the axial and shear deformations are usually small comparing with the flexural deformation. For simplic-ity, only the rotational deformation of the connection due to flexural action is considered in this book. Typical moment-rotation (M–θr) curves for several commonly used connections are shown in Figure 4.1. In order to incorporate the M–θr curves more systematically and efficiently into a frame analysis computer program, the moment-rotation relationship is usually modeled by using mathematical functions.

The connection behavior can be simplified as a set of M–θr relationships. Mathematically, these relations can be expressed in the general form:

M = f(θr) (4.1)

or, inversely:

θr = g(M) (4.2)

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80 Semi-rigid Connections Handbook

where f and g are some mathematical functions. M is the moment at the connection, and θr is the relative rotation of the connection.

4.2 Behavior Under Monotonic Loading4.2.1 Linear ModelThe linear model (Rathbun, 1936; etc.) utilizes the initial connection stiffness Rki to represent the connection behavior as shown in Figure 4.2a. Although this model is easy to apply the moment-rotation behavior, it overestimates the connection stiffness at finite rotation.

Tarpy and Cardinal (1981), Melchers and Kaur (1982), and Lui and Chen (1986) also proposed a bilinear model (see Figure 4.2b), in which the initial slope of moment-rotation curve is replaced by a shallower line at a certain transition moment.

Razzaq (1983) proposed a piecewise multilinear model (see Figure 4.2c), in which the nonlin-ear moment-rotation curve is approximated by a series of straight-line segments. Although these linear models are easy to use, the inaccuracies and the jumps in stiffness at the transition points make them undesirable.

rθ

M

Flush end-plate

Extended end-plate

Top- and seat-anglewith double web-angle

Top- and seat-angle

Header plate

Double web-angle

Single web-angle

"Pinned"

"Rigid"

M

θrcolumn

beam

FIGURE 4.1 Typical moment-rotation (M–θr) curves of the beam-to-column connections.

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Modeling of Connections 81

4.2.2 Polynomial ModelThe first mathematical model is proposed by Frye and Morris (1975), which is based on an odd-power polynomial factor to evaluate the moment-rotation behavior of several types of connec-tion. The Frye-Morris model was developed on the basis of a procedure formulated by Sommer (1969). They used the method of least squares to determine the constant of the polynomial. This model is represented by:

(4.3)

where K is a standardization parameter depending on the geometrical and mechanical properties of the connection, and C1, C2, and C3 are curve-fitting constants.

The main drawback of this formulation is that the tangent connection stiffness may become negative at some value of connection moment M. This is physically unacceptable, and the nega-tive stiffness may cause numerical difficulties in the analysis of frame structures if the tangent stiffness formulation is used.

Following the procedure of Frye-Morris (1975), Picard et al. (1976) and Altman et al. (1982) de-veloped prediction equations to describe the M–θr curve for strap-angle connections and top- and seat-angle connection with double web angles, respectively. Goverdhan (1983) re-estimated the size parameters in the standardization constant K for flush end-plate connections to get a good agree-ment with moment-rotation curves obtained from experimental results. The curve-fitting constants

θr C KM C KM C KM= + +1 23

35( ) ( ) ( )

rθ

M

rθ

M

rθ

M

rθ

M

a b

c d

FIGURE 4.2 Different mathematical representations for the moment-rotation curve: (a) linear model; (b) bilinear model; (c) multilinear model; (d) nonlinear model.

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Connection types Curve-fitting constants Standardization constants

Single web-angle connection

C1 = 4.28 × 10−3

C2 = 1.45 × 10−9

C3 = 1.51 × 10−16

K d t ga a=− −2 4 1 81 0 15. . .

Double web-angle connection

C1 = 3.66 × 10−4

C2 = 1.15 × 10−6

C3 = 4.57 × 10−8

K d t ga a=− −2 4 1 81 0 15. . .

Top- and seat-angle with double web-angle connection

C1 = 2.23 × 10−5

C2 = 1.85 × 10−8

C3 = 3.19 × 10−12

K d t t l g dc a b= −( )− − − −1 287 1 128 0 415 0 694 1 3502. . . . .

Top- and seat-angle connection

C1 = 8.46 × 10−4

C2 = 1.01 × 10−4

C3 = 1.24 × 10−8

K d t l da b=− − − −1 5 0 5 0 7 1 1. . . .

Extended end-plate connection without column stiffeners

C1 = 1.83 × 10−3

C2 = −1.04 × 10−4

C3 = 6.38 × 10−6

K d t dg p b=− − −2 4 0 4 1 5. . .

TABLE 4.1 (a) Curve-fitting constants and standardization constants for Frye-Morris polynomial model (all size parameters are in inches)

C1, C2, and C3 and the standardization constant K for each connection type are summarized in Table 4.1. The size parameters for each type of connection are shown in Figure 4.3.

4.2.3 Exponential ModelLui and Chen (1986) used an exponential function to curve-fit the experimental M–θr data. This model is a good representation of the monotonic nonlinear connection behavior. However, if there are some sharp changes in slope in the M–θr curve, this model cannot adequately represent it (Wu, 1989). Kishi and Chen (1986) refined the Lui-Chen exponential model to accommodate any sharp changes in slope in the M-θr experimental data (modified exponential model). The Kishi-Chen model can be used instead of the M–θr experimental data.

The modified exponential model is represented by a function of the following form:

(4.4)

where M0 is the initial connection moment, α is a scaling factor for the purpose of numerical sta-bility, Cj and Dk are curve-fitting parameters, θk is the initial rotation of the k-th linear component given from the experimental M–θr curve, and H[θ] is Heaviside’s step function (unity for θ ≥ 0, zero for θ < 0).

Using the linear interpolation technique for the original M–θr experimental data, the weight function for each piece of M–θr data is nearly equal. The constants Cj and Dk for the exponential and linear terms of the function are determined by linear regression analysis.

The instantaneous connection stiffness Rk at an arbitrary relative rotation |θr| can be evaluated by differentiating Equation 4.4 with respect to |θr|.

M M Cj

Djj

mr

kk

n

= + − −

+

= =∑ ∑0

1 11

2exp

θα

θθ θ θ θr k r kH−( ) −

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Extended end-plate connection with column stiffeners

C1 = 1.79 × 10−3

C2 = 1.76 × 10−4

C3 = 2.04 × 10−4

K d tg p=− −2 4 0 6. .

Header plate connection

C1 = 5.10 × 10−5

C2 = 6.20 × 10−10

C3 = 2.40 × 10−13

K t g d tp p w=− − −1 6 1 6 2 3 0 5. . . .

T-stub connection C1 = 2.10 × 10−4

C2 = 6.20 × 10−6

C3 = −7.60 × 10−9

K d t l dt b=− − − −1 5 0 5 0 7 1 1. . . .

TABLE 4.1 (b) curve-fitting constants and standardization constants for Frye-Morris polynomial model (all size parameters are in centimeters)

Connection types Curve-fitting constants Standardization constants

Single web-angle connection

C1 = 1.67 × 10−0

C2 = 8.56 × 10−2

C3 = 1.35 × 10−3

K d t ga a=− −2 4 1 81 0 15. . .

Double web-angle connection

C1 = 1.43 × 10−1

C2 = 6.79 × 101

C3 = 4.09 × 105

K d t ga a=− −2 4 1 81 0 15. . .

Top- and seat-angle with double web-angle connection

C1= 1.50 × 10−3

C2 = 5.60 × 10−3

C3 = 4.35 × 10−3

K d t t l g dc a b= −( )− − − −1 287 1 128 0 415 0 694 1 3502. . . . .

Top- and seat-angle connection

C1 = 2.59 × 10−1

C2 = 2.88 × 103

C3 = 3.31 × 104

K d t l da b=− − − −1 5 0 5 0 7 1 1. . . .

Extended end-plate connection without column stiffeners

C1 = 8.91 × 10−1

C2 = −1.20 × 104

C3 = 1.75 × 108

K d t dg p b=− − −2 4 0 4 1 5. . .

Extended end-plate connection with column stiffeners

C1 = 2.60 × 10−1

C2 = 5.36 × 102

C3 = 1.31 × 107

K d tg p=− −2 4 0 6. .

Header plate connection

C1 = 6.14 × 10−3

C2 = 1.08 × 10−3

C3 = 6.05 × 10−3

K t g d tp p w=− − −1 6 1 6 2 3 0 5. . . .

T-stub connection C1 = 6.42 × 10−2

C2 = 1.77 × 102

C3 = −2.03 × 104

K d t l dt b=− − − −1 5 0 5 0 7 1 1. . . .

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p1 = da

p2 = ta p3 = g

p5 = g

p2 = tp4 = la

p1 = d

p6 = db (fastener dia.)

p3 = tc

weldedp2 = tp

p1 = dg

p3 = db (fastener dia.) welded

columnstiffener p2 = tp

p1 = dg

welded

p1 = tp

p4 = tw

p3 = dp

p2 = g

p2 = t

p3 = ltp4 = db (fastener dia.)

p1 = d

p2 = tp4 = db (fastener dia.)

p1 = d

p3 = la

a b

c d

e f

g h

FIGURE 4.3 Size parameters for various connection types of the Frye-Morris polynomial model: (a) single web-angle connection; (b) double web-angle connection; (c) top- and seat-angle connection with double web-angle; (d) top- and seat-angle connection; (e) extended end-plate connection without column stiffener; (f) extended end-plate connection with column stiffener; (g) header plate connection; (h) T-stub connection.

84

p1 = da

p2 = ta p3 = g

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When the connection is loaded, we obtain the equation as follows:

(4.5)

When connection is unloaded, we have:

(4.6)

This model has the following merits:

1. The formulation is relatively simple and straightforward. 2. It can deal with connection loading and unloading for the full range of relative rotation in

a second-order structural analysis with secant connection stiffness. 3. The abrupt changes in the connection stiffness among the sampling data is only generated

from inherent experimental characteristics.

The curve-fitting and tangent connection stiffness values from the experimental data examined are calculated with m = 6 in Equations 4.4 through 4.6. The comparison between the Chen-Lui exponential model (1985) and the modified exponential model for the numerical example of the test data, including a linear component, is shown in Figure 4.4.

Yee and Melchers (1986) proposed a four-parameter exponential model:

(4.7)

where Mp is the plastic moment capacity, Rki is the initial connection stiffness, Kp is the strain-hardening stiffness, and C is a constant controlling the slope of the curve.

R R dMd

Cj j

D Hk ktr

j rk r k

r r

= = = −

+ −

=θ αθ

αθ θ

θ θ 2 2exp

==∑∑k

n

j

m

11

R R dMd

Cj

D Hk ktr

jk k

j

m

kr

= = = + = =

=∑θ α θθ 0 1 12

M MR K C

MKp

ki p r r

pp= − −

− +( )

+1 exp

θ θθθr

Experiment

Exponential model

Modified exponential model

M

θrFIGURE 4.4 Comparison between results using exponential and modified exponential models for M–θr data including a linear term.

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As with the other exponential model, the Wu-Chen model (1990) proposed a three-parameter exponential model in the form of:

(4.8)

where Mu is an idealized elastic-plastic mechanism moment, θ0 is a reference rotation (= Mu/Rki), Rki is the initial connection stiffness, and n is the shape parameter, which is determined empiri-cally from test data.

4.2.4 Power ModelSeveral power models have been developed for the different types of connection. Two or three parameters are required in their functions.

A two-parameter model (Batho and Lash, 1936; Krishnamurthy et al., 1979) has the simple form:

(4.9)

where a and b are two curve-fitting parameters with the condition a > 0 and b > 1.Colson and Louveau (1983) introduced a three-parameter power model function as:

(4.10)

where Rki is the initial connection stiffness, Mu is the ultimate capacity of connection moment, and n is the shape parameter.

Kishi and Chen (1987a, 1987b) proposed a similar model removing the strain-hardening stiff-ness of the Richard-Abott model (Richard and Abott, 1975):

(4.11a, b)

where Rki, Mu, and n are the same as those defined in the previous Equation 4.10, and θ0 is a ref-erence plastic rotation (= Mu/Rki). Equation 4.11 has the shape shown in Figure 4.5. From this figure, it is seen that the larger the power index n, the steeper the curve. The shape parameter n can be determined using the method of least squares for the differences between the predicted moments and the experimental test data (Kishi et al., 1993, 1995).

This power model is an effective tool for designers to execute the second-order nonlinear structural analysis quickly and accurately. This is because the tangent connection stiffness Rk and relative rotation θr can be determined directly from Equation 4.11 without iteration, in which the tangent connection stiffness Rk in Equation 4.11 is:

(4.12)

MM

nu

r= +

ln 10

θθ

θrbaM=

θrki u

n

MR M M

=−( )

1

1

θθ

θθ

r

kiu

n nki r

r

M

R MM

MR

=

−

=

+1 11

0

/ or

n n1/

R dMd

Rk

r

ki

r

n n n= =

+

+θ θθ

10

1( )/

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4.2.5 Bounding Line ModelAl-bermani et al. (1994) and Zhu et al. (1995) have proposed a bounding line model as shown in Figure 4.6. It requires four parameters and the concept of this model is to divide the curve into three segments, in which the first and third segments are linear elastic and plastic portions, respectively, and the second segment is a smooth transition portion. The form of this model is represented as:

(4.13)

where m1 = My + Rkpθr, m2 = Mc + Rkpθr, Rki and Rkp are the initial connection stiffness and the bounding connection stiffness, respectively, My is the yielding connection moment, and Mc is the bounding connection moment.

4.2.6 Ramberg-Osgood ModelThe Ramberg-Osgood model was originally proposed for nonlinear stress-strain relationships by Ramberg and Osgood (1943) and then standardized by Ang and Morris (1984). The M–θr curve of this model is represented as:

(4.14)

where (KM)0 and θ0 are constants defining the position of the intersection point A (see Figure 4.7), n is a parameter defining the sharpness of the curve, and K is a dimensionless factor depend-ing on the connection type and geometry.

M

R M m

RM mM M

R R m M m

R M

ki r

kic y

kp ki

kp r

=

<

+−−

−( ) ≤ <≥

θ

θ

1

11 2

mm

2

θθr

nKMKM

KMKM0 0 0

1

1= +

−

( ) ( )

1/

0

01+

M

Mu

θr

θr

θθ

θ r

R ki

M=

= Mu / Rki

M =

Rki

θrRki

n n

n = n1

n = 8

n = n2n = n3

n1 < n2 < n3

FIGURE 4.5 Three-parameter power model.

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4.2.7 Richard-Abbott ModelThe four-parameter power model is proposed by Richard and Abbott (1975) for modeling elastic-plastic stress-strain relation as shown in Figure 4.8. In the virgin loading path, the M–θr curve is written by the following expression:

(4.15)

where Rki is the initial connection stiffness, Rkp is the strain-hardening connection stiffness, n is the shape parameter, θ0 is the reference relative rotation [= M0/(Rki − Rkp)], and M0 is the reference connection moment.

MR R

Rki kp r

r

n nkp r=

−( )

+

+θ

θθ

θ

10

1

M

My

Mc

θrRki

Rkpm1

m2

1

Rkp1

FIGURE 4.6 Bounding line model.

M

(KM)0

(KM)0

θ r

θ0

θ0θ0

Rki Rki

n = n1n = n2

n1 < n2 < n3

n = n3A

2

=

FIGURE 4.7 Ramberg-Osgood model. Sa

mple

Chap

ter


4.3 Behavior Under Cyclic Loading4.3.1 Independent Hardening ModelThe independent hardening model (Chen and Saleeb, 1982) is a simple method for represent-ing behavior under cyclic loading. The characteristics of connection behavior are assumed to be unchanged following the virgin M–θr curve under the loading and reloading conditions. Since the loops of the M–θr relation in each cycle are independent, no hardening effect is considered. The cyclic M–θr relation based on this model is schematically shown in Figure 4.9. In the case of

0

M

M0

θrθ

R ki

=M0 /(Rki - Rkp)

M= θrRki

n = n1n = n2n = n3n = 8

n1 < n2 < n3

R kp

1

FIGURE 4.8 Richard-Abbott model.

M

θ r

R ki1

R ki1

R ki1

a

b

c

d

e

FIGURE 4.9 Independent hardening model.

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unloading from the point a on the initial loading curve, the connection unloads linearly down to M = 0 with the initial connection stiffness Rki. When the sign of the connection moment M changes at the point b in the unloading process, the connection enters into the reverse loading process. In this case, the connection behavior (excluding the residual plastic rotation) resulting from the previous loading process is assumed to coincide with that of the virgin connections under monotonic loading.

4.3.2 Kinematic Hardening ModelThe kinematic hardening model is a modified independent hardening model taking into account the effect of material hardening. The behavior of this connection model is illustrated in Figure 4.10, which is represented by the hardening line with the slope Rb. In the case of reversal unload-ing, the path of the M–θr curve moves along the line with the slope of the initial connection stiff-ness Rki (i.e., for line ab or cd) until it reaches the hardening line. For further reversal unloading, the path follows the virgin nonlinear M–θr curve of the connection under the monotonic loading (i.e., for line bc or de). If the hardening line has a zero slope (i.e., Rb = 0), the kinematic hardening model is exactly the same as the independent hardening model.

4.3.3 Bounding Surface Model with Internal VariablesThe previously mentioned independent and kinematic hardening models are simple to use in frame analysis. Although these models can handle the M–θr relation under one cycle loading, unloading, and reversal loading, the connection behavior for a repetition of this loading cycle cannot be expressed with acceptable accuracy. As shown in Figure 4.11, one problem associated with the independent and kinematic hardening models is that the M–θr relation is completely dif-ferent depending on whether the connection is reloaded from the region between a and b or from

1

1

1

b1θr

M

a

b

c

d

e

R ki

R ki

R

R

R

R

ki

b1

b1

FIGURE 4.10 Kinematic hardening model.

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the region between b and c. To overcome this deficiency, a bounding surface model with internal variables (Dafalias and Popov, 1976) may be used (Cook, 1983; Goto et al., 1991, 1993).

In this model, the M–θr relation is defined in the incremental form:

(4.16)

where Rkt is the tangent stiffness of the connections. The tangent connection stiffness Rkt is repre-sented in terms of the initial connection stiffness Rki and the plastic tangent connection stiffness Rkp as:

(4.17)

∆ = ∆M Rkt rθ

RR RR Rkt

kt kp

ki kp

=+

FIGURE 4.11 Problems associated with the kinematic hardening model: (a) reloading from the region between a and b; (b) reloading from the region between b and c.

a

M

θr

R ki

a

b

c

1

b

1

a

b

c

R ki

θr

M

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The plastic tangent connection stiffness Rkp is represented by using the plastic internal variables δ and δin

(4.18)

where h is the hardening shape parameter, Rb is the slope of the bounding line, δ is the distance of the current moment state from the corresponding bound, and δin is the value of δ at the initia-tion of each loading process. These quantities are schematically shown in Figure 4.12 using the moment-plastic rotation (M–θr

p) curve.

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R R hkp bin

= +−

δδ δ

1

R b

R b

1

M

θrp

R kp

R kp

R kp1

11

a

bc

bounding line

δin δin

δδ

inδ

δ

FIGURE 4.12 Bounding surface model.

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apter


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Documents

Sample - jrosspub.com · 80 Semi-rigid Connections Handbook where f and g are some mathematical functions.M is the moment at the connection, and θ r is the relative rotation of the