chap -monotonic stiffness modeling

8/11/2019 Chap -Monotonic Stiffness Modeling

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complex, incremental, and require a large number of computations. Simple com-

puter subroutines developed by the author were used for calculation of the force-

deformation and moment-rotation responses.

The first section of this chapter will address the flange deformation compo-

nent by proposing two models of different complexity and accuracy. In the second

section, deformation resulting from stem yielding and plasticity will be addressed.

In the third section, the final component, slip and bearing deformation, will be con-

sidered and a robust procedure will be proposed. Finally, a method of assembling

the various deformation components into the overall T-stub deformation will be

presented in Section 7.4, followed by a brief discussion of transforming a P- T-

stub response into an M-connection response.

7.1 Flange/Tension Bolt Model

A simple but accurate method of obtaining the force-deformation relationship

for a T-stub flange is critical to an accurate connection model. A model that uses

geometrical and mechanical properties consistent with the modified Kulak et al.

strength model was desired so as to make implementation easier by design engi-

neers. It was decided a priori that the model should incorporate the changing stiff-

ness of the tension bolts as a function of the force present in the bolts. It was also

determined that for the sake of simplicity, the model should yield a piecewise lin-

ear force-deformation relationship. Because of the emphasis placed on designing

ductility into connections, it was decided that the model must also be able to pre-

dict the response of the flange well into its plastic and strain hardening range with

a reasonable degree of accuracy. Finally, because of the length to depth ratio of

many T-stub flanges, shear deformation was considered significant enough to be

included in the formulation of the model.


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Figure 7-3: Half Model of a T-stub Flange

The system is loaded by applying a vertical displacement to the support A,

as is shown in Figure 7-4. The ratio of the vertical reaction at this support, T, to its

displacement, , is the stiffness of the flange. The value of the vertical reaction at

the pinned support at C is the prying force. The relationship between the prying

force and the displacement, , will be referred to as the prying gradient.

7.1.1 Bolt Stiffness

It was decided beforehand that the model should incorporate a variable bolt

stiffness that captures the changing behavior of the bolts as a function of the loads

that they are subjected to. Based on observations of T-stub component tests and

individual bolt tests, the bolt stiffness model shown in Table 7-1 was developed.

gt

0.5r

b a

b a

A B C


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Figure 7-4: Half Flange Model Loading

where,

B = Bolt force

Bo= Bolt pretension

Bn= Tensile capacity of the bolt

Bfract= Fracture load of the bolt

Kb= Elastic stiffness of the bolt

A graphical comparison of the model and experimental results is made in

Figure 7-5. The experimental results were taken from a direct tension bolt test.

The model is made up of four linear segments. The first segment models the bolt

Table 7-1: Bolt Stiffness Model

Bolt Force Bolt Stiffness

A

B

C

Q

T

b a

0 B Bo


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before its pretension is overcome, the second segment models the bolt during the

linear-elastic portion of its response, the third segment models the bolt after initial

yielding has started and the fourth segment models the bolt after it has reached a

plastic state. The force limits used to distinguish between the different bolt stiff-

nesses were based on the tests of individual bolts discussed in Chapter 3. The

limit of 85% of the tensile capacity is used to identify the onset of yielding. The

ultimate strength of the model is intentionally lower than that of the bolt subjected

to pure tension. This is because the bolt as it is loaded by the T-stub flange is

actually subjected to bending in addition to tension. The amount of bending is

dependent on the geometry of the flange and location of the bolt. This bending

acts to reduce the overall strength of the bolt. Another characteristic of the bolt

model is that it only extends to the point of maximum load on the experimental

curve. This is because the bolts will always be loaded in force control, regardless

of what type of loading is applied to the T-stub. As the bolts reach their point of

maximum resistance, they will elongate until fracture without displaying the

unloading shown on the experimental curve in Figure 7-5.


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Figure 7-5: Bolt Force-Elongation Model

Until the pretension in the bolts is overcome, they are assumed to be infi-

nitely rigid. The value of 1000Kbwas deemed a sufficiently high stiffness. The lin-

ear-elastic stiffness, Kb, governs the bolt response from the pretension force until

first yield, at which point the elastic stiffness is reduced by 90%. Finally, the plas-

tic portion of the bolts response is modeled by assuming a stiffness equal to 2%

of the elastic stiffness. A positive stiffness, even in the plastic range, is necessary

to ensure a stable flange system under load control. The elastic stiffness of the

bolt, Kb, was discussed in Section 3.8 and is calculated as (Barron et al., 1998b)

EQ 7-1

Pretension - Kb,1

Elastic - Kb,2

Yielding - Kb,3

Plastic - Kb,4

Bolt Fracture

0

20

40

60

80

100

120

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Bolt Elongation (in)

BoltForce(kip)

Experimental

Model

1Kb

fdbAbE

LsAbE

LtgAbeE

fdbAbeE


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where:

f = a stiffness correlation factor (Section 3.8)

db= the nominal diameter of the bolt

Ab= the nominal area of the bolt shank

Abe= the effective area of the threads

Ls= the shank length of the bolt

Ltg= the length of threads within the bolts grip

A consequence of using a plastic stiffness corresponding to 2% of the elasticstiffness for the bolt is that the model predicts very large bolt elongations at ulti-

mate. These exaggerated ultimate elongations are undesirable in a model in

which the accurate prediction of deformation capacity is required. It thus

becomes necessary to limit, or cap, the ultimate elongation of the bolt in the

model.

The ultimate elongation of the bolt is predicted as shown in Equation 7-2.

This prediction is based on the assumption that the shank of the bolt remains

elastic with the inelastic deformation concentrated in the threads that are included

in the grip. It is also recognized that a portion of the bolt inside the nut will deform

inelastically. As a result, two of these threads are included in the prediction.

Because of the way that the model is implemented, it is convenient to convert the

ultimate elongation to a fracture load. This load is referred to as Bfractand is calcu-

lated as shown in Equation 7-3.

EQ 7-2fract0.90BnLs

AbE fract Ltg 2nth


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EQ 7-3

where

Ab = the nominal or gross area of the bolt

Ls= the length of the bolt shank

Ltg= the length of the threaded portion included in the bolts grip

fract= the fracture strain of the bolt material

Bn= the tensile capacity of the bolt

Kb= the elastic stiffness of the bolt

nth= the number of threads per inch of the bolt

7.1.2 Elastic-Plastic Flange Model

The basic flange stiffness model considers only the limits of plastic hinges

forming at the K-zone and bolt line, leading to the formation of a plastic mecha-

nism. If a uniform beam is assumed, it can be shown that the plastic hinge will

always form at the K-zone before it forms at the bolt hole. It is important to recog-

nize, however, that a significant amount of material is removed from the flange

when the holes are drilled for the bolts. Because of this, it is possible for a plastic

hinge to form at the hole before one forms at the K-zone. Bearing this in mind, the

decision tree shown in Figure 7-6 presents the possible flange states. These var-

ious flange states are then supplemented by the four different bolt stiffnesses

resulting in a model that contains up to seven different stiffnesses.

Bfract fract0.85Bn

Kb

0.90 0.85( )Bn

0.10Kb

0.02Kb( ) 0.90Bn


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determined by using the product of the strain hardening modulus of the steel and

the moment of inertia of the flange divided by the length of the plastic hinge as

shown in Equations 7-8(h) and 7-8(i) (White, 1999; Douty, 1964). The hinge

length was assumed to be equal to the thickness of flange. Note also that the

model in its present form is purely mechanistic.

EQ 7-4(a)

EQ 7-4(b)

EQ 7-4(c)

EQ 7-5(a)

EQ 7-5(b)

EQ 7-5(c)

EQ 7-6(a)

Kee,k12EI 3EI Kb,k3( )

ee,k

Qee,k18EI Kb,kab2b 2EI( )

ee,k

ee,k 12EI1 Kb,k2

Kpe,k12EI 3EI Kb,ka2 Kh1( ) Kb,kKh1 3[ ]

pe,k

Qpe,k18EI 2EI Kb,kab Kh1( ) Kb,kKh1ab2b[ ]

pe,k

pe,k 12EI Kh11 Kb,k a3b2a a2b3b( ) 3EI4 ( ) Kb,kKh12

Kep,k12EI Kb,kKh23 3EI Kh2 Kb,ka2( )[ ]

ep,k


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EQ 7-6(b)

EQ 7-6(c)

EQ 7-7(a)

EQ 7-7(b)

EQ 7-7(c)

where,

EQ 7-8(a)

EQ 7-8(b)

EQ 7-8(c)

EQ 7-8(d)

Qep,k18EIKh2 Kb,kab2b 2EI( )

ep,k

ep,k 12EI Kh21 Kb,ka2b3b 3EIa2 ( ) Kb,kKh2 2

Kpp,kKh1Kh2 Kb,ka2 Kh1 Kh2( )

pp.k

Qpp,kKh2 Kb,kab Kh1( )

pp,k

pp,k Kh2 4 Kh1a Kb,ka2b2

1 b b3 3ab2 3a2b ( ) a3a

2 3a 2b4b2 4a 3b3ab

3 a3a 3a 2bb

4 a2 2a b b2


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EQ 7-8(e)

EQ 7-8(f)

EQ 7-8(g)

EQ 7-8(h)

EQ 7-8(i)

The stiffnesses and prying gradients were derived to be used in an incre-mental solution technique. The incremental applied load and prying force can be

calculated as shown in Equations 7-9 and 7-10. An engineer would begin by

determining the initial stiffness, Kee,1and initial prying gradient, Qee,1. Next, sev-

eral checks would be made to determine which limit will be reached first. Potential

limits include the bolt force limits that define which bolt stiffnesses are to be used,

moment limits at joints A and B, and total flange separation limits that are possible

when the prying gradient is negative. Incremental displacements are then calcu-

lated for each of the potential limits with the smallest value governing. Finally, the

moments at joints A and B, the prying force, the bolt force, the applied load, and

Ipt f3

12

a 1 12EIGptfa2

b 1 12EIGptfb2

Kh1EsItf

Kh2 1dhp

EsI

tf


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the new stiffness and prying gradient are calculated and the process is repeated

until the bolt force reaches Bfract

.

EQ 7-9

EQ 7-10

Considering force equilibrium of the system, the force in the bolt, B, after the

pretension has been overcome can be shown as the sum of the applied load, T,

and the prying force, Q. Moment equilibrium of the system yields the moments MA

and MBat joints A and B, respectively.

EQ 7-11

EQ 7-12

EQ 7-13

Incremental values are then calculated from these relationships.

EQ 7-14

T Kij,k

Q Qij,k

B T Q

MA Tb Qa

MB Qa

B T Q


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EQ 7-15

EQ 7-16

Substituting these values for the incremental applied load and prying force

into Equations 7-14 through 7-16 and solving for the incremental displacement,

, yields the bolt force and moment limits.

EQ 7-17

EQ 7-18

EQ 7-19

When the prying gradient is negative, the possibility of the T-stub flange separat-

ing completely from the column must also be checked.

EQ 7-20

MA Tb Qa

MB Qa

1 BKij,k Qij,k

2MA

Kij,kb Qij,ka

3MB

Qij,ka

4 QQij,k


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The bolt force limits are calculated as was described previously in Section

7.1.1. The moment limits are simply the plastic moments at the K-zone and bolt

line.

EQ 7-21

EQ 7-22

7.1.3 Elastic-Yielding-Plastic Flange Model

The elastic-plastic flange model yielded acceptable results for relatively flex-

ible flanges, but was less accurate when used on stiffer flanges having small ten-

sion bolt gages. The cause of the inaccuracy was believed to be attributable toyielding in the stiffer flanges prior to the formation of plastic hinges. The elastic-

plastic flange model is not able to detect any loss of stiffness due to these

sources. It was therefore determined that a partially plastic flange was needed in

the model.

Adding the partially plastic flange state to the model complicates the formu-

lation considerably. The elastic-plastic decision tree shown previously in Figure 7-

6 is revised to include the possible partially plastic states and is shown as Figure

7-7. Regardless of which path is followed through the tree, five different flange

states are possible. Combined with the four possible stiffness states of the ten-

sion bolts, a total nine different stiffnesses may be experienced in this model

MpAFypt f

2

4

MpB 1 dhp

Fypt f

2

4


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before failure. Although this may seem overly complicated, it is unlikely that even

the elastic-plastic model would ever be used extensively without being programed

into a simple computer subroutine. With this in mind, there seems little use for the

elastic-plastic model. In fact, the refined model is relatively efficient considering

the alternatives. Most other flange models that incorporate strain hardening and

shear deformation are iterative and no other models are known that incorporate a

changing bolt stiffness.

A complete derivation of the stiffnesses and prying gradients of the partially

plastic states was deemed too complicated for the present work. Furthermore,

the theoretical partially plastic stiffnesses would most definitely be nonlinear and

would thus yield nonlinear force-deformation relationships. For these reasons, a

rational system of weighted averages of the fully plastic states was used to deter-

mine the partially plastic stiffnesses and prying gradients. The results are shown

as Equations 7-23 through 7-27.

EQ 7-23

EQ 7-24

EQ 7-25

EQ 7-26

EQ 7-27

Kye k,Kee k, 3Kpe k,

4 Qye k,

Qee k, 3Qpe k,4

Key k,Kee k, 3Kep k,

4 Qey k,

Qee k, 3Qep k,4

Kpy k,Kpe k, 3Kpp k,

4 Qpy k,

Qpe k, 3Qpp k,4

Kyp k,Kep k, 3Kpp k,

4 Qyp k,

Qep k, 3Qpp k,4

Kyy k,Kee k, 3Kpp k,

4 Qyy k,

Qee k, 3Qpp k,4


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The partially plastic stiffnesses were weighted towards the more flexible of

the two plastic states because it was thought that the added deformation of these

states contributed more to the overall behavior during the yielding than the stiffer

states. A calibration of possible weights resulted in the ratio of 1:3 that was used

in Equations 7-23 through 7-27.

The solution technique is the same for the elastic-yielding-plastic flange

model as for the elastic-plastic model except that yield moment limits must be

checked in addition to the plastic moments. The yield moments are calculated as

EQ 7-28

. EQ 7-29

It should be noted that the reduced cross section resulting from the material

lost to the bolt holes is accounted for in the moment limits but is not accounted for

in the moment of inertia or shear stiffness factors. The influence on the overall

stiffness was not significant enough to warrant the added complexity.

A comparison of the model prediction to experimental results for representa-

tive T-stubs are show in Figures 7-8 through 7-13. The general response of the

model compares well with the experimental data. For those T-stubs that failed

with tension bolt fractures, the ultimate deformation is predicted with reasonable

accuracy. The response of stiffer flanges does not match as well as for more flex-

ible flanges. This is likely because of the complexities associated with partially

plastic hinges and shear deformations.

MyA23MpA

MyB23MpB


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Figure 7-10: TB-05 Flange Stiffness Model Comparison

Figure 7-11: TB-06 Flange Stiffness Model Comparison

0

100

200

300

400

500

600

700

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Uplift (inch)

AppliedLoad,

P

(kip)

Experimental

Model

Net Section Fracture

0

100

200

300

400

500

600

700

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Uplift (inch)

AppliedLoad,

P

(kip)

Experimental

Model

Net Section Fracture


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Figure 7-12: TC-04 Flange Stiffness Model Comparison

Figure 7-13: TC-12 Flange Stiffness Model Comparison

0

100

200

300

400

500

600

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Uplift (inch)

AppliedLoad,

P

(kip)

Experimental

Model

0

100

200

300

400

500

600

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Uplift (inch)

AppliedLoad,

P

(kip)

Experimental

Model


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7.1.3.1 Membrane Action

One limitation of the flange model is that in its current state it is not able

include effects of membrane action that can be important when thin, flexible

flanges are considered. To consider this mechanism, the plastic mechanism stiff-

ness was modified to include second order, nonlinear geometric effects for the

portion of the flange between the bolt line and stem. Equations 7-30(a) through 7-

30(c) replace the previously presented formulations for the plastic-plastic flange

mechanism stiffness and prying gradient shown as Equations 7-7(a) through 7-

7(c). Setting ABequal to zero in Equations 7-30(a) through 7-30(c) yields Equa-

tions 7-7(a) through 7-7(c).

EQ 7-30(a)

EQ 7-30(b)

EQ 7-30(c)

where

AB= the displacement of the flange near the K-zone minus the dis-placement at the bolt line (- )

AB= the axial elongation of the flange between the bolt line andstem given by

Kpp,kAB b( ) Kb,ka

2Kh1 Kh2( ) Kh1Kh2[ ] pt fABbE Kh2 Kb,ka( )

pp,k


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EQ 7-31

An examination of Equations 7-30(a) through 7-30(c) shows that, as would

be expected for a geometrically nonlinear problem, the stiffness and prying gradi-

ent are functions of the centerline flange displacement, , and the bolt line flange

displacement, . As a result, the solution of the problem is now dependent on the

increment size of the solution process.

None of the T-stubs tested had flange stiffnesses low enough to properly

illustrate and calibrate the model. As a result, the analytical model will be com-

pared to a finite element model that was developed specifically for the purpose. A

T-stub cut from a W16 x67 was chosen as a test case and a detail of the flange is

shown in Figure 7-14. Four different load-deformation responses are shown in

the figure. The curve labeled Analytical w/out Membrane and ABAQUS w/out

Membrane represent the model responses without including the effects of mem-brane action. The ABAQUS model was configured identically to the 2D flange

model discussed in Section 5.2. The curves labeled Analytical with Membrane

and ABAQUS with Membrane represent the model responses including the

effects of membrane action. The membrane effects were added to the ABAQUS

model by restraining the nodes on the bolt line through the thickness of the flange

against horizontal translation. Both ABAQUS models include simple tri-linear

material models. It was discovered that the response of the analytical model

grossly overestimated the membrane stiffness of the flange when a totally elastic

material model was assumed. As a result, a tri-linear material model was added

AB AB2

b2

b


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to incorporate axial yielding in the model. The analytical model including mem-

brane effects roughly predicts the same response as the ABAQUS model.

Figure 7-14: Detail of Thin, Flexible T-stub Flange Susceptible to Membrane Action

The model has limitations. First, the tri-linear material model used in the

analytical model to account for axial yielding does not interact with the material

model used to monitor the yielding caused by bending. The ABAQUS model

inherently included this interaction and this difference may be a cause of the dis-

crepancy between the two curves in Figure 7-15. Second, the clearance between

bolt and the bolt hole is not accounted for in any way. The actual flange will not be

totally restrained as was assumed here. Instead, the bolt line will be free to travel

horizontally until the bolt comes into contact with the bolt hole. At that point the

restraint will not be total but will instead be more of an elastic restraint dependent

on factors such bearing of the bolt on the bolt hole, local bending of the bolt, and

shear deformations of the bolt.

3/8"

11/16

"

7"

101/4"


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Figure 7-15: Analytical and Finite Element Model Results Including Membrane Action

One eventuality that was uncovered during the analysis of the example T-

stub flange is the introduction of significant shear forces into the tension bolts as a

direct result of the membrane action. Figure 7-16 shows the relationship between

the pseudo shear force introduced to a tension bolt and the total applied load.

The pseudo shear force is the horizontal reaction at the bolt line from the finite ele-

ment model. As the figure shows, the shear force increases as the load and dis-

placement increase. Figure 7-17 shows the bolt force response from the

ABAQUS model plotted on an interaction diagram for the bolt.1 As the figure

shows, the levels of shear force introduced into the bolts in the model are suffi-

cient to reduce the tensile capacity.

1. Interaction diagrams for bolts are discussed in Section 3.7.5

0

50

100

150

200

250

300

350

400

450

500

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Flange Uplift (in)

AppliedLoad(kip)

Analytical w/out MembraneAnalytical with Membrane

ABAQUS w/out Membrane

ABAQUS with Membrane


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7.2 Stem Model

As was previously noted, the load-deformation behavior of the stem is very

complex. The stress distributions revealed by the finite element analyses, shear

bolt interaction, and frictional forces combine to make a simple but rigorous treat-

ment of the stem response impossible. Consequently, a semi-rational bi-linear

model will be developed that satisfactorily predicts the initial stiffness, yield load,

plastic or secondary stiffness, and ultimate deformation.

7.2.1 Elastic Stiffness

To obtain the elastic stiffness of the stem, a tapered beam model similar to

that shown in Figure 7-18 was utilized. Lines were drawn from the center of the

first two bolt holes to the intersection of the tapered edges of the stem with the

gross section. The angle of these lines relative to the horizontal was limited to a

value no greater than defined in the discussion of the modified Whitmore

strength model (Section 6.2.1.3). The material outside of these lines was not con-

sidered to participate as far as stiffness is concerned. No account was taken of

the area of the stem lost during drilling or punching of the shear bolt holes. The

load was assumed to be distributed uniformly along the tapered length of the

stem. This idealization is not far from the actual condition when the effects of fric-

tion are considered. Only when the stem enters into its nonlinear range will this

assumption be grossly inaccurate. At that point, however, the secondary stiff-

ness, which is independent of the force transfer mechanism, governs. The elastic

stiffness of the stem can be written as

ef f


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. EQ 7-32

Figure 7-18: Stem Stiffness Model

7.2.2 Yield Load

The yield load, or load at which the load-deformation response of the stem

becomes non-linear, was predicted by multiplying the area of the net section by

the materials yield stress, as is shown in Equation 7-33. The actual stem does

not start yielding uniformly because of the stress concentrations that are created

by the holes and the effects of the taper. This approximation, though, provides

reliable and relatively accurate results.

Ke st em,4Lsb tsE ef ftan( )

2

2Lsb ef ftan gs gs2Lsb ef ftan gs

ln

gs

wT-stub

wbeam

Lsb

Lgrossk


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EQ 7-33

where

Pyield= the force required to initiate stem yielding

Fy= the yield strength of the T-stem base material

Weff= the effective width of the T-stem (Section 6.2.1.3)

dh,eff= the effective bolt hole diameter

7.2.3 Plastic Stiffness

The plastic stiffness was based on the assumption that the material between

the last two bolt holes, Figure 7-19, yields and starts to strain harden before the

rest of the cross section. This is consistent with observation from component test-

ing and finite element modeling. The length of the strain hardening area is taken

as 3db, which leads to a plastic stem stiffness of

. EQ 7-34

Pyield Fy Wef f 2dh,eff( )

Kp,stemgs dh eff,( )tsEs

3db


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Figure 7-19: Area of Concentrated Strain Hardening

7.2.4 Deformation at Fracture

As the yielding in the material progresses in the stem, a fracture initiates

between the last two bolt holes. The total deformation at fracture can be esti-

mated as the sum of the plastic deformation in this region, as is illustrated in Fig-

ure 7-20, and the elastic deformation in the rest of the stem. This is not entirely

consistent with the plastic stiffness determination because the plastic stiffness

was based on an assumed yield zone with a length of 3dbwhile an assumed yield

length of 1db is assumed here. Regardless, the following equation provides rea-

sonable results.

EQ 7-35

3db

stem,fract fractdh,effPyield

Ke stem,


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Figure 7-20: Net Section Fracture Initiation Zone

7.2.5 Results

A summary of the stem deformation data was compared to the values com-

puted using Equations 7-32 through 7-35. The results are presented in Table 7-2.

The yield load predictions were quite accurate. The elastic stiffness values were

somewhat lower than those observed during testing. Because of the stems high

elastic stiffness relative to other components, however, an error of 20 to 30% will

not greatly affect the predicted overall force-deformation behavior. The plastic

stiffness predictions are also somewhat lower than the actual stiffnesses. Finally,

the deformations at fracture predicted by Equation 7-35 are slightly nonconserva-

tive. The test data are not entirely reliable for this value though, because of local

bearing deformations near the mounting for displacement instrumentation.

db


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Table 7-2: Stem Deformation Model Accuracy

Figures 7-21 though Figure 7-25 show comparisons on the stem model with

experimental force deformation curves from the component tests. The T-stubs

that were chosen for comparison were those with narrow tension bolt gages and

substantial stem deformations. T-stubs with wider tension bolt gages did not

always sustain inelastic stem deformations. It should be noted, however, that the

stems of the T-stubs within a group were the same and the model will predict iden-

tical behavior. That is, the stem of T-stub TA-05 was identical to that of TA-07 and

as a result, the models predicted behavior is the same.

Figure 7-21: Model Comparison for T-stub TA-05

Average Standard% Error Deviation

Yield Load 0.2% 8.7%

Elastic Stifness -19.7% 32.7%

Plastic Stiffness -21.6% 41.2%

Fract Deformation 12.7% 35.9%

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Stem Deformation (in)

AppliedLoad(kip)

Experimental

Model


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0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40


AppliedLoad(kip)

Experimental

Model

T-bolt Fracture

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Stem Defomation (in)

AppliedLoad(kip)

Experimental

Model


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Figure 7-24: Model Comparison for T-stub TB-05

Figure 7-25: Model Comparison for T-stub TC-09

0

100

200

300

400

500

600

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45


AppliedLoad(kip)

Experimental

Model

0

100

200

300

400

500

600

700

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40


AppliedLoad(kip)

Experimental

Model


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7.3 Slip/Bearing Model

As was discussed earlier in the chapter, the bearing deformation of the T-

stem, bearing deformation of the beam flange, and relative slip between the beam

flange and T-stem will be treated together. Some treatment of the individual

mechanism is required, though, before they can be combined. Work by Rex and

Easterling (1996a, 1996b) will be used extensively to characterize the behavior of

both the slip and bearing deformation.

7.3.1 Bearing Mechanism

A modified version of the bearing deformation model developed by Rex and

Easterling (1996a) will be adopted for use in this work. In their work, Rex and

Easterling considered many existing bearing models including the model used in

the Eurocode (1993), a model developed by Tate and Rosenfeld (1946) and a

model developed by Vogt (1947). After testing single bolt lap splices and simple

bearing assemblages and conducting finite element analyses, Equation 7-36 was

developed to predict the deformation of a bolt bearing on a plate. The relationship

is an application of the Richard equation.1

EQ 7-36

1. A background of the Richard equation (three parameter power model) is not provided inthe present work. For a thorough background and rigorous treatment of applications of

the Richard equation, the reader is referred to Richard and Abbott (1975) and Rex andEasterling (1996a, 1996b)

PbearingRn bea ring,

1.741 0.5( )2

0.009


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where

Pbearing

= plate load

Rn,bearing= nominal bearing strength ( )

Le= end distance of plate

= normalized deformation ( )

bearing= bearing defromation or hole elongation

= steel correction factor

Ki,bearing= initial bearing stiffness

The % elongation is taken as 30% for typical steel which yields a steel cor-

rection factor of unity. The initial bearing stiffness required was given by Rex and

Easterling as

EQ 7-37

where

Kbr= bolt bearing stiffness

Kbe= bending stiffness

Kve= shear stiffness

The bearing stiffness in Equation 7-37, Kbr, was derived by considering a bolt

and hole in their deformed state as shown in Figure 7-26. The deformed material

shown in grey is assumed to have reached its ultimate stress. Thus the force

exerted on the plate by the bolt is the product of the materials ultimate stress and

Rn LetpFu 2.4dbtpFu

Ki,bearing Rn,bearing

30%% Elongation

Ki,bearing1

1Kbr

1Kbe

1Kve


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Comparison of bearing stiffnesses computed using Equation 7-40 with those

obtained experimentally and with finite element analyses showed that the relation-

ship between the bolt diameter, db, was not completely linear. The error was

attributed to simplifying assumptions made in the derivation of Kbrand a factor of

0.8 was added to db. The resulting relationship is

. EQ 7-41

The bending and shearing stiffnesses, Kband Kv, refer to the stiffness asso-

ciated with the end distance of a lap plate. The end of the plate was modeled as a

short, deep beam as shown in Figure 7-27. The bending and shear stiffnesses of

the fixed-fixed beam can be written in terms of the inverse of the beam slender-

ness as

, EQ 7-42

, EQ 7-43

and . EQ 7-44

Kbr 120Fytpdb0.8

Kbe 32EtphL

3

Kve 6.67GtphL

hL

Ledb 0.5


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259

the literature reviewed attempted to characterize the pre or post slip behavior.

Karsu tested a total of 61 lap plate connections and investigated the effects of

varying parameters such as plate thickness, end distance, edge condition, and

bolt diameter. Gillett conducted 75 lap plate tests, of which data is available for 66

tests. Varied parameters include plate thickness, end distance, steel grade, and

bolt diameter and grade. Caccavale (1975) documented 11 lap splice tests with

the plate thickness being the primary variable. Sarkar and Wallace (1992) docu-

mented 16 lap splice tests. Data supplied independently to Rex and Easterling

(1996b) contained 19 tests with varied parameters including the end distance,

plate thickness, and bolt type. Frank and Yura (1981) conducted 77 tests of steel

plates in double shear and reported slip-deformation relationships similar to that

shown in Figure 7-28. The load-slip curve shows a linear initial portion up to a slip

load followed by a degrading post slip relationship. This behavior was also noted

by Gillett (1978).

Figure 7-28: Slip Deformation

Slip Load

Applied

Load

Measured Slip


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The model proposed by Rex and Easterling characterizes the slip behavior

by using three parameters; the initial slip stiffness, Kfi

, the slip load, Pslip

, and the

post slip stiffness, Kfp. Using a constant stiffness for the post slip portion of the

curve implies a linear load-deformation relationship. Although this is clearly not

the case based on Figure 7-28, it was deemed sufficiently accurate. The three

parameter method or rational method as it is referred to by Rex and Easterling,

is shown graphically in Figure 7-29.

Figure 7-29: Proposed Slip Model

The slip load prediction was based on the LRFD and recommendations

given by Fisher et al and is given as

A

ppliedLoad

Measured Slip

slip

Kfp

Kfi

Pslip

fu


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EQ 7-45

where

nsb= number of shear bolts

slip= 1.0 for A325 bolts and 0.88 for A490 bolts

= coefficient of friction between the two plates

Fu= ultimate strength of the bolt material

Ab= nominal area of the bolt

The stiffnesses Kfiand Kfpwere determined as functions of the displacements slip

and fuas

EQ 7-46

. EQ 7-47

value of0.0076 in with a COV of 46% was determined for slipby conducting a

statistical analysis of data reported by Karsu and Gillett and fuwas determined to

be a function of the thickness of the joined plates as is shown in Table 7-3.

Pslip nsbslip 0.70Fu( ) 0.75Ab( )

KfiPslipslip

KfpPslip

slip fu


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7.3.2.1 Accuracy

Rex and Easterling compared the predicted values from several slip load

models with experimental test data to evaluate the models accuracy. The LRFD

(AISC, 1994) model had an average difference of -23.6% with a coefficient of vari-

ation of 23% while Rexs rational model showed and average difference of -8.3%

with a coefficient of variation 22.0%. Rex and Easterling also compared predic-

tions of the slip stiffnesses from their model with test data and obtained an aver-

age difference of -2.0% for Kfi with a coefficient of variation of 42.0% and an

average difference of 0.0% for Kfp with a coefficient of variation of 29.0%.

The slip loads from the T-stub tests conducted in this research were com-

pared to the prediction of the LRFD model and Rexs rational model. An average

difference of -22.1% with a standard deviation of 10.7% was obtained for LRFD

predictions and an difference of -7.4% with a standard deviation of 11.9% was

obtained using Rexs rational model. A class A surface was assumed for the T-

stub faying surfaces. No attempt was made to extract the load-slip relationship

from the T-stub test data for evaluation of the stiffnesses. The accuracy of theextracted data would not have justified the effort required.

7.3.3 Combining the Slip and Bearing Mechanisms

Before the bearing and slip models developed by Rex and Easterling can be

directly implemented into a the procedure, some minor changes and simplifica-

Table 7-3: Ultimate Slip Deformation Definition

(tp1+ tp2) fu(tp1+ tp2)


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tions are required. First, the bearing model was developed for the case of a single

bolt lap splice. As a result, some manipulation is required for application to a T-

stem containing multiple shear bolts. The most significant alteration concerns the

bending and shear stiffnesses used in the evaluation of the initial bearing stiff-

ness, Ki,bearing. Rexs model considered initial stiffness contributions from bolt

bearing, Kbr, bending of the material contained within the end distance, Kbe, and

the shear stiffness of the material contained within the end distance, Kve. In a typ-

ical T-stub stem, there are an even number of bolts with only one pair subjected to

these end conditions. The initial stiffness of the remaining bolts, though, will be

unaffected by the end conditions and can predicted by the bearing stiffness, K br,

directly. Because of the way the procedure is implemented, consideration of the

influence of Kbe and Kvefor only the two end bolts is excessively cumbersome. As

a result, the influence of Kbe and Kveon the two end bolts will be neglected and the

initial stiffness of all of the shear bolts will be set equal to Kbr.

The bearing model is implemented in displacement control (i.e. Equation 7-

36 yields a force as a function of a given displacement). It is more convenient to

implement the model in force control but Equation 7-36 cannot easily be solved to

provide a displacement as a function of the force. Additionally, the bearing defor-

mation model in its present form provides a continuous response. Since the

model being developed is multi-linear, the continuous response of the bearing

model will be represented by a series of straight lines. To accomplish this, a peak

load is sent to the routine. If the peak load is greater than the slip load, the differ-

ence between the peak load and slip load is divided into a predetermined number

of bearing load steps. The routine plots the points for the slip plateau and first

contact of the bolts with the holes. For each of the bearing load steps, an iterative


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process is used to find the bearing deformations of the stem and beam flange that

yield a bearing force that, when added to the frictional force, will equal the target

load.

After the connection has exceeded its slip load, the routine estimates a bear-

ing deformation for the stem and calculates the associated load. Next, a separate

iterative loop is used to find the bearing deformation of the beam flange that will

result in the load equal to the load developed by the stem bearing deformation.

The sum of the stem bearing deformation, the beam flange bearing deformation,

and the initial clear distance between the bolts and the holes is the total slip dis-

tance and is used to calculate to the frictional force from the slip model. The fric-

tional force is then added to the bearing force. If the total force is less than the

target load for the step, a larger bearing deformation for the stem is estimated and

the procedure is repeated. If the total load is larger than the target load for the

step, the deformation increment is reduced until sufficient accuracy is achieved.

The entire procedure is illustrated as a flowchart in Figure 7-30.


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Figure 7-30: Slip and Bearing Deformation Flowchart

Enter the routine with a targetforce, P

target, and select a trial

stem bearing deformation,

bearing,stem

Select a trial flange bearing

deformation, bearing,flange

Find the force associated withthe trial stem bearing

deformation, Pbearing,stem

Find the force associated withthe trial flange bearing

deformation, Pbearing,flange

Does Pbearing,flange

= Pbearing,stem

?

Yes

No

Adjustbearing,flange

total= clear slip+ bearing

bearing= bearing,stem+ bearing,flange

Find Pslip

from total

Ptotal

= Pslip

+ Pbearing,stem

Does Ptotal= Ptarget?

Yes

No

Adjust

bearing,stem

Load Step Finished


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Figure 7-31 shows a comparison of the model with experimental results for

T-stub TA-07. The predicted behavior differs significantly from the experimental.

Two factors contribute to the difference. First, the model assumes that there is no

interaction of the shear bolt bearing deformation with the stem deformation. In the

actual T-stub, however, stem deformations influence the behavior of the bolt bear-

ing by forcing more deformation into the last pair of bolts than the first. Secondly,

the model assumes that all of the bolt holes are drilled precisely where they were

supposed to be and that the bolts are located directly in the center of the holes. In

reality, the holes are drilled close to where theyre supposed to be but not exactly,

and the bolts are aligned in a relatively random manner. Because the bolts are

relatively free to slide where they want to, the effects of lack of fit, or the variation

of hole location, impact the slip behavior to a greater degree than the alignment of

the bolts, particularly when cyclic behavior is considered. In the model, though,

the bolts are not free to slide where they want to and it is more convenient to

account for the lack of fit issues by altering the alignment of the bolts than it is to

consider variable hole locations.


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Figure 7-31: Comparison of Uniform Bearing Model with Experimental Results for T-stub TA-07

Figure 7-32 shows the results obtained using a model based on the linear

and spread bolt alignments shown in Figure 5-35. The routine was modified to

include variable bolt locations. The initial bolt location, bearing deformation, and

bearing force are kept track of independently for each pair of bolts in the model.

This allows each pair of bolts to be located arbitrarily.

An additional source of error is related to the testing procedure and appara-

tus. Referring to Figures 4-8 and 4-3, the member that was used to load the T-

stubs, the force element, was a built up tee section of a WT section and con-

nected the upper header beam to the T-stub specimen. Each force element was

used to test several different T-stubs. As a result, the bolt holes on the end that

was bolted to the T-stem sustain cumulative bearing deformation. This accumu-

lated damage led to longer free slip distances. Measured bearing deformation up

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Slip Deformation (in)

AppliedLoad(kip)


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to 0.125 were noted. As a result of these deformations, T-stub tests that were the

first to be loaded with a given force element are used for comparison of the slip/

bearing deformation model. The comparisons are shown in Figures 7-33 through

7-36

Figure 7-32: Linear and Spread Slip/Bearing Models for T-stub TA-07

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50


AppliedLoad(kip)

Experimental

Linear Alignment

Spread Alignment


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Figure 7-33: Linear Slip/Bearing Model for T-stub TA-12

Figure 7-34: Linear Slip/Bearing Model for T-stub TB-08

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Slip Defomation (in)

AppliedLoad(kip)

0

50

100

150

200

250

300

350

400

450

500

550

600

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50


AppliedLoad(kip)


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Figure 7-35: Linear Slip/Bearing Model for T-stub TC-07

Figure 7-36: Linear Slip/Bearing Model for T-stub TC-15

0

50

100

150

200

250

300

350

400

450

500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50


AppliedLoad(kip)

0

50

100

150

200

250

300

350

400

450

500

550

600

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50


AppliedLoad(kip)


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7.4 Assembly of the Total Model

After each of the different deformation mechanisms have been examined,

they can be assembled into the total deformation response of the T-stub and

finally, into the moment-rotation curve of the connection. The total deformation

response is assembled by adding the deformations from the various mechanisms

together at common loads. Figure 7-37 illustrates the assembly process.

Because each of the mechanism responses has independent load points due to

specific local behavior, linear interpolation is required to insure that the total defor-

mation response includes all of the load points from all of the individual mecha-

nisms. Figures 7-39 though 7-44 show comparisons of the predicted and

experimental behavior for several T-stubs. When examining the comparisons, the

reader is reminded that the experimental results of T-stubs TA-05, TC-07, and TD-

01 include excessive bearing and slip deformations that resulted from cumulative

damage to the beam flanges of the components test series.


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Figure 7-37: Assembly of the Individual Deformation Components

Figure 7-38: Comparison of Predicted and Experimental Total Deformation for T-stub TA-05

Flange Stem Slip & Bearing

0

50

100

150

200

250

300

350

400

450

500

-0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Total Deformation (in)

Applied

Load

(kip)

Flange

Flange + Stem

Flange + Stem + Slip

0

100

200

300

400

500

600

-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

T-stub Deformation (in)

AppliedLoad(kip)

Experimental

Model


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0

50

100

150

200

250

300

350

400

450

500

-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80


AppliedLoad(kip)

Experimental

Model

0

50

100

150

200

250

300

350

400

450

500

-0.10 0.10 0.30 0.50 0.70 0.90 1.10 1.30


AppliedLoad(kip)

Experimental

Model


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Figure 7-41: Comparison of Predicted and Experimental Total Deformation for T-stub TB-08

Figure 7-42: Comparison of Predicted and Experimental Total Deformation for T-stub TC-07

0

100

200

300

400

500

600

-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


AppliedLoad(kip)

Experimental

Model

0

50

100

150

200

250

300

350

400

450

500

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45


AppliedLoad(kip)

Experimental

Model


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The moment-rotation response of the connection is computed by combining

two monotonic curves for the top and bottom T-stub. One T-stub is subjected to

monotonic tension while the second is subjected to monotonic compression. The

moment is calculated as the T-stub force multiplied by the beam depth and the

rotation is calculated by the sum of the displacements of the two T-stubs at a

given load divided by the beam depth. Figure 7-45 shows the deformation curves

of the two T-stubs (TA-07) that were used to create the monotonic moment-rota-

tion curve shown in Figure 7-46. The moment-rotation curve is based on a 24

deep beam.

Figure 7-45: Full Range (Tension/Compression) Force/Deformation Curve for T-stub TA-07

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Total Deformation (in)

AppliedLoad(kip)


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chap -monotonic stiffness modeling

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