Abstract— In order to dissipate input earthquake energy in the Moment Resisting Frame (MRF) and Concentrically Braced Frame (CBF), inelastic deformation in main structural members, requires high expense to repair or replace the damaged structural parts. The new proposed knee braced frame in which the diagonal brace provide most of the lateral stiffness and the knee anchor that is a secondary member, provides ductility through flexural yielding. In this case, the structural damages caused by an earthquake will be concentrated on these members, which can be easily replaced by reasonable cost.

In this investigation, using non-linear and linear static analysis of several knee Braced Frames (KBF), the seismic behavior of this system is assessed for controlling the vulnerability of the main and the secondary elements. Seismic parameters and mechanism of plastic hinges formation of both frame types are investigated by using the non-linear analysis.

Keywords—knee braced frame, seismic parameters, energy

dissipation.

I. INTRODUCTION HE seismic design of steel structures must satisfy two main criteria. These structures must have adequate

strength and stiffness to control interstory drift in order that prevent damage to the structural and non-structural elements during moderate but frequent excitations.

Under extreme seismic excitations, the structures must have sufficient strength and ductility to prevent collapse. MRF and CBF have been used as lateral load resisting structural systems in steel buildings, since stiffness and ductility are generally two opposing properties, neither of MRF or CBF, alone can economically fulfill these two criteria.

Although the MRF is good for ductility and the CBF is good for stiffness, by combining the good features of these two systems into a hybrid system, an economical seismic-resistant structural system can be obtained. One such system is Eccentric Braced Frame (EBF) proposed by Roeder and Popov [2].

Recently, Aristizabal Ochoa [3] has proposed an alternative system, the Knee Braced Frame (KBF). In this system, the knee element acts as a `ductile fuse` to prevent collapse of the structure under extreme seismic excitations by dissipating energy through flexural yielding. A diagonal brace with at least one end connected to the knee element provides most of the elastic lateral stiffness. In this system, however,

M. Naeemi was with Department of Civil Engineering, Iran University of

Science and Technology, Tehran, Iran., (e-mail: m.naeemi@dot-corp.com). M. Bozorg, is with Engineering Department, Tarbiat Modares University,

Tehran, Iran. (Phone: 21-88876283; e-mail: majidcivil2003@yahoo.com).

the brace was not designed for compression and thus allowed to buckle. Consequently, the hysteretic response of this structure will be very similar to that of CBF with pinching in the hysteretic loops, which is not a desirable feature for energy dissipation.

II. SAMPLE MODELS OF FRAMES In Fig. 1 different types of KBF systems are shown. They

are referred to as (a) K-knee braced frame (b) X-knee braced frame (c) knee braced frame with single brace and one knee element (d) knee braced frame with single brace and with two knee elements.

Fig. 1 Different knee brace frames: (a) K-KBF, (b) X-KBF, (c) KBF

with single brace and 1KE, (d) KBF with single brace and 2KEs The optimal shape of KBF is selected from the above

systems according to the elastic analysis results of them. And the optimal angle of the knee element achieved when the frame has the maximum stiffness, which the tangential ratio of (b/h)/(B/H) is nearly one, it means that the knee element should be parallel to the diagonal direction of the frame, and the diagonal element passes through the mid point of the knee element and the beam-column intersection, as shown in Fig. 2.

mbhb

HB

mhHh

0.133.1,33.134

75.025.0

=→===

=→=

Fig. 2 The selected shape and dimension of the sample frames In this study the framing systems with two equal side spans

4m long are braced and length of the middle span is 5m. The

Seismic Performance of Knee Braced Frame Mina Naeemi and Majid Bozorg

T

mBmH

43

==

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number of frame stories is selected so that investigates the rigid, semi-rigid, moderate and ductile structures. Therefore the frames are chosen in four levels 5-story, 10-story, 15-story, and 20-story. For instant, the 5-story frame is shown in Fig. 3

Fig. 3 An example of under-study frames

III. LOADING AND DESIGN The gravity loads include dead and live load of 600kg/m2

and 200kg/m2 respectively. To calculate the equivalent static lateral seismic loads Refer to “(1),” assume that the behavior factor R for Knee-bracing system is 7.

RIBAC

WCV..

.

=

= (1)

Where V is the base shear, A is design base acceleration

ratio (for very high seismic zone=0.35g), B is response factor of building (is depending on the basic period T), and I is the importance factor of building (is depending on the building performance considered 1.0 in this paper).

All of the frames are designed according to the AISC89, allowable stress design.

IV. NONLINEAR STATIC ANALYSIS (PUSHOVER) The most basic inelastic analysis method is the complete

nonlinear time history analysis, which at this time is considered overly complex and impractical for general use. Available simplified nonlinear analysis method referred to as nonlinear static analysis procedures. This method uses a series of sequential elastic analysis, superimposed to approximate a force-displacement capacity diagram of the overall structure.

The capacity curve is generally constructed to represent the first mode response of the structure based on the assumption that the fundamental mode of vibration is predominant response of the structure. This is generally valid for buildings with fundamental periods of vibration up to about one second,

for more flexible buildings with a fundamental period greater than one second; the analysis should be considered addressing higher mode effects. The higher mode effects maybe determined by loading progressively applied in proportion to a mode shape other than the fundamental mode shape.

The step by step procedures are as followed: 1) Create a computer model of the structure and apply

gravity loads. It is necessary to define bilinear model behavior for each member. (The bilinear models which represent the plastic joint behavior of SAP2000 defaults are used for beams and columns in this paper and the models which relate to plastic joint behavior of knee and diagonal elements represented in next section.)

2) Apply lateral story forces to the structure in proportion to the product of the mass and fundamental mode shape.

3) Increase the lateral force level until some element (or a group of elements) yields and revise the model using zero (or very small) stiffness for the yielding elements.

4) Apply new increment of lateral load to the revised structure such that other elements yields and the structure reaches an ultimate limit, such as: reaching the lateral displacement of control point (roof level) a limit state as defined follow for design earthquake:

hh

m

m

020.0025.0

<∆<∆

sec7.0sec7.0

≥<

TT

Where m∆ is inelastic displacement of the control point, h is the story height, and T is the first mode of structure. 5) Record the base shear and the roof-displacement so

that create the capacity curve which represents the nonlinear behavior of structure.

V. FORCE-DISPLACEMENT RELATION OF COMPONENTS Component behavior generally will be modeled using

nonlinear load-deformation relations defined by a series of straight line segments. Fig. 4 illustrates two kinds of representations which are used for computer modeling that is created according to modeling parameters and acceptance criteria for nonlinear approach in FEMA273.

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12

0/0y

Q/Q

CE

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(b)-1.5

-1

-0.5

0

0.5

1

1.5

2

-10 -5 0 5 10 15 20

Strain

Stre

ss

Fig. 4 Load- deformation relations for (a) a knee element,

BOX180x180x10, (b) a diagonal element, 2UNP100

VI. NONLINEAR ANALYSIS RESULTS Fig. 5 illustrates the pushover nonlinear results for KBF

system in the form of force-displacement curve of sample frames.

(a)

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12

Displacement(cm)

Bas

e Sh

ear(

KN

)

(b)

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14

Displacement(cm)

Bas

e Sh

ear(

KN

)

(c)

0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25

Displacement(cm)

Bas

e Sh

ear(

KN

)

(d)

0

200

400

600

800

1000

1200

0 5 10 15 20 25

Displacement(cm)

Bas

e Sh

ear(

KN

)

Fig. 5 Sample frames capacity curves, (a) 5-story, (b) 10-story,

(c) 15-story, (d) 20-story

VII. ESTIMATION OF SEISMIC PARAMETERS In order to investigate the seismic performance of sample

frames the seismic parameters such as: ductility, factor of behavior and formation of plastic hinge can be estimated by using the force-displacement curves and pushover analysis.

A. Ductility Effect in reducing strength factor, µR

Different relations are proposed to determine this factor, in each relation have been attempted to use most of the seismic effective components, the most comprehensive relation is proposed by Miranda, whereas his proposed equation includes some more effective components such as period of structure, soil properties and earthquake acceleration. Based on Miranda’s [10] assumption µR is calculated as in (2)

11+

−=

φµ

µR (2)

⎥⎦⎤

⎢⎣⎡ −−−

−+= 2)

53(ln2/3exp

21

1011 T

TTT µφ For rock earth

⎥⎦⎤

⎢⎣⎡ −−−

−+= 2)

51(ln2exp

52

1211 T

TTT µφ For residual soil

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−+= 2)

41(ln3exp

43

31

g

gg

TT

TT

TT

φ For soft soil

Where µ is ductility, T is period of structure, and gT is dominant period of earthquake.

B. Over strength factor, Ω In addition to laboratorial method the analytical method

such nonlinear static analysis can be used to calculate the Ω factor related to overall yielding of structure as the collapse mechanism yV , to the force in which the first plastic hinge is

formed in structure sV ; therefore the Ω factor can be found Refer to (3).

s

y

VV

=Ω0 (3)

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yV is the base shear related to point of the reduction

stiffness in equivalent bilinear force-displacement of structure. 0Ω is the nominal over strength factor which is adjusted by

multiplying some coefficient to consider the effect of yielding stress increase by the reason of strain rate increase in an earthquake, 1F , the difference between nominal and actual yielding stress of material, 2F , … so the actual over strength factor can be obtained as follows:

...210 ×××Ω=Ω FF

C. Factor of behavior, R The factor of behavior is calculated in two states according to the method which is used by every code to design structure.

uR is the factor of behavior based on ultimate limit stresses and WR is the factor of behavior based on allowable limit stresses, that the relation between WR and uR is defined by a dimensionless parameter uW RRY = , which is evaluated around 1.4 to 1.7 (the UBC-97 code has proposed 1.4 for this parameter) the seismic parameters for sample frames are calculated in Table 1.

From the above tables it can be found that for 5-story frame the Ω factor obtained about 3 and that of 10 to 20-story frames is about 2 to 2.5. The above values are compatible as mentioned in reference [10], which is evaluated 3 for short structures and 2 for tall structures. The displacement limitation code limits the maximum displacement of structure, for this reason the µR factor for 10 to 20-story frames is smaller than that of 5-story frame. Also the value of uR versus the height of structure is plotted in Fig. 6. As it can be found from this figure, the obtained values of uR for KB system is more than that of systems such as Eccentric or Centric Braced Frames, so more ductility is achieved as it is desired in this paper.

0

2

4

6

8

10

12

0 5 10 15 20 25

Number of story

Ru

valu

e

Fig. 6 Investigation of varying uR for KBF

VIII. THE COMPARISON OF NONLINEAR PERFORMANCE OF SAMPLE FRAMES

Fig. 7 illustrates the plastic hinge formation in one of the nonlinear analysis step for 5 and 15-story frames of KBF. By evaluating the results of the KBF system, it can be found that, as the lateral force increases the first plastic hinge forms in a knee element, so that most of the plastic hinge is concentrated in the knee elements, which is a secondary member of the KBF system. Therefore most of the structural damages caused by an earthquake will be occurred on the knee element and after earthquake the damaged members can be replaced more easily and at reasonable cost.

(a)

TABLE I SEISMIC PARAMETERS OF KNEE BRACING SAMPLE FRAMES

Number of stories 5-Story 10- Story 15- Story 20-Story

T(sec) 0.7 1.4 2.3 3.2

yV (KN) 576.2 1013.1 828.4 957.0

sV (KN) 215.0 482.4 380.0 435.0

0Ω 2.68 2.10 2.18 2.20

Ω 3.10 2.43 2.50 2.50 µ 3.40 2.46 2.70 2.10 φ 1.03 0.81 1.02 1.03

µR 3.33 2.80 2.67 2.07

uR 10.32 6.79 6.67 5.20

wR 14.45 9.50 9.34 7.28

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Fig. 7 Plastic hinge formation in one of nonlinear analysis

steps, for (a) 5-story and (b) 10-story frames

IX. CONCLUSION 1) In the KBF system the diagonal brace provides most of

the elastic lateral stiffness where the beams and columns are hinge-connected. The knee elements prevent collapse of the structure under extreme seismic excitations by dissipating energy through flexural yielding. Since the cost of repairing the structure is limited to replacing the knee members only.

2) The area under the force-displacement diagram of the KBF system shows the energy dissipating capacity.

3) According to the values of ductility effect reducing strength factor and over strength factor calculated in tables 1 and 2 for KBF system, it is assumed 51.2=µR and 476.2=Ω so for the ultimate limit stresses.

REFERENCES [1] Jinkoo Kim, Youngill Seo, “Seismic design of steel structures with

buckling-restrained knee braces”, Journal of Constructional steel research 59, p.1477-1497, July 2003.

[2] Roeder, C.W. and Popov, E. P. “Eccentrically braced steel frames for earthquakes”, J. Structural Div., ASCE 1978, 104, 391-411.

[3] Aristizabal-Ochoa, J. D., “Disposable knee bracing: improvement in seismic design of steel frames”, J. Structure. Engineering, ASCE, 1986, 112, (7), 1544-1552.

[4] Uang C.M, “Establishing R (or Rw) and Cd factors for building seismic provision”, J. of Structure. Eng., VOL, 117, No.1, January.

[5] Cosenza E. and Luco A.D. Fealla C. and Mazzolani F.M “On a simple evaluation of structural coefficients in steel structures”, 8th European

Conference on Earthquake Engineering, Lisbon, Portugal, September 1986.

[6] Nassar A.A. and Osteraos J.D and Krawinkler H. “seismic design based on strength and ductility demand”, Proceeding of the Earthquake Engineering 10th worth Conference, p.5861-5866, 1992.

[7] Thambirajah Balendra, Ming-Tuck Sam, Chih-Young Liaw and Seng-Lip Lee, “Preliminary studies into the behavior of knee braced frames subject to seismic loading”, Eng. Struct. 1991, Vol. 13, January.

[8] Thambirajah Balendra, Ming-Tuck Sam, Chih-Young Liaw, “ Diagonal brace with ductile knee anchor for a seismic steel frame”, Earthquake Engineering and Structural Dynamics, Vol. 19, p. 847-858 (1990).

[9] Thambirajah Balendra, Ming-Tuck Sam, Chih-Young Liaw, “Earthquake-resistant steel frames with energy dissipating knee element”, Engineering Structures, Vol. 17, No. 5, p.334-343, 1995.

[10] Miranda,E. and Bertero, V.V., “Evaluation of Strength Reduction Factors for Earthquake-Resistant Design”, Earthquake Spectra, 1994, Vol.10, No.2, pp.357-379.

(b)

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