BENDING – SHEAR INTERACTION IN RBS SHORT COUPLING BEAMS

EUROSTEEL 2014, September 10-12, 2014, Naples, Italy

BENDING – SHEAR INTERACTION IN RBS SHORT COUPLING BEAMS

Andrei Crisana, Dan Dubinab a Politehnica University Timisoara, Dept. of Steel Structures and Structural Mechanics, Romania

[email protected] b Romanian Academy, Timișoara branch, CCTFA

[email protected]

KEYWORDS: M-V interaction, plastic hinge, experimental, numerical, RBS

ABSTRACT

Eurocode [1] specifies that, for plastic hinges in beams it should be verified the full plastic moment of resistance and rotation capacity are not decreased by compression or shear forces. For current moment resisting frames, low values of shear to shear capacity or axial force to axial capacity were observed in comparison to bending moment ratio. For these cases, the influence of shear and axial force on bending moment and rotation capacity of moment plastic hinges can be ignored. However, there are some cases, when the lateral force-resisting system is composed by close-spaced columns rigidly connected by beams. These beams are long enough to allow for the development of moment plastic hinges at the ends, but they are also sufficiently short to create a significantly shear forces to influence the bending moment and rotation capacity of the beam. The paper presents a parametric study using an experimentally calibrated numerical model carried-out at Department of Steel Structures and Structural Mechanics of Politehnica University Timisoara on the purpose to observe and characterise the plastic mechanism of short steel beams with Reduced Beam Section (RBS) applied in Moment Resisting Frames. Such a solution was applied to design a multi-story building structure in Bucharest, were the structural designer aimed to get a good dissipative behaviour under seismic actions in the plastic hinges expected to form in the RBS zones of the short beams of the frames. An experimental confirmation of this solution was asked and validity of plastic hinge model, applied in global analysis, to be checked. In case, an interaction model between bending moment and shear force had to be calibrated. A synthesis of the main result of this investigation is presented, including main experimental and numerical investigations, and, at the end, a simplified interaction model, compatible with SAP2000 [2] is proposed, enabling for non-linear analysis in practical design.

CONCLUSIONS

Five existing interaction models were used as a starting point. Based on these models, a new approach was proposed, resulting a better M – V interaction model. This interaction model is based on EN1993-1-3 model, and it’s given in Eq. (1).

112

15.1

rpl

f

pl V

V

M

M

M

M (1)

where Mpl is the plastic bending capacity of the beam Mf is bending capacity of flanges, Vr is the shear resistance of the section, M and V are the design bending moment and shear force. In the next step, the EN1993-1-3 equation was modified order to further reduce the bending moment capacity due to high values of shear force, resulting a modified equation: The results of the modified EN1993-1-3 approach are presented in Fig. 1, for FLO_F15W20 and FLO_F15W20H cases. It can be observed that the proposed approach best describes the behaviour of the considered numerical model.

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mom

ent

[M/M

pl]

Shear [V/Vpl]

FLO_F15W20

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mom

ent

[M/M

pl]

Shear [V/Vpl]

FLO_F15W20H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

Fig. 1. Modified EN1993-1-3

It is well known that the use of general-purpose finite element analysis programs in structural analysis for design purposes is, generally, prohibited for being time and cost consuming. More common design programs model the beam elements as wire frames with given rigidity. This approach gives satisfactory results for sufficiently long beams. For present study, the results were not satisfactory from the initial rigidity point of view (see Fig. 2, SAP2000 – Initial model).

0

100

200

300

400

500

600

0 10 20 30 40 50 60 70

Bas

e sh

ear

forc

e [k

N]

Top displacement [mm]

ABAQUS - Reference model

SAP2000 - Proposed model

SAP2000 - Initial model

Fig. 2. Pushover curves for ABAQUS and SAP2000 model.

In order to solve this problem, the rigidity of the joint was computed using the initial rigidity equation of joints given in EN1993-1-8 [3], considering a full strength joint. The part of the beam between the longitudinal axis of the column and the column face was modelled as a rigid element with a spring at the end having the before computed rigidity. In Fig. 2 is presented the proposed model, depicted by the SAP2000 – proposed model curve. Based on the result of present study, it can be said that the modified EN1993-1-3 approach, can accommodate the loss of capacity for the considered sections. Further analyses are required in order to determine a more complex and adaptable interaction model. The wire-frame model for structural analysis used in most commercially available software, can underestimate the structural rigidity in case of structures with short coupling beams. The proposed model, implemented in SAP2000, can successfully simulate the M-V interaction and, in the same time, give a more reliable behaviour for considered frame from the rigidity point of view.

REFERENCES

[1] Eurocode 8, “Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings”, 2004

[2] Computers and Structures, Inc., SAP2000, Structural Analysis Program

[3] EN 1993-1-8 (2005), Eurocode 3. Design of steel structures Design of joints, CEN, EN 1993-1-8

EUROSTEEL 2014, September 10-12, 2014, Naples, Italy

BENDING – SHEAR INTERACTION IN RBS SHORT COUPLING BEAMS

Andrei Crisana, Dan Dubinab

a Politehnica University Timisoara, Dept. of Steel Structures and Structural Mechanics, Romania [email protected]

b Romanian Academy, Timișoara branch, CCTFA [email protected]

INTRODUCTION

In structural plastic analysis, beams and columns behaviour may be modelled using concentrated

plastic hinge models, distributed plastic hinge models, or other models whose behaviour has been

demonstrated to adequately represent the plastic behaviour of the element. The model shall be

capable of representing inelastic response along the component length, except where it is shown by

equilibrium that yielding is restricted to the component ends. Where nonlinear response is expected

in a mode other than flexure, the model shall be established to represent these effects [1].

Eurocode [2] specifies that, for plastic hinges in beams it should be verified the full plastic moment

of resistance and rotation capacity are not decreased by compression or shear forces. For current

moment resisting frames, low values of shear to shear capacity or axial force to axial capacity were

observed in comparison to bending moment ratio. For these cases, the influence of shear and axial

force on bending moment and rotation capacity of moment plastic hinges can be ignored. However,

there are some cases, when the lateral force-resisting system is composed by close-spaced columns

rigidly connected by beams. These beams are long enough to allow for the development of moment

plastic hinges at the ends, but they are also sufficiently short to create a significantly shear forces to

influence the bending moment and rotation capacity of the beam.

Such beams work like intermediate length links in eccentrically braced frames under interaction

between bending moment and shear, concentrated at the ends of the link. In order to reduce the risk

of brittle failure in beam-to-column connections, either strengthening of the connection or

weakening of the beam can be applied. First approach consists in providing sufficient over-strength

to connections, for instance by means of haunches or cover plates. The second approach can take

benefit from the "Reduced Beam Section" or "dog-bone" solution that involves the removal of a

portion of the beam flanges a short distance away from the column face. In this way, the plastic

hinge is moved away from the column-to-beam flange connection, allowing stable yielding of the

beam. This approach was proven to have a successful behaviour in numerous laboratory tests [3],

[4], [5].

Present paper focuses on the bending moment and shear force interaction in short coupling beams

with Reduced Beam Section. On this purpose, a parametric study was carried out using

experimentally calibrated numerical models. The finite element models were calibrated and

validated using experimental tests performed on four full-scale specimens at the CEMSIG

Laboratory of Politehnica University Timisoara, Romania. The general-purpose finite element

analysis program ABAQUS [6] is a powerful tool but its use in civil structural analysis is limited.

On this line, a simplified interaction model, compatible with SAP2000 [7] is proposed, enabling for

nonlinear analyses in practical design.

1 SUMARY OF EXPERIMENTAL INVESTIGATIONS

The study presented hereafter is connected with the design of a 18-story office building, located in

Bucharest, Romania. The building site is located in a high seismic area, which is characterized by a

design peak ground acceleration 0.30g for a returning period of 225 years, and soft soil conditions,

with TC=1.6 s. It is noteworthy the long corner period of the soil, which in this case may affect

flexible structures. For serviceability check, the returning period is 30 years, while for collapse

prevention it is 475 years.

Lateral force-resisting system intended to create a tube-in-tube structural layout with both perimeter

and core framings steel framing composed of closely spaced columns and short beams [8]. The

central core is also made of steel framing with closely spaced columns and short beams. The ratio of

beam length-to-beam height, L/h, varies from 3.2 to 7.4, which results in seven different types of

beams. Some beams are below the general accepted inferior limit (L/h=4). The moment frame

connections employee reduced beam section connections that are generally used for beams loaded

mainly in bending.

RBS short beams have been applied in order to control by design the development of plastic hinges,

aiming to create a classical moment frame plastic mechanism dissipative model. However, since

high shear forces have been identified in the preliminary design phase, an experimental

investigation was ordered to estimate that a classic plastic bending hinge model can be applied. The

experimental program together with the numerical model calibration and validation, presented in

detail in [8], will be summarized hereafter.

In Fig. 1 is presented the experimental setup for a full-scale frame module. Specimens were tested

under cyclic loading only.

a) b)

Fig. 1. Test setup a) and laboratory setup b) [8]

The laboratory tests clearly shown the plastic hinge which have been developed in the RBS zone of

the short beam are affected by shear (see the inclined strips in Fig. 2, b) and c))

a)

a

a +

10

0 m

m

HEA 800 HEA 800

100 300

790

b) c)

Fig. 2. RBS details a), deformations b) and experimental curve c)

Actuator

1000 kN

Test

Specimen

Out of plane

stability frame

Reaction

wall

The cyclic loading sequence was taken from the ECCS Recommendations [9]. According to the

ECCS procedure, the yielding displacement Dy and the corresponding yielding force Fy are obtained

from the monotonic force vs. displacement curve.

2 NUMERICAL SIMULATIONS AND INTERACTION MODELS

2.1 Calibration and validation of numerical models Using experimental tests as reference, the numerical models were calibrated and validated. The full

description of the numerical model is presented in [8] and summarized only hereafter.

All the components were modelled using solid elements. The engineering stress-strain curves

obtained from tensile tests were transformed into true stress – strain curves and used further for the

numerical model. The Young modulus was considered 210 000 N/mm2 and Poisson’s ratio as 0.3.

For the cyclic analysis, a combined isotropic/kinematic hardening model was used for the material,

containing the cyclic hardening parameters from [10].

In Fig. 3a) is presented the numerical model used for simulations [8], the simplified model Fig. 3, b)

and the behaviour curves comparison Fig. 3, c). A good agreement can be observed between the

experimentally tested specimen and the numerically simulated one.

For the purpose of parametrical analysis developed in present study, in order to reduce the

computation time, the numerical model was simplified. Since there no plastic deformations were

observed in the columns (neither in experimental tests, nor in numerical simulations) and the main

interest was the beams behaviour, the columns were no longer considered into the model and the

beam only model, was further used. For this simplified model, two rigid body constraints were

created to replace the columns. The control point was considered at the intersection of the columns

and beam longitudinal axes. The beam was simply supported and loaded with concentrated moment

at the control points.

b) Simplified numerical model

-900

-700

-500

-300

-100

100

300

500

700

900

-100 0 100 200 300 400

Fo

rce [

kN

]

Displacement [mm]

FEM_Original [8]

FEM_modified

RBS-S3

a) Original numerical model [8] c) Numerical models – original and modified

Fig. 3. Original [8] a), simplified b) numerical model and behaviour curves

It must be mentioned that for the simplified model, the material properties were not changed.

In order to validate this simplified numerical model, a different specimen, experimentally tested

was used. This specimen, the test procedure and the relevant experimental results are presented in

detail in [11]. Further, in Fig. 4a is presented the experimentally obtained curve from [11], while in

Fig. 4b is presented the global response curve for the simplified numerical model.

Once again, it can be observed that from the point of view of moment – rotation curves the

numerical model is capable to simulate the real behaviour of tested specimens presented in [8] and

[11].

-6

-4

-2

0

2

4

6

-0.04 -0.02 0 0.02 0.04

Mo

men

t (M

N-M

)

Plastic rotation (rad)

a. specimen DC-3 [11] b. FEM model

Fig. 4. Global response curves [8]

2.2 Parametric study The simplified calibrated model was further used for a parametric study to identify the interaction

between the bending moment and shear force in short coupling beams. For this purpose, the

material properties were constrained to an elastic – perfect plastic behaviour, to accommodate the

Eurocode [12] model for plastic design. For this study, the properties of S355 steel were considered.

The geometric dimensions that were varied for the simulated beam are the presented in Table 1.

Table 1. Geometric dimensions of simulated beams

Beam Flange thickness [mm]

Web thickness [mm]

Overall beam height [mm]

Length range [mm]

Flo_F15W15 15 15 450 500 – 1100

Flo_F15W20 15 20 450 500 – 1050

Flo_F20W15 20 15 450 500 – 1050

Flo_F20W20 20 20 450 500 – 1000

Flo_F15W15H 15 15 550 500 – 1200

Flo_F15W20H 15 20 550 500 – 1200

Flo_F20W15H 20 15 550 500 – 1000

Flo_F20W20H 20 20 550 500 – 1050

For the parametric study, taking as reference the dimensions of tested specimens, the beam section

and lengths have been modified in order to obtain different moment/shear ratios. The length of the

beam was measured form the face of the column to the symmetry point of the beam (actual

coupling beam resulting with double length). The dimensions of the beam section used for the

parametric study are presented in Fig. 2, a).

The shear force and bending moment resulted from the numerical simulations were taken in the

middle of the reduced section, while the resistant shear force and resistant bending moment were

repeatedly computed for the reduced section.

For this study, the actual model was considered to be a cantilever beam (half of the real coupling

beam length), with a fixed end (the control point of the rigid body constraint, at the intersection of

the axes of the column and the beam) and loaded at the free end.

2.3 Existing interaction models The following approaches were considered as the starting point for the interaction model of M-V:

− Basler model [13]

1

2

=−

−+

fp

f

uMM

MM

V

V (1)

where Mf is bending capacity of flanges, Mp is the plastic bending capacity of the beam, Vu is the

shear resistance of the section, M and V are the design bending moment and shear force.

− EN1993-1-3 approach [14]

112

1

2

=

−

−+

rpl

f

plV

V

M

M

M

M (2)

where Mpl is the plastic bending capacity of the beam Mf is bending capacity of flanges, Vr is the

shear resistance of the section, M and V are the design bending moment and shear force.

− EN1993-1-1 approach [12]

ply

w

w

yplVyMf

t

AWM ≤

−=

4

2

,,

ρ, for

RdVV 5.0> (3)

where My,V is the plastic bending capacity of the beam in the presence of shear force V, VRd is the

shear resistance of the section, V is the design shear force, Wpl,y is the plastic resistance modulus for

the section, Aw is the shear area (Aw = hw.tw), hw

and tw are the height and thickness of the web, ρ is

the reduction factor, given by ρ = (2VEd / Vpl,Rd – 1)2

− Modified AASHTO approach [15]

λ28.05.0 +=

pV

V, with

−

+−= 1

41

123

p

r

rM

Ma

aλ (4)

where ar = Aw / Af with Aw web area and Af flanges area, Mp is the capable plastic bending moment

of the section and M and V are the design bending moment shear force.

− AASHTO LRFD approach [15]

16.12.2 ≤−=

nnM

M

V

V (5)

where Mp and Vp are the capable bending moment and shear force, while M and V are the design

bending moment shear force.

3 PARAMETRIC STUDY RESULTS

3.1 Analysis results In Fig. 5 are presented the results of the numerical parametric simulations for the beam sections

presented in Table 1. The results are represented on M-V interaction graphs, together with the

considered initial interaction models. It can be observed that the AASHTO Modified approach is

the most suited for all considered sections. Even if in some cases, EN1993-1-1 and EN1993-1-3

approaches gives satisfactory results, there are certain section dimensions for which these

approaches are not suitable (e.g. Fig. 5, FLO_F15W20, FLO_F15W20H).

In the next step, the EN1993-1-3 equation (2) was modified order to further reduce the bending

moment capacity due to high values of shear force, resulting a modified equation:

112

1

5.1

=

−

−+

rpl

f

plV

V

M

M

M

M (6)

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W15

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W20

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F20W15

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F20W20

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W15H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W20H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F20W15H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F20W20H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

Fig. 5. Parametric study results

The results of the modified EN1993-1-3 approach are presented in Fig. 6, for FLO_F15W20 and

FLO_F15W20H sections. In the figure, it can be observed that the modified equation, (6), best

describes the behaviour of the numerical model.

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W20

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Mo

men

t [M

/Mpl]

Shear [V/Vpl]

FLO_F15W20H

BASLER

EN1993-1-1

EN1993-1-3

AASHTO

AASHTO Modified

FEM Results

Fig. 6. Modified EN1993-1-3

3.2 Plastic hinge model The use of general-purpose finite element analysis programs like ABAQUS [6] in structural analysis for design purposes is, generally, prohibited for being time and cost consuming. However,

to account for both bending and shear effects, when applying a bar members model, a more simple

approach can be used; this approach is presented hereafter. The SAP2000 [7] program allows the

definition of several types of plastic hinges. Plastic hinges for axial, shear and bending moment and

the coupling of axial force and bending moment about one or both axes. The program does not

allow the definition of combined bending moment with shear force hinges.

In order to complete the definition of coupled bending moment – shear force hinges, the following

steps were considered: i) definition of a shear force (force controlled) hinge with the capacity

computed from the numerical parametric simulations and ii) definition of a bending moment

(deformation controlled – FEMA) hinge having a the resisting bending moment computed using the

reduced section properties (minimum section into the dog-bone region).

The original ABAQUS model [8] (see Fig. 3a) was used to determine the pushover curve for the

frame. This model was calibrated against experimental tests and further validated for a new set of

experimental data. Material properties were changed in order to model an elastic – perfect plastic

material behaviour. This model was further used as benchmark for proposed models. For the

SAP2000 model, both types of hinges were assigned to beams in the position of minimum area

(centre of the RBS). As most usual structural analysis programs, SAP2000 model the beam

elements as wire frames with given rigidity. This approach gives satisfactory results in most cases

for sufficiently long beams. For present study, considering this approach, the results were not

satisfactory from the initial rigidity point of view (see Fig. 7, SAP2000 – Initial model).

0

100

200

300

400

500

600

0 10 20 30 40 50 60 70

Ba

se s

hea

r fo

rce [

kN

]

Top displacement [mm]

ABAQUS - Reference model

SAP2000 - Proposed model

SAP2000 - Initial model

Fig. 7. Pushover curves for ABAQUS and SAP2000 model.

In order to solve this problem, the rigidity of the joint was computed using the initial rigidity

equation of joints given in EN1993-1-8 [16], considering a full strength joint. The part of the beam

between the longitudinal axis of the column and the column face was modelled as a rigid element

with a spring at the end having the before computed rigidity. In Fig. 7 is presented the proposed

model, depicted by the SAP2000 – proposed model curve.

It can be observed that the final SAP2000 model, considering the computed rigidity of the joint and

an interactive model for hinges, is able to simulate the behaviour of the frame. It must be mentioned

that the computation time was greatly decreased (more than 10 times).

4 CONCLUSIONS

Present paper presents the bending moment and shear force interaction in short coupling beams

with Reduced Beam Section using a parametric study carried out using calibrated and validated

numerical models. Based on the results of this parametric study, it can be said that the considered

interaction models [12], [13], [14], [15] give unsatisfactory results. The modified EN1993-1-3

approach, given by eq. 6, can accommodate the loss of capacity for the considered sections. Further

analyses are required in order to determine a more complex and adaptable interaction model.

The wire-frame model for structural analysis used in most commercially available software, can

underestimate the structural rigidity in case of structures with short coupling beams.

The proposed model, implemented in SAP2000, can successfully simulate the M-V interaction and,

in the same time, give a more reliable behaviour for considered frame.

REFERENCES

[1] FEMA 356, „NEHRP Guidelines For The Seismic Rehabilitation Of Buildings”, 2000

[2] Eurocode 8, “Design of structures for earthquake resistance -Part 1: General rules, seismic actions and

rules for buildings”, 2004

[3] Chen, S. J., Yeh, C. H., and Chu, J. M. ‘‘Ductile steel beam-tocolumn connections for seismic

resistance.’’J. Struct. Eng.,122(11), 1996, 1292–1299,

[4] Plumier, A. (1990), “New Idea for Safe Structures in seismic Zones”, IABSE Symposium, Mixed

structures including new materials, Brussels 1990 (pp. 431 – 436).

[5] Zekioglu, A., Mozaffarian, H., and Uang, C.-M. ‘‘Moment frame connection development and testing

for the city of hope national medical center.’’Proc., Structure Congress XV, Vol. 1, ASCE, N.Y. 1997,

pp. 100-104

[6] Hibbit. D., Karlson, B. and Sorenso, P (2007), ABAQUS User’s Manual, Version 6.9.

[7] Computers and Structures, Inc., SAP2000, Structural Analysis Program

[8] Florea Dinu; Dan Dubina; Calin Neagu; Cristian Vulcu; Ioan Both; Sorin Herban, Dragos Marcu,

„Experimental and Numerical Evaluation of a RBS Coupling Beam For Moment Steel Frames in

Seismic Areas”, Steel Construction Volume 6, Issue 1, 2013, pp. 27–33

[9] ECCS 1986. European Convention for Constructional Steelwork, Technical Committee 1, Structural

Safety and Loadings; Working Group 1.3, Seismic Design. Recommended Testing Procedure for

Assessing the Behavior of Structural Steel Elements under Cyclic Loads, First Edition.

[10] Dutta, A., Dhar, S., Acharyya, S. K. “Material characterization of SS 316 in low-cycle fatigue loading”,

Journal of Materials Science, vol. 45, No. 7, 2010, pp. 1782–1789.

[11] Brandon Chi, Chia-Ming Uang, "Cyclic Response and Design Recommendations of Reduced Beam

Section Moment Connections with Deep Columns", Journal of Structural Engineering, Vol. 128, No. 4,

April 1, 2002

[12] EN 1993-1-1 (2005). Eurocode 3. Design of steel structures. General rules and rules for buildings,

CEN, EN 1993-1-1.

[13] Basler, K. 1961. “Strength of plate girders under combined bending and shear”. Journal of the Structural

Division, vol. 87, ST 7.

[14] EN 1993-1-3: Eurocode 3: Design of steel structures - Part 1-3: General rules - Supplementary rules for

cold-formed members and sheeting (2006)

[15] Sung C. Lee, Doo S. Lee and Chai H. Yoo "Flexure and Shear Interaction in Steel I-Girders", Journal of

Structural Engineering, Vol. 139, No. 11, November 1, 2013

[16] EN 1993-1-8 (2005), Eurocode 3. Design of steel structures Design of joints, CEN, EN 1993-1-8