Advance Analysis of Hybrid Frame Structures by Refined Plastic-hinge Approach

7/28/2019 Advance Analysis of Hybrid Frame Structures by Refined Plastic-hinge Approach

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ADVANCED ANALYSIS OF HYBRID FRAME STRUCTURES BYREFINED PLASTIC-HINGE APPROACH

S.L. CHAN, S.W. LIU, Y.P. LIU

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University,Hong Kong, China,

ABSTRACTHybrid frames composed of steel, concrete and composite members are widely used to-datedue to their structural efficiency, especially in high-rise buildings. The design of this form ofstructures is inconvenient as it needs several separate design codes for steel, concrete andcomposite elements. This paper proposes a nonlinear design method which only requiressection capacity check without the use of different codes for the hybrid frame structures. Byusing the pointwise-equilibrium-polynomial (PEP) element allowing for initial imperfection inconjunction with a robust nonlinear incremental-iterative procedure, the second-order effects ofindividual members and the structural system can be modeled. The sectional fibre approach isused to determine the section capacity of arbitrary shape reinforced concrete or compositemember subjected to axial force and biaxial bending. To fulfil the requirement of seismic design,progressive collapse analysis and advanced analysis, the refined plastic-hinge approach isutilized to model the plastic behaviour with strain-hardening effect. Once the initial and full yieldsurfaces are determined, the gradual yielding is simulated. Two examples are employed todemonstrate the validity and accuracy of the proposed method.

1. INTRODUCTION

The lateral and vertical resistance components of building structures are often provided byseveral structural forms. The part for lateral resistance is provided by a hybrid system whichcombines with steel with concrete to form the composite frames, while the vertical resistance ismainly reinforced-concrete or composite columns. Figure 1 shows one hybrid system consistedof composite and reinforced concrete columns and steel beams. The advantage of this form ofhybrid system is structural efficient and cost-effectiveness with optimal use of materialsaccording to their mechanical characteristics, such as concrete is compression and steel intension. Generally speaking, hybrid frames are more load resistant, lighter and stiffer than baresteel frame at the same total material cost and provides better ductile performance overreinforced-concrete frames.

4 th International Conference on Steel & Composite StructuresWednesday 21 Friday 23 July 2010

Sydney, Australia

Steel & Composite StructuresProceedings of the 4th International ConferenceEdited by Brian Uy, Zhong Tao, Fidelis Mashiri, Xinqun Zhu, Olivia Mirza & Ee Loon TanCopyright c 2010 ICSCS Organisers. Published by Research Publishingdoi:10.3850/978-981-08-6218-3 key-3 14


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Brian Uy, Zhong Tao, Fidelis Mashiri, Xinqun Zhu, Olivia Mirza & Ee Loon Tan

Figure 1: Hybrid steel-concrete frame system

The current design practise is to use separate codes for design of members made of differentmaterials, for example, Eurocode-2 [1] is used for reinforced concrete member, Eurocode-3 [2]for steel member and Eurocode-4 [3] for composite member. This brings much inconvenienceand sometimes complexity to the design engineer. This paper aims to develop a practicalnumerical approach for hybrid frames with the consideration of geometrical instability andmaterial nonlinearity that all structural members can be designed in a unified way with thesection capacity check carried out for all members to insure safety in stability and strength.

Although the description for the stability check in steel, concrete and composite codes may bedifferent, the requirement for consideration of second-order effects such as P- and P- effectsis conceptually and numerically similar. It is noted that the P- effect is commonly ignored inmost previous research and therefore the tedious member buckling strength design by code isstill needed. In this paper, by using the pointwise-equilibrium-polynomial (PEP) element [4]allowing for initial imperfection in a robust nonlinear incremental-iterative procedure, both the P- effect of individual members and the P- effect of the structural system can be simulated. Thenear exact analysis for regular and irregular reinforced concrete and composite membersubjected to axial force and biaxial bending is performed by the sectional fibre approach so thatthe section capacity can be calculated. Thus, a unified design method is developed in thepresent project and no tedious member design check to various codes is required.

The proposed method is further extended to inelastic analysis by using the plastic hingeapproach which is required in seismic design, advanced analysis and progressive analysis. Tocapture the gradual yield behaviour under the interaction of axial and bending effects, the firstand full yield surfaces for a hybrid section, which contains steel, reinforcement and concretematerials, are defined. The sectional fibre approach is adopted to calculate the stress resultantsof the concrete, the reinforcement and the structural steel. Both the strength reduction andstiffness deterioration can be represented in the proposed method.

In this paper, the basic element formulation for considering the geometric nonlinearity with initialimperfections is discussed in section 3, while the proposed refined plastic hinge approach for thematerial nonlinearities is discussed in section 4. The cross-section analysis procedure for

^

Z

15


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Steel & Composite Structures Proceedings of the 4th International Conference

generating the first and full yield surfaces is illustrated in section 5. Finally, two examples arepresented for demonstration of the validity, accuracy and advantages of the proposed method.

2. BASIC ASSUMPTIONS AND DEFINITIONS

In this paper, some basic assumptions are adopted in the cross-section analysis as: (1) Planesections before deformation remain plane after deformation, that means a linear straindistribution exists across the depth of the section; (2) the bond-slip between the concrete andsteel is not considered, it is assumed that full strain compatibility exists between the steel andthe surrounding concrete; (3) the tensile strength of concrete is neglected in computations; (4)steel reinforcement embedded in concrete does not buckle under compression; (5) compressivestresses and strains are taken to be positive and (6) the effect of concrete cracking or crushingon the torsional stiffness is ignored.

In the formulation of the beam-column element, the following assumptions are taken: (1) TheEuler-Bernoulli hypothesis is valid and warping is neglected; (2) strains are small but thedeflection can be large; (3) plane section normal to the centroid axis before deformation remainsplane after deformation and normal to the axis; (4) the concept of lumped plasticity is employed,i.e., the yielding of material is assumed to be concentrated at the both ends of beam-columnelement and (5) loads are conservative and shear distortions are negligible.

3. GEOMETRIC NONLINEARITY

The P and P effects are two principal parameters needed to be considered in the second-order or advanced analysis. In the topic of formulating a design element capturing the P effectof a member, Chan and Zhou [4-8] developed several elements with different features to simplifythe analysis procedure and make the advanced analysis practical. In this paper, the pointwiseequilibrating polynomial (PEP) element proposed by Chan and Zhou [4] is adopted. The PEPelement is capable of modelling the member initial curvature which is mandatory for bucklingdesign by second-order analysis in various codes. The basic force-displacement relations in anelement are illustrated below and more details about its formulation can be referred to theoriginal papers. The equilibrium condition can be stipulated as follows.

Figure 2 : The basic forces vs. displacements relations in an element

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121

0

..

)2

()( M x L

L M M

vvPv EI ++

++= (1)

in which E is the Youngs modulus of elasticity, I the second moment of area, L the memberlength, v the lateral displacement due to applied loads, v 0 the initial member deflection, P theaxial force, and M 1 and M 2 the nodal end moments. A superdot represents a differentiation withrespect to the distance x along an element.

The secant stiffness matrix, which relates the equilibrium equations between forces, moments,displacements and rotations, can be obtained by the energy principle. For incremental-iterativenonlinear procedure, the tangent stiffness matrix which relates the incremental forces, momentsto rotations and displacements is needed and can be formed by the second variation of the totalpotential energy functional.

The remarkable advantage of proposed advanced analysis by PEP element is its automaticcomputation of primary linear and secondary non-linear stresses such that the assumption of K-factor or effective length factor is avoided. The influence on member stiffness in the presence ofaxial load is also allowed for in the stress computation and analysis. The first eigenvaluebuckling mode shape is used to determine the direction of local member imperfection and theglobal frame imperfection due to out-of-plumbness which is normally taken in the range from1/1000 to 1/200 of the building height H. The member initial imperfection can be obtained fromcodes such as Table 5.1 in the Eurocode-3 [2] and Table 6.1 in the HKSC [9] and the laterimperfections are adopted herein.

4. MATERIAL NONLINEARITY

4.1. Stress-strain constitutive relationship

To consider the inelastic behaviour of hybrid section, the constitutive relationships of steel andconcrete shown in Figure 3 are adopted. The steel material is assumed to be ideally elastic-plastic, while the stress-strain curve of concrete by Saenz [10] is adopted to consider thematerial nonlinearity. The tensile strength of concrete is neglected as specified in many codes.

(a) Steel and reinforcement (b) Regular and confined concrete

Figure 3 : Stress vs. strain curve for steel and concrete

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The equation of the constitutive relationship for concrete is taken as,

in which 0 E , yield and yield are the initial youngs modulus, initial yield strain and yield stress.

4.2. Cross-section properties

To evaluate the stress resultants in any shape section, the section can be divided into 3components, i.e., the unconfined and confined concrete, steel and void areas. More details ofthe stress resultants of each component is discussed in section 5.

(a) Overview

=

(b) Concrete

+

(c) Steel

-

(d) Void

Figure 4 : Components of cross section

4.3. First and full yield surface

To reflect the gradual strength reduction and stiffness deterioration, the first and full yieldsurfaces should be identified for second-order inelastic analysis. Here, the generalized stressessuch as axial force (P) and moments (My and Mz) are monitored in the nonlinear incremental-iterative process to check the state of section capacity.

For reinforced concrete and steel-concrete composite sections, the first yield surface is defined

as the strain of external fibre reaching the initial yield strain yield of concrete, while the full yieldsurface is determined when the external fibre exceeds the strain cu of concrete.

The two yield surfaces divide the loading space into three zones, i.e., elastic, elasto-plastic andplastic hardening zone as shown in Figure 5. If the loading combination is within the first yieldsurface, the section can be ideally treated as elastic without reduction in either strength orstiffness. If the loaded coordinate falls into the elasto-plastic zone, the section is under graduallyyielding stage and its strength and stiffness are discussed in the followings. Once the loadingcombination exceeds the full yield surface, the sectional hardening appears.

02

01 ( 2)s yield yield

E

E E

= + +

( 2)

h

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Figure 5: First and full yield surfaces under given axial force

4.4. Plastic hinge method

For inelastic analysis, it is necessary to assume a yield function to monitor the gradualplastification of a section. This refined plastic hinge approach introduced by Chan and Chui [11]

is revised and adopted in this study. The nodal rotations of a deformed PEP element withpseudo-springs at the end nodes are shown in

Figure 6.

Figure 6: Internal forces of a PEP element with end-section springs

The zero-length spring elements belonging to the internal degrees of freedom of beam-columnelement can be eliminated by a standard static condense procedure such that the size of theelement stiffness matrix will not be increased. The final incremental stiffness relationships of thehybrid element can be formulated below and more details can be referred to references 12 to14.

^

^

D D

First yield surface

Full yield surface

D D

Plastic moment

Elastic moment

Elastic zone

Plastic hardening zone

Elasto-plastic zone

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21 1 1 22 2 1 2 12 1

22 1 2 21 1 2 11 1 2

/ 0 0

0 ( ) / /

0 / ( ) / e e

e e

P EA L L

M S S K S S S K

M S S K S S K S

= + +

(3)

in which S 1 and S 2 are the stiffness of the end springs, K ij are the flexural stiffness of the PEPelement considering the presence of axial force, P is the axial force increment, eM1 and eM2are the incremental nodal moments at the junctions between the spring and the global node andbetween the beam and the spring, L is the axial deformation increment, e and e are theincremental nodal rotations corresponding to these moments, and

11 1 1211 1 22 2 12 21

21 22 2

( )( ) 0K S K

K S K S K K K K S

+

= = + + >+ (4)

The section spring stiffness, S, can be calculated by the following equation,

6( ) pr

er

M M EI S

L M M

= +

( 5)

where EI is the flexural constant, L is the member length and M er and M pr are respectively thefirst and full yield moments reduced due to the presence of axial force and the represents thestrain-hardening parameter. From this equation, the section stiffness varies from infinity to asmall strain-hardening value which represents three sectional stages, i.e., elastic, partially plasticand fully plastic with strain-hardening stages.

5. CROSS-SECTION ANALYSIS

In this section, the numerical procedure of sectional analysis for arbitrary reinforced concreteand encased composite sections are discussed.

The arbitrary section studied here is shown in Figure 7. Two variables, which are the orientation n and the depth d n of neutral axis, should be determined to calculate the stress resultants in thecross section. By changing the n from 0 o to 360 o, a quasi-Newtion iterative procedure is

introduced to determine d n under a given angle n.

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Figure 7 Arbitrary shape cross-section

5.1. Coordinate systems Three coordinate systems are adopted to describe the analysis procedure namely as XCY, xoyand uov with the XCY system for description of cross section defined by designers. The xoy anduov systems have the same origin usually taken as the geometrical centroid of cross section.The whole iterative process involves coordinate transformation twice, i.e., the global XCYsystem transformed to the load-reference xoy system and then to the uov system of which the u-axis is parallel to the neutral axis.

5.2. Stress Resultants in Cross-section

As mentioned in section 4, the whole section can be divided into 3 parts, which are concrete,steel and void area. The stress resultants of each component will be discussed in the followings.

(1) Stress resultants in Concrete

To determine the first yield surface of reinforced concrete and encased composite section, theelastic limit strain of concrete is taken as yield. This assumption implies that the inconspicuous

nonlinear behaviour before the concrete strain reaching the yield is neglected and it is acceptablefor most engineering practice. The proposed method is to cut the concrete component intoseveral layers rather than small fibres (see Figure 8) and then stress resultants are calculated byintegrating the stresses at each layer as below.

Y

XC

y

xo

u

v

dn

n

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Figure 8 : Stress block of concrete in first yield stages

||||||

1 1)(

111 + +

=== ===

i

i

j

j

v L Lv

v

u

uc

in

j

n

i zci

n

i zc zc dudv N P N (6)

+ +

=====

1 1)(

111

i

i

j

j

v L Lv

v

u

uc

in

j

n

i

uci

n

i

uc vdudv M M (7)

+ +

=====

1 1)(

111

i

i

j

j

v L Lv

v

u

u

c

in

j

n

i

vci

n

i

vc ududv M M (8)

in which n L is the number of the compression layers, n v(i) is the number of iteration points in the

layers. is equal to +1 when0> zcP

and equal to -1 when0< zcP

.It should be pointed out that the strain of the external layer is taken as the crushing strain cu inthe full yielding stage. Here, the equivalent stress block (see Figure 9) is introduced to calculatethe stress resultants.

Figure 9 : Stress block of concrete in full yield stages

||||||1 )(

011 +

=====

i

i

ccu

u

uv

c

n

i zci

n

i zc zc vdud N P N (9)

22


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+

+====

1 )(

011

)]([i

i

ccu

u

uv

nc

n

iuci

n

iuc vdud vv M M (10)

+

====

1 )(

011

i

i

ccu

u

uv

c

n

ivci

n

ivc vudud M M (11)

in which n c is the number of vertices of the compression zone, ( )v u is the linear equation ofboundary line equal to v(u) v n with vn being the coordinate of the neutral axis in v-axis.

(2) Stress resultants in steel

Each rebar is treated as an individual fibre, while the structural steel will be meshed into severalfibres with rectangular areas. The stress resultants of steel section and reinforcement can becomputed as,

sk sk

ns

k rjrj

nr

j zs A A N

==+=

11(12)

sk sk sk

ns

k rjrjrj

nr

jus v Av A M

===

11(13)

sk sk sk

ns

k rjrjrj

nr

jvs u Au A M

==+=

11(14)

In which the subscript r and s represent respectively reinforcement and steel fibres.

(3) Void area in the cross-section

The negative area approach is used to remove the void due to steel, rebar components oropening in the section.

(4) Total force and moments sum up and transformation

The bending moments obtained from the above equations are summed and transformed to thexoy system by the following transformations.

uousucu M M M M += , vovsvcv M M M M += (15)

nvnu x M M M sincos = , nvnu y M M M cossin += (16)

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5.3. Iteration scheme

By rotating the orientation

n of neutral axis from 0o

to 360o

with the change of depth d n, thesectional capacity can be precisely determined as seen in Figure 7. The Regula-Falsi numericalmethod is utilized to satisfy the equilibrium, compatibility, and constitutive relationships.

The axial force capacity N z is iterated with respect to d n by the following equation with n keptunchanged.

)(''

, z zd z z

nnnk n N N N N

d d d d

+= (17)

where d n,k is the updated neutral axisi depth, d n and d n are the neutral axis depth where the

axial capacity is smaller and greater than the design value respectively, N z and N z are the axialforce capacity calculated at d n and d n and N zd is the current design axial loading.

The whole procedure of sectional analysis is illustrated in Figure 10.

Figure 10 : Flowchart of analysis procedure

y z Input data

Get the geometry centroid

Calculate N z, My and M z nd

z N zd N

adjust n

Initialize n

Initialize d n

es

n 0360

es

Output stress resultants

End

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6. VERIFICATION EXAMPLES

6.1. Initial and full yield surfaces verified by design codesIn this example, a reinforced concrete section and an encased composite section shown inFigures 11 (a) and 12 (a) respectively will be studied to verify the accuracy of the proposedsectional analysis method. The sections are subjected to axial force (P) and uniaxial bendingmoment (M). The P-M interaction curves obtained from the proposed method, HKCC [15] andHKSC [9] are shown in Figures 11 (b) and 12 (b).

(a) General dimensions (b) P-M interation curve

Figure 11 : Sectional analysis of reinforced-concrete section

(a) General dimensions (b) P-M interation curve

Figure 12 : Sectional analysis of encased composite section

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For reinforced concrete section, the proposed method produces almost the same results as thetedious hand calculation by code as seen Figure 11. It should be noted that the concrete areaoccupied by reinforcement is commonly neglected in the hand calculation and therefore aslightly unconservative result is obtained, especially for cases of high steel ratio. The proposedapproach uses the negative area method to remove the concrete areas occupied by rebar.

For encased composite section, P-M interaction curve from the proposed method is also veryclose to the result calculated by code as can be seen in Figure 12. It should be pointed out thatonly four key points (i.e. A, B, C and D in Figure 12) are given in HKSC [9] to form the design P-M interaction curve. The small difference of the maximum bending moment (see point D)between the proposed method and the design code is due to the use of different failure criteria.In the proposed method, the failure of the section is controlled by the concrete strain at extremefibre, which is the basic assumption in many concrete design codes. However, the designformula for the encased composite section in the steel code representing the failure of the

section is controlled by fully plastic state of all components. As a result, the proposed methodprovides a more conservative result for design.

6.2. Numerical analysis for three simple portal frame

In this example, a simple portal frame with three types of sections, i.e., bare steel, reinforced-concrete and composite, is studied. The geometrical layout and the section properties areshown in Figure 13 below.

Material Properties:

Concrete : C45 , f cu = 45 N/mm2

, m = 1.5Steel section: S355, f y = 355 N/mm 2, m = 1.0

Reinforcement: R460, f y = 460 N/mm 2, m = 1.15

Case Beam Column

1

2

3

Figure 13 : Geometry of portal frame and section propertities

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As discussed in Section 3, initial imperfections such as local member initial curvature and globalframe imperfection should be considered in a practical second-order analysis. Here, the initialmember imperfections of 1/300 and 1/400 of member length are assumed for columns andbeams respectively, while the global imperfection is taken as 1/500 of building height. Theequivalent axial stiffness (EA) e and (EI) e are calculated by the following equations.

( ) e c c s s r r EA E A E A E A= + + (18)

r r sscce I E I E I E EI ++= 5.0)( (19)

The load-deflection curves of the top of right column for three different types of frame obtainedfrom the proposed second-order inelastic analysis are plotted in Figure 14.

Figure 14 : Load-deflection curves of portal frames

From Figure 14, it can be seen that ultimate load resistance of composite frame is much higherthan the other two frames and the bare steel frame shows lowest load resistance. This figurealso indicates that both steel and composite frames experience excellent ductility whilereinforced concrete frame shows relative poor ductility with maximum lateral deflection at about175mm.

It should be emphasized that as both the P- and P- effects as well as initial imperfections havebeen taken into account in the proposed second-order analysis. Only the section capacity checkfor cross section of a structural member is needed without assumption of effective length.. Theprocedure leads not only to time-saving, but also safety as the error for assuming an effectivelength is eliminated. Also, the tedious member design by different codes can be avoided sincethe proposed sectional analysis method has the capability to exactly determine the yield andfailure surfaces automatically.

E

EE

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7. CONCLUSIONS

In this paper, a practical second-order analysis is proposed for design of hybrid frame structures.By using an initially curved beam-column element in the robust nonlinear incremental-iterativeprocedure, the second-order effects of individual members and the structural system can bemodelled. Further, a divergence-proof iterative procedure is used to exactly calculate the yieldsurfaces of arbitrary shape reinforced concrete and composite sections subjected to axial forceand biaxial bending. Thus, the proposed method checks the stability and section strength in theprocess of structural analysis and as a result no additional individual member design by code isneeded. The remarkable advantage is that assumption of effective length for member bucklingchecks is no longer required. Also, the member design by different codes is avoided and onlythe material stress-strain curve is needed to check the section capacity.

The proposed method is further extended to inelastic analysis so that this method can be

applied to advanced analysis, plastic design, seismic design and progressive collapse. Tocapture gradual yielding behaviour, the first and full yield surfaces of reinforced concrete andsteel-concrete composite section are computed and defined in the proposed method for asecond-order inelastic analysis.

8. ACKNOWLEDGMENT

The authors acknowledge the financial support by the Research Grant Council of the Hong KongSAR Government on the projects Second-order and Advanced Analysis and Design of SteelTowers Made of Members with Angle Cross-section (PolyU 5115/08E ) and the projectAdvanced analysis for progressive collapse and robustness design of steel structures (PolyU5115/07E).

9. REFERENCES

; Es d ^

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Advance Analysis of Hybrid Frame Structures by Refined Plastic-hinge Approach