Non-Linear Behavior of Composite Slim Floor Beams with ...

European Journal of Scientific Research

ISSN 1450-216X Vol.56 No.3 (2011), pp.311-325

© EuroJournals Publishing, Inc. 2011

http://www.eurojournals.com/ejsr.htm

Non-Linear Behavior of Composite Slim Floor

Beams with Partial Interaction

Eyad K. Sayhood

Asst. Prof. Department of Building and Construction

University of Technology, Baghdad- Iraq

E-mail: [email protected]

Mohammed Sh. Mahmood

Asst. Lecturer, College of Engineering, Diyala University, Iraq

E-mail: [email protected]

Abstract

In this research a non-linear finite difference models are presented to predict the

behavior of simply supported composite slim floor beams. The formulation is based on the

partial interaction theory of composite beams. For the purpose of analysis, a section of the

beam is divided into three main layers (first concrete layer, steel beam layer and second

concrete layer) each layer is divided into numbers of sub-layers. The materials of

composite slim floor beam are connected together by a natural shear bond without using

shear connectors. Equilibrium and compatibility are satisfied for forces and displacements

at an assumed element giving two differential equations of second order in terms of slip or

two differential equations of second order in terms of axial force. The equations are solved

numerically using finite difference method. The results of the current study are compared

with the available experimental and theoretical results of previous researches and they

show close agreement. A parametric study was made to show the effect of some parameters

on the interface slip, deflection and axial force.

Keywords: Non-Linear, Slim Floor, Partial Interaction, Shear Bond, Composite Action.

1. Introduction A composite construction consists of using two materials together in one structural unit and using each

material to its best advantage. A concrete slab connected to a steel beam forms a composite beam. The

concrete which is a good in compression takes the compressive forces. The steel, generally placed

toward the tension face of the beam, takes all the tensile forces. Composite construction, i.e., steel

beams or girders connected to concrete slabs by means of mechanical shear connectors are common in

building and bridges in many situations. This structural system is more advantageous to either

reinforced concrete or steel alone. Greatest economic and structural benefit is generally realized when

the concrete slab and the steel beam act under full composite action. Theoretically, the later can be only

realized if there is no slip at the interface of the steel beam flange. In practice, however, mechanical

shear connectors primarily transfer shear, due to their deformable nature, some slip always occurs. The

amount of the slip is directly related to the magnitude of shear forces and the strength and load-

deformation characteristics of the connectors.

Non-Linear Behavior of Composite Slim Floor Beams with Partial Interaction 312

The slim floor composite beams have shallow beam section where the slab is supported on the

lower flange of the steel beam, either by pre-welding a plate across this flange or by using an

asymmetric steel section, leaves only the lower face of the bottom flange exposed. In this system steel

beams are integrated into the concrete or composite slabs. So, the floor system has a small depth with

high stiffness and strength and therefore, besides architecture it is also interesting in view of economy.

Slim floor construction provides a steel floor system of minimum depth which competes, directly with

reinforced concrete flat slabs. There are two generic forms of slim floor construction, in situ concrete

slabs with or without precast concrete, and deep decking acting compositely with an in situ concrete

slab. The precast concrete units placed over the bottom flange at a bearing length not less than 75mm [1]

, may mean that the ends of the units should be chamfered to install them between the flanges of the

beam. In situ concrete with mesh reinforcement over the steel beam is recommended in order to avoid

the effect of out-of balance imposed loads when adjacent spans differ by 20% or when transfer of high

in-plane force is required [1]

.

Figure 1: Slim floor composite beam [1]

.

Steel deck

In situ concrete

Slim floor beam

Reinforcement

Figure 1 illustrates a slim floor construction that is similar to composite construction using

shallow deck profiles, in which the decking sits on the bottom flange of the beam. The decking

supports the loads during concreting, and resists the imposed loads subsequently as a composite slab.

This type of slim floor construction uses a beam called Asymmetric Slim floor Beam (ASB). It is often

found that the stiffness of the composite ASB section is 1.5 to 2 times that of the ASB section [1]

,

which greatly reduces its deflection under imposed loads. Economic assessments of the use of the ASB

sections have shown that the potential weight saving relative to conventional slim floor beams is of the

order of 15% to 25%, and the additional saving in fabrication cost is significant because it is not

necessary to weld a bottom plate to the section. With these economies, ASB slim floor construction

may be found to be cheaper than the conventional composite beam and slab, and reinforced concrete

flat slab construction in the same medium span range.

The shear bond strength has a design value of 0.6 N/mm2 as justified by full scale test for ASB

sections with their raised pattern rolled into the top flange. This property is of great significance in

structural design of flexural members. Moreover, the transferring of stresses between concrete and steel

has a great influence in limiting the space and the width of cracks. Effective bond strength creates the

composite action of steel with concrete. The design value of shear bond strength is obtained from the

approximate relationship based on the results on slim floor tests as below [1]

:

fg

StrengthbondShearDesign)f(StrengthBondShearMeasured

.

cuc

sbm

30150

×

=

(1)

Shear bond at the interface is:

f.Pf sbmsb = (2)

The longitudinal shear force transfer is assumed to occur by shear bond stresses acting

uniformly around the flanges and both sides of the web of the ASB section. For beams subjected to

313 Eyad K. Sayhood and Mohammed Sh. Mahmood

uniformly distributed load, the maximum compressive force,sbF , that can act in the slab at mid span of

the beam is obtained from consideration of elastic shear flow along the beam as follows:

L.fF sbsb4

= (3)

2. Assumptions In the present study the following assumptions are based on in the analysis of slim floor beams:

a. Plane sections are assumed to remain plane after bending.

b. At every section of a slim floor beam, each layer deflects the same amount, and no buckling of

layer occurs.

c. Friction effect between the layers is neglected.

d. The connection is assumed to have negligible thickness and possesses finite normal and

tangential stiffness and prevents separation.

e. The amount of slip permitted by the shear bond is directly proportional to the load transmitted

at any given load on the beam.

f. The distribution of the strains through the depth of the individual layers is linear.

g. The shear connection between layers is continuous along the length.

3. Non-Linear Modelling In non-linear analysis the material properties Ec, Es and fsb are all functions of strain or displacement

and the solution is to be obtained iteratively at each loading stage. This can be achieved by dividing the

cross sectional area of the concrete and steel beam into a number of strips as shown in figure 2, and

using the summation over the appropriate area[3]

. The appropriate values of E are the secant value

determined from the assumed stress- strain curves corresponding to strains in the center of each strip,

where the strain at center of each strip can be calculated.

3.1. Concrete

Different researchers have used several models for stress-strain curves of concrete in their works, in the

present study the adopted stress-strain relationship of concrete in compression is that proposed in the

BS 8110 [2]

where the variation of curved portion of the stress-strain relationship is given by:

...f. cu 103115500 26 εεσ ×−= (4)

With initial modulus of elasticity ( iE ) equal to:

f.E cui 5500= (5)

Where (cuf ) and (σ ) are in (N/mm

2).

3.2. Steel

Hot rolled steel sections are used extensively as structural material because of its important properties

such as high strength as compared to other building material, and the ductility which is the ability to

deform substantially in either tension or compression before failure. Steel has a similar stress-strain

curve in tension and compression. The mathematical model for representing the behavior of structural

steel to a practical strain limit consists of elastic, perfectly plastic and strain hardening portions

respectively [4]

. In the present study the strain hardening is not taken into account.


3.3. Shear Bond

The non-linearity of the shear bond strength can be done using the equation (7), which can be solved

numerically using finite difference method. In any loading stage the shear bond stress can be calculated

from equation (7) after substituting the appropriate values of the interface slip at each interface. The

shear bond stress is converted to shear bond strength using equation (2)[5]

.

Figure 2: Division of cross section [5]

.

First concrete layer

Second concrete layer

Steel layer

First interface

Second interface

Shear bond perimeter

Z c1 i A c1 i N.A (first concrete layer)

Z c2 k A c2

k

N.A (second concrete layer)

Z s j

A s j

N.A(steel layer)

4. Formulation An element of a composite slim floor beam, of length (δx) is s hown in figure 3. The slim floor

beam consists of three layers denoted by (Concrete1), (Steel) and (Concrete2) respectively, jointed

together by a medium of negligible thickness but has finite tangential stiffness. The three layers

subjected to moment (M), shear force (V) and axial force (F), (q) denotes the shear per unit length at

the interface, and (p) is the applied load.

Figure 3: Elements of slim floor composite beam.

h 1

Pδx

(Concrete1)

Vc1

Vc1 +δVc1

Fc 1 +δFc 1

Mc 1 +δMc 1 Mc1

Fc1

h 2 (Steel)

Vs

Vs+δVs

Fs+δFs

Ms+δMs Ms

Fs

q1δx

h 3 (Concrete2)

Vc2

Vc 2 +δVc 2

Fc2 2 +δFc 2

Mc 2 +δMc 2

Mc2

Fc2

q 2 δx

δx

d 1

d 2

Considering the element shown in figure 3 and satisfying Equilibrium and compatibility [6-7]

for

force and displacement at the assumed element arriving at two differential equations of second order in

terms of slip (first approach) or two differential equations of second order in terms of axial force

(second approach)[5]

, as follows:


0EI

d.NU.U.xx,U

0EI

d.NU.U.xx,U

214232

122111

=+λ−λ−

=+λ−λ− (6)

Where:

++

++

=

∑∑∑∑∑=====

21

11

3

22

21

1111111

2

111

.

1

.

1

...

n

j

j

s

j

s

n

i

i

c

i

c

n

k

k

c

k

c

n

j

j

s

j

s

n

i

i

c

i

c

sb

AEAEIEIEIE

d

L

fλ

( )

+ +

+ +

=

−

+ +

=

∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

= = = = =

= = = =

2 3

2 2

3

2 2

2 1

1 1

2 3

2 2

2 1

1 1

1 1 1 1 1

2 2 2

3

1 1 1 1

2 1 2 2

.

1

.

1

. . .

.

1

. . .

.

n

j

j s

j s

n

k

k c

k c

n

k

k c

k c

n

j

j s

j s

n

i

i c

i c

s b

n

j

j s

j s

n

k

k c

k c

n

j

j s

j s

n

i

i c

i c

s b

A E I E I E I E I E

d

L

f

A E I E I E I E

d d

L

f

λ

λ

( )

∑∑∑

∑∑∑∑

===

====

++=

−

++

=

3

22

21

11

23

22

21

11

111

1111

2114

...

.

1

...

.

n

k

k

c

k

c

n

j

j

s

j

s

n

i

i

c

i

c

n

j

j

s

j

s

n

k

k

c

k

c

n

j

j

s

j

s

n

i

i

c

i

c

sb

IEIEIEEI

AEIEIEIE

dd

L

fλ

2

2

322

211

hhd

hhd

+=

+=

Equation (6) represents the first approach in the present study, which will be solve numerically

using finite difference method to obtain the interface slip along the beam, then the shear bond stress

(fsbm ) can be calculated numerically using equation (7) after that the axial force can be determined

numerically using equation (8)[5]

.

)A.E

1

A.E

1(P.fx

sscc

sbm +−= (7)

U.L

fx,Fc

U.L

fx,Fc

2

2sb

2

1

1sb

1

−=

−=

(8)

EI

M.d.L

f

Fc.Fc.xx,Fc

EI

M.d.L

f

Fc.Fc.xx,Fc

tsb

tsb

0

0

22

12262

11

25111

=+−−

=+−−

λλ

λλ

(9)

Where:


( )

( )

++

++

=

−

++

=

∑∑∑∑∑

∑∑∑∑

=====

====

3

22

23

22

21

11

23

22

21

11

11111

3226

1111

3115

.

1

.

1

...

.

.

1

...

.

n

k

k

c

k

c

n

j

j

s

j

s

n

k

k

c

k

c

n

j

j

s

j

s

n

i

i

c

i

c

sb

n

j

j

s

j

s

n

k

k

c

k

c

n

j

j

s

j

s

n

i

i

c

i

c

sb

IEAEIEIEIE

dd

L

f

AEIEIEIE

dd

L

f

λ

λ

23 cs YYd −=

Equation (9) represents the second approach in the present study, which will be solve

numerically using finite difference method to obtain the axial force along the beam, after that the

interface slip can be determined numerically using equation (8), then the shear bond stress (fsbm ) can be

calculated numerically using equation (7). The deflection (W) of any node along the beam can be

determined numerically for any layer by using equation (10) because the layers deflect in same amount [5-6]

.

I.EI.EI.E

d.Fcd.FcMxx,W

n

k

kc

kc

n

j

js

js

n

i

ic

ic

t

3

22

21

11 111

3211

∑+∑+∑

−−=

=== (10)

5. Validity 5.1. First Example

Asymmetric slim floor (300 ASB) was tested to failure load of (136kN/m) by Corus construction center [1]

. The width of the concrete slab was limited to (1m), representing L/8 approximately. The material

properties of the beam are shown in table 1 and the geometric properties are shown in figure 4.

Table 1: Material Properties of 300 ASB.


Concrete density 1970 kg/m3

Initial modulus of elasticity 34785 N/mm2

Compressive strength (fcu) 40 N/mm2

300 ASB Yield strength (fy) 380 N/mm

2






Interface Shear bond strength 1.28 N/mm2

Figure 4: Geometric properties of 300 ASB (first example) [1]

.

20.4

231

88 21.5

30

198

1000

306

17.3

263 284.5

Section A-A (All dimensions in (mm))

136 kN/m

A

A

7500

Beam Elevation


Table 2: Comparison of maximum slip between the numerical and experimental results for 300 ASB.

Numerical solution (Present study) results. Experimental Maximum

slip results (mm)[1]

.

Number of nodes 29 49 99

2 Approach

First Second First Second First Second

Maximum slip (mm) 2.242 3.519 2.120 2.783 1.980 1.971

It can be shown from table 2 that the calculated value of the maximum slip using the first

approach with number of node equal to (99) is (1.980) which differs from the experimental value by

(1%), while the maximum slip with the same number of nodes for the second approach is (1.971)

which differs from the experimental value by (1.5%).

Midspan deflection values have been calculated at each loading stage using the two approaches.

Figure 5 shows that using a number of nodes equal to (99) gives results very close to the experimental

results. The maximum midspan deflection calculated using first approach is (149.262 mm), and

(146.511 mm) using the second approach, which means the first approach gives results closer to the

experimental result (152 mm) than those obtained by the second approach.

Figure 5: Comparison of present study (Midspan deflection) with experimental results for 300 ASB (first

example).

[1 ]

[1 ]

Midspan Deflection (mm) Midspan Deflection (mm)

0

20

40

60

80

100

120

140

0.00 E + 00 5.00 E + 01 1.00 E + 02 1.50 E + 02 2.00 E + 02 2.50 E + 02 Midspan Deflection (mm)

Applied Load (kN/m)

Experimental

[ 11 ] Second approach

( 29 Nodes

) Second approach

( 49 Nodes

) Second approach

( 99 Nodes

) 0

20

40

60

80

100

120

140

0 50 100

150

200

250

Midspan Deflection (mm)

Applied Load (kN/

Experimental [11] First approach (29 Nodes) First approach (49 Nodes) First approach (99 Nodes)

a. First approach b. Second approach

5.2. Second Example

Asymmetric slim floor (280 ASB) beam of span (7.5 m) was tested to failure load of (113 kN/m) by

Corus construction center [1]

. The width of the concrete slab was limited to (1m). The material

properties of the beam are shown in table 3 and the geometric properties are shown in figure 6.







2








Figure 6: Geometric properties of 280 ASB (second example) [1]

.

113kN/m

A

A

7500

Beam Elevation 18.3

210

80

16.7

30

183

1000

280

19 . 5

245

261. 7

Section A (All dimensions in (mm))

Table 4: Comparison of maximum slip between the numerical and experimental results for 280 ASB.

Numerical solution (Present study) results. Experimental Maximum

slip results (mm)[1]

.

Number of nodes 29 49 99

6

Approach

First Second First Second First Second

Maximum slip (mm) 6.181 9.83 6.122 8.322 5.910 5.901

A comparison between experimental results and present numerical solutions are shown in table

4. Midspan deflection values have been calculated at each loading stage using the two approaches,

figure 7. The same conclusions of the first example can be drawn for this example. Figure 7: Comparison of present study (Midspan deflection) with experimental results for 280 ASB (second

example).

0

20

40

60

80

100

120

0 50 100 150 200 250Midspan Deflection (mm)

Ap

plie

d L

oa

d (

kN

/m)

Experimental [11]

First Approach (29 Nodes)


First Approach (99 Nodes)0

20

40

60

80

100

120

0 50 100 150 200 250Midspan Deflection (mm)

Ap

plie

d L

oa

d (

kN

/m)

Experimental [11]

Second Approach (29 Nodes)




[1[1


5.3. Third Example

Asymmetric slim floor (300 ASB) beam of span (7.5 m) was analyzed by finite element method and

using ANSYS program [7]

. The failure load is (132 kN/m). The width of the concrete slab was limited

to (1m). The material properties of the beam are shown in table 5 and the geometric properties are

shown in figure 8.








2







Figure 8: Geometric properties of 300 ASB (third example)

[7].

20.4

231

88 21.5

32.4

190

1000

300

17.3

265.1

286..5

Section A-A

132 kN/m

A

A

7500

Beam Elevation

(All dimensions in (mm))

Figure 9: Comparison of present study (Midspan deflection) with F.E.M results for 300 ASB (third example).

0

20

40

60

80

100

120

140

0 50 100 150 200 250


Ap

plie

d L

oa

d (

kN

/m)

Finite Element Method [81]




0

20

40

60

80

100

120

140

0 50 100 150 200 250


Ap

plie

d L

oa

d (

kN

/m)

Finite Element Methd [81]





[7]

[7]


It can be shown from figure 9 that there is a relative competition between the finite element

results and numerical results (present study) in spite of that (2660) elements are used in finite element

method while in present numerical solution only (99) nodes along the span of the beam are required to

obtain results close to those of finite element solution.

6. Parametric Study The parametric study has been done on the composite slim floor beam shown in the first example of

the previous section. Many parameters can be changed in the models presented in this study to examine

the effect of each parameter on the model results. All results will be obtained using a number of nodes

(99) along the beam. The value of interface slip convergence is (0.0001) and numbers of strips (sub-


layers) are (8) for the first concrete layer, (4) for each flange of steel beam,(10) for the web of steel

beam and (14) for the second concrete layer.

6.1. Thickness of First Concrete Layer

Three different values of thickness of first concrete layer are used (hc1, 1.5hc1, and 2hc1). Figure 10

shows that the increase of thickness of the first concrete layer by (50%) leads to increase the maximum

values of interface slip by (4.7%) using the first approach and (5%) using the second approach.

Increasing the thickness of first concrete layer by (100%), causes an increase in the maximum values of

interface slip by (7.8%) using the first approach and (10%) using the second approach. The increase of

the values of slip comes from the increase of axial compressive strength in the concrete above the steel

beam.

Figure 10: Distribution of slip along the beam for different values of thickness of first concrete layer.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000

distance along the beam (mm)

Inte

rfa

ce s

lip

(m

m)

U1 @ hc1

U2 @ hc1

U1 @ 1.5hc1

U2 @ 1.5hc1

U1 @2 hc1

U2 @ 2hc1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000

Distance along the beam (mm)

Inte

rfa

ce s

lip (

mm

)

U1 @ hc1

U2 @ hc1

U1 @ 1.5 hc1

U2 @ 1.5hc1

U1 @ 2hc1

U2 @2hc1



Figure 11: Distribution of deflection along the beam for different values of thickness of first concrete layer.

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


De

flection

(m

m)

hc1

1.5hc1

2hc1

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


De

flection

(m

m)

hc1

1.5hc1

2hc1



Figure 11 shows that the increase of the thickness of the first concrete layer by (50%) gives a

decrease in the maximum values of deflection by (7%) using the first approach, and (5.5%) using the

second approach. Increasing the thickness of first concrete layer by (100%) gives a decrease in the

maximum values of deflection by (16%) using the first approach, and (12%) using the second


approach. The decrease in the values of deflection is due to the increase in the total moment of inertia

of the composite beam section which affects the value of deflection.

Figure 12: Distribution of axial force along the beam for different values of thickness of first concrete layer.

0

100

200

300

400

500

600

700

800

0 1000 2000 3000 4000 5000 6000 7000

Distance Along The Beam (mm)

Axia

l F

orc

e (

kN

)

Fc1 @hc1

Fc2 @hc1

Fc1 @1.5hc1

Fc2 @1.5hc1

Fc1 @2hc1

Fc2 @2hc1

0

100

200

300

400

500

600

700

800

0 1000 2000 3000 4000 5000 6000 7000


Axia

l fo

rce (

kN

)

Fc1 @ hc1

Fc2 @ hc1

Fc1 @ 1.5hc1

Fc2 @1.5 hc1

Fc1 @ 2hc1

Fc2 @2hc1



Figure 12 shows a decrease in maximum values of axial force by (14%) using the first approach

and (11%) using the second approach, with increasing the thickness of the first concrete layer by

(50%). Also a decrease in maximum values of axial force by (21%) using the first approach and (17%)

using the second approach, with increasing the thickness of the first concrete layer by (100%).

6.2. Width of Top Flange of Steel Beam

Three different values of width of top flange of steel beam (150 mm, 200 mm, and 250 mm) are

considered. Figure 13 shows that increasing the width of top flange of steel beam by (33%) gives a

decrease in the maximum values of interface slip by (12%) using first approach and (9%) using second

approach, while increasing the width of top flange of steel beam by (67%) gives a decrease in the

values of interface slip by (17%) using the first approach and (15%) using the second approach. This

can be attributed to the increase of bond strength due to the increase of contacting perimeter of steel

beam with concrete.

Figure 13: Distribution of slip along the beam for different values of width of top flange of steel beam.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000


Inte

rface

slip (

mm

)

U1 @ bt=150 mm

U2 @ bt=150 mm

U1 @ bt=200 mm

U2 @ bt=200 mm

U1 @ bt=250 mm

U2 @ bt=250 mm

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000

Diatance along the beam (mm)

Inte

rfa

ce s

lip

(m

m)

U1 @ bt=150mm

U2 @ bt=150mm

U1 @ bt=200mm

U2 @ bt=200mm

U1 @ bt=250mm

U2 @ bt=250mm



Figure 14: Distribution of deflection along the beam for different values of width of top flange of steel beam.

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


Defle

ctio

n (

mm

)

bt=150 mm

bt=200 mm

bt=250 mm

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


Defle

ctin

(m

m)

bt=150 mm

bt=200 mm

bt=250 mm



Figure 14 shows that the maximum values of the deflection decrease by (8%) using the first

approach and (7%) using the second approach with an increase in the width of top flange of steel beam

by (33%), and the maximum values of the deflection decrease by (17%) using the first approach and

(16%) using the second approach with an increase in the width of top flange of steel beam by (67%).

This is due to the increase in the total moment of inertia of the composite section which affects the

value of deflection.

Figure 15: Distribution of axial force along the beam for different values of width of top flange of steel beam.


0

100

200

300

400

500

600

700

800

900

1000

0 1000 2000 3000 4000 5000 6000 7000


Axi

al fo

rce

(kN

)

Fc1 @ bt=150 mm

Fc2 @ bt=150 mm

Fc1 @ bt=200 mm

Fc2 @ bt=200 mm

Fc1 @ bt=250 mm

Fc2 @ bt=250 mm

0

100

200

300

400

500

600

700

800

900

1000

0 1000 2000 3000 4000 5000 6000 7000


Axia

l fo

rce (

kN

)

Fc1 @ bt=150 mm

Fc2 @ bt=150 mm

Fc1 @ bt=200 mm

Fc2 @ bt=200 mm

Fc1 @ bt=250 mm

Fc2 @ bt=250 mm


Figure 15 shows that increasing the width of top flange of steel beam by (33%) produces an

increase in the maximum values of the axial force by (9%) using the first approach and (7%) using the

second approach. Increasing the width of top flange of steel beam by (67%) produces an increase in the

maximum values of axial force by (13%) using the first approach and (12%) using the second

approach. This is due to the increase of contacting perimeter, which causes an increase in bond strength

between the steel beam and the concrete.


6.3. Shear Bond Strength

Three different values of shear bond strength are used here (0.85, 1.28, and 1.70) N/mm2. Figure 16

indicates that the increase of bond strength by (50%) leads to decrease the maximum values of

interface slip by (4%) using the first approach and (2%) using the second approach, and increasing the

bond strength by (100%) lead to decrease the maximum values of interface slip by (9%) using the first

approach and (7%) using the second approach.

Figure 16: Distribution of slip along the beam for different values of shear bond strength.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000


Inte

rfa

ce

slip

(m

m)

U1 @ fsbm=0.85 MPa

U2 @ fsbm=0.85 MPa

U1 @ fsbm=1.28 MPa

U2 @ fsbm=1.28 MPa

U1 @ fsbm=1.70 MPa

U2 @ fsbm=1.70 MPa

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000

Distance along th beam (mm)

Inte

rfa

ce

slip

(m

m)

U1 @ fsbm=0.85 MPa

U2 @ fsbm=0.85 MPa

U1 @ fsbm=1.28 MPa

U2 @ fsbm=1.28 MPa

U1 @ fsbm=1.70 MPa

U2 @ fsbm=1.70 MPa



Figure 17 indicates that the increase of bond strength by (50%) gives a decrease in the

maximum values of deflection by (3%) using the first approach and (4%) using the second approach,

and the increase of bond strength by (100%) gives a decrease in the maximum values of deflection by

(5%) using the first approach and (7%) using the second approach.

Figure 17: Distribution of deflection along the beam for different values of shear bond strength.

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


De

fle

ctio

n (

mm

)

fsbm=0.85 MPa

fsbm=1.28 MPa

fsbm=1.70 MPa

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000 7000


De

fle

ctio

n (

mm

)

fsbm=0.85 MPa

fsbm=1.28 MPa

fsbm=1.70 MPa



Figure 18 shows that the maximum values of the axial force increase by (35%) using first

approach and (31%) using the second approach with the increase of bond strength by (50%), and the

maximum values of the axial force increase by (77%) using the first approach and (74%) using the

second approach with the increase of bond strength by (100%).


Figure 18: Distribution of axial force along the beam for different values of shear bond strength.

0

200

400

600

800

1000

1200

0 1000 2000 3000 4000 5000 6000 7000


Axia

l fo

rce

(kN

)

Fc1 @ fsbm=0.85 MPa

Fc2 @ fsbm=0.85 MPa

Fc1 @ fsbm=1.28 MPa

Fc2 @ fsbm=1.28 MPa

Fc1 @ fsbm=1.70 MPa

Fc2 @ fsbm=1.70 MPa

0

200

400

600

800

1000

1200

0 1000 2000 3000 4000 5000 6000 7000

Dstance along the beam (mm)

Axia

l fo

rce

(kN

)

Fc1 @ fsbm=0.85 MPa

Fc2 @ fsbm=0.85 MPa

Fc1 @fsbm=1.28 MPa

Fc2 @ fsbm=1.28 MPa

Fc1 @ fsbm=1.70 MPa

Fc2 @ fsbm=1.70 MPa

a. First approach


7. Conclusions The following conclusion can be drawn from the present study:

1. Comparing results of the present study with the available experimental and theoretical results

show that the present study can be used to investigate the full range behavior of composite slim

floor beams.

2. The finite difference method with an incremental-iterative solution technique is efficient to

predict the non-linear behavior of composite slim floor beams.

3. The adopted non-linear strategy gives a fast convergence for the nonlinear behavior even for

large load increments.

4. Increasing the thickness of the first concrete layer by (100%) leads to increase the maximum

value interface slip by (7.8%) using the first approach and (10%) using the second approach.

Decreasing the maximum value of deflection (16%) using first approach and (12%) using

second approach, decreases the maximum value of axial force (21%) using first approach and

(17%) using second approach.

5. Increasing the width of top flange of steel beam by (67%) gives a decrease in the maximum

value of interface slip (17%) using the first approach and (15%) using the second approach,

decreasing in the maximum value of deflection (17%) using first approach and (16%) using

second approach, and increasing in the maximum value of the axial force (13%) using first

approach and (12%) using second approach.

6. Increasing of bond strength by (100%) produces decreasing in the maximum value of the

interface slip (9%) using first approach and (7%) using second approach, decreasing in the

maximum value of deflection (5%) using first approach and (7%) using second approach, and

increasing in the maximum value of axial force (77%) using first approach and (74%) using

second approach.

7. Increasing of compressive strength of concrete by (33%) leads to decrease the interface slip

(8%) using the first approach and (7%) using the second approach, decrease the deflection (7%)

using first approach and (5%) using second approach, and increasing the axial force (10%)

using first approach and (13%) using second approach.

8. The interface slip is proportional directly with shear force in both cases of uniformly distributed

load and point load.

9. The interaction efficiency of composite slim floor beam subjected to uniformly distributed load

is less than that for case of point load.

10. The deflection and axial force are directly proportional to the bending moment.


References [1] Web site www.corusconstruction.com, Corus construction center

[2] BSI, “Code of practice for structural use of concrete" BS- 8110: Part 1, British Standard

Institute, London, 1997, p.p 163.

[3] R.I.M. Al-Amery, T.M. Roberts, "Nonlinear finite difference analysis of composite beams with

partial interaction" Computers and structures, Vol.35,, Issue 1, 1990, Pages 81-87.

[4] Chen, W. F. and Atsuta T., “Theory of beam column vol. 1, in plain behavior and design"

McGraw-Hill book company, 1976.

[5] Johnson, R.P. and May, I. M. “Partial interaction design of composite beams", The structural

engineer, Vol.53, No.8, 1975.

[6] Newmark, N. M., Seiss, C. P., and Viest, I. M., “Test and analysis of composite beams with

incomplete interaction" Processing of the society for experimental stress analysis, Vol. 9, No. 1,

1951.

[7] Jose, L. R., and Enrique, M. “Development and validation of a model for mixing beams of slim

floor", III International Congress of the Metallic Construction - III CICOM” - Ouro Preto, MG,

Brazil - April, 2006.

8. Nomenclature

Ac1 Cross-sectional area of first concrete layer (mm2)

Ac2 Cross-sectional area of second concrete layer (mm2)

As Cross-sectional area of steel layer (mm2)

b Effective flange width of composite section (mm)

d1, d2, d2 Constant as defined.

Ec1 Modulus of elasticity of first concrete layer (N/mm2)

Ec2 Modulus of elasticity of second concrete layer (N/mm2)

Es Modulus of elasticity of steel layer (N/mm2)

Fc1 Axial force in first concrete layer (N)

Fc2 Axial force in second concrete layer (N)

fcu Characteristic concrete compressive strength (N/mm2)

fsb Shear bond at the interface (N)

fsb1 Shear bond strength at first interface (N/mm)

fsb2 Shear bond strength at second interface (N/mm)

fsbm Measured shear bond strength (N/mm2)

fy Yield strength of steel (N/mm2)

gc Partial factor for concrete.

h1, hc1 Thickness of first concrete layer (mm)

h2, hs Thickness of steel layer (mm)

h3, hc2 Thickness of second concrete layer (mm)

Ic1 Moment of inertia of first concrete layer (mm4)

Ic2 Moment of inertia of second concrete layer (mm4)

Is Moment of inertia of steel layer (mm4)

L Beam span (mm)

M Moment subjected to the beam (N.m)

Mt Total moment of typical section (N.m)

N Total vertical shear force at distance x from support (N)

n1 Number of strips of first concrete layer.

n2 Number of strips of steel layer.

n3 Number of strips of second concrete layer.

P Contacting perimeter of the steel beam with concrete (mm)

p Applied load (N)

q Shear flow at interface (N/mm)

U1 Slip at first interface (mm)

U2 Slip at second interface (mm)

V Shear force (N)

W Deflection of beam (mm)

xδ Length of slim floor beam segment (mm)

61 λλ − Constant as defined.

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