GRILLAGE ANALYSIS OF COMPOSITE CONCRETE SLAB ON STEEL BEAMS WITH PARTIAL INTERACTION

Prof. Dr. Husain M. Husain1, Dr. Ali N. Attiyah2 and Jenan Ni’amah Yasser3

ABSTRACT: The present study is concerned with the behavior of a composite structure made up of a concrete slab connected to steel beams in two directions by shear connectors by taking into consideration the linear action of shear connectors in the force-slip relationship. The grillage or grid framework method as simplified method of analysis is used in this study to study slip, deflection and stresses caused by moments from applied normal loads. A method is suggested to derive the required section rigidities (the flexural and torsional rigidities) of the grillage members from the composite action of the individual grillage composite members. Design charts are constructed for estimating the percentage decrease in flexural rigidity of each composite member with partial shear connection. It was found for a composite structure analyzed by grillage members, the effective width of each member should be used to calculate the flexural rigidity of that member. Also Poisson’s ratio effect was included in the calculation of the flexural rigidities of the grillage members. Effect on deflections by transverse shearing forces was found to be small and thus it can be neglected (percentage differences is less than 11.8 %).

1. Introduction

Each different building material has a special prominent quality which distinguishes it

from other materials. There is no material that can provide all the structural

requirements. This is the reason of using different materials that can be arranged in an

optimum geometric configuration, with the aim that only the desirable property of

each material will be utilized by virtue of its designated position. The structure is then

known as a composite structure, and the relevant method of building as composite

construction.

The composite concrete slab-on-steel beam structure consists of three major structural

elements, namely a reinforced concrete slab resting on longitudinal and transverse

steel beams, which interact, compositely with the slab by means of mechanical shear

connectors. The analysis of composite beams and their behavior assuming linear and

nonlinear material and shear connector behavior has been in general based on an

approach initiated by Newmark, Siess and Viest in 1951[1]. The equilibrium and

compatibility equations for an element of the beam were reduced to a single second

1) University of Tikreet , College of Engineering , Civil Department.

2) University of Kufa , College of Engineering , Civil Department.

3) University of Kufa , College of Engineering , Civil Department.

1

order differential equation in terms of either the resultant axial force in the (concrete)

flange or the interface slip. Solution for the axial force or the interface slip was

substituted back into the basic equilibrium and compatibility equations, which could

then be solved to give the displacements and the strains throughout the beam. That

approach was initially based on linear material and shear connector behavior.

In the method suggested for the present study, the composite structure is idealized as a

grillage, the grillage mesh is assumed to be coincident with the center-lines of the

main steel beams. The concrete slab and the steel beams are assumed to behave in the

elastic range and the force- slip behavior of the shear connectors is linear. To use the

T-beam approach, the concept of the effective width is used which refers to a fictitious

width of the slab that when acted on by the actual maximum stress the slab would

have the same static equilibrium effect as the existing variable stress. The effective

width is affected by various factors, such as the type of loading, the boundary

conditions at the supports and the ratio of beam spacing to span B/L [2].

Johnson (1975)[3] proposed a partial interaction theory for simply supported composite

beams, in which the analysis was based on elastic theory. Kennedy, Grace, and

Soliman (1989)[4] presented an experimental study that was conducted on three

composite bridge models each subjected to one- vehicle load. Jasim (1994)[5] presented

a method of analysis which depended on elastic theory. In that analysis he adopted

same assumptions of Newmark[1].

2. Assumptions of the Grillage Analogy

The grillage analogy involves the representation of effectively a three- dimensional

composite structure by a two- dimensional assemblage of discrete one- dimensional

interconnected beams in bending and torsion. In analysis, the following assumptions

are introduced:

1- Concrete and steel are linearly elastic materials. The concrete slab is assumed

to be able to sustain sufficient tension such that no tensile cracks develop in

this part. The distribution of strains through the depth of each component is

linear.

2- The longitudinal and transverse steel beams are assumed rigidly connected

(welded connections).

2

3- The shear connection between the two components is continuous along the

length. The discrete deformable connectors with equal moduli and uniform

spacing are assumed to be replaced by a medium of negligible thickness.

Friction and bond effects between the two components are neglected.

4- The amount of slip permitted by the connector is directly proportional to the

force transmitted through the connector.

5- At every section of the composite beams, each component deflects the same

amount. No separation is assumed to occur.

3. Evaluation of Elastic Section Rigidities of Grillage Members

The idealization of a composite slab–beam structure by an equivalent grillage requires

the evaluation of the elastic section rigidities of the grillage members. The elastic

rigidities of these members should be derived from the section properties of the actual

composite slab–beam structure so that an adequate picture for the composite section

behavior under the applied loadings can be obtained from the equivalent grillage. The

elastic section rigidities required for the sections of the equivalent composite grillage

members are as follows:

1-Bending (or flexural) rigidity (EI). 2-Torsional rigidity (GJ). 3-Shearing

rigidity (GAv).

Herein, suggestions are presented for these quantities and adopted in this work.

3.1 Bending (or Flexural) rigidity:

Flexural rigidities of the equivalent grillage members play an important role in the

calculation of deflections and in the distribution of moments. In analyzing the

composite slab-beam structure by the grillage analogy, the flexural rigidities of the

composite members are derived from partial interaction theory. Generally, two factors

(besides the partial interaction effect) must also be considered in the calculation of the

flexural rigidity of the grillage members. These factors are due to the shear lag and

Poisson ُs ratio effects. Shear lag effects can be included by using the effective width

concept. The two–dimensional confining effect of Poisson’s ratio can be considered

by dividing the modulus of elasticity of concrete E1 by (1-υ2).The interaction

phenomenon can be illustrated from the discussion of the lower and upper limits of

behavior of composite beams, i.e., no interaction and complete (or full) interaction,

3

respectively. The analysis and flexural rigidity will be carried out on the basis of

elastic theory.

Usually, the interaction between steel and concrete is incomplete due to the

occurrence of slip. It produces a discontinuity in the strain distribution at the interface

where appreciable strain difference. The neutral axis of the slab is closer to the beam

and that of the beam is closer to the slab, when compared with the no- interaction

case. The result of the partial interaction is the partial development of the compressive

force in the concrete slab and tensile force in the steel beam. This leads to less

ultimate load than that resisted when complete interaction exists. Partial interaction is

the usual practical case in the design and analysis of composite structures.

A large number of research studies have been devoted to calculate the deflections of

composite beams with partial shear interaction.

The solution submitted by Jasim[5] for the final form of the governing equation for a

composite beam by using Fourier series method will be adopted in the present study to

calculate the flexural rigidity of composite sections for simply supported beams under

different loading cases.

Uniformly Distributed Load

For the case of uniformly distributed load on a simply supported beam the solution for

the maximum deflection is:

(1)

Where = the mid-span deflection of composite beam with full shear

connection ‚ w is the displacement in z-direction, L is the span length, E2 is the

modulus of elasticity of steel, I is the moment of inertia of the transformed fully

composite section about the elastic neutral axis assuming uncracked section, yp is the

mid-span deflection of composite beam with partial shear connection,

4

and .

(2)

Where C12 is the depth of center of gravity of steel beam below mid-plane of slab, I1,

I1t and I2 are the moments of inertia of concrete slab about its own centroid,

transformed area of concrete about its own centroid, steel beam about its own

centroid, respectively. A1 , A1t and A2 are the cross sectional area of concrete slab,

transformed area of concrete above interface, cross sectional area of steel beam,

respectively.h1 is the thickness of concrete slab and h2 is the depth of steel beam. is

the effective modulus of elasticity for concrete slab due to lateral confinement of slab

and E2 is the modulus of elasticity of steel. C1 is a factor found from

C1 = (3)

Where n is the number of connectors per row and p is the spacing of connectors along

the beam.

Since is the curvature , then the integration of this equation twice results

in (4)

where λ is a factor depending on the boundary conditions ( for simply

supported beams under uniform load w per unit length).

Thus (w is the applied load) or (5)

By substituting Eq. (5) into Eq. (1), then

(6)

Defining and substituting this into Eq.

(6) ,then this equation can be written as

5

(7)

Point Load at Mid-span

For the case of a point load at mid-span of a simply supported beam, the solution for

the maximum deflection is:

(8)

where

By using the same procedure, the pertinent equation is

(9)

Using the notation Eq. (9) reduces to

(10)

Point Load at ¼ Span

For this loaded case the maximum deflection is

(11)

Where

Defining , then

(12)

Distributed Load of Trapezoidal Shape

For this case of loading the pertinent equation is

(13)

where

6

Using notation

Eqs. (5) and (13) are combined to give

(14)

Boundary Conditions

Furthermore, the effect of two types of boundary conditions on the prediction of

flexural rigidity of a composite beam is studied. They are a beam with fixed ends and

a cantilever. The effect of different boundary conditions can be considered by

changing the beam effective length. This effect should be included in Eq.(2) by

replacing the beam span (L)with the beam effective length (Le) . For the fixed –

ended beam , the beam effective length is half its span, Le= 0.5 L. For the cantilever,

Le = 2L. Thus Eq. (2) may be rewritten as:

(15)

Effect of Load Pattern

The following three load patterns were studied: (1) a concentrated load at the beam

center; (2) a concentrated load at ¼ span; and (3) trapezoidal distributed load.

Comparisons were made between these types of load patterns with the uniformly

distributed load to find the flexural rigidity of composite beam with partial interaction

(EIp). Results are presented for a representative composite beam 8.6m in span with

universal steel section UB 305×127×37 and concrete flange 1500mm in width and

150mm in depth. The Young ُs moduli of steel and concrete were taken as 205000

N/mm2 and 25000 N/mm2, respectively. Connector stiffness k = 180000 N/mm and

spacing P = 520 mm. Tab.(1) shows the maximum difference between the uniformly

distributed load case and other pattern load cases for EIp value. In all cases, the

difference is less than 1.3%, thus Eq. (7) may be used for all loading cases to obtain

the flexural rigidity of a composite beam with partial interaction. This means that for

each value of factor C the values of D1, D2, D3, and D4 are almost equal for the

majority of K2 values. A discrepancy occasionally occurs in D1 and it is about 1%.

7

This leads to the conclusion that the same chart may be used for all types of loads

which in turn greatly simplifies the calculations needed in design [5].

Thus, Fig.(1) shows such a chart for various values of factor C and in terms of the

percentage increase in flexural rigidity of composite beam with partial shear

connection and the parameter K2 in this chart is for simply supported beams, Figs (2)

and (3) are design charts to find D1 for fix-ended beam and cantilevers respectively.

Tab.(1): Maximum difference in EIp between uniformly distributed load case and other load cases

Load

arrangement

(a) Simply supported beam (b) Beam with fixed ends

Uniform

load

Central

point

load

Point

load

at ¼

span

Trapezoidal

load

Uniform

load

Central

point

load

Point

load

at ¼

span

Trapezoidal

load

Maximum

difference

between

flexural

rigidities

(%)

0)

reference

value(

1.20 0.62 0

0

(reference

value)1.02 0.54 0

3.2 Torsional Rigidity of a Composite Section

It is hypothesized that the strength and the stiffness of composite sections under

torsion are to be considered as that of an open section consisting of two parts acting

independently, i.e., the upper part consisting of the reinforced concrete section with

the upper flange of the steel I-section firmly attached to it, and the lower part

consisting of the web and the lower flange of the steel I-section, as shown in

Fig.(4). Based on this hypothesis the stiffness of a composite section is evaluated in

the pre – cracked stage as follows [8]:

The upper part of the composite section is divided into three portions, two equal

concrete portions of dimensions (bce×h1) and a central composite portion of

dimensions (bs × (h1+tf)), as shown in Fig. (5). The torsional stiffness of the upper part

may then be estimated from the following Eq. for the interior composite beam

8

(16)

and for the edge beam

(17)

Where ‚

neq: equivalent Poisson ُs ratio of central portion of the upper part, ( )

Eeq: equivalent modulus of elasticity of central portion of the upper part of composite

section,

β2 is a coefficient is a function of (b/a)[9] and b is the longer dimension of the

rectangular cross section and a is the shorter dimension of the rectangular cross

section. The torsional stiffness of the lower part may be estimated as follows

a. Free to warp: (18)

b. Warping prevented (or restrained): (19)

Here ‚ ‚ ‚

Cw is the warping constant,

In this work, the case of warping being prevented will be used, and the torsional

rigidity of a composite section can be calculated from the following equation

(20)

This hypothesis is giving an experimental to theoretical ratio of (0.95)[10].

3.3 Shearing Rigidity

Distortion by transverse shearing forces is one of the modes of deformation that can

occur in a composite structure when it is subjected to a general loading. The vertical

(or transverse) shearing force across a composite section causes the flanges and webs

to bend independently out of plane (as a result of shearing deformation). It is known

9

that the transverse shearing deformation is usually small compared with deformation

due to bending. But in some cases, such as in short deep members subjected to high

shearing forces, it is necessary to consider the transverse shearing deformation in

order to obtain a more accurate description of the behavior of the beam. A shearing

rigidity (GAV) is assigned to the stiffness matrix of a grillage member to take into

account the effect of transverse shearing forces on the deformation of that member.

In the grillage analogy, the ability of the composite structure to resist distortion can be

approximately achieved by providing the grillage members an equivalent shear area

(AV). The independent bending moments, which are developed in the webs and in the

flanges are caused by the shearing forces generated in these components. However, in

the present work, the transverse shearing rigidity for a composite member will be

computed by two methods as follows

1- Shearing rigidity for the steel component only by calculating the shear area for the

steel web, Fig.(6a), and it can be stated as:

(21)

2- Shearing rigidity for concrete and steel components together because the depth of

concrete may take into account the shear area especially when it is not small.

Recognizing that the transformed section concept can be applied to the steel web as

shown in Fig. (6b), thus this method can be stated as:

(22)

Where m is the modular ratio = E2/E1

10

Fig. (1) Design chart for simply supported beams.

2.5

2

1.5

1

0.5

Val

ue o

f D

1

C=3.5

-0.5

0

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

C=3

C=2.5

C=2

C=1.5

C=1.25

C=3.5

Log10K2

Fig. (2) Design chart for fix- ended beams.

Val

ue o

f D

1

2.5

2

1.5

1

0.5-3 -2 -1 0 1 2 3

C=3.5

C=3

C=2.5

C=1.5C=1.25

C=2

Log10K2

4. Applications

A composite slab-beam structure is selected from the available reference to assess the

accuracy of the grillage method. The theoretical results of Kennedy model[4] were

derived by the finite element method using the orthotropic plate element; also an

experimental study was made for this model. The composite slab-beam model

considered here is simply supported at two opposite edges and being free at the

longitudinal edges. This type of construction is used in bridge decks. The structure

dimensions are shown in Fig.(7), and material properties are as follows

Upper Component (concrete slab)

Depth of concrete h1= 48 mm.

Compressive strength of concrete f´c = 35 N/ mm2

11

)b (Edge beam

h1

tf

bsbce

Fig. (5): Evaluation of pre – cracked stiffness for upper part division.

)a (Interior beamtf

bcebsbce

h1

Fig. (6): transverse shearing rigidity.

)a (Steel area

h2

h1

tw

)b (transformed area

mtw h2

h1

Fig. (3) Design chart for cantilevers.

Val

ue o

f D

1

-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5

2.5

2

1.5

1

0.5

Log10 K2

C=3.5

C=3

C=2.5

C=2

C=1.5

C=1.25

Fig. (4): Shear stress flow in composite sections.)b (Independent action

tf

h1

)a (Composite actiontf

20

Modulus of elasticity of concrete E1= 27806 N/ mm2 (calculated from

)

Poisson’s ratio of concrete υ1= 0.15

Shear modulus of elasticity of concrete G1= 12090 N/ mm2 (calculated from

G =E/2(1+ υ)).

Lower Components (Longitudinal and transverse steel Shear Connectors (stud shear connectors)Depth of steel beam h2 = 152.2 mm Length of shear connector = 38 mmFlange width of steel beam bs = 152.2 mm Diameter of shear connector = 12 mmThickness of flange of steel beam tf = 6.6 mm According to (OHBD) code Thickness of web of steel beam tw = 5.84 mm Number of connectors per row n =2Cross sectional area of steel beam A2 = 2858 mm2 Spacing P=180 mm.Moment of inertia of steel beam I2 =12112334.49 Strength of shear connector = 57000 N.

Modulus of elasticity of steel beam E2 = 200000 MPa Poisson’s ratio of steel beam υ2 = 0.3

Shear modulus of elasticity of steel beam G2= 76923 N/ mm2 (calculated from G =E/2(1+υ)).

Connector stiffness may be conservatively estimated as the secant stiffness at the

shear connector design strength with an equivalent slip of 0.8 mm [11], hence k =

57000/ 0.8 = 71250 N/ mm.

Evaluating the elastic rigidities for each grillage member as given in section(3)

1- For longitudinal members:

a-edge beams: (EIp= 0.6 EIf = 2.9×1012 N.mm2) , (GJ= 2.6 ×1011 N.mm2).

b-interior beams: (EIp= 0.5 EIf = 3.4×1012 N.mm2) , (GJ= 3.0×1011 N.mm2).

2- For transverse members in this model it is assumed that the flexural rigidity is the

average value between fully and zero interaction as follows, taking the effective of

the concrete slab in the longitudinal direction equal 0.5b as shown in Fig.(11)[4]:

EIp = 0.5(EIf + EIo)

But if there are shear connectors between the concrete slab and the transverse steel

beam, the value of the flexural rigidity must be estimated by the same method

represented in section 3.5.1, thus:

a-for edge beams: (EIp= 0.5( EIf + EIo)= 4 ×1012 N.mm2) ,

(GJ= 3.6 ×1011 N.mm2).

b-for interior beams: (EIp= 0.5( EIf + EIo)= 4 ×1012 N.mm2) ,

(GJ= 3.5 ×1011 N.mm2).

12

The shearing rigidity is constant for all grid members and it can be calculated as

shown in section 3.5.3, thus

GAv = 101.67 N (for transformed shear area) ,or: GAv = 68.37 N (for steel

shear area)

Two different loading conditions are considered. Point load of 89 kN is applied, the

position of this load is given in the following

1-A center load applied over the bridge (point no. 13, Fig.(7)).This is the first

loading condition.

2-An eccentric load applied over the edge of the bridge (point no. 3, Fig.(7)).

This is the second loading condition.

In Fig.(8), the vertical deflections at the mid- span cross- section (section A-A) are

plotted for the first loading condition. The corresponding values of the deflections for

the second loading condition are plotted in Fig. (9).Tab. (2) shows the comparisons of

the maximum deflections in the composite structure as calculated by the suggested

method for the two loading conditions. In the grillage analysis the maximum

deflections in both cases of loading are calculated for:

Case (I): without transverse shear effect. , Case (II): with transformed shear

area. , Case (III): with steel shear area only.

Tab. (2):Comparisons of maximum deflections (composite bridge model) (percentage differences

with respect to experimental results)

Method of analysis

1st loading 2nd loading

Max.

Deflection

(mm)

Percentage

Difference

(%)

Max.

Deflection

(mm)

Percentage

Difference

(%)

Grillage

analogy

Case (I) 3.30 +17.90 7.86 +10.0Case (II) 3.57 +27.50 8.27 +15.9Case (III) 3.69 +31.80 8.47 +18.8

Orthotropic plate method [20] 3.30 +17.90 7.50 +5.2Experimental result [4] 2.80 - 7.13 -

From the above comparison, it is clear that when the effect of transverse shear area

(Av) is ignored the deflections obtained by the grillage analogy are rather in acceptable

agreement with the experimental and finite element results (applied to the equivalent

13

orthotropic plate). Also this effect is shown in Figures (8) and (9), and it is well

known that an eccentric load on a bridge gives rise to twisting moments that are much

greater in magnitude than those caused by the same load applied at the center. Thus,

the concrete deck slab, with its significant torsional resistance, is able to distribute

transversely the eccentric load quite effectively in composite bridges. Comparisons

between the results are also given in Tabs. (3) and (4).

Comparisons between the variations of center deflection with an applied central load

shown in Fig. (10).

Tab. (3):Vertical deflections (in mm) at mid- span of bridge model under 1st.

loading condition (percentage differences with respect to experimental results)

Node

no.Exper. Ortho.

Perce.

Diff.

(%)

Grill.

case I

Perce.

diff.

(%)

Grill.

case II

Perce.

diff.

(%)

Grill.

case III

Perce.

diff.

(%)

23 2.54 1.91 -24.8 2.3 -9.5 2.38 -6.3 2.41 -5.1018 2.67 2.29 -14.2 2.9 +8.6 3.06 +14.6 3.12 +16.813 2.8 3.30 +17.9 3.3 +17.9 3.57 +27.5 3.69 +31.88 2.67 2.29 -14.2 2.9 +8.6 3.06 +14.6 3.12 +16.83 2.54 1.91 -24.8 2.3 -9.5 2.38 -6.3 2.41 -5.10

Tab. (4):Vertical deflections (in mm) at mid- span of bridge model under 2nd. loading

condition (percentage differences with respect to experimental results)

Node

no.

Exper. Ortho.

Perce.

Diff.

(%)

Grill.

case I

Perce.

diff.

(%)

Grill.

case II

Perce.

diff.

(%)

Grill.

case III

Perce.

diff.

(%)

23 -1.70 -1.50 +11.8 -1.28 24.7 -1.32 22.4 -1.34 21.218 -0.30 -0.29 -4.0 0.36 20.0 0.37 23.0 0.38 26.713 2.16 2.30 +6.5 2.30 6.5 2.38 10.2 2.41 11.68 4.33 4.69 +8.3 4.80 10.8 4.98 15.0 5.07 17.03 7.13 7.50 +5.2 7.86 10.0 8.27 15.9 8.47 18.8

5. Effect of Degree of Interaction

The degree of interaction between the concrete slab and the steel beams may be

increased by increasing the number of shear connectors or by increasing the connector

stiffness. This increase leads to increase in the (EIp / EIf) ratio. Thus, in this section

14

various values of this ratio are assumed to study its effect on the same bridge model,

without including the transverse shear effect.

In Figures (11) and (12), the vertical deflections at the mid- span cross- section are

plotted for the first and second loading conditions respectively. It is clear that the

values of the vertical deflection decreased when the degree of interaction increased.

This increase is obtained for longitudinal beams. From this result, it is found that the

composite structure resistance is more efficient for applied load when the degree of

interaction is increased. Also a comparison between the results is shown in Tabs. (5)

and (6).

Tab. (5): Influence of degree of interaction on vertical deflections (in mm)

for 1st. loading condition

Node no. EI= EIo EIp= 0.7 EIf EIp= 0.9 EIf EI= EIf

23 2.73 1.70 1.25 1.0918 3.38 2.26 1.78 1.6113 3.76 2.60 2.09 1.928 3.38 2.26 1.78 1.613 2.73 1.70 1.25 1.09

Tab. (6): Influence of degree of interaction on vertical deflections (in mm) for 2nd.

loading condition

Node no. EI= EIo EIp= 0.7 EIf EIp= 0.9 EIf EI= EIf

23 -1.45 -1.33 -1.09 -0.9918 0.49 0.04 -0.06 -0.08313 2.73 1.704 1.25 1.098 5.57 3.95 3.103 2.7993 8.99 6.80 5.55 5.09

15

Fig (8): Vertical deflections at mid-span section of bridge deck model under 1st.loading

condition

16

A

Fig. (7): Details of composite bridge model.(a) Plan view, (b) Section (A-A), (c) Section (B-B)

1. 2. 3. 4. 5.

10.9.8.7.6.

15.14.13.12.11.

20.19.18.17.16.

25.24.2322. .21.

x

y(a)

3050 mm

2290

m

m

A

BB

(b)

152.2 mm

(c)

.5b=267.2250.5b=267.2250

48 mm

48

152.2

254.17b=

534.45

Node number

Distance from left end (mm)

Deflection

(mm)

23Experimental [ 4]

Orthotropic plate [ 4]

Grillage case IGrillage case IIGrillage case III

1.5

2

2.5

3

3.5

4

4.5

381318

76.1 610.55 1145 1679.45 2213.9

Node number

Distance from left end (mm)

Deflection (mm)

Experimental [4]

Orthotropic plate[4]

Grillage case IGrillage case IIGrillage case III

10

23 18 13 8 3

76.1 610.55 1145 1679.45 2213.9

-4

-2

0

2

4

6

8

Fig. (9): Vertical deflections at mid-span section of bridge deck model under 2nd.loading

condition

Fig. (10): Load-deflection curve at center of Kennedy’s bridge deck model

Fig (11) Influence of degree of interaction on vertical deflections for 1st.loading condition

17

0102030405060708090

100

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9Deflection at center ( mm)

Load at center ( kN)

Node number

012345678

3813

18

23

Distance from left end (mm)

Deflection

(mm) EIp=0.9 EIf

EI=EIo

EIp=0.7 EIf

EI=EIf

Grill. case IExperimental [4]

76.1 610.55 1145 1679.45 2213.9

EI=EIoEIp=0.7EIfEIp=0.9EIfEI=EIfGrill. case IExperimental [4]

Node number

-4

-2

0

2

4

6

810

Distance from left end (mm)

Deflection (mm)

76.1 610.55 1145 1679.45 2213.9

23 18 13 8 3

0102030405060708090

100

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9Deflection at center ( mm)

Load at center ( kN)

Experimental

Grill. case IGrill. case IIGrill. case III

Fig (12) Influence of degree of interaction on vertical deflections for 2nd.loading condition

6. Conclusions

The main concluding remarks that have been achieved in this study may be

summarized as follow

1. Design charts are constructed for estimating the percentage decrease in flexural

rigidity of each composite member with partial shear connection. The charts are in

terms of the parameter k2, and were given for various values of the factors C.

,

2. The loss of interaction between the concrete slab and the steel beams leads to

considerable increase in deflection (as the sum of flexural rigidities of the two

separate components is considerably smaller than the value for the connected

components). Almost fully interacting components give stiffer structure.

3. To calculate the flexural rigidity of the equivalent grillage members the case of

uniformly distributed load can be used in place of any loading case because the

difference between the results from different load patterns is negligible (less than

1.3%).

4. In representing a composite structure by grillage members, the effective width of

each member should be used to calculate the flexural rigidity of that member. Also

Poisson’s ratio effect is to be included in the calculation of the flexural rigidities of

the grillage members.

5. Effect of transverse shearing forces on deflection is found to be small and thus it

can be neglected (percentage differences is less than 11.8 %).

References:1. Heins,C.P. and Fan,H.M., ”Effective Composite Beam Width at Ultimate

Load”, Journal of the Structural Division, Proc. of the ASCE, Vol.102, ST11, pp. 2163-2179, Nov.1976.

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2. Newmark,N.M.,Siess,C.P. and Viest,I.M., “Tests and Analysis of Composite Beams with incomplete interaction”, Proc. Soc. Experimental Stress Analysis, Vol.9, No.1, pp. 75-92 , 1951.

3. Johnson,R.P., “Composite Structures of Steel and Concrete: Vol.1”, Crosby Lockwood Staples, London , 210pp. , 1975.

4. Kennedy,J.B.,Grace,N.F. and Soliman,M., “Welded- versus Bolted-Steel I-Diaphrams in Composite Bridges”, Journal of the Structural Division, Proc. of the ASCE, Vol.115, ST2, pp. 417, Feb.1989.

5. Jasim,N.A., “The Effect of Partial Interaction on Behaviour of Composite Beams “, Thesis presented for the degree of Ph.D.,Department of Civil Engineering, College of Engineering, University of Basrah, Iraq, 188pp.,Oct.1994.

6. Hendry,A.W. and Jeager,L.G., “The Analysis of Grid Framework and Related Structures”, Chatto and Windus , London , 1958.

7. Gere,J.M.and Weaver,W.,”Analysis of Framed Structures”,Van Nostrand Co., New York,1958.

8. Hassan,F.M. and Kadhum,D.A.R., “Behaviour and Analysis of Composite Sections under Pure Torsion”, Engineering and Technology, Vol.7, No.1, pp. 67-97,1989.

9. Timoshenko, S., “Strength of Materials :Part II”, Van Nostrand Co., New York, 1958.

10. Frodin,J.G., Taylor, R. and Stark, J.W.,”A Comparison of Deflection in Composite Beams Having Full and Partial Shear Connection”, Proc.of Inst.of Civil Engineers, Part 2,Vol.41,pp. 307-322,June1978.

11. Wang,Y.C., “Deflection of Steel-Concrete Composite Beams with Partial Shear Interaction”, Journal of Structural Engineer,Vol.124,No.10,pp. 1159-1165,Oct.1998.

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للسقوف المشبك بطريقة الخرسانية التحليلالتداخل الحديدية والعتبات أسلوب باستخدام المركبة

الجزئي

الخلاصة من المتكونة المركبة المنشآت بسلوك تهتم الحالية الدراسة

باتجاهين حديدية عتبات إلى مربوط كونكريتي سقف الخطي الفعل االعتبار بنظر آخذة قص روابط بواسطة تم الدراسة هذه الجزئي. في التداخل في القص لروابط المشبكات طريقة باستخدام المركبة للمنشآت تحليل تقديم

من الناتجة واإلجهادات الهطول لدراسة مبسطة كطريقة القيم لحساب طريقة المسلطة. واقترحت األحمال

األعضاء من عضو لكل اللي وصالدة االنحناء لصالدة المطلوبةQ ضمن تصميمية جداول اشتقت ذلك على المشبك. وبناءا

التداخل بسبب االنحناء صالدة في النقصان مقدار لحساب العرض اعتبار من البد إنه الحالي البحث في الجزئي. ووجد

عضو لكل االنحناء صالدة اشتقاق في وإدخاله المؤثر الهطول قيم على القص قوى تأثير إن كذلك إنشائي. ووجد

Q قليل إهماله. % ويمكن11.8 يتجاوز وال جدا

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