CHAPTER - 6 - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/10105/13/13...Experimental...

69

CHAPTER - 6

ANALYSIS OF ULTIMATE STRENGTH OF CSCC BEAMS

UNDER COMBINED LOADING

6.0 GENERAL

Experimental investigation of 32 Confined Steel Concrete Composite beams (CSCC beams)

under pure bending, under pure torque, 30% of ultimate torque and bending till failure and 60%

of ultimate torque and bending till failure were reported in Chapter 5. Based on the behaviour of

these beams under increasing loads up to failure including torsion and flexure, modes of failure,

load deflection curves and torque twist curves, the fundamental concept and assumption and three

failure surfaces on which the analysis is based are presented in this chapter.For each failure

surface an equation for the ultimate strength of confined steel concrete composite beam is

derived. In addition the method used to solve these equations is also presented in this Chapter.

6.1 MATERIAL PROPERTIES

CSCC beams consist of various construction materials each with its unique strength properties.

The conventional behaviour of concrete, cold rolled sheet, shear connectors and reinforcing steel

are discussed here in order to understand their performance under composite action.

6.1.1 Conventional Behaviour of Concrete

Since concrete has poor tensional limitation its contribution in the tension zone is normally

neglected. In compression, the uniaxial behaviour of concrete found experimentally both for

confined and unconfined concrete cubes showed that the strength has increased to nearly 1.5

times for the confined concrete.

6.1.2 Conventional Behaviour of Cold Formed Sheet

The primary purpose of the sheet element in composite beam is to carry the tensile stresses. Cold

formed sheet is elastic for a certain region and then its stress strain relationship becomes nonlinear

as the onset of the plasticity is developed.

70

6.1.3 Conventional Behaviour of Shear Connectors

The strength of shear connectors depends on the ability of the connections to redistribute the loads

among themselves load slip characteristics of the connectors. Mechanical connectors can be

described as two different types namely ductile and brittle. A ductile connector is one which can

maintain their peak load carrying capacity over large displacements. It has large plastic plateau

whereas the mechanical connector can be described as brittle when their ability to resist the load

diminishes rapidly after their peak carrying capacity has achieved.

6.1.4 Conventional Behaviour of Reinforcing Steel

Steel bars were used in the reinforced concrete beam. For the prediction of Young’s modulus and

yield stress bilinear stress strain variations is used. In the present analytical study the steel

reinforcement is assumed to be elastic and perfectly plastic and the same stress-strain variation is

assumed throughout the investigation.

6.2 BASIS FOR ANALYSIS

The behaviour and ultimate strength of reinforced concrete members subjected to various

combination of transverse shear force axial force and bending moment have been investigated

rather extensively. However , the behaviour and ultimate strength of reinforced concrete members

subjected to a torsional moment combined with transverse shear, axial load or bending moment

have received less attention the reason for this are ;

(a) Torsion is usually a secondary effect in concrete structures

(b) It is usually possible to arrange the members in a structure so that they are subjected to

only a very small twisting moments.

(c) Torsion requires special testing equipments which is not readily available

Torsion is seldom taken into consideration in the design of concrete structures. Current practices

assume that if the members are arranged so that they are subjected to only very small twisting

moments the factor of safety will adequately provide for the effects of torsion. However, since the

knowledge of the behaviour of reinforced concrete has increased considerably in recent years and

since design procedure are continually being refined it is essential that the effect of torsion be

understood so that provision can be made for the design.

71

6.2.1 Crack Pattern under Pure Torsion 16

The cracks generated due to pure torsion follow the principal stress trajectories. The first cracks are

observed at the middle of the longer side. Next, cracks are observed at the middle of the shorter

side. After the cracks connect, they circulate along the periphery of the beam.

In structures, a beam is not subjected to pure torsion. Along with torsion it is also subjected to

flexure and shear. Hence, the stress condition and the crack pattern are more complicated than

shown before.

Fig. 6.1.Formation of Cracks in a Beam Subjected to Pure Torsion

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6.2.2 Components of Resistance for Pure Torsion

After cracking, the concrete forms struts carrying compression. The reinforcing bars act as ties

carrying tension. This forms a space truss. Since the shear stress is larger near the sides, the

compression in concrete is predominant in the peripheral zone. This is called the thin-walled tube

behaviour. The thickness of the wall is the shear flow zone where the shear flow is assumed to be

constant. The portion of concrete inside the shear flow zone can be neglected in calculating the

capacity.

The components in vertical and horizontal sections of a beam are shown below:

Fig. 6.2 Internal Force in Beam

The components can be denoted as below.

Tc = torsion resisted by concrete

Ts = torsion resisted by the longitudinal and transverse reinforcing bars.

The magnitude and the relative value of each component change with increasing torque.

6.2.3. Modes of Failure

For a homogenous beam made of brittle material, subjected to pure torsion, the observed plane of

failure is not perpendicular to the beam axis, but inclined at an angle. This can be explained by

theory of elasticity. A simple example is illustrated by applying torque to a piece of chalk.

73

Fig.6.3 Failure of a Piece of Chalk under Torque

For a beam of rectangular section, the plane of failure is further influenced by warping. Torsional

warping is defined as the differential axial displacement of the points in a section perpendicular to

the axis, due to torque.

For a reinforced concrete beam, the length increases after cracking and after yielding of the bars.

For a beam subjected to flexure and torsion simultaneously, the modes of failure are explained by

the Skew Bending Theory. The observed plane of failure is not perpendicular to the beam axis, but

inclined at an angle. The curved plane of failure is idealized as a planar surface inclined to the axis

of the beam.

The skew bending theory explains that the flexural moment (Mu) and torsional moment (Tu)

combine to generate a resultant moment inclined to the axis of the beam. This moment causes

compression and tension in a planar surface inclined to the axis of the beam.

The following figure shows the resultant moment due to flexural moment and torsion in a beam.

Fig. 6.4 Beam Subjected to Flexural Moment and Torsion

The modes of failure are explained based on the relative magnitudes of the flexural moment (M )

and torsional moment (T) at ultimate. Three discrete modes of failure are defined from a range of

74

failure. The idealized pattern of failure with the plane of failure and the resultant compression (C)

and tension (T) are shown for each mode

Modified bending failure (Mode 1): This occurs when the effect of M is larger than that of T

Fig. 6.5 Idealized Pattern for Mode 1 Failure

Lateral bending failure (Mode 2): This is observed in beams with thin webs when the effect of M

and T are comparable.

Fig. 6.6 Idealized Pattern for Mode 2 Failure

3) Negative bending failure (Mode 3): When the effect of T is large and the top steel is less, this

mode of failure occurs.

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Fig.6.7 Idealized Pattern for Mode 3 Failure

6.3. COMBINED BENDING AND TORSION

A large number of investigations were carried out on the behaviour of R.C. beams in pure

bending, pure torsion and combined bending & torsion under ultimate load conditions. Here,

bending, torsion and shear are inseparable effects in R.C. beams. Probably there is no structure

subjected only to pure torsion or to pure bending. When members subjected to combined bending

and torsion the strength mostly depends upon the interaction between bending and torsion

although other factors such as quality of the material, quantity of reinforcement etc, may also

affect the behaviour to some extent.

Failure under combined bending and torsion is sudden not giving any warning before failure

without the margin of safety. Earlier researches have suggested that bending moment exerted a

favourable influence on the torsional strength of beam. In modern structural configurations it is

necessary to study the combined bending and torsional behaviour and is an important practical

problem.

6.4. COMBINED BENDING AND TORSION FOR COMPOSITE BEAM

It is important that any method of analysis used to predict the ultimate strength of beam subjected

to combined loading be valid only at the limiting cases of pure bending and pure torsion therefore

a smooth transition from pure bending to pure torsion is obtained by a ratio of twisting moment

to bending moment for composite beams. As per earlier research and results16

, failure on a warped

plane near the top face of the beam is considered as Mode 1.the failure on warped plane with a

hinge located near one of the sides is denoted as Mode 2.Failure on warped plane with a hinge at

bottom as Mode-3.

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In general, the modes of failure depend on the deformation and the nature of failure surfaces

which in turn depends on the crack pattern. When a beam specimen is subjected to torsion

according to skew bending theory, it exhibits three modes of failure. Since bending and torsion

are inseparable, a trial is made in this chapter to account the effect of bending in addition to

torsion and hence the beam is analysed as per the three modes of failure.

The results presented in Chapter-V indicate that the specimen exhibited two different stages of

behaviour during the test. Before the concrete cracked the relationship between the torque and

twist was nearly linear. After cracking the slope of torque twist relationship became curved and

after cracking occurred the specimen appeared to assume a new configuration of equilibrium

which was depended mainly on the value of ‘ф‘which is the ratio of ultimate twisting moment

and ultimate bending moment, the reinforcement of the beam and the composite action between

the concrete and sheet and concrete and connector.

When failure occurred cracks on three faces of beam usually widened to define the failure plane.

The crack on two of these faces decreased in width as they approached the fourth face. The

rotation of beam at failure occurred about an axis located near the face consequently the failure

plane appeared to consist of tension zone on one side of neutral axis and the compression zone on

the other side, the neutral axis being the hinge or axis of rotation

A”Z” shaped crack often developed on the face of the beam adjacent to the hinge, with the central

portion of the “Z” being parallel to the longitudinal axis of the beam. However, this is not

sufficient justification to assume that the hinge was parallel to the axis of the beam. Furthermore,

rotation about a longitudinal hinge could not explain the vertical deflection observed at

failure.The neutral axis of a beam subjected to pure bending is perpendicular to the axis of the

beam. Adoption of an inclined hinge that is parallel to one of the faces of the beam but which is

inclined to the longitudinal axis of the beam was already suggested in the earlier literature. A

smooth transition from pure bending to pure torsion is obtained when this type of hinge is used.

The angle of the hinge with the longitudinal axis is a function of the ratio of twisting moment to

bending moment and the properties of the beam.

77

The assumptions and three failure surfaces on which the analysis is based are presented here.For

each failure surface an equation for the ultimate strength of the CSCC beam is derived in this

Chapter.

6.5 ASSUMPTIONS OF ANALYSIS

The method of analysis is based on the following assumptions:

1. Failure occurs on a warped plane. The boundaries of warped plane are defined on three

sides of the beam by spiral crack and on the fourth side by a rectangular compression zone

which joins the ends of the spiral crack. The crack defining the failure plane occurs after

the separation of sheet from concrete.

2. The crack defining the failure plane on three sides of beam is composed of three straight

lines spiralling along the beam at a constant angle. The angle between the cracks and the

longitudinal axis of beam is never less than 45°.

3. The concrete outside of the rectangular compression zone is cracked and carries more

tension.

4. The reinforcement near the face of the beam on which the compression zone is located is

neglected.

5. The reinforcement has a well defined yield plateau. All the reinforcement crossing the

failure plane outside the compression zone yields at failure.

6. The cross sectional area of sheet and shear connectors intersected by the failure plane is

constant per unit length of the beam.

7. No local loads are present within the length of failure plane.

8. At failure the characteristic of concrete in the compression zone are known and the

concrete reaches its strength in flexural compression.

9. The bond stress developed for the half the length of the beam.

The assumption 1 and 2 defines the failure plane. The three lines composing the spiral

crack are assumed to be straight for simplicity.

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The assumption 3, 4 are introduced to simplify the analysis. The assumption 8 is true only for

pure bending that is ф = 0. As torsion is added and value of ф in increased, the compression zone

is subjected to stress conditions which are complex and further complicated by inclined cracks

crossing the compression zone.

6.6 MODE-1 FAILURE

6.6.1 Equation for Ultimate Strength of CSCC Beam under Combined Loading

The failure surface of this mode of failure is illustrated in Fig.6.8. Three sides of the failure

surface are defined by spiral crack which is located on the bottom and the two vertical faces of the

beam. The ends of this spiral cracks are joined by a compression zone located at the top face of

the beam.

Fig. 6.8 Mode-1 Failure

The notations used in the equation are:

b = beam width

h = beam depth

L1 = length of the compression zone

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C1 = length of warped failure plane projected on the longitudinal axis of the beam

Cm1 = the maximum value that C1 can have

Ast1 = area of the longitudinal reinforcement at the bottom face

fst1 = actual stress in longitudinal reinforcement

ts = thickness of sheet

fs = actual stress in cold formed sheet

fBCS = actual bond stress between concrete and sheet

fBCC = bond stress between concrete and connector

Astud = area of connector = п ds (l1 +l2)

ds = diameter of stud

l1, l2 = lengths along vertical and horizontal direction of stud

x1 = distance from top face of the beam to the neutral axis

fc = compressive strength of concrete cubes of standard size.

K01 = the ratio between the length of the failure crack on the bottom face when

projected on the longitudinal axis of the beam and the total length of the failure

plane measured parallel to the longitudinal axis.

Mt the = ultimate theoretical twisting moment of the beam according to Mode-1 failure.

Mt exp = experimental value of ultimate twisting moment of beam.

Mb1the = ultimate theoretical bending moment of the beam according to Mode 1

Mb exp = experimental value of ultimate bending moment of the beam

k1 = a coefficient used to determine the depth of equivalent compression zone

A rectangular stress block in the concrete compression zone is used. A concrete stress intensity of

0.85 fc' is assumed to be uniformly distributed over an equivalent compression zone. This zone is

bounded by the edges of the cross section and a straight line parallel to the neutral axis at a

distance equal to k1x1 from the top face of the beam. The value of k1 is assumed as 0.85.

Equation for Mt1 and Mb1

External moments are due to

1. Bending

2. Twisting

Internal moments are due to

80

1. Component of tensile force in the bottom sheet

2. Component of tensile force along longitudinal reinforcement

3. Component of tensile force in the side sheet

4. Component of bond force at bottom between

a). The concrete and sheet

b) The concrete and connector

5. Component of bond force at sides between

a) The concrete and sheet


From the equilibrium of internal and the external moments about an axis parallel to the neutral

axis and located at mid-depth of the equivalent compression zone, the following equation is

obtained

)1.6(11111

2

111

2

111111

1mCqnrClpCzfAM

b

CM ytstbt

cdhh01

1

012 C

b

hb

bK

1z )2

( 1101

xkh

11

1

yst

ss

fA

ftbp

11

1

yst

BCCrstudstud

fA

fnnAq

11

12 ysy

BCS

fA

fbLr

2

11

2

01

11

xkh

b

Knl

b

xkh

m 2

11

1

81

6.6.2. Equation for Neutral Axis(x1)

Forces considered to derive x1 acting perpendicular to the plane which contains neutral axis and is

perpendicular to the top face of the beam are;

1) Component of tensile force in longitudinal reinforcement

2) Component of tensile force in bottom sheet

3) Component of bond force at bottom between



4) Force in concrete compression zone

From the equilibrium of forces acting perpendicular to the plane which contains the neutral axis

and is perpendicular to the top face of the beam, the following equation is obtained.

)2.6(85.0 2

11

1111

2

10111

1Lkfc

Cqrpb

CKbfA

x

yst

6.6.3 Equation for C1

To determine the value of C1 corresponding to the minimum value of Mt1, equation (6-1) is

differentiated with respect to C1 and equated to zero and solved for C1.

Since negative values of C1 have no physical significance the minimum value of C1 is zero. In

accordance with the second assumption of section 6-5 and neglecting the depth of compression

zone, the maximum value of C1 is:

bhCm 21

)3.6(

To solve these equations, the necessary properties of the section must be computed. The equation

may be then be solved by an iterative procedure. The initial value of x1 is assumed to be zero and

the initial values of z1 and y1 are computed. The value of C1 can be computed by the equation

(6.3). This value of C1 can be used in the equation (6-2) and to calculate a value for x1. .The

calculated value of x1 is then used to compute new values of z1 and y1 and the procedure is

repeated. With each iteration the difference between the initial and the computed value of x1

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decreases. The process is continued until the desired accuracy is obtained. The ultimate twisting

moment is then calculated using equation (6.1).

6.7. MODE-2 FAILURE

6.7.1 Equation for Ultimate Strength of CSCC beam under Combined Loading

The failure surface for this mode of failure is illustrated in Fig. 6.9. Three sides of failure surface

are defined by a spiral crack which is located on one of the vertical faces of beam and two of the

horizontal faces. The ends of spiral crack are joined by a compression zone located on one of the

vertical sides of the beam. The notation used is the same as per Mode-1 with the following

exception.

Fig. 6.9 Mode-2 Failure


b = beam width

h = beam depth

83




Ast = area of the longitudinal reinforcement at the bottom face

fst = actual stress in longitudinal reinforcement








x2 = distance from vertical face of the beam to the neutral axis


K02 = the ratio between the length of the failure crack on the vertical face when



Mt the = ultimate theoretical twisting moment of the beam according to Mode-2 failure.


Mb2the = ultimate theoretical bending moment of the beam according to Mode 2


k1 = a coefficient used to determine the depth of equivalent compression zone


0.85 fc' is assumed to be uniformly distributed over an equivalent compression zone. This zone is


distance equal to k1x1 from the top face of the beam. The value of k1 is assumed as 0.85.

The stress block in the compression zone is considered to be the same as for Mode-1. Here the

component of bending moment is zero. But moment due to transverse shear is considered.

84


External moments are due to;

1) Twisting

2) Moment due to transverse shear

Internal moments are due to;

1 Component of tensile force in the bottom sheet

2. Component of tensile force along longitudinal reinforcement

3 Component of tensile force in the side sheet

4. Component of bond force at bottom between



5. Component of bond force at sides between





obtained;

)4.6(222

2

222

2

22222222

2 CmqCnrClpzfAMh

CM ytStbt

2z )2

( 2102

xkb

cdbb02

22

2

yst

ss

fA

ftbp

22

2

yst

BCCrstudstud

fA

fnnAq

22

22 ysy

BCS

fA

fbLr

85

2

2202

2

xkb

bh

Kl

h

xkb

m 221

2

2

2102

2

xkb

bh

Kn

2

022 C

h

hb

hK

6.7.2. Equation for Neutral Axis (x2)


perpendicular to the vertical face of the beam are;

1) Component of tensile force in longitudinal reinforcement

2) Component of tensile force in bottom sheet





From the equilibrium of forces acting perpendicular to the plane which contains the neutral axis

and is perpendicular to the top face of the beam, the following equation is obtained

)5.6(85.0 2

22

2222

2

2

0222

2Lkfc

Cqrpb

CKhfA

x

ytst

86

6.7.3. Equation for C2

To determine the value of C2 corresponding to the minimum value of Mt2, equation (6.5) is

differentiated with respect to C2. Since negative values of C2 have no physical significance the

minimum value of C2 is zero. In accordance with the second assumption of section (6.5) and

neglecting the depth of compression zone, the maximum value of C2 is:

)6.6(22 hbC

6.8. MODE-3 FAILURE

6.8.1 Equation for Ultimate Strength of CSCC Beam under Combined Loading

The failure surface of this mode of failure is illustrated in Fig.6.10. Three sides of the failure

surface are defined by spiral crack which is located on the top and the two vertical faces of the

beam. The ends of this spiral cracks are joined by a compression zone located at the bottom face

of the beam.

The notation used is the same as for mode-1with following exception:

Fig. 6.10 Mode - 3 Failure

87


b = beam width

h = beam depth




Ast = area of the longitudinal reinforcement at the bottom face

fst = actual stress in longitudinal reinforcement








x3 = distance from bottom face of the beam to the neutral axis


K03 = the ratio between the length of the failure crack on the top face when



Mt the = ultimate twisting moment of the beam according to Mode-3 failure.


Mb3 the = ultimate bending moment of the beam according to Mode- 3 failure



0.85 fc ‘is assumed to be uniformly distributed over an equivalent compression zone. This zone is


distance equal to k1x1 from the top face of the beam. The value of k1 is 0.85.

88


External moments are due to;

1) Bending

2) Twisting

Internal moments are due to

1) Component of tensile force in the bottom sheet is zero.

2) Component of tensile force along longitudinal reinforcement is zero.

3) Component of tensile force in the side sheet is considered



b) The concrete and connector are zero so not considered

5) Component of bond force at sides between

a. The concrete and sheet

b. The concrete and connector is considered



obtained.

)7.6(33333

2

333

2

33333

3 CmqnrClpCfAMb

CM ytstbt

cdhh03

3

032 C

b

hb

bK

3z )2

( 31

03

xkh

ylsyl

ss

fA

ftbp3

ylsl

BCCrstudstud

fA

fnnAq r

3

ylsyl

BCS

fA

fbLr

23

89

3

2

03

34

1n

b

Kl

4

1 03

3

Km

6.8.2. Equation for Neutral axis (x3)


perpendicular to the vertical face of the beam are;

2) Component of tensile force in side sheet

3) Component of bond force at sides between




From The Equilibrium of forces acting perpendicular to the plane which contains the neutral axis

and is perpendicular to the top face of the beam, the following equation is obtained

)8.6(85.0

1

2

33

3333

2

3

0333

3Lkfc

Cqrpb

CKfA

x

ytst

6.8.3. Equation for C3

To determine the value of C3 corresponding to the minimum value of Mt3, equation (5-8) is

differentiated with respect to C3.Since negative values of C3 have no physical significance the

minimum value of C3 is zero. In accordance with the second assumption of section (6-5) and

neglecting the depth of compression zone, the maximum value of C3 is:

hbCm 23

)9.6(

6.9. Theoretical Ultimate Strength of CSCC Beams under Combined Loading

The values of x1, x2, x3 were calculated for the three modes of failure using the equations (6.2),

(6.5) and (6.8).The three theoretical ultimate strengths were obtained for each beams using the

equations (6.1), (6.4) and (6.7).The theoretical ultimate strength is equal to the smallest of the

three values. The predicted mode of failure is designated as the mode that yields the smallest

value of the theoretical ultimate strength.

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