Mechanics of Composite Beams

22
Mechanics of Composite Beams

Mehdi Hajianmaleki and Mohammad S. Qatu
Mississippi State University USA 1. Introduction
A structural element having one dimension many times greater than its other dimensions can be a rod, a bar, a column, or a beam. The definition actually depends on the loading conditions. A beam is a member mainly subjected to bending. The terms rod (or bar) and column are for those members that are mainly subjected to axial tension and compression, respectively. Beams are one of the fundamental structural or machine components. Composite beams are lightweight structures that can be found in many diverse applications including aerospace, submarine, medical equipment, automotive and construction industries. Buildings, steel framed structures and bridges are examples of beam applications in civil engineering. In these applications, beams exist as structural elements or components supporting the whole structure. In addition, the whole structure can be modeled at a preliminary level as a beam. For example, a high rise building can be modeled as a cantilever beam, or a bridge modeled as a simply supported beam. In mechanical engineering, rotating shafts carrying pulleys and gears are examples of beams. In addition, frames in machines (e.g. a truck) are beams. Robotic arms in manufacturing are modeled as beams as well. In aerospace engineering, beams (curved and straight) are found in many areas of the plane or space vehicle. In addition, the whole wing of a plane is often modeled as a beam for some preliminary analysis. Innumerable other examples in these and other industries of beams exist. This chapter is concerned with the development of the fundamental equations for the mechanics of laminated composite beams. Two classes of theories are developed for laminated beams. In the first class of theories, effects of shear deformation and rotary inertia are neglected. This class of theories will be referred to as thin beam theories or classical beam theories (CBT). This is typically accurate for thin beams and is less accurate for thicker beams. In the second class of theories, shear deformation and rotary inertia effects are considered. This class of theories will be referred to as thick beam theory or shear deformation beam theory (SDBT). This chapter can be mainly divided into two sections. First, static analysis where deflection and stress analysis for composite beams are performed and second dynamic analysis where natural frequencies of them are assessed. In many applications deflection of the beam plays a key role in the structure. For example, if an aircraft wig tip deflection becomes high, in addition to potential structural failure, it may deteriorate the wing aerodynamic performance. In this and other applications, beams can be subjected to dynamic loads. Imbalance in driveline shafts, combustion in crank shaft applications, wind on a bridge or a
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Advances in Composite Materials - Analysis of Natural and Man-Made Materials
structure, earthquake loading on a bridge or a structure, impact load when a vehicle goes over a pump are all examples of possible dynamic loadings that beam structures can be exposed to. All of these loads and others can excite the vibration of the beam structure. This can cause durability concerns or discomfort because of the resulting noise and vibration.
2. Stiffness of beams
Figure 1 shows a free body diagram of a differential beam element. Beams are considered as one dimensional (1D) load carriers and the main parameter for analysis of load carrier structures is stiffness.
Fig. 1. Free body diagram of a differential beam element In general for composite laminates, stiffness matrix composed of ABD parameters is used to relate the stress resultants to strains. Nx A N 11 y A12 N A xy = 16 M x B11 M y B12 B Mxy 16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26
B16 0 x 0y B26 B66 xy D16 x D26 y D66 2 xy
(1)
where regular ABD stiffness parameters for beams are defined as (Qatu, 2004).
Aij = bQ ij ( hk hk 1 )
N k k =1
Bij = bQij
N k k =1 N
(h
2 k
2 hk 1
(2)
Dij = bQij
k k =1
(h
2
3 k 3 hk 1
(3)
(4)
Note here that the above definitions are different from those used for general laminate analysis in the literature. The beam width is included in the definitions of these terms, while it is customary to leave this term out in general laminate analysis. In 1D analysis of beams, as we will see later, only parameters in x direction are considered and other parameters are
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ignored. So instead of 6X6 stiffness matrix for general laminate analysis we will have a 2X2 matrix for CBT and 3X3 matrix for SDBT. This formulation has the disadvantage of not accounting for any coupling. To overcome this problem, we propose that instead of normal definition of A11, B11, and D11, one can use equivalent stiffness parameters that include couplings. That is why we will deal with stiffness parameters first. 2.1 Equivalent modulus One approach for finding equivalent modulus for the whole laminate was proposed by finding the inverse of the ABD matrix (J matrix) (Kaw, 2005). The laminate modulus of elasticity is then defined as E= b I J 44 J = [ ABD]1 (5)
where J44 is the term in 4th row and 4th column of the inverse of the ABD matrix of the laminate and I is the moment of inertia. If one wants to use this approach for finding parameters A11, B11, and D11 the following formulas derived by authors should be used. A11 = B11 = D11 = b J 11 1 J 14 b J 44 (6)
(7)
(8)
2.2 Equivalent stiffness parameters by Rios and Chan Another approach using compliance matrix can be done by the following formulation (Rios and Chan, 2010).
A11 =
a11
1
2 b11 d11
(9)
B11 =
1 a d b11 11 11 b11 b2 d11 11 a11 1
(10)
D11 =
(11)
where a11, b11, and d11 are relevant compliance matrix terms. Similar to previous section we have a11=J11, b11=J14, d11=J44.
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2.3 Equivalent stiffness parameters by Vinson and Sierakowski Finding equivalent modulus of elasticity of each lamina and using normal definition of ABDs leads to the following formulation (Vinson and Sierakowski, 2002).
4 sin 4 ( k ) 1 cos ( k ) 1 2 = + 12 cos 2 ( k ) sin 2 ( k ) + k E11 E22 Ex G12 E11
k A11 = bEx ( hk hk 1 ) N k =1
(12)
Equivalent A11, B11 and D11 using these formulas would be (13)
k B11 = bEx N k =1 N k D11 = bEx k =1
(h
2 k
2 hk 1
(h
2
3 k 3 hk 1
(14)
(15)
3. Static analysis
In static analysis section we will consider composite beams loaded with classical loading condition and derive differential equations for displacements. Those equations would be solved with classical boundary conditions of both ends simply supported and both ends clamped. We will use the static analyses to find deflection and stress of composite beams under both CBT and SDBT.
3.1 Classical beam theory Applying the traditional assumptions for thin beams (normals to the beam midsurface remain straight and normal, both rotary inertia and shear deformation are neglected), strains and curvature change at the middle surface are: (Qatu, 1993, 2004)
0 =
u0 , 2 w = 2 x x
(16)
where u, w are displacements in x and z directions, respectively. Normal strain at any point would be
= 0 + z
(17)
Force and moment resultants are calculated using

N A11 M = B 11
B11 0 D11
(18)
The equations of motion are
2 M = pz x 2
(19)
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N = px x
where px and pz are external forces per unit length in x and z direction, respectively. The potential strain energy stored in a beam during elastic deformation is
PE = 1 1 l dV = ( N 0 + M ) dx 2 V 2 0
(20)
(21)
writing this expression for every lamina and summing for all laminate we have
PE = 1 l 2 A11 ( 0 ) + 2 B11 0 + D11 2 dx 0 2
(22)
substituting kinematic relations to equation (22) it will become

2 2 2 2w 1 l u0 u0 w PE = A11 2 + 2 B11 + D11 x 2 dx 2 0 x x x
(23)
The work done by external forces on beam would be

W= The kinetic energy for each lamina is KE =
k
1 l ( pxu0 + pz w ) dx 2 0
(24)
where ( ) is the lamina density per unit volume, and t is time. The kinetic energy of the entire beam is KE =
2 2 I 1 l u0 w dx + 2 0 t t
2 2 1 ( k ) l zk u0 w dx + b 0 zk 1 t 2 t
(25)
(26)
where I1 is the average mass density of the beam per unit length. These energy expressions can be used in an energy-based analysis such as finite element or Ritz analyses.
3.1.1 Euler approach Inserting displacement relations in equations of motion will result in (Vinson and Sierakowski, 2002)
A11
B11
3u 4w D11 4 + pz ( x ) = 0 3 x x
2u 3w B11 3 + px ( x ) = 0 2 x x
(27)
(28)
Solving these two equations for u and w will result in the following differential equations.
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2 4 A11D11 B11 B11 px ( x ) w 4 = pz ( x ) A A x x 11 11 2 3 A11D11 B11 u0 D11 px ( x ) 3 = pz ( x ) B B11 x x 11
(29)
(30)
Stress in the axial direction in any lamina can be found by the following equation
x = Q11 ( 0 + z ) = Q11
u0
2w x 2
(31)
Different loading and boundary conditions can be applied to these equations in order to find equations for u and w. These boundary conditions are Simply supported: w = 0, M = 0 dw =0 Clamped: w = 0, dx Free: V = 0, M = 0 where V and M are shear force and bending moment and are linearly dependent on third and second derivative of w respectively. Here, we propose solution for both ends simply supported and both ends clamped with constant loading q0. The reader is urged to apply other boundary conditions and find the equations for deflection. For specific case of simply supported boundary conditions at both ends and assuming u0(0)=0 we have
2 A11D11 B11 q0 l 4 w( x ) = 24 A11 3 x 4 x x 2 + l l l
(32)
3 2 2 A11D11 B11 q0 l 3 x x 4 6 u0 ( x ) = 24 B11 l l
(33)
x =
2 2 A11D11 B11
xz =
q0 l x z 1 z x dz = 2 1 Q11 ( B11 zA11 ) dz 2 l b h x 2 b A11D11 B11 h
q0 l 2
x 2 x Q11 ( B11 zA11 ) l l
(34)
(35)
For clamped boundary conditions at both ends we have

2 A11D11 B11 q0 l 4 w( x ) = 24 A11
3 2 2 A11D11 B11 q0 l 3 x x x 4 6 + 2 u0 ( x ) = 24 B11 l l l
3 2 x 4 x x 2 + l l l
(36)
(37)
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q0 l 2
x =
2 2 A11D11 B11
xz =
q0 l x z 1 z x dz = 1 Q11 ( B11 zA11 ) dz 2 2 l b h x 2 b A11D11 B11 h
x 2 x 1 + Q11 ( B11 zA11 ) l l 6
(38)
(39)
One should note that for simply supported boundary condition the maximum moment and consequently maximum stress occurs at middle of the beam, while for the clamped case maximum stress occurs at two ends.
3.1.2 Matrix approach Inserting the strain and curvature relations in the force and moment resultants equations and using those in the equations of motion, one can express the equations of motion in terms of displacements. Expressing those equations in matrix form we have
L11 L 21 L12 u0 px 0 + = L22 w0 pz 0
(40)
where L11 = A11
2 4 3 , L22 = D11 4 , L12 = L21 = B11 3 . 2 x x x The beam is supposed to have simply supported boundary condition. So we have on x=0, a.
w0 = N x = Mx = 0
(41)
The above equations of motion as well boundary terms are satisfied if one chooses displacements functions as
where m = m / a Fourier series in x
[u, w] = [ Am cos( mx ),C m sin( mx )]

M m=1
(42)
and a is the beam length. The external forces can be expanded in a
[ px , p z ] = pxm sin ( m x ) , pzm cos ( m x )

M m=1
(43)
Substituting these equations in the equations of motion we have the characteristic equation
C 11 C 12 Am pxm C + =0 21 C 22 C m pzm
(44)
4 3 2 D11 , C 21 = C 12 = m B11 . Stress in the axial direction would be A11 , C 22 = m where C 11 = m found using the following procedure.
Am C 11 C 12 pxm C = C m 21 C 22 pzm
1
(45)
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0 A11 = B 11
x = Q11 ( 0 + z )
B11 N D11 M
(46) (47)
3.2 Shear deformation beam theory The inclusion of shear deformation in the analysis of beams was first made in early years of twentieth century (Timoshenko, 1921). A lot of models have been proposed based on this theory since then. In this chapter a first order shear deformation theory (FSDT) approach is presented to account for shear deformation and rotary inertia (Qatu, 1993, 2004).
u = u0 + z , w = w0
(48)
Strains and curvature changes at the middle surface are:
0 =
where 0 is middle surface strain, is the shear strain at the neutral axis and is the rotation of a line element perpendicular to the original direction. Normal strain at any point can be found using equation 17. Force and moment resultants as well as shear forces are calculated using
N A11 M = B11 Q 0 B11 D11 0 0 0 0 A55
u0 w ,= , = + x x x
(49)
(50)
where for A55 we have (Vinson and Sierakowski, 2002).

A55 =
k
5 N 4 bQ ( hk hk 1 ) 3h 2 hk3 hk31 4 k = 1 55
(51)
The equations of motion considering rotary inertia and shear deformation would be
N = px x Q = pz x
(52)
(53)
M Q = 0 x
(54)
The potential strain energy stored in a beam during elastic deformation is

PE =
1 1 l + Q dx dV = N 0 + M 2 V 2 0 x
(55)
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Writing this expression for every lamina and summing for all laminate we have (Vinson and Sierakowski, 2002)
PE =
1 l 2 A11 ( 0 ) + 2 B11 0 + D11 2 + A55 2 dx 2 0
(56)
substituting kinematic relations to equation (56) it will become

PE =
2 1 l u0 u0 A 11 + 2 B11 0 2 x x x
w + D11 + A55 + x x
2
dx
(57)
The work done by external forces on beam is found by equation (24). Finding the kinetic energy for each layer and then summing for all layers yield the kinetic energy of the entire beam.
2 2 l u u0 w KE = I 1 0 + I 1 + 2I2 0 t t t t 2 dx + I 3 t
(58)
These energy expressions can be used in an energy-based analysis such as finite element or Ritz analyses.
3.2.1 Euler approach Inserting displacement relations in equations of motion will result in
A11
B11
dw 2u 2 + D11 2 A55 + =0 2 dx x x
2 w + 2 A55 + pz ( x ) = 0 x x
2 u0 2 + B11 2 + px ( x ) = 0 2 x x
(59)
(60)
(61)
Taking second derivative of equation (60) and solving for result in following equations.
3 from equations (59, 61) will x 3
px ( x ) A11 B11 4w 1 2 pz ( x ) p x = ( ) z 4 2 2 2 A55 x A11D11 B11 x A11D11 B11 x 3u0 B11 1 px ( x ) = p (x) 3 2 z A11 x x A11D11 B11 px ( x ) B11 A11 3 = p (x ) 3 2 2 z x x A11D11 B11 A11D11 B11
(62)
(63)
(64)
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For specific case of pz(x)=q0 with simply supported boundary conditions we have
w( x ) =
3 2 2 x 4 q0 l 4 A11 x x q 0 l x x 2 + + + 2 24 l l A11D11 B11 l 2 A55 l l
(65)
(x) =
x =
3 2 q0 l 3 ql A11 x x + 1 4 6 + 0 2 24 A11D11 B11 l l 2 A55
(66)
2 2 A11D11 B11
q0 l 2
x 2 x 2q l2 z Q11 ( B11 zA11 ) 0 A55 l l
(67)
maximum deflection would occur at middle of the beam and it would be
wmax =
q0 l 2 5q0l 4 A11 2 + 384 A11D11 B11 8 A55
(68)
The first term in equation (68) is deflection due to bending and the second term is due to shear. For clamped boundary condition one can use the term due to bending from CBT analysis and add the term due to shear.
3.2.2 Matrix approach Expressing equations of motion in terms of displacement we have in matrix form
2 2 2 2 where L11 = A11 , L22 = A55 , L33 = D11 A55 , L13 = L31 = B11 , x 2 x 2 x 2 x 2 L23 = L32 = A55 , L12 = L21 = 0 . The following simply supported boundary conditions are x used on x=0, a
L11 L12 L21 L22 L31 L32
L13 u0 px 0 L23 w0 + pz = 0 L33 0 0
(69)
w0 = N x =
The above equations would be satisfied if
m m=1
=0 x
(70)
[u0 , w0 , ] = [ Am cos( mx ),Cm sin( mx ), Bm cos( mx )]

Am C 11 C 12 C 13 pxm C = C m 12 C 22 C 23 pzm Bm C 13 C 23 C 33 0
1
(71)
(72)
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2 2 C11 = m A11 , C 22 = m A55 , where C 23 = C 32 = A55 m , C 21 = C 12 = 0 . 2 C 33 = m D11 A55 ,
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2 C 31 = C13 = m B11 ,
4. Dynamic analysis
To the knowledge of authors, there is no simple approach for dynamic analysis of composite beams considering all kinds of couplings. A review was conducted on advances in analysis of laminated beams and plates vibration and wave propagation (Kapania and Raciti, 1989. Another review was done on the published literature of vibrations of curved bars, beams, rings and arches of arbitrary shape which lie in a plane (Chidamparam and Leissa, 1993). Among FSDT works, some were validated for symmetric cross-ply laminates that have no coupling (Chandrashekhara et al., 1990; Krishnaswamy et al., 1992; Abramovich et al., 1994). In some other models, symmetric beams having fibers in one direction (only bendingtwisting coupling) were considered (Teboub and Hajela, 1995; Banerjee 1995, 2001; Lee at al., 2004). Some FSDT models were validated for cross-ply laminates that have only bendingstretching coupling (Eisenberger et al. 1995; Qatu 1993, 2004). Higher order shear deformation theories (HSDT) were also developed for composite beams to address issues of cross sectional warping and transverse normal strains. Some were validated for cross-ply laminates (Khdier and Reddy, 1997; Kant et al., 1998; Matsunaga, 2001; Subramanian, 2006). Other theories like zigzag theory (Kapuria et al. 2004) were used to satisfy continuity of transverse shear stress through the laminate and showed to be accurate for natural frequency calculations of beams with specific geometry and lay-up (symmetric or cross-ply laminates). Another theory was global-local higher order theory (Zhen and Wanji, 2008) that was validated for cross-ply laminates. In this section, classic and FSDT beam models will be evaluated for their accuracy in a vibration analysis using different approaches for stiffness parameters calculation. Their results will be compared with those obtained using a 3D finite element model for different laminates (unidirectional, symmetric and asymmetric cross ply and symmetric and asymmetric angle-ply). The accurate model presented would then be verified for composite shafts.
4.1 Classical beam theory Equations of motion for dynamic analysis of laminated beams would be
2 M 2w = pz I 1 x 2 t 2 N 2u = I 1 2 px x t
(73)
(74)
where I 1 = b ( k ) ( hk hk 1 ) . Expressing those equations in matrix form we have for free

N k =1
vibration
L11 L 21 L12 u0 I 1 + L22 w0 0 0 2 2 I1 t u0 0 w = 0 0
(75)
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The equations of motion as well as simply supported boundary terms are satisfied if one chooses displacements as
[u, w] = [ Am cos( mx ), Cm sin( mx )] sin(t )

M m=1
(76)
C 11 C 12 Am I1 + 2 C 21 C 22 C m 0 0 Am pxm + =0 I1 C m pzm
(77)
The nontrivial solution for natural frequency can be found by setting the determinant of characteristic equation of matrix to zero. One should note here that if the laminate is symmetric, the B11 term vanishes and the bending frequencies are totally decoupled from axial ones. As a result, the following wellknown formula for the natural frequencies of a symmetrically laminated simply supported composite beam can be applied:
where is density, is length and A is the cross section area of the beam. As we will see later it cannot be used for thick laminates and those that have any kind of coupling.
4.2 Shear deformation beam theory The equations of motion considering rotary inertia and shear deformation would be (Qatu, 1993, 2004)
n =
D11 A
(78)
N 2u 2 = I 1 2 + I 2 2 px x t t Q 2w = pz I 1 2 x t
(79)
(80)
where
1 2 1 3 2 3 So by expressing h k hk hk hk 1 , 1 . 2 3 k =1 equations of motion in terms of displacement we have in matrix form (for free vibration)
N
( I1 , I 2 , I 3 ) = b ( k ) ( hk hk 1 ) ,
L11 L 21 L31 L12 L22 L32
M 2u 2 Q = I2 2 + I3 2 x t t
) (
(81)
L13 u0 I 1 L23 w0 0 L33 I2
0 I1 0
I2 0 I3
2 t 2
u0 0 w = 0 0 0
(82)
Equations of motion as long as simply supported boundary condition would be satisfied if
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[u0 , w0 , ] = [ Am cos( mx ), Cm sin( mx ), Bm cos( mx )] sin(t )

M m=1
(83)
C 11 C 12 C 13 Am I1 2 C 12 C 22 C 23 C m + 0 C 13 C 23 C 33 Bm I2 I1 0 0 I 2 Am pxm 0 C m + pzm = 0 I3 Bm 0
(84)
The nontrivial solution for natural frequency can be found by setting the determinant of characteristic equation matrix to zero.
5. Case studies
5.1 Rectangular beam A rectangular cross section beam model having 1 m length, 0.025 m width, and 0.05 m height was considered and modeled in ANSYS finite element code. Solid elements were used to apply 3D elasticity. A convergence study was done and the convergent model had 8 elements in thickness, 4 elements in width direction and 160 elements in length direction. Ratio of length to height of 20 was selected to be at the border of thin beams. Figure 2 shows the model.
Fig. 2. 3D finite element model in ANSYS The simply supported boundary condition was modeled by applying constraint on z direction at middle line of end faces. The material properties are E1 = 138 GPa, E2 =8.96 GPa, 12=0.3, G12= 7.1 GPa, =1580 kg/m3. Both static and modal analyses are done and the results of CBT and SDBT with different stiffness parameters are compared with 3D FEM in order to find the most accurate model.
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5.1.1 Static analysis A load of 250000 N/m were applied to the beam and the resulting deflection for cross-ply and angle-ply laminates were assessed using different models. The simply supported beam maximum deflection using Euler approach and matrix approach are given in Tables 1 and 2. Maximum normal stress is also presented in Table 3. Since the stress due to shear is low the results for CBT is not presented. The maximum deflection and stress in clamped beam are presented in Tables 4 and 5.
Laminate [0]4 [0/90]s [45]4
CBT
SDBT
FEM 3D 0.1000 0.1116 0.7824
(S11) (S11)VS (S11)Kaw (S11)Chan (S11) (S11)VS (S11)Kaw (S11)Chan
0.0901 0.0906 0.0906 0.1019 0.1026 0.1022 0.2753 0.7980 0.7980
0.0906 0.0988 0.0993 0.0993 0.1022 0.1107 0.1113 0.1109 0.7980 0.2840 0.8067 0.8067
0.0993 0.1109 0.8067
Table 1. Maximum deflection of a SS beam (Euler aproach) CBT SDBT FEM 3D 0.1000 0.1116 0.7824
Laminate [0]4 [0/90]s [45]4
(S11) (S11)VS (S11)Kaw (S11)Chan (S11) (S11)VS (S11)Kaw (S11)Chan
0.0908 0.0913 0.1028 0.1034 0.2776 0.8046
NA* NA NA
0.0913 0.1002 0.1008 0.1031 0.1123 0.1129 0.8046 0.2870 0.8140
NA NA NA
0.1008 0.1125 0.8140
* Ill conditioning observed
Table 2. Maximum deflection of a SS beam (Matrix aproach) Maximum Stress (S11) 3.000E+08 3.397E+08 3.000E+08 (S11)VS 3.000E+08 3.397E+08 3.000E+08 matrix 3.000E+08 3.397E+08 3.000E+08 FEM 3.04E+08 3.42E+08 3.12E+08
Laminate [0]4 [0/90]s [45]4
Table 3. Maximum axial stress of a SS beam (at middle point) CBT

(S11) (S11)VS (S11)Kaw (S11)Chan (S11)
Laminate [0]4 [0/90]s [45]4
SDBT
(S11)VS (S11)Kaw (S11)Chan
FEM 3D
0.01801 0.01812 0.01812 0.01812 0.02673 0.02684 0.02684 0.02684 0.02658 0.02039 0.02051 0.02044 0.02044 0.02911 0.02923 0.02916 0.02916 0.05506 0.1596 0.1596 0.1596 0.06378 0.1683 0.1683 0.1683 0.0286 0.1654
Table 4. Maximum deflection of a clamped beam (Euler aproach)
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Laminate [0]4 [0/90]s [45]4
Maximum Stress (S11) 1.000E+08 1.132E+08 1.000E+08 (S11)VS 1.00E+08 1.132E+08 1.00E+08 matrix 1.000E+08 1.132E+08 1.000E+08 FEM 1.00E+08 1.16E+08 1.00E+08
Table 5. Axial stress at middle of a clamped beam The results show that Euler and matrix approaches have very close results. In general, using SDBT along normal ABD parameters can cause problems in laminates where coupling exists. However using equivalent ABDs from Vinson and Sierakowski or Chans formulation one can get the most accurate results for deflection. This formulation is valid for any laminate having bending-twisting coupling.
5.1.2 Dynamic analysis Different approaches for calculating the natural frequencies of the first 5 modes were evaluated. Five different stacking sequences were selected to cover different kinds of composite beams. These include unidirectional, symmetric cross-ply, asymmetric cross-ply, angle-ply and general laminates. The results are given in Table 6. The results show that the classic beam model using normal ABD parameters is only valid for 1st mode of cross-ply laminates. The effective length becomes less on higher modes and the thin beam assumption no longer applies leading to inaccurate results. Although the [45]4 laminate is symmetric; it has bending twisting coupling and using the normal ABD formulation leads to inaccurate results. The equivalent ABDs by equivalent stiffness parameters improve the classic approach for unsymmtric laminates but still not accurate enough for higher modes since the shear deformation is not included. Using FSDT approach for thick beams (Qatu, 1993, 2004) along Vinson and Sierakowski equivalent modulus of elasticity for calculation of ABD parameters (Eqs. 13-15) one can reach accurate results for higher modes. This approach does not have coupling problems and accurate results for all laminate is achieved. The overall range of error is about 1 percent. The other equivalent parameters defined by compliance matrix are not as accurate as Vinson and Sierakowski and even do not have real results in some cases. 5.2 Tubular beam Experimental results of a tubular boron/epoxy beam (Zinberg and Symonds, 1970) are used in this section to verify the accuracy of the model for tubular cross section. The laminate was [90/45/-45/06/90] from inner to outer layers. The following equations were used for stiffness parameters.
k 2 A11 = Ex rk rk2 1 k =1 N
D11 =
r 4 rk4 1 ) Exk ( k 4
N k =1
(85)
(86)
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[04]
n 1 2 3 4 5 n 1 2 3 4 5 n 1 2 3 4 5 n 1 2 3 4 5 n 1 2 3 4 5
(S11) 5.1433 20.5549 46.1795 81.9230 127.6498 (S11) 5.6613 22.6450 50.9513 90.5801 141.5314 (S11) 4.688 18.718 41.990 74.337 115.526 (S11) 9.302 37.207 83.716 148.829 232.546 (S11)* 9.898 39.593 89.084 158.372 247.457
CBT (S11)VS 9.869 39.477 88.824 157.909 246.733 CBT (S11)VS 9.275 37.098 83.472 148.394 231.865 CBT (S11)VS 4.674 18.663 41.867 74.120 115.188 CBT (S11)VS 3.3251 18.1610 40.8622 72.6439 113.5062 CBT (S11)VS 3.4403 13.7558 30.9295 54.9322 85.7212
(S11)Kaw 9.869 39.477 88.824 157.909 246.733
(S11)Chan 9.869 39.477 88.824 157.909 246.733 [0/90]s (S11)Chan 9.291 37.163 83.618 148.653 232.271 [02/902] (S11)Chan 4.680 18.688 41.924 74.219 115.343 [454]
(S11) 9.431 33.413 64.529 98.109 132.196
SDBT (S11)VS (S11)Kaw 9.406 9.406 33.343 33.343 64.428 64.428 97.996 97.996 132.082 132.082 SDBT (S11)VS (S11)Kaw 8.886 8.901 31.861 31.903 62.266 62.327 95.540 95.611 129.583 129.656 SDBT (S11)VS (S11)Kaw 4.624 NA 17.901 NA 38.319 NA 64.025 NA 93.316 NA SDBT (S11)VS (S11)Kaw 3.3033 4.4890 12.9626 17.3840 28.3035 37.2377 48.4101 62.2926 72.3163 90.9410 SDBT (S11)VS (S11)Kaw 3.4173 NA 13.3998 NA 29.2239 NA 49.9103 NA 74.4320 NA
(S11)Chan 9.406 33.343 64.428 97.996 132.082
FEM 3D 9.373 32.978 63.28 95.77 128.67 FEM 3D 8.873 31.651 61.51 93.91 126.88 FEM 3D 4.609 17.651 37.251 61.354 88.302 FEM 3D 3.3540 12.970 28.316 48.321 71.93 FEM 3D 3.4830 13.458 29.222 49.857 73.97
(S11)Kaw 9.291 37.163 83.618 148.653 232.271
(S11) 8.910 31.931 62.369 95.659 129.705
(S11)Chan 8.901 31.903 62.327 95.611 129.656
(S11)Kaw NA NA NA NA NA
(S11) 4.637 17.950 38.413 64.164 93.493
(S11)Chan 4.630 17.924 38.362 64.088 93.397
(S11)Kaw 4.5402 13.3005 29.9262 53.2021 83.1283
(S11)Chan (S11) 4.5402 5.5659 18.1610 21.2317 40.8622 44.5592 72.6439 72.9229 113.5062 104.2553 [302/602] (S11)Chan 4.8010 19.1791 43.0585 76.3072 118.7286 (S11) 5.0735 19.5074 41.3688 68.4318 98.7990
(S11)Chan 4.4890 17.3840 37.2377 62.2926 90.9410
Table 6. Nondimensional natural frequencies = a2 12 / E1h 2 of rectangular simply supported beams. a/h = 20, b/h =0.5, Graphite/Epoxy, E1/E2 = 15.4, G12/E2 = 0.79, 12 = 0.3 (subscript stands for formulation in deriving ABDs)
(S11)Kaw NA NA NA NA NA
(S11)Chan 4.7449 18.3329 39.1346 65.1941 94.7571
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A number of researchers have worked on this beam with different beam and shell models and their results are shown in Table 7.
Author Zinberg, Symonds, 1970 Method used Measured experimentally BernoulliEuler beam theory. Stiffness determined by shell finite elements Sanders shell theory Donnell shallow shell theory BresseTimoshenko beam theory Effective Modulus Beam Theory Continuum based Timoshenko Beam Finite element analysis using ABAQUS EulerBernoulli beam theory Finite element analysis using ANSYS CBT using V-S SDBT using V-S Frequency (Hz) 91.67
dos Reis et al., 1987 Kim and Bert, 1993 Bert and Kim, 1995 Singh and Gupta, 1996 Chang et al. 2004 Qatu and Iqbal, 2010 present study
82.37 97.87 106.65 96.47 95.78 96.03 95.4 102.47 95.89 96.12 94.71
Table 7. Tubular Boron-epoxy beam fundamental natural frequencies (Hz) by different authors (E11 = 211 GPa, E22 = 24 GPa, G12 = G13 = G23 = 6.9 GPa, = 0.36, density = 1967 kg/m3), length = 2470 mm, mean diameter = 126.9 mm, thickness = 1.321 mm. (90, 45, -45,0,0,0,0,0,0,90) laminate (from inner to outer) The results show that most of the models can predict the natural frequency of this beam with good accuracy. Only the models by dos Reis et al. predicted results that are far from those obtained by experiment. However the FSDT used in this paper is the most accurate model for this case. The effect of ply orientation on reduction of stiffness and consequently natural frequency of a graphite-epoxy tube is presented in Table 8 (Bert and Kim, 1995).
Theory Lamination angle 30 45 60 50.13 39.77 35.33 71.15 52.85 38.20 69.95 52.38 37.97 45.51 35.90 31.96 46.05 36.15 32.17 45.91 36.06 32.09
Sanders Shell Bernoulli-Euler Bresse-Timoshenko Present FEM analysis Present CBT approach Present SDBT approach
0 92.12 107.08 101.20 100.28 108.42 104.43
15 72.75 89.88 86.82 68.80 71.12 70.50
75 33.67 31.42 32.90 30.57 30.78 30.70
90 33.28 30.22 30.05 30.27 30.50 30.36
Table 8. Effect of lamination angle on fundamental natural frequencies of tubular a graphiteepoxy beam. (E11 = 139 GPa, E22 = 11 GPa, G12 = G13 = 6.05 GPa, G23 = 3.78 GPa, = 0.313, density = 1478 kg/m3) These results are for the first natural frequency of a graphite epoxy tubular beam with the same geometry of the previous one. Results of the present CBT, SDBT and FEM using shell elements are presented. Results show a good agreement between this study and the previous ones. It shows the decrease in natural frequency by lowering stiffness and also bending twisting coupling.
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6. Conclusion
Different approaches for static and dynamic analysis of composite beams were proposed and a modified FSDT model that accounts for various laminate couplings and shear deformation and rotary inertia was validated. The method was verified using 3D FEM model. The results showed good accuracy of the model for rectangular beams in static analysis for laminates having bending-twisting coupling and in dynamic analysis for all kinds of laminates. Also the model was verified for dynamic analysis of tubular cross section beams (or shafts) and the results were accurate compared to experimental ones and other models. This model provides an accurate approach for calculating the natural frequencies of beams and shafts with arbitrary laminate for engineers and scientists.
7. References
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Teboub, Y. & Hajela, P. (1995). Free vibration of generally layered composite beams using symbolic computation, Composite Structures, Vol.33, No.3, pp. 123-134, ISSN: 02638223. Banerjee, J. R. Williams F. W. (1995). Free Vibration of Composite Beamsan Exact Method Using Symbolic Computation, AIAA Journal of Aircraft, Vol. 32, No.3, pp. 636-642, ISSN 0021-8669. Banerjee, J. R. (2001). Explicit analytical expressions for frequency equation and mode shapes of composite beams, International Journal of Solids and Structures, Vol.38, No.14, pp. 2415-2426, ISSN 0020-7683. Li, J., Shen, R., Hua, H. & Jin, J. (2004). Bendingtorsional coupled dynamic response of axially loaded composite Timosenko thin-walled beam with closed cross-section, Composite Structures, Vol.64, No.1, pp. 2335, ISSN 0263-8223. Eisenberger, M., Abramovich, H. & Shulepov, O. (1995). Dynamic stiffness analysis of laminated beams using a first order shear deformation theory. Composite Structures, Vol.31, No.4, pp. 265-271, ISSN 0263-8223. Khdeir, A. A. & Reddy, J. N. (1997). Free And Forced Vibration Of Cross-Ply Laminated Composite Shallow Arches. Intl J Solids Structures, Vol.34, No.10, pp. 1217-1234, ISSN 0020-7683. Kant, T., Marur, S. R. & Rao, G. S. (1998). Analytical solution to the dynamic analysis of laminated beams using higher order refined theory. Composite Structures, Vol.40, No.1, pp. 1-9, ISSN 0263-8223. Matsunaga, H. (2001). Vibration and Buckling of Multilayered Composite Beams According to Higher Order Deformation Theories, Journal of Sound and vibration, Vol.246, No.1, pp. 47-62, ISSN 0022-460X. Subramanian, P. (2006). Dynamic analysis of laminated composite beams using higher order theories and finite elements, Composite Structures, Vol.73, No.3, pp. 342-353, ISSN 0263-8223. Kapuria, S., Dumir, P.C. & Jain, N. K. (2004). Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Composite Structures, Vol.64, No.3-4, pp. 31727, ISSN 0263-8223. Zhen, W. & Wanji, C. (2008). An assessment of several displacement based theories for the vibration and stability analysis of laminated composite and sandwich beams, Composite Structures, Vol.84, No.4, pp. 337-349, ISSN 0263-8223. Zinberg, H. & Symonds, M.F. (1970). The development of an advanced composite tail rotor driveshaft. In: Proceedings of 26th annual forum of the American Helicopter Society, Washington, USA. dos Reis, H. L. M., Goldman, R. B. & Verstrate, P. H. (1987). Thin-walled laminated composite cylindrical tubes. Part IIIcritical speed analysis, Journal of Composite Technology and Research, Vol.9, No.2, pp. 5862, ISSN 0884-6804. Kim, C. D. Bert, C. W. (1993). Critical speed analysis of laminated composite hollow drive shafts. Composite Engineering, Vol.3, No.7-8, pp. 63343, ISSN: 1359-8368. Bert, C. W. & Kim, C. (1995) Whirling of composite-material driveshafts including bending twisting coupling and transverse shear deformation, Journal of Vibration and Acoustics, Vol.117, No.1, pp. 1721, ISSN 1048-9002.
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Singh, S. P. & Gupta, K. (1996). Composite shaft rotordynamic analysis using a layerwise theory, Journal of Sound and Vibration, Vol. 191, No. 5, pp. 73956, ISSN 0022460X. Chang, M-Y., Chen, JK. & Chang, C-Y. (2004). A simple spinning laminated composite shaft model, International Journal of Solids and Structures, Vol.41, No.3-4, pp. 63762, ISSN 0020-7683. Qatu, M. S. & Iqbal, J. (2010). Transverse vibration of a two-segment cross-ply composite shafts with a lumped mass, Composite Structures, Vol.92, No.5, pp. 1126-1131, ISSN 0263-8223.
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Advances in Composite Materials - Analysis of Natural and ManMade Materials Edited by Dr. Pavla Tesinova
ISBN 978-953-307-449-8 Hard cover, 572 pages Publisher InTech
Published online 09, September, 2011
Published in print edition September, 2011 Composites are made up of constituent materials with high engineering potential. This potential is wide as wide is the variation of materials and structure constructions when new updates are invented every day. Technological advances in composite field are included in the equipment surrounding us daily; our lives are becoming safer, hand in hand with economical and ecological advantages. This book collects original studies concerning composite materials, their properties and testing from various points of view. Chapters are divided into groups according to their main aim. Material properties are described in innovative way either for standard components as glass, epoxy, carbon, etc. or biomaterials and natural sources materials as ramie, bone, wood, etc. Manufacturing processes are represented by moulding methods; lamination process includes monitoring during process. Innovative testing procedures are described in electrochemistry, pulse velocity, fracture toughness in macro-micro mechanical behaviour and more.
How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Mehdi Hajianmaleki and Mohammad S. Qatu (2011). Mechanics of Composite Beams, Advances in Composite Materials - Analysis of Natural and Man-Made Materials, Dr. Pavla Tesinova (Ed.), ISBN: 978-953-307-449-8, InTech, Available from: http://www.intechopen.com/books/advances-in-composite-materials-analysis-ofnatural-and-man-made-materials/mechanics-of-composite-beams
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