 Open Access
 Total Downloads : 778
 Authors : Emad Omar Ali, Hany Ahmed Elghazaly, Mohamed Sayed Gomaa, Mohamed Abo Elmaaty Amin
 Paper ID : IJERTV3IS100892
 Volume & Issue : Volume 03, Issue 10 (October 2014)
 Published (First Online): 29102014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Structural Analysis of Composite Laminated BoxBeams under Various Types of Loading
Effect of Fiber Orientation Angle and Number of Layers on the BoxBeam
Deformations
H. A. Elghazaly
Department of Civil Engineering Faculty of Engineering, Fayoum University
Fayoum, Egypt
M. S. Gomaa
Department of Civil Engineering Faculty of Engineering, Fayoum University
Fayoum, Egypt

Abou Elmaaty Amin
Department of Civil Engineering Faculty of Engineering, Fayoum University
Fayoum, Egypt
Emad Omar Ali
Department of Civil Engineering Faculty of Engineering, Fayoum University
Fayoum, Egypt
AbstractFiber reinforced polymer composite (FRP) is a new construction material, gradually gaining acceptance from civil engineers. In the past 15 years, experiments have been conducted to investigate the applicability of using FRP composite in bridge, and tunnel structures, including the applications of FRP composite beam, deck, and column. Beam is one of the most important structural elements in any structural system, so knowing the structural behavior of beams is very important. In this study an analytical solution for composite laminated beam with Boxsection has been developed. The solution includes the structural characteristics which are often ignored in the most published studies such as axial and bending stiffness. Also, a finite element model has been developed using ANSYS software to validate the results obtained from the analytical solution and it has been seen a good agreement between results. Moreover, a parametric study has been conducted using the developed finite element model. The parametric study includes the effect of fiber orientation angle for symmetric angle ply Box beam on the axial, bending, and torsional deformations. Furthermore, the effect of changing the number of layers in both the web and flange laminates on the formerly mentioned deformations (i.e. axial, bending, torsional deformations) has been studied.
KeywordsComposite laminated beams; Classical lamination theory; Fiber orientation angle; composite beam stiffness; Finite element method

INTRODUCTION
The fiberreinforced composite materials are ideal for structural applications where high strengthtoweight and stiffnesstoweight ratios are required. Composite materials can be tailored to meet the particular requirements of stiffness and strength by altering layup and fiber orientations. The ability to tailor a composite material to its job is one of the most significant advantages of a composite material over an ordinary material. A number of researches have been made to develop numerous solution methods in the recent 25 years.
Zang, et al. studied the stress and strain distribution numerically in the thickness direction in the central region of symmetric composite laminates under uniaxial extension and
inplane pure shear loading [1]. Anido, et al. carried out an experimental evaluation of stiffness of laminated composite rectangular beam under flexure. Three point bending tests were performed on layup angle ply [Â±45]s beam elements made of AS4/35016 carbonepoxy [2]. Brown presented a combined analytical and experimental study of fiber reinforced plastic composite bridges consisting of cellular box decks and wide flange Ibeam as stringer. The study included design, modeling, and experimental/numerical study of fiber reinforced composite decks and deckandstringer bridge systems [3]. Aktas introduced a deflection function of an orthotropic cantilever beam subjected to point and distributed load using anisotropic elasticity. The deflections at the free end of the beam were calculated numerically using the obtained formulations for different fiber directions. It was found that the free end deflection of the beam increased for angles ranging from 0Âº to 90Âº for both load cases due to decreasing of stiffness [4]. Song, et al. presented analytical solutions for the static response of anisotropic composite I beams loaded at their freeend, also the variation of the displacement quantities along the beam span was presented [5]. Zhou developed a systematic analysis procedure to investigate stiffness and strength characteristics of the multi cellular FRP bridge deck systems consisted of pultruded FRP shapes [6]. Cardoso performed an optimal design of thin walled composite beams against stress, displacement, natural frequencies, and critical load. The thickness of the laminates and ply orientations were considered as design variables. Equivalent beam stiffness and equivalent density properties were calculated [7]. Lee* and Lee1 presented a flexural torsional analysis of Ishaped laminated composite beams. A general analytical model was developed to thinwalled I section beams subjected to vertical and torsional load [8]. Hessabi studied the effect of stacking sequence on interlaminar stress distribution and the consequent change of the mode of failure experimentally [9]. Vo and Lee developed a general analytical model applicable to thinwalled box
section composite beams subjected to vertical and torsional load. This model was based on the classical lamination theory, and accounts for the coupling of flexural and torsional responses for arbitrary laminate stacking sequence configurations, i.e. symmetric, as well as, asymmetric [10]. Gopal carried out finite element analysis to perform static

CONSTITUTIVE EQUATION FOR LAMINATED
COMPOSITE BEAMS
The resultant forces and moments per unit length in the x y plane through the laminate thickness can be calculated from the following Equation:
analysis on a crossply laminated composite square plate based on the first order shear deformation theory. A finite
Nx A11
N
A12
A16
B11
B12
B16
x
o
o
y A12 A22 A26 B12 B22 B26 y
element program (MATLAB) was used to obtain the finite element solutions for transverse displacements, normal stresses and transverse shear stresses [11]. Nik and Tahani introduced an analytical method to study the bending behavior of laminated composite plates. The method is capable of analyzing laminate plates with arbitrary lamination
Nxy Mx
My
Mxy
= A16 A26 A66 B16 B26 B66
B11 B12 B16 D11 D12 D16
B12 B22 B26 D12 D22 D26
B16 B26 B66 D16 D26 D66
o
xy
kx
ky
kxy
1
and boundary conditions [12]. Schmalberger used a finite element software ANSYS to simulate the behavior of four layer symmetric laminate which was verified by showing the solutions for problems using the Classical Laminate Plate Theory (CLPT). The beam used in simulation was constructed as [/90]s layup for simplification. The response
Where, [A], [B], and [D] matrices are called the
extensional, coupling, and bending stiffness matrices, respectively. Also [] and [k] are the midplane strains and curvatures of the laminate.
Equation (1) can be written as,
of the beam was investigated as a function of the orientation
of fibers in outer layers [13]. Mokhtar, et al. carried out a
o a
x
o 11
a12
a16
b11
b12
b16 Nx
N
y a12 a22 a26 b12 b22 b26 y
survey on plate bending of crossply laminate by using the finite element method using ANSYS software. Two types of
o a a a b b b Nxy
xy 16 26 66 16 26 66
=
2
modeling were proposed: the first was modeling using a type of shell element, shell 99 and the secon was an approach based on a type of solid element, solid 46 [14]. Jitech developed an analytical method for stress analysis of
kx
ky
kxy
b11 b12 b16 d11 d12 d16
b12 b22 b26 d12 d22 d26
b16 b26 b66 d16 d26 d66
Mx
My
Mxy
composite Ibeam. This method included the structural response due to symmetrical of laminates, as well as, unsymmetrical Ibeam cross section. The analytical expressions for the sectional properties such as centroid, axial and bending stiffness of composite Ibeam were derived. A
finite element model was created using ANSYS software to
In structural analysis the beams are divided to wide and
narrow beam depending on the width to depth ratio. For a narrow beam, there are no forces and moments in ydirection
[15] hence, (2) can be modified as,x = a11 b11 Nx 3
verify the results and excellent agreement was found with
analytical results [15]. Or
In this paper an analytical solution for composite laminated beam with Boxsection has been developed based
kx
Nx
b11 d11 Mx
x
= 1 1 4
on the Classical Laminate Plate Theory (CLPT) and the analytical solution of composite laminated Ibeam developed by Jitech [15]. The solution includes the structural
Mx kx
1 1
Where,
characteristics which are often ignored in the most published studies such as axial and bending stiffness. Also, a finite
= 1
1
2
element model has been developed using ANSYS software to validate the results obtained from the analytical solution and it has been seen a good agreement between results.
11
11
11
= 1
Moreover, a parametric study has been conducted using the developed finite element model. The parametric study
1
includes the effect of fiber orientation angle for symmetric angle ply Box beam on the axial, bending, and torsional
1
11
11 11
11
1
deformations. Furthermore, the effect of changing the number of layers in both the web and flange laminates on the
=
2
11
11
1 1 1
11
(5)
formerly mentioned deformations (i.e. axial, bending, and torsional deformations) has been studied. Also for each loading condition, the optimum fiber orientation angle and the optimum number of layers in web and flange laminates have been determined.
, , and refer to the axial, coupling and bending stiffnesses of the beam.

ANALYTICAL SOLUTION OF COMPOSITE LAMINATED BEAM WITH BOXSECTION
SUBJECTED TO AXIAL FORCE AND BENDING
For the top bottom laminate
2 = + ,2 (11)
MOMENT
1,2
,2
1,1

Geometry of Laminated BoxSection
2 = + ,2 (12)
The laminated composite BoxSection is divided into four sublaminated, two flanges and two webs as shown in Fig. 1.
1,1
For the web laminate
,2
1,1
The two flanges have different number of layers, thickness of layers, and laminate sequence, but the two webs have the same properties.
= (13)
1, ,
= (14)
1, ,
Because the load is applied at the centroid then there is no curvature for all flange and web laminates so,
,1 = ,2 = , = 0 (15)
Moreover, since the strain for all flange and web laminates are equal along the Xaxis then,
= = = (16)
,1
,2
,
x
Where, c is the strain at the centroid in the Xdirection.
Using Equations from (9) to (16) in (7), then the total force in Xdirection can be modified as,
= + + 2 (17)
1,1
1,2
1,
Fig. 1. Composite laminated boxsection geometry.

Centroid of the Composite Laminated BoxSection
The centroid of the Boxsection showed in Fig. 1, can be calculated as,
= 1 1 + 22 + 2 33 (6)
Where,
For another expression of the total force in Xdirection using axial stiffness, then the resultant force can be written as,
= (18)
Using (17) in (18), then the axial stiffness can be
calculated as,
= + + 2 (19)
=
+
+ 2
(7)
1,1
1,2
1,
1
2
3
If the top and bottom flanges have the same properties,
Nx1, Nx1, and Nx3 are the axial forces per unit width of the upper flange, lower flange, and the web along Xdirection and
N x is the resultant axial force acting on the Boxsection in the
then (19) can be modified as,
= 2 + (20)
1, 1,
Xdirection. Using (4) and (7) in (6) then, the centroid
distance from YAxis can be calculated as,
Where, the first term of (19) represents the axial stiffness of the beam flanges and the second term represents the axial
+
+ 2
stiffness of the beam webs.
=
1,1 1
1,2 2
1, 3
8
+ + 2
3.4 Equivalent Bending Stiffness,
1,1
1
Where A is defined in (5).
1,2
1,
If a moment is applied at the centroid of the Boxsection in the Xdirection then the applied moment can be related to the bending stiffness as,

Equivalent Axial Stiffness, E A
From (4) the axial force and bending moment per unit width for each sublaminate in the cross section can be
= (21)
x
Where, Kc is the radius of curvature at the centroid in X
calculated as,
For the top flange laminate
1 = + ,1 (9)
direction.
1,1
,1
1,1
1 = + ,1 (10)
1,1
,1
1,1
Also the applied moment can be calculated from another expression as,
For web laminate, the force in Xdirection per unit width can be calculated as,
=
+
+
+
, = = + (28)
1
1
1
2
2
2
1,
,
1,
2
+
+ 2 , (22)
Using (25), then (28) can be deduced as,
, = (29)
1,
2
Where, z1c , z1c , and hwc are the distances between the midplanes of upper flange, lower flange, and web laminates from the centroid as shown in Fig. 2.
Then the third term of (22) can be calculated as,
+
2
2 +
2 , = 2
(30)
2
1,
2
2
+
2
= 2 1 3 +
2 (31)
2
,
1,
12
Fig. 2. Distances between midplanes of sublaminates from the centroid.
The first term of (22), for the upper flange laminate can be calculated as,
Substituting (26), (27), and (31) in (22), then the moment can be written as,
2 + 2 1 +
1,1 1
1,1
1,1
+ ,1 +
+ 2 + 2
+
1 + 1 1 =
1,1
,1
1,1
(23)
=
1,2
2
1,2
2
1,2
(32)
+ ,1 1 1
1,1
,1
1,1
+ 2
3 +
2
1,
12
And the midplane strain in Xdirection for the upper flange can be written as,
Substituting (32) in (21), then the bending stiffness can be calculated as,
2 + 2 1 +
1,1 1
1,1
1,1
+ 2 + 2 2 +
,1
=
=
1,2
2
1
1,2
3
1,2
2
(33)
= +
(24)
+ 21,
12
+
Where,
,1
1
,1


FINITE ELEMENT MODEL and VALIDATION
= 0 (25)
Then (23) can be deduced as,
1 + 1 1 = 2 + 2 1 (26)

Finite Element Model
The finite element commercial package ANSYS 14.5 has been used for the finite element analysis using ANSYS SHELL99 element in the modeling. SHELL99 is a linear layered structural shell element providing up to 250 layers.
The element has 8 nodes with six degree of freedom for each
1,1 1
1,1
1,1
node as shown in Fig. 3.
Similarly, the second term of (22), for the lower flange laminate can be calculated as,
2 + 2 2 = 2 + 2 2 (27)
1,2
2
1,2
1,2
Fig. 3. Shell99 linear layered structural element geometry (ANSYS, 2009).

Validation of the Developed Analytical Expression Example1: Fixed Cantilever Beam with BoxSection Subjected to Axial Force 20 t
A fixed cantilever composite beam with Boxsection has length and cross section as shown in Fig. 4. The beam is subjected to an axial load equal 20 ton with stacking sequence [0/90/0] for all flange and web laminates. The material properties used in the analysis are shown in table1.
Fig. 4. Fixed Cantilever Composite Beam with Boxsection Subjected to Axial Load 20 t.
TABLE1. ORTHOTROPIC PROPERTIES FOR UNIDIRECTIONAL GRAPHITE/EPOXY LAMINA [16]
Material Properties
(t/cm2)
(t/cm2)
(t/cm2)
Value
1810
103
71.7
0.28
The axial displacement at the free end of the beam has been calculated from the developed analytical expression and obtained from the FEM. Also the stress in Xdirection of the upper flange laminate has been calculated from the developed analytical expression and obtained from the FEM as following:
Results from the FEM
the FEM results are shown in Fig. 5.
Fig. 5. Axial Displacement of Composite Beam with Boxsection Subjected to Axial Force 20 t.
Comparison between Results
The results from the developed analytical expression and the FEM have been compared and it has been shown that the results are in good agreement as indicated in table2.
TABLE2. COMPARISON BETWEEN FEM AND THE DEVELOPED ANALYTICAL EXPRESSION FOR BOXSECTION SUBJECTED TO AXIAL FORCE 20T
Results
Analytical expression
FEM
Error (%)
Axial Displacement (cm)
0.03059
0.03061
0
Example2: Fixed Cantilever Beam with BoxSection Subjected to Pure Moment 100 t.cm.
A fixed cantilever composite beam with Boxsection has length and cross section as shown in Fig. 6. The beam is subjected to pure moment 100 t.cm with stacking sequence [0/90/0] for all flange and web laminates. The material properties used in the analysis are the same in example1.
Fig. 6. Fixed Cantilever Composite Beam with Boxsection Subjected to Pure Moment 100 t.cm.
The deflection at the free end of the beam and the strain of the upper flange have been calculated from the developed analytical expression and obtained from the FEM. Also the stress in Xdirection of the upper flange laminate has been calculated from the developed analytical expression and obtained from the FEM as following:
Results from the FEM
the FEM results are shown in Figs. 7 and 8.
Fig. 7. Deflection of Composite Beam with Boxsection Subjected to Pure Moment 100 t.cm.
Fig. 8. Strain in Xdirection of Composite Beam with Boxsection Subjected to Pure Moment 100 t.cm.
Comparison between Results
The results from the developed analytical expression and the FEM have been compared and it has been shown that the results are in good agreement as indicated in table3.
TABLE 3. COMPARISON BETWEEN FEM AND THE DEVELOPED ANALYTICAL EXPRESSION FOR BOXSECTION SUBJECTED TO PURE MOMENT 100 T.CM
Results
Analytical Expression
FEM
Error (%)
Deflection (cm)
0.1639
0.159
3
Strain in Xdirection
0.132×103
0.136×103
2.9


PARAMETRIC STUDY

Studied Cases and Beams Configurations
All the studied cases and beams configurations are shown in Figs. 9 to 13. The beams configurations included the beam cross section, beam length, boundary conditions, and the loading type.
Fig. 9. Case (1) Fixed Cantilever BoxBeam Subjected to Axial Force.
Fig. 10. Case (2) Fixed Cantilever BoxBeam Subjected to Pure Bending Moment.
Fig. 11. Case (3)Simply Supported BoxBeam Subjected to Bending Moment and Shear Force due to Concentrated Load.
Fig. 12. Case (4) Simply Supported BoxBeam Subjected to Bending Moment and Shear Force due to distributed Load.
Fig. 13. Case (5) Fixed Cantilever BoxBeam Subjected to Pure Torsion Moment.

Material Properties of the Studied Cases
The material used in the analysis is T300/976 Graphite Epoxy [4]. The T300/976 Graphite/Epoxy properties are shown in table4.
TABLE4. T300/976 GRAPHITE/EPOXY PROPERTIES
Material
E1(t/cm2)
E2(t/cm2)
G12(t/cm2)
u12
T300/976
Graphite/Epoxy
1560
130
70
0.23

Effect of Fiber Orientation Angle (FOA) on the Beam Deformations
In this section the effect of FOA on the Axial, bending, and torsional deformations of the beam has been studied. For each studied case, the number of layers in both web and flange laminates are four layers. Furthermore, the laminate stacking sequence has been taken as [f/f]s for flange laminate and [w /w ]s for web laminate. The flange FOA (f) has been increased from 0Âº to 90Âº with 10Âº step for each web FOA (w ) which also has been increased from 0Âº to 90Âº with 10Âº.
Case (1) Fixed Cantilever BoxBeam Subjected to Axial Load In this case, the section has been subjected to axial force. The resulting axial displacements at the free end of the beam have been obtained for the studied angles and plotted together
in chart as shown in Fig. 14.
Fig. 14. The Relation between the Axial Displacement and the FOA for Box Beam Subjected to Axial Force.
For Fig. 14, it is shown that, by increasing the flanges FOA, the axial displacement increases. Also, by increasing the web FOA, the axial displacement increases. This increase is a gradual one when the web FOA ranges between 0Âº and 20Âº. On the contrary, a drastic change in the rate of increase is shown when the web FOA increases from 20Âº to 70Âº specially when the web FOA exceeds 30Âº, then the rate of increase changes to be a gradual one again. Moreover, it can be observed that the lowest axial displacement occurred at FOA equals 0Âº for the flanges and web laminates because the fiber direction in the same direction of the force.
Case (2) Fixed Cantilever BoxBeam Subjected to Pure Bending Moment
In this case, the section has been subjected to pure bending moment. The resulting deflections at the free end of the beam have been obtained for the studied angles and plotted together in chart as shown in Fig. 15.
Fig. 15. The Relation between the Deflection and the FOA for BoBeam Subjected to Pure Bending Moment.
For Fig. 15, it is shown that, the deflection increases by increasing the flanges FOA. Also, by increasing the web FOA, the deflection increases. Also it can be illustrated that the lowest deflection occurred at FOA equals 0Âº for the flanges and web laminates because the fiber direction in the same direction of the forces resulting from the normal stresses for both the flanges and web laminates.
Case (3) Simply Supported BoxBeam Subjected to Bending Moment and Shear Force due to Concentrated Load
In this case, the section has been subjected concentrated load at the beam midspan. The resulting deflections at the midspan of the beam have been obtained for the studied angles and plotted together in chart as shown in Fig. 16.
It is shown that from Fig. 16, the deflection increases by increasing the flanges FOA. Also, the deflection increases by increasing the web FOA except for flange FOA equals 0Âº and 10Âº, the deflection decreases when the web FOA increases from 0Âº to 20Âº then the deflection increases again for the other web FOA. It can be observed that the lowest deflection occurred at FOA equals 0Âº for the flange laminate and 20Âº for web laminate because the influence of the shear force on the web laminate.
Fig. 16. The Relation between the Deflection and the FOA for BoxBeam Subjected to Bending Moment and Shear Force due to Concentrated Load. Case (4) Simply Supported BoxBeam Subjected to Bending Moment and Shear Force due to Distributed Load
In this case, the section has been subjected distributed load. The resulting deflections at the midspan of the beam have been obtained for the studied angles and plotted together in chart as shown in Fig. 17.
Fig. 17. The Relation between the Deflection and the FOA for BoxBeam Subjected to Bending Moment and Shear Force due to Distributed Load.
For Fig. 17, it is shown that the deflection increases by increasing the flanges FOA. Also, the deflection increases by increasing the web FOA except for flange FOA equals 0Âº,10Âº, and 20Âº, the deflection decreases when the web FOA increases from 0Âº to 10Âº then the deflection increases again for the other web FOA. It can be observed that the lowest deflection occurred at FOA equals 0Âº for the flange laminate and 10Âº for web laminate because the influence of the shear force on the web laminate which is less effectiveness than the concentrated load.
Case (5) Fixed Cantilever BoxBeam Subjected to Torsion Moment at Its Free End
In this case, the section has been subjected to torsion moment. The resulting rotation angles at the free end of the beam have been obtained for the studied angles and plotted together in chart as shown in Fig. 18.
Fig. 18. The Relation between the Rotation Angle and the FOA for Box Beam Subjected to Torsion Moment.
It is shown that from Fig. 18, the Rotation Angle decreases by increasing the FOA for both the flange and web laminates until FOA equals 45Âº then it increases when the FOA increases from 45Âº to 90Âº. Also, the rotation angels are symmetric around flange and web FOA equals 45Âº. It can be observed that the lowest Rotation angle occurred at FOA equals 45Âº for the flange and web laminates because of the shear stress resulting from the torsion moment.

Effect of Number of Layers on the Beam Deformations
In this section the effect of increasing number of layers for both flange and web laminates on the minimum deformations obtained from the previous section has been studied. Furthermore, the effect of increasing number of layers for both flange and web laminates on the optimum FOA obtained from the previous section has been studied. The number of layers has been increased from four layers to twelve layers for all studied cases. The relation between the beam deformations and the number of layers has been shown in the Figs. 19 to 23.
Fig. 19. The Relation between the Minimum Displacement and the Number of layers for Fixed Cantilever BoxBeam Subjected to Axial Force.
Fig. 20. The Relation between the Minimum Deflection and the Number of layers for Fixed Cantilever BoxBeam Subjected to Pure Moment.
Fig. 21. The Relation between the Minimum Deflection and the Number of layers for Simply Supported BoxBeam Subjected to Concentrated Load.
Fig. 22. The Relation between the Minimum Deflection and the Number of layers for Simply Supported BoxBeam Subjected to Distributed Load.
Fig. 23. The Relation between the Minimum Rotation angle and the Number of layers for Fixed Cantilever BoxBeam Subjected to Torsion Moment.
It is illustrated from Figs. 19 and 20, that the minimum axial displacements due to axial force and the minimum deflections due to pure bending moment are constants and not affected by increasing the number of layers so using three layers is enough to decrease the cost. But, for Figs. 21 and 22 it is shown that the minimum deflections due to concentrated or distributed load change as the number of layers change. Also, the difference ratio between the minimum deflections is less than 1.5% and the maximum value occurs at three layers so, using four layers is enough to decrease the cost.
For Fig. 23 it is observed that the minimum rotation angles due to torsion moment change as the number of layers change. Also, the difference ratio between the minimum rotation angles is less than 8.5% and the maximum value occurs at three and six layers. Furthermore, the minimum rotation angle occurs at number of layers equal 4n where, n = 1, 2, 3, .etc. So, using four layers is enough to decrease the cost.


CONCLUSIONS


Summary
A simplified analytical method has been developed based on the Classical Laminate Plate Theory (CLPT) and the analytical solution of composite laminated Ibeam developed by Jitech [15] to calculate sectional properties; equivalent axial stiffness, and equivalent bending stiffness. Also, a Finite Element Model (FEM) has been developed to validate the results obtained from the analytical solution and it has been shown that the results obtained from the analytical solution are in a good agreement with the results obtained from the FEM. Furthermore, the FEM has been used in the parametric study to know the effect of the fiber orientation angle and the number of layers on the Boxbeam deformations.

Conclusion Points
The main conclusions obtained from the present study have been presented as following:
Effect of Fiber Orientation Angle (FOA)

The minimum axial displacement occurs at Fiber Orientation Angle (FOA) equal 0Âº for both flange and web laminates.

For symmetric angle ply Boxbeam subjected to pure bending moment the minimum deflection occurs at flange FOA equal 0Âº and web FOA equal 0Âº.

For symmetric angle ply Boxbeam subjected to bending moment due to concentrated load at its mid span, the minimum deflection occurs at flange FOA equal 0Âº and web FOA equal 20Âº.

For symmetric angle ply Boxbeam subjected to bending moment due to distributed load, the minimum deflection occurs at flange FOA equal 0Âº and web FOA equal 10Âº.

For symmetric angle ply Boxbeam subjected to pure torsion moment the minimum rotation at its free end occurs at flange FOA equal 45Âº and web FOA equal 45Âº.
Effect of Number of Layers

Increasing the number of layers does not affect the minimum axial displacement or the optimum FOA for the flanges and web laminates, so using three layers is enough to decrease the cost.

For symmetric angle ply Boxbeam subjected to pure bending moment, Increasing the number of layers does not affect the minimum deflection or the optimum FOA for the flanges and web laminates, so using three layers is enough to decrease the cost.

For symmetric angle ply Boxbeam subjected to bending moment due to concentrated or distributed load, the minimum deflection changes as the number of layers chages and The difference between the minimum deflections is less than 1.5%, so using four layers is enough to decrease the cost.

For symmetric angle ply Boxbeam subjected to pure torsion moment the minimum rotation changes as the number of layers changes and the minimum rotation angle occurs when the number of layers equal 4n where, n = 1, 2, 3, .etc. So, it is enough to use 4 layers only to decrease the cost. Also the difference ration between the minimum rotation angles is less than 8.5%.

ACKNOWLEDGMENT
The support of this research by the Department of Civil engineering, Fayoum University is gratefully acknowledged.
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