Exact Solution of Thin walled Open Section Beam using a Coupled Field - - PDF document

SLIDE 1

18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

1 Abstract The exact static solutions of shear flexible thin walled laminated I-beams are derived using a coupled field formulation. The formulation accommodates the effect of elastic couplings due to material anisotropy, shell wall thickness, warping shear, transverse shear deformation and constrained warping effects. The governing equations are first derived in terms of forces or stress resultants. The spatial distribution of beam forces and the displacements along the length of the beam are derived in closed form. Examples of isotropic and laminated composite I-beam subjected to bending and torsion forces are studied and compared favorably with available numerical results. 2 General Introduction Composites are being extensively used as a material

• f choice in the aircraft industry since they offer a

high strength to weight ratio, increased fatigue life and improved damage tolerance performance. Thin walled structures are an integral component of a typical aero structure. Structures like rotor blades, wing spars can be modeled as one dimensional beam since their cross sectional dimensions are much small compared to its length. Additionally they are being increasingly used in aircraft structures as stiffeners whose primary objective is to improve panel stability. Thin walled composite beams (TWCB) demonstrate very complex behavior under the application of bending and twisting loads. Several non-classical effects like material coupling, transverse shear and restrained torsional warping must be included in developing an analytical model for TWCB. Composites have a very low shear modulus to extensional modulus and hence transverse shear deformation has a significant influence on their response to applied shear, bending and twisting

• loads. Also the end restraints cause a non-uniform
• ut-of-plane warping and estimates of torsional

stiffness based

• n

Saint-Venant theory are

• inaccurate. The effect is even more significant in
• pen sections and Vlasov theory is normally adopted

to incorporate restraint warping effect. A 1-D mathematical model is usually used to analyze a TWCB. The kinematics of the beam is derived by expressing the local displacements in the thin-walled shells in terms of generalized beam displacements which include extension, bending in two directions, shear in two directions, and the twist. The twisting includes the component of both St. Venant torsional moment and a bi-moment which arises due to restrained warping effect. Jung et.al [1] has compiled an extensive survey of existing numerical and analytical composite beam theories. Chandra and Chopra [2] included the extension bending - coupling stiffness, transverse shear effects and generalized the theory to accommodate material coupling due to unsymmetric laminate stacking

• sequence. Jung et.al [3] developed a mixed method

applicable to coupled composite beams, with arbitrary cross section. Numerical simulations were carried out and showed good accuracy with experimental results. Most of the numerical simulations for TWCB up to now have been carried out using finite element (FE) method due to its versatility. A displacement based 1-D FE model for flexural torsional buckling of composite I-beams was developed by Lee and Kim [4]. Jaehong Lee [5] presented a shear deformable beam theory and applied it for the flexural analysis

• f TWCB using a FE analysis. He introduced several

non-classical effects displayed by them like transverse shear, warping shear, material coupling,

Exact Solution of Thin walled Open Section Beam using a Coupled Field Formulation

Srinivasan Ramaprasad 1*, Darsi Nagendra Kumar 2

1 Center of Excellence Aerospace and Defense, Mahindra Satyam Computer Services Ltd.,

Bangalore,

2 Lead Engineer - Aircraft Systems, Cassidian Air Systems, EADS DS India Pvt. Ltd., Bangalore

* Corresponding author (srinivasan_ramaprasad @ mahindrasatyam.com)

Keywords: Composite Beams, Warping, Flexural, Torsional, Coupling, Anisotropic

SLIDE 2

restrained warping effects etc. Mira Mitra et.al [6] presented a new super convergent TWCB element for box beam analysis. They used the shape functions which satisfied the governing equations and demonstrated superior convergence in FE results. Analytical solutions for static analysis of composite beams have also been made by researchers. Jung and Lee [7] derived a closed form solution for the static response of both symmetric and anti-symmetric lay- up I-beam with transverse shear coupling and included most of the non-classical effects in their

• formulation. Most of the analytical solutions have

been achieved for simplified laminate stacking

• sequence. Obtaining the exact solution for the static

behavior or arbitrary laminated TWCB and including all the non-classical effects is very difficult due to the complexities arising from the coupling effects of extensional, flexural and torsional deformations. D.K. Shin et.al [8] presented the development of exact stiffness matrix for TWCB with arbitrary lamination from the solution of spatially coupled ordinary differential equations that arise in the solution process. The exact stiffness matrix was used to derive closed form expressions to symmetric laminated TWCB with various boundary

• conditions. From the foregoing literature it is evident

that closed form analytical solutions for thin walled composite beams are very few. The

• bjective
• f

the present study is to systematically develop the exact solutions to TWCB incorporating arbitrary lamination sequence and all non-classical effects. The solution is based on a new method called coupled field formulation (CFE) introduced by S. Ramaprasad [9] for the analysis of planar laminated curved beams. The CFE method [9] is particularly attractive for deriving exact solutions for 1-D problems since it not only satisfies the governing equations of motion, but also satisfies all the boundary conditions of the problem. In this work, the TWCB problem is formulated in terms of 8 generalized beam forces and six rigid body displacements for a total of 14 field variables. Explicit simple closed form expressions for the displacements and forces are derived which are applicable even for a highly coupled lamination

• sequence. The simplicity of the solutions developed

enables any aircraft designer to explore the influence

• f different design parameters on the fully coupled

response of the TWCB. The solutions can be used to develop closed form expressions for any arbitrary cross section provided the corresponding stiffness’s are correctly derived from the geometry of the cross

• section. The developed solutions are validated by

studying the bending and torsional response of isotropic and laminated I-beams with NASTRAN results. 3 Theoretical Formulation A Thin walled beam is characterized as a flexible body whose length is much larger than its cross sectional dimensions. The kinematics of the thin walled beam is quite complex and is developed based on some simplifying assumptions [10] as mentioned below.

• 1. The contour of a cross section does not deform

in its own plane.

• 2. A general plate segment of the beam is

modeled as a thin plate but with shear

• deformation. Hence, transverse shear strains

and warping shear are introduced and assumed uniform over the cross section.

• 3. The tangential stress (“s”) is negligible and

stiffness of the plate is derived based on plane stress reduction from the 3D Constitutive equations.

• 4. Each plate element in a cross section is

governed by the First Order shear Deformation theory.

• 5. The Shear strain in the Cross sectional plane is

assumed to be zero. The beam coordinate system is shown in fig. 1 Fig.1. Beam Coordinate System The shell displacements are denoted by “v”, “w” and “u” respectively along the “s”,” n” and “x”

• directions. The beam displacements in the element

Coordinate system are denoted by U, V, and W along the beam axis and transverse to the beam axis

SLIDE 3

3 EXACT SOLUTION OF THIN WALLED OPEN SECTION BEAM USING A COUPLED FIELD FORMULATION

respectively. The rotations along the beam coordinate axes are represented by βx, βy, βz. The warping degree of freedom is introduced to capture the twist due to material couplings accurately. Based on the assumptions made, the kinematics of the beam, are derived [10] as (1) Where the rotations are given by (2) The displacements in (2) represent the mid surface displacements of the contour and the variation of the displacements along the thickness of the shell is represented using first order shear deformation

• theory.  is the sectorial coordinate or warping

function defined by



 ) (s r 

(3) The transverse shear deformation in the shells can be expressed in terms of the generalized beam shear strains , and warping shear

w

 as [8]

(4)

The displacements in (2) represent the mid surface

displacements of the contour and the variation of the displacements along the thickness of the shell is represented using first order shear deformation theory. (5)

Where , and are mid plane displacements and , are rotations of the shell mid plane about the x and s axis respectively. The shell rotations can be expressed in terms of beam rotations as

(6) and

s v

s

     (7)

The mid-plane strains and the associated curvatures can now be written as [10]

q n r nq n z n y

w xz xy nx sx w xz xy sx w y z x x

                                   ) cos( ) sin( ) sin( ) cos( ) ( )) cos( ( )) sin( (

(8)

Where the following membrane strains and bending curvatures are defined as

w x w w x sx w w z z y y x

x x x x x x U                                  ; ; ; ; (9) Constitutive equations for Shells

The normal lamina stress strain equations are applicable to TWCB as well. However the shell can be assumed to be in a state of either plane stress (

s

 =0) or be in a state of plane strain (

S

 =0) as

                                

nx sx x nx sx x

Q Q Q Q Q      

* 55 * 66 * 16 * 16 * 11 * * *

(10) For plane stress (

s

 =0) condition

22 2 26 66 * 66 22 26 12 16 * 16 22 2 12 11 * 11

; Q Q Q Q Q Q Q Q Q Q Q Q Q       (11a)

For plane strain (

S

 =0) condition we have

66 * 66 16 * 16 11 * 11

, , Q Q Q Q Q Q    (11b)

             

s r x W x V w s q x W x V v s z y x U u

x x w y z

                    sin cos cos sin ) (                               x x W x V

x w w

• xz

y

• xy

z

       ; ; ) ( cos sin ) ( sin cos s q s r

w xz xy nx w xz xy sx

                 

               

x s n x s w s n x w x s v s n x v x s n x s u s n x u

s x

, , , , , , , , , , ,        u v w

x



s



q

w y z x

         cos sin

0 , xz xy 



SLIDE 4

Variational Formulation The principle of minimum potential energy is used for the variational formulation to derive the stress resultants associated with beam

• kinematics. The total potential energy of the

system is calculated as

 

2 1          



V U dV V U

V nx nx sx sx x x

         (12) The variation of the strain energy and the potential energy yields respectively dx M T V V M M M N U

l sx t w xz z xy y w w y y z z x x



                         (13)

and

dx m m t W q V q U q V

l z z y y x x z y x

                        

(14) Where

z y x z y x

m m t q q q , , , , ,

are the distributed generalized loadings in the three translational and three rotational directions respectively. The stress resultants in (13) are defined as

             

       

             

A sx t A nx sx A nx sx z A nx sx y A x w A x y A x z A x x

dn ds n M dn ds q r T dn ds V dn ds V dn ds q n M dn ds n z M dn ds n y M dn ds N                   cos sin sin cos cos sin (15) Substituting equations (8) to (10) into (15) and integrating we get the stress resultants as

    

        



E R

• r

E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E M T V V M M M N

sx w xz xy w y z x t z y y z x

                                                                               

88 78 68 58 48 38 28 18 78 77 67 57 47 37 27 17 68 67 66 56 46 36 26 16 58 57 56 55 45 35 25 15 48 47 46 45 44 34 24 14 38 37 36 35 34 33 23 13 28 27 26 25 24 23 22 12 18 17 16 15 14 13 12 11

(16) The compliance matrix relating the beam strains with the beam stress resultants is obtained by inverting the [E} given by.

        

R C R E  

1



(17) Governing Equations By integrating the derivatives of the varied quantities by parts and collecting the coefficients

• f

w x z y

W V U        , , , , , ,

, the following governing equations and the associated essential and natural boundary conditions can be derived as

z z z y y y x x x

V W q x V V V q x V N U q x N : ; : ; : ;             (18)

z z y y z t x t

M m V x M T M t x T M : ; : ; ) (           

  

  M T M x M M m V x M

t y y z z y

: ; : ;         

The first six governing equations in (18) can be integrated exactly considering only concentrated forces to yield

SLIDE 5

5 EXACT SOLUTION OF THIN WALLED OPEN SECTION BEAM USING A COUPLED FIELD FORMULATION

5 2 4 1 3 2 1

; ; ; ; F x F M F x F M F T M F V F V F N

y z t z y x

        

(19) Since there are eight generalized forces and we have

• nly seven equations, we must augment the above

equilibrium equations with

• ne

compatibility

• equations. Using (9) the compatibility equation can

be written as:

w sx w

x x         

(20) Using the compliance relation (17), substituting in (20) and combining with the last of governing equations in (18), two coupled first order ordinary differential equation are obtained:

                                   x P P F M M C C C C M M x

t w t w 1 3 78 88 77 44

2 2 2

Where Po and P1 are functions of force variables and the compliance coefficients. The General Solution of equation (21) can be written as

78 88 77 44 1 1 3

2 4 2 2 2 2 2 C C C C where P x P F P e g e f M M

x x t w

                                                          



       

 

(21) Substituting equation (19) and (22) into (17) and integrating we derive the following equations for the displacement components.

x x y x x z x x x x x x

e g D e f C x B x A c e g D e f C x B x A b e g D e f C x B x B x c A e W e g D e f C x B x B x b A d V e g D e f C x B x A a U

         

 

    

                            

3 3 2 3 3 2 2 2 2 2 6 6 3 3 2 6 6 5 5 3 2 2 5 5 1 1 2 1 1

2 2 6 2 ) ( 6 2 ) ( 2

(22a)

x x x x x

e g D e f C x A Q e g D e f C x B x A h

    

 

 

        

7 7 7 4 4 2 4 4

2

(22b) In equation (22), the constants Ai, Bi, are functions

• f force variables Fo to F5 and the compliance

coefficients while Ci and Di are functions of compliance coefficients and the decay parameter  . 4 Results and Discussions Numerical investigations has been made to validate the present formulation and also to assess the influence of non-classical structural effects such as warping restraint and transverse shear deformation

• n the response of TWCB under bending and

twisting loads. The I-beams tested by Chandra and Chopra [2] are considered for present study. The material properties of the beam are given in table 1.

Properties Values E11, psi (GPa) 20.59E+06 (141.9) E22, psi (GPa) 1.42E+06 (9.78) G12=G13, psi (GPa) 0.89E+06 (6.13) G23, psi (GPa) 0.696E+06 (4.80) 0.42 Material Properties of AS4/3501-6 graphite epoxy lamina

12

 Table1: Material properties of AS4/3501-6 graphite epoxy lamina

The beam has a length of 762 mm with 25.4 mm x 12.7 mm section and is detailed in [7].The beam is clamped at its root and restrained from warping from both ends. The bending-torsion coupled I-beam is symmetric with respect to the beam axis. The lay-up

• f

the top and bottom flanges are [(0/90)2/(90/0)/and the lay-up of web is [0/90]2s.

Fig.1 Comparison of bending slopes for bending- torsion coupled beam under tip shear load

SLIDE 6

• Fig. 2 Bending slope variation with fiber angles for

bending-torsion coupled beam under tip shear load

• Fig. 3 Comparison of twist angles for bending-torsion

coupled beam under tip torque

• Fig. 1 shows the bending slope distribution for the

bending torsion coupled I –beam (under a unit tip shear load. The plot shows the comparison

• f the bending slope from the current theoretical

prediction with those obtained using NASTRAN

• results. The NASTRAN results were obtained by

modeling the I-section beam with 2D CQUAD shell

• elements. It can be seen that the prediction from the

current formulation agrees very well with NASTRAN results. In fig. 3 comparison is made for the variation of the bending slope with fiber

• rientation angles and again the correlation with

NASTRAN results is very good.

• Fig. 3 presents the twist distribution along the beam

length (subjected to a unit tip torque load. The correlation with NASTRAN 2D results is excellent as is seen from fig.3

• 5. Conclusions

The exact solutions of a shear flexible TWCB of

• pen section is formulated using a coupled field
• formulation. The developed solution accommodates

all the non-classical effects associated with the bending and twisting behavior of a TWCB. The developed solutions are generic and can handle arbitrary lamination sequence and different boundary conditions. The formulation is validated by comparing the bending and twisting response with available results and NASTRAN.

• 6. References

[1] S. N. Jung, V. T. Nagaraj, and I. Chopra “Assessment of Composite Rotor Blade Modeling Techniques”. Vol. 44, (3), Journal of American Helicopter Society, pp 188-205, 1999. [2] R. Chandra, and I. Chopra “Experimental and Theoretical Analysis of Composite I Beams with elastic couplings”. AIAA Journal, Vol. 29, No. 12, pp 2197-2206, 1991. [3] S. N. Jung, V. T. Nagaraj, and I. Chopra “Assessment of Composite Rotor Blade Modeling techniques”, Vol. 44, (3), Journal of American Helicopter Society, pp 188-205, 1999. [4] J. Lee, and S. Kim, “Flexural Torsional Buckling of Thin Walled I section Composites”. Vol. 79, No. 2, Computers and Structures, pp 987-995, 2001. [5] Jaehong Lee, “Flexural Analysis of Thin Walled Composite Beams using a Shear Deformable Theory”, Vol. 70, Composite Structures, pp 212-222, 2005. [6] Mira Mitra, S. Gopalakrishnan, and M. Seetharama Bhat, “A New Super Convergent Thin Walled Composite Beam Element for Analysis of Box Beam Structures”. Vol 41, International Journal of Solids and Structures, pp 1491-205, 1518, 2004. [7] S. N. Jung, and J. Y. Lee, “Closed Form Analysis of Thin Walled Composite I-Beams considering non- classical Effects”, Vol. 60, Composite Structures, pp. 9-17, 2003. [8] D. K. Shin, Nam -II- Kim, and Moon-Young Kim “Exact Stiffness Matrix

• f

Mono-symmetric Composite I-Beams with Arbitrary Lamination”.

• Vol. 79, Composite Structures, pp 467-480, 2007.

[9] S. Ramaprasad “Exact Solution of Shear Flexible Laminated Curved Beams Using a Coupled Field Formulation”. International Conference

• n

Composites for 21st Century: Current and Future Trends”, ICC-CFT2011, Jan 4-7, IISC, Bangalore, India,, 2011.