An elasticity approach for simply-supported isotropic and orthotropic stiffened plates

International Journal of Mechanical Sciences 89 (2014) 21–30

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

An elasticity approach for simply-supported isotropic and orthotropic stiffened plates K. Bhaskar n, Anup Pydah Aerospace Engineering Department, Indian Institute of Technology, Madras, Chennai 600 036, India

art ic l e i nf o

a b s t r a c t

Article history: Received 2 August 2013 Received in revised form 4 August 2014 Accepted 13 August 2014 Available online 26 August 2014

Static deflections and natural frequencies of vibrations are obtained for blade-stiffened plates using a three-dimensional model for the plate and a two-dimensional (plane stress) model for the stiffener with simply supported edge/end conditions. These are used as a benchmark for assessing the approach based on the classical hairbrush hypothesis. Results obtained by using the rigorous elasticity model for the plate alone or the stiffener alone are also presented. These results indicate the greater importance of non-classical effects in the analysis of stiffened plates as compared to unstiffened plates. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Stiffened plates Elasticity solution Shear deformation

1. Introduction The widespread use of stiffened plates as primary load bearing structures is testament to their inherent structural efficiency and to the advent of modern state-of-the-art manufacturing and materials technology. Composite stiffened plates, in particular, provide designers with the opportunity to tailor strength, stiffness and other properties as per structural requirements. This has led to prolific use of such structures in the aerospace, civil, automobile, naval and other high performance industries. The analysis of stiffened plates can be carried out by different approaches. The orthotropic plate “smeared-out” idealization replaces the plate–stiffener system with an equivalent homogeneous plate with orthotropic properties [1]. While this idealization simplifies the analysis to a great extent, it only provides accurate results when the stiffeners are of uniform size, are closely spaced and their rigidities do not dominate the plate rigidity. Furthermore, this smearing-out of stiffener properties leads to a loss in the discrete nature of the plate–stiffener system and hence in capturing the influence of different geometric parameters on its response. The plate–beam discrete idealization involves isolating the plate from the beam, modelling them using various simplified theories (usually based on Kirchhoff–Love hypothesis) and maintaining compatibility at the interfaces. However, because of the difficulty

n

Corresponding author. Tel.: þ 91 44 22574010; fax: þ 91 44 22574002. E-mail addresses: [email protected] (K. Bhaskar), [email protected] (A. Pydah). http://dx.doi.org/10.1016/j.ijmecsci.2014.08.013 0020-7403/& 2014 Elsevier Ltd. All rights reserved.

in developing general closed-form analytical solutions for this system, increased emphasis was laid on the development of various computer based approximate and numerical schemes using energy principles [2,3], the constraint method [4], BEM and FEM [5–7]. For composite stiffened plates, the effect of shear deformation on the plate behaviour cannot be neglected. This is because the shear stiffness of such materials is small compared to their bending and membrane stiffness ðEL =GLT ¼ 10  50Þ unlike for metals where these stiffnesses are comparable ðE=G ¼ 2ð1 þ νÞ⋍2:6Þ. With regard to the unstiffened plate, Pagano [8] and Srinivas et al. [9] formulated 3D elasticity solutions for laminates for capturing the shear deformation completely. As an alternative, shear deformation effects may also be accounted for in a 2-D formulation by assuming appropriate displacement fields, as has been illustrated by Carrera [10]. Deb et al. [11] developed an approximate shear deformation theory for stiffened plates based on the Reissner–Mindlin plate theory and Timoshenko beam theory and the smeared-out idealization. Mukherjee et al. [12], Sadek et al. [13], Biswal et al. [14] and Ghosh et al. [15] presented finite elements based on a higher order shear deformation theory (HSDT) for static and vibrational analysis of laminated stiffened plates. Bhar et al. [16] carried out a comparison of the finite element results of composite stiffened plates based on first order shear deformation theory (FSDT) and HSDT. They strongly advocate the use of HSDT over the Classical Plate Theory and even FSDT specially when the panels become thick. Sapountzakis et al. [17] presented an optimized model based on the classical approach, which accounts for the inplane forces and displacements at the interface of the plate and the beam. By comparing their results with a number of finite element models, they bring out the importance of considering the inplane shear

22

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

forces for a more accurate description of the behaviour of the stiffened plate. Qing et al. [18] developed a 3D solution for the free vibrations of stiffened plates based on the variational approach, which uses finite elements to solve state vector equations. The model automatically considered transverse shear deformations and rotary inertia. In this context, the aim of the current work is to present an analytical elasticity solution for blade stiffened plates wherein the plate is modelled as a 3D solid and the stiffener as a plane stress problem, so that non-classical effects such as transverse shear deformation and rotary inertia are automatically accounted for; all the plate edges as well as the ends of the stiffeners are taken to be simply supported. By comparing this elasticity solution with various approximate models, an attempt will be made to quantify the individual contributions of the plate and beam to the total shear deformation of the structure for different geometric parameters and material properties. Both static deflections and free vibrational frequencies will be studied.

2. Formulation Consider a rectangular plate of sides a, b and uniform thickness h (Fig. 1) with simply supported edges. The plate is integrally stiffened by a single eccentric rectangular beam of height H and breadth B, attached to only one side, say, the bottom surface of the plate along the central line y¼b/2. The ends of the stiffener are also taken to be simply supported. The elasticity solution is formulated by isolating the stiffener from the plate and taking into account the continuity conditions at the interface. The plate is modelled using the equations of 3D elasticity while the stiffener is modelled using a plane stress formulation. With regard to the interface tractions, it is assumed that they remain constant over the width B of the stiffener. In the present work, attention is focused on flexure of the plate symmetrically about the stiffener, and hence the torsional behaviour of the stiffener does not come into picture. 2.1. Analysis of the stiffener Assuming that the stiffener is specially orthotropic with respect to the x–z coordinates with the plane stress constitutive law 8 9 2 9 38 ϵ = Q 11 Q 12 0 > > < σx > = < x > 0 7 σz ¼ 6 4 Q 12 Q 22 5 ϵz > > :τ > ; :γ > ; 0 0 Q xz

55

xz

the equations of motion can be written in terms of the displacements u and w along the x and z directions respectively, as Q 11 u;xx þ Q 55 u;zz þ ðQ 12 þ Q 55 Þw;xz ¼ ρu;tt ðQ 12 þ Q 55 Þu;xz þ Q 55 w;xx ; þ Q 22 w;zz ¼ ρw;tt

ð1Þ

where ρ is the mass density of the stiffener. Selection of displacement functions  mπ  1 x eiωt uðx; z; tÞ ¼ ∑ UðzÞ cos a m¼1 1

wðx; z; tÞ ¼ ∑ WðzÞ sin m¼1

mπ  x eiωt a

where ω is the natural frequency ensures that the shear diaphragm type simple support conditions at x ¼ 0; a;

w ¼ 0;

σx ¼ 0

are satisfied a priori. Substitution of the above displacement functions into (1) reduces them to a 4th order system of linear ordinary differential equations in z. Following the standard procedure of seeking solutions for U(z) and W(z) as )   ( U0 U ¼ esz W0 W one gets the auxiliary equation as A0 s4 þ B0 s2 þ C 0 ¼ 0

ð2Þ

where A0 ¼ Q 22 Q 55 B0 ¼ p2 Q 212  p2 Q 11 Q 22 þ 2p2 Q 12 Q 55 þ Q 22 ρω2 þ Q 55 ρω2 C 0 ¼ p4 Q 11 Q 55  p2 Q 11 ρω2  p2 Q 55 ρω2 þ ρ2 ω4 and p ¼ mπ =a The nature and multiplicity of the roots of (2) depend on the material properties and the assumed initial value of ω and this dictates the final solution. For example, in the case of real and distinct roots, the final solution is of the form: " # !(  )   1 4 cos maπ x C 1i s z u mπ  eiωt ¼ ∑ ∑ ð3Þ ei sin a x w m ¼ 1 i ¼ 1 C 2i Of the 8 constants C1i, C2i (for each harmonic m), only 4 are independent. The inter-relationships are established by substituting

Fig. 1. Eccentrically stiffened plate.

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

the above displacement field in the ordinary differential equations and equating the coefficients of esi z (i¼1–4) in each equation to 0. The stiffener is subjected to the following interface tractions: 1. A transverse normal traction Q int ðx; tÞ assumed to be constant across the breadth B and taken as  mπ  1 x eiωt ð4Þ Q int ¼ ∑ Q m sin a m¼1 2. An in-plane shear traction Sint ðx; tÞ assumed to be constant across the breadth B and taken as mπ  1 x eiωt ð5Þ Sint ¼ ∑ Sm cos a m¼1

These are as shown in Fig. 2 when the stiffener bends symmetrically about mid-span. The 4 independent constants of (3) can be determined in terms of the unknowns Qm and Sm by enforcing the conditions: 1. At the top surface z ¼  H=2,

σ z ðx; tÞ ¼  Q int ðx; tÞ and τxz ðx; tÞ ¼ Sint ðx; tÞ

23

where ρ is the mass density of the plate material, taken to be the same as that of the stiffener. The displacement functions mπ  nπ  1 1 x sin y eiωt uðx; y; z; tÞ ¼ ∑ ∑ UðzÞ cos a b m¼1n¼1 1

vðx; y; z; tÞ ¼ ∑

1

∑ VðzÞ sin

m¼1n¼1 1

wðx; y; z; tÞ ¼ ∑

1

 mπ  nπ  x cos y eiωt a b

∑ WðzÞ sin

m¼1n¼1

 mπ  nπ  x sin y eiωt a b

satisfy the shear diaphragm conditions at x ¼ 0; a;

w ¼ 0;

v ¼ 0;

σx ¼ 0

at y ¼ 0; b;

w ¼ 0;

u ¼ 0;

σy ¼ 0

a priori, and reduce the problem to a 6th order system of linear ordinary differential equations. Proceeding as done for the stiffener by starting with 9 8 9 8 > < U0 > = = > V ¼ V 0 esz > > > ; : ; :W > W 0 one gets the auxiliary equation as

2. At the bottom surface z ¼ H=2,

As6 þ Bs4 þCs2 þD ¼ 0

σ z ðx; tÞ ¼ 0 and τxz ðx; tÞ ¼ 0

2.2. 3D elasticity formulation for the plate The simply supported plate of dimensions a  b  h is taken to be orthotropic with respect to the x–y–z axes and governed by the following constitutive law: 9 2 9 8 38 ϵx > σx > C 11 C 12 C 13 0 0 0 > > > > > > > > > > 6 7 > > > > > ϵx > σy > 0 0 0 7> > > > > 6 C 12 C 22 C 23 > > > > > > > > 6 7 > = 6 C 13 C 23 C 33 = < σz > < ϵx > 7> 0 0 0 6 7 ¼6 7 γ τ 0 0 0 0 0 C > > > > 6 7 yz yz 44 > 6 > > > > > > 7> > > > > > > > 6 0 7> γ τ > > > > 0 0 0 0 C xz xz 55 > 4 > > > 5> > > > > > ; ; : τxy > : γ xy > 0 0 0 0 0 C 66

ð7Þ

where A, B, C and D are functions of m, n, the material properties Cij and ω. For the case of real and distinct roots, the final solution is of the form    9 8 9 0 2 3 18 cos maπ x sin nbπ y > > K 1i > > = < =    > 1 1 6 B 6 7 C v ¼ ∑ ∑ @ ∑ 4 K 2i 5esi z A sin maπ x cos nbπ y eiωt     > > > > > > : ; m¼1n¼1 i¼1 K mπ nπ ; : sin a x sin b y w 3i ð8Þ

C 11 u;xx þ C 66 u;yy þ C 55 u;zz þ ðC 12 þ C 66 Þv;xy þ ðC 13 þ C 55 Þw;xz ¼ ρu;tt

wherein only 6 constants (for each harmonic set fm; ng) are independent. Solution forms for the other cases have been discussed in Pagano [8] and Srinivas and Rao [9]. For the purpose of enforcing lateral surface conditions, the interface tractions Q int and Sint acting over the strip 0 r x r a and ðb=2  B=2Þ r y rðb=2 þ B=2Þ need to be expressed in a double series as mπ  nπ  1 1 Q int ¼ ∑ ∑ Q mn sin x sin y eiωt ð9Þ a b m¼1n¼1

C 66 v;xx þ C 22 v;yy þ C 44 v;zz þ ðC 12 þ C 66 Þu;xy þ ðC 23 þ C 44 Þw;yz ¼ ρv;tt

where

The equations of motion in terms of the displacements u, v and w along the x, y and z directions respectively are

C 55 w;xx þ C 44 w;yy þC 33 w;zz þ ðC 13 þ C 55 Þu;xz þ ðC 23 þ C 44 Þv;yz ¼ ρw;tt ð6Þ

Q mn ¼

2 b

Z

ðb=2 þ B=2Þ

ðb=2  B=2Þ

Fig. 2. Interface tractions on the stiffener.

Q m sin

nπ  y dy b

24

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

3. Simpler models based on classical assumptions

and  mπ  nπ  x sin y eiωt Sint ¼ ∑ ∑ Smn cos a b m¼1n¼1 1

1

ð10Þ

where Smn ¼

2 b

Z

ðb=2 þ B=2Þ

ðb=2  B=2Þ

Sm sin

nπ  y dy b

The lateral boundary conditions are (Fig. 3)

As stated earlier, the purpose of this work is to verify the accuracy of the simple classical approach by comparison with the elasticity solution. The kinematic assumptions of the classical approach may be employed to model the plate alone, the stiffener alone, or both. While doing so, it is important to shift the interface tractions to the centre line of the stiffener or the mid-plane of the plate, as described below with respect to the eccentrically stiffened plate of Fig. 1.

1. At the top surface (z ¼  h=2),

σ z ðx; y; tÞ ¼ 0;

τxz ðx; y; tÞ ¼ 0 and τyz ðx; y; tÞ ¼ 0

2. At the bottom surface (z ¼ h=2),

σ z ðx; y; tÞ ¼ Q int ;

τxz ðx; y; tÞ ¼ Sint and τyz ðx; y; tÞ ¼ 0

and these yield the 6 unknown constants in terms of Qm and Sm.

3.1. Analysis of the stiffener using the classical beam theory The interface tractions of Fig. 2, when shifted to the centre line of the beam, give rise to an additional distributed moment per unit length as shown in Fig. 5. Rewriting all the generalized forces as loads per unit length of the beam as required by the onedimensional beam theory, one has qðx; tÞ ¼ Q int ðx; tÞ  B;

sðx; tÞ ¼ Sint ðx; tÞ  B

and ms ðx; tÞ ¼

2.3. Analysis of the plate–stiffener combination With the continuity of interface tractions already satisfied as explained above, the only step left is to enforce the displacement continuity along the interface patch. In the present analysis with the stiffener modelled as a plane stress problem where both the stresses and the displacements are taken to be uniform over the width B, it is not possible to enforce interface displacement continuity over the entire width. Instead, as an approximation shown later to be reasonably valid for thin stiffeners, continuity for u and w displacements is enforced only along y¼b/2, the centre line of the stiffener. When this is done and the corresponding Fourier terms are compared, one gets a set of homogeneous linear equations in Qm and Sm leading to an eigenvalue problem yielding the natural frequencies ω. If the stiffened plate were to be loaded statically by a given transverse loading, the above formulation is still valid except that the time dependent terms are absent and the specified load is accounted for while enforcing the lateral boundary conditions. In such a case, the interface displacement continuity conditions yield Qm and Sm in terms of the transverse load. If the plate were to be stiffened by a concentric stiffener (Fig. 4), the symmetry of deformation about the mid-plane of the plate may be accounted for a priori while modelling it as a 3D solid. Thereafter, the analysis is similar to the above.

H  sðx; tÞ 2

The net effect of these can be captured by starting with a displacement field including a centre line displacement u0 as wðx; z; tÞ ¼ wðx; tÞ uðx; z; tÞ ¼ u0 ðx; tÞ  z  w;x The corresponding equations of motion are N;x ¼ s

ð11Þ

V;x  ρ  H  B  w;tt ¼  q

ð12Þ

M;x þ ms ¼ V

ð13Þ

where axial and rotary inertia terms are neglected and N is the axial force resultant.

Fig. 4. Concentric stiffener configuration.

Fig. 3. Interface tractions on the plate (bottom view).

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

25

Fig. 5. Shifting of interface tractions to the mid-line of the beam.

The corresponding equations of motion are

Eliminating V from (12) and (13) yields M;xx þ ms ;x þq ¼ ρ  H  B  w;tt

Nx ;x þ N xy;y þ Sint ¼ 0

ð15Þ

Expressing the beam constitutive law in terms of Young's modulus in the x direction as

Nxy ;x þ Ny;y ¼ 0

ð16Þ

σ x ¼ Ex ϵx ¼ Ex ðu0 ;x  zw;xx Þ

M x ;x þ M xy;y þmp ¼ Q x

ð17Þ

and obtaining the generalized force–displacement relations by appropriate integrations over the cross-sectional area A, the governing equations can be derived from the equilibrium equations as

M xy ;x þ M y;y ¼ Q y

ð18Þ

Q x ;x þ Q y;y  Q int  ρ  h  w;tt ¼ 0

ð19Þ

ð14Þ

Ex  A  u0 ;xx ¼ s Ex  I yy w;xxxx þ ρ  H  B  w;tt ¼ ms ;x þ q where Iyy is the area moment of inertia of the cross-section about its centroidal axis. For Q int ðx; tÞ and Sint ðx; tÞ expressed in Fourier series as given earlier (see Eqs. (4) and (5)), and with the shear diaphragm boundary conditions at the ends now expressed as at

x ¼ 0; a;

w¼M¼N¼0

i:e: w ¼ w;xx ¼ u0 ;x ¼ 0; a solution can be sought as mπ  1 x eiωt wðx; tÞ ¼ ∑ W m sin a m¼1 1

uðx; tÞ ¼ ∑ U m cos m¼1

 mπ  x eiωt a

Substitution of this in the governing equations yields Um and Wm in terms of Qm and Sm.

wherein axial and rotary inertia terms are neglected and N, M and Q are the plate stress resultants. Eliminating Qx and Qy from (19) using (17) and (18), we have M x ;xx þ 2M xy ;xy þ M y ;yy  ρ  h  w;tt ¼ Q int  mp ;x

Using the plane stress reduced constitutive law for the plate given by 8 9 2 38 ϵ 9 σ = 0 > Q 11 Q 12 > < x> < x > = 6 7 σ y ¼ 4 Q 12 Q 22 0 5 ϵy > > > > :τ ; 0 0 Q 66 : γ xy ; xy and obtaining the generalized force displacement relations by appropriate integration over the thickness of the plate, the governing equations can be derived as A11 u0 ;xx þ A12 v0 ;xy þ A66 ðu0 ;yy þ v0 ;xy Þ ¼  Sint A66 ðu0 ;xy þ v0 ;xx Þ þ A12 u0 ;xy þ A22 v0 ;yy ¼ 0 D11 w;xxxx þD22 w;yyyy þ ð2D12 þ 4D66 Þw;xxyy þ ρ:h:w;tt ¼ Q int þ mp ;x where ½A and ½D are symmetric matrices defined as Z h=2 ½Q ð1; z2 Þ dz ½A; ½D ¼  h=2

3.2. Analysis of the plate using the classical plate theory As in the case of the stiffener, shifting the interface tractions to the mid-plane of the plate results in an additional distributed moment mp per unit area, acting over the area a  B, besides Q int ðx; y; tÞ and Sint ðx; y; tÞ as given by Eqs. (9) and (10). Hence, one has  mπ  nπ  h h 1 1 ∑ ∑ Smn cos x sin y eiωt mp ðx; y; tÞ ¼  Sint ðx; y; tÞ ¼ 2 2m¼1n¼1 a b

A solution of the form 1

wðx; y; tÞ ¼ ∑

1

u0 ðx; y; tÞ ¼ ∑

where u0, v0 and w are mid-plane displacements is sufficient to capture the effect of the loading system.

1

∑ U mn cos

m¼1n¼1 1

v0 ðx; y; tÞ ¼ ∑

1

∑ V mn sin

m¼1n¼1

 mπ  nπ  x sin y eiωt a b  mπ  nπ  x sin y eiωt a b

 mπ  nπ  x cos y eiωt a b

ensures that the shear diaphragm type simple support conditions: at

x ¼ 0; a;

w¼0

and

v¼0

at

y ¼ 0; b;

w¼0

and

u¼0

uðx; y; z; tÞ ¼ u0 ðx; y; tÞ  z  w;x vðx; y; z; tÞ ¼ v0 ðx; y; tÞ  z  w;y

1

∑ W mn sin

m¼1n¼1

A displacement field given by wðx; y; z; tÞ ¼ wðx; y; tÞ

ð20Þ

are satisfied a priori. Substitution of the displacements in the governing equations yields Umn, Vmn and Wmn in terms of Qm and Sm.

26

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

Subsequent steps involving enforcement of displacement continuity along the beam–plate interface to determine Qm and Sm are similar to the ones explained earlier.

4. Results and discussion Before proceeding further with numerical studies, it is appropriate to verify whether the plane stress idealization of the stiffener is acceptable, along with the associated restrictions that the interface tractions be taken as constant along its width and that the interface displacement continuity be imposed only along the centre line y¼ b/2. For this purpose, a finite element study is conducted wherein both the plate and the stiffener are discretized using 20 noded solid elements; ANSYS is used with Solid 186 elements. A mesh involving a total of 21 000 elements with 6 elements across the plate and stiffener depths and 6 elements across the stiffener width is found to yield convergent results. The problems considered are those of two eccentrically stiffened, simple supported square plates, one made up of an isotropic material with ν ¼ 0.3 and the other, a unidirectional fibre composite with the fibres along the x direction and with properties given by EL =ET ¼ 25, GLT =ET ¼ 0:5, GTT =ET ¼ 0:2 and νLT ¼ νTT ¼ 0:25. The height of the stiffeners is chosen such that the central deflection of the stiffened plates is nearly half of that of the corresponding unstiffened plate. In all cases, the breadth B of the stiffener is taken to be equal to height h of the plate. The plates are subjected to a uniform load q0 distributed equally between the top and bottom surfaces. The maximum deflection and bending stress results of this finite element study are compared with those of the present elasticity solution in Tables 1 and 2 wherein each model is named after the idealization used to characterize the plate and beam respectively, with PS denoting plane stress. As can be seen, both the maximum deflections and maximum inplane stresses are predicted accurately, indicating that the plane stress idealization is acceptable even for fairly wide stiffeners with B=H ¼ 1=0:6 C 1:7. Fig 6 shows a comparison of 2 the variation of the non-dimensionalized inplane stress σ x h =q0 a2 and transverse shear τxz h=q0 a, through the thickness of the eccentrically stiffened orthotropic plate. Clearly, the present analytical solution is able to precisely predict the stress fields even for thick orthotropic stiffened plates. Static and dynamic results derived from the elasticity solution are now presented for isotropic and orthotropic square plates of

various thicknesses. These are compared to the three other approximate models described earlier. Each model is named after the kinematic assumption used to characterize the plate and the beam respectively (Classical Plate Theory (CPT), 3-Dimensional theory for the plate (3D), Euler–Bernoulli beam theory (EB) and plane stress assumption for the beam (PS)). The following cases are considered: 1. For static analysis (uniform load applied equally at the top and bottom surfaces): (a) Unstiffened isotropic and orthotropic plates (Table 3). (b) Isotropic plate with a single central concentric isotropic stiffener (Table 4). (c) Isotropic plate with a single central eccentric isotropic stiffener (Tables 5 and 10). (d) Isotropic plate with a central orthogonal system of concentric isotropic stiffeners (Table 6). (e) Orthotropic plate with a single, central concentric orthotropic stiffener running parallel to the direction of the fibres of the plate (Table 7). (f) Orthotropic plate with a single, central concentric orthotropic stiffener running perpendicular to the direction of the fibres of the plate (Table 8). (g) Orthotropic plate with a single central eccentric orthotropic stiffener running parallel to the direction of the fibres of the plate (Tables 9 and 11). 2. For dynamic analysis: (a) Unstiffened isotropic and orthotropic plates (Tables 12 and 13). (b) Isotropic plate with a single central concentric isotropic stiffener (Table 14). (c) Orthotropic plate with a single, central concentric orthotropic stiffener running parallel to the direction of the fibres of the plate (Table 15). The first 5 modes which are symmetric about the stiffener are considered. In each case, the heights of the stiffeners are adjusted so as to bring down the central deflection of the stiffened plate to half of its unstiffened counterpart when subjected to uniform load; this is done using the results derived from the classical formulation. The breadth B of the stiffener is always taken to be equal to the

Table 1 Single central eccentric stiffener – isotropic. a/h

10 20 a

H/h

1.4 1.8

σ x ða=2; a=2; H=2Þ  a 2 q0 h

3

wcentre E h  103 q0 a4

σ y ða=2; a=4; h=2Þ  a 2 q0 h

3D–PS

3D–3D

% Errora

3D–PS

3D–3D

% Error

3D–PS

3D–3D

% Error

25.70 24.08

25.13 23.87

2.3 0.9

0.323 0.430

0.320 0.428

0.9 0.5

0.172 0.169

0.175 0.171

 1.7  1.2

Error of 3D–PS with respect to 3D–3D.

Table 2 Single central eccentric stiffener – orthotropic. a/h

10 20

H/h

0.6 0.9

σ x ða=2; a=2;  h=2Þ  a 2 q0 h

3

wcentre ET h  104 q0 a4

10σ y ða=2; a=4;  h=2Þ  a 2 q0 h

3D–PS

3D–3D

% Error

3D–PS

3D–3D

% Error

3D–PS

3D–3D

% Error

67.20 45.47

65.60 44.96

2.4 1.1

0.711 0.867

0.716 0.871

 0.7  0.5

 0.564  0.515

 0.559  0.514

0.9 0.2

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

27

Fig. 6. Stress fields for an eccentrically stiffened orthotropic plate with a/h¼ 10 and H/h ¼0.6.

Table 3 Unstiffened plate – statics. a/h

Table 5 Single central eccentric stiffener.

Isotropic plate

Orthotropic plate

3

a

H/h

3

wcentre E h  103 q0 a4

10 20 35 50 100

a/h

wcentre ET h  104 q0 a4

CPT

3D

% Errora

CPT

3D

% Errora

44.35 44.36 44.36 44.36 44.36

46.39 44.87 44.53 44.44 44.38

 4.4  1.1  0.4  0.2  0.1

64.90 64.95 64.95 64.95 64.95

94.70 72.51 67.43 66.17 65.26

 31.5  10.4  3.7  1.8  0.5

10 20 35 50 100

1.4 1.8 2.3 2.6 3.3

3

wcentre E h  103 q0 a4 CPT–EB

3D–EB

CPT–PS

3D–PS

% Error

22.61 22.90 21.97 22.22 22.66

24.05 23.32 22.13 22.29 22.68

24.27 23.66 22.41 22.50 22.78

25.70 24.08 22.56 22.57 22.79

 12.0  4.9  2.6  1.6  0.6

Error of CPT with respect to 3D. Table 6 Orthogonal central concentric stiffeners.

Table 4 Single central concentric stiffener. a/h

H/h

10 20 35 50 100 a b

0.9 1.25 1.6 1.85 2.45

a/h

3

H/h

wcentre E h  103 q0 a4

3

wcentre E h  103 q0 a4 CPT–EB

3D–EB

Diffa

CPT–PS

3D–PS

Diffa

% Errorb

22.42 22.38 22.39 22.55 22.62

24.22 22.89 22.58 22.65 22.64

1.81 0.51 0.19 0.09 0.02

23.32 22.81 22.63 22.71 22.69

25.10 23.33 22.81 22.80 22.71

1.78 0.52 0.19 0.09 0.02

 10.7  4.1  1.8  1.1  0.4

Estimates of shear deflections of the plate alone. Error of CPT–EB with respect to 3D–PS.

10 20 35 50 100

1. The shear deflection of the plate alone can be calculated as the difference between 3D–EB and CPT–EB results or between 3D–PS

3D–EB

CPT–PS

3D–PS

% Error

21.98 21.44 22.16 22.34 23.57

23.66 21.90 22.33 22.42 23.59

22.41 21.66 22.28 22.41 23.61

24.07 22.13 22.45 22.49 23.62

 8.7  3.1  1.3  0.7  0.2

Table 7 Single central concentric stiffener (along the fibres). a/h

thickness h of the plate. For isotropic plates, ν is taken to be 0.3, while for orthotropic plates, the properties are taken as given earlier for the case of Table 2. From these results, the following observations may be noted:

0.65 0.95 1.2 1.4 1.8

CPT–EB

10 20 35 50 100

H/h

0.35 0.5 0.75 0.9 1.3

3

wcentre ET h  104 q0 a4 CPT–EB

3D–EB

Diff

CPT–PS

3D–PS

Diff

% Error

34.94 36.53 33.62 33.56 32.16

61.63 44.43 36.57 35.02 32.48

26.69 7.91 2.95 1.46 0.32

38.09 38.24 34.96 34.53 32.68

64.60 45.98 37.83 35.96 32.99

26.51 7.75 2.87 1.43 0.32

 45.9  20.6  11.1  6.7  2.5

28

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

and CPT–PS results. It is interesting to see that these two estimates are more or less identical for the different cases considered in Tables 4–9. (They are explicitly shown in columns 5 and 8 of Table 8 Single central concentric stiffener (perpendicular to the fibres). a/h

H/h

10 20 35 50 100

3

wcentre ET h  104 q0 a4

0.45 0.7 0.9 1.1 1.5

CPT–EB

3D–EB

CPT–PS

3D–PS

% Error

33.90 32.31 33.49 32.46 32.60

64.15 43.44 38.23 35.03 33.21

38.84 35.61 35.36 33.87 33.28

66.78 45.64 39.75 36.29 33.87

 49.2  29.2  15.7  10.6  3.7

Table 9 Single central eccentric stiffener (along the fibres). a/h

H/h

10 20 35 50 100

0.6 0.9 1.2 1.4 1.8

3

wcentre ET h 4 n10 q0 a4 CPT–EB

3D–EB

CPT–PS

3D–PS

% Error

36.77 34.90 33.37 32.99 33.65

60.70 41.58 35.80 34.18 33.91

42.88 38.97 36.04 34.89 34.51

67.20 45.47 38.36 36.04 34.77

 45.3  23.3  13.0  8.5  3.2

Tables 4 and 7.) This shows that the plate shear deflection can be captured accurately by using the rigorous three-dimensional model for the plate alone and this is unaffected by the choice of the stiffener model. The above conclusion also implies that the stiffener shear deflection alone can be captured as the difference between either CPT–PS and CPT–EB or between 3D–PS and 3D–EB results, and thus depends only on the accurate modelling of the stiffener. 2. For all the cases of Tables 4–9, the total shear deflection of the stiffened plate is given by the difference between the 3D–PS and CPT–EB results. As a consequence of observation 1, it is clear that this total shear deflection can be decomposed as the sum of the plate shear deflection alone and the stiffener shear deflection alone. Thus, if the results of the three simpler models CPT–EB, CPT–PS and 3D–EB are available, one can obtain those of the most rigorous 3D–PS model without actually carrying out the corresponding tedious analysis. 3. Considering the relative magnitudes of the plate shear deflection and the stiffener shear deflection for the cases of Tables 4– 9, one can see that the former is more significant for thicker plates (low a/h) and the latter for thinner plates (high a/h). 4. The error of CPT–EB with respect to 3D–PS yields the total shear deformation effect and is shown in the last columns of Tables 4–9.Comparing these with the CPT errors of the corresponding unstiffened plates (Table 3), it is immediately seen that the shear deformation effect is more significant for stiffened plates irrespective of the type of stiffening. The effect is always more pronounced for orthotropic plates than for isotropic plates of the same thickness ratio as expected.

Table 10 Single central eccentric stiffener – isotropic. a/h

H/h

10

1.4

50

a

σ x ða=2; a=2; H=2Þ  a 2 q0 h

σ y ða=2; a=4; h=2Þ  2 q0 ha

τxy ð0; 0;  h=2Þ a 2 q0 h

τxz ð0; a=2; h=2Þ a  q0 h

CPT–EB 3D–EB CPT–PS 3D–PS % Errora

0.342 0.345 0.319 0.323 5.7

0.165 0.167 0.170 0.172  3.9

0.122 0.124 0.128 0.130  6.3

– 1.014 0.924 0.930 –

CPT–EB 3D–EB CPT–PS 3D–PS % Error

0.610 0.610 0.604 0.604 1.0

0.163 0.164 0.164 0.165  0.7

0.121 0.121 0.122 0.122  0.9

– 3.848 3.225 3.778 –

2.6

Error of CPT–EB with respect to 3D–PS.

Table 11 Single central eccentric stiffener (along the fibres). a/h

H/h

10

0.6

50

σ x ða=2; a=2; H=2Þ  a 2 q0 h

10σ y ða=2; a=4;  h=2Þ  a 2 q0 h

10τxy ð0; 0;  h=2Þ  a 2 q0 h

τxz ð0; a=2; h=2Þ a  q0 h

CPT–EB 3D–EB CPT–PS 3D–PS % Error

0.778 0.891 0.586 0.711 9.3

 0.503  0.574  0.484  0.564  10.9

0.406 0.542 0.419 0.555  26.9

– 1.009 0.830 0.835 –

CPT–EB 3D–EB CPT–PS 3D–PS % Error

1.319 1.318 1.247 1.249 5.6

 0.518  0.519  0.511  0.513 0.9

0.402 0.409 0.406 0.413  2.5

– 4.096 3.371 3.857 –

1.4

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

Table 12 Unstiffened isotropic plate – dynamics. a/h

ω

Table 14 Single central concentric stiffener.

rffiffiffiffi 2 ρa  10 E h

a/h

H/h

Modea

CPT

3D

% Errorb

10

0,0 1,0 0,2 2,0 1,2

59.7 149.3 298.7 298.7 388.3

57.8 138.1 258.7 258.7 324.9

3.3 8.1 15.5 15.5 19.5

10

20

0,0 1,0 0,2 2,0 1,2

59.7 149.3 298.7 298.7 388.3

59.2 146.2 286.6 286.6 368.3

0.8 2.1 4.2 4.2 5.4

a b

Nodal lines parallel to y and x, respectively. Error of CPT with respect to 3D.

Table 13 Unstiffened orthotropic plate – dynamics. a/h

29

Modea

CPT–EB

3D–EB

CPT–PS

3D–PS

% Errorb

0.9

0,0 1,0 0,2 1,2 2,0

71.4 219.9 268.6 388.4 402.6

69.5 202.3 266.6 368.4 377.3

68.0 191.7 234.3 320.3 329.7

66.4 180.1 233.1 310.2 315.1

7.5 22.1 15.2 25.2 27.8

20

1.25

0,0 1,0 0,2 1,2 2,0

74.4 224.0 275.8 402.3 405.8

73.5 215.9 275.1 392.2 394.3

73.3 214.2 265.1 377.3 381.6

72.4 207.2 264.5 369.0 372.7

2.8 8.1 4.3 9.0 8.9

35

1.6

0,0 1,0 0,2 1,2 2,0

76.3 226.5 281.6 402.6 418.5

75.8 222.4 281.3 397.8 411.9

75.9 222.8 277.8 393.4 409.2

75.4 218.9 277.6 388.9 403.2

1.2 3.5 1.4 3.5 3.8

50

1.85

0,0 1,0 0,2 1,2 2,0

77.1 227.3 285.0 402.6 424.8

76.8 224.6 284.9 399.5 420.4

76.9 225.4 283.1 398.0 420.1

76.6 222.8 283.0 395.0 415.9

0.7 2.0 0.7 1.9 2.1

100

2.45

0,0 1,0 0,2 1,2 2,0

78.6 229.0 290.6 403.0 435.7

78.5 227.9 290.6 401.8 433.8

78.6 228.6 290.2 401.9 434.5

78.4 227.5 290.1 400.6 432.6

0.3 0.7 0.2 0.6 0.7

rffiffiffiffiffiffi 2 ρ a n10 ω ET h Mode

CPT

3D

% Error

10

0,0 0,2 1,0 1,2 0,4

152.3 323.3 578.3 681.6 761.4

124.8 255.0 336.8 410.8 496.7

22.0 26.8 71.7 65.9 53.3

20

0,0 0,2 1,0 1,2 0,4

152.3 323.3 578.3 681.6 761.4

143.6 300.6 472.6 561.8 654.2

6.1 7.6 22.4 21.3 16.4

a b

Nodal lines parallel to y and x, respectively. Error of CPT–EB with respect to 3D–PS.

Table 15 Single central concentric stiffener (along the fibres). a/h

5. The net shear deformation effect, for the same degree of stiffening in that the maximum deflection is halved with respect to the unstiffened case, depends on the actual stiffener configuration. Tables 4–6 show that this is more significant for the isotropic plate with a single concentric stiffener than that with a set of concentric orthogonal stiffeners, and further more for a plate with a single eccentric stiffener. Similarly, from Tables 7 to 9, it is seen that the net shear deformation effect is more significant for the eccentrically stiffened orthotropic plate than the concentric stiffened plate with the stiffener running along the fibre direction in both cases; further that it is more significant for the plate stiffened perpendicular to the fibres than along the fibres, with the stiffener being concentric with respect to the plate in both cases. 6. Considering the total stiffener weight to achieve the same degree of stiffening so as to halve the maximum deflection, it can be seen from Tables 4 and 6 that a single concentric stiffener is a more efficient configuration than a set of orthogonal concentric stiffeners; for example, for a/h ¼10, the total 3 stiffener volume is given by 2aHB ¼ 18h for the single stiffener 3 and 4aHB ¼ 26h for the orthogonally stiffened case. Similarly, Tables 4 and 5 show that the eccentrically stiffened configuration is more efficient than the concentrically stiffened one. The corresponding observations for the stiffened orthotropic plate from Tables 7 to 9 are that eccentric stiffening is more efficient than concentric stiffening and also that a concentric stiffener in the fibre direction is more efficient than one perpendicular to the fibres.

rffiffiffiffi 2 ρa ω n10 E h

H/h

ω

rffiffiffiffiffiffi 2 ρ a n10 ET h

Mode

CPT–EB

3D–EB

CPT–PS

3D–PS

% Error

10

0.35

0,0 0,2 1,0 0,4 1,2

181.0 325.0 621.5 731.8 797.8

175.4 320.7 612.2 729.3 729.9

138.2 249.9 357.1 426.4 477.2

135.7 248.6 349.1 413.0 476.6

33.4 30.7 78.0 77.2 67.4

20

0.5

0,0 0,2 1,0 0,4 1,2

180.2 328.7 620.2 738.2 795.0

177.4 326.4 615.8 737.2 758.2

164.8 301.6 502.6 608.7 633.8

162.7 300.3 498.2 592.1 633.3

10.8 9.5 24.5 24.7 25.5

35

0.75

0,0 0,2 1,0 0,4 1,2

185.8 335.1 621.6 743.1 813.1

183.5 333.0 619.1 742.3 786.9

179.2 324.2 574.4 702.2 727.1

177.3 322.5 571.8 701.6 707.6

4.8 3.9 8.7 5.9 14.9

50

0.9

0,0 0,2 1,0 0,4 1,2

186.6 337.6 621.7 747.1 815.8

185.0 336.0 620.0 746.5 797.4

183.3 332.0 597.2 725.9 770.0

181.7 330.6 595.4 725.4 754.2

2.7 2.1 4.4 3.0 8.2

100

1.3

0,0 0,2 1,0 0,4 1,2

190.0 343.8 622.4 754.3 825.3

189.1 342.8 621.6 754.0 816.4

189.2 342.3 616.0 748.8 813.9

188.4 341.4 615.3 748.4 805.3

0.8 0.7 1.2 0.8 2.5

30

K. Bhaskar, A. Pydah / International Journal of Mechanical Sciences 89 (2014) 21–30

7. The maximum values of bending stresses (σ x , σy and τxy Þ and the transverse shear stress (τxz) as calculated by all 4 models are shown in Tables 10 and 11 for central stiffened isotropic and orthotropic plates. It is interesting to note that the errors between the stress predictions by CPT–EB model and the 3D– PS model are much lesser than the maximum displacement errors for the same stiffened plates (refer to Tables 5 and 9). Here too, the errors are larger for the orthotropic stiffened plate than for the isotropic stiffened plate and reduce as the plates become thinner. 8. Tables 14 and 15 give the first five natural frequencies for concentrically stiffened isotropic and orthotropic plates based on the four hierarchical models chosen; as stated earlier, only modes symmetric about the stiffener are considered here. The mode shape is purely sinusoidal in the direction of the stiffener (x-direction) but not in the perpendicular direction. So the mode shapes are presented in terms of the number of nodal lines (straight or slightly curved) in the y and x directions, given in that order in column 3. The height of the stiffener is taken to be the same as for the corresponding static load cases considered in Tables 4 and 7. The last column presents the error of the classical CPT–EB model with respect to the rigorous 3D-PS model which completely accounts for transverse shear deformation, rotary inertia and thicknessstretch effects. As a first observation, one can immediately see that the total influence of the above mentioned non-classical effects is more significant for the stiffened plate than for the unstiffened plate (see last columns of Tables 12 and 13). Further, as expected, all the three simpler models of Tables 14 and 15 yield overpredictions of the natural frequencies as compared to the 3D– PS model. Also, as seen for the static analysis, the error of the classical model decreases with a/h and is greater for the orthotropic plate than for the isotropic case. 9. Another observation worth noting is regarding the errors of the classical theory estimates of the higher frequencies. From the extensive literature on unstiffened isotropic plates, it is generally known that these errors are larger than those of the lower modes; this trend can also be seen here in Table 12. However, both for unstiffened orthotropic plates (Table 13) and for stiffened plates (Tables 14 and 15), this trend does not hold good; for example, in Table 14, the error is larger for the lower (1,0) mode than for the higher (0,2) mode. This anomalous trend cannot be explained easily because the stiffness and inertia contributions of the plate or the stiffener depends not only on the theoretical model used for their idealization but also on the mode shapes. Thus for any of the simpler models, the total error is a combination of that in the prediction of the individual stiffness and inertia contributions, and hence an anomalous trend is certainly possible.

5. Conclusion Four hierarchical models have been employed for the analysis of simply supported plates stiffened by a single central blade stiffener or a set of orthogonal blade stiffeners. Results have been obtained using an exact analytical solution of the governing equations. These results have been used to highlight the importance of the rigorous models and the errors of the commonly employed classical approach, which have been shown to be significantly higher for stiffened plates than for corresponding unstiffened plates. References [1] Troitsky MS. Stiffened plates: bending, stability and vibration. New York: Elsevier Science Ltd.; 1976. [2] Kukreti AR, Cheraghi E. Analysis procedure for stiffened plate systems using an energy approach. Comput Struct 1993;46:649–57. [3] Bedair OK, Troitsky MS. A study of the fundamental frequency characteristics of eccentrically and concentrically simply supported stiffened plates. Int J Mech Sci 1997;39:1257–72. [4] Rossow MP, Ibrahimkhail AK. Constraint method analysis of stiffened plates. Comput Struct 1978;8:51–60. [5] Deb A, Booton M. Finite element models for stiffened plates under transverse loading. Comput Struct 1988;28:361–72. [6] Prusty BG, Satsangi SK. Analysis of stiffened shell for ships and ocean structures by finite element method. Ocean Eng 2001;28:621–38. [7] Tanaka M, Matsumoto T, Oida S. A boundary element method applied to the elastostatic bending problem of beam-stiffened plates. Eng Anal Bound Elem 2000;24:751–8. [8] Pagano NJ. Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 1970;4:20–34. [9] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 1970;6: 1463–81. [10] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 2003;10(3):215–96. [11] Deb A, Deb M, Booton M. Analysis of orthotropically modeled stiffened plates. Int J Solids Struct 1991;27:647–64. [12] Mukherjee A, Menghani LC. Displacement and stress response of laminated beams and stiffened plates using a high-order element. Compos Struct 1994;28:93–111. [13] Sadek EA, Tawfik SA. A finite element model for the analysis of stiffened laminated plates. Comput Struct 2000;75:369–83. [14] Biswal KC, Ghosh AK. Finite element analysis for stiffened laminated plates using higher order shear deformation theory. Comput Struct 1994;53:161–71. [15] Ghosh AK, Biswal KC. Free-vibration analysis of stiffened laminated plates using higher-order shear deformation theory. Finite Elem Anal Des 1996;22: 143–61. [16] Bhar A, Phoenix SS, Satsangi SK. Finite element analysis of laminated composite stiffened plates using fsdt and hsdt: a comparative perspective. Compos Struct 2010;92:312–21. [17] Sapountzakis EJ, Katsikadelis JT. A new model for slab and beam structures – comparison with other models. Comput Struct 2002;80:459–70. [18] Qing G, Qiu J, Liu Y. Free vibration analysis of stiffened laminated plates. Int J Solids Struct 2006;43:1357–71.