FINITE ELEMENT STATIC, DYNAMIC AND STABILITY ANALYSES OF ARBITRARY STIFFENED PLATES A thesis submitted to Indian Institute of Technology, Kharagpur for the award of the degree of Do˝or of Philosophy in Engineering by Manoranjan Barik Department of Ocean Engineering and Naval Architecture Indian Institute of Technology Kharagpur - 721 302, India January, 1999
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Finite Element Static,Dynamic and Stability Analyses of Arbitary Stiffened Plates - jan Barik
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FINITE ELEMENT STATIC, DYNAMIC ANDSTABILITY ANALYSES OF ARBITRARY
STIFFENED PLATES
A thesis submitted to Indian Institute of Technology, Kharagpur
for the award of the degree of
Do˝or of Philosophyin
Engineering
by
Manoranjan Barik
Department of Ocean Engineering and Naval ArchitectureIndian Institute of TechnologyKharagpur - 721 302, India
January, 1999
With dedication toMy Ailing Mother
With love toMy wife Trushna
And little ones Trushita and MonishaWho endured all the sufferings silently
And looked for this day patiently
* INDIAN INSTITUTE OF TECHNOLOGYKHARAGPUR 721302, INDIA
DEPARTMENT OF OCEAN ENGINEERINGAND NAVAL ARCHITECTURE
This is to certify that the thesis entitled ‘FINITE ELEMENT STATIC,DYNAMIC AND STABILITY ANALYSES OF ARBITRARY STIFFENEDPLATES’ being submitted to the Indian Institute of Technology, Kharag-pur by Mr. Manoranjan Barik for the award of the degree of Doctor ofPhilosophy in Engineering is a record of bonafide research work carriedout by him under my supervision and guidance, and Mr. Barik fulfills therequirements of the regulations of the degree. The results embodied inthis thesis have not been submitted to any other University or Institutefor the award of any degree or diploma.
Madhujit Mukhopadhyay
Acknowledgements
I express my sincere gratitude toProfessor L. R. RahejaandProfes-sor S. C. Mishra, the Ex-Heads, Department of Ocean Engineering andNaval Architecture, Indian Institute of Technology, Kharagpur, who dur-ing their headships extended all the computational and other related fa-cilities of the Department to make my progress of work smooth.
I owe a lot toProfessor S. K. Satsangi, Professor and Head, De-partment of Ocean Engineering and Naval Architecture, whose out of theway provisions and help made my work to see the end at last, amidst allthe obstacles.
The authority ofRegional Engineering College, Rourkelagrantedme the study leave and theMinistry of Human Resources and Develop-ment, Government of Indiaprovided me the Fellowship to carry out theresearch work. Their provisions are highly acknowledged with thanks.
I extend my heartfelt thanks toDr. O. P. Sha, Assistant Professor,Department of Ocean Engineering and Naval Architecture, who virtuallymade me the owner of his Personal Computer.
Dr. A. H. Sheikh, Assistant Professor of the Department was alwayseager to extend his analytical ability without any hesitation. My sincerethanks are due to him.
With all the humbleness, I gratefully acknowledge the valuable sug-gestions received on various occasions fromProfessor S. Majumdar,Professor, Department of Civil Engineering, Indian Institute of Technol-ogy, Kharagpur.
Special thanks are due toMr. Parimol Kumar Roy without whosehelp the completion of the thesis would have been delayed considerably.
v
vi ACKNOWLEDGEMENTS
I acknowledge the help received in various forms from all the facul-ties and staff members of the Department whose excellent cooperationmade my stay here a homely, pleasant and enjoyable one.
Krishna andPrusheth, the LATEX lovers did marvelous jobs to han-dle in my own way, the commas and semicolons of the LATEX. I sincerelyacknowledge their invaluable help.
The works could not have seen such a happy ending without thelov-
ing cooperation ofAbhinna, who helped me in taking the final prints. Ifeel short of words to thank him.
I express my sincere thanks to myChurch Members at RourkelaandKharagpur and the IIT Christian Fellowship Members, particularlyJamesandPatrick who held me up through their fervent prayer supportthroughout my research work.
The sweet presence of my co-scholars,Satish, Asokendu, Sushanta,Murthy, Chaitali, Sangita ... made my stay at the Institute a memo-rable one, for together we suffered the moments of trauma, together wetriumphed over the success, together we shared the moments of joy andhappiness and the greatest of all was that we understood each other betterthan any body else.
Above all, I express my deep sense of gratitude tomy ProfessorandsupervisorProfessor Madhujit Mukhopadhyay, Professor, Departmentof Ocean Engineering and Naval Architecture, Indian Institute of Tech-nology, Kharagpur, whose constant encouragement, guidance and thetime I spent along with him was invaluable to me. There were momentswhen he pushed me forward, enough to stumble, so that I may rise upand stand upright on my own on firm ground. And often he dragged meforward just to enable me to reach my goal. I adore him for his many ex-cellent qualities and feel myself blessed to work under him, for workingwith him was never a burden, rather a pleasure, the moments of which Iwill be carrying along with me throughout my life’s journey.
Indian Institute of TechnologyKharagpur (Manoranjan Barik)
Plates are in extensive use as one of the important structural elementsin the modern day structures in civil, marine, aeronautical and mechan-ical engineering. These plates may assume arbitrary shapes dependingon their structural behaviour, the area of application and the type of ser-vices they are put to. Though they have wide applications without anyrib reinforcement, but various engineering structures demand economyin weight with enhancement of strength through stiffening of the platedstructures. When the arbitrarily shaped bare and the stiffened plates arein service, they are subjected to the static lateral load, the dynamic loadand the inplane load. To investigate the actual behaviour of the platesunder these loads, rigorous analysis is required to assess the strength andstability under various boundary conditions.
In the present era of super speed number crunching machines, numer-ical methods have found their way into the structural analysis because ofthe non-amenability of analytical solutions for complex structural prob-lems such as arising out of the arbitrary shape of the plates. Among thesenumerical tools, the finite element method has been proved to be the mostversatile and powerful one because of its generality and capability to han-dle structural and geometrical complexities with ease. Several numberof commercial softwares and in-house codes have been developed us-ing the finite element method for carrying out the structural analyses.But most of these packages have inadequate facility for efficient stiff-ener modelling, improper specification of boundary conditions in case ofa curved boundary and loss of generality in the mesh division processbecause of the stiffener position in the plate. Moreover, these codes are
xv
xvi PREFACE
not susceptible to easy modification in case of need. On the other hand,though there are plenty of elements developed so far in the finite elementdomain, many have been found to be inadequate and inefficient in someway or other for analyzing plates of arbitrary geometrical configurations.The present investigation is an attempt to accommodate the unstiffenedand the stiffened plate problems of arbitrary shapes by developing newefficient elements.
A plate bending element has been developed following the philoso-phy of isoparametric element to enable the analysis of arbitrarily shapedbare plates. The basic element considered for the development of this ele-ment is the simplest 12 degrees of freedom rectangular plate bending ele-ment popularly known asACM element. Bare plates of many geometricalconfigurations have been analyzed for static, dynamic and stability mak-ing use of this new element. For analyzing arbitrary stiffened plates, an 8degrees of freedom rectangular plane stress element has been combinedto the basicACM element. The stiffener modelling has been done con-sidering a curved general element which can be placed anywhere withinthe plate element which removes the restraint of positioning the stiffen-ers along the nodal lines. The static, free vibration and buckling analyseshave been performed on arbitrary plates with eccentric and concentricstiffeners using this stiffened plate bending element.
The thesis has been presented in six chapters. It also includes thebibliography section showing the important references concerned withthe present investigation.
Chapter 1 includes the general introduction and the scope of presentinvestigation.
The review of literature confining to the scope of the study has beenpresented in theChapter 2. The general methods of analysis of the stiff-ened plates have been briefly addressed in this chapter.
The Chapter 3 comprises the mathematical formulation of the twoelements. The elastic and the geometric stiffness matrices and the mass
PREFACE xvii
matrix for the plate element and the stiffener element have been formu-lated separately. The boundary conditions have been implemented byconsistently formulating the stiffness matrices of the boundary line andadding them to the global stiffness matrix.
TheChapter 4 briefly describes the computer programme implemen-tation of the theoretical formulation presented in Chapter 3. The differentfunctions and the associated variables which have been used in writingthe codes inC++ language have been presented in brief. A few numbersof flowchart of the computer codes have been illustrated.
Several numerical examples which include the static, the free vibra-tion and the stability analyses of bare and the stiffened plates of variousgeometries have been presented in theChapter 5 to validate the formu-lation of the proposed method. Attempt has been made to include a widespectrum of problems of diverse geometrical plate shapes. The resultshave been compared wherever possible and the discrepancies in themhave been discussed.
TheChapter 6 sums up and concludes the present investigation. Anaccount of possible scope of extension to the present study along with alist of publications has been appended to the concluding remarks.
At the end, some important publications and books referred duringthe present investigation have been listed in theBibliography section.
List of Symbols
Although all the principal symbols used in this thesis are defined in thetext as they occur, a list of them is presented below for easy reference. Onsome occasions, a single symbol is used for different meanings depend-ing on the context and thus its uniqueness is lost. The contextual expla-nation of the symbol at its appropriate place of use is hoped to eliminatethe confusion.
English
As cross sectional area of the stiffener[Bp] strain matrix for plate element of stiffened plate[Bs] strain matrix for stiffener element of stiffened plate[Bu] strain matrix for bare plate elementdx, dy element length in x and y-directiondv volume of the element[Du] rigidity matrix of bare plate element[Dp] rigidity matrix of stiffened plate element[Ds] rigidity matrix of stiffener elementE modulus of elasticity{f} acceleration field vector{FI} nodal inertia force parameter{fku} reaction component per unit length of bare plateFx, Fy, Fxy inplane forcesFS axial force in the stiffenerG modulus of rigidityIs second moment of area of the stiffener
xix
xx LIST OF SYMBOLS
|J | jacobianJs torsional constant of the stiffener|Jst| jacobian of the stiffener[K] global elastic stiffness matrix[KG] global geometric stiffness matrix[Ku]e elastic stiffness matrix of the bare plate bending element[KuG]e geometric stiffness matrix of the bare plate bending element[Ku] global elastic stiffness matrix of the bare plate[KuG] global geometric stiffness matrix of the bare plate[Kp]e elastic stiffness matrix of the stiffened plate element[KpG]e geometric stiffness matrix of the stiffened plate element[Ks]e elastic stiffness matrix of the stiffener element[KsG]e geometric stiffness matrix of the stiffener element[Kp]b stiffness matrix of the boundary of stiffened plate[Ku]b stiffness matrix of the boundary line of the bare plateku, kv, kw translational restraint coefficientkα, kβ rotational restraint coefficient[M ] global consistent mass matrix[Mu]e consistent mass matrix of the bare plate bending element[Mp]e consistent mass matrix of the stiffened plate element[Ms]e consistent mass matrix of the stiffener elementMs bending moment of the stiffenerMx,My,Mxybending moments of the plateNi(s, t) cubic serendipity shape functionsNu, Nv, Nw finite element shape functionsNθξ
, Nθη finite element shape functions{P} global load vector{P}e element load vectorq load intensitys-t axis system of the plate in the mapped domains1 length along the boundary
xxi
Ss first moment of area of the stiffenert thickness of the plateTs torsional moment of the stiffeneru, v inplane displacementsu´, v´, w´ displacements at midplane of the platew out of plane displacement{w} acceleration vector in z-directionxi, yi Cartesian nodal coordinatesx, y, z global axis systemx1-y1 local axes at the point of a curved boundaryx´-y´ local axes at any point of a curved stiffener
Greek
α angle between thex´ -y´ andx-y axes systemβ angle between thex1-y1 andx-y axes system{δ}u nodal displacement vector of bare plate{δ}p nodal displacement vector of stiffened plate{δ}u nodal acceleration vector of bare plateξ-η axis system of the element in the mapped domain{σ}u stress resultant vector of bare plate{σ}p stress resultant vector of stiffened plate{σ}s stress resultant vector of stiffenerσx, σy, τxy stresses at a point{ε}u generalized strain vector of bare plate{ε}uG geometric strain vector of bare plate{ε}p strain vector of stiffened plate element{ε}pE elastic plate strain vector{ε}pG geometric plate strain vector{ε}s strain vector of the stiffener{ε}sE elastic stiffener strain vector{ε}sG geometric stiffener strain vector
xxii LIST OF SYMBOLS
εx, εy, γxy bending strainsν Poisson´ s ratio∂
∂x,
∂
∂ypartial derivatives with respect tox andy
ρ mass density of the materialθn, θt slopes normal and transverse to the boundaryλ stiffener direction in mapped domainλ1 boundary line direction in mapped domain{ψ} normalized vectorω frequency of vibration
Subscripts
u for bare plateG for geometric stiffness matrixb for boundaryp for the plate element of the stiffened plates for the stiffener element of the stiffened plate
Operators
( ) first derivative with respect to time( ) second derivative with respect to time[ ]−1 inverse of the matrix[ ]−T transpose of the matrix
5.16 Buckling parameter k = λb2/π2D for uniformly com-pressed all edges simply supported rectangular plates145
5.17 Buckling parameter k = λb2/π2D for uniformly com-pressed all edges clamped rectangular plates. . . . . 146
5.18 Buckling parameter k = λb2/π2D for all edges clampedrectangular plates with biaxial uniform compression . 147
5.19 Buckling parameter k = λb2/π2D for uniaxially com-pressed all edges simply supported rectangular plateswith triangular load i.e; α = 1 in the expressionNx =
5.22 Deflection at the centre of simply supported squarestiffened plate(×104 mm.) . . . . . . . . . . . . . . . 154
5.23 Convergence of deflection(w), plate moment(My) andplate stress(σx) of the eccentrically stiffened squareplate at its centre with different mesh divisions.. . . . 154
5.28 Convergence of moments at the centre. . . . . . . . . 175
5.29 Deflection (mm.) at inner edge and inner girder . . . 176
5.30 Deflection (mm.) at outer girder and outer edge . . . 177
5.31 Frequency in Hz of a clamped square plate with a cen-tral concentric stiffener . . . . . . . . . . . . . . . . . 181
5.32 Frequency in Hz of a clamped square plate with a cen-tral eccentric stiffener . . . . . . . . . . . . . . . . . . 183
5.33 Frequency parameter[ω(a/π)2√
ρt/D] of square cross-stiffened plate with concentric stiffeners having all edgesclamped . . . . . . . . . . . . . . . . . . . . . . . . . .184
5.34 Frequency in Hz of a simply supported rectangularplate with a central L-shaped eccentric stiffener in theshorter span direction . . . . . . . . . . . . . . . . . . 184
5.35 Frequency parameter[ω(a/R)2√
ρt/D] of simply sup-ported multi-stiffened rectangular plate with concen-tric stiffeners in one direction . . . . . . . . . . . . . . 186
5.36 Frequency parameter [ω(a/R)2√
ρh/D] of a simplysupported multi-stiffened skew plate having concen-tric stiffeners in one direction . . . . . . . . . . . . . . 189
5.37 Frequency in Hz of all edges clamped trapezoidal platewith a central concentric stiffener . . . . . . . . . . . 192
5.40 Frequency (Hz/Parameter) of all edges clamped cir-cular stiffened plates . . . . . . . . . . . . . . . . . . . 198
5.41 Frequency in Hz of a simply supported elliptical platewith a central eccentric stiffener in the shorter spandirection . . . . . . . . . . . . . . . . . . . . . . . . .201
5.42 Buckling parameter k = λb2/π2D for square platewith a central concentric stiffener subjected to uniax-ial and uniform compression in the stiffener direction 203
5.44 Buckling parameter k = λb2/π2D for rectangular platewith a central concentric stiffener subjected to uniax-ial and uniform compression in the stiffener direction 205
5.46 Buckling parameter k = (Nr)cr a2/D for uniformlycompressed circular plates with concentric stiffenersalong the diameter with varying flexural and torsionalstiffness parameters of the stiffener . . . . . . . . . . 207
List of Figures
3.1 Mapping of the arbitrarily shaped plate . . . . . . . . 50
3.2 Mapping of the element . . . . . . . . . . . . . . . . . 51
3.3 Coordinate axes at a typical point of a curved boundary 62
5.10 Simply supported square plate with a central stiffener 152
xxvii
xxviii LIST OF FIGURES
5.11 Variation of deflection along centrelines of simply sup-ported square plate with a central stiffener . . . . . . 153
5.12 Simply supported rectangular plate with a central stiff-ener in each direction . . . . . . . . . . . . . . . . . . 156
5.13 Deflection at x = 190.5 mm. and x = 381.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
5.14 Deflection at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
5.15 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .158
5.16 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .158
5.17 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .159
5.18 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .159
5.19 Deflections at x = 190.5 mm. and x = 381.0 mm. forplate with two stiffeners under concentrated load. . . 160
5.20 Moment Mxx at y = 381.0 mm. and y = 762.0 mm.for plate with two concentric stiffeners under concen-trated load . . . . . . . . . . . . . . . . . . . . . . . .160
5.21 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two eccentric stiffeners under concentratedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .161
LIST OF FIGURES xxix
5.22 Moment Myy at x = 190.5 mm. and x = 381.0 mm.for plate with two concentric stiffeners under concen-trated load . . . . . . . . . . . . . . . . . . . . . . . .161
5.23 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners under concentratedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
The axial displacement of a point, sayP1 in the stiffener at a depthz
(Fig. 3.6) from the reference middle plane of the plate and normal to it is
expressed as
u′ = u′ − z∂w′
∂x′(3.4.61)
3.4 Arbitrary Stiffened Plate Element Formulation 83
Pz
Figure 3.6:Sectional view of a typical stiffener
The corresponding strain in the stiffener is
εsl =∂u′
∂x′− z
∂2w′
∂x′ 2(3.4.62)
and the normal stress is
σsl = E
[∂u′
∂x′− z
∂2w′
∂x′ 2
](3.4.63)
The axial force in the stiffener is given by
Fs =
∫
As
σsl dAs = E∂u′
∂x′
∫
As
dAs − E∂2w′
∂x′ 2
∫
As
z dAs
= EAs∂u′
∂x′− ESs
∂2w′
∂x′ 2(3.4.64)
whereAs is the cross sectional area andSs is the first moment of area of
the stiffener with respect to the middle plane of the plate. The value of
Ss depends on the disposition of the stiffener.
The bending moment is
Ms =
∫
As
σsl z dAs (3.4.65)
Substituting the value ofσs from Eq.(3.4.63) in Eq.(3.4.65), it yields
Ms = ESs∂u′
∂x′− EIs
∂2w′
∂x′ 2(3.4.66)
84 MATHEMATICAL FORMULATION
whereIs is the second moment of area (reference plane being the mid-
plane of the plate) of the stiffener.
It is observed from Eqs.(3.4.64) and (3.4.66) that the eccentricity of
the stiffener produces coupling between the axial and the flexural effects.
The torsional moment is given by
Ts = −GJs∂2w′
∂x′ ∂y′(3.4.67)
whereG is the modulus of rigidity andJs is the torsional constant of the
stiffener.
Combining Eqs.(3.4.64), (3.4.66), and (3.4.67) the generalized stress-
strain relationship of the stiffener in the local axis system at the point P
is expressed as
{σs} = [Ds] {εs} (3.4.68)
where
{σs} = [Fs Ms Ts]T (3.4.69)
{εs} =
[∂u′
∂x′− ∂2w′
∂x′ 2− ∂2w′
∂x′ ∂y′
]T
(3.4.70)
and
[Ds] =
E As E Ss 0
E Ss E Is 0
0 0 G Js
(3.4.71)
3.4.3.3 Strain-Displacement Relationship
The local displacements and coordinate parameters in the generalized
strain vector of the stiffener given in Eq.(3.4.70) are replaced by the
global parameters using Eqs.(3.4.58) and (3.4.60) and it becomes
{εs} = [Ts] {εs} (3.4.72)
3.4 Arbitrary Stiffened Plate Element Formulation 85
where
[Ts] =
cos2α sin2α1
2sin2α 0 0 0
0 0 0 cos2α sin2α −1
2sin2α
0 0 0 −1
2sin2α
1
2sin2α −1
2cos2α
(3.4.73)
and
{εs} =
[∂u
∂x
∂v
∂y
(∂u
∂y+
∂v
∂x
)− ∂2w
∂x2− ∂2w
∂y22
∂2w
∂x ∂y
]T
(3.4.74)
Once the strain vector of the stiffener is expressed in terms of the dis-
placement components at the mid-plane of the plate, the same displace-
ment shape function of the plate element is used which yields the stiff-
ness matrix of the stiffener in terms of the nodal parameters of the plate
element and by this process, the compatibility between the plate and the
stiffener element is retained.
It may be observed from Eqs.(3.4.74) and (3.4.10) that
{εs} = {ε(x, y)p} (3.4.75)
Hence using the same interpolation functions given in Eq.(3.4.4) and
Eq.(3.3.4), the Eq.(3.4.74) can be expressed with the help of Eq.(3.3.22)
as
{εs} = [Bp]{δp} (3.4.76)
Combining Eqs.(3.4.72) and (3.4.76), yields
{εs} = [Ts] [Bp]{δp} = [Bs] {δp} (3.4.77)
where
[Bs] = [Ts][Bp] (3.4.78)
86 MATHEMATICAL FORMULATION
ξ
η(-1,+1)
(-1,-1)
(+1,+1)
(+1,-1)
λ
Figure 3.7:Stiffener orientation in the mapped domain
3.4.3.4 Elastic Stiffness Matrix of the Stiffener Element
Following the steps mentioned earlier for the plate element, the elastic
stiffness matrix of the stiffener element is given by
[Ks]e =
∫[Bs]
T [Ds][Bs] dl (3.4.79)
Herel is taken along the stiffener axis inx-y plane. This can be rewritten
as
[Ks]e =
∫[Bs]
T [Ds][Bs] |Jst| dλ (3.4.80)
whereλ is in the direction of the stiffener axis in theξ-η plane as shown
in the Fig. 3.7 and the Jacobian|Jst| is given by
|Jst| = dl
dλ(3.4.81)
The Jacobian is calculated by the ratio of the actual length to the length
on the mapped domain considering any segment of the stiffener and is
3.4 Arbitrary Stiffened Plate Element Formulation 87
constant when a straight line or a circular arc in thex-y plane is mapped
into a straight line. But in case of a complex mapping, the ratiodl
dλmay change from point to point. The integration is carried out along the
stiffener axis which isλ in the ξ-η plane by taking Gauss points on the
mapped stiffener axis.
3.4.3.5 Consistent Mass Matrix of the Stiffener Element
The consistent mass matrix for the arbitrarily oriented stiffener element
is formulated following the steps similar to that of the plate element and
it can be written as:
[Ms] = ρ
∫[N ]T [L]T [Ts]
T [Ps][Ts][L][N ] |Jst| dλ (3.4.82)
where
[Ps] =
Ascos2α Assinαcosα 0 −Sscosα 0
Assinα cosα Assin2α 0 −Sssinα 0
0 0 As 0 0
Sscosα −Sssinα 0 Is 0
0 0 0 0 Js
(3.4.83)
88 MATHEMATICAL FORMULATION
[Ts] =
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 cosα sinα
0 0 0 −sinα cosα
(3.4.84)
and the matrices[L] and [N ] are given by the Eqs.(3.4.21) and (3.4.22)
respectively.
3.4.3.6 Geometric Stiffness Matrix of the Stiffener Element
The stiffener strain for an x-directional stiffener is expressed as:
{εs} =
∂U
∂x
∂V
∂x
∂W
∂x
+
1
2
(∂W
∂x
)2
+1
2
(∂U
∂x
)2
0
0
= {εsE}+ {εsG}
(3.4.85)
The field variables are expressed as:
{f} =
U
V
W
=
−z∂w
∂x
−z∂w
∂y
w
(3.4.86)
3.4 Arbitrary Stiffened Plate Element Formulation 89
Substituting the values of U, V and W in the expression for{εsG}, yields
{εsG} =
z2
2
(∂2w
∂x2
)2
+1
2
(∂w
∂x
)2
0
0
=1
2{θs}T {θs}
(3.4.87)
where
{θs}T =
{−z
∂2w
∂x2
∂w
∂x
}(3.4.88)
It can be shown that
δ {εsG} = {θs}T δ {θs} (3.4.89)
{θs} can be written as:
{θs} = [HsG] {εsG} (3.4.90)
where
[HsG] =
1 0
0 −z
(3.4.91)
and
{εsG}T =
{∂w
∂x
∂2w
∂x2
}(3.4.92)
{εsG} can be expressed as:
{εsG} =
∂w
∂x
∂2w
∂x2
= [BsG] {δp} (3.4.93)
90 MATHEMATICAL FORMULATION
Hence
{θs} = [HsG] [BsG] {δp} (3.4.94)
The internal work done by the distributed internal stresses can be ex-
pressed as:
δW =
∫∫∫{εsG}T σx dx dy dz
= δ
∫∫∫{δp}T [BsG]T [HsG]T σx [HsG] [BsG] {δp} dx dy dz
(3.4.95)
In the case of the stiffener along the x-direction, the integration with
respect toy and z can be performed with the innermost integral only,
puttingy-coordinates in the[BsG] matrix.
[HsG]T σx [HsG] =
1 0
0 −z
σx
1 0
0 −z
= σx
1 0
0 z2
(3.4.96)
Hence,
∫∫[HsG]T σx [HsG] dy dz =
σxAs 0
0 σxIs
= [σs]
(3.4.97)
If the stiffener coordinate axis is at an angle ofα with respect to the
globalx-axis (Fig. 3.5) then with the help of the Eq.(3.4.58) the deriva-
tives with respect to thex-y and thex′-y′ coordinates are related as:
3.4 Arbitrary Stiffened Plate Element Formulation 91
∂w
∂x′
∂2w
∂x′2
= [TsG1]
∂w
∂x
∂w
∂y
−∂2w
∂x2
−∂2w
∂y2
2∂2w
∂x ∂y
(3.4.98)
where
[TsG1] =
cos α sin α 0 0 0
0 0 − cos2 α − sin2 α sin α cos α
(3.4.99)
The derivatives with respect to thex-y coordinates can be expressed in
terms ofξ-η coordinates such as:
∂w
∂x
∂w
∂y
−∂2w
∂x2
−∂2w
∂y2
2∂2w
∂x ∂y
= [TsG2]
∂w
∂ξ
∂w
∂η
∂2w
∂ξ2
∂2w
∂η2
∂2w
∂ξ ∂η
(3.4.100)
92 MATHEMATICAL FORMULATION
where
[TsG2] =
[J ]−1 [0]
[TF1] [TF2]
(3.4.101)
and[J ] is the Jacobian and[TF1], [TF2] being given by the Eq.(3.3.18).
Hence from the Eq.(3.4.93)
{εsG} = [TsG1] [TsG2]
∂Nw
∂ξ
∂Nw
∂η
∂2Nw
∂ξ2
∂2Nw
∂η2
∂2Nw
∂ξ ∂η
{δp}
= [TsG1] [TsG2][BsG
] {δp}
= [BsG] {δp}
(3.4.102)
The geometric stiffness matrix is given by
[KsG]e =
∫[BsG]T [σs] [BsG] dx
=
∫[BsG
]T[TsG2]
T [TsG1]T [σs] [TsG1] [TsG2]
[BsG
] |Jst| dλ
(3.4.103)
3.4 Arbitrary Stiffened Plate Element Formulation 93
3.4.4 Boundary Conditions for the Stiffened Plate
The boundary conditions for the arbitrary stiffened plate are considered
following the same procedure as adopted in the case of the arbitrary bare
plate. Referring to the Fig. 3.3, the relationship between the in-plane
displacements in the local and the global coordinates at the pointP is
given by:
u1
v1
=
cosβ sinβ
−sinβ cosβ
u
v
(3.4.104)
whereu1 andv1 are the displacements along the direction ofx1 andy1
respectively.
The displacements at the pointP which may be restrained can be
expressed as
{fbp} =
u1
v1
w
θn
θt
=
u1
v1
w
∂w
∂x1
∂w
∂y1
(3.4.105)
whereθn andθt represent the slopes which are normal and transverse to
the boundaries respectively as in the case of bare plate.
Substituting from Eqs.(3.3.50) and (3.4.104), the Eq.(3.4.105) can be
94 MATHEMATICAL FORMULATION
written as
{fbp} =
cosβ sinβ 0 0 0
−sinβ cosβ 0 0 0
0 0 1 0 0
0 0 0 cosβ sinβ
0 0 0 −sinβ cosβ
u
v
w
∂w
∂x
∂w
∂y
(3.4.106)
Expressing Eq.(3.4.106) in terms of the shape functions;
{fbp} = [Nbp] {δp} (3.4.107)
where
[Nbp] =
cosβ sinβ 0 0 0
−sinβ cosβ 0 0 0
0 0 1 0 0
0 0 0 cosβ sinβ
0 0 0 −sinβ cosβ
[Nu]
[Nv]
[Nw]
∂[Nw]
∂x
∂[Nw]
∂y
(3.4.108)
3.4 Arbitrary Stiffened Plate Element Formulation 95
The reaction components per unit length along the boundary line due
to the elastic springs corresponding to the possible boundary displace-
ments given in Eq.(3.4.105) can be expressed as
{fkp} =
fku
fkv
fkw
fkα
fkβ
=
kuu1
kvv1
kww
kαθn
kβθt
(3.4.109)
whereku, kv, kw, kα andkβ are the spring constants or restraint coeffi-
cients corresponding to the direction ofu1, v1, w, θn andθt respectively.
The Eq.(3.4.109) can be rewritten by combining the Eqs.(3.4.105),
(3.4.106) and (3.4.107) as
{fkp} = [Nkp]{δp} (3.4.110)
96 MATHEMATICAL FORMULATION
where
[Nkp] =
ku cosβ ku sinβ 0 0 0
−kv sinβ kv cosβ 0 0 0
0 0 kw 0 0
0 0 0 kα cosβ kα sinβ
0 0 0 −kβ sinβ kβ cosβ
[Nu]
[Nv]
[Nw]
∂[Nw]
∂x
∂[Nw]
∂y
(3.4.111)
Following the procedure similar to the case of bare plate the stiffness
of the boundary for the stiffened plate can be expressed as
[Kbp] =
∫[Nbp]
T [Nkp] |Jb| dλ1 (3.4.112)
3.4.5 Stresses in the Stiffener
Once the nodal displacements of the stiffened plate are known, the stress
resultants of the stiffener as expressed in the Eq.(3.4.69) can be obtained
with the help of the Eq.(3.4.68) and Eq.(3.4.67).
The stresses in the stiffener at a depthz from the mid-plane of the
plate can be computed as:
σz =Fs
As
− Fse
Is
z +Ms
Is
z (3.4.113)
3.5 Consistent Load Vector 97
whereFs, Ms, As andIs denote the axial force, bending moment, cross-
sectional area and moment of inertia respectively of the stiffener ande is
the eccentricity of the stiffener with respect to the plate mid-plane.
3.5 Consistent Load Vector
The nodal load vector for an element when subjected to a uniformly dis-
tributed load of intensityq(x, y) can be obtained by the expression
{P}e =
∫ ∫[Nj]
T q dx dy
=
∫ +1
−1
∫ +1
−1
[Nj]T q |J | dx dy
(3.5.1)
where[Nj] is the displacement function for thej-th node and|J | is the
determinant of the Jacobian. The global load{P} can be obtained by
assembling the nodal load vector{P}e of each of the elements. When
concentrated load is present at any of the nodal points, the load value is
added to the corresponding degree of freedom of that particular node.
3.6 Solution Procedures
The solution procedures adopted in the analysis of the arbitrary stiffened
and bare plates for static, dynamic and stability analyses are presented in
this section.
3.6.1 Static Analysis
The elastic stiffness matrices of the plate as well as the stiffener elements
are computed and they are assembled into the global elastic stiffness ma-
trix [K] which is stored by adopting the skyline storage (Zienkiewicz and
98 MATHEMATICAL FORMULATION
Taylor [198]) technique. In this process of storage, the matrix is stored
in a single array eliminating the zero entries if any within the band thus
reducing the storage requirement of the computer. The equation of equi-
librium for the static analysis given by the Eq.(3.1.1) is solved following
the Cholesky decomposition method by adopting the algorithm of Corr
and Jennings [39].
3.6.2 Free Vibration Analysis
The equilibrium equation for undamped free vibration is given by the
Eq.(3.1.2).
Considering the motion as harmonic the solution of the equation (3.1.2)
is
{δ} = H{ψ}eiωt (3.6.1)
where{ψ} is a normalized vector of the order of{δ}, H is the weight-
ing parameter of{ψ} andω is the frequency of vibration in radians per
second. On substitution the equilibrium equation becomes
[K]{ψ} = ω2[M ]{ψ} (3.6.2)
This is a generalized eigenproblem and is solved by the simultaneous it-
eration algorithm of Corr and Jennings [39] and its solution is the eigen-
valueω2 and the eigenvector{ψ}. The same skyline storage scheme as
earlier is adopted for the global elastic stiffness matrix[K] and the mass
matrix [M ].
3.6 Solution Procedures 99
3.6.3 Stability Analysis
The equilibrium equation for the stability analysis is given by the Eq.(3.1.3).
Since the matrix[K] is positive definite, it can be decomposed as:
[K] = [L][L]T (3.6.3)
where[L] is a lower triangular matrix. Hence Eq.(3.1.3) can be rewritten
as:
[L]−1 [KG] [L]−T [L]T{δ} =1
λ[L]T{δ} (3.6.4)
The above equation represents a standard eigenvalue problem which is
solved by the simultaneous iteration algorithm of Corr and Jennings [39]
and the eigenvalues corresponding to the lowest buckling loads are ob-
tained.
Chapter -4
COMPUTERIMPLEMENTATION
4.1 Introduction
The finite element method has been established as a powerful numeri-
cal tool because of its broad spectrum of generality and its ease of appli-
cability to rather more complex and difficult problems showing greater
efficacy in its solution than that of any other existing similar techniques.
This advantage of the method over others has led various research orga-
nizations and modern industries to endeavour the development of general
purpose software packages and other in-house codes for solving practical
problems of more complex nature. In an effort to make the method more
powerful and to address more complicated problems, the finite element
analysis programmes themselves become extremely complex and com-
putationally involved. These programmes are available as black box
modules which are to be used with the help of CAD programmes. These
conventional programmes cannot easily be modified to perform a desired
task necessitating redesign and rebuild of finite element libraries to suit
one’s need. Hence there is a requirement for finite element analysis pro-
grammes to be easily modifiable to introduce new analysis procedures
101
102 COMPUTER IMPLEMENTATION
and new kinds of design of structural components or even emerging tech-
nology of new materials whenever needed. In the present investigation,
the computer codes have been generated with such modularity which is
amenable to easy modification whenever the need arises.
Throughout all these years the finite element codes have been de-
veloped employing procedural language such as FORTRAN which is
unstructured in its nature. Now there is a trend to pay attention to the
verification, portability and reusability of the computer programmes dur-
ing the process of their development and to the possibility of the use of
other software products . However, FORTRAN does not have the pro-
vision to meet all these requirements. TheC++ language, apart from its
object-oriented programming approach allows for more efficient software
development, because it includes the dynamic memory allocation, decla-
ration of datatypes, modularization and the pointer concept. Though,
the benefit of object-oriented programming has not been utilized in the
present investigation, but the other advantages of the language as men-
tioned above have been fully utilized. Few of the utilities are of utmost
importance which are provided by Press et al. [150] and are used exten-
sively in the codes generated for the present investigation. Apart from
these, for efficiency of the finite element programmes, advance features
like automatic mesh generation, automatic nodal connectivity and skyline
storage scheme have been implemented in the computer codes.
4.2 Application Domain
The Computer Programmes have been developed in the present investi-
gation by making use of theC++ programming language to include a
wide spectrum of application domain. They have the analytical modules
to solve the following types of problems:
4.2 Application Domain 103
1. Static analysis of arbitrary bare plates - to evaluate displace-
ments and stress resultants at salient points of the bare plates of
arbitrary shape.
2. Free vibration analysis of arbitrary bare plates - to extract the
natural frequencies (eigenvalues) and the corresponding mode shapes
(eigenvectors) of the structure.
3. Stability analysis of arbitrary bare plates - to estimate the elastic
buckling load and the buckled mode shapes of the structure from
the eigenvalue solution.
4. Static analysis of arbitrary stiffened plates - to determine the
displacements and stress resultants at various points of the plate
skin and to evaluate stiffener stresses at different sections of the
eccentric and concentric stiffeners.
5. Free vibration analysis of arbitrary stiffened plates - to deter-
mine the natural frequencies of the stiffened plated structures along
with its corresponding mode shapes.
6. Stability analysis of arbitrary stiffened plates - to assess the
elastic buckling load of the structures and their buckling mode
shapes.
Computer programme codes have been written to incorporate vari-
ous boundary and loading conditions of the structures. The modularity
of the programme development has been retained by employing differ-
ent modules in the shape of differentC++ functions performing specific
functionalities of the programmes.
104 COMPUTER IMPLEMENTATION
4.3 Description of the Programme
The finite element procedure involves three basic steps in terms of the
computation carried out which may be termed as:
• Preprocessor
• Processor
• Postprocessor
The different functions of these steps have been elaborated in the Fig. 4.1.
4.3.1 Preprocessor
This module of the programme reads the necessary information about the
geometry and boundary conditions of the plate, material properties, load-
ing configuration and its magnitude, stiffener orientation and its prop-
erties etc. Also in this module, all the nodal coordinates and the nodal
connectivity are generated. The differentfunctions which are used in
this module are described briefly in the subsequent sections. A flowchart
of the preprocessor unit has been shown in the Fig. 4.2.
4.3.1.1 functioninput()
The following variables are used in the functioninput() to generate the
data required for the analysis of the bare and the stiffened plates.
4.3 Description of the Programme 105
Sole the Equations for Different Analyses
PREPROCESSORRead the Input DataGenerate the meshGenerate Nodal ConnectivityRead the Stiffener Position and Orientation
Echo the Input Data
POSTPROCESSOR
print the Output
END
Assemble the Matrices to Global Matrix
5. Free Vibration Analysis of Stiffened Plates
3. Stability Analysis of Bare Plates
1. Static Analysis of Bare Plates
4. Static Analysis of Stiffened Plates
Solve the Equations for different Analyses
PROCESSORGenerate Element Matrices for the PlateGenerate Element Matrices for the Stiffener if requiredGenerate Boundary Stiffness Matrices for the Boundaries
2. Free Vibration Analysis of Bare Plates
6. Stability Analysis of Stiffened Plates
START
Figure 4.1:Basic Elements of the Computer Programmes
106 COMPUTER IMPLEMENTATION
bpoin : Number of points on the boundary for the mapping
of the plate geometries
bcord : Cartesian coordinates of the boundary points
nnode : Number of node in the element
ndofn : Number of degrees of freedom per node
ngaus : Number of gauss points
nxi : Number of mesh ins-direction in the mapped domain
neta : Number of mesh int-direction in the mapped domain
nelem : Number of elements
npoin : Number of nodal points generated for the mapping
of the elements
nodes : Number of nodes for the analysis
tdof : Number of total degrees of freedom
young : Young’s modulous of elasticity
poiss : Poisson’s ratio
thick : thickness of the plate
ntype : Shape of the plate geometry
=1 Square plate
=2 Rectangular plate
=3 Annular plate
=4 Circular plate
=5 Skew plate
=6 Sector plate
=7 Elliptical plate
=8 Trapezoidal plate
=9 Triangular plate
4.3 Description of the Programme 107
stif : Index for bare or stiffened plate
=0 Bare plate analysis
=1 Stiffened plate analysis
soln : Index for type of analysis
=1 Static analysis
=2 Free vibration analysis
=3 Buckling analysis
4.3.1.2 functionnodgen()
The different variables used in the functionnodgen()which is used to
generate the peripheral nodes in each of the element’s boundary are as
presented below.
mnods : Nodal numbers in the element boundary
ielem : Element counter
inode : Node counter
4.3.1.3 functionstcod()
The functionstcod() is used to generate the nodal coordinates in the
mappeds-t domain of the plate. The different variables used in the func-
tion are:coord : Coordinates in the mapped domain
xi-divn : Length of element ins-direction in mapped domain
eta-divn : Length of element int-direction in mapped domain
xi-small :1
3of xi-divn
eta-small :1
3of eta-divn
108 COMPUTER IMPLEMENTATION
input()bpoin, bcord, nnode, ndofn, ngaus, nxi,
neta, nelem, npoin, nodes, tdof, young,
poiss, thick, ntype, stif, soln
nodgen()mnods, ielem, inode
xycod()xynod,xi,eta
rgdplt()dmatx1, young,poiss, thick
coord, xi-divn,
stcod()
eta-divn, xi-small,eta-small
connect()lnods
stifin()w, d, e
band()hband, sky
rgdstf()As, Ss, Is, GJs, dmatx2
shape1, deriv1
sfr1()
Figure 4.2:Preprocessor unit of the computer codes
4.3 Description of the Programme 109
4.3.1.4 functionconnect()
The functionconnect()generates the nodal connectivity in the elements.
The variables used along with others is:
lnods : Node numbers associated with the element
4.3.1.5 functionband()
The functionband() computes the half bandwidth of the matrix and the
skyline value for the skyline storage scheme. It has the following vari-
ables:hband : Half bandwidth of the matrix
sky : Skyline value for the skyline storage
4.3.1.6 functionxycod()
The functionxycod() generates all the nodalx-y coordinates of the ele-
ments. The variables used are:xynod : Cartesian coordinates of the node
xi : s-coordinate of the node in mapped domain
eta : t-coordinate of the node in mapped domain
4.3.1.7 functionsfr1()
The functionsfr1() calculates the cubic serendipity shape functions, their
derivatives and elements of the Jacobian matrix. The different variables
in this function are:shape1 : Cubic serendipity shape functions
deriv1 : Shape function derivatives
110 COMPUTER IMPLEMENTATION
4.3.1.8 functionrgdplt()
The functionrgdplt() has been used for computing the rigidity matrix of
the plate element. It comprises the following variables:
dmatx1 : Elements of the plate rigidity matrix
young : Young’s modulous of elasticity
poiss : Poisson’s ratio
thick : Plate thickness
4.3.1.9 functionstifin()
The functionstifin() reads the necessary information of the stiffeners. It
has the following variables:
w : Width of the stiffener
d : Depth of the stiffener
e : Eccentricity of the stiffener
4.3.1.10 functionrgdstf()
The functionrgdstf() calculates the different elements of the rigidity ma-
trix of the stiffener. The following variables are used in the function:
As : Cross-sectional area of the stiffener
Ss : First moment of area of the stiffener
Is : Second moment of area of the stiffener
G : Modulus of rigidity of the stiffener
Js : Torsional constant of the stiffener
dmatx2 : Elements of the stiffener rigidity matrix
4.3 Description of the Programme 111
4.3.2 Processor
This module of the programmes performs the following tasks:
1. Generation of the element matrices.
2. Assembly of the element matrices into global matrices.
3. Imposition of the boundary conditions.
4. Solution of the algebraic equations for static analysis of plates to
obtain nodal unknowns and the computation of the stress resultants
for the skin and the stiffener at all the nodes.
5. Determination of eigenvalues and eigenvectors for the free vibra-
tion and buckling analyses using simultaneous vector iteration tech-
nique.
A flowchart showing the processor unit is presented in the Fig. 4.3.
The various modular functions which are used in this processor unit are
briefly presented herein.
4.3.2.1 functionform-stif-mass-geom()
The functionform-stif-mass-geom()calls the other functions in turn for
the processing of each of the elements.
4.3.2.2 functionelm-stif-mass-geom()
This function generates the elastic and geometric stiffness and mass ma-
trices for the plate element.
112 COMPUTER IMPLEMENTATION
global() global-stif()b
nd
-sti
f()
1
1 2 3
end
sfr1() jacob1() sfr2() bmat()
Computes element matrices
for the plate
elm-stif-mass-geom() elm-stf-mass-geom()Computes element matrices
for the stiffener
global-stif-mass-geom() global-stf-mass-geom()
Bare PlateAnalysis
stif Stiffened PlateAnalysis
soln
AnalysisStatic Free Vibration
Analysis AnalysisBuckling
stop
0
stfin()
Figure 4.3:Processor unit of the computer codes
4.3 Description of the Programme 113
4.3.2.3 functionelm-stf-mass-geom()
This function generates the elastic and geometric stiffness matrices and
the mass matrices of the stiffener elements.
4.3.2.4 functionglobal-stif-mass-geom()
The assembly of all the element matrices of the plate elements into the
global ones are carried out through this function.
4.3.2.5 functionglobal-stf-mass-geom()
The assembly of all the element matrices of the stiffener elements into
the global mtrices are performed through this function.
4.3.2.6 functionglobal()
This is a common function called by the individual functions to assemble
all the element matrices into global matrix.
4.3.2.7 functionelm-load()
This function calculates the consistent element load vector and takes into
account any application of concentrated load on the plate.
4.3.2.8 functiongbl-load()
The generated element load vectors are assembled into global load vector
using this function.
114 COMPUTER IMPLEMENTATION
4.3.2.9 functionbnd-stif()
The stiffness of the boundary lines of the plate element if it happens to
be one of the elements in the periphery is computed in this function.
4.3.2.10 functionsfr2()
This module calculates the displacement shape functions and their deriva-
tives.
4.3.2.11 functionbmat()
This function evaluates the matrix for the strain-displacement relation-
ship.
4.3.2.12 functiondbmat()
The stress resultants are computed in this function.
4.3.2.13 functionsolve()
The functionsolve()is used to solve the simultaneous algebraic equations
generated in the process of analysis. The equilibrium equations are in the
form of [A]{X} = {B}, where[A] is the global stiffness matrix,{B}is the global load vector and{X} is the nodal unknown vector whose
solution is sought. This has been solved using Choleski factorization by
performing decomposition, forward elimination and backward substitu-
tion with the help of the functionsdecomp(), forsol(), andbacksol().
4.3.2.14 functionr8usiv()
This function is used for the eigenvalue solution. Using this module, a
simultaneous iteration algorithm has been adopted for the free vibration
4.3 Description of the Programme 115
and buckling analyses. The input data to this function are the global elas-
tic stiffness matrixgstif, the global geometric stiffness matrixgbl-geom,
the global mass matrixgbl-massand the corresponding pointer vectors.
Through this function the eigenvalues and the corresponding eigenvec-
tors are extracted. The required number of modes of vibration or buckling
is to be specified by the user. The function requires three arraysu, v and
w, of size(n,m) wheren is the total degrees of freedom andm is a value
higher than the number of modes. The numerical value ofm has been
considered as 1.5 times the number of modes in the present programme.
The tolerance value has been set to10−6 and the maximum number of
iterations to 40. The initial trial vectors are generated from a random
number generator. Ther8usiv() module consists of a number of func-
tions which are presented below with brief descriptions of their function-
alities and sequence in which they are called inside the functionr8usiv().
functionr8ured() : decomposes a symmetric matrix into lower
triangular matrix
functionr8uran() : generates random trial vectors
functionr8uort() : orthonormilises the vectors by the Schmidt
process
functionr8ubac() : solves the equation[l]T{v} = {u} by
backward substitution
functionr8upre() : performs premultiplication in the form
{v} = {l}{u}functionr8ufor() : solves equation{l}{v} = {u} by forward
substitution
116 COMPUTER IMPLEMENTATION
functionr8udec() : sorts the vectors{u} and{v} according to the
descending order of eigenvalue prediction
functionr8uran() : generates trial vectors in{w}functionr8uort() : orthonormalizes{w}functionr8uerr() : estimates the vector errors in successive trials
A flowchart of this module is shown in the Fig. 4.4
4.3.3 Postprocessor
In this part of the programme , all the input data are echoed to check for
their accuracy. The functionprint-disp() is used to print the output data
in terms ofdisplacements, moments, stresses, eigenvaluesetc. depend-
ing on the type of analysis carried out. The results are stored in a series
of separate output files for each category of problems analyzed and those
from 0◦ to 60◦ with all edges simply supported and clamped are pre-
sented in the Table 5.20 along with the results of other investigators such
as Mukhopadhyay and Mukherjee [135], Mizusawa et al. [118], Fried
and Schmitt [52], Yoshimura and Iwata [193], Wittrick [190], Durvasula
and Nair [49]. The results are in good agreement. Best agreement is
obtained with Durvasula.
5.3 Arbitrary Stiffened Plates 151
5.2.3.6 Buckling of Uniformly Compressed Bare CircularPlates
The buckling loads for the simply supported and the clamped bare circular
plates are computed and the results are presented in the form of the pa-
rameterk = (Nr)cra2/D where(Nr)cr is the critical compressive force
uniformly distributed around the edge of the plate,a is the radius of the
circular plate andD is the flexural rigidity of the plate. The results are
presented in the Table 5.21 for various mesh divisions of the whole plate
to study the convergence of the buckling parameter and they are com-
pared with the analytical values of Timoshenko and Gere [184]. There is
excellent agreement between the results.
5.3 Arbitrary Stiffened Plates
In this section the problems relating to the arbitrary stiffened plates are
analyzed when they are subjected to static, dynamic and buckling loads
and are presented in the subsequent subsections.
5.3.1 Static Analysis of Arbitrary Stiffened Plates
Stiffened plates of different shapes with eccentric as well as concentric
stiffeners with different boundary conditions and loadings are analyzed
and results are compared with those available. The results are presented
in tabular or graphical forms depending on the suitability for the purpose
of comparing them with those of others.
152 NUMERICAL EXAMPLES
5.3.1.1 Square Plate with a Central Stiffener
A simply supported square plate with a central stiffener as shown in
the Fig. 5.10 is considered. The problem is solved both for eccentri-
cally and concentrically placed stiffeners. The plate and the stiffeners
have the same material properties with elastic modulusE = 1.1721 ×105 N/mm2(17× 106 psi) and Poisson’s ratioν = 0.3. The plate is sub-
jected to a uniformly distributed load of6.89476×10−3 N/mm2(1.0 psi).
The plate is analyzed by the present method using various mesh divisions
and the analysis is carried out with the mesh division of16 × 16 for the
whole plate. The deflection curves along the two centre lines are com-
pared in Table 5.22 and Fig. 5.11 with the results obtained by Rossow
and Ibrahimkhail [157] who applied the constraint method of the finite
element analysis. In Table 5.22, results from NASTRAN and STRUDL
E = 11721 x 10 N/mm5 2
ν = 0.3
0.254
0.2542
.54
Y
X
25.4
A A
25
.4
All dimensions are in mm.
SECTION AT A-A
Figure 5.10: Simply supported square plate with a centralstiffener
5.3 Arbitrary Stiffened Plates 153
Figure 5.11:Variation of deflection along centrelines of sim-ply supported square plate with a central stiff-ener
are also indicated. The agreement between the results is excellent.
To test the convergence of the results obtained by the present method,
the deflection, moment and top fiber stress of the plate skin at the central
location of the eccentrically stiffened square plate for varying number of
mesh divisions are computed and presented in the Table 5.23 from which
the attainment of good convergence for all is evident.
154 NUMERICAL EXAMPLES
Table 5.22: Deflection at the centre of simply supportedsquare stiffened plate(×104 mm.)
Distributed LoadSource
Eccentric Concentric
Rossow and Ibrahimkhail [157] 34.722 115.722
NASTRAN 37.846 -
STRUDL - 115.291
Present Method 34.696 115.875
Table 5.23: Convergence of deflection(w), plate moment(My) and plate stress(σx) of the eccentricallystiffened square plate at its centre with differentmesh divisions.
Mesh Deflection(w) Moment(My) Stress(σx)
Division ×104(mm.) ×103(N −mm/mm) (N/mm2)
2× 2 32.614 6.005 4.532
4× 4 34.163 21.943 9.339
6× 6 34.493 24.412 9.992
8× 8 34.595 25.275 10.189
10× 10 34.646 25.697 10.271
12× 12 34.671 25.933 10.3097
14× 14 34.671 26.075 10.3297
16× 16 34.696 26.169 10.3394
5.3 Arbitrary Stiffened Plates 155
Table 5.24:Central deflection of rectangular cross-stiffenedplate (×103 mm.)
Concentrated Load Distributed LoadSource
Eccentric Concentric Eccentric Concentric
Rossowand
Ibrahimkhail[157]
32.258 87.986 224.790 611.505
NASTRAN 31.496 - 221.336 -
Chang [32] 31.648 87.986 228.498 611.556
STRUDL - 87.960 - 612.648
Presentmethod
31.445 87.986 226.873 611.454
5.3.1.2 Cross Stiffened Rectangular Plate
A simply supported rectangular plate with a central stiffener in each di-
rection shown in Fig. 5.12 is analyzed for a uniform pressure of6.89 ×10−2 N/mm2 (10.0 psi) as well as for a concentrated load of4.448 kN
(1.0 kip) at the centre of the plate with a mesh division of16 × 16. The
material properties areE = 2.0684 × 105 N/mm2 (30 × 106 psi) and
ν = 0.3 for both the plate and the stiffener. The same problem is solved
by Rossow and Ibrahimkhail [157] by applying the constraint method to
the finite element analysis and by Chang [32] using conventional finite
element method. The deflection and bending moments along the differ-
ent lines for eccentrically as well as concentrically stiffened plates sub-
jected to uniformly distributed load as well as concentrated load obtained
by different methods are compared in Figs. 5.13-5.23. Additionally a
156 NUMERICAL EXAMPLES
6.35
12.7
12
7
Section at A-A
E = 2.0684 x 10 N/mm5 2
ν = 0.3
6.35
12.7
76
.2
Section at B-B
Sti
ffen
ers
A
762
y
B B
A
15
24
x
All dimensions are in mm.
Figure 5.12:Simply supported rectangular plate with a cen-tral stiffener in each direction
comparison of the central transverse displacements obtained by various
methods is made in the Table 5.24.
5.3 Arbitrary Stiffened Plates 157
Figure 5.13: Deflection at x = 190.5 mm. and x = 381.0 mm.for plate with two concentric stiffeners, underdistributed load
Figure 5.14: Deflection at x = 190.5 mm. and x = 381.0 mm.for plate with two eccentric stiffeners, underdistributed load
158 NUMERICAL EXAMPLES
Figure 5.15: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two concentric stiffeners,under distributed load
Figure 5.16: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two eccentric stiffeners,under distributed load
5.3 Arbitrary Stiffened Plates 159
Figure 5.17: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two concentric stiffeners,under distributed load
Figure 5.18: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two eccentric stiffeners,under distributed load
160 NUMERICAL EXAMPLES
Figure 5.19:Deflections at x = 190.5 mm. and x = 381.0 mm.for plate with two stiffeners under concentratedload
Figure 5.20: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two concentric stiffenersunder concentrated load
5.3 Arbitrary Stiffened Plates 161
Figure 5.21: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two eccentric stiffenersunder concentrated load
Figure 5.22: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two concentric stiffenersunder concentrated load
162 NUMERICAL EXAMPLES
Figure 5.23: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two eccentric stiffenersunder concentrated load
5.3.1.3 Rectangular Multi-Stiffened Plate
A rectangular steel plate stiffened with nine number of equispaced T-
stiffeners along the short span as shown in the Figure 5.24 is analyzed
by the present method with a mesh division of16 × 16 for the whole
plate. The plate is subjected to a concentrated load of 10 kN at its centre.
The long edges are considered as simply supported and the short ones
free. The deflections and bending moments along the central line in the
longitudinal direction are computed and the results are compared graph-
ically in the Figures 5.25-5.26 with those of Smith [179] who solved the
problem based on generalized slab beam analysis.
5.3 Arbitrary Stiffened Plates 163
E = 2.06843 x 10 N/mm
ν = 0.3
5 2
y
A A
25
40
x
76.2
Section at A-A
14
.22
4
15
2.4
All dimensions are in mm.
4.7752
508
Figure 5.24:Rectangular multi-stiffened plate
164 NUMERICAL EXAMPLES
Figure 5.25: Variation of deflection along the centre line ofthe rectangular multi-stiffened plate
Figure 5.26:Variation of plate moment Mx along the centreline of the rectangular multi-stiffened plate
5.3 Arbitrary Stiffened Plates 165
5.3.1.4 Rectangular Slab with Two Edge Beams
Three numbers of specimens of slabs having edge beams at two opposite
edges made of Araldite as shown in the Figure 5.27 are considered for the
analysis using the present method employing a mesh division of16× 16
for the whole plate. Each of the specimens is considered to be free at the
edges where the beams are placed and simply supported along the edges
transverse to the beams. Two concentrated loads of equal magnitude are
applied at the centre of each of the beams. The Table 5.25 describes the
details of the geometrical and the material properties of the specimens
along with the magnitude of the concentrated loads applied to each one
of them. The deflections along different sections and the normal stresses
at the bottom of the beam along its length are compared graphically with
the finite element results of Mukhopadhyay and Satsangi [136] in the
Figures 5.28-5.31. Additionally, the numerical values of the deflections
and the stresses at some typical points are compared with the theoretical
and experimental results of Allen [5] along with the finite element results
of Mukhopadhyay and Satsangi [136] and the spline finite strip results
of Sheikh [172] in the Table 5.26. The agreement between the results is
found to be reasonably close.
166 NUMERICAL EXAMPLES
Table 5.25: Geometrical and material properties of thespecimens of the rectangular slab with edgebeams
(Ref. Figure 5.27)
Specimen a b d L
No. (mm) (mm) (mm) (mm)
SPEC1 138.58 4.52 19.10 131.78
SPEC2 138.58 4.52 15.24 131.78
SPEC3 138.58 4.52 11.43 131.78
Specimen t E ν P
No. (mm) (N/mm2) (N )
SPEC1 4.445 2977 0.35 176.59
SPEC2 4.445 2977 0.35 161.03
SPEC3 4.445 2977 0.35 61.39
5.3 Arbitrary Stiffened Plates 167
t
C
A
A
B
B
C
d
a
L
x
y
b
Figure 5.27:Rectangular slab with two edge beams
Figure 5.28: Deflection along A-A of the slab with edgebeams
168 NUMERICAL EXAMPLES
Figure 5.29: Deflection along B-B of the slab with edgebeams
Figure 5.30: Deflection along C-C of the slab with edgebeams
5.3 Arbitrary Stiffened Plates 169
Figure 5.31: Stress at the beam soffit of the slab with edgebeams
Table 5.26: Deflection and stress at the beam soffit of therectangular slab with edge beams
Source Specimen No.
of Result 1 2 3
Deflection Present Method 0.4216 0.7214 0.5842
(mm.) Sheikh [172] 0.4191 0.7290 0.6045
at 25.4 mm. Experimental [5] 0.4521 0.6477 0.5232
from the Theoretical [5] 0.4064 0.7087 0.5918
beam centre Mukhopadhyay 0.4572 0.6477 0.6756
and Satsangi [136]
Stress Present Method 14.5410 20.1534 12.3485
(N/mm2) Sheikh [172] 15.5615 21.7323 13.5758
at the Experimental [5] 15.3339 25.0568 12.9759
beam centre Theoretical [5] 13.3000 18.3883 11.4798
Mukhopadhyay 14.8858 20.1396 12.5071
and Satsangi [136]
170 NUMERICAL EXAMPLES
5.3.1.5 Stiffened Skew Bridge Deck
A perspex scale model of a450 skew bridge deck (Fig. 5.32) having five
equispaced longitudinal beams and two transverse edge beams supported
at the ends of each of the equally spaced longitudinal beams, the vertical
deflection and movement in the direction of the axes of the transverse
beams at these points being restrained and subjected to a concentrated
vertical load of 100N acting at the midpoint of one of the longitudinal
free edges is analyzed with a mesh of16× 16 applied to the entire plate.
The transverse and the longitudinal beams are having depths and widths
of 22.0mm and 6.0mm respectively. The Young’s modulus and Pois-
son’s ratio for the 3-mm thick slab are 3354.0N/mm2 and0.390 respec-
tively and those for the beams are 3176.0N/mm2 and 0.335 respectively.
The vertical deflections along the loaded free edge and along the centre
line in the transverse direction are compared in Fig. 5.33-5.34 with the
experimental results and also with those of Just [74]. The agreement is
reasonably good.
5.3 Arbitrary Stiffened Plates 171
Y
jY
j
A A645o
3
3
300
>
ULongitudinal Beams
:yTransverse Beams
¸
®
¸
¸
¸
®
®
®
®¸
®¸
¸¸X
X-
-
Y
Y
?
?
6
¾- 6
3
22
Section at X-X
6
?
?
¾- 6
Section at Y-Y
3
22
All dimensions are in mm.
9090
9090
45
45
B
B
Figure 5.32: Skew bridge deck with beams in both direc-tions
172 NUMERICAL EXAMPLES
Figure 5.33: Deflection along A-A of the stiffened skewbridge deck
Figure 5.34: Deflection along B-B of the stiffened skewbridge deck
5.3 Arbitrary Stiffened Plates 173
5.3.1.6 Stiffened Curved Bridge Deck
A perspex model of a horizontally curved bridge deck (Fig. 5.35) with
two curved girders having free curved boundaries and simply supported
on straight edges subjected to a concentrated load of100 N applied at dif-
ferent points of the deck is analyzed with different mesh sizes. The con-
vergence study for this problem for the load placed at position 4 and 7 for
the deflection at various positions on the outer girder and the outer edge
and that of the moments at the centre are presented in Table 5.27-5.28
respectively. The convergence for the deflection and moment is found to
be good. The deflections obtained with a mesh division of16 × 16 at
the inner edge, inner girder, outer girder and outer edge are compared
with the theoretical and experimental results of Joshipara [73], finite
element results of Kalani et al. [75] and Satsangi and Mukhopadhyay
[163] in Table 5.29-5.30. The elastic modulus and Poisson’s ratio are
3.6 × 104Kg/cm2 and 0.38 respectively. At all places except the inner
edge, present results compare favorably with the experimental ones.
174 NUMERICAL EXAMPLES
Figure 5.35: Curved bridge deck with two circumferentialgirders
5.3 Arbitrary Stiffened Plates 175
Table 5.27: Convergence of deflection at outer girder andouter edge
Figure 5.44: Annular sector plate with eccentrically placedcircumferential stiffeners
196 NUMERICAL EXAMPLES
Table 5.39: Frequency parameter [ωa2√
ρt/D] of annularsector plate with eccentrically placed circumfer-ential edge stiffeners
Mode sequences
Stif
fene
r
Met
hod
1 2 3 4 5 6
A 23.445 30.041 44.436 56.250 77.176 81.324
S1 P 23.172 29.276 42.981 55.975 75.013 81.895
A 19.466 26.978 41.382 51.955 74.442 77.792
S2 B 19.011 27.514 40.779 50.196 73.203 76.433
P 19.041 26.359 40.671 49.283 72.866 75.635
P - Present;
A - Mukherjee and Mukhopadhyay [123];
B - Sheikh and Mukhopadhyay [173]
5.3 Arbitrary Stiffened Plates 197
5.3.2.10 Free Vibration of Circular Stiffened Plates
Three different square stiffened plates attempted in the earlier sections
are extended to circular stiffened plates keeping all the properties of the
corresponding plate and the stiffener same but the geometry of the plate
being changed from a square to a circular one keeping the diameter equal
to one of the sides of the square stiffened plate. In the first case the
clamped square stiffened plate having the central concentric stiffener pre-
sented in the section 5.3.2.1 is extended to the circular one (Fig. 5.45).
Similarly in the second case, the same square plate but having an eccen-
tric rectangular stiffener (20mm×3mm) considered by Rao et al. [154]
and presented in the section 5.3.2.2 is extended to the circular stiffened
plate. Lastly, a clamped square cross-stiffened plate with concentric
stiffeners analyzed by Mizusawa et al. [116] and attempted in the sec-
tion 5.3.2.3 is extended to the desired circular one. The results of the first
seven natural frequencies of all the circular stiffened plates (C1, C2, C3)
are presented in Table 5.40 along with the corresponding original square
stiffened plate results (S1, S2, S3) for comparison of the effects of the
curved boundaries on the frequencies/frequency parameters. There is in-
crease in the frequencies in all the cases for all the modes because of the
curved boundaries which is expected, as the circular plate having the di-
ameter equal to the side of the square plate is stiffer than the square one.
The circular stiffened plate results are presented for the first time.
5.3.2.11 Free Vibration of Elliptical Stiffened Plate
The simply supported rectangular plate with a central eccentric L-shaped
stiffener presented in the previous example in the Section 5.3.2.4 is ex-
tended to an elliptical stiffened plate by retaining all its properties but
198 NUMERICAL EXAMPLES
Table 5.40:Frequency (Hz/Parameter) of all edges clampedcircular stiffened plates
C1: Diameter = 600 mm, plate thickness = 1.0 mm,
ν = 0.34 ρ = 2.78× 10−6 Kg/mm3,
E = 6.87× 107 N/mm2, Is = 2290 mm4,
As = 67.0 mm2, Js = 22.35 mm4
(Nair and Rao [137]) (concentric stiffener)
C2: Material properties and plate dimensions same as
CIR-1 but with an eccentric rectangular stiffener
of size20 mm× 3 mm
(Rao et al. [154])
C3: Diameter = a, EIs/Da = 10.0, As/ah = 0.1
(Mizusawa et al. [116])
T Mode sequences
1 2 3 4 5 6 7
S1 H 50.152 63.410 74.132 84.243 111.990 118.338
C1 58.475 72.467 96.724 111.189 139.549 149.083
S2 H 54.371 65.101 79.808 84.584 116.531 118.676
C2 63.216 74.660 104.865 111.880 145.922 156.153
S3 P 11.35 11.70 12.65 12.65 17.42 24.81 24.96
C3 13.22 14.84 16.58 16.58 23.25 28.71 30.08
T - Type of plate
H - Frequency in Hz
P - Frequency parameter[ω(a/π)2√
ρt/D]
5.3 Arbitrary Stiffened Plates 199
A A
600 mm
1mm
3 mm
20
mm
SECTION AT A-A
xρ = 2.78 10 ν = 0.34
Kg/mm3
E = 6.87 x 10 N/mm7 2
-6
Figure 5.45:Circular plate with a central stiffener
changing the geometrical shape of the plate to an ellipse by keeping the
major and minor axes of the ellipse equal to the length and width of the
rectangular plate respectively (Fig. 5.46). The results of the first six fre-
quencies for the elliptical stiffened plate (ELP) as well as the correspond-
ing rectangular stiffened plate (REC) are presented in the Table 5.41.
While comparing the results of the elliptical stiffened plate with those of
the rectangular stiffened plate an increase in the frequencies is observed
in the second, fourth and the fifth mode of the elliptical plate but the first
200 NUMERICAL EXAMPLES
45
0 m
m
650 mm
3 mm
10 mm
20
mm
3 mm4.98 mm
SECTION AT A-A
11 2E = 2.051 x 10 N/mm
ρ = 7825 Kg / m3
ν = 0.3
Figure 5.46:Elliptical plate with a central stiffener
and the third mode frequencies are found to be lower whereas there is
hardly any change in the frequency of the sixth mode. The results are
presented for the first time.
5.3 Arbitrary Stiffened Plates 201
Table 5.41: Frequency in Hz of a simply supported ellipti-cal plate with a central eccentric stiffener in theshorter span direction
ν = 0.3, ρ = 7825 Kg/m3, E = 2.051× 1011 N/m2
Plate Mode sequences
Type 1 2 3 4 5 6
REC 142.76 174.85 334.83 352.82 367.67 517.48
ELP 129.10 234.75 241.30 383.25 423.16 518.72
5.3.3 Stability Analysis of Arbitrary Stiffened Plates
Stability analysis for the stiffened plates with various configurations and
boundary conditions is carried out and the buckling parameters are tabu-
lated and compared with the published results of the other investigators
wherever possible.
5.3.3.1 Buckling of Square Stiffened Plates
A number of square stiffened plates with a concentric central stiffener
have been analyzed for various stiffener sizes and flexural rigidities, and
the buckling parameters are presented in Table 5.42 for plates with dif-
ferent boundary conditions. The ratio of the cross-sectional area of the
stiffener to that of the plate (As/bt) is varied from0.05 to 0.20 and the
ratio of the bending stiffness of the stiffener to that of the plate (EIs/bD)
202 NUMERICAL EXAMPLES
is varied from5 to 25. The torsional inertia of the stiffener has been ne-
glected in the analysis. The results are compared with the semi-analytic
finite difference results of Mukhopadhyay [131] and they agree fairly
well.
5.3.3.2 Buckling of Simply Supported Rectangular StiffenedPlates
A series of simply supported rectangular stiffened plates with a concen-
tric central stiffener have been analyzed for various proportions of the
plate and of the stiffener and the buckling parameters are presented along
with those of other investigators in Table 5.43. The ratio of the cross-
sectional area of the stiffener to that of the plate (As/bt) is varied from
0.05 to 0.20 and the ratio of the bending stiffness of the stiffener to that
of the plate (EIs/bD) is varied from5 to 20. The torsional inertia of
the stiffener has been neglected in the analysis. To analyze this prob-
lem Mukhopadhyay [131] used the semi-analytic finite difference method
whereas Mukhopadhyay and Mukherjee [135] used the finite element
method. Good agreement of the results is obtained when compared with
the results of Timoshenko and Gere [184] and those of others.
5.3.3.3 Buckling of Rectangular Stiffened Plates with Dif-ferent Boundary Conditions
Rectangular stiffened plates with a concentric central stiffener have been
analyzed for two aspect ratios of the plate and for different proportions
and rigidities of the the stiffener and the buckling parameters are pre-
sented in Table 5.44. As before the ratio of the cross-sectional area of the
stiffener to that of the plate (As/bt) is varied from0.05 to 0.20 and the
ratio of the bending stiffness of the stiffener to that of the plate (EIs/bD)
5.3 Arbitrary Stiffened Plates 203
Table 5.42: Buckling parameter k = λb2/π2D for squareplate with a central concentric stiffener sub-jected to uniaxial and uniform compression inthe stiffener direction
ν = 0.3
Boundary Condition
CCCC CSSC
EI s
/bD
As/b
t
Present [131] Present [131]
0.05 24.25 25.46 17.35 17.32
5 0.10 24.25 25.46 17.15 17.05
0.20 24.25 25.46 16.41 16.27
0.05 24.25 - 17.94 -
10 0.10 24.25 - 17.93 -
0.20 24.25 - 17.90 -
0.05 24.25 25.46 18.03 18.36
15 0.10 24.25 25.46 18.03 18.36
0.20 24.25 25.46 18.02 18.34
0.05 24.25 25.46 18.070 -
20 0.10 24.25 25.46 18.068 -
0.20 24.25 25.46 18.064 -
0.05 24.25 - 18.09 18.46
25 0.10 24.25 - 18.09 18.46
0.20 24.25 - 18.09 18.46
204 NUMERICAL EXAMPLES
Table 5.43:Buckling parameter k = λb2/π2D for uniformlycompressed all edges simply supported rectan-gular stiffened plates
ν = 0.3
Aspect Ratio (a/b)
1 2
EI s
/bD
As/b
t
Present [184] [135] Present [184] [131]
0.05 11.84 12.0 11.72 7.93 7.96 7.93
5 0.10 11.02 11.1 10.93 7.27 7.29 7.28
0.20 9.64 9.72 9.70 6.24 6.24 6.24
0.05 15.73 16.0 16.0 10.16 10.20 10.16
10 0.10 15.73 16.0 16.0 9.33 9.35 9.33
0.20 15.49 15.8 15.44 8.02 8.03 8.02
0.05 15.73 16.0 16.0 12.36 12.4 -
15 0.10 15.73 16.0 16.0 11.36 11.4 -
0.20 15.73 16.0 16.0 9.77 9.80 -
0.05 15.73 16.0 - 14.52 14.6 -
20 0.10 15.73 16.0 - 13.36 13.4 -
0.20 15.73 16.0 - 11.51 11.6 -
5.3 Arbitrary Stiffened Plates 205
Table 5.44: Buckling parameter k = λb2/π2D for rectangu-lar plate with a central concentric stiffener sub-jected to uniaxial and uniform compression inthe stiffener direction
ν = 0.3
Boundary Condition
CSCS SCSC
Aspect Ratio Aspect Ratio
EI s
/bD
As/b
t
1 2 1 2
0.05 18.98 13.54 21.77 18.03
5 0.10 18.98 12.61 21.40 16.41
0.20 18.98 11.03 16.47 13.85
0.05 18.98 16.64 21.76 21.18
10 0.10 18.98 16.64 21.76 21.25
0.20 18.98 16.66 21.76 19.63
0.05 18.98 16.66 21.76 21.17
15 0.10 18.98 16.66 21.76 21.16
0.20 18.98 16.66 21.76 21.26
0.05 18.98 16.66 21.76 21.15
20 0.10 18.98 16.66 21.76 21.15
0.20 18.98 16.66 21.76 21.22
is varied from5 to 20. The torsional inertia of the stiffener has been
neglected in the analysis. These results are presented for the first time.
206 NUMERICAL EXAMPLES
Table 5.45: Buckling parameter k = λb2/π2D for skewstiffened plate
(Aspect Ratio = 1.0)EIs/bD = 10.0; GJs/bD = 0.0;
As/bt = 0.1; ν = 0.3
Boundary Condition Skew Angle Present [118] [135]
0 16.0 16.0 16.0
All Edges Simply Supported 30 19.96 20.28 20.90
45 27.68 28.68 29.89
0 24.24 24.89 30.8
All Edges Clamped 30 32.41 33.74 36.9
45 47.97 51.62 56.3
5.3.3.4 Buckling of Skew Stiffened Plates with DifferentBoundary Conditions
Skew stiffened plates with a concentric central stiffener and having dif-
ferent boundary conditions have been analyzed for different skew angles
and the buckling parameters are presented in Table 5.45. The present
results agree well with the finite element results of Mukhopadhyay and
Mukherjee [135] and those of Mizusawa et al. [118] who analyzed the
problem usingB-spline functions.
5.3.3.5 Buckling of Uniformly Compressed DiametricallyStiffened Circular Plates
The buckling loads for the all edges simply supported (SS) and clamped
(CC) circular plates with concentric stiffeners along the diameters are
computed with varying flexural and torsional stiffness parameters of the
5.3 Arbitrary Stiffened Plates 207
Table 5.46: Buckling parameter k = (Nr)cr a2/D for uni-formly compressed circular plates with concen-tric stiffeners along the diameter with varyingflexural and torsional stiffness parameters of thestiffener
As/at = 0.1; ν = 0.3
EIs
aD
GJs
aDk
Single Stiffener Cross Stiffeners
SS CC SS CC
0.0 0.0 4.20 14.72 4.20 14.72
2.5 0.0 7.09 26.64 10.79 44.34
5.0 0.0 9.66 26.65 16.67 44.35
7.5 0.0 11.92 26.65 22.21 44.35
10.0 0.0 13.19 26.65 27.22 44.35
0.0 0.0 4.20 14.72 4.20 14.72
0.0 2.5 4.27 14.97 4.63 16.28
0.0 5.0 4.27 14.97 4.63 16.29
stiffener and the results are presented in the form of buckling parameter
k = (Nr)cra2/D where(Nr)cr is the critical compressive force uniformly
distributed around the edge of the plate,a is the radius of the circular plate
andD is the flexural rigidity of the plate . The results are presented in
the Table 5.46. These results are presented for the first time.
Chapter -6
CONCLUSIONS
6.1 Summary
The investigation carried out in this thesis may be summarized as fol-
lows:
• A new four-noded plate bending element has been proposed in the
manner of isoparametric element for the analysis of the bare plates,
which is derived from the simplest rectangular basic plate bend-
ing element having 12 degrees of freedom largely known asACM
element. The new element has all the advantages of an isoparamet-
ric plate bending element by which it is capable of accommodating
the arbitrary geometrical configuration of a plate but without the
disadvantages of the shear-locking problem inherent in the isopara-
metric element.
• A stiffened plate bending element has been proposed for the anal-
ysis of the stiffened plates by combining the 12 degrees of free-
dom rectangular basicACM element and a four-noded rectangu-
lar plane stress element of 8 degrees of freedom. For this stiffened
plate element, a general curved stiffener element has been proposed
209
210 CONCLUSIONS
as a discrete element whose direction and orientation are arbitrary
inside the plate element facilitating its placement and the mesh
generation in the plate independent of the nodal lines. The com-
patibility between the stiffener element and the plate skin is main-
tained by expressing the stiffness matrix of the stiffener element
in terms of the nodal degrees of freedom of the plate element.
Through this compatibility, the contribution of the stiffener to the
stiffness of the plate bending element is truly reflected by its lo-
cation and orientation inside the element, contributing more to the
nodes at the vicinity and less to the far ones.
• A consistent formulation has been carried out for the derivation of
the stiffness matrix for a curved boundary assuming it to be sup-
ported on elastic springs continuously spread along the boundary
line. The springs are considered in the directions of the possible
displacements and rotations of the boundary line. Any specific
boundary condition can be attained from this general one by as-
signing an appropriate value to the spring constants.
• Using these elements, static, free vibration and stability analyses
of bare plates and the stiffened plates have been carried out. In
the extensive numerical examples presented, an attempt has been
made to include various plate geometries such as square, rectangu-