Structural Analysis of Stiffened Plates and Shells Analysis & Design Center for Advanced Vehicles 5...

1 Multidisciplinary Analysis & Design Center for Advanced Vehicles

Buckling Analysis of Stiffened plates with straight and curvilinear stiffener(s)

Wei Zhao

Dec,13, 2013

ESM 6044: Theory of plates and shells

Department of Aerospace and Ocean Engineering


Outline

Background

Analytical solutions to straightly stiffened plate

Numerical solutions to curvilinearly stiffened plate

Summary


Background Panels in wing box can be modeled as plate element, compressive stress,

shear stress, generated by the aerodynamics, can be applied at the edges

of these panels. Local and global buckling can be occurred at such loads or

combination of loads.

Stiffeners, straight or curve, will be added on both top and bottom skin of the

wing to stiff the panel to resist local mode and buckling.

Fuselage or submarine can be modeled as cylindrical shell, buckling of

cylindrical shell can occur when they are subjected to the action of axial

compression, circumference pressure , torque, or combination of these

loads.

Hoop stiffeners or axial stiffener or both can be added on the cylindrical

shell to resist the buckling

stiffener Stiffened plate with stiffeners;

(a). Straight stiffeners;

(b). Curvilinear stiffeners

(a).

(b).

stiffener


Elastic stability of flat plate

Consider the forces applied at the edges of the

plate, shown in Fig. 1, acting in the middle plane,

equilibrium equation:

𝐷𝛻4w = q + Nx𝜕2𝑤

𝜕𝑥2+ Ny𝜕2𝑤

𝜕𝑦2+ 2Nxy

𝜕2𝑤

𝜕𝑥𝜕𝑦

Consider the case in Fig.2, simply supported

conditions, the deflection of the plate can be

represented as:

𝐷𝜋4𝑚2

𝑎2+𝑛2

𝑏2− 𝑁𝑥𝜋

2𝑚2

𝑎2𝑤𝑚𝑛 sin

𝑚𝜋𝑥

𝑎sin𝑛𝜋𝑦

𝑏

∞

𝑛=1

∞

𝑚=1

= 0

𝑁𝑥𝑐𝑟 =𝜋2𝐷

𝑏2𝑚

𝛽+ 𝛽𝑛2

𝑚

2

, 𝛽 =𝑎

𝑏

Approximate critical buckling stress:

Fig. 1

Fig. 2 𝜎𝑥,𝑐𝑟 = 𝑁𝑥,𝑐𝑟/ℎ

𝜎𝑥,𝑐𝑟 =𝜋2𝐸

3 1 − 𝜈2ℎ

𝑏

2

Nx here is applied load, it’s NOT stress! Be

careful on the total potential strain energy

when applied with initial stress.


Critical buckling parameter and buckling mode

𝐾𝑐𝑟 =𝑚

𝛽+ 𝛽1

𝑚

2

E.g. 𝛽 = 1, m=1k=4;

𝑁𝑥,𝑐𝑟 = 4𝜋2𝐷

𝑏2~ℎ3 (width and length are fixed)


Elastic stability of flat plate with equidistance stiffeners

Fig. 1

Fig. 2

Assumptions:

1). The torsion rigidity GJ of stiffeners can be neglected during buckling, stiffeners

are mainly used to increase flexural rigidity of stiffened structures.

2). Global buckling of stiffened structure is considered in this problem, not the

case of local buckling of plate between two stiffeners.


Equilibrium Equation Method

Compatibility conditions:

The plate buckles together with stiffener, buckling mode is symmetric about

the line y=0; 𝜕𝑤

𝜕𝑦= 0.

The intensity of shear loading transmitted from plate to stiffener contributes to

stiffener equilibrium.

Outward bending deflections of stiffener and plate are equal.

𝑤 𝑥 = 𝐹 𝑦 sin𝑚𝜋𝑥

𝑎

𝑑4𝐹 𝑦

𝑑𝑦4− 2𝑚𝜋

𝑎

2 𝑑2𝐹 𝑦

𝑑𝑦2+𝑚𝜋

𝑎

2 𝑚𝜋

𝑎

2

+𝑁𝑥𝐷𝐹 𝑦 = 0

𝛽 =𝑎

𝑏 , 𝛾 =𝐸𝐼𝑖𝐷𝑏, 𝛿 =𝐴𝑖𝑏ℎ

𝑁𝑥,𝑐𝑟 =𝜋2𝐷

𝑏21 + 𝛽2 2 + 2𝛾

𝛽2 1 + 2𝛿


Energy Method

Taking

𝑤 = 𝐴𝑚𝑛 sin𝑚𝜋𝑥

𝑎sin𝑛𝜋𝑦

𝑏

∞

𝑛=1

∞

𝑚=1

Strain energy 𝑈𝑝 ,𝑈𝑠 and potential energy 𝑇𝑝 and 𝑇𝑠

Total strain energy:

𝑇 =−𝑁𝑥2

𝑎𝑏

4

𝑚2𝜋2

𝑎2 𝐴𝑚𝑛2

∞

𝑛=1

∞

𝑚=1

−𝑁𝑥,𝑖𝐴𝑖ℎ

𝜋2

4𝑎 𝑚2

𝑚

𝐴𝑚1 sin𝜋𝑐𝑖𝑏+ 𝐴𝑚2 sin

2𝜋𝑐𝑖𝑏+ ⋯

2

𝑈 =𝜋4𝐷

2

𝑎𝑏

4 𝐴𝑚𝑛

2𝑚2

𝑎2+𝑛2

𝑏2

2∞

𝑛=1

∞

𝑚=1

+𝜋4𝐸𝐼𝑖4𝑎3 𝑚4 𝐴𝑚1 sin

𝜋𝑐𝑖𝑏+ 𝐴𝑚2 sin

2𝜋𝑐𝑖𝑏+⋯

2∞

𝑚=1

Π = 𝑈 + 𝑇 Differentiate Π with respect to 𝐴𝑚𝑛 equating to zero to obtain linear

algebraic homogenous equations, find the critical stress by equating

the determinate of this system to zero. 𝜕Π

𝜕𝐴𝑚𝑛= 0

𝑈𝑝 𝑈𝑠

𝑇𝑝 𝑇𝑠


Elastic stability of flat plate with one central straight stiffener

Introduce some parameters:

𝛽 =𝑎

𝑏= 1, 𝛾𝑖 =

𝐸𝐼𝑖𝐷𝑏= 0.4 , 𝛿𝑖 =

𝐴𝑖𝑏ℎ= 0.02, 𝑐𝑖 =

𝑏

𝑛𝑢𝑚𝑠 + 1

The stiffened plate buckles into one half-wave and we can take m=1;

(buckling mode is perpendicular to force-direction). Here stiffeners have

same EI and size.

One stiffener, 𝑐𝑖 =𝑏

2, critical compressive force


𝑏21 + 𝛽2 2 + 2𝛾

𝛽2 1 + 2𝛿= 4.62

𝜋2𝐷

𝑏2

Two stiffeners, 𝑐𝑖 =1

3𝑏,2

3b


𝑏21 + 𝛽2 2 + 3𝛾

𝛽2 1 + 3𝛿= 4.91

𝜋2𝐷

𝑏2

𝑖 stiffeners, 𝑐𝑖 =𝑏

𝑖+1, ……


𝑏2

1 + 𝛽2 2 + 2 𝛾𝑖𝑖 sin2 𝜋𝑐𝑖𝑏

𝛽2 1 + 2 𝛿𝑖 sin2 𝜋𝑐𝑖𝑏𝑖

= 𝐾𝑐𝑟𝜋2𝐷

𝑏2


Critical buckling parameter of unstiffened plate and stiffened plate

Suppose the stiffeners have same size and EI, then we can obtain the critical

buckling parameter K_cr increases as the number of stiffeners increase, shown

in the below plot.

For this case, we can increase

the critical buckling

compressive force by

increasing ratio of bending

rigidity 𝛾, and decreasing the

area ratio 𝛿 when the number

of stiffener is fixed. Aspect ratio

𝛽 of this plate is fixed in this

case.

𝑁𝑥,𝑐𝑟 = 𝐾𝑐𝑟 ∗𝜋2𝐷

𝑏2

𝐾𝑐𝑟 =1 + 𝛽2 2 + 2 𝛾𝑖𝑖 sin

2 𝜋𝑐𝑖𝑏

𝛽2 1 + 2 𝛿𝑖 sin2 𝜋𝑐𝑖𝑏𝑖


Weight increment via Critical buckling parameter

Global buckling can be resisted using stiffeners with less weight

increment than that from increasing thickness of the plate;

It’s efficient and economical configuration;

Structural optimization.(bending rigidity ratio, and area ratio)

Increase thickness

of the plate

Increase No.

of stiffeners

Difference of

increment of weight.

Weight increment

=Δ𝑊 ∗ 𝑎𝑏ℎ


Buckling analysis of plates — Foppl Von Karman plate theory

Flopp Von Karman plate theory

Green-Lagrangian stress tensor

𝜖𝑖𝑗 =1

2

𝜕𝑢𝑖

𝜕𝑢𝑗+𝜕𝑢𝑗

𝜕𝑢𝑖+𝜕𝑢𝑘

𝜕𝑢𝑖

𝜕𝑢𝑘

𝜕𝑢𝑗

Initial stress 𝜎0 is in-plane stress:

𝜎0 =𝜎𝑥0 𝜏𝑥𝑦

0

𝜏𝑥𝑦0 𝜎𝑦

0

Plates with initial in-plane stress, i.e. residual stress or thermal stress,

assume is 𝜎0.

Total potential energy U

𝑈 =1

2 𝜖𝑇𝐷𝑓𝜖𝑉

𝑑𝑉 +𝛼

2 𝛾𝑇𝐷𝑠𝛾𝑉

𝑑𝑉 + 𝜎0 𝑇𝜖𝑁𝐿

𝑉

𝑑𝑉

𝐾𝐺 = 𝐾𝐺𝑠 + 𝐾𝐺𝑏 𝐾

𝑲− 𝜆𝑲𝑮 𝒒 = 0

𝒖 = 𝑢, 𝑣, 𝑤, 𝜃𝑥, 𝜃𝑦


Buckling Eigenvalue (Load Factor)

𝜎0 =1 11 1

∗ 10−3 Pa

ℎ

𝑎= 0.001;

𝑎 = 1; 𝐸 = 69𝑒7 𝑃𝑎 𝜈 = 0.3


Stiffened plate with curvilinear stiffener(s)

𝑎

𝑥

𝑦 𝑆. 𝑆

𝑆. 𝑆

𝜎𝑥 𝜎𝑥

𝑛

𝑡

𝑏

stiffener

𝑎

𝑥

𝑦

𝑆. 𝑆

𝑆. 𝑆

𝜎𝑥

𝜎𝑥

𝑛 𝑡

𝑎 𝑡

𝑛

stiffener

Rectangular plate with one curvilinear stiffener

Square plate with two curvilinear stiffeners

Finite element model of stiffened plate

(8-node plate element & 3-node beam element)

(a)

(b) (c)


Relevant theories-Plate

Theory of plate: Foppl-Von Karman plate theory

Displacement field

8-noded isoparametric plate element

𝑈 𝑥, 𝑦, 𝑧 = 𝑢 𝑥, 𝑦 + 𝑧 ∗ 𝜃𝑥 𝑥, 𝑦

𝑉 𝑥, 𝑦, 𝑧 = 𝑣 𝑥, 𝑦 + 𝑧 ∗ 𝜃𝑦 𝑥, 𝑦

𝑊 𝑥, 𝑦, 𝑧 = 𝑤 𝑥, 𝑦

𝒙𝑝 = 𝑁𝑖𝑥𝑖

8

𝑖=1

𝒖𝑝 = 𝑵𝑖𝒖𝑝𝑖

8

𝑖=1

𝑈𝑝/𝑠 =1

2 𝜖𝑃

𝐿𝑇𝐷𝑝𝜖𝑝𝐿

Ω

𝑑Ω =1

2 𝑢𝑝

𝑇𝐿𝑃𝐿 𝑇𝐷𝑝𝐿𝑝

𝐿

Ω

𝑢𝑝𝑑Ω

𝐺𝑝/𝑠 =1

2 𝜎𝑝𝜖𝑝

𝑁𝐿

Ω

𝑑Ω =1

2 𝑢𝑝

𝑇𝐿𝑝𝑁𝐿𝑇𝜎𝑝𝐿𝑝

𝑁𝐿𝑢𝑝Ω

𝑑Ω

𝒖𝑝 = 𝑢, 𝑣, 𝑤, 𝜃𝑥 , 𝜃𝑦𝑇

Strain energy:

Geometric

strain energy:


Geometric stiffness matrix

For the analysis of the buckling behavior, the action of the in-

plane loads causing bending strains is considered by which the

stiffness matrix is modified by another matrix 𝐾𝑝𝐺 .

Stretching of an element

Beam

Foppl-Von Karman plate

theory (learned from class)


Relevant theories-Beam

Stiffener- Timoshenko Beam Theory

Uniform cross section, homogenous, isotropic and linearly elastic

material.

𝒓𝑠 = 𝑥𝑠, 𝑦𝑠

Local coordinates are defined by 𝑡, 𝑛 and 𝑏 in

tangential, normal and binormal directions

Three-node isoparametric beam element

𝒓𝑠 = 𝑁𝑗𝑟𝑠𝑗

3

𝑗=1

Displacement field

𝒖𝑠 = 𝑢𝑡, 𝑣𝑛, 𝑤𝑏 , 𝜃𝑡, 𝜃𝑏 , 𝒖𝑠 = 𝑁𝑗𝒖𝑠𝑗

3

𝑗=1

n

t

y

x

𝛼


Relation of two displacement fields

C

A

B

Displacement Compatibility Conditions:

Point A, B and C are three nodes in beam element, also

they are located in the plate. So, for isoparametric

element, we can obtain:

Coordinates of three points:

𝒓𝑠,𝐴 = 𝑵𝑖,𝐴 ∗ 𝒓𝑖,𝑝

8

𝑖=1

, 𝒓𝑠,𝐵 = 𝑵𝑖,𝐵 ∗ 𝒓𝑖,𝑝

8

𝑖=1

,

𝒓𝑠,𝐶 = 𝑵𝑖,𝐶 ∗ 𝒓𝑖,𝑝

8

𝑖=1

Displacement field of three points A,B and C in plate:

𝒖𝑠,𝐴 = 𝑵𝑝,𝐴 ∗ 𝒖𝑖,𝑝

8

𝑖=1

, 𝒖𝑠,𝐵= 𝑵𝑝,𝐵 ∗ 𝒖𝑖,𝑝

8

𝑖=1

, 𝒖𝑠,𝐶= 𝑵𝑝,𝐶 ∗ 𝒖𝑖,𝑝

8

𝑖=1

𝒓𝑖,𝑝 , 𝒖𝑖,𝑝

Displacement field of stiffener:

𝒖𝑠= 𝑵𝑗 ∗ 𝒖𝑗𝑗=𝐴,𝐵,𝐶

= 𝑵𝑗

3

𝑗=1

𝑵𝑖𝒖𝑖,𝑝

8

𝑖=1

= 𝑵𝑝𝑠𝒖𝑝


Total potential energy

Coordinates and displacement fields of plate and stiffeners

can be represented by interpolation of coordinates and

displacement of plate, respectively. Obtain the total strain

energy of stiffener and plate, 𝑈𝑠and 𝑈𝑃 ,and Geometry strain

energy (derive from the nonlinear strain term), 𝐺𝑃 and 𝐺𝑠.

Employ Hamilton principle to obtain

𝑲+ 𝜆𝑐𝑟𝑲𝑮 𝒒 = 0


Square plate with one curvilinear stiffener

E = 69000MPa,

ρ = 2823kg/m3,

μ = 0.3,

a = b = 120mm

t = 1.2mm

w = 1.1447mm

h = 12.5794mm

First buckling mode shape, 𝜆𝑐𝑟 = 31.158

𝜎0 =1 00 1

∗ MPa


Rectangle plate with 2 curvilinear stiffeners

E = 73GPa,

ρ = 2837kg/m3,

μ = 0.33,

a = 606.9mm,

b = 711.2mm,

t = 4mm,

w = 10.525mm,

h = 34.598mm

First buckling mode shape, 𝜆𝑐𝑟 = 19.053

𝜎0 =1 00 1

∗ MPa


Summary

Stiffener can help to resist local buckling without much

more weight increment of whole structure;

Different kinds of stiffeners, straight or curve, can be

selected based on the types of loads applied, structural

optimization can be employed to obtain the optimal

stiffened structures.

Highlight on current FEA of stiffened plate is saving

meshing step in commercial software, ANSYS, ABQUAS,

NASTRAN, etc., which increase the efficiency of

structural optimization of stiffened structure.


References

P.S.Bulson, The stability of flat plates, Chatto&Windus, London,

1970

J.S.Rao, Dynamics of Plates, Narosa Publishing House, 1999

Irving H. Shames and Clive L.Dym, Energy and Finite Element

Methods in Structural Mechanics. New Age International LtD, 1991

Stephen P. Timoshenko and James M. Gere, Theory of Elastic

Stability, McGRAW-Hill Book Company, INC, 1961.

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