Lecture 8.1: Introduction to Plate

Behavior and Design

OBJECTIVE/SCOPE

To introduce the series of lectures on plates, showing the uses of plates to resist in-plane

and out-of-plane loading and their principal modes of behaviour both as single panels and

as assemblies of stiffened plates.

SUMMARY

This lecture introduces the uses of plates and plated assemblies in steel structures. It

describes the basic behavior of plate panels subject to in-plane or out-of-plane loading,

highlighting the importance of geometry and boundary conditions. Basic buckling modes

and mode interaction are presented. It introduces the concept of effective width and

describes the influence of imperfections on the behavior of practical plates. It also gives

an introduction to the behavior of stiffened plates.

1. INTRODUCTION

Plates are very important elements in steel structures. They can be assembled into

complete members by the basic rolling process (as hot rolled sections), by folding (as

cold formed sections) and by welding. The efficiency of such sections is due to their use

of the high in-plane stiffness of one plate element to support the edge of its neighbour,

thus controlling the out-of-plane behavior of the latter.

The size of plates in steel structures varies from about 0,6mm thickness and 70mm width

in a corrugated steel sheet, to about 100mm thick and 3m width in a large industrial or

offshore structure. Whatever the scale of construction the plate panel will have a

thickness t that is much smaller than the width b, or length a. As will be seen later, the

most important geometric parameter for plates is b/t and this will vary, in an efficient

plate structure, within the range 30 to 250.

2. BASIC BEHAVIOUR OF A PLATE PANEL

Understanding of plate structures has to begin with an understanding of the modes of

behaviour of a single plate panel.

2.1 Geometric and Boundary Conditions

The important geometric parameters are thickness t, width b (usually measured transverse

to the direction of the greater direct stress) and length a, see Figure 1a. The ratio b/t, often

called the plate slenderness, influences the local buckling of the plate panel; the aspect

ratio a/b may also influence buckling patterns and may have a significant influence on

strength.

In addition to the geometric proportions of the plate, its strength is governed by its

boundary conditions. Figure 1 shows how response to different types of actions is

influenced by different boundary conditions. Response to in-plane actions that do not

cause buckling of the plate is only influenced by in-plane, plane stress, boundary

conditions, Figure 1b. Initially, response to out-of-plane action is only influenced by the

boundary conditions for transverse movement and edge moments, Figure 1c. However, at

higher actions, responses to both types of action conditions are influenced by all four

boundary conditions. Out-of-plane conditions influence the local buckling, see Figure 1d;

in-plane conditions influence the membrane action effects that develop at large

displacements (>t) under lateral actions, see Figure 1e.

2.2 In-plane Actions

As shown in Figure 2a, the basic types of in-plane actions to the edge of a plate panel are

the distributed action that can be applied to a full side, the patch action or point action

that can be applied locally.

When the plate buckles, it is particularly important to differentiate between applied

displacements, see Figure 2b and applied stresses, see Figure 2c. The former permits a

redistribution of stress within the panel; the more flexible central region sheds stresses to

the edges giving a valuable post buckling resistance. The latter, rarer case leads to an

earlier collapse of the central region of the plate with in-plane deformation of the loaded

edges.

2.3 Out-of-plane Actions

Out-of-plane loading may be:

• uniform over the entire panel, see for example Figure 3a, the base of a water tank.

• varying over the entire panel, see for example Figure 3b, the side of a water tank.

• a local patch over part of the panel, see for example Figure 3c, a wheel load on a

bridge deck.

2.4 Determination of Plate Panel Actions

In some cases, for example in Figure 4a, the distribution of edge actions on the panels of

a plated structure are self-evident. In other cases the in-plane flexibilities of the panels

lead to distributions of stresses that cannot be predicted from simple theory. In the box

girder shown in Figure 4b, the in-plane shear flexibility of the flanges leads to in-plane

deformation of the top flange. Where these are interrupted, for example at the change in

direction of the shear at the central diaphragm, the resulting change in shear deformation

leads to a non-linear distribution of direct stress across the top flange; this is called shear

lag.

In members made up of plate elements, such as the box girder shown in Figure 5, many

of the plate components are subjected to more than one component of in-plane action

effect. Only panel A does not have shear coincident with the longitudinal compression.

If the cross-girder system EFG was a means of introducing additional actions into the

box, there would also be transverse direct stresses arising from the interaction between

the plate and the stiffeners.

2.5 Variations in Buckled Mode

i. Aspect ratio a/b

In a long plate panel, as shown in Figure 6, the greatest initial inhibition to buckling is the

transverse flexural stiffness of the plate between unloaded edges. (As the plate moves

more into the post-buckled regime, transverse membrane action effects become

significant as the plate deforms into a non-developable shape, i.e. a shape that cannot be

formed just by bending).

As with any instability of a continuous medium, more than one buckled mode is possible,

in this instance, with one half wave transversely and in half waves longitudinally. As the

aspect ratio increases the critical mode changes, tending towards the situation where the

half wave length a/m = b. The behavior of a long plate panel can therefore be modeled

accurately by considering a simply-supported, square panel.

ii. Bending conditions

As shown in Figure 7, boundary conditions influence both the buckled shapes and the

critical stresses of elastic plates. The greatest influence is the presence or absence of

simple supports, for example the removal of simple support to one edge between case 1

and case 4 reduces the buckling stress by a factor of 4,0/0,425 or 9,4. By contrast

introducing rotational restraint to one edge between case 1 and case 2 increases the

buckling stress by 1,35.

iii. Interaction of modes

Where there is more than one action component, there will be more than one mode and

therefore there may be interaction between the modes. Thus in Figure 8b(i) the presence

of low transverse compression does not change the mode of buckling. However, as

shown in Figure 8b(ii), high transverse compression will cause the panel to deform into a

single half wave. (In some circumstances this forcing into a higher mode may increase

strength; for example, in case 8b(ii), pre-deformation/transverse compression may

increase strength in longitudinal compression.) Shear buckling as shown in Figure 8c is

basically an interaction between the diagonal, destabilizing compression and the

stabilizing tension on the other diagonal.

Where buckled modes under the different action effects are similar, the buckling stresses

under the combined actions are less than the addition of individual action effects. Figure

9 shows the buckling interactions under combined compression, and uniaxial

compression and shear.

2.6 Grillage Analogy for Plate Buckling

One helpful way to consider the buckling behaviour of a plate is as the grillage shown in

Figure 10. A series of longitudinal columns carry the longitudinal actions. When they

buckle, those nearer the edge have greater restraint than those near the centre from the

transverse flexural members. They therefore have greater post buckling stiffness and

carry a greater proportion of the action. As the grillage moves more into the post buckling

regime, the transverse buckling restraint is augmented by transverse membrane action.

2.7 Post Buckling Behaviour and Effective Widths

Figures 11a, 11b and 11c describe in more detail the changing distribution of stresses as a

plate buckles following the equilibrium path shown in Figure 11d. As the plate initially

buckles the stresses redistribute to the stiffer edges. As the buckling continues this

redistribution becomes more extreme (the middle strip of slender plates may go into

tension before the plate fails). Also transverse membrane stresses build up. These are self

equilibrating unless the plate has clamped in-plane edges; tension at the mid panel, which

restrains the buckling, is resisted by compression at the edges, which are restrained from

out-of-plane movement.

An examination of the non-linear longitudinal stresses in Figures 11a and 11c shows that

it is possible to replace these stresses by rectangular stress blocks that have the same peak

stress and same action effect. This effective width of plate (comprising beff/2 on each

side) proves to be a very effective design concept. Figure 11e shows how effective width

varies with slenderness (λp is a measure of plate slenderness that is independent of yield

stress; λp = 1,0 corresponds to values of b/t of 57, 53 and 46 for fy of 235N/mm2,

275N/mm2 and 355N/mm

2 respectively).

Figure 12 shows how effective widths of plate elements may be combined to give an

effective cross-section of a member.

2.8 The Influences of Imperfections on the Behavior of Actual Plates

As with all steel structures, plate panels contain residual stresses from manufacture and

subsequent welding into plate assemblies, and are not perfectly flat. The previous

discussions about plate panel behavior all relate to an ideal, perfect plate. As shown in

Figure 13 these imperfections modify the behavior of actual plates. For a slender plate the

behavior is asymptotic to that of the perfect plate and there is little reduction in strength.

For plates of intermediate slenderness (which frequently occur in practice), an actual

imperfect plate will have a considerably lower strength than that predicted for the perfect

plate.

Figure 14 summarizes the strength of actual plates of varying slenderness. It shows the

reduction in strength due to imperfections and the post buckling strength of slender

plates.

2.9 Elastic Behavior of Plates under Lateral Actions

The elastic behavior of laterally loaded plates is considerably influenced by its support

conditions. If the plate is resting on simple supports as in Figure 15b, it will deflect into a

shape approximating a saucer and the corner regions will lift off their supports. If it is

attached to the supports, as in Figure 15c, for example by welding, this lift off is

prevented and the plate stiffness and action capacity increases. If the edges are encastre

as in Figure 15d, both stiffness and strength are increased by the boundary restraining

moments.

Slender plates may well deflect elastically into a large displacement regime (typically

where d > t). In such cases the flexural response is significantly enhanced by the

membrane action of the plate. This membrane action is at its most effective if the edges

are fully clamped. Even if they are only held partially straight by their own in-plane

stiffness, the increase in stiffness and strength is most noticeable at large deflections.

Figure 15 contrasts the behavior of a similar plate with different boundary conditions.

Figure 16 shows the modes of behavior that occur if the plates are subject to sufficient

load for full yield line patterns to develop. The greater number of yield lines as the

boundary conditions improve is a qualitative measure of the increase in resistance.

3. BEHAVIOUR OF STIFFENED PLATES

Many aspects of stiffened plate behavior can be deduced from a simple extension of the

basic concepts of behavior of un-stiffened plate panels. However, in making these

extrapolations it should be recognized that:

• "smearing" the stiffeners over the width of the plate can only model overall

behaviour.

• stiffeners are usually eccentric to the plate. Flexural behaviour of the equivalent

tee section induces local direct stresses in the plate panels.

• local effects on plate panels and individual stiffeners need to be considered

separately.

• the discrete nature of the stiffening introduces the possibility of local modes of

buckling. For example, the stiffened flange shown in Figure 17a shows several

modes of buckling. Examples are:

(i) plate panel buckling under overall compression plus any local compression arising

from the combined action of the plate panel with its attached stiffening, Figure 17b.

(ii) Stiffened panel buckling between transverse stiffeners, Figure 17c. This occurs if the

latter have sufficient rigidity to prevent overall buckling. Plate action is not very

significant because the only transverse member is the plate itself. This form of buckling

is best modeled by considering the stiffened panel as a series of tee sections buckling as

columns. It should be noted that this section is mono-symmetric and will exhibit different

behavior if the plate or the stiffener tip is in greater compression.

(iii) Overall or orthotropic bucking, Figure 17d. This occurs when the cross girders are

flexible. It is best modeled by considering the plate assembly as an orthotropic plate.

4. CONCLUDING SUMMARY

• Plates and plate panels are widely used in steel structures to resist both in-plane

and out-of-plane actions.

• Plate panels under in-plane compression and/or shear are subject to buckling.

• The elastic buckling stress of a perfect plate panel is influenced by:

⋅ plate slenderness (b/t).

⋅ aspect ratio (a/b).

⋅ boundary conditions.

⋅ interaction between actions, i.e. biaxial compression and compression and shear.

• The effective width concept is a useful means of defining the post-buckling

behaviour of a plate panel in compression.

• The behaviour of actual plates is influenced by both residual stresses and

geometric imperfections.

• The response of a plate panel to out-of-plane actions is influenced by its boundary

conditions.

• An assembly of plate panels into a stiffened plate structure may exhibit both local

and overall modes of instability.

5. ADDITIONAL READING

1. Timoshenko, S. and Weinowsky-Kreiger, S., "Theory of Plates and Shells" Mc

Graw-Hill, New York, International Student Edition, 2nd Ed.

Lecture 8.2: Behavior and Design of Un-

stiffened Plates

OBJECTIVE/SCOPE

To discuss the load distribution, stability and ultimate resistance of unstiffened plates

under in-plane and out-of-plane loading.

SUMMARY

The load distribution for un-stiffened plate structures loaded in-plane is discussed. The

critical buckling loads are derived using Linear Elastic Theory. The effective width

method for determining the ultimate resistance of the plate is explained as are the

requirements for adequate finite element modeling of a plate element. Out-of-plane

loading is also considered and its influence on the plate stability discussed.

1. INTRODUCTION

Thin-walled members, composed of thin plate panels welded together, are increasingly

important in modern steel construction. In this way, by appropriate selection of steel

quality, geometry, etc., cross-sections can be produced that best fit the requirements for

strength and serviceability, thus saving steel.

Recent developments in fabrication and welding procedures allow the automatic

production of such elements as plate girders with thin-walled webs, box girders, thin-

walled columns, etc. (Figure 1a); these can be subsequently transported to the

construction site as prefabricated elements.

Due to their relatively small thickness, such plate panels are basically not intended to

carry actions normal to their plane. However, their behaviour under in-plane actions is of

specific interest (Figure 1b). Two kinds of in-plane actions are distinguished:

a) those transferred from adjacent panels, such as compression or shear.

b) those resulting from locally applied forces (patch loading) which generate zones of

highly concentrated local stress in the plate.

The behavior under patch action is a specific problem dealt with in the lectures on plate

girders (Lectures 8.5.1 and 8.5.2). This lecture deals with the more general behaviour of

un-stiffened panels subjected to in-plane actions (compression or shear) which is

governed by plate buckling. It also discusses the effects of out-of-plane actions on the

stability of these panels.

2. UNSTIFFENED PLATES UNDER IN-PLANE

LOADING

2.1 Load Distribution

2.1.1 Distribution resulting from membrane theory

The stress distribution in plates that react to in-plane loading with membrane stresses

may be determined, in the elastic field, by solving the plane stress elastostatic problem

governed by Navier's equations, see Figure 2.

where:

u = u(x, y), v = v(x, y): are the displacement components in the x and y directions

νeff = 1/(1 + ν) is the effective Poisson's ratio

G: is the shear modulus

X = X(x, y), Y = Y(x, y): are the components of the mass forces.

The functions u and v must satisfy the prescribed boundary (support) conditions on the

boundary of the plate. For example, for an edge parallel to the y axis, u= v = 0 if the edge

is fixed, or σx = τxy = 0 if the edge is free to move in the plane of the plate.

The problem can also be stated using the Airy stress function, F = F(x, y), by the

following biharmonic equation:

∇4F = 0

This formulation is convenient if stress boundary conditions are prescribed. The stress

components are related to the Airy stress function by:

; ;

2.1.2 Distribution resulting from linear elastic theory using Bernouilli's hypothesis

For slender plated structures, where the plates are stressed as membranes, the application

of Airy's stress function is not necessary due to the hypothesis of plane strain

distributions, which may be used in the elastic as well as in the plastic range, (Figure 3).

However, for wide flanges of plated structures, the application of Airy's stress function

leads to significant deviations from the plane strain hypothesis, due to the shear lag

effect, (Figure 4). Shear lag may be taken into account by taking a reduced flange width.

2.1.3 Distribution resulting from finite element methods

When using finite element methods for the determination of the stress distribution, the

plate can be modelled as a perfectly flat arrangement of plate sub-elements. Attention

must be given to the load introduction at the plate edges so that shear lag effects will be

taken into account. The results of this analysis can be used for the buckling verification.

2.2 Stability of Unstiffened Plates

2.1.1 Linear buckling theory

The buckling of plate panels was investigated for the first time by Bryan in 1891, in

connection with the design of a ship hull [1]. The assumptions for the plate under

consideration (Figure 5a), are those of thin plate theory (Kirchhoff's theory, see [2-5]):

a) the material is linear elastic, homogeneous and isotropic.

b) the plate is perfectly plane and stress free.

c) the thickness "t" of the plate is small compared to its other dimensions.

d) the in-plane actions pass through its middle plane.

e) the transverse displacements w are small compared to the thickness of the plate.

f) the slopes of the deflected middle surfaces are small compared to unity.

g) the deformations are such that straight lines, initially normal to the middle plane,

remain straight lines and normal to the deflected middle surface.

h) the stresses normal to the thickness of the plate are of a negligible order of magnitude.

Due to assumption (e) the rotations of the middle surface are small and their squares can

be neglected in the strain displacement relationships for the stretching of the middle

surface, which are simplified as:

εx = ∂u/∂x , γxy = ∂u/∂y + ∂v/∂x (1)

An important consequence of this assumption is that there is no stretching of the middle

surface due to bending, and the differential equations governing the deformation of the

plate are linear and uncoupled. Thus, the plate equation under simultaneous bending and

stretching is:

D∇4w = q

-kt{σx ∂

2w/∂x2 + 2τxy ∂2w/∂x∂y + σy ∂

2w/∂y2} (2)

where D = Et3/12(1 - ν2

) is the bending stiffness of the plate having thickness t, modulus

of elasticity E, and Poisson's ratio ν; q = q(x,y) is the transverse loading; and k is a

parameter. The stress components, σx, σy, τxy are in general functions of the point x, y of

the middle plane and are determined by solving independently the plane stress

elastoplastic problem which, in the absence of in-plane body forces, is governed by the

equilibrium equations:

∂σx/∂x + ∂τxy/∂y = 0, ∂τxy/∂x + ∂σy/∂y = 0 (3)

supplemented by the compatibility equation:

∇2 (σx + σy) = 0 (4)

Equations (3) and (4) are reduced either to the biharmonic equation by employing the

Airy stress function:

∇4 F = 0 (5)

defined as:

σx = ∂2F/∂y2 , σy = ∂2F/∂x2

, τxy = -∂2F/∂x∂y

or to the Navier equations of equilibrium, if the stress displacement relationships are

employed:

∇2 + [1/(1- )] ∂/∂x {∂u/∂x + ∂v/∂y} = 0

∇2 + [1/(1- )] ∂/∂y {∂u/∂x + ∂v/∂y} = 0 (6)

where = ν/(1 + ν) is the effective Poisson's ratio.

Equation (5) is convenient if stress boundary conditions are prescribed. However, for

displacement or mixed boundary conditions Equations (6) are more convenient.

Analytical or approximate solutions of the plane elastostatic problem or the plate bending

problem are possible only in the case of simple plate geometries and boundary

conditions. For plates with complex shape and boundary conditions, a solution is only

feasible by numerical methods such as the finite element or the boundary element

methods.

Equation (2) was derived by Saint-Venant. In the absence of transverse loading (q = 0),

Equation (2) together with the prescribed boundary (support) conditions of the plate,

results in an eigenvalue problem from which the values of the parameter k, corresponding

to the non-trivial solution (w ≠ 0), are established. These values of k determine the

critical in-plane edge actions (σcr, τcr) under which buckling of the plate occurs. For these

values of k the equilibrium path has a bifurcation point (Figure 5b). The edge in-plane

actions may depend on more than one parameter, say k1, k2,...,kN, (e.g. σx, σy and τxy on

the boundary may increase at different rates). In this case there are infinite combinations

of values of ki for which buckling occurs. These parameters are constrained to lie on a

plane curve (N = 2), on a surface (N = 3) or on a hypersurface (N > 3). This theory, in

which the equations are linear, is referred to as linear buckling theory.

Of particular interest is the application of the linear buckling theory to rectangular plates,

subjected to constant edge loading (Figure 5a). In this case the critical action, which

corresponds to the Euler buckling load of a compressed strut, may be written as:

σcr = kσ σE or τcr = kτ σE (7)

where σE = (8)

and kσ, kτ are dimensionless buckling coefficients.

Only the form of the buckling surface may be determined by this theory but not the

magnitude of the buckling amplitude. The relationship between the critical stress σcr, and

the slenderness of the panel λ = b/t, is given by the buckling curve. This curve, shown in

Figure 5c, has a hyperbolic shape and is analogous to the Euler hyperbola for struts.

The buckling coefficients, "k", may be determined either analytically by direct integration

of Equation (2) or numerically, using the energy method, the method of transfer matrices,

etc. Values of kσ and kτ for various actions and support conditions are shown in Figure 6

as a function of the aspect ratio of the plate α =a/b. The curves for kσ have a "garland"

form. Each garland corresponds to a buckling mode with a certain number of waves. For

a plate subjected to uniform compression, as shown in Figure 6a, the buckling mode for

values of α < √2, has one half wave, for values √2 < α < √6, two half waves, etc. For α =

√2 both buckling modes, with one and two half waves, result in the same value of kσ .

Obviously, the buckling mode that gives the smallest value of k is the decisive one. For

practical reasons a single value of kσ is chosen for plates subjected to normal stresses.

This is the smallest value for the garland curves independent of the value of the aspect

ratio. In the example given in Figure 6a, kσ is equal to 4 for a plate which is simply

supported on all four sides and subjected to uniform compression.

Combination of stresses σσσσx, σσσσy and ττττ

For practical design situations some further approximations are necessary. They are

illustrated by the example of a plate girder, shown in Figure 7.

The normal and shear stresses, σx and τ respectively, at the opposite edges of a subpanel

are not equal, since the bending moments M and the shear forces V vary along the panel.

However, M and V are considered as constants for each subpanel and equal to the largest

value at an edge (or equal to the value at some distance from it). This conservative

assumption leads to equal stresses at the opposite edges for which the charts of kσ and kτ

apply. The verification is usually performed for two subpanels; one with the largest value

of σx and one with the largest value of τ. In most cases, as in Figure 7, each subpanel is

subjected to a combination of normal and shear stresses. A direct determination of the

buckling coefficient for a given combination of stresses is possible; but it requires

considerable numerical effort. For practical situations an equivalent buckling stress σcreq

is found by an interaction formula after the critical stresses σcreq

and τcro , for independent

action of σ and τ have been determined. The interaction curve for a plate subjected to

normal and shear stresses, σx and τ respectively, varies between a circle and a parabola

[6], depending on the value of the ratio ψ of the normal stresses at the edges (Figure 8).

This relationship may be represented by the approximate equation:

(9)

For a given pair of applied stresses (σ, τ) the factor of safety with respect to the above

curve is given by:

= (10)

The equivalent buckling stress is then given by:

σcreq

= γcreq

√{σ2 + 3τ2

} (11)

where the von Mises criterion has been applied.

For simultaneous action of σx, σy and τ similar relationships apply.

2.2.2 Ultimate resistance of an unstiffened plate

General

The linear buckling theory described in the previous section is based on assumptions (a)

to (h) that are never fulfilled in real structures. The consequences for the buckling

behaviour when each of these assumptions is removed is now discussed.

The first assumption of unlimited linear elastic behaviour of the material is obviously not

valid for steel. If the material is considered to behave as linear elastic-ideal plastic, the

buckling curve must be cut off at the level of the yield stress σy (Figure 9b).

When the non-linear behavior of steel between the proportionality limit σp and the yield

stress σy is taken into account, the buckling curve will be further reduced (Figure 9b).

When strain hardening is considered, values of σcr larger than σy, as experimentally

observed for very stocky panels, are possible. In conclusion, it may be stated that the

removal of the assumption of linear elastic behavior of steel results in a reduction of the

ultimate stresses for stocky panels.

The second and fourth assumptions of a plate without geometrical imperfections and

residual stresses, under symmetric actions in its middle plane, are also never fulfilled in

real structures. If the assumption of small displacements is still retained, the analysis of a

plate with imperfections requires a second order analysis. This analysis has no bifurcation

point since for each level of stress the corresponding displacements w may be

determined. The equilibrium path (Figure 10a) tends asymptotically to the value of σcr for

increasing displacements, as is found from the second order theory.

However the ultimate stress is generally lower than σcr since the combined stress due to

the buckling and the membrane stress is limited by the yield stress. This limitation

becomes relevant for plates with geometrical imperfections, in the region of moderate

slenderness, since the value of the buckling stress is not small (Figure 10b). For plates

with residual stresses the reduction of the ultimate stress is primarily due to the small

value of σp (Figure 9b) at which the material behavior becomes non-linear. In conclusion

it may be stated that imperfections due to geometry, residual stresses and eccentricities of

loading lead to a reduction of the ultimate stress, especially in the range of moderate

slenderness.

The assumption of small displacements (e) is not valid for stresses in the vicinity of σcr as

shown in Figure 10a. When large displacements are considered, Equation (1) must be

extended to the quadratic terms of the displacements. The corresponding equations,

written for reasons of simplicity for a plate without initial imperfections, are:

(12)

This results in a coupling between the equations governing the stretching and the bending

of the plate (Equations (1) and (2)).

(13a)

(13b)

where F is an Airy type stress function. Equations (13) are known as the von Karman

equations. They constitute the basis of the (geometrically) non-linear buckling theory.

For a plate without imperfections the equilibrium path still has a bifurcation point at σcr,

but, unlike the linear buckling theory, the equilibrium for stresses σ > σcr is still stable

(Figure 11). The equilibrium path for plates with imperfections tends asymptotically to

the same curve. The ultimate stress may be determined by limiting the stresses to the

yield stress. It may be observed that plates possess a considerable post-critical carrying

resistance. This post-critical behaviour is more pronounced the more slender the plate, i.e.

the smaller the value of σcr.

Buckling curve

For the reasons outlined above, it is evident that the Euler buckling curve for linear

buckling theory (Figure 6c) may not be used for design. A lot of experimental and

theoretical investigations have been performed in order to define a buckling curve that

best represents the true behaviour of plate panels. For relevant literature reference should

be made to Dubas and Gehri [7]. For design purposes it is advantageous to express the

buckling curve in a dimensionless form as described below.

The slenderness of a panel may be written according to (7) and (8) as:

λp = (b/t) √{12(1−ν2)/kσ} = π√(Ε/σcr) (14)

If a reference slenderness given by:

λy = π√(Ε/fy) (15)

is introduced, the relative slenderness becomes:

p = λp/λy = √(σy/σcr) (16)

The ultimate stress is also expressed in a dimensionless form by introducing a reduction

factor:

k = σu /σy (17)

Dimensionless curves for normal and for shear stresses as proposed by Eurocode 3 [8] are

illustrated in Figure 12.

These buckling curves have higher values for large slendernesses than those of the Euler

curve due to post critical behaviour and are limited to the yield stress. For intermediate

slendernesses, however, they have smaller values than those of Euler due to the effects of

geometrical imperfections and residual stresses.

Although the linear buckling theory is not able to describe accurately the behaviour of a

plate panel, its importance should not be ignored. In fact this theory, as in the case of

struts, yields the value of an important parameter, namely p, that is used for the

determination of the ultimate stress.

Effective width method

This method has been developed for the design of thin walled sections subjected to

uniaxial normal stresses. It will be illustrated for a simply-supported plate subjected to

uniform compression (Figure 13a).

The stress distribution which is initially uniform, becomes non-uniform after buckling,

since the central parts of the panel are not able to carry more stresses due to the bowing

effect. The stress at the stiff edges (towards which the redistribution takes place) may

reach the yield stress. The method is based on the assumption that the non-uniform stress

distribution over the entire panel width may be substituted by a uniform one over a

reduced "effective" width. This width is determined by equating the resultant forces:

b σu = be σy (18)

and accordingly:

be = σu.b/σy = kb (19)

which shows that the value of the effective width depends on the buckling curve adopted.

For uniform compression the effective width is equally distributed along the two edges

(Figure 13a). For non-uniform compression and other support conditions it is distributed

according to rules given in the various regulations. Some examples of the distribution are

shown in Figure 13b. The effective width may also be determined for values of σ < σu. In

such cases Equation (19) is still valid, but p, which is needed for the determination of

the reduction factor k, is not given by Equation (16) but by the relationship:

p = √(σ/σcr) (20)

The design of thin walled cross-sections is performed according to the following

procedure:

For given actions conditions the stress distribution at the cross-section is determined. At

each subpanel the critical stress σcr, the relative slenderness p and the effective width

be are determined according to Equations (7), (16) and (19), respectively. The effective

width is then distributed along the panel as illustrated by the examples in Figure 13b. The

verifications are finally based on the characteristic Ae, Ie, and We of the effective cross-

section. For the cross- section of Figure 14b, which is subjected to normal forces and

bending moments, the verification is expressed as:

(21)

where e is the shift in the centroid of the cross-section to the tension side and γm the

partial safety factor of resistance.

The effective width method has not been extended to panels subjected to combinations of

stress. On the other hand the interaction formulae presented in Section 2.2 do not

accurately describe the carrying resistance of the plate, since they are based on linear

buckling theory and accordingly on elastic material behaviour. It has been found that

these rules cannot be extended to cases of plastic behaviour. Some interaction curves, at

the ultimate limit state, are illustrated in Figure 15, where all stresses are referred to the

ultimate stresses for the case where each of them is acting alone. Relevant interaction

formulae are included in some recent European Codes - see also [9,10].

Finite element methods

When using finite element methods to determine the ultimate resistance of an unstiffened

plate one must consider the following aspects:

• The modelling of the plate panel should include the boundary conditions as

accurately as possible with respect to the conditions of the real structure, see

Figure 16. For a conservative solution, hinged conditions can be used along the

edges.

• Thin shell elements should be used in an appropriate mesh to make yielding and

large curvatures (large out-of-plane displacements) possible.

• The plate should be assumed to have an initial imperfection similar in shape to the

final collapse mode.

The first order Euler buckling mode can be used as a first approximation to this shape. In

addition, a disturbance to the first order Euler buckling mode can be added to avoid snap-

through problems while running the programme, see Figure 17. The amplitude of the

initial imperfect shape should relate to the tolerances for flatness.

• The program used must be able to take a true stress-strain relationship into

account, see Figure 18, and if necessary an initial stress pattern. The latter can

also be included in the initial shape.

• The computer model must use a loading which is equal to the design loading

multiplied by an action factor. This factor should be increased incrementally from

zero up to the desired action level (load factor = 1). If the structure is still stable at

the load factor = 1, the calculation process can be continued up to collapse or even

beyond collapse into the region of unstable behaviour (Figure 19). In order to

calculate the unstable response, the program must be able to use more refined

incremental and iterative methods to reach convergence in equilibrium.

3. UNSTIFFENED PLATES UNDER OUT-OF-PLANE

ACTIONS

3.1 Action Distribution

3.1.1 Distribution resulting from plate theory

If the plate deformations are small compared to the thickness of the plate, the middle

plane of the plate can be regarded as a neutral plane without membrane stresses. This

assumption is similar to beam bending theory. The actions are held in equilibrium only

by bending moments and shear forces. The stresses in an isotropic plate can be calculated

in the elastic range by solving a fourth order partial differential equation, which describes

equilibrium between actions and plate reactions normal to the middle plane of the plate,

in terms of transverse deflections w due to bending.

∇ 4w =

where:

q = q(x, y) is the transverse loading

D = Et3/12(1-

2) is the stiffness of the plate

having thickness t, modulus of

elasticity E, and Poisson's ratio

υ .

is the biharmonic operator

In solving the plate equation the prescribed boundary (support) conditions must be taken

into account. For example, for an edge parallel to the y axis, w = ∂w/∂n = 0 if the edge is

clamped, or w = ∂w2/∂n2

= 0 if the edge is simply supported.

Some solutions for the isotropic plate are given in Figure 20.

An approximation may be obtained by modeling the plate as a grid and neglecting the

twisting moments.

Plates in bending may react in the plastic range with a pattern of yield lines which, by

analogy to the plastic hinge mechanism for beams, may form a plastic mechanism in the

limit state (Figure 21). The position of the yield lines may be determined by minimum

energy considerations.

If the plate deformations are of the order of the plate thickness or even larger, the

membrane stresses in the plate can no longer be neglected in determining the plate

reactions.

The membrane stresses occur if the middle surface of the plate is deformed to a curved

shape. The deformed shape can be generated only by tension, compression and shear

stains in the middle surface.

This behaviour can be illustrated by the deformed circular plate shown in Figure 22b. It is

assumed that the line a c b (diameter d) does not change during deformation, so that a′ c′ b′ is equal to the diameter d. The points which lie on the edge "akb" are now on a′ k′ b′ , which must be on a smaller radius compared with the original one.

Therefore the distance akb becomes shorter, which means that membrane stresses exist in

the ring fibres of the plate.

The distribution of membrane stresses can be visualised if the deformed shape is frozen.

It can only be flattened out if it is cut into a number of radial cuts, Figure 22c, the gaps

representing the effects of membrane stresses; this explains why curved surfaces are

much stiffer than flat surfaces and are very suitable for constructing elements such as

cupolas for roofs, etc.

The stresses in the plate can be calculated with two fourth order coupled differential

equations, in which an Airy-type stress function which describes the membrane state, has

to be determined in addition to the unknown plate deformation.

In this case the problem is non-linear. The solution is far more complicated in

comparison with the simple plate bending theory which neglects membrane effects.

The behaviour of the plate is governed by von Karman's Equations (13).

where F = F(x, y) is the Airy stress function.

3.1.2 Distribution resulting from finite element methods (FEM)

More or less the same considerations hold when using FEM to determine the stress

distribution in plates which are subject to out-of-plane action as when using FEM for

plates under in-plane actions (see Section 2.1.3), except for the following:

• The plate element must be able to describe large deflections out-of-plane.

• The material model used should include plasticity.

3.2 Deflection and Ultimate Resistance

3.2.1 Deflections

Except for the yield line mechanism theory, all analytical methods for determining the

stress distributions will also provide the deformations, provided that the stresses are in the

elastic region.

Using adequate finite element methods leads to accurate determination of the deflections

which take into account the decrease in stiffness due to plasticity in certain regions of the

plate. Most design codes contain limits to these deflections which have to be met at

serviceability load levels (see Figure 23).

3.2.2 Ultimate resistance

The resistance of plates, determined using the linear plate theory only, is normally much

underestimated since the additional strength due to the membrane effect and the

redistribution of forces due to plasticity is neglected.

An upper bound for the ultimate resistance can be found using the yield line theory.

More accurate results can be achieved using FEM. The FEM program should then

include the options as described in Section 3.1.2.

Via an incremental procedure, the action level can increase from zero up to the desired

design action level or even up to collapse (see Figure 23).

4. INFLUENCE OF THE OUT-OF-PLANE ACTIONS

ON THE STABILITY OF UNSTIFFENED PLATES

The out-of-plane action has an unfavourable effect on the stability of an unstiffened plate

panel in those cases where the deformed shape due to the out- of-plane action is similar to

the buckling collapse mode of the plate under in-plane action only.

The stability of a square plate panel, therefore, is highly influenced by the presence of

out-of-plane (transversely directed) actions. Thus if the aspect ratio α is smaller than ,

the plate stability should be checked taking the out-of-plane actions into account. This

can be done in a similar way as for a column under compression and transverse actions.

If the aspect ratio α is larger than the stability of the plate should be checked

neglecting the out-of-plane actions component.

For strength verification both actions have to be considered simultaneously.

When adequate Finite element Methods are used, the complete behaviour of the plate can

be simulated taking the total action combination into account.

5. CONCLUDING SUMMARY

• Linear buckling theory may be used to analyse the behaviour of perfect, elastic

plates under in-plane actions.

• The behaviour of real, imperfect plates is influenced by their geometric

imperfections and by yield in the presence of residual stresses.

• Slender plates exhibit a considerable post-critical strength.

• Stocky plates and plates of moderate slenderness are adversely influenced by

geometric imperfection and plasticity.

• Effective widths may be used to design plates whose behaviour is influenced by

local buckling under in-plane actions.

• The elastic behaviour of plates under out-of-plane actions is adequately described

by small deflection theory for deflection less than the plate thickness.

• Influence surfaces are a useful means of describing small deflection plate

behaviour.

• Membrane action becomes increasingly important for deflections greater than the

plate thicknesses and large displacement theory using the von Karman equations

should be used for elastic analysis.

• An upper bound on the ultimate resistance of plates under out-of-plane actions

may be found from yield live theory.

• Out-of-plane actions influence the stability of plate panels under in-plane action.

6. REFERENCES

[1] Bryan, G. K., "On the Stability of a Plane Plate under Thrusts in its own Plane with

Application on the "Buckling" of the Sides of a Ship". Math. Soc. Proc. 1891, 54.

[2] Szilard, R., "Theory and Analysis of Plates", Prentice-Hall, Englewood Cliffs, New

Jersey, 1974.

[3] Brush, D. O. and Almroth, B. O., "Buckling of Bars, Plates and Shells", McGraw-

Hill, New York, 1975.

[4] Wolmir, A. S., "Biegsame Platten und Schalen", VEB Verlag für Bauwesen, Berlin,

1962.

[5] Timoshenko, S., and Winowsky-Krieger, S., "Theory of Plates and Shells", Mc Graw

Hill, 1959.

[6] Chwalla, E., "Uber dés Biégungsbeulung der Langsversteiften Platte und das Problem

der Mindersteifigeit", Stahlbau 17, 84-88, 1944.

[7] Dubas, P., Gehri, E. (editors), "Behaviour and Design of Steel Plated Structures",

ECCS, 1986.

[8] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and

rules for buildings, CEN, 1992.

[9] Harding, J. E., "Interaction of direct and shear stresses on Plate Panels" in Plated

Structures, Stability and Strength". Narayanan (ed.), Applied Science Publishers, London,

1989.

[10] Linder, J., Habermann, W., "Zur mehrachsigen Beanspruchung beim"

Plattenbeulen. In Festschrift J. Scheer, TU Braunschweig, 1987.

Lecture 8.3: Behaviour and Design of

Stiffened Plates

OBJECTIVE/SCOPE

To discuss the load distribution, stability and ultimate resistance of stiffened plates under

in-plane and out-of-plane loading.

SUMMARY

The load distribution for in-plane loaded unstiffened plate structures is discussed and the

critical buckling loads derived using linear elastic theory. Two design approaches for

determining the ultimate resistance of stiffened plates are described and compared. Out-

of-plane loading is also considered and its influence on stability discussed. The

requirements for finite element models of stiffened plates are outlined using those for

unstiffened plates as a basis.

1. INTRODUCTION

The automation of welding procedures and the need to design elements not only to have

the necessary resistance to external actions but also to meet aesthetic and serviceability

requirements leads to an increased tendency to employ thin-walled, plated structures,

especially when the use of rolled sections is excluded, due to the form and the size of the

structure. Through appropriate selection of plate thicknesses, steel qualities and form and

position of stiffeners, cross-sections can be best adapted to the actions applied and the

serviceability conditions, thus saving material weight. Examples of such structures,

shown in Figure 1, are webs of plate girders, flanges of plate girders, the walls of box

girders, thin-walled roofing, facades, etc.

Plated elements carry simultaneously:

a) actions normal to their plane,

b) in-plane actions.

Out-of-plane action is of secondary importance for such steel elements since, due to the

typically small plate thicknesses involved, they are not generally used for carrying

transverse actions. In-plane action, however, has significant importance in plated

structures.

The intention of design is to utilise the full strength of the material. Since the slenderness

of such plated elements is large due to the small thicknesses, their carrying resistance is

reduced due to buckling. An economic design may, however, be achieved when

longitudinal and/or transverse stiffeners are provided. Such stiffeners may be of open or

of torsionally rigid closed sections, as shown in Figure 2. When these stiffeners are

arranged in a regular orthogonal grid, and the spacing is small enough to 'smear' the

stiffeners to a continuum in the analysis, such a stiffened plate is called an orthogonal

anisotropic plate or in short, an orthotropic plate (Figure 3). In this lecture the buckling

behavior of stiffened plate panels subjected to in-plane actions will be presented. The

behavior under out-of-plane actions is also discussed as is the influence of the out-of-

plane action on the stability of stiffened plates.

Specific topics such as local actions and the tension field method are covered in the

lectures on plate girders.

2. STIFFENED PLATES UNDER IN-PLANE

LOADING

2.1 Action Distribution

2.1.1 Distribution resulting from membrane theory

The stress distribution can be determined from the solutions of Navier's equations (see

Lecture 8.2 Section 2.1.1) but, for stiffened plates, this is limited to plates where the

longitudinal and transverse stiffeners are closely spaced, symmetrical to both sides of the

plate, and produce equal stiffness in the longitudinal and transverse direction, see Figure

4. This configuration leads to an isotropic behavior when the stiffeners are smeared out.

In practice this way of stiffening is not practical and therefore not commonly used.

All deviations from the "ideal" situation (eccentric stiffeners, etc.) have to be taken into

account when calculating the stress distribution in the plate.

2.1.2 Distribution resulting from linear elastic theory using Bernouilli's hypothesis

As for unstiffened plates the most practical way of determining the stress distribution in

the panel is using the plane strain hypothesis. Since stiffened plates have a relatively

large width, however, the real stress distribution can differ substantially from the

calculated stress distribution due to the effect of shear lag.

Shear lag may be taken into account by a reduced flange width concentrated along the

edges and around stiffeners in the direction of the action (see Figure 5).

2.1.3 Distribution resulting from finite element methods

The stiffeners can be modeled as beam-column elements eccentrically attached to the

plate elements, see Lecture 8.2, Section 2.1.3.

In the case where the stiffeners are relatively deep beams (with large webs) it is better to

model the webs with plate elements and the flange, if present, with a beam-column

element.

2.2 Stability of Stiffened Plates

2.2.1 Linear buckling theory

The knowledge of the critical buckling load for stiffened plates is of importance not only

because design was (and to a limited extent still is) based on it, but also because it is used

as a parameter in modern design procedures. The assumptions for the linear buckling

theory of plates are as follows:

a) the plate is perfectly plane and stress free.

b) the stiffeners are perfectly straight.

c) the loading is absolutely concentric.

d) the material is linear elastic.

e) the transverse displacements are relatively small.

The equilibrium path has a bifurcation point which corresponds to the critical action

(Figure 6).

Analytical solutions, through direct integration of the governing differential equations

are, for stiffened plates, only possible in specific cases; therefore, approximate numerical

methods are generally used. Of greatest importance in this respect is the Rayleigh-Ritz

approach, which is based on the energy method. If Πo, and ΠI represent the total potential

energy of the plate in the undeformed initial state and at the bifurcation point respectively

(Figure 6), then the application of the principle of virtual displacements leads to the

expression:

δ(ΠI) = δ(Πo + ∆Πo) = δ(Πo + δΠo + δ2Πo + ....) = 0 (1)

since ΠI is in equilibrium. But the initial state is also in equilibrium and therefore δΠo =

0. The stability condition then becomes:

δ(δ2Πo) = 0 (2)

δ2Πo in the case of stiffened plates includes the strain energy of the plate and the

stiffeners and the potential of the external forces acting on them. The stiffeners are

characterized by three dimensionless coefficients δ, γ, υ expressing their relative rigidities

for extension, flexure and torsion respectively.

For rectangular plates simply supported on all sides (Figure 6) the transverse

displacements in the buckled state can be approximated by the double Fourier series:

(3)

which complies with the boundary conditions. The stability criterion, Equation (2), then

becomes:

(4)

since the only unknown parameters are the amplitudes amn, Equations (4) form a set of

linear and homogeneous linear equations, the number of which is equal to the number of

non-zero coefficients amn retained in the Ritz-expansion. Setting the determinant of the

coefficients equal to 0 yields the buckling equations. The smallest Eigenvalue is the so-

called buckling coefficient k. The critical buckling load is then given by the expression:

σcr = kσσE or τcr = kτσE (5)

with σE =

The most extensive studies on rectangular, simply supported stiffened plates were carried

out by Klöppel and Scheer[1] and Klöppel and Möller[2]. They give charts, as shown in

Figure 7, for the determination of k as a function of the coefficients δ and γ, previously

described, and the parameters α = a/b and ψ =σ2/σ1 as defined in Figure 6a. Some

solutions also exist for specific cases of plates with fully restrained edges, stiffeners with

substantial torsional rigidity, etc. For relevant literature the reader is referred to books by

Petersen[3] and by Dubas and Gehri[4].

When the number of stiffeners in one direction exceeds two, the numerical effort required

to determine k becomes considerable; for example, a plate panel with 2 longitudinal and

2 transverse stiffeners requires a Ritz expansion of 120. Practical solutions may be found

by "smearing" the stiffeners over the entire plate. The plate then behaves orthotropically,

and the buckling coefficient may be determined by the same procedure as described

before.

An alternative to stiffened plates, with a large number of equally spaced stiffeners and the

associated high welding costs, are corrugated plates, see Figure 2c. These plates may also

be treated as orthotropic plates, using equivalent orthotropic rigidities[5].

So far only the application of simple action has been considered. For combinations of

normal and shear stresses a linear interaction, as described by Dunkerley, is very

conservative. On the other hand direct determination of the buckling coefficient fails due

to the very large number of combinations that must be considered. An approximate

method has, therefore, been developed, which is based on the corresponding interaction

for unstiffened plates, provided that the stiffeners are so stiff that buckling in an

unstiffened sub-panel occurs before buckling of the stiffened plate. The critical buckling

stress is determined for such cases by the expression:

σvcr = kσ Z1s σE (6)

where σE has the same meaning as in Equation (5).

s is given by charts (Figure 8b).

Z1 =

kσ , kτ are the buckling coefficients for normal and shear stresses acting independently

For more details the reader is referred to the publications previously mentioned.

Optimum rigidity of stiffeners

Three types of optimum rigidity of stiffeners γ*, based on linear buckling theory, are

usually defined[6]. The first type γI*, is defined such that for values γ > γI* no further

increase of k is possible, as shown in Figure 9a, because for γ = γI* the stiffeners remain

straight.

The second type γII*, is defined as the value for which two curves of the buckling

coefficients, belonging to different numbers of waves, cross (Figure 9b). The buckling

coefficient for γ < γII* reduces considerably, whereas it increases slightly for γ > γII*. A

stiffener with γ = γII* deforms at the same time as the plate buckles.

The third type γIII* is defined such that the buckling coefficient of the stiffened plate

becomes equal to the buckling coefficient of the most critical unstiffened subpanel

(Figure 9c).

The procedure to determine the optimum or critical stiffness is, therefore, quite simple.

However, due to initial imperfections of both plate and stiffeners as a result of out of

straightness and welding stresses, the use of stiffeners with critical stiffness will not

guarantee that the stiffeners will remain straight when the adjacent unstiffened plate

panels buckle.

This problem can be overcome by multiplying the optimum (critical) stiffness by a factor,

m, when designing the stiffeners.

The factor is often taken as m = 2,5 for stiffeners which form a closed cross-section

together with the plate, and as m = 4 for stiffeners with an open cross-section such as flat,

angle and T-stiffeners.

2.2.2 Ultimate resistance of stiffened plates

Behaviour of Stiffened Plates

Much theoretical and experimental research has been devoted to the investigation of

stiffened plates. This research was intensified after the collapses, in the 1970's, of 4 major

steel bridges in Austria, Australia, Germany and the UK, caused by plate buckling. It

became evident very soon that linear buckling theory cannot accurately describe the real

behaviour of stiffened plates. The main reason for this is its inability to take the following

into account:

a) the influence of geometric imperfections and residual welding stresses.

b) the influence of large deformations and therefore the post buckling behaviour.

c) the influence of plastic deformations due to yielding of the material.

d) the possibility of stiffener failure.

Concerning the influence of imperfections, it is known that their presence adversely

affects the carrying resistance of the plates, especially in the range of moderate

slenderness and for normal compressive (not shear) stresses.

Large deformations, on the other hand, generally allow the plate to carry loads in the

post-critical range, thus increasing the action carrying resistance, especially in the range

of large slenderness. The post-buckling behaviour exhibited by unstiffened panels,

however, is not always present in stiffened plates. Take, for example, a stiffened flange of

a box girder under compression, as shown in Figure 10. Since the overall width of this

panel, measured as the distance between the supporting webs, is generally large, the

influence of the longitudinal supports is rather small. Therefore, the behaviour of this

flange resembles more that of a strut under compression than that of a plate. This

stiffened plate does not, accordingly, possess post-buckling resistance.

As in unstiffened panels, plastic deformations play an increasingly important role as the

slenderness decreases, producing smaller ultimate actions.

The example of a stiffened plate under compression, as shown in Figure 11, is used to

illustrate why linear bucking theory is not able to predict the stiffener failure mode. For

this plate two different modes of failure may be observed: the first mode is associated

with buckling failure of the plate panel; the second with torsional buckling failure of the

stiffeners. The overall deformations after buckling are directed in the first case towards

the stiffeners, and in the second towards the plate panels, due to the up or downward

movement of the centroid of the middle cross-section. Experimental investigations on

stiffened panels have shown that the stiffener failure mode is much more critical for both

open and closed stiffeners as it generally leads to smaller ultimate loads and sudden

collapse. Accordingly, not only the magnitude but also the direction of the imperfections

is of importance.

Due to the above mentioned deficiencies in the way that linear buckling theory describes

the behaviour of stiffened panels, two different design approaches have been recently

developed. The first, as initially formulated by the ECCS-Recommendations [7] for

allowable stress design and later expanded by DIN 18800, part 3[8] to ultimate limit state

design, still uses values from linear buckling theory for stiffened plates. The second, as

formulated by recent Drafts of ECCS-Recommendations [9,10], is based instead on

various simple limit state models for specific geometric configurations and loading

conditions. Both approaches have been checked against experimental and theoretical

results; they will now be briefly presented and discussed.

Design Approach with Values from the Linear Buckling Theory

With reference to a stiffened plate supported along its edges (Figure 12), distinction is

made between individual panels, e.g. IJKL, partial panels, i.e. EFGH, and the overall

panel ABCD. The design is based on the condition that the design stresses of all the

panels shall not exceed the corresponding design resistances. The adjustment of the linear

buckling theory to the real behaviour of stiffened plates is basically made by the

following provisions:

a) Introduction of buckling curves as illustrated in Figure 12b.

b) Consideration of effective widths, due to local buckling, for flanges associated with

stiffeners.

c) Interaction formulae for the simultaneous presence of stresses σx, σy and τ at the

ultimate limit state.

d) Additional reduction factors for the strut behaviour of the plate.

e) Provision of stiffeners with minimum torsional rigidities in order to prevent lateral-

torsional buckling.

Design Approach with Simple Limit State Models

Drafts of European Codes and Recommendations have been published which cover the

design of the following elements:

a) Plate girders with transverse stiffeners only (Figure 13a) - Eurocode 3 [11].

b) Longitudinally stiffened webs of plate and box girders (Figure 13b) - ECCS-TWG 8.3,

1989.

c) Stiffened compression flanges of box girders (Figure 13c) - ECCS [10].

Only a brief outline of the proposed models is presented here; for more details reference

should be made to Lectures 8.4, 8.5, and 8.6 on plate girders and on box girders:

The stiffened plate can be considered as a grillage of beam-columns loaded in

compression. For simplicity the unstiffened plates are neglected in the ultimate resistance

and only transfer the loads to the beam-columns which consist of the stiffeners

themselves together with the adjacent effective plate widths. This effective plate width is

determined by buckling of the unstiffened plates (see Section 2.2.1 of Lecture 8.2). The

bending resistance Mu, reduced as necessary due to the presence of axial forces, is

determined using the characteristics of the effective cross-section. Where both shear

forces and bending moments are present simultaneously an interaction formula is given.

For more details reference should be made to the original recommendations.

The resistance of a box girder flange subjected to compression can be determined using

the method presented in the ECCS Recommendations referred to previously, by

considering a strut composed of a stiffener and an associated effective width of plating.

The design resistance is calculated using the Perry-Robertson formula. Shear forces due

to torsion or beam shear are taken into account by reducing the yield strength of the

material according to the von Mises yield criterion. An alternative approach using

orthotropic plate properties is also given.

The above approaches use results of the linear buckling theory of unstiffened plates

(value of Vcr, determination of beff etc.). For stiffened plates the values given by this

theory are used only for the expression of the rigidity requirements for stiffeners.

Generally this approach gives rigidity and strength requirements for the stiffeners which

are stricter than those mentioned previously in this lecture.

Discussion of the Design Approaches

Both approaches have advantages and disadvantages.

The main advantage of the first approach is that it covers the design of both unstiffened

and stiffened plates subjected to virtually any possible combination of actions using the

same method. Its main disadvantage is that it is based on the limitation of stresses and,

therefore, does not allow for any plastic redistribution at the cross-section. This is

illustrated by the example shown in Figure 14. For the box section of Figure 14a,

subjected to a bending moment, the ultimate bending resistance is to be determined. If the

design criterion is the limitation of the stresses in the compression thin-walled flange, as

required by the first approach, the resistance is Mu = 400kNm. If the computation is

performed with effective widths that allow for plastic deformations of the flange, Mu is

found equal to 550kNm.

The second approach also has some disadvantages: there are a limited number of cases of

geometrical and loading configurations where these models apply; there are different

methodologies used in the design of each specific case and considerable numerical effort

is required, especially using the tension field method.

Another important point is the fact that reference is made to webs and flanges that cannot

always be defined clearly, as shown in the examples of Figure 15.

For a box girder subjected to uniaxial bending (Figure 15a) the compression flange and

the webs are defined. This is however not possible when biaxial bending is present

(Figure 15b). Another example is shown in Figure 15c; the cross-section of a cable stayed

bridge at the location A-A is subjected to normal forces without bending; it is evident, in

this case, that the entire section consists of "flanges".

Finite Element Methods

In determining the stability behaviour of stiffened plate panels, basically the same

considerations hold as described in Lecture 8.2, Section 2.2.2. In addition it should be

noted that the stiffeners have to be modelled by shell elements or by a combination of

shell and beam-column elements. Special attention must also be given to the initial

imperfect shape of the stiffeners with open cross-sections.

It is difficult to describe all possible failure modes within one and the same finite element

model. It is easier, therefore, to describe the beam-column behaviour of the stiffeners

together with the local and overall buckling of the unstiffened plate panels and the

stiffened assemblage respectively and to verify specific items such as lateral-torsional

buckling separately (see Figure 16). Only for research purposes is it sometimes necessary

to model the complete structure such that all the possible phenomena are simulated by the

finite element model.

3. STIFFENED PLATES UNDER OUT-OF-PLANE

ACTION APPLICATION

3.1 Action Distribution

3.1.1 Distribution resulting from plate theory

The theory described in Section 3.1.1 of Lecture 8.2 can only be applied to stiffened

plates if the stiffeners are sufficiently closely spaced so that orthotropic behaviour occurs.

If this is not the case it is better to consider the unstiffened plate panels in between the

stiffeners separately. The remaining grillage of stiffeners must be considered as a beam

system in bending (see Section 3.1.2).

3.1.2 Distribution resulting from a grillage under lateral actions filled in with

unstiffened sub-panels

The unstiffened sub-panels can be analysed as described in Section 3.1.1 of Lecture 8.2.

The remaining beam grillage is formed by the stiffeners which are welded to the plate,

together with a certain part of the plate. The part can be taken as for buckling, namely the

effective width as described in Section 2.2.2 of this Lecture. In this way the distribution

of forces and moments can be determined quite easily.

3.1.3 Distribution resulting from finite element methods (FEM)

Similar considerations hold for using FEM to determine the force and moment

distribution in stiffened plates which are subject to out-of-plane actions as for using FEM

for stiffened plates loaded in-plane (see Section 2.1.3) except that the finite elements used

must be able to take large deflections and elastic-plastic material behaviour into account.

3.2 Deflection and Ultimate Resistance

All considerations mentioned in Section 3.2 of Lecture 8.2 for unstiffened plates are valid

for the analysis of stiffened plates both for deflections and ultimate resistance. It should

be noted, however, that for design purposes it is easier to verify specific items, such as

lateral-torsional buckling, separately from plate buckling and beam-column behaviour.

4. INFLUENCE OF OUT-OF-PLANE ACTIONS ON

THE STABILITY OF STIFFENED PLATES

The points made in Section 4 of Lecture 8.2 also apply here; that is, the stability of the

stiffened plate is unfavourably influenced if the deflections, due to out-of-plane actions,

are similar to the stability collapse mode.

5. CONCLUDING SUMMARY

• Stiffened plates are widely used in steel structures because of the greater

efficiency that the stiffening provides to both stability under in-plane actions and

resistance to out-of-plane actions.

• Elastic linear buckling theory may be applied to stiffened plates but numerical

techniques such as Rayleigh-Ritz are needed for most practical situations.

• Different approaches may be adopted to defining the optimum rigidity of

stiffeners.

• The ultimate behaviour of stiffened plates is influenced by geometric

imperfections and yielding in the presence of residual stresses.

• Design approaches for stiffened plates are either based on derivatives of linear

buckling theory or on simple limit state models.

• Simple strut models are particularly suitable for compression panels with

longitudinal stiffeners.

• Finite element models may be used for concrete modelling of particular situations.

6. REFERENCES

[1] Klöppel, K., Scheer, J., "Beulwerte Ausgesteifter Rechteckplatten", Bd. 1, Berlin, W.

Ernst u. Sohn 1960.

[2] Klöppel, K., Möller, K. H., "Beulwerte Ausgesteifter Rechteckplatten", Bd. 2, Berlin,

W. Ernst u. Sohn 1968.

[3] Petersen, C., "Statik und Stabilität der Baukonstruktionen", Braunschweig: Vieweg

1982.

[4] Dubas, P., Gehri, E., "Behaviour and Design of Steel Plated Structures", ECCS, 1986.

[5] Briassoulis, D., "Equivalent Orthotropic Properties of Corrugated Sheets", Computers

and Structures, 1986, 129-138.

[6] Chwalla, E., "Uber die Biegungsbeulung der langsversteiften Platte und das Problem

der Mindeststeifigeit", Stahlbau 17, 1944, 84-88.

[7] ECCS, "Conventional design rules based on the linear buckling theory", 1978.

[8] DIN 18800 Teil 3 (1990), "Stahlbauten, Stabilitätsfalle, Plattenbeulen", Berlin: Beuth.

[9] ECCS, "Design of longitudinally stiffened webs of plate and box girders", Draft 1989.

[10] ECCS, "Stiffened compression flanges of box girders", Draft 1989.

[11] Eurocode 3, "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules

and rules for buildings, CEN, 1992.

Lecture 8.6: Introduction to Shell

Structures

OBJECTIVE/SCOPE

To describe in a qualitative way the main characteristics of shell structures and to discuss

briefly the typical problems, such as buckling, that are associated with them.

SUMMARY

Shell structures are very attractive light weight structures which are especially suited to

building as well as industrial applications. The lecture presents a qualitative interpretation

of their main advantages; it also discusses the difficulties frequently encountered with

such structures, including their unusual buckling behaviour, and briefly outlines the

practical design approach taken by the codes.

1. INTRODUCTION

The shell structure is typically found in nature as well as in classical architecture [1]. Its

efficiency is based on its curvature (single or double), which allows a multiplicity of

alternative stress paths and gives the optimum form for transmission of many different

load types. Various different types of steel shell structures have been used for industrial

purposes; singly curved shells, for example, can be found in oil storage tanks, the central

part of some pressure vessels, in storage structures such as silos, in industrial chimneys

and even in small structures like lighting columns (Figures 1a to 1e). The single curvature

allows a very simple construction process and is very efficient in resisting certain types of

loads. In some cases, it is better to take advantage of double curvature. Double curved

shells are used to build spherical gas reservoirs, roofs, vehicles, water towers and even

hanging roofs (Figures 1f to 1i). An important part of the design is the load transmission

to the foundations. It must be remembered that shells are very efficient in resisting

distributed loads but are prone to difficulties with concentrated loads. Thus, in general, a

continuous support is preferred. If it is not possible to have a foundation bed, as shown in

Figure 1a, an intermediate structure such as a continuous ring (Figure 1f) can be used to

distribute the concentrated loads at the vertical supports. On occasions, architectural

reasons or practical considerations impose the use of discrete supports.

As mentioned above, distributed loads due to internal pressure in storage tanks, pressure

vessels or silos (Figures 2a to 2c), or to external pressure from wind, marine currents and

hydrostatic pressures (Figures 2d and 2e) are very well resisted by the in-plane behaviour

of shells. On the other hand, concentrated loads introduce significant local bending

stresses which have to be carefully considered in design. Such loads can be due to vessel

supports or in some cases, due to abnormal impact loads (Figure 2f). In containment

buildings of nuclear power plants, for example, codes of practice usually require the

possibility of missile impact or even sometimes airplane crashes to be considered in the

design. In these cases, the dynamic nature of the load increases the danger of

concentrated effects. An everyday example of the difference between distributed and

discrete loads is the manner in which a cooked egg is supported in the egg cup without

problems and the way the shell is broken by the sudden impact of the spoon (Figure 2g).

Needless to say, in a real problem both types of loads will have to be dealt with either in

separate or combined states, with the conceptual differences in behaviour ever present in

the designer's mind.

Shell structures often need to be strengthened in certain problem areas by local

reinforcement. A possible location where reinforcement might be required is at the

transition from one basic surface to another; for instance, the connections between the

spherical ends in Figure 1b and the main cylindrical vessel; or the change from the

cylinder to the cone of discharge in the silo in Figure 1c. In these cases, there is a

discontinuity in the direction of the in-plane forces (Figure 3a) that usually needs some

kind of reinforcement ring to reduce the concentrated bending moments that occur in that

area.

Containment structures also need perforations to allow the stored product (oil, cement,

grain, etc.) to be put in, or extracted from, the deposit (Figure 3b). The same problem is

found in lighting columns (Figure 3c), where it is general practice to put an opening in

the lower part of the post in order to facilitate access to the electrical works. In these

cases, special reinforcement has to be added to avoid local buckling and to minimise

disturbance to the general distribution of stresses.

Local reinforcement is also often required at connections between shell structures, such

as commonly occur in general piping work and in the offshore industry. In these cases

additional reinforcing plates are used (Figure 3d), which help to resist the high stresses

produced at the connections.

In contrast to local reinforcement, global reinforcement is generally used to improve the

overall shell behaviour. Because of the efficient way in which these structures carry load,

it is possible to reduce the wall thickness to relatively small values; the high value of the

shell diameter to thickness ratios can, therefore, increase the possibility of unstable

configurations. To improve the buckling resistance, the shell is usually reinforced with a

set of stiffening members.

In axisymmetric shells, the obvious location for the stiffeners is along selected meridians

and parallel lines, creating in this way a true mesh which reinforces the pure shell

structure (Figure 4a). On other occasions, the longitudinal and ring stiffeners are replaced

by a complicated lattice (Figure 4b), which gives an aesthetically pleasing structure as

well as mechanical improvements to the global shell behaviour.

2. POSSIBLE FORMS OF BEHAVIOUR

There are two main mechanisms by which a shell can support loads. On the one hand, the

structure can react with only in-plane forces, in which case it is said to act as a

membrane. This is a desirable situation, especially if the stress is tensile (Figure 5a),

because the material can be used to its full strength. In practice, however, real structures

have local areas where equilibrium or compatibility of displacements and deformations is

not possible without introducing bending. Figure 5b, for instance, shows a load acting

perpendicular to the shell which cannot be resisted by in-plane forces only, and which

requires bending moments, induced by transverse deflections, to be set up for

equilibrium. Figure 5c, however, shows that membrane forces only can be used to

support a concentrated load if a corner is introduced in the shell.

It is worthwhile also to distinguish between global and local behaviour, because

sometimes the shell can be considered to act globally as a member. An obvious example

is shown in Figure 6a, where a tubular lighting column is loaded by wind and self-weight.

The length AB is subjected to axial and shear forces, as well as to bending and torsion,

and the global behaviour can be approximated very accurately using the member model.

The same applies in Figure 6b where an offshore jacket, under various loading

conditions, can be modelled as a cantilever truss. In addition, for certain types of vault

roofs where the support is acting at the ends, the behaviour under vertical loads is similar

to that of a beam.

Local behaviour, however, is often critical in determining structural adequacy. Dimpling

in domes (Figure 7a), or the development of the so-called Yoshimura patterns (Figure 7b)

in compressed cylinders, are phenomena related to local buckling that introduce a new

level of complexity into the study of shells. Non- linear behaviour, both from large

displacements and from plastic material behaviour, has to be taken into account. Some

extensions of the yield line theory can be used to analyse different possible modes of

failure.

To draw a comparison with the behaviour of stiffened plates, it can be said that the global

action of shell structures takes advantage of the load-diffusion capacity of the surface and

the stiffeners help to avoid local buckling by subdividing the surface into cells, resulting

in a lower span to thickness ratio. A longitudinally-stiffened cylinder, therefore, behaves

like a system of struts-and-plates, in a way that is analagous to a stiffened plate. On the

other hand, transverse stiffeners behave in a similar manner to the diaphragms in a box

girder, i.e. they help to distribute the external loads and maintain the initial shape of the

cross-section, thus avoiding distortions that could eventually lead to local instabilities. As

in box girders, special precautions have to be taken in relation to the diaphragms

transmitting bearing reactions; in shells the reaction transmission is done through saddles

that produce a distributed load.

3. IMPORTANCE OF IMPERFECTIONS

As was explained in previous lectures, the theoretical limits of bifurcation of equilibrium

that can be reached using mathematical models are upper limits to the behaviour of actual

structures; as soon as any initial displacement or shape imperfection is present, the curve

is smoothed [2]. Figures 8a and 8b present the load-displacement relationship that is

expected for a bar and a plate respectively; the dashed line OA represents the linear

behaviour that suddenly changes at bifurcation point B (solid line). The plate has an

enhanced stiffness due to the membrane effect. The dashed lines represent the behaviour

when imperfections are included in the analysis.

As can be seen in Figure 8c, the post-buckling behaviour of a cylinder is completely

different. After bifurcation, the point representing the state of equilibrium can travel

along the secondary path BDC. Following B, the situation is highly dependent on the

characteristics of the test, i.e. whether it is force-controlled or displacement controlled. In

the first case, after the buckling load is reached, a sudden change from point B to point F

occurs (Figure 8c) which is called the snap-through phenomenon, in which the shell

jumps suddenly between different buckling configurations.

The behaviour of an actual imperfect shell is represented by the dashed line. Compared

with the theoretically perfect shell, it is evident that true bifurcation of equilibrium will

not occur in the real structure, even though the dashed lines approach the solid line as the

magnitude of the imperfection diminishes. The high peak B is very sharp and the limit

point G or H (relevant to different values of the imperfection) refers to a more realistic

lower load than the theoretical bifurcation load.

The difference in behaviour, compared with that of plates or bars, can be explained by

examining the pattern of local buckling as the loading increases. Initially, buckling starts

at local imperfections with the formation of outer and inner waves (Figure 9a); the latter

represent a flattening rather than a change in direction of the original curvature and set up

compressive membrane forces which, along with the tensile membrane forces set up by

the outer waves, tend to resist the buckling effect. At the more advanced stages, as these

outer waves increase in size, the curvature in these regions changes direction and

becomes inward (Figure 9b). As a result, the compressive forces now precipitate buckling

rather than resist it, thus explaining why equilibrium, at this stage, can only be

maintained by reducing the axial load.

The importance of imperfections is such that, when tests on actual structures are carried

out, the difference between theoretical and experimental values produces a wide scatter

of results (see Figure 10). As the imperfections are unavoidable, and depend very much

on the quality of construction, it is clear that only a broad experimental series of tests on

physical models can help in establishing the least lower-bound that could be used for a

practical application. Thus it is necessary to choose:

1. The structural type, e.g. a circular cylinder, and a fixed set of boundary

conditions.

2. The type of loading, e.g. longitudinal compression.

3. A predefined pattern of reinforcement using stiffeners.

4. A strict limitation on imperfection values.

In consequence, the experimental results can only be used for a very narrow band of

applications. In addition, the quality control on the finished work must be such that the

experimental values can be used with confidence.

To allow for this, Codes of Practice [3] use the following procedure:

1. A critical stress, σcr or τcr, or a critical pressure, pcr, is calculated for the perfect

elastic shell by means of a classical formula or method in which the parameters

defining the geometry of the shell and the elastic constants of the steel are used. 2. σcr, τcr or pcr is then multiplied by a knockdown factor α, which is the ratio of the

lower bound of a great many scattered experimental buckling stresses or buckling

pressures (the buckling being assumed to occur in the elastic range) to σcr, τcr or

pcr, respectively. α is supposed to account for the detrimental effect of shape

imperfections, residual stresses and edge disturbances. α may be a function of a

geometrical parameter when a general trend in the set of available test points,

plotted with that parameter as abscissa, points to a correlation between the

parameter and α; such a trend is visible in Figure 10, where the parameter is the

radius of cylinder, r, divided by the wall thickness, t.

4. CONCLUDING SUMMARY

• The structural resistance of a shell structure is based on the curvature of its

surface.

• Two modes of resistance are generally combined in shells: a membrane state in

which the developed forces are in-plane, and a bending state where out-of-plane

forces are present.

• Bending is generally limited to zones where there are changes in boundary

conditions, thickness, or type of loads. It also develops where local instability

occurs.

• Shells are most efficient when resisting distributed loads. Concentrated loads or

geometrical changes generally require local reinforcement.

• Imperfections play a substantial role in the behaviour of shells. Their

unpredictable nature makes the use of experimental methods essential.

• To simplify shell design, codes introduce a knock-down factor to be applied to the

results of mathematical models.

5. REFERENCES

[1] Tossoji, Ei., "Philosophy of Structures", Holden Day 1960.

[2] Brush, D.O., Almroth, B.O., "Buckling of Bars, Plates and Shells", McGraw Hill,

1975.

[3] European Convention for Constructional Steelwork, "Buckling of Steel Shells",

European Recommendation, ECCS, 1988.

Lecture 8.7: Basic Analysis

of Shell Structures

OBJECTIVE/SCOPE:

To describe the basic characteristics of pre- and post-buckling shell behaviour and to

explain and compare the differences in behaviour with that of plates and bars.

SUMMARY:

The combined bending and stretching behaviour of shell structures in resisting load is

discussed; their buckling behaviour is also explained and compared with that of struts and

plates. The effect of imperfections is examined and ECCS curves, which can be used in

design, are given. Reference is also made to available computer programs that can be

used for shell analysis.

1. INTRODUCTION

Lecture 8.6 introduced several aspects of the structural behaviour of shells in an

essentially qualitative way. Before moving on to consider design procedures for specific

applications, it is necessary to gain some understanding of the possible approaches to the

analysis of shell response. It should then be possible to appreciate the reasoning behind

the actual design procedures covered in Lectures 8.8 and 8.9.

This Lecture, therefore, presents the main principles of shell theory that underpin the

ECCS design methods for unstiffened and stiffened cylinders. Comparisons are drawn

with the behaviour of columns and plates previously discussed in Lectures 6.6.1, 6.6.2

and 8.1.

2. BENDING AND STRETCHING OF THIN SHELLS

The deformation of an element of a thin shell consists of the curvatures and normal

displacements associated with out-of-surface bending and the stretching and shearing of

the middle surface. Bending deformation without stretching of the middle surface, as

assumed in the small deflection theory for flat plates, is not possible, and so both bending

and stretching strains must be considered.

If the shape and the boundary conditions of a shell and the applied loads are such that the

loads can be resisted by membrane forces alone, then these forces may be found from the

three equilibrium conditions for an infinitely small element of the shell. The equilibrium

equations may be obtained from the equilibrium of forces in three directions; that is, in

the two principal directions of curvature and in the direction normal to the middle

surface. As a result, the three membrane forces can be obtained easily in the absence of

bending and twisting moments and shear forces perpendicular to the surface. An example

is an unsupported cylindrical shell subjected to uniform radial pressure over its entire

area (Figure 1). Obviously, the only stress generated by the external pressure is a

circumferential membrane stress. The assumption does not hold if the cylinder is

subjected to two uniform line loads acting along two diametrically opposed generators

(Figure 2). In this case, bending theory is required to evaluate the stress distribution,

because an element of the shell cannot be in equilibrium without circumferential bending

stresses. Circumferential bending stresses are essential to resist the external loads, and

because the wall is thin and has very little flexural resistance, they greatly affect its load

carrying resistance [1].

Significant bending stresses usually only occur close to the boundaries, or in the zone

affected by other disturbances, such as local loads or local imperfections. Locally, the

resulting stresses may be quite high, but they generally diminish at a small distance from

the local disturbance. Bending stresses may, however, cause local yielding which can be

very dangerous in the presence of repeated loadings, since it can result in a fatigue

fracture.

It is normally more structurally efficient if a shell structure can be configured in such a

way that it carries load primarily by membrane action. Simpler design calculations will

usually also result.

3. BUCKLING OF SHELLS - LINEAR AND NON-

LINEAR BUCKLING THEORY

Buckling may be regarded as a phenomenon in which a structure undergoes local or

overall change in configuration. For example, an originally straight axially loaded

column will buckle by bowing laterally; similarly a cylinder may buckle when its surface

crumples under the action of external loads. Buckling is particularly important in shell

structures since it may well occur without any warning and with catastrophic

consequences [2-4].

The equations for determining the load at which buckling is initiated, through bifurcation

on the main equilibrium path of a cylindrical shell, may be derived by means of the

adjacent equilibrium criterion, or, alternatively, by use of the minimum potential energy

criterion. In the first case, small increments (u1, v1, w1) are imposed on the pre-buckling

displacements (u0, v0, w0)

u = u0 + u1

v = v0 + v1 (1)

w = w0 + w1

The two adjacent configurations, represented by the displacements before (u0, v0, w0) and

after the increment (u, v, w) are analysed. No increment is given to the load parameter.

The function represented by (u1, v1, w1) is called the buckling mode. As an alternative,

the minimum potential energy criterion can be adopted to derive the linear stability

equations. The expression for the second variation of the potential energy of the shell in

terms of displacements is calculated. The linear differential equations for loss of stability

are then obtained by means of the Trefftz criterion. Readers requiring a more detailed

coverage of shell buckling are advised to consult [4].

In practice, for some problems, the results obtained by these analyses are adequate and in

accordance with experiment. In other cases, such as the buckling of an axially

compressed cylinder, the results can be positively misleading as they may substantially

overestimate the actual carrying resistance of the shell. The use of these methods leads to

the following value for the axial buckling load of a perfect thin elastic cylinder of

medium length:

(2)

Assuming ν = 0,3 for steel gives σcr = 0,605 E

This buckling load is derived on the assumption that the pre-buckling increase of the

radius due to the Poisson effect is unrestrained and that the two edges are held against

translational movement in the radial and circumferential directions during buckling, but

are able to rotate about the local circumferential axis. These edge restraints are usually

called "classical boundary conditions".

Equation (2) is of little use to the designer because test results yield only 15-60% of this

value. The reason for the big discrepancy between theory and experimental results was

not understood for a long time and has been the subject of many studies; it can be

explained as follows:

The boundary conditions of the shells have a significant effect and can, if modified, give

rise to lower critical loads. Many authors have investigated the effects of the boundary

conditions on the buckling load of cylindrical shells. The value given by Equation (2)

refers to a real cylinder only if the edges are prevented from moving in the

circumferential direction, i.e. v = 0 (Figure 3). If this last condition is removed and

replaced by the condition nxy = 0 (i.e. free displacement but no membrane stress in the

circumferential direction) a critical value of approximately 50% of the classical buckling

load is obtained. This boundary condition is quite difficult to obtain in practice and

cylinders with such edge restraints are much less sensitive to imperfection than cylinders

with classical boundary conditions; they are, therefore, not of primary interest to the

designer. If, instead, the top edge of the cylinder is assumed to be free, the critical

buckling load drops down to 38% of the critical value given by Equation (2). In general,

it can be stated that if a shell initially fails with several small local buckles, the critical

load does not depend to any great extent on the boundary conditions, but, if the buckles

involve the whole shell, the boundary conditions can significantly affect the buckling

load.

The critical buckling load may also be reduced by pre-buckling deformations. To take

these deformations into account, the same boundary conditions in both the pre- and post-

buckling range must be included. The consequence is that during the compression prior to

buckling, the top and bottom edges cannot move radially (Poisson's ratio not being zero)

and, therefore, the originally straight generators become curved. The post-buckling

deformations are not infinitely small and the critical stress is reduced.

The complete understanding of the reason for the large discrepancies between theoretical

and experimental results in the buckling of shells has caused much controversy and

discussion, but now the explanation that initial imperfections are the principal cause of

the phenomenon is generally accepted.

4. POST-BUCKLING BEHAVIOUR OF THIN

SHELLS

The starting point for this illustrative study of the post-buckling behaviour of a perfect

cylinder, under axial compression (Figure 3), is Donnell's classical equations [2]. A

suitable function for w (trigonometric) may be assumed and introduced into the

compatibility equation, expressed in terms of w and of an adopted stress function F. The

quadratic expressions can be transformed to linear ones by means of well known

trigonometric relations. Then the stress function F, and as a consequence the internal

membrane stresses, may be computed. The expression for the total potential energy can

then be written, and minimized, to replace the equilibrium equation. The solution is

improved by taking more terms for w.

In Figure 4, the results obtained by using only two buckling modes are shown and

compared with the curves obtained later, i.e. with a greater number of modes. The results

show that the type of curve does not change by increasing the number of modes, but the

lowest point of the post-buckling path decreases and can attain a value of about 10% of

the linear buckling load. In the limiting case, i.e. where the number of terms increases to

infinity, the lowest value of the post-buckling path tends to zero, while the buckling

shape tends to assume the shape of the Yoshimura pattern (Figure 5). It is the limiting

case of the diamond buckling shape that can be described by the following combination

of axi-symmetric and chessboard modes.

(3)

It is worth noting that the buckling load associated with either the combination or the two

single modes is the same and is given by Equation (2).

A comprehensive overview of post-buckling theory is given in [5]. As will be discussed

later, a realistic theory for shell buckling has to take into account the unavoidable

imperfections that appear in real structures. Figure 6 shows the influence of imperfections

on the strength of a cylinder subject to compressive loading and Figure 7 shows typical

imperfections.

5. NUMERICAL ANALYSIS OF SHELL BUCKLING

Simple types of shells and loading are amenable to treatment by analytical methods. The

buckling load of complex shell structures can, however, be assessed only be means of

computer programs, many of which use finite elements and have a stability option.

CASTEM, STAGS, NASTRAN, ADINA, NISA, FINELG, ABAQUS, ANSYS, BOSOR

and FO4BO8 are some of the general and special purpose programs available. Correct

use of a complicated program requires the analyst to be well acquainted with the basis of

the approach adopted in the program.

The stability options and the reliability of the numerical results depend on the method of

analysis underlying each specific program, and on the buckling modes considered.

Analysis of various types may be performed:

1. Geometrical changes in the pre-buckling range are ignored, the pre-buckling

behaviour of the structure is thus assumed to be linear, and the buckling stress

corresponds to that at the bifurcation point B which is found by means of an

eigenvalue analysis (Figure 8a). Applied to a simple shell, this procedure yields

the classical critical load. w denotes the lateral deflection of the shell wall at some

representative point.

2. Non-linear collapse analysis enables successive points on the non-linear primary

equilibrium path to be determined until the tangent to the path becomes horizontal

at the limit point (Figure 8b). At that stage, assuming weight loading, as is

normally the case for engineering structures, non-linear collapse ("snap-through")

occurs.

3. Investigating bifurcation buckling from a non-linear pre-buckling state involves a

search for secondary equilibrium paths (corresponding to different buckling

modes, e.g. different numbers of buckling waves along the circumference of an

axi-symmetric shell) that may branch off from the non-linear primary path at

bifurcation points located below the limit point (Figure 8c). The lowest

bifurcation point provides an estimate of the buckling load.

4. General non-linear collapse analysis of an imperfect structure consists of

determining the non-linear equilibrium path and the limit point L for a structure

whose initial imperfections and plastic deformations are taken into account

(Figure 8d). The limit load, which is the ordinate of L, causes the structure to

"snap-through".

The four load-deflection diagrams given in Figure 8 may relate, for example, to a

spherical cap subjected to uniform radial pressure acting towards the centre of the sphere;

in this case the critical failure mode depends on the degree of shallowness of the cap.

6. BUCKLING AND POST-BUCKLING BEHAVIOUR

OF STRUTS, PLATES AND SHELLS

Equilibrium paths, for a perfectly straight column, a perfectly flat plate supported along

its four edges, and a perfectly cylindrical shell, presented in the preceding Lecture 8.6,

are repeated here (Figure 9) for completeness.

In each diagram, σ represents the uniformly applied compressive stress, σcr its critical

value given by classical stability theory, and U the decrease in distance between the ends

of the members.

Each point on the solid or dashed lines represents an equilibrium configuration which is

at least theoretically possible, in the sense that the conditions for equilibrium between

external and internal forces are met.

Simple elastic shortening, according to Hooke's law, is reflected by the three straight

lines OA. They represent the pre-buckling, primary, or fundamental state of equilibrium,

in which the column, the plate and the shell remain perfectly straight, flat and cylindrical,

respectively.

As long as σ < σcr, the primary equilibrium is stable, i.e. if a minute accidental

disturbance (a very small lateral force, for example) causes a slight transverse

deformation of the member, the deformation disappears when its cause is removed, and

the member returns of its own accord to its previous configuration. Any point of the line

OA, which is located above B, represents, however, unstable equilibrium, i.e. the effect

of a disturbance, even an infinitely small one, does not disappear with its cause, but

instantaneously increases and the member is set in (violent) motion, deviating further and

irreversibly from its previous equilibrium configuration. Some minor cause of

disturbance always exists, for example, in the form of an initial shape imperfection or of

an eccentricity of loading. A state of unstable equilibrium, therefore, although

theoretically possible, cannot occur in real structures.

When the stress reaches its critical value, σcr, a new equilibrium configuration appears at

point B. This configuration is quite different from the primary one and features lateral

deflections and bending of the strut, the plate, or the wall of the shell.

If the new configuration is characterised by displacements with respect to the primary

state of equilibrium which increase gradually from zero to high (theoretically infinite)

values, the post-buckling states of equilibrium are represented by points on a secondary

equilibrium path which intersects with the primary path at the bifurcation point B.

In fact, B is the lowest of an infinite number of bifurcation points, but the paths branching

off from all the others represent highly unstable equilibrium and have no practical

significance.

The great difference between the strut, the plate and the cylinder is embodied in their

post-buckling behaviour. In the case of the column (Figure 9a), the secondary path, BC,

is very nearly horizontal, but in reality it curves imperceptibly upwards; the equilibrium

along BC is almost neutral (it is, strictly speaking, weakly stable). For the plate (Figure

9b) the secondary path, BC, climbs above B, although less steeply than before; the plate

deflects laterally, more and more under a gradually increasing load, but the equilibrium at

points on BC is stable.

After bifurcation, the point representing the state of equilibrium of an axially loaded

cylinder (Figure 9c), in theory, can travel along the secondary path BDC. The equilibrium

at points located below B on the solid curve is, however, highly unstable and, hence,

cannot really exist. What would happen after point B is reached, if it were possible to

manufacture a perfect cylinder from material of unlimited linear elasticity and to support

and load or deform it in the theoretically correct manner, depends on the loading method.

When displacements, u, of one plate of a supposedly rigid testing machine with respect to

the other plate are imposed in a controlled manner, buckles suddenly appear in the wall of

the cylinder. The compressive stress drops at once from σcr to the ordinate of point E

(only a fraction of σcr), while the shortening of the cylinder remains equal to ucr, the

abscissa of B. In contrast with bifurcation, finite displacements are involved in the

transition between the equilibrium configurations represented by points B and E; such an

occurrence is called snap-through. The buckling process is further complicated by the

existence of different intersecting equilibrium paths, which correspond to different

numbers of circumferential buckling waves and which have the same general shape as

BDC. Some parts of these paths represent stable equilibrium, while other parts represent

unstable equilibrium; after the initial snap-through from state B to state E, the shell can

jump repeatedly from one buckling configuration to another.

When the load, rather than the displacement, u, is controlled a different effect occurs; if,

for example, a load = 2πrtσcr is imposed, the overall shortening of the cylinder almost

instantly increases from ucr to the abscissa of point F, and its wall suddenly exhibits deep

buckles, while the average compressive stress remains equal to σcr. It should be noted that

this "snap-through" has dynamic characteristics which are not considered in this

description.

Esslinger and Geier [5] explain the fundamentally different behaviour of columns, plates

and shells by the following argument illustrated by Figures 10, 11 and 12.

The differential equation

expresses the lateral equilibrium of any element of an axially loaded strut when

bifurcation occurs (Figure 10a) by stating that the deflecting force per unit length, due to

the external loads, Fcr, given by the first term, cancels out the restoring force per unit

length, due to the internal bending stresses, given by the second term. Both the deflecting

forces (Figure 10b) and the restoring forces (Figure 10c) are proportional to the lateral

deflection. Consequently, equilibrium of the column is independent of the magnitude of

the transverse deformation and of u, for a given constant axial force Fcr.

The restoring forces which balance the lateral forces (Figure 11b) deflecting a buckling

plate (Figure 11a), are due not only to longitudinal and transverse bending moments

(Figure 11c), but also to transverse membrane forces (Figure 11d). The restoring forces

due to membrane action are zero, as long as the plate is flat, but they then increase

proportionately to the square of its lateral deflection. As a result, the compressive

external load required for equilibrium increases together with the lateral deformation and

with the plate shortening u.

Figure 12a shows the radial component of the buckling pattern of a compressed cylinder

at the bifurcation point. Outward displacements of a curved surface cause tensile

membrane forces, while inward displacements generate compressive membrane forces.

Figure 12b gives a more accurate picture of an inward buckle of very small amplitude; it

is seen that the original sign of the circumferential curvature of the shell wall is not

reversed at the start of buckling. The radial forces arising from the combination of the

membrane forces with the curvature of the deformed cylinder, which still has its initial

sign, are shown in Figure 12c. These radial forces all tend to counteract buckling. Hence

the high resistance of a perfect cylinder to the initiation of buckling, given by Equation

(2). Increasing inward displacements cause the change of circumferential curvature to

exceed the magnitude, 1/r, of the original curvature of the cylinder, as shown in an

exaggerated manner in Figure 12d, and more realistically in Figure 12e. In the region of

the inward buckles, the wall of the cylinder is now curved inwards and, as a result, the

compressive membrane forces in these areas no longer resist the appearance of dents, but

precipitate them (Figure 12f). Hence, the total restoring effect of the membrane forces

has now weakened substantially compared with the state prevailing at the bifurcation

point. The upshot is that, once buckling has started, equilibrium is conceivable only under

decreasing axial load.

7. IMPERFECTION SENSITIVITY

The behaviour of actual imperfect components differs from the theoretical behaviour

described above and is represented by the dotted curves in Figure 9. They show that true

bifurcation of equilibrium does not actually occur in the case of real structural members.

However, the solid lines provide an approximate picture - the smaller the initial

imperfections, the truer the picture is - of the behaviour of the component, and therein

lies the significance of the bifurcation buckling concept.

The dotted lines in Figures 9a and 9b, have been drawn for a column and a plate with

slight initial curvature. It can be seen that the carrying resistance of the strut is not much

lower than the theoretical buckling load, provided that the imperfection is not too great.

One can conclude from Figure 9b that the equilibrium path of an imperfect plate may not

exhibit any discontinuity when the compressive stress increases beyond σcr, and also that

the plate may possess a considerable post-buckling strength reserve. If it is thin, this

reserve may be considerably greater than the bifurcation buckling load. Raising the stress

beyond σcr for the perfect plate does not bring about immediate ultimate failure. Both the

column and the plate finally fail by yielding caused by excessive bending.

Owing to the imperfection of a real cylinder, the dotted equilibrium path does not display

the very sharp high peak B which is a feature of the theoretical equilibrium path OBDC.

The culminating point G or H (Figure 9c) of the dotted line, called a limit point, is at a

much lower level than the bifurcation point, even when the amplitude of the initial

deviations from the perfect cylindrical shape is minute. The lower dotted curve is the

equilibrium path for a cylinder with somewhat larger imperfections. When the loading is

due to weight and happens to correspond to the limit point, the curve must jump

horizontally from G or H towards the right hand branch of the curve. The concomitant

shortening, u, of the steel shell is so large, and due to buckles which are so deep, that

normally part of the wall material is strained into the plastic range and so the buckling

phenomenon, in this case a snap-through or non-linear collapse, is almost always

catastrophic.

One should not infer from the description in the preceding paragraph that only imperfect

structural components display behaviour characterized by a limit point. Due to gradual

changes in the geometry of a perfect structure, its primary equilibrium path may be non-

linear from the outset of loading and, indeed, feature a limit point.

As a summary two points can be established:

1. The real collapse stress, σuG or σuH (Figure 9c), is much lower than the theoretical

critical stress, σcr, for the perfect shell, even though the imperfections may be

hardly perceptible.

2. Nominally identical shells collapse under markedly different loads because the

unintentional actual imperfections of such shells, as erected, are different in

magnitude and in distribution, and because an appreciable decrease in ultimate

load may result from slightly larger imperfections.

A sweeping generalization to the effect that all shells are always very sensitive to

deviations from the perfect shape would be unwarranted. The imperfection sensitivity

depends on the type of shell and loading. It may vary from slight to extreme, even for the

same kind of shell under different loading conditions. For example, the imperfection

sensitivity of cylindrical shells under uniform external pressure is quite low, whilst the

same shells are highly imperfection sensitive when they are compressed in the meridional

direction. The difference relates to the buckling mode; under axial load, the buckling

modes are characterised by waves which, compared to the diameter, are short in both the

longitudinal and the circumferential direction. Small initial imperfections, which may

occur anywhere on the surface of the cylinder and which are likely to have roughly the

same shape as some of the critical buckles, tend to deepen under increasing load and to

trigger off a snap-through at an early loading stage. The buckling pattern under external

pressure, however, consists of buckles which are long in the meridional direction, and

less numerous in the hoop direction, and therefore probably of considerably larger size

than the principal initial dents and bulges.

Another factor that should be mentioned as contributing to the imperfection sensitivity of

axially loaded cylinders is the multiplicity of different buckling modes associated with

the same bifurcation load. Any realistic theoretical treatment of the buckling problem is

complicated further by the existence of residual stresses due to cold or hot forming and/or

due to welding. Behaviour is also affected by the appearance of plastic deformations in

the steel and, in some cases, by the presence of stiffeners. The non-linear structural

behaviour of the shell may be due to the latter, as well as to changes in the geometry

resulting from the deformation of the shell.

In conclusion, imperfections are the main cause of the large difference between the

ultimate load obtained in tests and the theoretical buckling load. A wide scatter of results

for nominally identical shells can be seen in Figure 13a where the ratio of experimental

buckling loads, Fu, against the theoretical values, Fcr, for axially loaded cylinders are

given for different r/t ratios. Figure 13b gives the factors proposed by ECCS to reduce the

theoretical buckling load to values appropriate for design.

8. CONCLUDING SUMMARY

• Bending and stretching are the modes by which shell structures carry loads.

• For shell structures, in industrial applications, buckling may be the critical limit

state due to slenderness effects.

• Imperfections are the main cause of the very significant difference between the

theoretical and the experimental buckling load.

• There are fundamental differences in initial buckling behaviour between shells

and plates.

• In practice, shell buckling analysis can be applied only to special structures which

have been manufactured/constructed using strict quality control procedures that

minimise imperfections.

9. REFERENCES

[1] Timoshenko, S. and Woinowsky-Krieger, S., "Theory of Plates and Shells", McGraw-

Hill, New York and Kogakusha, Tokyo, 1959.

[2] Flügge, W., "Stresses in Shells", Springer-Verlag, New York, 1967.

[3] Bushnell, D., "Computerised Buckling Analysis of Shells", Martinus Nijhoff

Publishers, Dordrecht, 1985.

[4] Timoshenko, S. and Gere, J.M., "Theory of Elastic Stability", McGraw-Hill, New

York and Kogakusha, Tokyo, 1961.

[5] Esslinger, M. T., and Geier, B. M., "Buckling and Post Buckling Behaviour of Thin-

Walled Circular Cylinders", International Colloquium on Progress of Shell Structures in

the last 10 years and its future development, Madrid, 1969.

10. ADDITIONAL READING

1. Koiter, W.T., "Over de Stabilitteit van het Elastisch Evenwicht Diss.", Delft,

H.J.Paris, Amsterdam, 1945.

Lecture 8.8: Design of Unstiffened

Cylinders

OBJECTIVE/SCOPE

To describe the buckling behaviour of cylindrical shells subjected to two different types

of external loading, axial compression and external pressure, acting independently or in

combination. To identify the key parameters influencing behaviour and to present a

design procedure, based on the European recommendations.

SUMMARY

The buckling behaviour of a cylindrical shell depends on several key parameters, such as

geometry, material characteristics, imperfections and residual stresses, boundary

conditions and type of loading. A reliable and economic procedure for the design of

cylindrical shells against buckling should take into account all the above parameters,

clearly specifying the range over which the design predictions are valid. The lecture

presents the relevant design procedure contained in the ECCS Shell Buckling

Recommendations [1] and briefly discusses alternative methods and identifies the

important differences.

1. INTRODUCTION

In Lectures 8.6 and Lecture 8.7 several aspects that influence the structural behaviour of

shells have been introduced and the main principles of shell theory have been presented.

In particular, it is worth recalling the following points:

i. The critical (bifurcation) buckling load of a perfect thin elastic cylinder under idealised

loading, such as uniform axial compression, may be calculated using classical methods.

ii. The boundary conditions affect the critical buckling load and, for the same shell and

type of loading, certain sets of boundary conditions can result in significantly lower

buckling loads compared to those corresponding to other sets.

iii. Geometric imperfections caused by manufacturing are the main cause of the

significant differences between critical buckling loads calculated using classical methods

and experimental buckling loads. Even very small imperfections can cause a substantial

drop in the buckling load of the shell.

iv. The sensitivity to imperfection depends primarily on the type of shell and type of

loading, and, to some extent, on the boundary conditions. It may vary from moderate to

extreme, even for the same shell geometry under different loading or boundary

conditions. For example, a cylinder under axial compression is extremely sensitive to

imperfections whilst the same shell under external pressure exhibits much lower

imperfection sensitivity.

This lecture deals with the design of unstiffened cylinders. The buckling behaviour under

two different types of loading, axial compression and external pressure, is first described

qualitatively. An appropriate design procedure based on the ECCS Shell Buckling

Recommendations [1] is then presented. The interaction behaviour for the two types of

loading is also discussed.

2. UNSTIFFENED CYLINDERS UNDER AXIAL

COMPRESSION

General Considerations

When a cylindrical shell is subjected to uniform axial compression (Figure 1) buckling

can occur in two possible modes:

• Overall column buckling, if the l/r ratio is large. This type of buckling does not

involve local deformation of the cross-section and can be analysed using methods

for columns, e.g. the Perry-Robertson formula. It will not be examined further in

this lecture.

• Shell buckling, which involves local deformation of the cross-section, and can, in

general, be either:

axisymmetric, where the displacements are constant around any circumferential section,

Figure 2(a), or

asymmetric (also called chessboard), where waves are formed in both axial and

circumferential directions, Figure 2(b).

It can be shown theoretically that both modes of shell buckling correspond to the same

critical buckling load. Assuming simply-supported boundary conditions (w = 0 and mx =

0, see Figure 3) that also preclude tangential displacements at both edges (v = 0) the

critical elastic buckling stress, σcr, is given by:

σcr = (1)

where

E is the elastic modulus

t is the cylinder thickness

r is the cylinder radius

ν is Poisson's ratio

It is worth noting that the critical buckling stress is independent of the length of the

cylinder.

Axisymmetric buckling is more often encountered in short and/or relatively thick

cylinders. Asymmetric buckling is more common in thin and/or relatively long cylinders.

If one of the cylinder ends is free (w ≠ 0) the critical buckling stress drops to 38% of that

given by Equation (1). If, however, the cylinder is clamped at both ends, rather than

being simply supported, the increase in the critical buckling stress is not that significant

from a design point of view.

On the other hand, the cylinder is sensitive to the tangential displacement at the

boundaries [2,3]. If it is not prevented (v ≠ 0), the critical stress drops to about 50% of the

value given by Equation (1).

Equation (1) cannot be used directly for design because cylindrical shells are extremely

sensitive to imperfections under axial compression. Imperfection sensitivity is taken into

account in design codes by introducing a "knock-down" factor, α, so that the product

ασcr represents the buckling load of the imperfect shell. In addition plasticity effects,

which are important for a certain range of cylinder geometries, must be taken into

account.

The "knock-down" factor is in general a function of shell geometry, loading conditions,

initial imperfection amplitude and other factors and is normally evaluated from

comparison with experimental results. The "knock-down" factor is selected so that a high

percentage of experimental results (for example, 95%) should have buckling loads higher

than the corresponding loads predicted by the design method. Figure 4 shows a typical

distribution of test data for cylinders subjected to axial compression together with a

typical design curve.

Due to the high sensitivity to imperfections, the design method should specify the

maximum allowable level of imperfections. These tolerances are related to the

imperfection amplitudes measured in the tests used in determining the appropriate

"knock-down" factors. Clearly, the tolerances should not be so strict that they cannot be

achieved using normal manufacturing processes. It should be noted that the use of

experimental databases containing a large number of test specimens which are not

representative of full-scale manufacturing, may lead to erroneous "knock-down" factors.

Ideally, the design method should also enable a designer to evaluate the buckling load of

a cylinder with imperfections which exceed the allowable limits. Currently, very few

design methods are valid for larger imperfections and the importance of adhering to the

stated tolerances cannot be over-emphasized.

The experimental databases used by various codes in estimating "knock-down" factors

can vary substantially. It is true to say that some design proposals are based on old,

limited or inappropriate data. For this reason, design predictions for the buckling load of

nominally identical cylinder geometries can vary substantially. References [4,5] present

comparisons between various codes and attempt to explain the reasons for the observed

differences.

The ECCS Design Procedure

The general philosophy adopted in the ECCS recommendations [1] is given in [6]. The

design method for axially compressed cylinders is presented below, together with some

clarifying comments.

The proposed method is valid for cylinders that satisfy w = v = 0 at the supports, i.e.

radial and tangential displacements prevented, see Figure 3, and also for geometries that

do not exceed the following geometric limit:

(2)

This limit is imposed to preclude the possibility of overall column buckling interacting

with shell buckling.

The cylinder should also satisfy the imperfection tolerances. They should be checked

anywhere on the surface of the shell, using either a straight rod or a circular template, as

shown schematically in Figure 5.

The length of the rod or template, lr, is related to the size of the potential buckles [1]. The

allowable imperfection, , is given by:

= 0,01 lr (3)

The strength requirement for cylinders under uniform axial compression is given by:

σd ≤ σu (4)

where σd is the applied axial compressive stress (characteristic load effect).

σu is the design value of the buckling stress (characteristic resistance).

Thus, the objective is to determine the value of σu, or, equivalently, the value of the ratio

σu/fy, where fy is the specified characteristic yield stress.

The proposed method is schematically shown in Figure 6, where σu/fy is plotted against a

slenderness parameter, λ. This parameter is defined as:

λ = (5)

where σcr is the elastic critical buckling stress of a perfect cylinder given by Equation (1)

α is the "knock-down" factor, which accounts for the detrimental effect of imperfections,

residual stresses and edge disturbances.

As can be seen from Figure 6, two regions are defined; the first, for which λ ≥ ,

defines the region of elastic buckling, whereas λ ≤ defines the region of plastic

buckling.

For λ ≥ or, equivalently, for ασcr ≤ 0,5fy (i.e. when the buckling stress of the

imperfect shell is less than half of the characteristic value of the yield stress), it is deemed

that elastic buckling governs and the design curve is given by

σu/fy = (1/γ) (1/λ2)(6a)

where 1/γ is an additional safety factor introduced for this type of geometry and loading

to account for the extreme sensitivity to imperfections and the unfavourable post-

buckling behaviour; a value of 3/4 is recommended for l/γ .

For λ ≤ (ασcr ≥ 0,5fy), material non-linearities also play a role (plastic buckling) and

the design curve is given by:

σu/fy = 1 - 0,4123 λ1,2 (6b)

The "knock-down" factor α, which appears in Equation (5), has been derived from

comparisons with experimental results and is determined from the following equations,

which are plotted in Figure 7. These equations are applicable if the amplitude of the

imperfections anywhere on the shell is less than or equal to the value given by Equation

(3). It is worth pointing out the dependence of α on cylinder slenderness, r/t.

α = for r/t < 212 (7a)

α = for r/t > 212 (7b)

Details of the experimental database used in deriving these equations are presented in [7].

It is also worth noting in Figure 6 that σu/fy approaches unity for very stocky cylinders (λ

close to 0) and, furthermore, that there is a smooth transition from elastic to plastic

buckling at the change-over of formulae, as would be expected from a physical point of

view.

If the maximum amplitude of the actual cylinder imperfections is twice the value given

by Equation (3) then the value of the "knock-down" factor given by Equations (7a) and

(7b) is halved. When 0,01 lr ≤ ≤ 0,02 lr linear interpolation between α and α/2 gives the

required "knock-down" factor.

Although a slight, practically unavoidable unevenness of the supports of the cylinder is

covered by the "knock-down" factor α, care must be exercised to introduce the

compressive forces uniformly into the cylinder and to avoid edge disturbances. The

design method does not cover cylinders loaded over part of their circumference, e.g. a

cylinder resting on a number of point supports.

Finally, the above procedure covers shell buckling design of cylinders satisfying the limit

imposed by Equation (2). However, very short cylinders fail by plate-type buckling

which depends on the length of the cylinder, rather than by shell buckling. In this case,

meridional strips of the cylinder wall buckle like strips of a "wide" plate under

compression and do not exhibit the sensitivity to imperfections associated with shell-type

behaviour.

In this case σu is given by:

For σE ≤ 0,5 fy (elastic buckling)

σu = σE (8a)

For σE ≥ 0,5 fy (plastic buckling)

σu = fy (8b)

where σE is the elastic critical buckling stress of a "wide" plate of length l and thickness t,

given by

σE = (9)

In general, for any particular geometry, both Equations (6) and (8) should be evaluated

and the higher value taken for σu. However, Equation (8) gives a value higher than

Equation (6) only for very short cylinders. It is quite easy to work out that for elastic

buckling, i.e. comparing Equation (6a) and Equation (8a), this is true when

< (10)

3. UNSTIFFENED CYLINDERS UNDER EXTERNAL

PRESSURE

General Considerations

External pressure may be applied either purely radially, as "external pressure loading"

(Figure 8(a)), or it may be applied all round the cylindrical shell, that is both radially and

axially (Figure 8(b)) as "external hydrostatic pressure loading".

For external lateral pressure, the hoop stress σθ , in the pre-buckling state away from the

supports is related to the applied pressure, p, by a simple expression:

σθ = (r/t) p (11)

In this case, the axial stress σx = 0.

For external hydrostatic pressure, the hoop stress is again given by Equation (11) but the

axial stress σx = 0,5σθ .

In the latter case, the cylinder is, in fact, subjected to combined axial and pressure

loading. This case will be examined in Section 4, where the complete interaction

behaviour is described.

Assuming classical simply supported conditions (v = w = 0 and nx = mx = 0, at the

supports) and neglecting the effect of boundary conditions on the pre-buckling state, it is

possible to derive the elastic critical buckling pressure of the cylinder. This is known as

the von Mises critical pressure, pcr

pcr = (12)

As can be seen, pcr depends on shell geometry, and in particular the ratios l/r and t/r, the

material characteristics, E and ν, and the number of complete circumferential waves, n,

forming at buckling. Thus, for a given geometry and material, Equation (12) must be

minimised with respect to n in order to obtain the lowest pcr.

Simpler expressions for pcr have also been developed, for example using the so-called

Donnell equations for "shallow" shells giving good results provided that the value of n

minimising pcr is greater than two [3].

For reasons similar to those mentioned in Section 2, namely sensitivity to imperfections

and plasticity effects, the value obtained from Equation (12) is not directly suitable for

design.

The sensitivity to imperfections of cylindrical shells under external pressure is not as

severe as under axial compression. This difference has been demonstrated both

theoretically and experimentally and is reflected in the "knock-down" factors used in

design.

Despite the reduced imperfection sensitivity, considerable differences exist between

various codes [5], possibly due to the different theoretical approaches adopted (see also

[4]).

The ECCS Design Procedure

The procedure described below applies only to uniform external pressure. Pressure due to

wind loading is not covered by this method. Furthermore, cylinders should be circular to

within 0,5% of the radius measured from the true centre. At least 24 equally spaced

points around the circumference should be chosen in order to establish whether the

cylinder satisfies this out-of-roundness tolerance. Finally, the procedure is valid for

cylinders having the "classical" simple supports during buckling (i.e. nx = v = w = mx =

0) and does not apply to cylinders which have one or two free edges.

The strength requirement is given by:

pd ≤ pu (13)

where pd is the applied uniform external pressure (characteristic load effect).

pu is the design pressure (characteristic resistance).

The first step consists of calculating the pressure py, at which the hoop stress at cylinder

mid-length reaches the yield stress. From Equation (11), it follows that

py = (t/r)fy (14)

Clearly, when pu = py the cylinder is so stocky that buckling plays no part in determining

the design pressure.

Then, pcr, the elastic critical buckling pressure is calculated using a formula, which is a

slight modification of the von Mises equation (12), as discussed in [8]. The equation has

the following form:

pcr = E (t/r) βmin (15)

where βmin is the minimum value of β with respect to n. The expression for β is given by

β =

+ (16)

A chart is also given in [1] which can be used in the evaluation of βmin. Furthermore, βmin

may be approximately estimated from

βmin = (17)

for l/r ≥ 0,5.

Having calculated py and pcr, an appropriate slenderness parameter is defined by

λ = (18)

If λ ≥ 1, elastic buckling takes place and the design pressure is determined from

pu/py = α/λ2 (19)

where α is the "knock-down" factor to account for the imperfection sensitivity.

From the results of about 700 tests, α = 0,5. Notice that, in contrast to the "knock-down"

factor relevant to axial compression, in this case, α is independent of shell slenderness,

r/t.

In the plastic region, 0 ≤ λ ≤ 1, the value of pu/py is obtained from the curve shown in

Figure 9.

This curve is closely approximated by the following expression:

pu/ py = 1/(1+λ2) (20)

4. UNSTIFFENED CYLINDERS UNDER AXIAL

COMPRESSION AND EXTERNAL PRESSURE

Buckling of unstiffened cylinders under combined loading is of considerable interest to

designers. For example, combined axial compression and lateral pressure is often

encountered in offshore design practice.

The available design recommendations are mostly empirical. They are based on a much

more limited number of experiments compared to the number used for the two basic

cases reviewed in Sections 2 and 3.

Buckling under combined axial and pressure loading can be quite sensitive to

imperfections, especially in cases where the loading is dominated by axial compression.

The ECCS recommendations [1], in common with some other codes, adopt a simple

piece-wise linear interaction based on the buckling strength under axial compression or

external pressure acting alone. The cylinder must be constrained so that v = w = 0 at both

ends. Furthermore, the cylinder must comply with the imperfection tolerances outlined in

Sections 2 and 3.

The interaction curve is shown in Figure 10.

Any combination of σd/σu and pd/pu falling within the region defined by the two straight

lines is safe. Thus, the applied loads (σd, pd) should satisfy the following two conditions:

I: σd ≤ σu (21a)

II: ≤ 1 if σd ≤ (21b)

or

≤ 1 if σd > (21c)

Equation (21b) states that if the applied axial compression is less than or equal to that

produced by applying uniform hydrostatic pressure, then the design pressure

corresponding to uniform external pressure may be achieved. A reduction due to the

simultaneous presence of axial compression is not deemed necessary. If however, the

axial compression is higher than this limiting value then the design strength is reduced

linearly as expressed by Equation (21c).

Experimental evidence has shown that this interaction diagram is conservative.

5. CONCLUDING SUMMARY

• The buckling behaviour of unstiffened cylinders under axial compression and

external pressure has been described and the key parameters have been identified.

• A good design procedure must consider:

⋅ the imperfection sensitivity, appropriate to the geometry, loading and boundary

conditions and hence, derive appropriate "knock-down" factors using reliable

experimental data.

⋅ the limitations that must be imposed on allowable imperfections due to available

experimental data and the characteristics of the manufacturing processes.

⋅ the interaction between elastic buckling and yielding.

⋅ the effect of boundary conditions.

• The procedure proposed by the ECCS recommendations has been presented. The

main steps for the two single loading cases are similar and can be summarised as

follows:

1. Determine the elastic critical buckling stress of the perfect shell.

2. Calculate the "knock-down" factor and, hence, the buckling stress of the imperfect

shell.

3. Depending on shell slenderness, modify the value from step (2) to account for plastic

buckling.

• The designer should be fully aware of the idealisations made in the design models

and of their limitations in respect of loading, boundary conditions and

imperfections.

• In general, design recommendations are limited to shells of revolution only,

having uniform thickness and subjected to idealised loading distributions. In

practical applications, various problems arise which are not covered in any of the

current design codes [6]. Much work remains to be done in this area and shell

designers should try to keep in touch with new developments.

6. REFERENCES

[1] ECCS - European Convention for Constructional Steelwork - Buckling of Steel Shells

European Recommendations, Fourth Edition, 1988.

[2] Timoshenko S P and Gere J M, Theory of Elastic Stability, McGraw-Hill, 1982.

[3] Brush D O and Almroth B O, Buckling of Bars, Plates and Shells, McGraw-Hill,

1975.

[4] Beedle L S, Stability of Metal Structures: A World View, Structural Stability

Research Council, 1991.

[5] Ellinas C P, Supple W J and Walker A C, Buckling of Offshore Structures, Granada,

1984.

[6] Samuelson L A, "The ECCS Recommendations on Shell Stability; Design Philosophy

and Practical Applications", in "Buckling of Shell Structures, on Land, in the Sea and in

the Air", J F Jullien (ed), Elsevier Applied Science, 1991, pp. 261-264.

[7] Vandepitte D and Rathe J, "Buckling of Circular Cylindrical Shells under Axial Load

in the Elastic-Plastic Region", Der Stahlbau, Heft 12, 1980, S. 369-373.

[8] Kendrick, S B, "Collapse of stiffened cylinders under external pressure", Paper

C190/72 in proc. Conf. on Vessels under Buckling Conditions, Instn of Mech. Engrs.,

London, 1972.

Previous | Next | Contents

Lecture 8.9: Design of Stringer-Stiffened

Cylindrical Shells

OBJECTIVE/SCOPE

To describe the buckling behaviour of stiffened shells and to analyse the different types

of failure. A practical design procedure, based on the European recommendations, for

stringer stiffened cylinders subject to axial load is presented.

SUMMARY

The buckling behaviour of stiffened shell structures is presented and the different types of

failure are discussed. The shell has to be designed with respect to local shell buckling

(limited to the shell panel between the stiffeners) and the stiffened panel buckling (or bay

instability) in which both the shell panel and the stringers participate. Buckling of the

stringers themselves must also be prevented. The design procedure relevant to this

problem, as proposed by ECCS recommendations [1], is presented.

1. INTRODUCTION

In Lectures 8.6 and Lecture 8.7 several aspects that influence the structural behaviour of

shells have been introduced and the main principles of the shell theory have been

presented. In particular it has been shown how the buckling strength of shell structures is

influenced by residual stresses, geometric imperfections, and in some cases by the

eccentricity of the load and by the boundary conditions. For these reasons axially

compressed cylindrical shells often fail at a buckling strength considerably lower than the

theoretical elastic value.

The buckling strength of cylindrical shells is often improved by the use of circumferential

and/or longitudinal stiffeners. Their size, spacing and position on the outside or inside of

the cylinder surface are factors that complicate the buckling behaviour of the shell.

In this lecture the general aspects of the buckling behaviour of stiffened shells are

presented and, as an example of the application of practical design procedures, the

buckling behaviour of stringer stiffened axially compressed cylinders is treated.

2. BUCKLING OF STIFFENED SHELLS

The shell may fail by overall buckling, by local buckling, or by a combination of the two.

If the critical loads relevant to the first two kinds of buckling differ from each other, there

is no interaction and, of course, the dominant mode of failure is the one relevant to the

lowest buckling load. If the two phenomena occur at about the same load the interaction

of the two types of buckling can theoretically cause a considerable reduction of the

critical load. The buckling modes interact because of the non-linear relations governing

post-buckling and cause a sharp drop in the post-buckling load bearing resistance [2,3].

In general, the design procedure of a stiffened shell (Figure 1) subjected to axial

compression and bending must consider the following types of failure:

• Overall column buckling or yielding (if L/r ratio is large).

• Local buckling between stiffeners (panel buckling) (Figure 2).

• Local buckling encompassing several stiffeners (stiffened panel buckling) (Figure

3).

• Local buckling of the individual stiffeners (Figure 4).

• Local yielding of the shell or of the stiffeners.

The longitudinal stiffeners (stringers), which may be placed on the outside of the shell

wall or on the inside, are frequently used to increase the axial or bending resistance of

cylinders. In what follows the procedure proposed in the ECCS Recommendations [1] for

the case of a cylindrical shell with longitudinal stiffeners and subject to meridional

compression is examined. Ring stiffened or orthogonally stiffened shells are not treated

here but the relevant design recommendations are similar to those presented here for

stringer stiffened shells and may be found in [1].

3. CYLINDRICAL SHELLS WITH LONGITUDINAL

STIFFENERS AND SUBJECTED TO MERIDIONAL

COMPRESSION

In the ECCS design rules for a circular cylindrical shell with longitudinal stiffeners and

subjected to axial compression and/or bending, it is assumed that the stringers are

distributed uniformly along the circumference of the cylinder (Figure 1). The properties

of the stiffeners are:

As is the cross-section area of the stiffener only

EIs is the bending stiffness of the stiffener only about the centroidal axis parallel to the

cylinder wall

GCs is the torsional stiffness of the stiffener only

es is the distance between the shell midsurface and the stiffener centroid, (positive for an

outside stiffener).

Cs may be evaluated by the formula for open sections consisting of flat strips

(1)

The following recommendations apply when

As < 2bt, Is < 15 bt3, GCs < 10bt

3E/[12(1-ν2

)] (2)

4. LIMITATION OF THE IMPERFECTIONS

The initial imperfections of the shell panel between the stiffeners must be limited. The

limitations are similar to those stated in Lecture 8.8 for unstiffened cylinders.

The inward and outward lack of straightness of the stiffener in the radial direction shall

not exceed the following values:

≤ 0,0015 lg when ≥ 0,06 (3)

≤ 0,0015 lg and ≤ 0,01 lr when 0 ≤ ≤ 0,06 (4)

See (3) for details.

The limits given for also apply to the initial circumferential out-of-straightness which

is the lateral misalignment of the stiffeners attachment to the shell (Figure 5). Also the

initial tilt of the stringer web and of the stringer flange (Figure 6) shall be limited by:

(5)

5. STRENGTH CONDITIONS

The design value of the extreme meridional acting stress is obtained from:

(6)

where:

ts = t +

The shell must be designed with respect to local shell bucking (subscript l) which is

limited to the shell panels between the stringer (Figure 2), and stiffened panel buckling

(Figure 3) or bay instability (subscript p) in which both the shell panel and the stringers

participate. Buckling of webs or flanges of the stiffeners themselves should be prevented

by limiting the ratio of certain stringer cross-section dimensions (Figure 4).

The value of the compressive stress causing local shell buckling is denoted by σul while

the value of the stress relevant to the stiffened panel buckling is denoted by σup. The

design value σd of the highest meridional acting stress shall not exceed any of the two

buckling stresses

σd ≤ σul and σd ≤ σup (7)

6. LOCAL PANEL BUCKLING

The elastic critical stress, σcr, l, for a perfect shell panel between stringers may be taken

equal to the higher of the two critical stresses for

a perfect complete cylinder σcr, l, = 0,605E(t/r) [b/√(rt) > 2,44] (8)

a perfectly flat plate σcr, l = 3,6 E(t/b)2 [b/√(rt) ≤ 2,44] (9)

The use of Equation (9) implies the neglect of the torsional rigidity of the stringers and of

the post-buckling reserve of resistance of the supposedly flat panel.

The elastic local shell buckling stress, σul, for an imperfect panel is the higher of the

stresses σul1 and σul2, provided that neither 4σul1 /3 nor σul2 exceeds 0,5fy; σul1 and σul2 are

the stresses σcr, l reduced to account for imperfections and, in the case of cylindrical panel

buckling, also for imperfection sensitivity. They are in fact the values determined from

Equations (8) and (9) reduced to (αl σcr, l / γ) by a factor αl and a partial safety factor γ, where αl accounts for the imperfections and γ accounts for the imperfection sensitivity.

For a cylindrical panel, αl is given by Equation (13) of [1], halved if the imperfection is

equal to 0,02 lr and obtained by linear interpolation if is in the range between 0,01 lr and

0,02 lr, while γ is taken equal to 4/3.

For plane panels the following factors are assumed:

αl = 0,83 and γ = 1,0

Thus the following design values of the elastic shell buckling stress are obtained for an

imperfect panel:

σul1 = ¾αo × 0,605E(t/r) (10)

σul2 = 0,83 x 3,6 E (t/b)2 (11)

When either 4σul1 /3 or σul2 exceeds 0,5 fy, plastic deformation comes into play and σul is

the higher of the stresses σul1 and σul2 obtained from:

if 0,605 αo E > 0,5 fy (12)

if 0,83 x 3,6 E > 0,5 fy (13)

7. STIFFENED PANEL BUCKLING

The elastic critical stress for a perfect stiffened cylindrical shell is given by:

(14)

for n = 0 or n ≥ 4 and m ≥ 1

where ts is defined in Equation (6) and the quantities A11 to A33 are defined in the

following way:

(15)

;

;

The half wave number in the longitudinal direction, m, and the full wave number in the

circumferential direction, n, must be chosen so that expression (14) is minimised. m and

n are integers, but decimal values for m and n may be allowed in the minimizing

processes. n = 0 represents axisymmetric buckling. n = 1 represents column buckling.

When n = 2 or 3 the results of the minimization process may contain an error of the order

of 20 to 25% on the unsafe side.

The critical stress resulting from Equation (14) is valid only if the stringers are spaced so

closely that their number, ns, fulfils the condition

ns ≥ 3,5 n (16)

Although comparisons with experiments have shown that Equation (14) yields safe

results even for numbers ns well below the limit of Equation (16), Equation (14) should

be used with circumspection. Evaluation of the torsional stiffness, GCs, of the stringers,

which has a marked influence on σcr, p, is not always straightforward. Setting Cs = 0 in

Equation (15), indicates the effective width of the panel (Figure 7). This concept was

introduced first within the theory of buckling of plane panels (Lecture 8.1). The stress

distribution in plan or curved panels becomes non-linear when the load exceeds the

buckling limit (Figure 7a). The most common way of considering the post-buckling

strength is to substitute an idealised stress distribution for the actual one such that the

maximum stress and the average stress are conserved. See for example Figure 7b where

the maximum stress σmax is the same as in Figure 7a and the dashed areas have the same

resultant. In practice, the effective panel width, be, may be obtained, but not explicitly,

from

1,9t√(Ε/fy) ≤ be = b√(αl σcr,l/σcr,p) ≤ b (17)

where αl σcr, l is the higher of the values 0,605 αo E(t/r) and 0,83x3,6E(t/b)2. Because in

practical applications the stringers are spaced rather closely, only the influence of local

shell buckling and of yielding on the effective width is considered in Equation (17).

The lower limit of the effective width (Equation 17) was originally proposed by von

Karman. If b ≤ 1,9 t , be should be set equal to b. When b>1,9t , σcr, p and be

must be determined through the iterative procedure depicted in Figure 8 which can easily

be implemented into a computer program. As a starting trial value of be take b, then the

quantities A11 to A33 may be calculated, expression (14) minimized and, after the

introduction of the tentative value σcr, p into Equation (17), a new value of be may be

obtained. The iteration process is stopped when successive values of be and σcr,p are

almost equal.

Example

Evaluate σcr, p for a mild steel cylinder having the following characteristics:

r/t = 1000

l/r = 1,6

ns = ns(min) outside flat bar stringer

E = 205000 Nmm-2

fy = 240 Nmm-2

As /(bt) = 0,5 hw /tw = 10

Applying the iterative procedure of Figure 8, which includes the effective width be

according to Equation (17), yields the minimum σcr, p = 202 Nmm-2 for m=1, n=11. The

final effective width gives be = 0,4 b which, in this case, is von Karman's lower bound of

Equation (17).

An alternative procedure for determining σcr,p, which may be used when the stringers are

flat bars, may be found in [1]. It is based on charts and contains also a non-iterative

method to evaluate σcr, p.

The results obtained so far are referred to the perfect panel. They must be corrected to

include the effect of imperfections. The stiffened panel buckling stress for an imperfect

stiffened cylinder may be obtained from:

σup = αsp σcr, p (18)

where:

αsp = 0,65 when > 0,2

αsp = αo when < 0,06 and also when < 60

αsp may be obtained by linear interpolation in the intermediate range of provided that

αsp σcr, p ≤ 0,5 fy. αo is the knockdown factor for an unstiffened cylinder of radius r and

thickness t (see Lecture 8.8).

As the stringer stiffeners decrease the imperfection sensitivity of meridionally

compressed cylinders, the value αsp is higher than αo when the stiffening effect is fairly

substantial.

It has been shown [4] that external stiffeners make the shell more sensitive to the

imperfections.

In the case of elastic-plastic buckling Equation (18) must be replaced by:

σup = fy {1 - 0,4123[fy /(αsp σcr, p)]0,6

} if αsp σcr, p > 0,5 fy (19)

8. LOCAL BUCKLING OF STRINGERS

To prevent the local buckling of stringers (Figure 4), the ratios of the stringers cross-

section dimensions shall be limited as follows:

hw /tw ≤ 0,35 for flat bar stiffeners with tw ≅ t

hw /tw ≤ 1,1 and bf /tf ≤ 0,7 for flanged stiffeners.

9. CONCLUDING SUMMARY

• The buckling behaviour of stiffened shells has been examined and the different

types of failure discussed.

• The design procedure must prevent:

a. local shell buckling (limited to the shell panel between the stiffeners)

b. stiffened panel buckling (in which the stiffeners and the panel participate)

c. bucking of the stiffeners themselves.

The procedure proposed by the ECCS [1] has been discussed in detail for stringer

stiffened cylinders.

10. REFERENCES

[1] European Convention for Construction Steelwork: "Buckling of Steel Shells -

European Recommendations", Fourth Edition, ECCS, 1988.

[2] Samuelson, L. A., Vandepitte, D. and Paridaens, R., "The background to the ECCS

recommendations for buckling of stringer stiffened cylinders", Proc. of Int. Coll. on

Buckling of Plate and Shell Structures, Ghent, pp 513-522, 1987.

[3] Ellinas, C. P. and Croll, J. G. A., "Experimental and theoretical correlations for elastic

buckling of axially compressed stringer stiffened cylinders", J Strain An., Vol. 18, pp 41-

67, 1983.

[4] Hutchinson, J. W. and Amazigo, J. C., "Imperfection Sensitivity of Eccentrically

Stiffened Cylindrical Shells", AIAA J, Vol. 5, No. 3, pp. 392-401, 1967.