Lecture 8.1 : Introduction to Plate Behaviour and Design Top 1. INTRODUCTION 2. BASIC BEHAVIOUR OF A PLATE PANEL 2.1 Geometric and Boundary Conditions 2.2 In-plane Actions 2.3 Out-of-plane Actions 2.4 Determination of Plate Panel Actions 2.5 Variations in Buckled Mode 2.6 Grillage Analogy for Plate Buckling 2.7 Post Buckling Behaviour and Effective Widths 2.8 The Influences of Imperfections on the Behaviour of Actual Plates 2.9 Elastic Behaviour of Plates Under Lateral Actions 3. BEHAVIOUR OF STIFFENED PLATES 4. CONCLUDING SUMMARY 5. ADDITIONAL READING
Lecture 8.1: Introduction to Plate Behaviour 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.
PREREQUISITES None.
RELATED LECTURES Lecture 8.2: 8.2: Behaviour and Design of Unstiffened Plates Lecture 8.3: 8.3: Behaviour and Design of Stiffened Plates
SUMMARY This lecture introduces the uses of plates and plated assemblies in steel structures. It describes the basic behaviour 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 behaviour of practical plates. It also gives an introduction to the behaviour 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 behaviour 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 th e greater direct stress) and length a, see Figure 1a. The ratio b/t, often called the plate slenderness, influences influences the local l ocal buckling of the plate panel; the aspect ratio a/b may also in fluence buckling patterns and may have a significant 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 inplane 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 important to differentiate 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 th e 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 selfevident. In other cases the in-plane flexibilities 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 inplane 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 n ot have shear coincident with the longitudinal compression. 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. 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 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 long itudinally. itudinally. As the aspect ratio increases the critical mode changes, tending towards the situation where the half wave length a/m = b. The behaviour of a long plate panel can therefore be modelled 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 between the modes. Thus in Figure 8b(i) the presence of low tran sverse compression compression does not n ot 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), predeformation/transverse compression may increase strength in longitudinal compression.) Shear buckling as shown in Figure 8c is basically an interaction between the diagonal, destabilising compression and the stabilising tension on the other diagonal.
Where buckled modes under the different action effects are sim ilar, 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 th e centre from the transverse flexural members. They therefore have gr eater 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 a ugmented 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 f y of 235N/mm2, 275N/mm2 and 355N/mm2 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 Behaviour 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 behaviour all relate to an ideal, perfect plate. As shown in Figure 13 these imperfections modify the behaviour of actual plates. For a slender plate the behaviour 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 per fect plate.
Figure 14 summarises the strength of actual plates of varying slenderness. It shows the reduction in strength due to imperfections imperfections and the post buckling strength of slender plates.
2.9 Elastic Behaviour of Plates Under Lateral Actions The elastic behaviour 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 i s 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 behaviour of a similar plate with different boundary conditions. Figure 16 shows the modes of behaviour that occur if the plates are subject to sufficient load for full yield line patterns to develop. The greater number of yield lin es as the boundary conditions improve is a qualit ative measure of the increase in resistance.
3. BEHAVIOUR OF STIFFENED PLATES Many aspects of stiffened plate behaviour can be deduced from a simple extension of the basic concepts of behaviour of unstiffened plate panels. However, in making these extrapolations it should be recognised 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 a ttached stiffening, 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 modelled by considering the stiffened panel as a series of tee sections buckling as columns. It should be noted that this section is monosymmetric and will exhibit different behaviour if the plate or the stiffener stiffener tip is in greater compression. (iii) overall or orthotropic bucking, Figure 17d. This occurs when the cross girders are flexible. It is best modelled 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 compression and an d 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 actions is influenced by its boundary conditions. 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 International Student Edition, 2nd Ed.
Lecture 8.2: Behaviour and Design of Unstiffened Plates OBJECTIVE/SCOPE To discuss the load distribution, stability and ultimate resistance of unstiffened plates under in-plane and out-ofplane loading.
PREREQUISITES Lecture 8.1: 8.1: Introduction to plate behaviour and design
RELATED LECTURES Lecture 8.3: 8.3: Stiffened Plates Lectures 8.4: 8.4: Plate Girder Behaviour and Design I an d II Lecture 8.6: 8.6: Introduction to Shell Structures
SUMMARY The load distribution for unstiffened 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 modelling 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 r equirements for strength and serviceability, 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 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: distinguished: a) those transferred from adjacent panels, such as compression compression or shear. b) those resulting from locally applied forces (patch loading) which generate zones of highly concentrated local stress in the plate. The behaviour under patch action is a specific problem dealt with in the lectures on plate girders (Lectures (Lectures 8.5.1 and 8.5.2). 8.5.2 ). This lecture deals with the more general behaviour of unstiffened 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 problem governed by Navier's equations, see Figure 2.
where: u = u(x, y), v = v(x, y): are the displacement components in 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 t he 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 th e Airy stress function by:
;
;
2.1.2 Distribution resulting from linear elastic theory u sing 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 Br yan 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 (Kirchhoff's th eory, 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 m agnitude.
Due to assumption (e) the rotations of the middle surface are small and their squares can be neglected in the strain displacement relationships relationships for the stretching stretching of the middle surface, which are simplified as:
εx = ∂u/ ∂x ,
2
2
P E/1
gxy
= ∂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 an d uncoupled. Thus, the plate equation under simultaneous bending and stretching is: 4
-kt
2
2
2
2
2
DN w = q {σx ∂ w/ ∂x + 2τxy ∂ w/ ∂x∂y + σy ∂ w/ ∂y }
(2)
where D = Et 3 /12(1 - n2) 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: equation:
Ν2 (σx + σy) = 0 (4) Equations (3) and (4) are r educed either to the biharmonic equation by employing the Airy stress function:
Ν4 F = 0 (5) defined as:
σx =
2
∂
2
2
2
2
F/ ∂y , σy = ∂ F/ ∂x , τxy = -∂ F/ ∂x∂y
or to the th e Navier equations of equilibrium, if the stress displacement relationships are employed:
Ν2 + [1/(1- )] ∂ / ∂x {∂u/ ∂x + ∂v/ ∂y} = 0 Ν2 + [1/(1- )] ∂ / ∂y {∂u/ ∂x + ∂v/ ∂y} = 0
(6)
= ν /(1 + ν) is the effective Poisson's ratio.
where
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 bifurcation point (Figure 5b). The edge in-plane actions may depend on more than one parameter, say k 1, k 2,...,k N, (e.g. σx, σy and τxy on the boundary may increase at different rates). In this case there are infinite combinations of values of k i for which buckling occurs. These parameters are constrained to li e 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 = k σE or τcr = k σE (7) σ
where σE =
τ
(8)
and k σ, k τ are dimensionless buckling coefficients. 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 k σ and k τ 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 k σ 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 k σ . Obviously, the buckling mode that gives the smallest value of k is the decisive one. For practical reasons a single value of k σ is chosen for plates subjected to n ormal 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, k σ is equal to 4 for a plate which is simply supported on all four sides and subjected to uniform compression.
Combination of stresses σx, σy and τ
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 k σ and k τ 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 subjected to a combination combination of normal and shear stresses. stresses. A direct determination determination of the t he buckling coefficient for a given combination of stresses is p ossible; but it r equires 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: equation:
(9) For a given pair of applied stresses (σ, τ) the factor of safety with respect to the above curve is given by:
= The equivalent buckling stress is then given by:
σcreq = γcreq √{σ2 + 3τ2} (11)
(10)
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 behaviour 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 behaviour 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 a symptotically symptotically to the value of σcr for increasing in creasing displacements, 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, imperfections, in the r egion 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 behaviour 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, imperfections, are:
(12)
This results in a coupling between 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) (geometrically) non-linear buckling theory. For a plate without imperfections the equilibrium path still has a bifurcation point at σcr, but, unlike the l inear 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 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−ν )/ k k } = π√(Ε/σcr) (14) 2
σ
If a reference slenderness given by:
λy = π√(Ε/ f fy ) (15) is introduced, the r elative 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 th e yield stress. For intermediate slendernesses, however, 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 str uts, 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 simply-supported plate subjected subjected to t o uniform compression (Figure 13a).
The stress distribution which is in itially 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 un iform one over a reduced "effective" width. This width is determined by equating the r esultant forces:
b σu = be σy (18) and accordingly: 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 th e relationship:
p
= √(σ/σcr) (20)
The design of thin walled cross-sections cross-sections is performed according to the following 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 d istributed 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 resistance.
γm the partial safety factor of
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 imperfect shape should relate t o 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 deflections w due to bending.
∇ 4w = where: q = q(x, y) D = Et 3 /12(1-
is the transverse loading 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 modelling the plate as a grid and n eglecting 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 n eglected 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 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 m embrane stresses 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 Airytype 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 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 determination of the deflections deflections which take in to 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 resistanc resistance e The resistance of plates, determined using the linear plate theory only, is normally much underestimated since the additional strength due to the membrane effect 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 unstiffened plate pan el 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 inplane 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 outof-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 component.
the stability of the plate should be checked neglecting the out-of-plane out-of-plane actions
For strength verification both actions h ave to be considered simultaneously. 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 s lenderness are adversely influenced by geometric imperfection imperfection and plasticity. Effective widths may be used to design plates whose behaviour is influenced by local buckling under inplane 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.
PREREQISITES Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lecture 8.2: 8.2: Behaviour and Design of Unstiffened Plates
RELATED LECTURES Lecture 8.4.1: 8.4.1: Plate Girder Behaviour & Design I Lecture 8.6: 8.6: Introduction to Shell Structures
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 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 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 gi rders, 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 a ctions. 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 stiffeners may be of open or of torsionally rigid closed sections, as shown in Figure 2. When th ese stiffeners stiffeners are arranged in a regular orthogonal grid, and the spacing is small enough to 'smear' the stiffeners to a cont inuum 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 behaviour of stiffened plate panels subjected to in-plane actions will be pr esented. The behaviour under out-of-plane actions is also discussed as is the influence of the out-of-plane 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 i sotropic behaviour behaviour when the stiffeners are smeared out. In practice this way of stiffening is not practical and therefore therefore not n ot 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 u sing 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 r elatively 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 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 modelled as beam-column elements eccentrically attached to the plate elements, see Lecture 8.2,, Section 2.1.3. 8.2 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 t o 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 ar e 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 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 then becomes:
δΠo = 0. The stability condition
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 a ll sides (Figure 6) the transverse displacements 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: 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 a mn retained in the Ritzexpansion. 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 = k σE or τcr = k σE (5) σ
τ
with σE = The most extensive studies on r ectangular, ectangular, 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 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, 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, 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 th e expression:
σvcr
= k σ Z1s σE (6)
where σE has the same meaning as in Equation (5). s is given by charts (Figure 8b).
Z1 = k σ , k τ are the buckling coefficients for normal and shear stresses stresses acting independently For more details the reader is r eferred 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 increase of k is possible, as shown in Figure 9a, because for γ = γI* the stiffeners stiffeners remain straight.
The second type γII*, is defined a s the value for which two curves of the buckling coefficients, belonging to different numbers of waves, cross (Figure 9b). The buckling coefficient 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 unstiffened subpanel (Figure 9c). The procedure to determine the optimum or critical stiffness is, therefore, quite simple. However, due to initial imperfections imperfections of both plate and stiffeners stiffeners as a s 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 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 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 pa nel; 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 investigations on stiffened stiffened panels ha ve 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, 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. 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 th e following provisions: a) Introduction of buckling curves as i llustrated 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 th e 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 th e proposed models is presented h ere; for more details reference should be made to Lectures 8.4,, 8.5 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). 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 subjected to compression can be determined using th e 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 V cr, 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 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, M u 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 always be defined clearly, as shown in the examples of Figure 15.
For a box gird er subjected subjected to uniaxial uni axial 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, 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 th e 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 lateraltorsional 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 simulated by the th e 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 c onsidered 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 d escribed in Section 3.1.1 of Lecture of Lecture 8.2. 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 separately from plate buckling and beam-column 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. 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 s tiffeners. 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 m ay 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 Stabilität der Baukonstruktionen", 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, Structures, 1986, 129138. [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, "Stahlbauten, Stabilitätsfalle, Plattenbeulen", Plattenbeulen", Berlin: Beuth. [9] ECCS, "Design of longitudinally stiffened webs of plate and box girders", Draft 1989. [10] ECCS, "Stiffened compression flanges of box g irders", 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 8.4.1: Plate Girder Behaviour and Design I OBJECTIVE To introduce basic aspects of the behaviour and design of plate girders. To explain how the typical proportions employed influence the types of behaviour that must be addressed in design, and to identify the various buckling considerations involved, involved, as a preparation for subsequent consideration of the design approaches of Eurocode 3 [1].
PREREQUISITES None
RELATED LECTURES Lectures 3.2: 3.2: Erection Lecture 7.2: 7.2: Cross-section Cross-section Classification Classification Lecture 7.3: 7.3: Local Buckling Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lecture 8.4.2: 8.4.2: Plate Girder Behaviour and Design - II Lecture 8.4.3: 8.4.3: Plate Girder Design - Special Topics Lecture 11.8: 11.8: Splices in Buildings Lecture 14.4: 14.4: Crane Runway Girders Lecture 15B.3: 15B.3: Plate Girder and Beam Bridges
SUMMARY Modern plate girders are introduced by explaining typical usage, types and the reasons for their inherent slender proportions. Their behaviour is described with particular emphasis on the different forms of buckling that can occur. The general basis of plate girder design is discussed in a simplified way as a prelude to a more detailed presentation in Lectures 8.4.2 and Lecture 8.4.3. 8.4.3. Post-buckling and tension field action are introduced and the roles of the main components in a plate girder identified.
1. INTRODUCTION Modern plate girders are normally fabricated by welding together two flanges and a web plate, as shown in Figure 1. Such girders are capable of carrying greater loads over longer spans than is generally possible using standard rolled sections or compound girders. Plate girders are typically used as long-span floor girders in buildings, as bridge girders, and as crane girders in industrial structures.
Plate girders are at their most impressive in modern bridge construction where main spans of well over 200m are feasible, with corresponding cross-section cross-section depths, ha unched over the supports, in the range of 5-10m. Because plate girders are fabricated separately, each may be designed individually to resist the applied actions using proportions that ensure low self-weight and high load r esistance. esistance. For efficient design it is usual to choose a relatively deep girder, thus minimising the required area of flanges for a given applied moment, Msd. This obviously entails a deep web whose area will be minimised by reducing its thickness to the minimum required to carry the applied shear, Vsd. Such a web may be quite slender (i.e. a high d/t w ratio) and may be prone to local buckling (see Lecture 7.3) 7.3) and shear buckling (see below). Such buckling problems have to be given careful consideration in plate girder design. One way of improving the load carrying resistance of a slender plate is to employ stiffeners (Lecture ( Lecture 8.1); 8.1); the selection of appropriate forms of stiffening is an important aspect of plate girder design.
1.1 Types There are several forms of plate girder; Figure 2 illustrates three different types - unstiffened, transversely stiffened, and transversely and longitudinally stiffened. The three girders shown have bisymmetric I-profile cross-sections, although flanges of different size are sometimes used, as already shown in Figure 1. Other types of cross-section (see Figure 3) are monosymmetric monosymmetric I-profiles, which are popular in composite construction with the smaller flange on top (see Lecture 10.2), 10.2), or as crane girders (see Lecture 14.4) 14.4) with the larger flange on top. Figure 3 also shows two other (less common) variations - the "delta girder" and the tubular-top-flange girder - both being possible solutions in cases of lon g laterally-unsupported top compression flanges prone to lateral-torsional lateral-torsional buckling (see Lecture 7.9.1 and 7.9.2 7.9.2). ).
There is also considerable scope for variation of cross-section in the longitudinal direction. A designer may choose to reduce the flange thickness (or breadth) in a zone of low applied moment, especially when a field-splice facilitates the change. Equally, in a zone of high shear, the designer might choose to thicken the web plate (see Figure 4). Alternatively, higher grade Fe E355 steel might be employed for zones of high applied moment and shear, while standard grade Fe E235 would be used elsewhere. So-called "hybrid" girders with different strength material in the flanges and the web offer another possible means of more closely matching resistance to requirements. More unusual variations are adopted in special circumstances, circumstances, such as bridgework bridgework (see Lecture 15B.4) 15B.4) e.g. tapered girders, cranked girders, haunched girders (see Figure 5), and of course, plate girders with web holes to accommodate services, see Figure 6.
1.2 Proportions Since the designer, in principle, is quite free to choose all the dimensions of a plate girder, some indication of the more usual proportions is now given (see also Figure 7):
Depth: Overall girder depth, h, will usually be in the range L o /12 ≤ h ≤ Lo /8, where Lo is the length between points of zero moment. However, for plate girder bridges the range will extend to approximately Lo /20. Flange breadth: The breadth, b, will usually be in the range h/5 flats' may be used un less the flange is very wide.
≤ b ≤ h/3, b being in multiples of 25mm. 'Wide
Flange thickness: The flange thickness, tf , will usually at least satisfy the requirements of Eurocode 3 (Table 5.3.1) for Class 3 (semi-compact) sections, i.e. c/t f ≤ 14ε. The thickness will usually be chosen from the standard plate thicknesses. Web thickness: Web thickness, tw, will determine the exact basis for web design, depending on whether the web is classified with regard to shear buckling as "thick" or "thin" (see later). Thin webs will often require stiffening; this may take the form of transverse stiffeners, longitudinal stiffeners or a combination, see Figure 2. Longitudinally stiffened girders are more likely to be found in large bridge construction where high d/t w ratios are appropriate, e.g. 200 ≤ d/tw ≤ 500, due to the n eed to minimise self-weight.
Clearly, depending on the particular loading pattern, and on depth and breadth restrictions, one can expect wide variations within all the above limits which should be regarded as indicative only.
2. DESIGN CONCEPTS Under static loading, ultimate limit states such as strength and stability will normally govern most plate girder design, with serviceability limit states such as deflection or vibration being less critical. Some absolute limits on plate slenderness are advisable so as to ensure sufficient robustness during erection. A generally accepted method [2] for designing plate girders (which is permitted by Eurocode 3) subject to a moment M ad and a coincident shear Vad is to proportion the flanges to carry all the moment with the web taking all the shear. This provides a particularly convenient means for obtaining an initial estimate of girder proportions. Thus, at any particular cross-section along a laterally-restrained plate girder, subject to specific values of bending moment and shear force, the flange and web plates can be sized separately. The required flange plate area may readily be obtained as follows: Af = M/[(h - t f )f y / γMO] ≅ M/(hf y / γMO) (1) (An iteration or two may be required depending on an assumed value of tf and its corresponding f y value from Table 3.1, Eurocode 3). Because the (normally) slender web will prevent the plastic moment of resistance of the crosssection from being attained, the flange b/tf ratio need only comply with the Eurocode 3 (Table 5.3.1) requirements for a Class 3 (semi-compact) (semi-compact) flange. The cross-sectional moment of resistance may then be ch ecked using: Mf.Rd = b tf (h - tf )f y / γMO (2) Unfortunately, Unfortunately, economic sizing of the web plate is not quite as straightforward, although where a thick web (defined later) is acceptable it can be rapidly sized by assuming uniform shear stress τy over its whole area. The web-toflange fillet welds must be designed to transmit the longitudinal shear at the flange/web interface.
3. INFLUENCE OF BUCKLING ON DESIGN Provided that the individual plate elements in a girder are each kept sufficiently sufficiently stocky, the design may be based on straightforward yield strength considerations. Economic and practical considerations will, however, dictate that not all of these conditions will always be met. In most cases various forms of buckling must be taken into account. Figure 8 lists the different phenomena.
3.1 Shear Buckling of the Web Once the d/t w value for an unstiffened web exceeds a limiting figure (69 ε in Eurocode 3) the web will buckle in shear before it reaches its full shear capacity A wτy. Diagonal buckles, of the type shown in Figure 9(a), resulting from the diagonal compression associated with the web shear will form. Their appearance may be delayed through the use of vertical stiffeners, see Figure 9(b) since the load at which shear buckling is initiated is a function of both d/tw and panel aspect ratio a/d.
3.2 Lateral-Torsional Buckling of the Girder This topic is covered in Lecture 7.9.1 and 7.9.2 7.9.2..
3.3 Local Buckling of the Compression Flange Provided that outstand proportions c/tf are suitably restricted, local buckling will have no effect on the girder's load carrying resistance.
3.4 Compression Buckling of the Web Webs for which d/t w ≤ 124ε and which are not subject to any axial load will permit the full elastic moment resistances of the girder to be attained. If this limit of d/t w (or a lower one if axial compression in the girder as a whole is also present) is exceeded, then moment resistance must be reduced accordingly. If it is desired to reach the girder's full plastic moment resistance a stricter limit will be appropriate.
3.5 Flange Induced Buckling of the Web If particularly slender webs are used, the compression flange may not receive enough support to prevent it from buckling vertically rather like an isolated strut buckling about its minor axis. This possibility may be eliminated by placing a suitable limit on d /tw. Transverse stiffeners also assist in resisting this form of buckling.
3.6 Local Buckling of the Web Vertical loads may cause buckling of the web in the region directly under the load as for a vertical strut. The level level of loading that may safely be carried before this happens will depend upon the exact way in which the load is transmitted to the web, the web proportions, and the level of overall bending present.
4. POST-BUCKLING STRENGTH OF WEB Owing to the post-buckling behaviour (see Lecture 8.3) 8.3) plates, unlike struts, are often able to support loads considerably in excess of their initial buckling load. In plate girder webs a special form of post-buckling termed "tension field action" is possible. Tension field action involves a change in the way in which th e girder resists shear loading from the development of uniform shear in the web at low shear loads, to the equivalent truss action, shown in Figure 10, at much higher loads. In this action the elements equivalent to truss members are: the flanges, which form the chords; the vertical stiffeners which form the struts; and the diagonal tension bands which form the ties. The compressive resistance of the other diagonal of each web panel is virtually eliminated by the shear buckling. The way in which this concept is utilized in design is explained in Lecture 8.4.2. 8.4.2.
5. DESIGN CONSIDERATIONS The principal functions of the main components found in plate girders may be summarised as follows: Flanges
resist moment
Web
resists shear
Web/flange welds
resist longitudinal shear at interface
Vertical stiffeners stiffeners
improve shear buckling resistance
Longitudinal stiffeners stiffeners
improve shear and/or bending resistance.
6. CONCLUDING SUMMARY
• • • •
The main components in a plate girder have been identified and their principal functions noted. Initial sizing may be made on the basis that the flanges carry all of the moment and the web takes all of the shear. Shear buckling is likely to prevent the full web shear resistance from being attained in slender webs. Its appearance need not imply failure since additional load may be carried through tension field action. Web stiffeners (transverse and/or longitudinal) enhance both initial buckling and post-buckling resistance.
7. REFERENCES [1] Eurocode 3: "Design of Steel Structures": European Prestandard ENV1993-1-1: Part 1, General rules and rules for buildings, CEN, 1992. [2] Narayanan, R. (ed)., "Plated Structures; Stability and Strength", Applied Science Publishers, London, 1983. Chapter 1 covers basic aspects of pla te girder behaviour and design.
8. ADDITIONAL READING 1.
Dubas, P. and Gehri, E. (eds), "Behaviour and Design of Plated Steel Structures", Publication No 44, ECCS, 1986.
Chapters 4 and 5 provide more detailed accounts of the main features of plate girder behaviour and design.
Lecture 8.4.2: Plate Girder Behaviour and Design II OBJECTIVE/SCOPE To present the basic design methods for plate girders subjected to either shear or moment, or a combination of both.
PREREQUISITES Lecture 8.4.1: 8.4.1: Plate Girder Behaviour and Design I
RELATED LECTURES Lecture 7.3: 7.3: Local Buckling Lecture 7.8.1: 7.8.1: Restrained Beams I Lecture 7.8.2: 7.8.2: Restrained Beams II Lecture 7.9.1: 7.9.1: Unrestrained Beams I Lecture 7.9.2: 7.9.2: Unrestrained Beams II Lecture 8.4.3: 8.4.3: Plate Girder Design - Special Topics
SUMMARY The design methods for plate girders subject to bending and shear, according to the methods of Eurocode 3[1], are presented. For shear loading two methods are described: the "simple post-critical method", and the "tension field method"; interaction diagrams can be used with both m ethods to allow for the effect of coincident moments.
1. INTRODUCTION Any cross-section of a plate girder is normally subjected to a combination of shear force and bending moment. The primary function of the top and bottom flange plates of the girder is to resist the axial compressive and tensile forces arising from the applied bending moment. The primary function of the web plate is t o resist the applied shear force. Plate girders are normally designed to support heavy loads over long spans in situations where it is necessary to produce an efficient design by providing girders of high strength to weight ratio. The search for an efficient design produces conflicting requirements, particularly particularly in the case of the web plate. To pr oduce the lowest axial flange force for a given bending moment, the web depth (d) must be made as large as possible. To reduce the self weight, the web thickness (tw) must be reduced to a minimum. As a consequence, in many instances the web plate is of slender proportions and is therefore prone to buckling at relatively low values of applied shear. A similar conflict may exist for the flange proportions. The required flange area is defined by the flange force and material yield stress. The desire to increase weak axis second moment of area encourages wide, thin flanges. Such flanges are prone to local buckling. Plate elements do not collapse when they buckle; they can possess a substantial post-buckling reserve of resistance. For an efficient design, any calculation relating to the ultimate limit state should take the post-buckling action into account. This is particularly so in the case of a web plate in shear where the post-buckling resistance arising from tension field action can be ver y significant.
Thus, in designing a plate girder it is necessary to evaluate the buckling and post-buckling action of webs in shear, and of flange plates in compression. The design of plate girder flanges largely follows procedures already discussed in Lecture 7.8, 7.8, Lecture 7.9.1, 7.9.1, and Lecture 7.9.2 for beams. However, the design of web p lates operating in the postbuckling range is very different and will be discussed here in some detail. The lecture will start by concentrating upon the resistance of plate girders to predominantly shear loading. The effects of high co-existent bending moments will be considered. The lecture will concentrate only on the main aspects of girder design assuming a basic cross-section. In particular, it is assumed that: 1. 2. 3. 4. 5.
Only transverse web stiffeners are present (i.e. there are no longitudinal stiffeners). Transverse web stiffeners possess sufficient stiffness and strength to resist the actions transmitted to them by the web. An appropriate means is available to anchor the tension field. No vertical patch loads are applied between the positions of the transverse web stiffeners. Only solid webs are considered (i.e. there are no web openings or holes).
Lecture 8.4.3 considers other important cases that do not comply with the above assumptions.
2. SHEAR BUCKLING RESISTANCE A typical transversely stiffened plate girder is shown diagrammatically in Figure 1, which also defines the notation used. The shear buckling resistance of the web depends mainly on the depth to thickness ratio (d/t w), and upon the spacing (a) of the tran sverse web stiffeners. stiffeners.
Intermediate transverse stiffeners are normally employed to increase the shear buckling resistance of the web, although designers may sometimes choose to use a thicker web plate rather than incur the additional fabrication costs arising from the use of intermediate stiffeners. Girders without intermediate stiffeners are normally termed "unstiffened" "unstiffened" girders, even though they will normally have stiffeners at points of support and possibly at the position of load application. Web buckling should be checked in all cases where the depth to thickness ratio, (d/t w), of the web exceeds 69ε . Eurocode 3 then offers the choice of 2 methods for plate girder design. The methods are: a) the simple post-critical method , which may be applied to both stiffened and unstiffened girders and is therefore of general application. b) the tension field method, which may only be applied to girders with intermediate transverse stiffeners. Even for such girders its range of application is limited to a range of stiffener stiffener spacing defined by:
1,0 ≤ a/d ≤ 3,0 There is now considerable evidence [2] that tension field action does develop in girders where the stiffener spacing lies outside this range, and also in unstiffened girders; such evidence, however, has yet to be presented in a form that is suitable for inclusion in a design code. The simple post-critical method is seen as a general-purpose method which can be applied to the design of all girders. The tension field method, on the other hand, can be applied to a certain range of girders only, but will lead to considerably more efficient designs for these girders, because it takes full account of the post-buckling post-buckling reserve of resistance. Each method will now be discussed.
2.1 Calculation of the Shear Buckling Resistance by the Simple Post-Critical Method This simple approach allows the design shear buckling resistance (Vba.Rd) to be determined directly as follows: Vba.Rd = d tw τba / γM1 (1) where all the terms in the expression are familiar, except the post-critical shear strength, τba. The calculation of this term depends upon the slenderness of the web which may be conveniently expressed by the following parameter:
(2) Here, k τ is a shear buckling factor calculated from elastic buckling theory [3]. For simplicity, it is conservatively assumed in this calculation that the boundaries of the web panel are simply supported, since the true degree of restraint offered by the flanges and adjacent web panels is not known. The resulting expression obtained for the shear buckling factor is dependent upon the spacing of the transverse transverse web stiffeners as follows: for closely spaced intermediate stiffeners (a/d < 1,0) :
k τ = 4 + for widely spaced in termediate stiffeners stiffeners (a/d
≥ 1,0) :
k τ = 5,34 + for unstiffened webs: k τ = 5,34 Knowing the shear buckling factor, the slenderness parameter parameter is determined from Equation (2) and the calculation of the post-critical shear strength then depends, as illustrated in Figure 2, upon whether the web is: a) stocky or thick ( w ≤ 0,8 , region AB in Figure 2) in which case the web will not buckle and the shear stress at failure will reach the shear yield stress of the web material:
τba = f yw yw /
where f yw yw is the tensile yield strength b) intermediate (0,8 < w < 1,2, region BC in Figure 2) which represents a transition stage from yielding to buckling action with the shear strength being evaluated empirically from from the following: following:
τba = [1 - 0,625 (
w-
0,8)] (f yw yw /
)
c) slender or thin ( w ≥ 1,2, region CD in Figure 2) in which case the web will buckle before it yields and a certain amount of post-buckling action is taken into account empirically:
τba = The calculation of the shear buckling resistance by the simple post-critical method is then completed by substitution of the appropriate value of τba into Equation (1).
2.2 Calculation of the Shear Buckling Resistance by the Tension Field Method For transversely stiffened girders where the transverse stiffener spacing lies within the range 1,0 ≤ a/d ≤ 3,0, full account may be taken of the considerable reserve of post-buckling resistance. This reserve arises from the development of "tension field action" within the girder. Figure 3a shows the development of tension field action in the individual web panels of a t ypical girder. Once a web panel has buckled in shear, it loses its resistance to carry additional compressive stresses. In this post-buckling range, a new load-carrying mechanism is developed, whereby any additional shear load is carried by an inclined tensile membrane stress field. This tension field anchors against the top and bottom flanges and against the transverse stiffeners on either side of the web panel, as shown. The load-carrying action of the plate girder than
becomes similar to that of the N-truss in Figure 3b. In the post-buckling range, the resistance offered by the web plates is analogous to that of the diagonal tie bars in the truss.
The total shear buckling resistance for design (Vbb.Rd) is calculated in Eurocode 3, by superimposing the postbuckling resistance upon the initial elastic buckling resistance as follows: follows: Total shear resistance = elastic buckling resistance + post-buckling post-buckling resistance r esistance:: Vbb.Rd = (d tw τbb)/ γM1 + 0,9 (gtw σbb sin φ)/ γM1 (3) The basis for this a ssumed behaviour is shown diagrammatically in Figure 4.
Figure 4a shows the situation prior to buckling, as represented by the first term in Equation (3). At this stage, equal tensile and compressive principal stresses are developed in the web. The shear buckling strength, τbb, is calculated from elastic buckling theory and leads to equations similar, but not identical, to those given earlier in Section 2.1 for the simple post-critical shear strength τba. Thus, the calculation of the shear buckling resistance again depends, as shown in Figure 5, upon whether the web is: a) stocky or thick ( stress is again taken:
w
= 0,8, region AB in Figure 5) in which case the web will not buckle and the shear yield
τbb = f yw yw / where f yw yw is the tensile yield strength b) intermediate (0,8 <
τbb = [1 - 0,8 (
w
< 1,25, region BC in Figure 5) where, in the transition from yield to buckling:
- 0,8)] (f yw / yw
c) slender or thin (
τbb = [1/
w
w
)
≥ 1,25, region CD in Figure 5) where the web will buckle and, from elastic buckling theory:
√3]
2 w ][f yw yw /
Thus, knowing τbb, the first term of th e expression in Equation (3) can be evaluated. The evaluation of the second term, corresponding to the post-buckling action, is more complex although it may still be reduced to a convenient design procedure, as described below.
In the post-buckling range, as shown in Figure 4b, an inclined tensile membrane stress field is developed, at an inclination φ to the horizontal. h orizontal. Since the flanges of the girder are flexible, they will begin to bend inwards under the pull exerted by the tension field. Further increase in the load will result in yield occurring in the web under the combined effect of the membrane stress field and the shear stress at buckling. The value of the tension field stress ( σbb) at which yield will occur, termed the "strength of the tension field" in Eurocode 3, may be determined by applying the Von Mises-Hencky yield criterion [2]. This results in the following expression for the strength of the tension field: 2 2 2 σbb = [f yw yw - 3τ bb + ψ ] 0.5 - ψ
where the term ψ = 1,5 τbb sin 2φ is introduced for convenience only. Once the web has yielded, final failure of the girder will occur when the mechanism comprising 4 plastic hinges has formed in the flanges, as shown in Figure 4c. A detailed analysis of this collapse mechanism, by considering the internal forces developed in the web and imposed upon the flanges (see [2]) allows the width (g) of th e tension field, which appears in the second term of Equation (3), to be evaluated: g = d cos
φ - (a - sc - st) sin φ
where, as in Figure 4c, sc and st denote the positions at which the plastic hinges form in the compression and tension flanges respectively. respectively. The hinge positions are calculated [2] from the knowledge that the hinges will form at the point of maximum moment, and therefore zero shear, in the flanges; the appropriate expression is as follows: S = [2/sin θ][MNf.Rk /tw σbb]1/2 ≤ a
(4)
where MNf.Rk is the plastic moment of resistance of the flange, i.e. 0,25 bt f 2 f yf yf . When high bending moments are applied to the girder, in addition to shear, then axial forces (Nf.Sd) will be developed in the flanges. Such axial forces will, of course, reduce the plastic moment of resistance of the flanges. Their effects can be calculated from standard plasticity theory as: 2 MNf.Rk = 0,25 btf 2 f yf yf {1 - [Nf.Sd / (btf f yf yf / γmo )] } (5)
All the terms required for the calculation of the total shear resistance from Equation (3), oth er than the inclination φ of the tension field, are now known. Unfortunately, the value of φ cannot be determined directly and an iterative procedure has to be adopted in which successive values of φ are assumed and the corresponding shear resistance evaluated in each case. The process is repeated until the value of φ providing the maximum, and therefore the required, value of the shear resistance has been established. The variation of the shear resistance with φ is not very rapid. The correct value of φ lies between a minimum of θ /2 and a maximum of θ, where θ is the slope of the panel diagonal tan-1(d/a), as shown in Figure 6. A parametric study [2] has established that, for girders of normal proportions, the value of φ which produces the maximum value of shear r esistance is approximately given given by:
φ = θ /1,5
The assumption of this value of φ will lead either to the correct value or to an underestimation of the shear resistance. It will therefore give a safe approximation and will also give a g ood starting value of φ if a more accurate process of iteration is to be carried out. The correct value of φ is that which gives the maximum value of Vbb.Rd.
3. INTERACTION BETWEEN SHEAR AND BENDING In general, any cross-section of a plate girder will be subjected to bending moment in addition to sh ear. As discussed in [2], this combination combination makes the stress conditions in the girder web considerably more complex. In the first place, the stresses from the bending moment will combine with the shear stresses to give a lower buckling load. Secondly, in the post-buckling range, the bending stresses will influence the magnitude of the tension field membrane stresses required to produce yield in the web. Finally, as already discussed with reference to Equation (5), the axial flange forces arising from the bending moment will reduce the plastic moment of resistance of the flanges. The proper evaluation of all these effects is complex but, as discussed in [2], certain assumptions may be made about the interaction of moment and shear to produce a simple and effective design procedure. In Eurocode 3, the procedure for allowing for moment/shear interaction naturally depends upon whether the simple post-critical method of Section 2.1 or the tension field method of Section 2.2 is being used to calculate the shear buckling resistance. Each case will now be considered separately. separately.
3.1 Interaction between Shear and Bending in the Simple Post-Critical Method The interaction between shear and bending can be conveniently represented by the diagram shown in Figure 7a (Figure 5.6.4a of Eurocode 3) where the shear resistance of the girder is plotted on the vertical axis and the m oment resistance is plotted horizontally. The interaction represents a failure envelope, with any point lying on the curve defining the co-existent values of shear and bending that the girder can just sustain.
The interaction diagram can be considered in 3 regions. In region AB, the applied bending moment M ad is low and the girder can then sustain a shear load Vad that is equal to the full value of the shear buckling resistance calculated from the simple post-critical method, as in Equation (1). Thus, in this region: Msd ≤ Mf.Rd Vsd ≤ Vba.Rd (6) The moment that defines the end of the range at point B (Mf.Rd) is the plastic moment of resistance of the crosssection consisting of the flanges only, i.e. neglecting any contribution from the web. In this calculation it is necessary to appreciate that the plates of the compression flange may buckle and, if necessary, to take this into account by adopting an effective width beff for the flange. The calculation of this effective width is as described in Lecture 7.3 for an outstand element in compression. At the other extreme of the interaction diagram in region CD, the applied shear V ad is low. Provided it does not exceed the limiting value of 0,5 Vba at point C then the plastic moment of resistance of the complete cross-section Mpl.Rd need not be reduced to allow for shear.
In the intermediate region BC the co-existent applied moment M Sd and shear VSd values must satisfy the following relationship: MSd ≤ Mf.Rd + (Mpl.Rd - Mf.Rd) [1 - (2VSd /Vba.Rd - 1)2] (7) The complete range of moment/shear interaction has thus been defined for the simple post-critical method.
3.2 Interaction between Shear and Bending in the Tension Field Method The procedure for the tension field method follows that described described above for the simple post-critical method. It leads to the construction of a similar, though not identical, interaction diagram, see Figure 7b (Figure 5.6.4b of Eurocode 3). In the low moment region AB, again defined by values of the applied moment less than Mf.Rd, the girder can sustain a shear load Vad that is equal to the "web only" shear resistance Vbw.Rd calculated from tension field theory. Thus: MSd ≤ Mf.Rd VSd ≤ Vbw Rd The "web only" shear resistance is the specific value of the total shear resistance V bw.Rd calculated from Equation (1), for the case when M Nf.Rk in EC3=0 in Equation (5). This is, in effect, a conservative approach which neglects the contribution of the flanges to the tension field action. At the other extreme in region CD, the procedure remains as for the simple post-critical method except that the limiting value of shear at point C is now taken as 0,5V bw. Similarly, the procedure for the intermediate region BC remains as before except that the substitution substitution of the ten sion field value Vbw for Vba in Equation (7) gives: MSd ≤ Mf.Rd + (Mpl.Rd - Mf.Rd) [1 - (2VSd / Vbw.Rd - 1)2] (8) The complete range of moment/shear interaction is thus defined for the tension field method.
4. CONCLUDING SUMMARY
• • • • •
Procedures for the design of plate girders subject to shear utilize varying degrees of post-buckling resistance and correspond to either the "simple post-critical" or "tension field" methods of Eurocode 3. Moment resistance of plate girders may normally be based on the plastic moment resistance of a cross section consisting of the flanges only. Design for coincident shear and moment should be undertaken using an interaction diagram. The simplest approach consists of designing the web to carry the whole of the shear, with the flanges resisting the moment. The "tension field" method is more restricted in application than the "simple post-critical" method, but gives higher strengths. Other aspects of design (stiffeners, etc.) are discussed in Lecture 8.4.3. 8.4.3.
5. REFERENCES [1] Eurocode 3 "Design of Steel Structures": European Prestandard ENV1993-1-1: Part 1.1, General rules and r ules for buildings, CEN, 1992. [2] Narayanan, R. (ed), "Plated Structures; Stability and Strength", Applied Science Publishers, London 1983.
Chapter 1 covers basic aspects of pla te girder behaviour and design. [3] Bulson, P. S. "The Stability of Flat Plates", Chatto & Windus, London, 1970. General coverage of plate buckling and explanation of k τ values for numerous cases.
6. ADDITIONAL READING 1.
Dubas, P. and G ehri, E. (eds)., "Behaviour and Design of Plated Steel Structures", Structures", Publication No. 44, ECCS, 1986.
Chapters 4 and 5 provide a detailed coverage of plate girder design, taking the reader well beyond the content of this lecture. They also refer to numerous original sources. 2.
Galambos, T. V. (ed)., "Guide to Stability Design Criteria for Metal Structures", 4th Edition, John Wiley, 1988.
Lecture 8.4.3: Plate Girder Design - Special Topics OBJECTIVE/SCOPE To extend the coverage of plate girder design previously given in Lectures 8.4.1 and Lecture 8.4.2. 8.4.2. To include the design of transverse web stiffeners and end posts and consideration of patch loading. To outline design procedures for longitudinally stiffened girders and for girders with large web openings.
PREREQUISITES Lecture 8.4.1: 8.4.1: Plate Girder Behaviour and Design I Lecture 8.4.2: 8.4.2: Plate Girder Behaviour and Design II
RELATED LECTURES Lectures 3.1: 3.1: Fabrication Fabrication Lectures 3.2: 3.2: Erection Lecture 3.5: 3.5: Fabrication/Erection of Buildings
SUMMARY The detailed design of particular elements of plate girders is considered in this lecture. The structural action of web panels, designed as described in earlier lectures, imposes stringent requirements on adjacent boundary elements. This lecture considers the design of transverse web stiffeners and end posts according to Eurocode 3 [1] and also considers the particular problems caused by patch loading. Two other aspects of design, not currently covered by Part 1.1 of Eurocode 3, viz. the design of longitudinally stiffened girders and girders with large web openings, are also discussed.
1. INTRODUCTION The two previous lectures on plate girders, Lectures 8.4.1 and Lecture 8.4.2, 8.4.2, have concentrated upon the main aspects of the structural behaviour upon which the design principles are based. The two design approaches proposed in Eurocode 3 [1] h ave been outlined; these are the "simple post-critical" method, which is generally applicable, and the "tension field" method which gives significantly higher load resistances by taking the post-buckling resistance of the girders into account. This lecture seeks to complete the discussion of plate girder design by considering further aspects of detailed design. For example, the development of post-buckling action in a web plate, assumed in the previous lectures, can only occur when the elements at the boundary of that web plate are able to provide an adequate anchorage for the tension field forces developed within the plate. This lecture will consider the design of these boundary elements. These elements may be in the form of intermediate transverse stiffeners or end posts. Girders may be subjected to h igh loads in localised regions, away from stiffener positions, creating a possibility that crippling of the web plate may occur. An example of this occurs in crane gantry girders subjected to a vertical loading which travels along the flange. The effects of such "patch loading" must be carefully taken into account in design. This aspect is very th oroughly treated in Eurocode 3 [1]. This lecture outlines the relevant design principles.
Two other important aspects of plate girder design are the treatment of girders with longitudinal web stiffeners, and of girders with large openings in the web plates. Openings are frequently required, particularly in building construction, to allow access for service ducts, etc. Neither of these two situations is covered in Part 1.1 of Eurocode 3. This lecture discusses good practice in r elation to the two situations.
2. TRANSVERSE WEB STIFFENERS To achieve an effective design, i.e. a plate girder of high strength/weight ratio, it is usually necessary to provide intermediate transverse web stiffeners. Eurocode Eurocode 3 [1] only allows the application of the tension field method, which has been shown in earlier lectures to give a significantly enhanced load resistance, when the web is stiffened. The Eurocode also specifies that such stiffeners must be spaced such that the stiffener spacing/web depth ratio (a/d) is within the following range: 1,0 ≤ a/d ≤ 3,0 Transverse stiffeners play an important role in allowing the full ultimate load resistance of a plate girder to be achieved. In the first place they increase the buckling resistance of the web; secondly they must continue to remain effective after the web buckles, to provide anchorage for the tension field; finally they must prevent any tendency for the flanges to move towards one another. The satisfactory performance of a transverse stiffener can best be illustrated by comparing the girders shown, after testing, in Slides 1 and 2. In Slide 1 the stiffeners have remained straight and have clearly fulfilled the function of vertical struts in the simplified N-truss model of the post-buckling action discussed in Lecture 8.4.2, 8.4.2, see Figure 1. In Slide 2 the stiffener has failed and has been unable to limit the buckling to the adjacent sub-panels of the girder; instead, the buckle has run through the stiffener position extending over both panels. Consequently, significant reduction in the failure load of th e girder occurred.
Slide 1
Slide 2 The requirements to ensure adequate stiffener performance are given in Section 5.6.5 of Eurocode 3. First, the stiffener must be of adequate rigidity in the direction perpendicular to the plane of the web to prevent web buckling. This condition is satisfied provided the stiffener has a second moment of area I s that satisfies the following following empirical formulae: Is ≥ 1,5 d3tw3 /a2 when a/d < Is ≥ 0,75 dtw3 when a/d ≥ Secondly, the buckling resistance of the vertical "stiffener "stiffener strut" must be sufficient to support the tension field forces shown in Figure 2a (which has been reproduced from Fig. 5.6.3a of Eurocode 3). It must also resist the resultant axial compressive force Ns that is imposed upon it. This force is calculated as follows: Ns = Vsd - dtw τbb / γM1 where τbb is the initial shear buckling resistance of the web panels, calculated as given in Lecture 8.4.2. 8.4.2. When the two web panels adjacent to the particular stiffener being designed are not identical, the lower value of τbb for the two panels should be taken. As previously, Vsd is the design value of the shear force.
The buckling resistance of the stiffener strut to this axial compressive force is then calculated using Section 5.7.6 of Eurocode 3. Since the stiffener is attached to the web plate, a portion of the web acts effectively with the stiffener in resisting the axial compression. It is difficult to calculate the extent of this portion of the web but experimental observations have allowed an empirical effective web width of 30εtw to be established, as shown in Figure 3 (which has been reproduced from Figure 5.7.4 of Eurocode 3). Having established the effective cross-section cross-section of the stiffener strut in this way, its buckling resistance is determined as for any other compression member according to Section 5.5.1 of the Eurocode.
For a "load bearing" stiffener, stiffener, i.e. a transverse stiffener stiffener located at a position where an external load is applied to the girder, an additional consideration is necessary. The resistance of the effective cross-section of the load bearing stiffener should also be checked at a position close to the loaded flange.
3. END PANELS AND POSTS The requirement for adequate boundary members to support the loading imposed by the post-buckling tension field is particularly onerous in the case of the end panel of the girder. The situation of the transverse stiffener at the end of the girder, i.e. the "end post", is very different from that of an intermediate stiffener, compare Figure 2b to Figure
2a. At the end of the girder, the forces imposed by the tension field in the end panel have to be resisted entirely by the end post without support from any further adjacent panels. Design procedures for end panels and posts are given in Clauses 5.6.4.3 and 5.6.4.4 of Eurocode 3 [1]. Basically, they allow the designer two options. Firstly, the designer may choose not to design an end post that will provide adequate anchorage for the tension field. As a consequence, the end panel of the web must be designed according to the simple post-critical method so that a tension field does not develop within it. This option offers a simple design procedure but has the disadvantage that the calculated shear resistance of the end panel will be significantly lower lower than that of the internal web panels in the girder. Since it is probable that the applied shear in the end region will be higher than at any point on the span, this procedure will not provide an effective design solution if the stiffener spacing remains constant over the complete length of the girder. As shown in Figure 4a, the designer should then reduce the spacing of the stiffeners bounding the end panel so that the shear resistance of that panel as calculated by the simple post-critical method becomes equal to that calculated by th e tension field method for the internal panels.
The more effective, but more complex, option is to design the end post to provide an adequate anchorage for the web tension field. The end panel of the web can then be designed according to the tension field method, so that the design shear buckling resistance (Vbb.Rd ) can be calculated as described in Lecture 8.4.2 for internal web panels, i.e. Vbb.Rd = [(d tw τbb ) + 0,9 (g t w σbb sin φ)]/ γM1 The slight difference for the end panel arises in the calculation of the width g of the tension field. For an internal panel the width is gi ven by: g = d cos
φ - (a - sc - st) sin φ
where, sc and st denote the lengths over which the tension field anchors onto the compression and tension flanges, see Figure 2a. For an end panel, the failure mechanism may be different since, as shown in Slide 3, a plastic hinge may also form in the end post.
Slide 3 This hinge affects the anchorage length for the compression flange which must now be calculated as:
sc = where Mpl.1 is the reduced plastic moment of the flange at the internal hinge position, allowing for the presence of the axial force (Nf1) at that position. The other plastic hinge will form either at the end of the flange, as in the case of an internal panel, or in the end post. The location of the hinge, as defined by ss in Figure 2b, will depend upon which of these two elements has the lower lower plastic moment of resistance. Mpl.2 takes the lesser of these two values. In this way, Clause 5.6.4.3 of Eurocode 3 allows the geometry of the tension field developed in the end panel to be fully defined, see Figure 2b. The design shear buckling resistance Vbb.Rd of the panel can then be calculated t ogether with the horizontal component Fbb of the anchorage force of the tension field imposed on the end post: Fbb = tw ss σ bb cos2
φ
The end post resists this force by acting as a vertical beam spanning between the two flanges. For this purpose it must satisfy the following following criterion: Mpl.2 + Mpl.3 ≥ 0,5 Fbb ss where the reduced plastic moment of the end post: 2 Mpl.3 = 0,25 bs ts2 f ys ys {1 - [Ns3 /(bs ts f ys ys )] }
allows for the effect of the axial force in the end post: Ns3 = Vsd - τ bb tw (d - ss )
If it proves difficult to provide an end post in the form of a single plate to resist these forces, then the designer may consider providing an end arrangement such as that shown in Figure 4b. In this case, two transverse stiffeners are used. These two stiffeners and the portion of the web projecting beyond the end support form a rigid end post to provide the necessary anchorage for the tension field in the end panel. The disadvantage of such an arrangement is that adequate space must be available to allow the girder to project beyond its end support.
4. WEB CRIPPLING There are many situations where it is not possible to provide transverse web stiffeners at all points where vertical loads are applied to the girder. For example, a crane gantry girder is subjected to a vertical loading that travels along the flange; also, girders may be launched during construction so that the flange actually moves over the fixed point of support. In such cases, special consideration must be given to the design of the unstiffened unstiffened web in the local region underneath, or above, the applied point or "patch" loading to prevent "web crippling". The webs of all beams must be checked for this possible local failure. Plate girders are particularly susceptible to this form of failure because of the slenderness of the web plates that are normally used in their construction. "Web crippling" is discussed in Section 5.7 of Eurocode 3. It distinguishes between the two different loading cases that are shown in Figure 5 (taken from Figure 5.7.1 of the Eurocode). In Figure 5a, the force is applied to one flange only and is therefore resisted by shear forces developed within the web plate. In this case the web plate has to be checked for its "crushing" and for its "crippling" resistance. In the other case, shown in Figure 5b, the force is applied to one flange, transmitted directly by compressive forces developed in the web, and resisted by a reactive force on the other flange. The web must again be checked for its "crushing" resistance. The "buckling" "buckling" resistance of the web must also be considered in this case. There are, therefore, three types of web resistance that must be calculated. In each case the resistance is dependent upon the length over which the applied force is effectively distributed on the flange. This is termed the "stiff bearing length" (ss ). It is calculated on the assumption of a dispersion of load through solid steel material at a slope of 1: 1. The terms "crushing", "crippling" and "buckling" resistance are introduced to differentiate between the phenomena being considered. In each case the appropriate resistance is calculated from empirical formulae: "Crushing" resistance, where crushing is local yielding of the web without any buckling, is given by: Ry.Rd = (ss + sy) tw f yw yw / γM1 "Crippling" resistance, where crippling is localised buckling of the web in the presence of plasticity, is given by: 1/2 Ra.Rd = 0,5 tw2 (E f yw [tf /tw )1/2 + 3 (tw /tf ) (ss /d)]/ γM1 yw )
"Buckling" of the web occurs with out-of-plane deformation over most of the depth of the web. The "buckling" resistance (Rb.Rd ) for the compressive loading situation illustrated in Figure 5b is determined simply by considering the web plate as a vertical compression member. First it is necessary to determine the breadth of the web "strut" (beff ) that is effective effective in r esisting the compression. compression. This breadth may be calculated as: beff = [h2 + ss2 ]1/2 where: h is the overall depth of the girder. ss is the stiff bearing length discussed above.
The buckling resistance of this idealised strut is then determined as for any other compression member according to Section 5.5.1 of Eurocode 3.
5. LONGITUDINAL WEB STIFFENERS To increase the strength/weight ratio of plate girders, slender webs may be reinforced by longitudinal, as well as transverse, stiffeners. stiffeners. A typical longitudinally stiffened stiffened girder is shown after failure in Slide 4. The main function of the longitudinal stiffeners is to increase the buckling resistance of the web with respect of both shear and bending loads. An effective stiffener will remain straight, thereby sub-dividing the web panel and limiting limiting the buckling to the smaller sub-panels. The resulting increase in the ultimate resistance of the girder can be significant.
Slide 4 The design of webs with longitudinal stiffeners stiffeners is not covered in Part 1 of Eurocode 3. It will be addressed in Part 2, for bridges. Because of the greater need for high strength/weight ratios in bridges, girders with longitudinal stiffeners are more commonly encountered in bridge than in building construction. construction. Design of longitudinal stiffeners is usually based on a number of empirical design curves derived from the results of a parametric study employing numerical modelling techniques. The design procedure is relatively straightforward, although somewhat conservative. Additional information on the behaviour of longitudinally stiffened girders has been presented [2] which will assist the designer to gain a better understanding understanding of the structural action. action.
6. GIRDERS WITH OPENINGS IN SLENDER WEBS Holes often have to be cut in the webs of plate girders used in building construction to provide access for service ducts, etc. No mention of such openings is made in Part 1 of Eurocode 3. Such holes have a particular influence on the behaviour of slender webs because the hole interrupts the tension field. (Design methods are available for stocky webs with web openings - see Reference3). Detailed fundamental work by Narayanan [2] has shown that girders with slender webs and web open ings possess a post-buckling reserve of resistance. The collapse mechanism for such girders, illustrated in Slide 5, is similar to the shear sway mechanism that is characteristic of all plate girders, as discussed in Lecture 8.4.2. 8.4.2. However, some codes adopt a conservative approach. They do not tak e account of such post-buckling action in any girder which has a web opening with any dimension exceeding some percentage of the minimum dimension of the web panel in which it is located. To provide a simple design procedure, the shear resistance of the perforated panel is calculated as the buckling resistance.
Slide 5 The disadvantage of such a procedure is that the shear resistance of the perforated panel will then be substantially lower than that calculated allowing for the full post-buckling reserve of resistance in adjacent unperforated panels. The designer should therefore reduce the spacing of the transverse stiffeners either side of the web opening so that the initial buckling resistance of the resulting narrow perforated panel is approximately equal to the full postbuckling resistance of adjacent panels.
7. CONCLUDING SUMMARY • To develop post-buckling action, the elements bounding the web plate of a girder must be designed to provide adequate anchorage for the tension field forces developed in the web panel. • The tension field method can only be applied when transverse web stiffeners are provided such that the stiffener spacing/web depth ratio lies within the range: 10,0 ≤ a/d ≤ 3,0. The stiffeners must have adequate rigidity in the direction perpendicular to the plane of the web to prevent web buckling. The buckling resistance of the vertical stiffener strut must also be sufficient to support the tension field forces. • Careful consideration must be given to the design of the end post. The designer has the option of either not allowing the development of tension field action in the end panel or, for a more efficient design, providing an end post of sufficient rigidity and strength. • The possibility of web crippling, web crushing and web buckling must be considered in those localised areas where patch loads are applied to the girder flange. • Longitudinal web stiffeners allow girders of higher strength/weight ratios to be designed. They are particularly relevant in bridge construction. They are not considered in Eurocode 3: Part 1.1. • Large web openings are frequently necessary in building construction to allow the passage of services. Plate girders with openings are not considered in Eurocode 3 Part 1.1. 8. REFERENCES [1] Eurocode 3 "Design of Steel Structures". European Prestandard ENV1993-1-1: ENV1993-1-1: Part 1.1, General rules and rules for buildings, CEN, 1992. [2] Narayanan, R (Editor), "Plated Structures; Stability and Strength", Applied Science Publishers, London, 1983. This reference gives detailed information, including the experimental background to the structural action covered in the clauses of Eurocode 3 r eferred to in this lecture. [3] Lawson, R. M., Design for Openings in the Webs of Composite Beams, SCI Publication 068. The Steel Construction Institute, 1987.
9. ADDITIONAL READING 1.
"European Recommendations for the Design of Longitudinally Stiffened Webs and of Stiffened Compression Flanges", Publication Publication 60, ECCS, 1990.
Lecture 8.5.1: Introduction to Design of Box Girders OBJECTIVE/SCOPE To describe the main features and advantages of box girders; to introduce the methods of global analysis; to describe aspects of behaviour particular to box girders.
PREREQUISITES Lecture 6.1: 6.1: Concepts of Stable and Unstable Elastic Equilibrium Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lectures 8.4: 8.4: Plate Girder Behaviour and Design
RELATED LECTURES Lecture 8.5.2: 8.5.2: Advanced Design of Box Girders
SUMMARY A general overview is given of the form and behaviour of box girders. Typical configurations are illustrated and the advantages of box girders over plate girders are highlighted. The structural behaviour of box sections is described; global analysis is discussed; the particular features of the design of webs and flanges are introduced; the function and form of cross sectional r estraints. including including diaphragms, are described.
1. INTRODUCTION A box girder is formed when two web plates are joined by a common flange at both the top and the bottom. The closed cell which is formed has a much greater torsional stiffness and strength than an open section and it is this feature which is the usual reason for choosing a box girder configuration. configuration. Box girders are rarely used in buildings (box columns are sometimes sometimes used but these ar e axially loaded rather than in loaded in bending). They may be used in special circumstances, such as when loads are carried eccentrically to the beam axis.
Steel and composite box girders are used for highway bridges, though they are most expensive than plate girders (because fabrication requires more time and effort). The use of box girders for bridges offers the following advantages over plate girders:
• • • •
Very good torsional rigidity Wide flanges may be used Clean external surfaces A non-rectangular cross-section can be used.
Torsional rigidity is particularly advantageous when the girder needs to be curved in plan. A box section can carry the torsion resulting from vertical loading without the need for lateral bracing. The use of wide flanges facilitates the choice of shallow construction depth (that is, a large span-to-depth ratio) which may be desirable when space or headroom is restricted. Shallower girders will be heavier but this may be a lesser consideration in such circumstances. circumstances. Any stiffening needed to the webs and flanges can usually be arranged inside the box section. The clear external surfaces offer a neater, more pleasing, appearance and avoid the corners and crevices which are difficult to protect adequately against corrosion. The use of inclined webs (closer together at the bottom than at the top) reduces the width of the bottom flange plate (often advantageous to structural performance, particularly when the flange is in compression). Inclined faces are frequently considered to give a better appearance than vertical faces. Inclined webs also offer better aerodynamic performance. The cross-sectional shape of the box girders of many long-span cable-stayed and suspension bridges are chosen after wind-tunnel testing to find the shape which offers minimum drag and optimum dynamic response. Box girder bridges are constructed with single, twin or multiple box girders. The deck of the bridge may be of reinforced concrete or it may be a stiffened steel deck. When reinforced concrete is used the steel girders may be closed box sections or may be open sections (U-shaped) which are closed when the slab is cast across the top. A selection of configurations is shown in Figures 1 - 3. The method of erection is often an important factor in ch oosing a cross-section cross-section for a box girder bridge. The torsional stability of the box section avoids the need for temporary bracing (such as is needed with plate girders). The bridge can be built by lifting consecutive pre-assembled units of the complete cross section and joining them end-to-end. This is particularly suitable for cantilever erection of long spans.
2. MAIN FEATURES OF BOX GIRDE GIRDERS RS Many of the features of box girders illustrated in Figures 1, 2 and 3 are similar to those of plate girders, though the proportions may be different. There are a few features which are particular to box girders.
Web plates carry shear forces and bending moments. Thin webs need transverse stiffeners to achieve adequate resistance, in the same way as they do in plate girders. Inclined webs are deeper (in their planes) and may therefore require more stiffening. Flange plates connecting two webs are wider than those of corresponding I-beam girders. Consideration must be given to shear lag. When the flange plate is in compression its stability (against out-of-plane buckling) must be considered; longitudinal and transverse stiffeners are frequently necessary. necessary. When open steel sections are used, made into composite box girders by a concrete deck slab, separate (and relatively relatively small) flange plates are provided at the top of each web. These flanges need to be stabilised laterally by bracing during construction. Shear connection on the top of the flanges is similar to that on the flanges of I-beam girders. When closed steel boxes are used, overlaid by a reinforced concrete slab to form a composite section, shear connection is required over the full width of the top flanges. When the bridge deck is steel, the top flanges are stiffened orthotropically to carry wheel loads from traffic as well as acting as a box girder flange. This stiffening usually takes the form of longitudinal trough stiffeners supported at regular intervals by transverse beams. At supports plated diaphragms are provided. At each support there are one or two bearings, located between the webs and directly below the diaphragms. The diaphragms serve to transfer the load from the webs to the bearings (generally acting as a deep beam) and to prevent distortion distortion of the section. In all except smaller boxes restraint against distortion is also required at intermediate positions; positions; this can be achieved with braced cross-frames, stiff ring frames or plated diaphragms. Access inside box sections is necessary during construction and during the life of the structure. Manholes must be provided in support and intermediate diaphragms, to permit passage along the length of the box; the size and location of the holes is taken into account in the design of the diaphragm.
3. GLOBAL ANALYSIS As for I-beam girders, global analysis determines the bending moments and shears in the main beams due to the applied loading. Since the principal loads are vertical, greatest attention is given to moments and shears in the vertical plane, though horizontal loading and effects must also be considered. However, when box girders are used, two additional effects must be considered, torsion and distortion. As we shall see later, consideration of distortional effects may be limited to local regions between intermediate diaphragms. Torsional effects must be determined by the global analysis.
For a single straight uniform section girder any simple line beam analysis would suffice for bending, shear and torsional effects, but in general the most appropriate model for evaluating the effects due to vertical loading is the grillage analogy. Beam elements in a grillage model have three degrees of freedom - vertical deflection, rotation about a transverse axis and rotation about a longitudinal axis - and are thus able to determine directly the three principal effects which are to be considered in the stress analysis. Computer programs are available specifically for grillage analysis, though many designers make use of general purpose space-frame pr ograms (in which the elements have the full 6 degrees of freedom). In very wide flanges shear lag effects must be taken into account. When the axial load is fed into a wide flange by shear from the webs the flange distorts in its plane; plane sections do not remain plane. The resulting stress distribution in the flange is not uniform (see Figure 4). This effect is n ot taken into account in a grillage analysis analysis and must be separately determined.
The calculation of distortional effects, which include transverse bending stresses and longitudinal warping stresses can be carried out by methods based on beam-on-elastic-foundation analogy. Where stiff intermediate cross-frames or diaphragms are provided the stresses are usually quite small. In very complex configurations finite element analysis might be used. Shell elements are connected to model the complete cross-section along the whole length of the girder. Provided that the mesh is sufficiently fine and the element behaviour is appropriate, effects such as warping, distortion and shear lag are determined at the same time as the principal bending, shear and torsion effects. Folded plate analysis is sometimes used, but it is only appropriate when the box section is uniform along its length, when there are no intermediate cross-frames and when the loading can be represented by harmonic series. It is difficult to apply to ordinary design situations. situations.
4. TORSION AND DISTORTION The general case of an eccentric load applied to a box girder is in effect a combination of three components bending, torsion and distortion. As a first step, the force can be separated into two components, a pair of symmetric vertical loads and a force couple, as shown in Figure 5(a). However, torsion is in fact resisted in a box section by a shear flow around the whole perimeter and the couple should in turn be separated into two parts, representing pure torsion and distortion, as shown in Figure 5(b).
The first two components, vertical bending loads and a torsional shear flow, are externally applied forces, and they must be resisted in turn at the supports or bearings. The third component, distortional forces, comprises an internal set of forces, statically in equilibrium, which do not give rise to any external r eaction. Distortional Distortional effects depend on the behaviour of the structure between the point of application and the nearest positions where the box section is restrained against distortion.
4.1 Torsion and Torsional Warping The theoretical behaviour of a thin-walled box section subject to pure torsion is well known and is treated in many standard texts. For a single cell box, the torque is resisted by a shear flow which acts around the walls of the box. This shear flow (force/unit length) is constant around the box and is given by q = T /2 A, where T is the torque and A is the area enclosed by the box. (In Figure 2 the torque is QB /2 and the shear flow is Q /4 D.) The shear sh ear flow produces shear stresses and strains in the walls and gives rise to a twist per unit length, θ , which is given by the general expression:
or, where J is the torsion constant. However, it is less well appreciated that this pure torsion of a thin walled section will also produce a warping of the cross-section, unless there is sufficient symmetry in the section. This is illustrated in Figure 6 for a rectangular section that is free to warp at its ends. However, in practice boxes are n ot subject to pure torsion; wherever there is a
change of torque (at a point of application of load or at a torsional restraint) there is restraint to warping, because the 'free' warping displacements due to the different torques would be different (restraint is high, for example. over intermediate supports where torsion is restrained). Such restraint gives rise to longitudinal warping stresses and associated shear stresses in each wall of th e box.
4.2 Distortion When torsion is applied directly around the perimeter of a box section, by forces exactly equal to the shear flow in each of the sides of th e box, there is n o tendency for the cross section to change its shape. Torsion can be applied in this manner if, at the p position osition where the force couple is applied, a diaphragm or stiff frame frame is provided to ensure that the section remains square and that torque is in fact fed into the box walls as a shear flow around the perimeter. Provision of such diaphragms or frames is practical, and indeed necessary, at supports and a t positions where heavy point loads are introduced. But such restraint can only be provided at discrete positions. When the load is distributed along the beam, or when point loads can occur anywhere along the beam such as concentrated axle loads from vehicles, the distortional effects must be carried by other means. The distortional forces shown in Figure 5(b) are tending to in crease the length of one diagonal and shorten the other. This tendency is resisted in two ways, by in-plane bending of each of the wall of the box and by out-of-plane bending. this is illustrated in Figure 7.
In general the distortional behaviour depends on interaction between the two sorts of bending. The behaviour has been demonstrated to be analogous to that of a beam on an elastic foundation (BEF), and this analogy is frequently used to evaluate the distortional effects. If the only resistance to transverse distortional bending is provided by out-of-plane bending of the flange plates there were no intermediate restraints to distortion, the distortional deflections in most situations would be significant and would affect the global behaviour. For this reason it is usual to provide intermediate cross-frames or diaphragms; consideration of distortional displacements displacements and stresses can then be limited to the len gths between cross-frames. cross-frames.
5. FLANGE DESIGN 5.1 Tension Flanges Tension flanges are designed mainly on the basis of longitudinal bending stresses, in the same way as for plate girders. Torsional and distortional effects and the effects of shear lag do need to be taken into account in some circumstances. The strength is taken as the yield stress. At the Serviceability Limit State elastic behaviour is normally specified. Then the stresses due to the restraint of torsional warping, distortional distortional warping stresses and the variation of str ess across the flange due to shear lag must be calculated. Stresses are highest adjacent to the web. At the Ultimate Limit State, plastic behaviour is n ormally accepted. accepted. Then the stresses due to th e restraint of torsional warping and shear lag can be neglected, since they are secondary effects.
5.2 Compression Flanges In addition to considering the load effects in relation to yield strength, the stability of the compression flange must also be considered. Relatively narrow flanges may be unstiffened. The strength of the flange plate then depends on ordinary panelbuckling resistance. It is convenient to express this in design by the determination of an effective width of compression flange; this is the width which has the same resistance, at yield stress, as the buckling resistance of the full panel. Typically a panel is considered fully effective up to a width of 24t (where t is the flange thickness), but thereafter the effective width is less than th e actual width. Wider flanges are provided with longitudinal stiffeners to provide stability and these in turn are supported at intervals by transverse stiffeners, cross-frames, or diaphragms. diaphragms. Usually the longitudinal stiffeners can be designed using rules which effectively treat them as struts. For this purpose the transverse members restraining them must be sufficiently stiff. If, when the flanges are particularly wide, the transverse stiffeners are not sufficiently stiff, the flange would have to be treated as a panel stiffened in two directions and the overall buckling strength determined; determined; this is too c omplex for most design purposes.
5.3 Orthotropic Steel Decks In some bridges, notably long-span bridges and movable bridges, where minimum weight is desirable, the roadway is carried on an all-steel deck. This form of deck is stiffened longitudinally; transverse members or diaphragms support the longitudinal stiffeners. The design of such a deck, to carry the direct loads from the traffic wheels, is an extremely complex matter. Connection details are subject to onerous fatigue loading. Configurations currently in use are the result of many years development, analysis, testing and feedback from previous designs.
6. WEB DESIGN The determination of the strength of webs in bending and shear follows the same gen eral rules as for plate girders. Shear buckling resistance of thin webs is improved by the presence of intermediate stiffeners. Tension field action can develop in the web in the same way as in plate girders. However, the further increase in tension field action on account of the bending stiffness stiffness of th e flange plate is not normally achievable.
7. CROSS SECTIONAL RESTRAINTS 7.1 General Function and Description The main functions of cross sectional restraints are:
• • • • •
to preserve the shape of the box against distortion distortion to transfer an externally applied torque to the box walls through shear flow to provide transverse support to longitudinal stiffeners stiffeners under compression to support traffic loads directly (from an orthotropic deck) to transmit forces from box walls to the supports.
7.2 Support Diaphragms At support positions a plate diaphragm is normally provided, see Figure 8, although occasionally a heavily braced frame can be used. Plate diaphragms may be thick and unstiffened in fairly small boxes; in larger boxes vertical stiffeners are provided over the bearings and sometimes horizontal stiffening stiffening is also provided.
The main purpose of support diaphragms is to transfer the large shear forces from the webs to the bearings. They must also transmit horizontal forces when the bearing is guided or fixed in position into shear flow along the web of the box section.
For large box sections, with large support forces, the use of a finite element program is recommended for the design of support diaphragms. 7.3 Intermediate restraints
For large deep box girders, intermediate restraint against distortion is usually provided by cross frames, see Figure 9. The frame comprises a ring of four members, each being an effective section comprising a width of flange or web and the transverse stiffener attached to it. The corners of the ring need to be designed to provide moment continuity. Diagonal bracing is connected to the frame as necessary, usually either across the section diagonal or as a V to the midpoint of a flange stiffener.
7.4 Load-carrying transverse stiffeners When a box girder has a stiffened steel deck, or a composite box girder is arranged with cross-girders, such that the deck slab designed to span l ongitudinally, ongitudinally, the transverse stiffeners stiffeners must also carry the traffic loads and transfer them to the webs. Such configurations usually also have cantilevers on the outer faces. Moment continuity must be provided between the transverse stiffeners, the cantilevers and the cross-girders. cross-girders. Transverse flange stiffeners which are required to provide support must usually be arranged to coincide with transverse web stiffeners, so that the load can be transferred into the web.
8. ARTICULATION The arrangement of the bearings which support the girder, known as the articulation arrangement, should take account of the high degree of torsional rigidity provided by the box girders. For a relatively short bridge of a few spans it is not necessary to provide two bearings at each intermediate support. Twin bearings at the ends will restrain the box against twist; single bearings on the box centreline are sufficient at intermediate supports. If the bridge is h ighly curved, single bearings may be sufficient at all supports; restraint against twist is provided by the combined effects of torsional rigidity and geometrical arrangement of the group of bearings.
9. CONCLUDING SUMMARY
• • • • • • •
Box girders are used because of their good resistance to torsion box girders can be designed to have a good aerodynamic shape the clear external surfaces and the use of inclined webs gives a good appearance the grillage method is sufficiently accurate for global analysis shear lag and the stability of the compression flange both require consideration in wide flanges design of webs is generally similar to that for plate girders distortional effects must be considered
10. ADDITIONAL READING 1. 2. 3. 4. 5. 6. 7. 8. 9.
Eurocode 3: Design of steel structures: European Prestandard, ENV1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992. Dubas, P. and Gehri, E., Behaviour and Design of Steel Plated Structures, Technical Committee 8 Group 8.3, ECCS-CECM-EKS No44, 1986. Johnson, R. P. and Buckby, R. J., Composite Structures of Steel and Concrete, Volume 2: Bridges, Collins, London, 1986. British Standard 5400: Part 3: Steel, Concrete and Composite Bridges, Part 3: Code of Practice for Design of Steel Bridges, British Standards Institution, 1982. Horne, M.R., CIRIA Guide 3, Structural action in steel box girders, Construction Industry Research and Information Association, London, 1977 Kollbrunner, C. F. and Basler, K.: Torsion, Spes/Bordas Lausanne/Paris, 1955. Stahlbau Handbuch: Stahlbau Handbuck für Studium und Praxis, BandI, Stahbau Verlag, Köln, 1982. Dalton, D. C. and Richmond, B., Twisting of Thin Walled Box Girders, Proceedings of the Institution of Civil Engineers, January 1968. Iles, D.C., Design Guide for Composite Box Girder Bridges, Bridges, The Steel Construction Institute, Ascot, 1994
Lecture 8.5.2: Advanced Design of Box Girders OBJECTIVE/SCOPE To introduce methods of global analysis, methods of determining cross-section distortion, and shear lag in box girder bridges.
PREREQUISITES None.
RELATED LECTURES Lecture 8.5.1: 8.5.1: Introduction to Design of Box Girders
SUMMARY Global analysis may be made by the grillage, orthotropic plate, folded plate and finite element methods. Distortion of the box may have to be controlled by diaphragms or cross frames. Simple or refined methods are available for the calculation of the forces in the diaphragms or cross frames. In very wide flanges, shear lag effects have to be taken into account.
1. INTRODUCTION Although steel or steel-concrete composite box girders are usually more expensive per tonne than plate girders, because they require more fabrication time, they can lead to a more economic solution overall. For bridges, box girders have several advantages over plate girders which make their use attractive:
• •
• • • •
a very high torsional stiffness. (In closed box sections torque is resisted mainly by Saint Venant shear stresses and the torsional stiffness is normally much greater than that of open s ections.) closed steel boxes provide torsional stiffness during their erection. (They thus avoid the need for the expensive temporary bracing which is required with plate girders and which also interferes with the construction of the concrete slab. For highly curved spans torsional stiffness is almost always essential during their construction.) wide flanges can be used. (This allows large span to depth ratios without resource to very thick material.) box girders have a neat appearance. (The stiffening can remain unseen inside the box.) the facility to choose a good aerodynamic shape. (This is particularly important for large suspension or cable-stayed bridges.) bridges.) box girders use fewer bearings. (Usually torsional restraint need be provided at only one position along a continuous box; single bearings can be used at all other supports. Further, with a highly curved box girder, single bearings at all supports is often sufficient.)
Box girders are sometimes used in building structures, but this is not common. This lecture deals mainly with box girders as used in bridges, both all-steel construction and composite construction with a reinforced concrete deck slab; most of the general remarks are applicable to box girders used in buildings
2. GLOBAL ANALYSIS AN ALYSIS METHODS Global analysis determines the load effects, bending moments, shear forces, torsional moments, etc., which occur in all parts of the structure as a result of the applied load. From this analysis stresses are determined, for comparison with the calculated strengths. Methods of analysis for composite bridge decks fall into one of three groups.
• • •
those that treat the bridge as a series of interconnected beams those that treat s eparately eparately the various parts of the box section s ection (flanges, (flanges, webs, diaphragms) diaphragms) those that treat the bridge deck as a continuum
Those in the first group are the simplest to analyse, since beam theory can be used for the behaviour of the individual elements. For a single straight girder a line-beam analysis can be used, provided this takes account of torsional effects as well as bending effects, but in general a grillage model is needed. Such an analysis gives good results for the distribution of moments and forces in multiple girder structures and when a curved single beam is modelled as a series of stra ight elements. However, However, simple beam theory does not t ake account of the distortion of the cross section or of shear lag effects and these must be determined separately. Analysis in the second group is by use of finite element techniques and inevitably involves the use of a powerful computer program. Provided suitable elements are available within the computer program, the analysis is able to give results which include most of the structural effects, including distortion and shear lag, but choice of element type and size r equires much experience, and interpretation of the results also requires careful consideration. The third group applies more exact theoretical modelling techniques. Examples are treatment of the whole deck as an orthotropic plate and analysis of folded plate models. However such techniques can only be properly applied when there is uniformity throughout the structure and for distributed loading. They are also only able to represent separately some aspects of the behaviour: the loading therefore needs to be divided into components such as uniform bending, uniform torsional, warping torsion, and distortion.
3. GRILLAGE 3.1 General In a grillage analysis, the structure is idealised as a number of longitudinal and transverse beam elements in a single horizontal plane, rigidly interconnected at nodes. Transverse beams may be orthogonal or skewed with r espect to the longitudinal beams, so that skew, curved, tapering or irregular decks can be analysed. In a simple grillage analysis, each beam is allotted a flexural stiffness in the vertical plane and a torsional stiffness. Vertical loads are applied only at the nodes. Computer software is used to carry out a matrix stiffness analysis to determine the displacements (rotations about the two horizontal axes an d the vertical displacement) at each n ode and the forces (bending moments, torsional moments and vertical shear forces) in the beams connected to each node. Grillage analysis does not determine warping and distortional effects, nor the effects of shear lag. Local effects under point loads (wheel loads) can only be studied with a grillage by the use of a fine mesh of beams locally to th e load; local effects are usually determined separately and added to global results as required.
3.2 Grillage Modelling for Box Girder Bridges The global structural action of a box girder bridge can be seen as the essentially separate actions of a reinforced concrete slab (or an ortotropically stiffened steel deck) which bends transversely and a series of longitudinal beams which deflect vertically and twist. The slab (or steel deck) bends as a result of being supported along several lines
which deflect by different amounts and in a manner which which varies along the span. The global analysis therefore needs to model accurately the way in which these support lines deflect, so that the interaction between longitudinal and transverse bending is properly established. The slab is effectively supported along each web line. The vertical deflection of each web line depends on a combination of the vertical and torsional deflections of the box girder of which it is a part. The best way to model these effects is to create a torsionally stiff beam element along the centreline of each box (i.e. the shear centre) and to connect it to the slab at the web positions. To do this, short 'dummy' transverse beams are needed; they do not physically represent any particular part of the structure and the forces in them do not need to be analyzed, but they must be given sufficient stiffness that their bending is insignificantly small. This form of model for a twin-box bridge with cantilevers is illustrated in Figure 1 (note that, for clarity, the dummy beams and longitudinal beams are shown slightly below the slab, whilst they would actually be treated in the analysis as co-planar).
3.3 Longitudinal Grillage Elements The main longitudinal beams are assigned the flexural properties of the full section of each girder (including the slab or deck). In multi-girder structures it is usual to consider the slab to be divided midway between boxes and for the full width of the cantilever to be included with the outer box. Strictly this is not exact, since it would introduce a discontinuity in the level of the n eutral axis, but the inaccuracy is negligible. The longitudinal elements representing the slab (shown dotted in Figure 1) are not strictly necessary, as they are much more flexible than the main girders, though they may be helpful in the application of distributed loads. They are shown here to illustrate the division of the slab. The longitudinal edge elements may be added to represent the edge beam. They do not have a major effect on overall performance but are often helpful in the application of load on the cantilevers. cantilevers.
3.4 Transverse Grillage Elements Where there are no transverse beams, the transverse elements simply represent a width of slab equal to the node spacing. Where there are transverse beams, including cross-beams and diaphragms inside the box, the elements should represent the stiffness stiffness of the effective effective transverse member. The slab elements are supported only on the dummy elements, they are not connected directly to the longitudinal beams. There is no moment continuity between between slab elements and the dummy beams.
3.5 Torsional Rigidities For an open box section, the torsional stiffness K is given by the general expression:
Where A is the area of the box and t is the thickness of element ds. When the section is composite, the concrete slab should be transformed into an equivalent thickness of steel by dividing by the modular ratio. For a strip of solid slab the torsional stiffness is given by:
Where t is the thickness and b is the width of the strip. However, in the grillage model only half this stiffness should be assigned to the transverse elements, since no Saint Venant shear stress flux goes around the perimeter of the strip's cross-section. Similarly, for an orthotropic steel deck, only the value H should be used for the torsional rigidity, not 2H.
3.6 Skew Bridges Skewed arrangements of multiple boxes can be devised, provided that support diaphragms can remain essentially square to the box centrelines and that there are either no cross-girders between boxes or the cross-girders are square to the boxes. Similarly, Similarly, grillage analysis with skew cross-members is difficult to interpret and gives un certain results for all except small skews.
3.7 Interpretation of the Output Output of a Grillage Analysis Computer software usually gives values of the vertical shear, bending moments and torsional moment for each grillage member at each joint in the grillage. Because the continuous structure has been idealised into discrete elements this discontinuity is unreal. A slightly 'better' value of moments in the main longitudinal members can be obtained by smoothing, as shown in Figure 2, though the di fference fference is usually very small.
4. ORTHOTROPIC PLATE ANALYSIS In orthotropic plate analysis, the deck structure is 'smoothed' across its length and breadth and treated as a continuum. The elastic properties of an orthotropic plate are defined by the two flexural rigidities Dx and Dy and a plate torsional rigidity H. The governing equation relating deflection w to load P acting normal to the plane of the plate is:
= p(x, y) Design charts for decks that can be idealised as orthotropic plates have been derived from series solutions. They give deflections and longitudinal and transverse moments due to a point load, and so provide a rapid method for distribution analysis. Their applicability is limited to simply supported decks of skew not exceeding 20° whose elastic properties can be represented solely by length, breadth, and the three quantities Dx, Dy and H. In composite structures, they can be used for beam-and-slab decks with not less than five equally spaced longitudinal members of uniform diaphragms over the supports.
5. FINITE ELEMENT ANALYSIS The finite element method is used increasingly in civil engineering. It is the most versatile of the matrix stiffness methods of elastic analysis, and can, in principle, approach the solution of almost any problem of global analysis of a bridge deck. In box girders, the finite element method allows the study of shear lag and the computation of effective flange breadths. It can also analyse local effects in slabs. To do this the webs, flanges and diaphragms are each divided into a suitable mesh of elements; the detail of the effects which can be revealed (for example the variation in stress across a flange due to shear lag) depends on the fineness of the mesh and the capabilities of the element types provided by the program. The disadvantage of finite element analysis is its cost, especially because of the high level of expert time required for the idealisation of the structure. The expert's know-how is n eeded in selecting an appropriate pattern of elements, in selecting the right type of element and in determining the right limit conditions for boundary nodes along the
supports. The interpretation of results also requires experience. The choice of inappropriate elements can be misleading in regions of steep stress gradient, because the conditions of static equilibrium are not then necessarily satisfied. The selection of the discretisation density level, or of the material behaviour, may have serious repercussions on the accuracy of the r esults. Nevertheless, for complex situations, or for complex portions of a major structure, there is no better substitute for a finite element analysis.
6. FOLDED PLATE ANALYSIS The folded plate method is normally limited to assemblages of rectangular plates. It is not applicable to skew decks due to coupling between the harmonics. The orthotropic plates may extend over several spans but must be simplysupported at the extreme ends, with rigid diaphragms over the end supports. When folded plate diaphragms are used to represent the transverse frames, the advantages are that it can give a complete and accurate solution in less computer time than is needed for th e finite element method, and it can accept a wide variety of types of loading and both displacement and force boundary conditions. To apply the method to a double cellular box-girder bridge with one single internal web, the distortion must be divided into symmetric and asymmetric deformations. For boxes with more internal webs, it is possible to divide th e deformations of the cross-section cross-section into eigenvalue functions of deformation.
7. TORSIONAL WARPING Pure torsion of a thin walled section will also produce a warping of the cross-section, unless there is sufficient symmetry in the section. To illustrate how warping can occur, consider what would happen to the four panels of a rectangular box section subject to torsion. Assume that the box width and depth are B and D respectively, respectively, and that the flange and web thicknesses are t f and t w. Under a torque T , the shear flow is given by q=T /2 BD. Consider first the flanges. The shear stress in the flanges is given by τ f = q/t f =T /2 BDt f . Viewing the box from above, each flange is sheared into a parallelogram, with a shear angle φ =τ f /G; if the end sections were to remain plane, the relative horizontal displacement between between top and bottom corners would be φ L at each end (see Figure 3a), and thus there would be a twist between the two ends of 2 φ L/D = 2τ f L/DG = TL/BD2Gt f .
By a similar argument, viewing the box from the side and considering the shear displacements of the webs, if the end sections were to remain plane the twist of the section would be TL/B2 DGt w. As the twist must be the same irrespective of whether we consider the flanges or the webs, it is clear that the end sections can only remain plane if 2 2 TL/BD Gt f = TL/B DGt w, i.e. Dt f = Bt w. If this condition is not met, the end sections cannot remain plane; instead there will be a slight counter-rotation in their planes of the two flanges and of the two webs, and a consequent warping of the section. Typical warping for this example is shown in Figure 3b.
Of course, for a simple uniform box section subject to pure torsion this warping is unrestrained and does not give rise to any secondary stresses. But if, for example, a box is supported and torsionally restrained at both ends and then subjected to applied torque in the middle, warping is fully restrained in the middle by virtue of symmetry and torsional warping stresses are generated. Similar restraint occurs in continuous box sections which are torsionally restrained at intermediate supports. This restraint of warping gives r ise to longitudinal warping stresses and a ssociated shear stresses in the same manner as bending effects in each wall of the box. The shear stresses effectively modify slightly the uniformity of the shear stress calculated by pure torsion theory, usually reducing the stress near corners and increasing it in mid-panel. Because maximum combined effects usually occur at the corners, it is conservative to ignore the warping shear stresses and use the simple uniform distribution. The longitudinal effects are, on the other hand greatest at the corners. They need to be taken into account when considering the occurrence of yield stresses in service and the stress range under fatigue loading. But since the longitudinal stresses do not actually participate in the carrying of the torsion, the occurrence of yield at the corners and the consequent relief of some or all of these warping stresses would not reduce the torsional resistance. In simple terms, a little plastic redistribution can be accepted at the ultimate limit state (ULS) and therefore there is no need to include torsional warping stresses in the ULS checks.
8. CROSS-SECTION DISTORTION When torsion is applied directly around the perimeter of a box section, by forces exactly equal to the shear flow in each of the sides of the box, there is no tendency for the cross section to change its shape. If torsion is not applied in th is manner, there is effectively a set of forces which is trying to extend the length of one diagonal across the section and reduce the other (see Figure 4). Diaphragms or frames can be provided to restrain distortion where large distortional forces occur, such as at support positions, and at intervals along a box, but in general the distortional effects must be carried by other means.
To illustrate how distortion occurs and is carried between effective restraints, consider a simply supported box with diaphragms only at the supports and which is subject subject to a point load over one web at mi dspan. Under the distortional forces, each side of the box bends in its own plane an d, provided there is moment continuity around the corners, out of its plane as well. The deflected shape is shown in Figure 5.
The in-plane bending of each side gives rise to longitudinal stresses and strains which, because they are in the opposite sense in the opposing faces of the box, produce a warping of the cross section (in the example shown the end diaphragms warp out of their planes, whilst the central plane can be seen to be restrained against warping by symmetry). The longitudinal stresses are therefore known as distortional warping stresses. The associated shear stresses are known simply as distortional shear stresses. The bending of the walls of a box, as a result of the distortional forces, produces transverse distortional bending stresses in the box section. The introduction of stiff intermediate cross-frames will restrict distortional effects to the lengths between frames (rather than between supports). but they must be stiff enough for this purpose. In general the distortional behaviour depends on interaction interaction between the two sorts of behaviour, the warping and the transverse distortional distortional bending. The behaviour has been demonstrated to be analogous to that of a beam on an elastic foundation (BEF), with the beam stiffness representing the warping resistance and the elastic foundation representing the transverse distortional bending resistance. A comprehensive description of the analogy is given in a paper by Wright [1]. A diagrammatic illustration of the distortional behaviour of a box with a single intermediate diaphragm is given in Figure 6.
9. SHEAR LAG When the axial load is fed into a wide flange by shear from the webs the flange distorts in its plane; plane sections do not remain plane. The resulting stress distribution in the flange is not uniform In very wide flanges, shear lag effects have to be taken into account for the verification of stresses, especially for short spans, since it causes the longitudinal stress at a flange/web intersection to exceed the mean stress in the flange. Shear lag can be allowed for in the elementary theory of bending, by using an effective effective flange breadth (less than the real breadth) such that the stress in the effective breadth equals the peak stress in the actual flange (see Figure 7). This effective flange breadth depends on the ra tio of width to span.
For a simply supported beam, for example, the effective breadth of the portion between the webs is Φe.b , where Φe , the effective breadth ratio, is given in Ta ble 1. b/L
Mid-span
Quarter-span
Support
α=0 α=1 α=0 α=1 α=0 α=1 0,00 1,00
1,00
1,00
1,00
1,00
1,00
0,05 0,98
0,97
0,98
0,96
0,84
0,77
0,10 0,95
0,89
0,93
0,86
0,70
0,60
0,20 0,81
0,67
0,77
0,62
0,52
0,38
0,30 0,66
0,47
0,61
0,44
0,40
0,28
0,40 0,50
0,35
0,46
0,32
0,32
0,22
0,50 0,38
0,28
0,36
0,25
0,27
0,18
0,75 0,22
0,17
0,20
0,16
0,17
0,12
1,00 0,16
0,12
0,15
0,11
0,12
0,09
Table 1: Effective breadth ratio
where b is the distance between webs. L is the span of the beam
Φe
for simply supported beams
α
=
Φe is the elastic effective breadth ratio. Fortunately, Fortunately, in most m ost situations the span/breadth ratio is not sufficiently large to cause more than 10-20% increase in peak stress, on account of shear lag.
10. DIAPHRAGMS At supports, forces are transferred from the box girder, through bearings, to the substructure below. Principally, these forces are vertical, though lateral restraint also has to be provided at certain selected positions. Where there is only a single bearing under the box and it offers little resistance to transverse rotation (e.g. elastomeric pot bearings), there will be no torsional restraint; the loads transferred from the two webs will be equal (pr esuming that the bearing is on the centreline). When there are two bearings, under or close to each of th e webs, torsional restraint is provided to the box; the load from each web will be different, and there will be a transfer of torsional shear from the flanges. Whenever there is lateral restraint there will be an associated torque, because the restraint will not be at the level of the shear centre of the box. The principal function of a support diaphragm is to provide an adequate load path to transfer shear forces from the webs to the bearings below the box. In doing so it also resists distortional forces. Plated diaphragms are normally provided at supports, since they provide these functions most easily, although, strictly, an adequately braced cross-frame could also do so. Clearly, full diaphragms close the box section, yet a ccess into the box is necessary for completion of fabrication and for future inspection and maintenance. Openings are usually provided to permit access along the box, but the effect of these openings on the performance of the diaphragm has to be carefully considered; the size and position of any opening needs to be limited. This can be a particular problem with small boxes, because the m inimum hole size may be a large proportion of the diaphragm size. Diaphragms are usually provided with vertical stiffeners above the bearings because of the large forces involved, though with small boxes a th ick unstiffened diaphragm may on occasion be appropriate. A diaphragm behaves essentially as a deep beam, with the diaphragm plate acting as its web and an effective width of each of the box flanges acting as its top and bottom flange. An example of an intermediate diaphragm in a large box girder of a cable stayed bridge is shown in Figure 8.
11. CONCLUDING SUMMARY
• • • • •
Grillage analysis analysis is most m ost often used for grillage grillage analysis. It allows a simple idealisation of the structure, and a sure interpretation of the output. Finite element analysis can be used for complex situations. It is the most versatile of the matrix stiffness methods of elastic analysis. Orthotropic plate analysis and folded plate analysis have a limited application. Eccentric loading of the girder section causes distortion which may have to be controlled by the provision of intermediate diaphragms or cross frames. In very wide flanges shear lag effects have to be taken into account.
12. REFERENCE 1.
Wright, R N, Abdel-Samad, S R and Robinson, A R, BEF Analogy for analysis of box girder bridges, Proc. ASCE, vol 94, ST7, 1968.
13. ADDITIONAL READING 1. 2. 3. 4. 5. 6. 7. 8. 9.
Eurocode 3: "Design of Steel Structures", ENV1993-1-1: Part 1.1, General rules and rules for buildings, CEN, 1992. Dubas, P. and Gehri, E., Behaviour and Design of Steel Plated Structures, Technical Committee 8 Group 8.3, ECCS-CECM-EKS, No44, 1986. Johnson, R. P. and Buckby, R. J., Composite Structures of Steel and Concrete, Volume 2: Bridges, Collins London, 1986. British Standard 5400: Part 3: Steel, Concrete and Composite Bridges, Part 3: Code of Practice for Design of Steel Bridges, British Standards Institution, 1982. Horne, M.R., CIRIA Guide 3, Structural action in steel box girders, Construction Industry Research and Information Association, London, 1977 Kollbrunner, C. F. and Basler, K., Torsion in Structures - An Engineering Approach (translated from the German), Springer Verlag, Berlin 1969. Stahlbau Handbuch: Stahlbau Handbuck für Studium und Praxis, BandI, Stahbau Verlag, Köln, 1982. Dalton, D. C. and Richmond, B., Twisting of Thin Walled Box Girders, Proceedings of the Institution of Civil Engineers, January, 1968. Iles, D.C., Design Guide for Composite Box Girder Bridges, Bridges, The Steel Construction Institute, Ascot, 1994
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 a ssociated with them.
PREREQUISITES None.
RELATED LECTURES Lecture 6.1: 6.1: Concepts of Stable and Unstable Elastic Equilibrium Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lecture 8.4.1: 8.4.1: Plate Girder Behaviour and Design I Lecture 8.5.1: 8.5.1: Design of Box Girders
SUMMARY Shell structures are very attractive light weight structures which are especially suited to building a s 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 th e 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 r esisting esisting distributed loads but ar e 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 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 ver y 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 in creases 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 th e 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; 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 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 hi gh 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 th e shell diameter to thi ckness 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, 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 l oad acting perpendicular to th e 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 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 th eoretically perfect 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 th at, 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 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 n ecessary to choose: 1. 2. 3. 4.
The structural type, e.g. a circular cylinder, and a fixed set of boundary conditions. The type of loading, e.g. longitudinal compression. A predefined pattern of reinforcement using stiffeners. A strict limitation on im perfection perfection 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 experimental values can be used with confidence.
To allow for this, Codes of Pra ctice [3] use the following procedure: 1.
A critical stress, σcr or τcr, or a critical pressure, p cr, 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. σ ,
cr
τ
cr
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
σ ,τ cr
cr
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.
PREREQUISITES Lecture 8.6: 8.6: Introduction to Shell Structures
RELATED LECTURES: Lecture 6.1: 6.1: Concepts of Stable and Unstable Elastic Equilibrium Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lecture 8.4.1: 8.4.1: Plate Girder Behaviour & Design I Lectures 8.5.1: 8.5.1: Introduction to Design of Box Girders
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 p lates. The effect of imperfections 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 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, 6.6.1, 6.6.2 and 8.1 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 pr incipal directions of curvature and in the direction normal to th e 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 th e boundaries, or in the zone affected by other disturbances, 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 danger ous 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 N ON-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, criterion, or, alternatively, by use of the minimum potential energy criterion. In the first case, small increments (u 1, v1, w1) are imposed on the prebuckling displacements (u0, v0, w0) u = u0 + u1 v = v0 + v1 (1) w = w0 + w1 The two adjacent configurations, represented by the displacements before (u0, v 0, w0) and after the increment (u, v, w) are analysed. No increment is given to the load parameter. The function represented by (u 1, v 1, 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 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 sign ificant 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, direction, i.e. v = 0 (Figure 3). If this last condition is removed and r eplaced 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 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 in cluded. 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, 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 between theoretical and experimental results in the buckling of shells has caused much controversy and discussion, but now the explanation that initial imperfections imperfections are the th e 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 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 expr ession 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 m odes.
(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 imperfec i mperfections. tions.
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 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.
3.
4.
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 ("snapthrough") occurs. 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. 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 "snapthrough".
The four load-deflection diagrams given in Figure 8 ma y relate, for example, to a spherical cap subjected to uniform radial pressure acting towards the centre of the sphere; in th is 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, 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 n ew equilibrium configuration configuration appears a ppears at point B. This configuration is quite different from the primary one and features lateral deflections deflections and bending of th e 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 pra ctical 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 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 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 c ylinder. 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 snapthrough from state B to state E, the shell can jump r epeatedly 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 = 2πrt σcr is imposed, the overall shortening of the cylinder almost instantly increases from u cr 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 characteristics which are n ot considered in this description. Esslinger and Geier [5] explain the fundamentally different 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 forces due to membrane action are zero, as long as th e 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 th e circumferential curvature curvature of the sh ell 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 t o 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, imperfections, the truer th e 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 n ot 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 th e 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. 2.
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. 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 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 therefore probably of con siderably 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 th e 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, F cr, 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, McGraw-Hill, New York and Kogakusha, Tokyo, 1961. [5] Esslinger, M. T., and Geier, B. M., "Buckling and Post Buckling Behaviour of Th in-Walled Circular Cylinders", 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 Cyli nders 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.
PREREQUISITES None.
RELATED LECTURES Lecture 8.6: 8.6: Introduction to Shell Structures Lecture 8.7: 8.7: Basic Analysis of Shell Structures
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. differences.
1. INTRODUCTION In Lectures 8.6 and Lecture 8.7 several aspects that influence the structural behaviour of sh ells 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 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 imperfections whilst the same shell under external pressure exhibits much lower imperfection imperfection sensitivity. 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 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, directions, Figure 2(b).
It can be sh own theoretically that both modes of shell buckling correspond to the same critical buckling load. Assuming simply-supported 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 n oting 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 r elatively long 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 "knockdown" 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 "knockdown" 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. imperfections. These tolerances are related to the imp erfection erfection 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 "knockdown" 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 imperfections and the importance of adhering to the stated tolerances cannot be over-emphasised. 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. substantially. References [4,5] present comparisons between various codes and attempt to explain the r easons 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 th e 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, l r, 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 a xial compressive stress (characteristic load effect).
σu is the design value of th e buckling stress (characteristic resistance). resistance). Thus, the objective is to determine the value of σu, or, equivalently, the value of the ratio specified characteristic yield stress.
σu /f y, where f y is the
The proposed method is schematically shown in Figure 6, where This parameter is defined as:
λ=
σu /f y is plotted against a slenderness parameter, 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 stresses and edge disturbances.
As can be seen from Figure 6, two regions are defined; the first, for which buckling, whereas λ ≤
λ≥
, defines the region of elastic
defines the region of plastic buckling.
For λ ≥ or, equivalently, for ασcr ≤ 0,5f y (i.e. when the buckling stress of th e 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 /f y = (1/ γ) γ) (1/ λ ) (6a) 2
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 by:
λ≤
(ασcr
≥ 0,5f y), material non-linearities also play a role (plastic buckling) and the design curve is given
σu /f y = 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 anywhere on the sh ell 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=
α=
for r/t < 212 (7a)
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 /f y 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 l r interpolation between α and α /2 gives the required "knock-down" factor.
≤
≤ 0,02 lr linear
Although a slight, practically unavoidable unevenness of the supports of the cylinder is covered by the "knockdown" factor α, care must be exercised to in troduce the compressive forces uniformly uniformly into the c ylinder 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 th e 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 associated with shell-type behaviour. In this case σu is given by: For σE ≤ 0,5 f y (elastic buckling)
σu = σE (8a) For σE ≥ 0,5 f y (plastic buckling)
σu = f y
(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 Equation (6a) and Equation (8a), this is true when
<
(10)
3. UNSTIFFENED CYLINDERS UNDER EXTERNAL PRESSURE General Considerations Considerations
External pressure may be applied either purely radially, as "external pressure loading" (Figure 8(a)), or it may be applied all round the c ylindrical shell, that is both radially and a xially (Figure 8(b)) as "external hydrostatic pressure loading".
For external lateral pressure, the hoop stress applied pressure, p, by a simple expression:
σ , in the pre-buckling state away from the supports is related to the θ
σ = (r/t) p (11) θ
In this case, the axial stress σx = 0. For external hydrostatic pressure, the hoop stress is a gain given by Equation (11) but th e 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 interaction behaviour is described. 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, p cr
pcr =
(12)
As can be seen, p cr 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 p cr. Simpler expressions for p cr 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 n ot 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, 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). resistance). The first step consists of calculating the pressure py, at which the h oop stress at cylinder mid-length reaches the yield stress. From Equation (11), it follows that py = (t/r)f y (14) Clearly, when pu = py the cylinder is so stocky th at 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 discussed in [8]. The equation has the following 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, estimated from
βmin =
βmin may be approximately
(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 imperfection sensitivity. 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: 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 encountered in offshore design pra ctice. 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)
≤ 1 if σd ≤
II:
(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. 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 perfect shell. 2. Calculate the "knock-down" factor and, hence, the buckling stress of th e 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. imperfections. In general, design recommendations are limited to shells of revolution only, having uniform thickness and subjected to idealised loading distributions. distributions. In practical applications, various problems 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. developments.
6. REFERENCES [1] ECCS - European Convention for Constructional Steelwork - Buckling of Steel Shells European Recommendations, 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.
Lecture 8.9: Design of Stringer-Stiffened
Cylindrical Shells
OBJECTIVE/SCOPE To describe the buckling behaviour of stiffened shells and to analyse th e different types of failure. A practical design procedure, based on the European recommendations, for stringer stiffened cylinders subject to axial load is presented.
RELATED LECTURES Lecture 8.1: 8.1: Introduction to Plate Behaviour and Design Lecture 8.6: 8.6: Introduction to Shell Structures Lecture 8.7: 8.7: Basic Analysis of Shell Structures Lecture 8.8: 8.8: Design of Unstiffened Cylinders
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 sh ells 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 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 t wo types of buckling can theoretically 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 between the th e 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 < 10bt3E/[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
≤ 0,0015 lg and
≥ 0,06 (3)
≤ 0,01 lr when 0 ≤
≤ 0,06 (4)
See (3) for details.
The limits given for also apply to the initial circumferential circumferential out-of-straightness 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 th e 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 (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 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 BUCKLI BUCKLING NG The elastic critical stress, critical stresses for
σcr, l, for a perfect shell panel between stringers may be taken equal to the higher of the two
σcr, l, = 0,605E(t/r) σcr, l = 3,6 E(t/b)2
a perfect complete cylinder a perfectly flat plate
[b/ √( √(rt) > 2,44]
(8)
√(rt) ≤ 2,44] [b/ √(
(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 0,5f y; 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], ha lved 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 following factors are a ssumed:
αl = 0,83 and γ = 1,0 Thus the following design values of th e elastic shell buckling stress are obtained for an imperfect 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 f y, plastic deformation comes into play and σul is the higher of the stresses σul1 and σul2 obtained from:
if 0,605 αo E
if 0,83 x 3,6 E
> 0,5 f y (12)
> 0,5 f y (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 th e longitudinal direction, m, and the full wave number in the circumferential direction, 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 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 C s = 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 (Lecture 8.1). 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, b e, may be obtained, but not explicitly, from 1,9t√(Ε/ f 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 th e following following characteristics: r/t = 1000 l/r = 1,6 ns = ns(min) outside flat bar stringer E = 205000 Nmm-2 As /(bt) = 0,5
f y = 240 Nmm-2 h w /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. imperfections. The stiffened stiffened panel buckling stress for an imperfect stiffened cylinder may be obtained from:
σup =
αsp σcr, p (18)
where:
αsp = 0,65 when
αsp = αo when
> 0,2
< 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 f y. αo is the knockdown factor for an un stiffened stiffened cylinder of radius r and thickness t (see Lecture 8.8). 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. imperfections. In the case of elastic-plastic buckling Equation (18) must be replaced by:
σup = f y {1 - 0,4123[f y /(αsp σcr, p)]0,6} if αsp σcr, p > 0,5 f y
(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
hw /tw ≤ 1,1
for flat bar stiffeners stiffener s with tw ≅ t
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 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 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 513522, 1987. [3] Ellinas, C. P. and Croll, J. G. A., "Experimental and theoretical correlations for elastic buckling of axially compressed stringer stiffened 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.
