Ct 5144 Structural stability February 2003 Prof. ir. A.C.W.M. Vrouwenvelder F G u ϕ Faculty of Civil Engineering and Geosciences Mec...

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Structural stability

February 2003 Prof. ir. A.C.W.M. Vrouwenvelder

F G u

ϕ

Faculty of Civil Engineering and Geosciences Mechanics, Materials and Constructions Section of Structural Mechanics

Preface Course CT5144 is a Civil Engineering Masters Course in the field of Structural Stability for building types of structures. It is assumed that the student is already familiar with the linear theory of elasticity and has an elementary knowledge of the Eulerian column buckling problem and elementary plasticity. The aim of the course is primarily to show a number of practical as well as theoretical aspects of the buckling phenomenon. The emphasis is on the use of the principle of minimum potential energy. This method may be considered as the most rigorous way of analyzing the (pre and post) buckling behavior. However, the method is sometimes quite cumbersome. For this reason also the direct formulation of the nonlinear equilibrium conditions is shown. As an intermediate step the method of Virtual Work is demonstrated for the sake of completeness. The methods are introduced for simple rigid-bar-spring-systems (chapter 2) and the classical Euler beam (chapter 3). In the later chapters 6 and 7 the method is applied to more complex phenomena as torsional buckling and lateral buckling. The more practical parts in this course are the chapters 4 and 5 where framework structures are considered. A method suitable for hand calculation of moderate sized frames for elastic and elastic-plastic buckling is presented. Furthermore attention is being paid to the methods applied in codes of practice and in computer programs. In these lecture notes use has been made of a mix of material from earlier courses at Delft University. The more rigorous parts have been inspired by the courses of Prof. Dr. ir. W. Koiter in the seventies, for the more practical parts use has been made of course material by Prof. ir. J. Witteveen and Prof. dr. ir. J. Blaauwendraad. I would like to thank ir. Madelon Burgmeijer and ir. Cox Sitters for assisting in getting a better text, good English and nice figures.

A. Vrouwenvelder Delft, 2003

Table of contents Preface 1

General theory 1.1 Introduction 1.2 Calculation of the equilibrium path and stability 1.3 Buckling

9 9 14 18

2

Discrete spring systems 2.1 Pin-ended rigid column supported by a horizontal spring, second order analysis 2.2 Pin-ended rigid column with higher order terms 2.3 Pin-ended rigid column with exact kinematic relation 2.4 Telescopic-spring 2.5 Pin-ended rigid column supported by an inclined spring

21 21 24 26 26 28

3

Eulerian beam 3.1 Buckling behaviour 3.2 Post-buckling behaviour and stability analysis 3.3 Eulerian beam subjected to lateral loading

30 30 33 36

4

Frames 4.1 Elastic buckling load 4.1.1 Euler column, fixed-end member, one degree of freedom 4.1.2 Euler column, fixed-end member, two degrees of freedom 4.1.3 A simple frame 4.1.4 Lower bound approximation 4.1.5 Example two-story frame 4.1.6 Arch structure 4.1.7 Frame girder, spatial stability 4.2 Geometrical non-linear rigid-plastic behaviour 4.2.1 Introduction 4.2.2 Simple framework 4.2.3 Formal second-order virtual work procedure 4.2.4 Proof of the formal second-order virtual work equation 4.3 Verification with respect to the regulations 4.3.1 Introduction / single column 4.3.2 General approach for the framework analysis 4.3.3 Final check

38 38 38 39 41 43 46 47 49 51 51 53 55 57 60 60 61 66

5

Finite element calculations 5.1 First-order stiffness matrix 5.2 Second-order stiffness matrix 5.3 Examples

70 70 71 73

6

Torsional buckling 6.1 Torsional buckling of cross-shaped section 6.2 Torsional buckling of an I section 6.3 Torsional buckling in general

79 79 81 85

7

Lateral buckling of an I beam 7.1 The load case of pure bending 7.2 Cantilever beam with point load at the free end

91 91 93

Appendix I Theory of deformation Appendix II Partial integration Appendix III Upper bound approximation of the buckling load

101 105 107

Reading list

111

The second term in the numerator is caused by the fact that the column in the deformed state makes an angle ϑ = v L with the vertical. Thus, in the column a horizontal force component is present, which can be in equilibrium with the spring force only (in this example, it is assumed that the spring always remains horizontal). For reasonably small values of v , the horizontal component of the normal force in the column can be approximated by F v L .

v=

F

Fb = k L Fp =

αF k

v=

n αF n −1 k

Sp

α

F=

Sp

1 α 1+ v α L v

Fig. 1.2: Load-displacement diagram of the spring-supported column. As a result the displacement is increasing more than proportionally with the load (see Fig. 1.2). Relation (1.2) can be rewritten as: F ⎞ αF ⎛ v ⎜1 − ⎟= k L k ⎝ ⎠

→

v=

k L αF kL − F k

(1.3)

The factor α F k is the previously obtained linear relation, the other factor in the right-hand side is the amplification factor caused by the non-linearities. This amplification factor is often written as n (n − 1) , i.e.: v=

n αF n −1 k

; n=

kL F

(1.4)

From the calculation it can be concluded that the load cannot exceed the value: Fmax = k L

Even when the horizontal force is absent this limiting value really exists. In that case the column remains vertical under increasing load, but becomes unstable at the value F = k L and sidesway starts to occur. This phenomenon is called buckling and F = k L is called the buckling load:

Fb = k L The line representing this function is indicted in Fig.1.2 as well. The deviation from the linear relationship and the buckling phenomenon are geometrically non-linear phenomena. This type

If the geometrically non-linearity is neglected and the physical non-linearity is taken onto account, the behaviour is initially linear. The limiting value is now reached for:

αF = S p

(1.5)

From a theoretical point of view, this value may be higher or lower than the buckling load. For civil engineering structures it can be assumed in most cases that the buckling load is (much) higher than the plastic collapse load. In reality of course, both forms of non-linearity occur at the same time. When in the plastic analysis the geometrically non-linearity is taken into account, the following equation can be derived:

αF +

Fv = Sp L

→ v=

L ( S p − αF ) F

(1.6)

This is a hyperbola starting at S p and gradually decreasing to zero (see Fig. 1.2). The actual structure will follow the elastic non-linear curve until it intersects the non-linear plastic curve. At that position the maximum load bearing capacity is reached. For larger displacements, equilibrium can be obtained only for reducing loads. Whether this will happen in reality depends on the load case, which may be load or displacement directed. For a displacement directed situation the load may indeed reduce. In the case of a dead-weight load, the structure starts to move with increasing speed and will collapse. So, an interesting point is the intersection of the plastic and the elastic non-linear lines. The load for this point is called the critical load Fc . This load can be found by elimination of the displacement v from the relations (1.4) and (1.6):

α Fc

kL = Sp k L − Fc

Taking the reciprocal of both the left-hand and right-hand side leads to: 1 1 1 − = α Fc α k L Sp

or:

α 1 1 1 1 = + = + Fc S p k L Fp Fb

(1.7)

This is the well-known formula of Merchant, which gives in a simple way the critical physical and geometrically non-linear failure load as a function of the buckling load and the plastic collapse load. However, this formula is valid only for this type of simple structures. In most cases the critical load Fc is overestimated. This will be demonstrated in the next example.

λF 1 4

1 4

λF

λF

1.5 λF

1.5λF

λF

λF

λF

λF

1 4

λF

λF

1.5 λ F

1.5 λ F 1 4

linear deflection

buckling mode

λF

λF

λF

λF

λF

λF

λF

elementary failure mode (linear mechanism)

4

6 3

7

5

1

6

8 2

order of the plastic hinges

3

8

2

ultimate failure mechanism

Fig. 1.4: Deflection and failure modes of a two-story frame. The various results have been displayed in Fig. 1.5. However in this case, the maximum load bearing capacity cannot be obtained from the intersection point of both geometrically nonlinear curves. The fact is that the plastic calculation already shows a different pattern. Somewhere in the structure the first plastic hinge develops. After that the load can be increased further, but with a changed slope in the load-deflection diagram. For a given load a second hinge develops, etc. In the origin of the load-deflection diagram, the geometrically non-linear calculation will coincide with the linear one. Under increasing load just like in the first example a growing deviation develops, which is determined completely by the elastic buckling load. However, after the creation of the first plastic hinge the structure has become less rigid, which means that the buckling load of the changed structure will be lower than the elastic buckling load. This means that the non-linear curve does not approach asymptotically the elastic buckling load of the original system anymore, but approaches the elastic buckling load of the structure with a plastic hinge in the lower beam close to the right column. And because this load is smaller, the deviation increases faster: Fb reduces and n (n − 1) increases. After the creation of the third hinge, the structure has become so weak

λ

Eulerian buckling mode

3.37

geometrically linear, elastic

geometrica lly non-linear, elastic

geometrically linear, plastic

1

geometrically non-linear, plastic (geometrically linear mechanism)

0.777 1

2 3

4

5

6 7 8 geometrically non-linear mechanism v

Fig. 1.5: Load-deflection diagram of a two-story frame.

maximum load bearing capacity is reached. Notice that a mechanism in the sense of the classical geometrically linear failure analysis has not been developed yet. In this case the term “frame instability” is applicable. When the load is displacement directed, the load-deflection diagram can be obtained for larger deflections as well and finally a full mechanism is able to develop, but for a lower load. In Fig. 1.5 the positions are indicated, where the plastic hinges are created. Remark Because of the creation of the sixth hinge, “the fifth hinge closes itself” and disappears. This means that the first, fifth and seventh hinge will not be present in the final mechanism (see Fig. 1.4).

1.2 Calculation of the equilibrium path and stability Under influence of loads working on a structure, it transforms into a certain state, which is characterised by a set of displacements as function of the time. When the loads are applied gently and the structure possesses an amount of internal damping as well, the velocities and accelerations will be small. Then the dynamic effects can be neglected. In that case, the

unloading. In case of plastic materials one can distinguish ideal plastic behaviour, rigid plastic behaviour, materials with hardening behaviour or softening behaviour, etc. Also the geometrically non-linearities can be classified further. In the first place, non-linear relations can be based on small strains or large strains. In this course, only small strains will be considered of say less than 0.01. For most of the structural application this is sufficient.

F

F

v

a

F

F

b

v

v

c

F

v

d

F

e

v

g

v

F

v

f

a b c d e f g

linear elastic non-linear elastic plastic elastic ideal-plastic rigid plastic with hardening with softening

Fig. 1.6: Load-displacement diagrams of physical linear and non-linear materials. Large deformations, which for example occur in rubbers or sometimes in soils require another theory. The general non-linear expression for small strains is given by (see appendix I): 1⎛ εij = ⎜ u i, j + u j, i + 2 ⎜⎝

∑

⎞ u h, iu h, j ⎟ ⎟ h=1 ⎠ 3

(1.10)

where the indices i and j receive the values 1, 2 and 3 successively (or x, y and z ). The comma represents differentiation with respect to the variable behind it. For example, for ε xx it

2ε xx = u x, x + u x, x + u x, xu x, x + u y, xu y, x + u z, xu z, x with u x, x = or:

εxx = ux , x +

∂u x ∂x

, etc

1 ( ux, x ux ,x + u y, x u y, x + uz, x uz, x ) 2

The first term is the usual linear term, the term between brackets represents the non-linear correction as long as the strains are small. The first one of the quadratic terms is often neglected. The geometrically non-linear relations appear in the first and third step of the calculation. The non-linearities can be incorporated exactly but also approximations are possible of the second, third, fourth, etc order. In this terminology “first order” is equal to linear. So with the above provided equations, different points of the equilibrium path can be calculated. However, the nature of the equilibrium itself cannot be investigated (stable, neutral, unstable), in other words it cannot be determined whether a small disturbance causes the structure to move and to leave the equilibrium path. In order to provide an answer to this type of questions, the principle of minimum potential energy forms a better basis for the analysis. The essence is that a structure the potential energy of which is minimal cannot transform potential energy into kinetic energy. So, if the structure is at rest, this state of rest is conserved (assuming that the structure is also in equilibrium from a thermal point of view). A small disturbance will only lead to a (damped) movement around the state with the minimum of potential energy. Naturally, the approach of the problem via the principle of minimum potential energy has its shortcomings too: both the structure and the load must possess a certain amount of potential energy. However, an important class of problems meets this condition, these are the structures with a linear-elastic material behaviour subjected to deadweight loads. With certain mutations, the theory can also be used for other circumstances. The potential energy for the considered system is given by:

P = Pelastic + Pload

(1.11)

The elastic energy is a quadratic function of the strains and the potential energy of the load is a linear function of the displacements, so the potential energy becomes:

P=

1 2

∫∫∫ ∑∑∑∑ E

ijkl

ε ij ε kl dV −

∑F u h

h

(1.12)

The summations take place over all indices, the integration concerns the volume of the structure. The condition now is that the potential energy for the displacement u = ui ( x, y, z) is a minimum. For this to happen it should hold:

P (u + δ u ) − P (u ) > 0 for all arbitrary δ u different from zero

In order to investigate this condition, variations of different orders are considered:

In the theory of buckling the following aspects can be distinguished (see Fig. 1.8): a) the pre-buckling behaviour or basic state with the basic mode b) the bifurcation point with the buckling load and buckling mode c) the post-buckling behaviour F

b

c

a

a: pre-buckling behaviour or basic mode (rigid, linear, non-linear) b: bifurcation point c: post-buckling behaviour or buckling mode (non-linear, 2nd - 4th order)

u, v Fig. 1.8: Buckling behaviour. The purpose of the buckling analysis is to determine the buckling load. Thus, this is the load where the basic state is left and buckling into the buckling mode may occur. The transition point from basic mode to buckling mode is called the bifurcation point. The pre-buckling behaviour can be modelled in three ways: − rigid − linear − non linear In these lecture notes the rigid pre-buckling behaviour is taken as a starting point. This means that the deformations due to the load up to the bifurcation point are neglected. This is not entirely correct, but leads in most cases to the correct estimation of the buckling load. In chapter 2 this topic will be discussed again. Non-linear pre-buckling behaviour is important for stability problems such as snapping-through. Another interesting theoretical example in which pre-buckling behaviour is important is the telescopic spring (see chapter 2). In all cases, the buckling and post-buckling behaviour has to be described by non-linear equations. During the elaborations the exact geometrical relations can be used, but also the second-order or higher-order approximations. Then the resulting post-buckling behaviour may be (see Fig. 1.9): − neutral − linearly increasing or decreasing − quadratically increasing or decreasing In the bifurcation point, a direct relation is present between the nature of the equilibrium and the type of post-buckling behaviour. The post-buckling behaviour is determining for the socalled imperfection-sensitivity of the structure. A structure with linearly decreasing post-

buckling behaviour is very imperfection sensitive. In extreme cases, it even may happen that the buckling load is not reached at all, because of small imperfections. This especially may occur, when at the same time large numbers of buckling modes are active, such as for example in shell structures. For beam structures this problem is of less importance.

linear non-linear neutral non-linear Fig. 1.9: Post-buckling behaviour. In order to analyse the buckling problem with the principle of minimum potential energy, firstly the expression for the potential energy is formulated. Subsequently the buckling load can be found from the condition that the potential energy is stationary, i.e. from the condition that δ P = 0 . This variational equation can be worked out further (later it will be shown how), which leads to the equilibrium equations in the deformed state. As mentioned before, this also can be done by direct formulation of the equilibrium, with or without the help of the virtual work equation. However, in that case it is not possible to investigate the stability of the equilibrium systematically. In the next chapter, the several methods will be compared for a number of examples. In each of the following chapters, only one of the methods is chosen.