Advances in Steel Structures - part 34 ppt

If a structure undergoes large displacements as the load is applied incrementally, then the stiffness matrix will not be constant during the loading process.. The new stiffness matrix fo

310 M Saidani et al

were observed: (a) weld fracture; (b) tube yield failure; (c) local fracture of the tube wall at the vicinity of the weld

Failure mode (a) would only occur if the weld were the weakest part of the joint This could be avoided by carefully controlling the quality (and size) of the weld Failure mode (b) is the more general one, occurring in almost all the tests If the welds in the cap plate-tube connection and the cleat plate-tube connection are strong, then failure mode (c) may take place especially for connections with thinner plates As can be seen from Figure 5, the axial deformation remains linear for load of up to 230kN (estimated first yield point) In this case, the failure load was 313kN This gives a ratio of ultimate strength to yield strength of 1.36 In reality the ultimate load could be higher since the test was stopped as soon as the machine ceased taking any more loads The stresses

at mid-height of the tube (Figure 6) were linear up to about 120kN and thereafter non-linear This was typical in all the specimens tested Good agreement is obtained with the finite element modelling as shown in the companion paper, Karadelis et al (1999)

Load (kN)

35O

30O

250

2OO

150

100

50

0

O v e r a l l axial d e f o r m a t i o n ( m m )

Figure 5: Load vs axial deformation (LVDT1)

Load (kN) 35O 3O0 25O 20O

150

100 5O

Stress in tube (N/mm 2) Figure 6: Load vs axial stress in the tube (SG2/3)

35o ] Load (kN)

250 f f

150

100

Out-of-plane bending moment (kN.m)

350

300

250

200

150

100

50

0

In-plane bending moment (kN.m)

Figure 7" Load vs out-of-plane bending Figure 8: Load vs in-plane bending

It is also evident from Figures 6 (and in fact in other specimens), that extensive stress redistribution and strain hardening were taking place Examination of Figures 7 and 8 show that the in-plane and

Behaviour of T-End Plate Connections to RHS Part I 311 out-of-plane bending moments were small and could therefore be ignored As the load approaches the failure load, the deformations in the specimen become more important resulting in a sharp increase in in-plane and out-of-plane bending moments Again, this was characteristic in all the joints tested

CONCLUSIONS AND FUTURE WORK

The behaviour of welded T-end plate connections has been investigated through a series of tests Numerical models have also been used to predict their behaviour It was found that, apart from any weld defects, the mode of failure of the joint could be by generalised tube yielding or local fracture

of the tube wall It is suspected that as the cap plate gets thicker (more than 25mm), the capacity of the joint is reduced suggesting that joints with excessively thicker plates are less stronger than would normally be expected The results also suggest that considerable stress re-distribution and strain hardening were taking place after the first yield More tests are under way for 'true' rectangular hollow section tubes (as compared to square) The effect of changing the orientation of the cap plate

in relation to the tube will be examined The finite element model will be further refined and benchmarked The results will be used to produce design guideline for this type of connection, based on this work but also on previous published work

A K N O W L E D G E M E N T S

The authors are very grateful for the generous support and contribution from British Steel, pipes and tubes (Corby, UK), especially to Mr Eddie Hole, sales manager, and Noel Yeomans, technical manager Thanks also to Mr John Griffiths for his assistance is preparing and testing the specimens

REFERENCES

Cran J.A (1977) World wide applications of structural hollow sections, the sky's the limit

Symposium on tubular structures, Delft, The Netherlands, 23.1-23.14

Comite International pour le Developpement et l'Etude de la Construction Tubulaire, British Steel, and the Commission of the European Communities (1984) Construction with hollow sections,

Wellingborough, Northants, UK

Packer J.A., Wardenier J., Kurobane Y., Dutta D., and Yeomans N (1992) Design guide for rectangular hollow section (RHS) joints under predominantly static loading Verlag TUV,

Rheinland, Koln, Germany

Kitipornchai S and Traves W.H (1989) Welded T-end connections for circular hollow tubes

Structural Engineering, ASCE 115:12, 3155-3170

Stevens N.J and Kitipornchai S (1990) Limit analysis of welded tee end connections for hollow tubes Structural Engineering, ASCE 116:9, 2309-2323

Granstrom A (1979) End plate connections for rectangular hollow sections The Swedish Steel

Construction Institute Report 15:15

Karadelis J.N., Saidani M., Omair M.R (1999) Behaviour of end-plate connection to rectangular hollow section Part II: Numerical modelling ICASS99', Hong Kong, PRC

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THE BEHAVIOUR OF T-END PLATE CONNECTION TO SHS

PART II: A NUMERICAL M O D E L

J N Karadelis, M Saidani, M Omair

School of the Built Environment, Coventry University,

Coventry, CV 1 5FB, UK

ABSTRACT

The behaviour and performance of a family of structural connections made of square hollow sections (SHS) has been investigated in the laboratory and a series of data have been collected and presented

in a graphical form In parallel, a rigorous finite element model was developed capable of analysing the system of the SHS, the cap-plate, the cleat-plate and its surrounding weld Evidence of non- linearity and deviation from the classical linear elastic theory led to a more complex numerical solution to fit more closely the experimental data A specific methodology is presented, as it applies

to analyses involving plasticity and large deflections (deformations) Test results obtained in the laboratory were compared with computed values from the finite element analysis and are presented graphically in the last pages of this paper Satisfactory agreement was obtained between recorded and computed strains and displacements The paper includes extensive discussion of the above results and the conclusions drawn from them A brief account of directly related future research work is also given

KEYWORDS

SHS, Non-linear, FE-Analysis, ANSYS, Stresses, Strains, Displacements

INTRODUCTION

There is no doubt that the description of non-linear phenomena inevitably lead to non-linear equations which immediately render classical methods of mathematical analysis inapplicable No method is yet known for finding the exact solution to a system of non-linear equations, such as the one shown below

Where:

[ke(8,F)] = stiffness matrix of the element which is a function of {8} and {F}

{8 e} = displacement vector of the element

{F e} = load vector of the element

313

314 J.N Karadelis et al

Non linear structural behaviour arises from a number of phenomena, which can be grouped into three main categories such as, changing status, geometric non-linearities and material non-linearities

THEORY

Due to their nature, non-linear problems require special solution techniques The established Newton-Raphson (N-R) method, Zienkiewicz O C et at (1994) is a series of successive linear approximations (iterations) with corrections and can be used in special algorithms to solve non-linear problems The stiffness matrix [K] and the restoring force {F nr} vary with the applied load Each linear approximation requires at least one iteration through the equation solver The stiffness matrix [K] may be updated in every iteration, occasionally, or not at all Accordingly, the method is called full, modified, or initial-stiffness (N-R) procedure

GEOMETRIC NON-LINEARITIES (LARGE DISPLACEMENTS APPROACH)

If a structure undergoes large displacements as the load is applied incrementally, then the stiffness matrix will not be constant during the loading process When a small tensile load is applied at the centre of the cap-plate welded around the periphery of a square hollow steel member (Figure 1, end

of paper), the strain energy stored in the material is due to bending of the plate only However, when the load becomes large enough to bow the plate significantly, the area of the plate around the line of application of the load will undergo further deformation to accommodate the additional strain Therefore, the stiffness of the plate at this central region will increase From the finite element point

of view, geometric non-linearities are not difficult to deal with, provided that stresses do not affect the stiffness matrices significantly (that is, the stiffness matrices formed are not stress dependant)

In a typical case like the above, the load was divided into a series of sufficiently small increments (steps) and these were applied one at a time ANSYS strongly recommends that for large displacements analyses the loads specified in the load steps should be 'stepped up' (as opposed to 'ramped on') This means that the value of a particular load step will be reached during the first iteration and will be kept constant during the remaining iterations, until the end of the load step This contributes to faster convergence After each increment the deflections caused were calculated by using the linear version of the equation 1, above That is, it was assumed that the stiffness matrix was constant during the application of each load increment The initial stiffness matrix was used to generate the equations for the next increment and so on, until the process was completed The original co-ordinates of the nodes were then shifted by an amount equal to the values of the displacements calculated The new stiffness matrix for the deformed plate was re-calculated and the process was repeated until the total load was reached The matrix notation of the incremental procedure, using the linear version of the equation 1 above, is shown below

{AF, }= [K (i-1)]{ A ai}, V i ~ 9t N :i = 1,2,3,4, (2) (i= positive integer representing stage of incremental loading)

The initial Tangent Modulus, tE0, was taken from a non-linear stress-strain curve obtained from experimental observations The initial stiffness matrix [K0], was then computed from the tangent modulus tE0 and the Poisson's Ratio, v Note that v could also be defined as Tangent Poisson's ratio tv0 and can be obtained by evaluating the first derivative of the volumetric strain with respect to the axial strain curve However, this was not found to be appropriate at this stage

The Behaviour of T-End Plate Connections to SHS Part H 315

It is important to keep the load increments small, so that the increments in displacement cause negligible changes in the stiffness matrix at each step On the other hand, increments should be sufficiently large otherwise the cost of computer running may become excessive These displacements were added together and gave the total displacement after the final increment Stresses and strains corresponding to the above displacements were treated in the same manner

ANSYS activates the large deflection analysis within the static analysis using the NLGEOM, ON

option It can be summarised as a three step process for each element

1 Determination of the updated transformation matrix [Tn] for the element

2 Extraction of the deformation displacement {Und}, from the total element displacement {Un}, in order to compute the stresses and the restoring force {Fenr}

3 After the rotational increments in {Au} are computed, node rotations must be updated

Any desired loading can be applied However, the program must be allowed to iterate until the solution converges Intermediate iterations are not in equilibrium and do not represent valid solutions In order to obtain valid solutions at intermediate load levels, and/or to observe the structure under different loading configurations, multiple load steps may be desired

A synoptical flow chart containing the main steps for large displacements analysis and demonstrating the presence of stress stiffening effects is shown below, in Figure 2 Simply stated, stress stiffening looks at the state of stress in a FE-model and calculates a stiffness matrix, [S], based upon it Matrix [S] is then added to the usual (initial) matrix [K] and a new set of displacements are calculated

Form

K

Calculate u0 and stresses

Calculate S from stress state and add to K

V3

First iteration

Second and subsequent iterations

316 J.N Karadelis et al

Calculate displacements and stresses using

Yes

J

STOP

Second and subsequent iterations

)

MATERIAL NON-LINEARITIES

Material non-linearities occur when the stress is a non-linear function of the strain The relationship

is also path dependant, that is, the stress depends on the strain history as well as the strain itself ANSYS accounts for several types of material non-linearities Rate independent plasticity will be utilised here as it is characterised by irreversible straining, once a certain level of stress is reached

Elasto-plastic behaviour General approach

ANSYS theory for elasto-plastic analysis, provides the user with three main elements: The yield criterion, the flow rule and the hardening rule

The yield criterion determines the stress level at which yielding is initiated For two and three dimensional stress systems this can be interpreted through the equivalent stress (~eq), MASE G E (1970) The flow rule determines the direction of plastic straining (ie: which direction the plastic strains flow) relative to x,y,z axes Finally, the hardening rule describes the changes the yield surface undergoes with progressive yielding so that the various states of stress for subsequent yielding can be established For an assumed perfectly plastic material the yield surface does not change during plastic deformation and therefore the initial yield condition remains the same For a strain hardening material, however, plastic deformation is generally accompanied by changes in the yield surface Two hardening rules are available and these are isotropic (work) hardening and kinematic hardening

The Behaviour of T-End Plate Connections to SHS Part H

Multilinear Isotropic Hardening

317

For materials with isotropic plastic behaviour, the assumption of isotropic hardening under loading conditions postulates that, as plastic strains develop, the yield surface simply increases in size and maintains its original shape ANSYS, uses the von Mises yield criterion with the associated flow rule and isotropic (work) hardening When the equivalent stress is equal to the current yield stress the material is assumed to undergo yielding The yield criterion is known as the 'work hardening hypothesis' and assumes that the current yield surface depends only upon the amount of plastic work done

The solutions of non-linear elastic and elasto-plastic materials are usually obtained by using the linear solution, modified with an incremental and iterative approach The material is assumed to behave elastically before reaching yield as defined by Hooke's low If the material is loaded beyond yielding, then additional plastic strains will occur They will accumulate during the iteration process and after the removal of the load will leave a residual deformation

In general:

~ ~n(ela) "q- iCn(pla) nt" ~ (3) where:

en = total strain for the current iteration

lgn(ela) = elastic strain for the current iteration

Agn(pla) = additional plastic strain obtained from the same iteration

gn-l(pla) = total, previously obtained plastic strain

Convergence is achieved when Agn(pla)/gn(ela) is less than a criterion value, Walz J E et al (1978) This means that very little additional plastic strain is accumulating and therefore the theoretical curve is very close to the actual one

For the case of uniaxial tension, it is necessary to define the yield stress and the stress-strain gradients after yielding For the uni-axial stress governing the SHS member alone, these are simply Crxx and exx When Oxx becomes greater than the uniaxial yield stress then yielding takes place The yield condition

is the well established von Mises yield criterion for one dimensional state of stress

Typically, a set of flow equations (flow rule) can be derived from the yield criterion which imply that the plastic strains develop in a direction normal to the surface (associative rule) The associative flow rule for the von Mises yield criterion is a set of equations called the incremental Prandtl-Reuss flow equations MASE G E (1970) That is, the strain increment is split into elastic and plastic portions and the plastic strain increments are dependant on the deviator stresses Hence, it is necessary to apply the total load on the structure in increments These load steps need only start after the FE-model is loaded beyond the point of yielding The size of the subsequent load steps depends on the problem The load increments will continue until the total load has been reached or until plastic collapse of the structure has occurred

318 J.N Karadelis et al

As the load increases the plastic region spreads in the structure and the non-linear problem is approached using the full Newton-Raphson procedure The stiffness used in the N-R iterations is the tangent stiffness and reflects the softening of the material due to plasticity It should be noted that in general, the flatter the plastic region of the stress-strain curve, the more the iterations needed for convergence

As plasticity is path dependant or a non-conservative phenomenon, it requires that in addition to multiple iterations per load step the loads be applied slowly, in increments, in order to characterise and model the actual load history Therefore, the load history needs to be discretised into a number of load steps with the presence of convergence tests in each step ANSYS recommends a practical rule for load increment sizes such as the corresponding additional plastic strain does not exceed the order

of magnitude of the elastic strain That is, the plasticity ratio:

Cn(ela)

In order to achieve inequality 9, the following loading sequence was adopted:

Load step one was chosen so that to produce maximum stresses approximately equal to yield stress The yield stress was estimated from the experimental stress-strain curves and validated by performing a linear run with a unit load and by restricting the stresses to the critical stress of the material This was found to be approximately equal to 200 kN Successive load steps were chosen such as to produce additional plastic strain of the same magnitude as the elastic strain or less This was achieved by applying additional load increments no larger that the load in step one, scaled by the ratio ET]E Such as:

E T

P~+, = -E P~ Vn~9~ N (6)

where:

E = Elastic slope

Ev = Plastic slope

with: Ev/E not less than 0.05

Table 1 below summarises the plasticity theory that characterises the elasto-plastic response of a certain type of materials However, the basic steps characterising a non-linear elasto-plastic analysis are shown in a brief flow chart, in Figure 3

Material Option

Multi-linear

Isotropic

Hardening

Yield Criterion

von Mises

Flow Rule

Associative (Prandtl-Reuss equations)

Hardening Rule Isotropic

Material Response Multilinear

Table: 1 Summary of the theories involved in a material with multi-linear isotropic hardening behaviour

(

The Behaviour of T-End Plate Connections to SHS Part H

( S T A R T )

Form

K

Calculate

u0~ (5"0, ~0

No

(Linear elastic)

STOP -')

Load Step 1

\ 7 5 /

~ Yes (proceed with nonlinear analysis)

AUl, AG1 and AE:I

" and add to

u0~ (Y0~ ~;0

Calculate displacements and stresses and add to previous values

Yield Criterion

~ ~ Load=Tgt.Load

~ Yes

319

Incremental Procedure and repeated Iterations

Figure: 3 The basic steps of a non-linear, elasto-plastic analysis