_{1}

^{*}

The 8-node iso-parametric thin shell element was employed in the study of stress concentrations in the welded tubular “K” joint. Element equilibrium equations were derived using isoparametric formulation based on thin shell theory. After assembly, the resulting system equations were solved using existing fortran programs. Numerical experiments were conducted to isolate and locate ideal gap (positions) for the two braces of the “K” joint. The nominal stresses were calculated from which stress concentration factors were obtained. The resulting stress concentration factors were presented both as tables and as figures. A good agreement between our solutions and those for model joints in the literature is good and acceptable. It was found that the wider apart the brace spacing is, the weaker the strength of the joint. It was also found that the best location for the braces occurs when the stress level changes sign either from positive to negative or vice versa at a critical sampling point.

Lapped joints generally, and lapped tubular joints in particular are known to be stronger than gapped joints following a heuristic reasoning. However, for the purpose of bracing, it is not always feasible or even possible to brace a structural system with lapped joint throughout. Hence a strong case arises from using gapped tubular joints which are in many cases welded on site due to ease of construction and transportation. As usual, one of the problems with welded steel construction is the development of stress concentration or ‘hot spot’ stresses. The hot spots affect the strength of both lapped and gapped joints.

The problem of hot spot stresses or stress concentrations has been investigated by Kuang et al. [

It can be seen from the above review that the effect of stress concentrations on strength of tubular joints has been found to be a problem generally [5,6].

However, the particular problem of stress concentrations in lapped and gapped tubular joints with respect to effective location of the spacing of the braces has not hitherto been properly addressed.

The purpose of this study is to investigate the effect of effective location of braces (brace spacing) on the strength of the gapped tubular “K” joints. The effect of section geometry on the distribution of stress concentration factors in hollow section (HS) or tubular joints is also investigated. The finite element method of analysis was used in the investigation of the static stress concentrations.

The usual curved thin shell iso-parametric element was used for the present investigation (e.g. see Hinton and Owen [

The element total potential energy functional is given as :

where is the strain energy of the element is the external work done by the element during deformation.

In a compact formulation Equation (15) can be written as:

In which is the element stiffness matrix is the element boundary loads, such that:

After assembly we solve the system equations of Equation (20) for displacements.

The finite element stresses are calculated from

in which

The system equilibrium equations for the present problem were solved after application of boundary conditions using the frontal solution algorithm and code developed by Hinton and Owen [

The finite element discretization and loading of the joint for purpose of stress analysis is shown in

and the nominal shear stresses are calculated from:

_{CFs} for “K” tubular joints subjected to inplane loads.

_{CF} at corner of top chord (Brace angles = 50/40 deg).

in which Q is applied shear force, R is shell radius and t is shell thickness. The gap between the two braces in the first case was kept constant at 50 mm and stress concentration factor was obtained as

or

where is finite element stress in the longitudinal axis of the chord is shear stress in the chord. is

_{CF} at corner of top chord (Brace angles = 50/50 deg).

nominal stress given as.

Geometric parameters for the “K” joint.

Example 1.

For the purpose of comparison with our solution the dimensions of the “K” joint studied by So and So [

We have also:

t/T = Brace of chord thickness ratio = 1.0D/T = Chord diameter to thickness ratio = 25.50d/D = Brace of chord diameter ratio = 0.52L/D = Chord length to diameter ratio = 6.18g/D = Brace separation to chord diameter ratio = 0.015 From g/D = ratio of 0.015, we calculate g to be, say =5 mm.

And, we calculate L used by So and So [

We next subject the joint with the above details to in-plane loading for the purpose of bending stress analysis. The result of analysis from our finite element code is compared with those of So and So [

Our result for the chord is 0.53% lower than that of Efthymiou and Durkin and 4.37% higher than that of So and So [

Example 2.

In this example we investigate the effect of damage and brace spacing/ separation on the strength of the “K” tubular joint. The geometric details of the joint is as follows:

Chord:

Brace:

The result of the peak (adverse) stress concentrations obtained mainly from top chord of the “K” joint are presented here in Tables 2 and 3.

Bracings in structural systems provide strength to the systems. However, there is no universal method of selecting locations for a selected bracing system. Usually the designer selects brace separations or gaps based on previous experience or heuristic reasoning. In this work we have used the finite element method as a tool to carry out some numerical experiments on locations of effective brace separations or spacings rather than use usual rule of thumb to select the desired spacings or gap between braces. By varying the gap between two braces and making computer runs produce a set of results. A close look at the results produced shows a trend in degradation of stresses as the gap is increased. This can give the analyst or the designer some idea of which brace separation to choose in the design.

In Tables 2 and 3 we have presented the results of our study for three brace separations e.g. 50 mm, 60 mm and 70 mm respectively. We see in the tables that from 50 mm to 60 mm spacing/separation there is no alternating stress concentration factors S_{CFs} i.e. from positive to negative; that is, we have monotonic stress pattern and there is no stress reversal. However, as we increase the separation gap to 70 mm, we see the alternating pattern of stress concentration factors. This marks the limit of brace separation for the joint details above.

From our study of gapped tubular “K” joints we conclude as follows:

1) Brace separation distance for “K” tubular joint should be carefully selected so as to reduce over stress concentrations.

2) For best result, the joint should be lapped.

3) However, if the joint must be spaced, then based on our findings in Tables 2 and 3 and Figures 3 and 4, 50 mm gap should be the limit of the spacing or gap. This is because after the 50 mm brace spacing or gap, adverse stress concentrations are observed when the spacing is increased beyond the 50 mm mark. In any case, the geometric parameters of the joint should be considered in order to avoid joint over stress.

4) We recommend or propose here that the finite element method be used as an ideal tool for the location of best brace spacing or gap (g).

5) Using our example joint studied here, we propose as a guide in the location result, that the ratio of the spacing

_{ }

or gap g to the chord diameter D should not be greater than 1, that is, g/D ≤ 1.