**No. R684, January, 1994**

Showkati, H and Ansourian, P

Influence of Primary Boundary Conditions on the Buckling of Shallow Cylindrical Shells

**Abstract**

Slender cylinders of intermediate length subjected to external pressure normally buckle into sinusoidal waves in the circumferential and meridional directions; the wave numbers are a function of the edge restraints, aspect ratio L/R and wall slenderness R/t. the influence of edge restraints diminishes with increasing L/R, and disappears in very long cylinders. All instability formulae purporting to evaluate the lowest eigenvalue require minimisation with respect to the circumferential wave number n; any ‘exact’ formulation requires the a-priori evaluation of n. Therefore, in any investigation of the effect of boundary conditions on the buckling pressure q_{cr}, the evaluation is crucial.

The classical solution for the estimation of n involves ‘standard’ end conditions which consist only of radial restraints, and is derived from a minimisation for example of the Southwell equation with respect to n. For other sets of boundary conditions, the governing equations become almost intractable unless reduced theories such as long wavelength theory is used, and resort must be had to numerical methods such as Rayleigh-Ritz, finite differences or finite elements. to the authors’ knowledge, a comprehensive theoretical or numerical study of the effect of boundary conditions on the buckling wavenumber and the critical pressure is not available in the literature.

In this paper, a numerical study is presented of the influence of 11 separate sets of boundary conditions on the circumferential wavenumber n and the buckling pressure q_{cr}. Geometric ratios are considered in the practical range 0.5£ L/R £ 5.0 and 300£ R/t £ 3000. Finite element analysis is carried out using refined conditions, simple expressions are derived for n and q_{cr}. It is found that, as the restraint of the cylinder wavenumber is not necessarily an integer value, and therefore the concept of an effective length similar to that of columns is appropriate. It is also found in this investigation that for the higher wavenumbers, the follower load effect on the buckling mode is small.

**No. R685, March, 1994**

*Pi, YL and Trahair, N**Plastic Collapse Analysis of Torsion*

**Abstract**

This paper is concerned with the plastic collapse analysis of structural steel I-section members subject to torsion. It proposes and exemplifies a method of plastic collapse analysis which is much simpler than elastic analysis. the load factor at plastic collapse is obtained from the sum of the independent load factors for uniform torsion plastic collapse and warping torsion plastic collapse.

This plastic analysis method allows a method of plastic design to be used for torsion which is much more simple and more economical than first yield design. the use of plastic analysis and design will facilitate the design of steel torsion members and lead to more economical structures.

**No. R686, March, 1994**

Pi, YL and Trahair, NTorsion and Bending Design of Steel Members

**Abstract**

This paper is concerned with the design of structural steel members subject to torsion and bending. It proposes methods of plastic collapse analysis and design for torsion which are much simpler and more economical than those of elastic analysis and first yield design. the load factor at plastic collapse is obtained from the sum of the independent load factors for uniform torsion plastic collapse and warping torsion plastic collapse. It is proposed that plastic analysis and design for torsion be permitted for ductile compact I-section members.

For other members, elastic torsion analysis may be used. For non-ductile compact members, a first hinge method of torsion design is proposed, while first yield and local buckling torsion design methods are proposed for non-compact and slender members. These methods are consistent with the corresponding methods of AS4100 for designing steel beams against bending.

Torsion often occurs in real structures in conjunction with bending actions. While the torsion actions are often secondary, in some cases flexural members may be required to transfer some of their loads by torsion actions. Simple interaction equations are proposed for members subject to combined bending and torsion actions.

**No. R687, March, 1994**

*Xiao, B, Carter, JP and Alehossein, H**Elastoplastic Analysis of Jointed Rocks Using A Coupled Finite Element and Boundary Element Method*

**Abstract**

This paper presents a coupled, elastoplastic, finite element and boundary element method for the two-dimensional, non-linear analysis of anisotropic jointed rock. the non-linear and anisotropic behaviour of a jointed rock mass is simulated by representing the mass as an equivalent anisotropic, elastoplastic continuum, so that the influence of the jointing system is “smeared” across the continuum, i.e. the individual joints are not modelled as discrete entities. Numerical examples have been solved to verify the capability, accuracy and efficiency of the present technique. the proposed technique has also been applied to the analysis of tunnel excavation problems in plane strain. the effects of anisotropy and non-linearity of the jointed rock mass during excavation have been investigated in some detail.

**No. R688, April, 1994**

*Rousch, CJ and Hancock, GJ**A Non-Linear Analysis Model for Simply-Supported and Continuous Purli*

**Abstract**

A non-linear elastic analysis has been developed for determining the lateral deflections of, and stresses in, the unconnected flanges of simply-supported and continuous channel and Z-section purlins screw-fastened to sheeting and subject to either wind uplift or gravity loading. the analysis incorporates a model, based on a combination of those developed by Peköz and Soroushian, and Thomasson, which depicts the unconnected purlin flange as a beam-column. the restraint against lateral deflection is provided primarily by the sheeting, and is represented by a linear extensional spring of stiffness k located at the level of the unconnected purlin flange.

To verify the model, lateral deflections and failure stresses obtained from vacuum rig tests can be compared with those determined by the non-linear analysis. In this report, results from vacuum rig tests on continuous Z-section purlins screw-fastened to sheeting and subject to simulated wind uplift and gravity loading are compared.

**No. R689, May, 1994**

Poulos, HGPiles Subjected to Externally-Imposed Soil Movements

**Abstract**

There are several circumstances in which piles are subjected to loading by the movement of the surrounding soil past the pile. Such circumstances include:

- the consolidation of soft clays around a pile, leading to negative friction forces;
- the expansion or contraction of reactive clay around a pile, leading to either tensile or compressive additional forces;
- the lateral movement of unstable or creeping slopes past piles, leading to additional bending moments and shears;
- the lateral movement of soil during an earthquake, again inducing additional bending moment shears.

The basis of analysing piles subjected to such movements is described briefly. the behaviour of piles subject to the above sources of soil movement are then described, and the implications for pile design are discussed.

**No. R690, May, 1994**

Rasmussen, KJR and Young, BStainless Steel Tubular Joints – Tests and Design of X- and K-Joints in Square Hollow Sections

**Abstract**

The report describes a test program on welded stainless steel X- and K-joints fabricated from square hollow section brace members and chords. the X-joints were tested in compression and tension using different ratios of brace width to chord width. the K-joints were tested by varying the ratio of brace width to chord width, the angle between chord and brace members, and the preload applied to the chord. A total of 23 tests were performed. the report includes load deflection curves for all tests, as well as profiles of the sidewall deflection.

Design rules are proposed for X- and K-joints by adopting the rules of the CIDECT Recommendations for *carbon* steel tubular structures and replacing the yield stress in these recommendations by a proof stress. Based on a reliability analysis using a target safety index of b _{r} = 3.0, it is shown that the 0.2% proof stress, as determined from the finished tube, can be used to determine the ultimate strength using the CIDECT design rules. This approach uses a resistance factor of F = 0.9. It is also shown that the serviceability limit state corresponding to joint deformations of 1% of the chord width will not be reached if the CIDECT strength rules are adopted. Furthermore, it is investigated whether the CIDECT Recommendations can be used when the design strength is based on the 0.5% proof stress. In this case, the resistance factor is selected as 0.75.

**No. R691, May, 1994**

Rasmussen, KJR and Hasham, ASStainless Steel Tubular Joints -Tests and Design of X- and K-Joints in Circular Hollow Sections

**Abstract**

The report describes a test program on welded stainless steel X- and K-joints fabricated from circular hollow section brace members and chords. the X-joints were tested in compression and tension using different ratios of brace width to chord width. the K-joints were tested by varying the ratio of brace width to chord width and the angle between chord and brace members. A total of 15 tests were performed. the report includes load deflection curves for all tests, profiles of the sidewall deflection, and load versus strain diagrams.

Design rules are proposed for X- and K-joints by adopting the rules of the CIDECT Recommendations for *carbon* steel tubular structures and replacing the yield stress in these recommendations by a proof stress. Based on a reliability analysis using a target safety index of b _{r} = 3.0, it is shown that the 0.2% proof stress, as determined from the finished tube, can be used to determine the ultimate strength using the CIDECT design rules. This approach uses a resistance factor of F = 0.9. It is also shown that the serviceability limit state corresponding to joint deformations of 1% of the chord width will not be reached if the CIDECT strength rules are adopted. Furthermore, it is investigated whether the CIDECT Recommendations can be used when the design strength is based on the 0.5% proof stress. It is shown that the same resistance factor (F = 0.9) as that obtained using the 0.2% proof stress can be used when the design strength is based on the 0.5% proof stress.

**No. R692, May, 1994**

Poulos, HGAn Approximate Numerical Analysis of Pile-Raft Interaction

**Abstract**

This paper presents an approximate method of numerical analysis of piled raft foundations in which the raft is modelled as a thin plate and the piles as interacting springs of appropriate stiffness. Allowance is made for the development of limiting pressures below the raft and of the ultimate axial load capacity of the piles.

Comparisons between this analysis and existing solutions verify that, despite the approximations involved, the analysis can provide solutions of adequate accuracy for the settlement and pile load distribution within a piled raft.

Comparisons are also made with the results of a series of centrifuge tests and with measurements of the performance of a full-scale piled raft. In both cases, the analysis predicts very well the settlement and proportion of load carried by the piles.

**No. R693, May, 1994**

*Zhao, Xiao-Ling and Hancock, GJ**Square and Rectangular Hollow Sections Under End Bearing Force*

**Abstract**

In this report, tests on a range of cold-formed square and rectangular hollow section members subject to end bearing force are described. the concentrated force was applied by means of a bearing plate, which acted across the full flange width of the section. the parameters varied in the tests included (1) the bearing length, (2) the slenderness of the member and (3) the shape of the section (RHS or SHS).

The results are compared with the Australian Steel Structures Standard AS4100-1990 and the cold-formed standards in America, Australia, Canada and Europe. the results are also compared with the formulae given by Zhao and Hancock (1992), which were based on tests of SHS and RHS sections under interior bearing force. A design model is proposed for SHS and RHS sections under end bearing force. the reliability analysis method is used to calibrate the existing and proposed design formulae.

**No. R694, June, 1994**

*Gao, H, Kwok, Kcs and Samali, B**Optimisation of Tuned Liquid Column Dampers for Suppressing Structural Vibrations*

**Abstract**

The application of tuned liquid column damper to civil engineering structures was first proposed in 1989. Further analytical and experimental research has been conducted since on its efficiency and improved performance. the studies so far have mainly concentrated on the original tuned liquid column damper (TLCD-I) proposed by Sakai et al. In contrast, little has been done to come up with a comprehensive mathematical modelling reflecting variations to the basic mathematical description given by Sakai et al. This paper describes the damping mechanism of tuned liquid column dampers and its variations through a numerical procedure, and tries to determine how damper characteristics influence the structural response, and to identify optimum damper parameters in order to maximise its efficiency. the second type original tuned liquid column damper (TLCD-II) and the variational tuned liquid column dampers (TLCD-VI and TLCD-V2) proposed in this report have a higher capacity to control stronger vibrations compared to TLCD-I, hence extending the range of applications for a tuned liquid column damper designed to suppress the structural vibrations induced by environment loads.

**No. R695, July, 1994**

*Pi, Yong Lin, Papangelis, JP and Trahair, NS**Prebuckling Deformations and Flexural-Torsional Buckling of Arches*

**Abstract**

When the ratio of the minor axis flexural stiffness is not small, classical analysis may lead to an inaccurate prediction of the flexural-torsional buckling loads of an arch, because prebuckling in-plane deformations change the curvature of the arch. A systematic treatment of the effects of prebuckling in-plane deformations on the elastic buckling of monosymmetric arches is presented in this paper. Nonlinear relationships between the displacements and the strains are obtained using position vectors. Energy equations are formulated for the elastic flexural-torsional buckling of monosymmetric arches which include the effects of prebuckling in-plane deformations, and closed form solutions are obtained for arches in uniform bending. the effect of the arch slenderness on flexural-torsional buckling is investigated. Comparisons with the existing theoretical solutions are made.

**No. R696, July, 1994**

*Sully, RM and Hancock, GJ**Behaviour of Cold-Formed SHS Beam-Columns*Full Report in PDF

**Abstract**

This research report describes a test programme conducted into the behaviour of cold-formed, compact square hollow section beam-columns. the tests were conducted in a purpose built test rig capable of applying load and moment in a constant ratio. the test specimens were pin-ended specimens loaded, with varying load/moment ratios, at two different ratios of end moment. the results of the tests were closely simulated using a finite element programme thereby validating its use in predicting cold-formed tubular beam-column behaviour. the finite element programme was then used to find the maximum load and moment capacity, for the section tested, at varying ratios of end moment, from which the corresponding maximum second order elastic moments were calculated. the results of this numerical investigation are compared with the relevant interaction design rules from AS4100 (Standards Australia (1990)) and the AISC-LRFD specification (1986).

**No. R697, September, 1994**

*Stark, RF and Booker, JR**A Numerical Procedure for Calculating the Flexibility Matrix of A Non-Homogeneous Elastic Half-Space Subjected To Uniform Surface Tractions Part I – Arbitrarily Shaped Foundations*

**Abstract**

A numerical technique is presented for the analysis of surface displacements of a non-homogeneous elastic half-space subjected to vertical and/or horizontal surface loads uniformly distributed over an arbitrarily shaped area. the non-homogeneity considered is some form of exponential variation of Young’s modulus with depth. Since this exponent which determines the degree of non-homogeneity may vary between zero to unity, both the homogeneous half-space and the Gibson soil may be included as limiting cases in a single numerical scheme. In order to account for the arbitrary shape of the loading, the boundary of the loaded area is piecemeal linearized. This enables to model any load pattern according to the desired order of accuracy. Consequiently, with respect to finite element applications, this technique allows to calculate the flexibility matrix or stiffness matrix of the subsoil on the basis of the actual shape of finite elements emanating from the discretization process of the foundation structure. Special attention is focused to the integration scheme, since the singularity becomes progressively sharp when the non-homogeneity parameter approaches unity. the performance of the numerical procedure is studied using analytical solutions for rectangular shaped areas. Further comparison with known solutions based on integral transform techniques for a uniformly distributed load acting over a circular area of the non-homogeneous soil mass shows excellent agreement as well.

**No. R698 September, 1994**

*Stark, RF and Booker, JR**A Numerical Procedure for Calculating the Flexibility Matrix of A Non-Homogeneous Elastic Half-Space Subjected To Uniform Surface Tractions Part II – Rectangular Shaped Foundations*

**Abstract**

An alternative approach for calculating the surface displacements of a non-homogeneous half-space acted upon by a vertical and/or horizontal, uniformly distributed loading over a rectangular area, is presented in this part of the paper. the flexibility coefficients are computed by integrating the fundamental solution of the non-homogeneous half-space. However, as will be shown, for this special loading pattern, the displacements can be found without numerical integration making this procedure extremely efficient. Comparisons with solutions for rectangular loaded areas on Boussinesq and Gibson type soil show perfect agreement. Results of a parametric study are presented for the surface displacements of a non-homogeneous half-space subject to vertical and horizontal loading. In this study both the material properties of the soil mass, i.e. Young’s modulus and Poisson’s ratio and the aspect ratio of the loading is varied. These results are presented in the form of influence charts which may readily be used in hand calculations for estimating the displacements of footings on a non-homogeneous soil.

**No. R699, October, 1994**

*Zhao, Xiao-Ling, Hancock, GJ and Trahair, NS**Lateral Buckling Tests of Cold-Formed Rhs Beams*

**Abstract**

In this report, tests on the lateral buckling of cold-formed RHS beams under uniform bending moment are described. the section size chosen for testing is 75mm x 25mm x 2.5mm, being the largest depth/width ratio manufactured in Australia by Tubemakers of Australia Limited. the beam spans varied from 2000mm to 7000mm to produce a large range of beam slenderness ratios. Carefully designed loading and support systems were used in the tests to apply gravity load through the centroid of the section and to ensure that simply supported end conditions were achieved. the test results are compared with design rules in existing hot-rolled standards and cold-formed standards. Comparisons are also made with the predictions given by Pi and Trahair using the finite element method. *An improved design rule is given for the lateral buckling of cold-formed RHS beams.*

**No. R700, October, 1994**

*Facioni, RJ, Kwok, KCS and Samali, B**&Nbsp;A Wind Tunnel Investigation of the Control of Wind-Induced Tall Building Vibrations By An Active Tuned Mass Damper*

**Abstract**

A wind tunnel investigation was conducted to investigate how effective an Active Tuned Mass Damper (ATMD) was in suppressing the wind-induced vibrations of tall buildings. the model ATMD and model building were developed at a 1:100 scale and tested in a wind tunnel simulating natural wind flow over a suburban terrain at a 1:100 scale. the ATMD model was developed so that parameters such as mass, stiffness and damping could be altered. Consequently, the ATMD was set up and tested with five various parameter configurations. the ATMD was driven by a sub-optimal control algorithm which used displacement feedback. the optimal phase shift of the control algorithm was determined to be at 110° which was used for all the wind tunnel tests. the wind tunnel investigations demonstrated that the ATMD was effective in reducing both the along-wind and cross-wind responses of the building.

**No. R701, October, 1994**

*Vodenitcharova, T and Ansourian, P**Influence of Boundary Conditions On the Bifurcation Static Instability of Circular Cylindrical Shells Subject To Uniform Lateral Pressure*

**Abstract**

This study is devoted to a theoretical investigation on buckling of complete cylindrical shells under uniform external lateral pressure loading.

The governing Flügge’s stability equations in coupled form are employed which are valid for all types of boundary conditions and lead to a great accuracy of the numerical results. the numerical process gives the buckling pressure for a selected circumferential buckling mode, material, shell geometry and boundary conditions.

The influence of seventeen different homogeneous boundary conditions placed on the displacements , and on the slope has been investigated.

A comparison is made with some analyses in the literature. the results indicate that the buckling pressure is in good agreement with that obtained by Donnell’s theory or by finite element analysis. It was found that, contrary to the widespread understanding that the critical pressure for a free cylinder is the same as for a ring, the present model obtains a slightly lower buckling pressure which depends on the length.

Cylindrical shells with geometric parameters in the wide range of L/R = 0.5-5 and R/h = 300-3000, are investigated, opposed to the seventeen types of boundary conditions.

**No. R702, November, 1994**

*Zhao, Xiao-Ling, and Hancock, GJ**Tests and Design of Butt Welds and Fillet Welds in Duragal Rhs Members*

**Abstract**

The report describes welded connection tests which were performed on DuraGal C450L0 Rectangular Hollow Sections. the thickness of the RHS sections varied from 1.6 mm to 3.0 mm. Butt welds, transverse fillet welds and longitudinal fillet welds were tested. the test configuration was the same as that used by the authors in a previous research project on welds in C350L0 RHS members. the test results are compared with existing Australian and American design formulae. the test results are also compared with the proposed design rules for welds in RHS members derived from the previous research project on C350L0 RHS members. the reliability analysis method is used to calibrate the existing and proposed design rules.

**No. R703, November, 1994**

*Pi, Yong-Lin and Trahair, NS**Three Dimensional Nonlinear Analysis of Elastic Arches*

**Abstract**

This paper presents a curved finite element model for the three dimensional nonlinear analysis of elastic arches. the model can be used in a structural analysis computer program in conjunction with the total Lagrange or the updated Lagrange procedure for the three dimensional large deflection and twist rotation analysis.

The model includes higher order curvatures which make the order of the bending strains consistent with that of the membrane strains, and the same low order polynomials are used for all the displacements. As a result, membrane locking problems are avoided. Neither selective reduced integration nor higher order interpolation polynomials are needed. No approximations are made for the twist rotations, and the nonlinear strains and the incremental stiffness matrix include the exact coupling due to the twist rotations. All signicant terms, some of which might be lost when improper approximations are made, are retained. Without these terms, superimposed rigid body motions may lead to overstiff solutions due to the development of internal strains. the model also includes warping of the cross section and the Wagner effect.

Application of the model to several numerical examples and comparison with the existing experimental and analytical results demonstrate that the model is much more effective and efficient than other methods in terms of accuracy, the number of elements needed for convergence, and the ability to pass maximum load points. the proposed model is also used to investigate the elastic flexural-torsional buckling and the elastic post-buckling behaviour of arches with different support conditions.