Ulf Arne Girhammar

Boundary-Layer Effect in Composite Beams with Interlayer Slip

An apparent analytical peculiarity or paradox in the bending behavior of elastic-composite beams with interlayer slip, sandwich beams, or other similar problems subjected to boundary moments exists. For a fully composite beam subjected to such end moments, the partial composite model will render a nonvanishing uniform value for the normal force in the individual subelement. This is from a formal mathematical point of view in apparent contradiction with the boundary conditions, in which the normal force in the individual subelement usually is assumed to vanish at the extremity of the beam. This mathematical paradox can be explained with the concept of boundary layer. The bending of the partially composite beam expressed in dimensionless form depends only on one structural parameter related to the stiffness of the connection between the two subelements. An asymptotic method is used to characterize the normal force and the bending moment in the individual subelement to this dimensionless connection parameter. The outer expansion that is valid away from the boundary and the inner expansion valid within the layer adjacent to the boundary (beam extremity) are analytically given. The inner and outer expansions are matched by using Prandtl’s matching condition over a region located at the edge of the boundary layer. The thickness of the boundary layer is the inverse of the dimensionless connection parameter. Finite-element results confirm the analytical results and the sensitivity of the bending solution to the mesh density, especially in the edge zone with stress gradient. Finally, composite beams with interlayer slip can be treated in the same manner as nonlocal elastic beams. The fundamental differential equation appearing in the constitutive law associated with the partial-composite action in a nonlocal elasticity framework is discussed. Such an integral formulation of the constitutive equation encompassing the behavior of the whole of the beam allows the investigation of the mechanical problem with the boundary-element method.

Lateral-Torsional Buckling of Partially Composite Horizontally Layered or Sandwich-Type Beams under Uniform Moment

This paper is devoted to the analytical and numerical modeling of the lateral-torsional stability of horizontally layered composite beams. Composite beams are classified as horizontally layered beams with interlayer slip or sandwich beams with a weak shear core. The governing differential equations of the out-of-plane behavior of horizontally layered composite beams are supported by variational arguments. In the theoretical analysis, a distinction is made between the influence of the shear connection at the interface with respect to the in-plane or transversal deformations and to the out-of-plane or lateral deformations, respectively. Some engineering results are presented for a partially composite beam under pure bending moment. In the case of noncomposite in-plane action (orthotropic connection), a simple closed-form solution is derived for the lateral-torsional buckling moment, and it is shown that the exact dimensionless buckling moment depends only on two structural parameters for beams composed of two identical subelements. The results are analogous to those obtained for the in-plane buckling of partially composite or sandwich-type beams, where the buckling moment increases with the stiffness of the shear connection. Prandtl’s valid solution for lateral-torsional buckling of ordinary beams is also found for composite beams in the case of noncomposite action in both the transversal and lateral directions. A generalization of Prandtl’s valid solution for composite beams with partial composite action in the lateral direction and noncomposite action in the transversal direction is derived. It is shown that the lateral-torsional buckling formulas are strongly affected by the kinematics of the connected shear layer. Also, the lateral-torsional buckling of partially composite beams with both in-plane and out-of-plane slip behavior is analyzed using the Rayleigh-Ritz method. This mathematical problem leads to a system of differential equations with nonuniform coefficients. An approximated solution is derived for the isotropic connection with isotropic noncomposite actions, whereas an exact solution is presented for the orthotropic connection with noncomposite in-plane action. Finally, the Rayleigh-Ritz approach is compared with some numerical results associated with the exact resolution of the differential equations with nonuniform coefficients. The Rayleigh-Ritz approach appears to be efficient to capture the main phenomena, including the nonmonotonic dependence of the buckling load to the connection parameter.

Fracture Mechanics Models for Brittle Failure of Bottom Rails due to Uplift in Timber Frame Shear Walls

In partially anchored timber frame shear walls, hold-down devices are not provided; hence the uplift forces are transferred by the fasteners of the sheathing-to-framing joints into the bottom rail and via anchor bolts from the bottom rail into the foundation. Since the force in the anchor bolts and the sheathing-to-framing joints do not act in the same vertical plane, the bottom rail is subjected to tensile stresses perpendicular to the grain and splitting of the bottom rail may occur. This paper presents simple analytical models based on fracture mechanics for the analysis of such bottom rails. An existing model is reviewed and several alternative models are derived and compared qualitatively and with experimental data. It is concluded that several of the fracture mechanics models lead to failure load predictions which seem in sufficiently good agreement with the experimental results to justify their application in practical design.

On the Shear Buckling of Clamped Narrow Rectangular Orthotropic Plates

This paper deals with stability analysis of clamped rectangular orthotropic thin plates subjected to uniformly distributed shear load around the edges. Due to the nature of this problem, it is impossible to present mathematically exact analytical solution for the governing differential equations. Consequently, all existing studies in the literature have been performed by means of different numerical approaches. Here, a closed-form approach is presented for simple and fast prediction of the critical buckling load of clamped narrow rectangular orthotropic thin plates. Next, a practical modification factor is proposed to extend the validity of the obtained results for a wide range of plate aspect ratios. To demonstrate the efficiency and reliability of the proposed closed-form formulas, an accurate computational code is developed based on the classical plate theory (CPT) by means of differential quadrature method (DQM) for comparison purposes. Moreover, several finite element (FE) simulations are performed via ANSYS software. It is shown that simplicity, high accuracy, and rapid prediction of the critical load for different values of the plate aspect ratio and for a wide range of effective geometric and mechanical parameters are the main advantages of the proposed closed-form formulas over other existing studies in the literature for the same problem.

Influence of Geometrical Parameters on Behaviour of Reticulated Timber Domes

Domes are very efficient structural systems for long clear span buildings. The introduction of laminated timber highlighted the economic advantages of this material and led to the use of timber domes even for very large spans. In this paper, reticulated timber domes of triangular network shape with decking and bottom tension ring are considered. These types of domes have high stiffness in all directions along the surface and are kinematically stable. The dome is subjected to uniformly distributed load over entire structure. The dome model is generated with a preprocessor program called DOME-IN and analysed with ABAQUS. The focus of this paper is to evaluate the behaviour of reticulated timber domes with respect to: (a) height-diameter ratio of the dome, H/D; (b) straight relative curved types of main beams in the triangular networks versus mesh density; (c) perpendicular relative parallel types of purlin arrangements in the triangular networks versus the number of purlins; and (d) mesh density of main network members or size of triangular networks. The influence of these parameters on relative maximum deflection of the dome, relative critical pressure for global buckling of the dome, relative maximum axial forces and relative maximum bending moments in the network members, is evaluated. It is found that: (a) the optimum height-diameter ratio is H/D ≈ 0.3; (b) that straight beams in the triangular networks are preferred to curved beams, especially for higher mesh densities;(c) that purlins perpendicular to the latitudinal beams are preferable to purlins parallel to the latitudinal beams if there is a decking that provides bracing to the system; and (d) mesh densities with six sectors (n = 6) (lowest number of practical interest) and as many divisions of the arc beam and of the sector of the ring beam (m = k > 4), which is necessary in order to meet the design criteria for the dome, can be recommended.

Tests and Analyses of Slotted-In Steel-Plate Connections in Composite Timber Shear Wall Panels

The authors present an experimental and analytical study of slotted-in connections for joining walls in the Masonite flexible building (MFB) system. These connections are used for splicing wall elements and for tying down uplifting forces and resisting horizontal shear forces in stabilizing walls. The connection plates are inserted in a perimeter slot in the PlyBoard™ panel (a composite laminated wood panel) and fixed mechanically with screw fasteners. The load-bearing capacity of the slotted-in connection is determined experimentally and derived analytically for different failure modes. The test results show ductile postpeak load-slip characteristics, indicating that a plastic design method can be applied to calculate the horizontal load-bearing capacity of this type of shear walls.

A Comparison of Exact and Approximate Analyses of Partially Interacting Composite Beam-Columns

Solutions of the static Euler-Bernoulli equations of composite beam-columns with interlayer slip have been compared with an approximate theory. The inter-layer force was taken to be proportional to ...

Brittle Failures in Timber Beams Loaded Perpendicular to Grain by Connections

AbstractA state-of-the-art review of simple analytical fracture mechanics models for calculation of the splitting capacity of timber beams loaded perpendicular to the grain direction by connections is presented. It is shown that most of the already available models are closely related and appear naturally as special cases of the most general model available. A new model, which is a semiempirical extension of an existing model based on a beam-on-elastic-foundation theory, is proposed. The so-called van der Put model, which forms the theoretical basis for the splitting equations used in the European and Canadian timber design codes, appears as a special case of the proposed model. The treatment of the splitting problem in some major timber design codes is reviewed and discussed based on the theoretical models and new test results. The approach used in the European timber design code where the maximum shear force on either side of a connection is considered rather than the total load applied on a connection ...

A complete timber building system for multi-storey buildings

The Masonite Flexible Building (MFB) system is a complete timber building system for commercial and residential multi-storey houses. The system is for tall and large buildings with long floor spans. The MFB system uses prefabricated wall, floor and roof elements which are delivered in flat packages and erected on the construction site. The MFB system might be classified as a panel construction, where the load-carrying structure consists of composite lightweight timber I-beams mechanically integrated with a composite laminated wood panel called PlyBoard T. The I-beams and the panel form a strong and rigid carcass for wall and floor elements, making the system well suited for high rise construction. A key feature of the MFB system is the connection technique which enables swift erection of the system units on site. The PlyBoard T panels are provided with a continuous slot along the periphery. The slot is used as a general connection interface for the joining of the wall elements. The floor elements are suspended and hooked onto the bearing walls using sheet steel hangers, allowing swift assembling of the floor deck and enabling direct vertical wall-to-wall load transfer parallel to grain. The paper presents the construction principles, system components and units, erection technique, functional and architectural aspects of the Masonite Building System.

Effect of Ring Beam Stiffness on Behaviour of Reticulated Timber Domes

Domes are efficient structural systems for long clear-span buildings. The introduction of laminated timber highlighted the economic advantages of this material and led to the use of timber domes even for very large spans. In this paper, reticulated timber domes of triangular network shape with decking and bottom tension ring are considered. These types of domes have high stiffness in all directions along the surface and are kinematically stable. The dome is subjected to uniformly distributed load over the entire structure. The dome model is generated with a preprocessor program called DOME-IN and analysed with ABAQUS. The focus of this paper is to evaluate the behaviour of reticulated timber domes with respect to different stiffnesses of the bottom ring beam, here defined as a non-dimensional ring beam area parameter Ar*, which is shown to be a very well adapted design parameter for the ring beam. As far as global buckling is concerned, the critical pressure is sensitive to the bottom ring beam stiffness only if the latter is within a certain range. In terms of design, the stiffness of the ring beam should exceed A* > 2 in order to utilise the full buckling load capacity of the dome system itself. The maximum deflection, normal forces and bending moments versus the ring beam area parameter are also evaluated. The maximum values of the deflection and the internal actions next to the bottom ring are very sensitive to the bottom ring beam stiffness only if the latter is less than about Ar* < 10. A recommended value for the design of the bottom ring beam is A* > 20.