Interfacial stresses in plated beams
Introduction
An existing beam can be retrofitted by bonding a fibre-reinforced plastic (FRP) or steel plate to its soffit (Fig. 1). This plate bonding technique has been used widely to retrofit reinforced concrete (RC) beams, and has also been used to retrofit beams of other materials. The technique has numerous advantages such as increasing the strength and stiffness of an existing beam with minimal interference to the surrounding environment. Consequently, many studies have been carried out on the behaviour and strength of such plated beams (e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]). In such retrofitted beams, debonding of the soffit plate from the beam is an important failure mode as it prevents the full ultimate flexural capacity of the retrofitted beam from being achieved. It is thus important to be able to predict the debonding failure load. Debonding failures depend largely on the interfacial shear and normal stresses between the beam and the bonded plate.
The determination of interfacial stresses has thus been researched for the last decade for beams bonded with either steel or FRP plates. In particular, several relatively simple approximate closed-form solutions for interfacial stresses have been developed [2], [3], [4], [9], [14], [16] based on a simple assumption for the adhesive layer as discussed later. Despite all of these studies, one striking fact is that the relationship between these existing solutions has not been established clearly in the existing literature. This paper therefore starts with a review of these existing solutions, identifying their assumptions and limitations, thereby clarifying the differences between them. This review also establishes the need for a more accurate solution of the same type, which is subsequently presented in the paper. This new solution is intended for application to beams made of all kinds of materials bonded with a thin plate, while all existing solutions have generally been developed focusing on the strengthening of RC beams, which allowed the omission of certain terms. Finally, numerical comparisons between the existing solutions and the present new solution enable a clear appreciation of the effects of various parameters.
It should be noted that the type of analysis discussed in this paper does not satisfy the zero shear stress condition at the end of the adhesive layer. This drawback is known to have only a limited effect in a very small zone near the end of the plate [3]. For this condition to be satisfied, a higher-order analysis has to be carried out. The first such analysis has just appeared [20]. This higher-order solution however does not provide explicit expressions for the interfacial stresses, so numerical results are not easily obtainable, which makes it difficult for exploitation in developing a design rule. The correctness of this analysis has also been questioned [21]. In another higher-order analysis from the authors' group yet to be published [21], explicit expressions have been obtained, but these are much more complex than the expressions from the type of analysis discussed in this paper. The simple approximate closed-form solutions discussed in this paper provide a useful but simple tool for understanding the interfacial behaviour and for exploitation in developing a design rule. The present paper is concerned only with this simple type of analysis and, hereafter, no further reference to higher-order interfacial stress analysis is made.
Section snippets
Assumptions and approaches
All existing solutions are for linear elastic materials only. The key assumption in all of these solutions is that the adhesive layer is subject to shear and normal stresses that are constant across the thickness of the adhesive layer. It is this key assumption that enables relatively simple closed-form solutions to be obtained, although this assumption is somewhat hidden in some of the solutions.
In the existing solutions, two different approaches have been employed. Roberts [3] and Roberts and
Assumptions of the new solution
The derivation of the new solution below is described in terms of adherends 1 and 2, where adherend 1 is the beam and adherend 2 is the soffit plate. Adherend 2 can be either steel or FRP but not limited to these two. The assumptions adopted in the present solution are summarised below.
Linear elastic behaviour of adherends 1 and 2, as well as of the adhesive layer, is assumed. Deformations of adherends 1 and 2 are due to bending moments, axial and shear forces. The adhesive layer is assumed to
Interfacial normal stress: governing differential equation
The governing differential equation for the interfacial normal stress is derived in this section. When the beam is loaded, vertical separation occurs between adherends 1 and 2. This separation creates an interfacial normal stress in the adhesive layer. The normal stress, σ(x), is given aswhere v1(x) and v2(x) are the vertical displacements of adherends 1 and 2, respectively. The equilibrium of adherends 1 and 2, neglecting second-order terms, leads to the following
General solutions for the interfacial shear and normal stresses
The governing differential equations for the interfacial shear and normal stresses [, ] are coupled and hence a solution is not easily found. To uncouple the equations, the effects of shear deformations in both adherends are now neglected. The governing differential equation for the interfacial shear stress then reduces toFor simplicity, the general solutions presented below are limited to loading which is
Application of boundary conditions
Having derived the general solutions for the interfacial shear and normal stresses, three load cases are now considered. A simply supported beam is investigated which is subjected to a uniformly distributed load, an arbitrarily positioned single point load, and two symmetric point loads as shown in Fig. 3. This section derives the expressions of the interfacial shear and normal stresses for each load case by applying suitable boundary conditions.
Comparison of analytical solutions
A comparison of the interfacial shear and normal stresses from the different closed-form solutions reviewed earlier is undertaken in this section. Two example problems are considered. The first is an RC beam bonded with a glass-fibre-reinforced plastic (GFRP), CFRP or steel soffit plate. The second is a hollow aluminium (AL) beam with a bonded CFRP soffit plate. In both examples, the beams are simply supported and subjected to a central point load or a uniformly distributed load. A summary of
Conclusions
This paper has been concerned with the prediction of interfacial shear and normal stresses in beams strengthened by an externally bonded plate. Such interfacial stresses provide the basis for understanding debonding failures in such beams and for the development of suitable design rules. Six existing approximate closed-form solutions for interfacial stresses in such beams have been reviewed, identifying their assumptions and limitations, thereby clarifying the differences between these
Acknowledgements
Both authors wish to thank The Hong Kong Polytechnic University for the provision of a postdoctoral fellowship to the first author.
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