4. Analytical and numerical solution

Plate is one of the important elements which may influence the overall performance of the combined support. To date, a little attention was devoted by the industry to evaluate the performance of the plate in the roof and rib blot system. The strength of the head plate can be part of a roof or rib bolt system component. However roof bolting system that uses pre-tension for reinforcing purposes may not always achieve full encapsulation, the head plate is in fact a vital component of the system. Thus, at this stage, different and critical buckling modes in a steel plate with hole in the centre of the plate were presented. A number of researchers [18–22] theoretically and experimentally reported buckling of square plates with central circular holes subjected to compression in the plane. The presented theoretical methods by most of the researchers [18–21] calculate the critical loads in a plate with hole in the centre were extracted from the minimum energy method [23]. However, the available methods buckling and post-buckling solutions, which are mathematically discussed, are limited to small hole sizes, and are not able to study the effects of different boundary conditions of the plate on the strengths of buckling plates with holes of arbitrary size. By inventing powerful computers and advantaging from simulating Finite Element software, it is now possible to calculate the stresses of post-buckling for rectangular plates with any aspect ratio, any shapes and sizes of cuts, and in different boundary conditions. Figure 2 illustrates the applied boundary conditions and the applied pressure on the top and bottom edges. Initially, a fundamental method is presented for computing the compressive buckling load of a simply supported elastic rectangular plate having a central circular hole.

Subsequently, a three-dimensional Finite Element model is developed to comprehensively compute the critical buckling load in a rectangular plate having a central circular hole. The developed models will be extended to simulate the postbuckling analysis in the future research reports. An energy method to determine the buckling load of r rectangular plates of constant thickness under compressive loads is developed by Timoshenko [17]. A review of this derivation shows that the method is also applicable to plates of variable thickness. In this case, Timoshenko's integrals I1, I2 are [17]:

$$I\_1 = \iint\limits\_{\text{Surface}} D \left\{ \left( \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{x}^2} + \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{y}^2} \right)^2 - 2 \times (\mathbf{1} - \boldsymbol{\mu}) \times \left[ \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{x}^2} \times \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{y}^2} - \left( \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{x} \partial \mathbf{y}} \right)^2 \right] \right\} d\mathbf{x} d\mathbf{y} \tag{1}$$
 
$$\boldsymbol{\mu} = \left[ \left( \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{x}^2} + \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{y}^2} \right)^2 - 2 \left( \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{y}^2} + \frac{\partial^2 \boldsymbol{\alpha}}{\partial \mathbf{y}^2} \right) \right]$$

$$I\_2 = \iint\limits\_{\text{Surface}} h \left[ \frac{\sigma\_x}{S} \times \left( \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}} \right)^2 + \frac{\sigma\_\mathcal{V}}{S} \times \left( \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{y}} \right)^2 + 2 \times \frac{\tau\_{xy}}{S} \times \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}} \times \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{y}} \right] d\mathbf{x} d\mathbf{y} \tag{2}$$

where.

h : is the plate thickness (function of x and y), x, y : is the rectangular coordinates with origin at centre of plate and x-axis in direction of load, and ω : is the lateral deflection of plate.

$$D = \frac{E \times h^3}{12 \times (1 - \mu^2)}\tag{3}$$

Figure 2. A simply supported plate subjected to compression.

where D : is the flexural rigidity of plate (function of x and y); μ ¼ 0:3 : is the Poisson's ratio; σ<sup>x</sup> : is the tensile stress in x direction; σ<sup>y</sup> : is the tensile stress in y direction; τxy : is the shear stress; and S : is the tensile stress in x direction far from hole.

This study investigates the effect of holes on the critical buckling of the web plates of studs subjected to pure compressive load along the length of the plate. The web of the studs is considered to be simply supported along the edges that intersect with the flanges. The buckling analysis is performed using the commercially available finite element software ABAQUS/Standard. In the current research, the general purpose of three-dimensional, stress/displacement, reduced integration with hourglass control, shell element S4R (available in the ABAQUS/Standard element library), is considered to model the plates. S4R is involved in four nodes (quadrilateral), with all six active degrees of freedom per node. S4R allows transverse shear deformation, and the transverse shear becomes very small as the shell thickness decreases.

### 4.1 Simulation of the behaviour of the steel mesh under dynamic impact loading

A same trend to simulated dynamic behaviour of the steel mesh due the applied dynamic loading was taken into account. Thus, the mesh steel reinforcement sizes 20 mm diameter which they arranged by 100 mm distance centre to centre of the steel bars were tested under free fall of the dropped hammer. The same dropped hammer, which was used in the last section, was considered for the current simulation. It is indicated that a 110 kg hammer was dropped at a velocity of 0.2 m/s on top of the steel reinforcement.

### A New Concept to Numerically Evaluate the Performance of Yielding Support under Impulsive… DOI: http://dx.doi.org/10.5772/intechopen.79643

Figures 3 and 4 illustrates the simulated experimental set up which was used to simulate the structural behaviour of the steel mesh under impact loading. It should be noted that steel mesh can play a significant role as a part of the yielding support in a coal mine, as it can mitigate the effect of the destructive released kinetic energy due to a possible coal burst. In the coal mines, it was observed that both rock bolt and cable bolt might be losing the initial bond stiffness at the early stages of the applied dynamic loading due to the failure and separation of the anchored zone in the cable and rock bolt inside embedded coal. The anchorage length in a posttensioned member and the magnitude of the transverse forces (both tensile and compressive), that act perpendicular to the longitudinal prestressing force, depend on the magnitude of the prestressing force and on the size and position of the

#### Figure 3. The testing set up for simulating the behaviour of the steel mesh under impact loading.

Figure 4. Simulation of the behaviour of the steel mesh under dynamic impact loading.

anchorage hooks. Both single and multiple anchorages are commonly used in the coal mining. Prestressing force anchors transfer large forces to the coal in concentrated areas. Furthermore, coal is a very brittle material. This can cause localised bearing failure or split open the end of members. Thus, the steel mesh can considerably reduce the effect of the induced dynamic loading due to the coal burst. In the current simulation, the tensile stress for the steel mesh was fy = 500 MPa and the ultimate stress for the steel mesh fu = 700 MPa was taken into account. The postfailure of the steel mesh which may result in the rupturing of the steel bars was also defined. The ductile damage function was determined to simulate the post-failure of the steel mesh. Also, the rupturing strain εrupture = 0.3% was assumed. Weld properties of the steel mesh can also influence the overall deformation and energy absorption of the yielding support.

## 4.2 Simulation under dynamic shear loading

After calibrating the numerical models under static loading, the structural behaviour of the simulated models under dynamic loading was also studied. Since preparing the laboratory experiments to simulate the behaviour of cable bolts under dynamic loading is demanding, a validated and novel numerical simulation was developed. In order to simulate the behaviour of the cable bolts under impact loading, a 110 kg mass at velocity of 0.2 m/s was dropped on top of the concrete blocks. Figure 5 presents the structural behaviour of the cable bolts under impact loading. As illustrated, the momentum energy from the dropped mass would initially be transferred to the concrete surfaces. The transmitted energy due to the impulsive loading will reach the cable bolt. Figure 5 demonstrates the failure process of the cable bolt under the impact loading starting with the initial deformed shape followed by the brittle shear failure. The computation time was

Figure 5. Cable bolt under impact loading.

A New Concept to Numerically Evaluate the Performance of Yielding Support under Impulsive… DOI: http://dx.doi.org/10.5772/intechopen.79643

around t = 5e-3 seconds, which is very short to simulate the effect of the impulsive loading.
