Prostaglandin E2 manufacturer Recently Asik and Tezcan published a

Recently, Asik and Tezcan [12] published a mathematical model of laminated glass beams, which is based on nonlinear strain–displacement relationship. The model was used to investigate the linear and nonlinear behavior of symmetric triplex glass beams in comparison with LG plate’s behavior. Ivanov [13] presented a finite Prostaglandin E2 manufacturer model for LG beams. On his model, the distribution of strain and stress through the beam thickness and along its axis was obtained as a result of linear finite element analysis. He developed a mathematical model of triples glass beam, consisting of a bending curvature differential equation and a differential equation of PVB-interlayer shear interaction.
The objectives of the this paper can be summarized as following:
A higher order finite element method (FEM) with nine nodes quadrilateral element was used in the analyses [14,15]. This FEM is designed for large deflections and rotation analysis of LG considering the effect of interlayer material under a different range of temperatures. The results obtained from the experimental study are compared with the corresponding FEM results and pronounced conclusions are obtained. El-Shami et al. [16] studied the structural behavior of glass plates other than rectangular shapes. They used a higher-order finite element model to analyze several examples with trapezoidal, rectangular, triangular, and hexagonal shaped glass plates (monolithic and LG).

Mathematical model
Since the theory of mindlin plate and solution techniques have previously been published in textbooks e.g., Cook et al. [17] and in papers e.g., Vallabhan et al. [9] and El-Shami et al. [16], only a brief account of the theory is presented here. The effect of interlayer material, on the laminated plate element is considered as shear strains and (Fig 1). The values of and can be calculated as:where and are the displacements in the x and y directions, for the upper and lower plates; respectively, w, and w, are the first derivatives of deflection with respect to x and y; respectively and t1,t2, and h are the thicknesses of the upper and lower plates and the interlayer; respectively. The degrees of freedom will be increased by two for each node. Then the total number of degrees of freedom will be 63 per element; 7 d.o.f per node. To calculate the stiffness matrix, the relation between the displacements and the strains should be calculated. This relation is called [B] matrix [18]. This matrix can be calculated as;where denotes the shape function, and denotes the displacement vector for ith node in which i=1,2,…,9. Similarly,Then,
The stiffness matrix for the interlayer can be calculated as:where [] denotes the 63×63 interlayer stiffness matrix, denotes the modulus of rigidity for the interlayer, and [I] denotes the identity matrix.
Because LG consists of two plates, this solution employs two different linear membrane stiffness matrices, two different nonlinear membrane stiffness matrices, and one linear bending stiffness matrix. The nonlinear stiffness matrix will be modified due to the fact that there are two plates. The total stiffness matrix for plates can be written as:where [1] and [2] denote the linear membrane stiffness matrices for the upper and lower plate; respectively. [1] and denote the nonlinear membrane stiffness matrices. [] denotes the linear bending stiffness matrix and [] denotes the nonlinear stiffness matrix due to the membrane actions, i.e., the initial stress matrix [18].
here is calculated in relation to , where is given as;
Consequently, the authors must use a special transformation matrix [T] to make the plate stiffness matrix match with the displacement vector U , then
The final stiffness matrix [K] for LG takes the following form:

Experimental procedure
Glass Research and Testing Laboratory (GRTL) staff tested different LG geometries, mainly two groups. Table 1 summarizes the geometries of group A specimens. The specimens have a PVB interlayer thickness of 0.762mm. All the specimens were tested within a temperature range of 20–60°C. The specimens of group B have a HG/MD interlayer thickness of 2.67mm. Only one geometry was used as shown in Table 2. All specimens were tested under a temperature of 35°C. A special type of strain gauge rosettes (CEA-060-062UR-350) was used. The strain gauges were temperature compensating and had an operating range of −75 to +175°C. M-Bond AE 10 epoxy adhesive, with an operating range between −195 and 95°C, bonded the gauges to the glass. The strain gauges were connected to the LG at the center and at the corner (50mm from the edge). The test chamber, constructed of a 12.7mm steel plate with four 203mm steel channels welded onto it to form an open rectangular box, provided four sides support for the specimen. This test chamber provided support condition corresponding to ASTM E998 [19]. The steel chamber had four openings connected to the vacuum machine. As the vacuum was applied, the plate was loaded with negative pressure. The heating system consists of heating chamber, space heaters, heated blowers, and appropriate ductwork to distribute warm air over the surfaces of the test specimen. Four space heaters were used to warm the air inside the heating box. A heated blower was used to blow the pre-heated air (into the heated box) to the heating chamber. To start the proceeding, The GRTL staff removed the specimen from the crate. Then, they measured and record the thickness of the glass specimen at different locations and obtain an average thickness. The staff mounted the spacemen on the test frame. To control the temperature, GRTL staff mounted the heating chamber over the test chamber containing the glass specimen. They connected the heating box to the chamber and the heated blower to the box to start the heating processes. Two small holes with sliding covers, one located near to the center and another near the top of the heating chamber, allowed the staff to measure temperature. The hot air blown on the glass surface allowed the glass to reach the required temperature. Then, the researchers loaded the specimen to one-half the design load and the measurements were recorded. Finally they loaded the specimen to the design load and record the measurements. They repeated this procedure at different temperatures. Once the staff completed the test, they removed the specimen from the test chamber. Fig. 2 shows a sketch for the experimental setup.