Effects of Forces Applied in the Middle Plane on Bending of Medium-Thickness Band

This paper examines the problem of calculating a plate of medium thickness uploaded by transverse loads and forces in the middle plane. We use variant equations from the theory of medium thickness plates as inputs, as proposed by B.F. Vlasov. The paper also considers the presence of tensile or compressive forces in the plane of the plate. The paper offers an example of calculation for a semi-infinite uploaded at the end of the transverse load and pivotally supported on the longitudinal edges. It is demonstrated that in the middle plane, the forces significantly influence the plate’s deflections and bending forces. There are material differences between results obtained with the theory of plates of medium thickness and those of the classical the theory of bending of plates.


Introduction
A rectangular plate with various conditions on the contour, installed on an elastic base, is a widely used element in building structures.In certain cases, when calculating plates, particularly ones that rest on an elastic base, one needs to factor in some forces applied to the middle plane, in addition to the transverse loads.Such extra forces can be caused by seasonal and daily fluctuations of temperature, pre-tensed reinforcement steel, impacts from production equipment, and pressure of the enclosing walls on the foundation slab.
The stressed and deformed state of a medium thickness plate bending from the action of a transverse load, is described with these equations, as proposed by B.F. Vlasov [ .(1) Here, D is the cylindrical rigidity of the plate, h is its thickness.The deflection w(x, y) and the rotation angles of the plate t x (x,y), t y (x,y) are associated with desired functions t(x,y), ) , ( y x M as relations: where G -the shear modulus, Q -Poisson's ratio.
We can write the following dependence: , 2

Analytical method of solving
Let us assume that the longitudinal side edges of the plate have a hinged support (Fig. 2).N y q(y)

Fig. 2. Statement of the problem of calculation of the plate
Then our desired functions can be represented as: We substitute decomposition (9) in equation ( 8), due to the orthogonality of trigonometric functions, for each of n get an ordinary differential equation of the fourth order, which in the case of only longitudinal forces N x will appear as: and in the case of longitudinal forces N y : , where Equations ( 11) and ( 12) can be represented using the standard form, typical for a problem of the bending of the beam located on an elastic base: Depending on the relation of the coefficient values r n and s n we find the type of roots of the respective characteristic equations, and therefore the type of particular integrals for its solution.
First examine a case of the plate under the action of forces N y , distributed evenly over its longitudinal edges (Fig. 2).By comparing equations ( 12) and ( 13) we can see that in this case, the coefficients of equation ( 13) will be values: For n n r s ! the roots k n of the characteristic equation is determined by the formula: where 2 while the general solution of differential equation ( 11) is written as: ) If the plate is under the action of forces N x coefficients of equation ( 13) appear: From the relations (20) we can see that this time, in the case of stretching plate ) 0 ( ! x N n n r s , and in the case of compression plate ) 0 ( x N n n r s ! ,the solutions to the problem will be represented respectively as expressions ( 19) or (17).
Us now examine the problem of the bending of a semi-infinite plate under the action of transverse load q(y) in its initial section and longitudinal compressing forces N y , evenly distributed over its longitudinal edges (Fig. 2).
In this case, if the transverse load is represented as y q q y E cos 0 , where b / S E , then even first term of decomposition ( 9) present an accurate solution, and so index n can be omitted everywhere.Assuming the plate's infinite length, solution to equation ( 13) can be written as: (21) To completely solve the problem of the bending of a medium-thickness plate, we also need to add to expression (19) the integral of the second equation from (1), which in the light of decomposition of (10) and the infinite length of the plate, can be written as: , where To find the three integration constants C 1 , C 2 , D 1 at the initial section (x = 0), one can set the following boundary conditions: Revealing these conditions using equations (21) and ( 22), we obtain a system of three algebraic equations: . 2 .
(25) To find the constants of integration, boundary conditions (23) and expressions ( 22) and (25) allow to obtain the following equation: .
Having solved the equation systems ( 24) and ( 26) and defining the constants C 1 , C 2 , D 1 , we can determine our calculated values using the formulas of the theory of plate middlethickness [1].For example, the deflection w(x,y) and the bending moment M y (x, y) are determine as: (27) where function W(x) that is part of ( 27) and (28) appears as (21) when the plate is compressed, and as (25) when the plate is stretched.In the diagram, solid lines 1 and 2 relate respectively to the cases compressed and stretched of the plate middle-thickness for h/b = 0.2, while dotted lines 3 and 4 describe the cases for the thin compressed and stretched plate.

Conclusions
Can see that an increase in longitudinal force N y causes material changes of deflection and bending moments: they are higher in the case of compression, and lower in the case of stretching.Expressed as percentage, deflections vary more than bending moments.It should also be noted that the effect of longitudinal forces on the plate middle-thickness is somewhat stronger than on a thin plate, and it is greater with compression than with stretching.Lines 1 and 3, nearly straight at low values of N y , become curves tending to rise abruptly as N y increases, approaching their respective critical values.

) 2 Fig. 1 .
Fig. 1.Efforts in the middle plane of the plate Notably, such forces are interrelated with equations of equilibrium: .0 , 0 w w relations between which depend on the sign of forces N y : in the case of stretching plate )

Fig. 3 2 Fig. 3 .
Fig. 3 has graphs representing the dependency of dimensionless deflection 3 0 0 / ) 0 , 0 ( b p D w w in the middle of the plate's initial section on dimensionless longitudinal forces D b N N y y / 2

Fig. 4 Fig. 4 .
Fig. 4. Diagrams of dependence of the bending moments on longitudinal forces )