and Nyx per unit length. As was done in. Section 2.3, the incremented forces are approximated by a truncated. Taylor expansion; these are denoted using single ...
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7 EFFECT OF IN-PLANE FORCES ON STATIC FLEXURE, DYNAMICS AND STABILITY
When a plate is subjected to in-plane forces, such forces have a stiffening or slackening effect on flexural behaviour and hence on the static deflections due to any applied transverse loading and also the flexural vibration frequencies. Further, a plate subjected to in-plane forces alone may become unstable in the flat configuration and undergo buckling. Design for stability is very important in current engineering practice because the availability of high-strength materials has led to the use of very thin plate structures. These aspects are discussed in this chapter.
7.1 GOVERNING EQUATIONS FOR COMBINED BENDING AND STRETCHING While deriving the equilibrium equations for the problem of flexure due to transverse loads alone (Section 2.3), the undeformed plate configuration was taken as the reference. To account for the effect of in-plane forces on flexural behaviour, one has to consider the equilibrium of the bent configuration. It should be noted that this will be done within the purview of the linear small deformation theory and with the assumption that the intensities of the in-plane forces at any point of the plate do not vary as the plate bends.
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dx 0
x
Nx
w, x
( w, x + w, xx dx)
z
N x*
0
N yx
x
Ny
N xy
Nx
N x*
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y
N *yx*
* N xy
N *y*
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( w, y + w, yx dx) * N xy
Fig.7.1 In-plane forces on a deformed plate element
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Consider a small element of the plate, of dimensions dx x dy, as shown in Fig.7.1, where attention is focussed on the effect of the internal inplane forces, namely, the normal forces Nx and Ny per unit length, and the shear forces Nxy and Nyx per unit length. As was done in Section 2.3, the incremented forces are approximated by a truncated Taylor expansion; these are denoted using single and double stars as
( )* = ( ) +
∂( ) dx ∂x
( )** = ( ) +
∂( ) dy ∂y
All the forces shown in Fig.7.1 are positive as per the sign convention adopted here. The plate element is in equilibrium under the action of these in-plane forces along with the transverse forces and moments (q, Qx , Qy , Mx , My and Mxy , and their incremented counterparts) as shown earlier in Fig.2.3. However, unlike in Fig.2.3, all the forces acting on the deformed element here have small inclinations with respect to the corresponding coordinate axes, as is clearly shown in Fig.7.1 for Nx and Nxy. These inclinations are also incremented using a truncated Taylor expansion in the x and y directions. For force equilibrium in the x-direction
( N x + N x, x dx)dy cos( w, x + w, xx dx) − N x dy cos w, x + [( N yx + N yx, y dy )dx − N yx dx]cos( w, x + w, xx
dx )=0 2
Putting cosθ ≈ 1 for small deformations, one obtains
N x, x + N yx, y = 0
(7.1)
While summing the above forces in the x-direction, the small contributions of Qx, Qx*, Qy, Qy*, which are all slightly inclined with respect to the z-direction, have been neglected. Similarly, by summing forces in the y-direction, one gets
N xy , x + N y , y = 0
(7.2)
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Satisfaction of moment equilibrium about z-axis yields, after neglect of higher-order terms,
N xy = N yx
(7.3)
For force equilibrium in the z-direction, in addition to transverse load q and the shear forces at the edges, one has to include the z-components of the in-plane forces. These additional quantities are: (a) ( N x + N x, x dx )dy sin( w, x + w, xx dx ) − N x dy sin w, x which, for small deformations (i.e. sin θ ≈ θ ), and after neglect of 2 the term involving dx dy, becomes ( N x w, xx + N x, x w, x )dxdy ; (b) ( N y w, yy + N y , y w, y )dxdy obtained similar to (a) above; (c) ( N xy + N xy , x dx)dy sin( w, y + w, xy dx) − N xy dy sin w, y which becomes ( N xy w, xy + N xy , x w, y )dxdy ; (d) ( N yx w, xy + N yx, y w, x )dxdy obtained similar to (c) above. Thus, due to the net additional effect of the in-plane forces, the z-direction equilibrium equation gets modified from Eqn.(2.14) to
Qx, x + Q y , y + q + N x w, xx + N y w, yy + 2 N xy w, xy + N x, x w, x + N y , y w, y + N xy, x w, y + N xy, y w, x = 0 By virtue of Eqns.(7.1) and (7.2), the coefficients of w,x and w,y turn out to be zero, and hence the above equation reduces to
Qx, x + Qy , y + q + N x w, xx + N y w, yy + 2 N xy w, xy = 0
(7.4)
The moment equilibrium equations with respect to the x and y axes are not affected by the in-plane forces because the contributions due to these forces are of higher-order and hence negligible compared to the other quantities considered in Section 2.3. For the sake of completeness, these two equations are reproduced below.
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M xy , x + M y , y − Qy = 0
(7.5)
M x, x + M xy , y − Qx = 0
(7.6)
Eqns.(7.1)-(7.3) pertain to the in-plane problem, i.e. the problem of determining the internal in-plane force field (Nx , Ny , Nxy). This problem can be solved once the in-plane tractions on the edges of the plate are specified; this is done without considering the transverse deformation at all because the in-plane force field is taken to be unaffected by it. Once the in-plane force field is obtained, its influence on the flexural behaviour of the plate is accounted for by solving Eqns.(7.4)-(7.6). Thus, the present small deformation formulation accounts for just one-way coupling between the in-plane forces and transverse deflections. (In contrast, for a problem involving finite deformations, bending is accompanied by in-plane stretching and the in-plane forces increase as the plate bends more and more, thus leading to a nonlinear theory. This will be discussed in Chapter 14). For a rectangular plate (0 ≤ x ≤ a, 0 ≤ y ≤ b) with (a) uniform Nx applied along x=0 and x=a, or (b) uniform Ny applied along y=0 and y=b, or (c) uniform Nxy applied along all the four edges, or (d) any combination of (a)-(c), the solution of Eqns.(7.1)-(7.3) is simply a uniform internal force field over the entire area; only such cases are considered in the present chapter. (Varying internal force fields due to non-uniform edge loading will be considered later in Chapter 13.) The bending problem can be reduced, as was done in Section 2.4, to a single governing equation in w by eliminating Qx and Qy from Eqns.(7.4)-(7.6) and by using the moment-curvature relations. This final governing equation is given by
D∇ 4 w = q + N x w, xx + N y w, yy + 2 N xy w, xy
(7.7)
and is the mathematical statement of the equilibrium of the deflected plate due to any combination of transverse and in-plane forces.
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When the in-plane forces are absent, the above equation reduces to the biharmonic equation derived earlier. When in-plane forces are present besides transverse loads, the equation accounts for the stiffening or slackening influence of the in-plane forces, and yields correspondingly lower or higher deflections. When in-plane forces alone are present, a deflected equilibrium configuration is possible only if the plate has buckled, and hence the governing equation pertains to the buckling problem. While the equilibrium configurations corresponding to the first two cases mentioned above (i.e. q alone, and q along with Nx , Ny and Nxy)
are stable equilibrium configurations, the last case (Nx , Ny and Nxy alone) is one of neutral equilibrium. The governing equation is nonhomogeneous as long as q is present yielding a unique solution (equilibrium problem), while it becomes homogeneous when q is zero, in which case w is indeterminate and is non-trivial only for certain specific values of the in-plane forces (eigenvalue problem). 7.2 ANALYSIS FOR STABILITY We shall start with the stability problem and illustrate the application of Navier and Levy methods to determine critical loads for certain rectangular plate problems. EXAMPLE 7.1:
BUCKLING OF A SIMPLY (NAVIER’S METHOD)
SUPPORTED
PLATE
Let the plate be subjected to biaxial compression with uniform compressive force Nx along the edges x=0 and x=a, and uniform compressive force Ny along the edges y=0 and y=b. Let Nxy be absent. (If Nxy is present, the problem can be solved only by an approximate method - see Chapter 8). Nx and Ny are increased simultaneously, keeping their relative proportion constant, till buckling occurs. Let the critical values be Nxcr and Nycr and let
N xcr = − k1Pcr N ycr = − k2 Pcr
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The conditions
Wm = Wm, y = 0 at y = ±b/2 lead to
Am cosh
αb
αAm sinh
2
+ Cm cos
αb
βb 2
− βCm sin
=0
βb
=0 2 2 αb βb Bm sinh + Dm sin =0 2 2 αb βb =0 αBm cosh + βDm cos 2 2 where the decoupling of the y-symmetric and the y-antisymmetric buckled mode shapes can be seen. Subsequent steps are similar to those of the vibration problem in Example 6.2; thus, one obtains the values of Pcr corresponding to the roots of the characteristic equation. The lowest Pcr and the corresponding mode shape have to be identified only by trial and error (see Problem 3).
7.3 STATIC FLEXURE The analysis for static deflections due to combined in-plane and transverse loading is illustrated below with reference to a simple problem. EXAMPLE 7.3: COMBINED BENDING AND STRETCHING SIMPLY SUPPORTED PLATE
OF
A
The simplest problem in this category is that of a uniaxially stressed simply supported plate subjected to transverse sinusoidal loading
q = q11 sin
πx a
sin
πy b
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Taking a=b and the in-plane force to solution for Eqn.(7.7) can be taken as
w = W11 sin
πx a
sin
be
Nx, the
πy a
to yield
W11 =
2
q11
π 4π 2 D a 2 a 2
+ Nx
Expressed in terms of the counterpart without the inplane force, the above equation becomes
wmax =
wmax without N x N xa2 1 + 2 4π D
Noting that the uniaxial compressive buckling load Pcr for the square plate is expressed as
wmax =
4π 2 D a2
, the above result can be
wmax without N x Nx 1 + P cr
This equation clearly shows that the influence of the in-plane force on the transverse deformation of the plate - stiffening it when tensile and slackening it when compressive would be significant if its magnitude is a fairly large fraction of the corresponding critical value. Though this has been arrived at with reference to the above simple problem, this is in general true for other in-plane force fields and transverse loads.
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