BENDING OF A PLATE SUPPORTED BY ELASTIC I-BEAMS UNDER HEATING

M. G. PINSKII

(Presented by the scientific seminar of the Department of Strength of Materials).

Consider a plate reinforced on opposite sides by I-beams of identical stiffness; the other two sides are freely supported.

The problem of the bending of a square plate supported along its edges by elastic beams was studied by K. K. Chalyshev [1], S. P. Timoshenko [2], B. G. Galerkin [3], E. Müller [4], and A. S. Kalmanok [9]. In 1953, Fuchs [5] and [8] investigated the bending of a rectangular plate on elastic supports.

In all these works, the case of the connection of the plate with the beam was investigated in which the plate lies on the beams (Fig. 1a).

Fig. 1

We shall consider the case of a connection with the plate arbitrary over the height of the beam (Fig. 2).

Under the action of a load, the plate is deformed (Fig. 3). Denoting the displacements in the directions of the axes \(x\), \(y\), respectively, by \(u\), \(v\), \(w\), from Fig. 3 one may write the following relations:

\[ u(z)=u_s-z\,\frac{\partial w_s}{\partial x}, \]

\[ v(z)=v_s-z\,\frac{\partial w_s}{\partial y}, \tag{1} \]

where \(u_s\), \(v_s\), and \(w_s\) are the displacements in the middle plane of the plate.

Fig. 2

Fig. 3

From Cauchy’s equations it follows that:

\[ \varepsilon_{xx}=\frac{\partial u}{\partial x}=(\varepsilon_{xx})_s-z\frac{\partial^2 w_s}{\partial x^2}, \]

\[ \varepsilon_{yy}=\frac{\partial v}{\partial y}=(\varepsilon_{yy})_s-z\frac{\partial^2 w_s}{\partial y^2}, \tag{2} \]

\[ \varepsilon_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} =(\varepsilon_{xy})_s-2z\frac{\partial^2 w_s}{\partial x\partial y}. \]

Fig. 4

Considering the equilibrium of an infinitesimally small element of the plate, we introduce into consideration the forces \(N_x\), \(N_y\), and \(N_{xy}\) [2, p. 90], referred to unit length (Fig. 4). The equation of bending of the plate has the form [2]:

\[ \Delta\Delta w=\frac{q}{D}, \tag{3} \]

where

\[ \Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \]

is the Laplace operator.

Projecting the forces lying in the middle plane onto the \(x\)- and \(y\)-axes and assuming that there are no body forces in these directions, we obtain the following equilibrium equation

\[ \frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=0, \]

\[ \frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}=0. \tag{4} \]

Equations (4) are analogous to the differential equations of equilibrium in the plane problem of the theory of elasticity [6]; consequently, equations (4) are identically satisfied if one sets

\[ N_x=\frac{\partial^2\varphi}{\partial y^2}, \]

\[ N_y=\frac{\partial^2\varphi}{\partial x^2}, \tag{5} \]

\[ N_{xy}=-\frac{\partial^2\varphi}{\partial x\partial y}, \]

where \(\varphi\) is the stress function, determined by the biharmonic equation

\[ \Delta\Delta\varphi=0. \tag{6} \]

Thus, the stressed state in the plate is characterized by equations (3) and (6), from which \(w\) and \(\varphi\) can be determined. The distance of the beam axis from the middle plane of the plate is \(e\) (Fig. 2). We assume that the beams resist bending in the vertical plane, without resisting tor-

tion and bending in the horizontal plane. In this respect the solution is approximate. Strictly speaking, the bending-torsional deformation of the beam should have been taken into account, which would have led to very complicated calculations.

At each edge \(x=\pm \dfrac{b}{2}\) one can write the following boundary conditions, proceeding from our assumptions (Fig. 5):

\[ \text{1. } M_x=0,\qquad \frac{\partial^2 w}{\partial x^2}+\mu \frac{\partial^2 w}{\partial y^2}=0. \tag{7} \]

\[ \text{2. } N_x=0,\qquad -\left.\frac{\partial^2\varphi}{\partial y^2}\right|_{x=\pm \frac{b}{2}}=0. \tag{8} \]

1. The pressure in the direction of the \(Z\)-axis is the vertical load on the beam [2, p. 96]

\[ - V_x = D\frac{\partial}{\partial x} \left[ \frac{\partial^2 w}{\partial x^2} + (2-\mu)\frac{\partial^2 w}{\partial y^2} \right] \tag{9} \]

and the differential equation of the deflected axis of the beam takes the form:

\[ EJ_z \frac{\partial^4 w}{\partial y^4} = D\frac{\partial}{\partial x} \left[ \frac{\partial^2 w}{\partial x^2} + (2-\mu)\frac{\partial^2 w}{\partial y^2} \right]. \tag{10} \]

Forces \(N_b\) are applied at the center of gravity of the beams; they can be found from the equation \(\Sigma Y=0\)

\[ 2N_b+\int_{-\frac{b}{2}}^{+\frac{b}{2}} N_y\,dx=0. \tag{11} \]

Whence, taking (5) into account,

\[ 2N_b+\int_{-\frac{b}{2}}^{+\frac{b}{2}} \frac{\partial^2\varphi}{\partial x^2}\,dx=0, \tag{12} \]

\[ N_b=-\frac{\partial\varphi}{\partial x}. \]

We represent the bending moment in the beam as the sum of two terms:

\[ M_b=M(q)+M(N_b), \tag{13} \]

where \(M(q)\) is the bending moment due to the load \(q\), and \(M(N_b)=-N_b e\) is the bending moment due to the force \(N_b\).

It is known that for the elastic axis of the beam one can write:

\[ EJ_z\frac{\partial^2 w}{\partial y^2}=-M_b. \tag{14} \]

Differentiating (14) twice with respect to \(y\), taking into account (13) and

\[ \frac{d^2 M(q)}{dx^2}=-V_x \]

we can write equation (14) in the form:

\[ EJ_z \frac{\partial^4 w}{\partial y^4} = -\frac{\partial^2 M_\sigma}{\partial y^2} = -\frac{\partial^2 M(q)}{\partial y^2} -\frac{\partial^2 M(N_\sigma)}{\partial y^2}. \tag{15} \]

\[ EJ_z w^{\mathrm{IV}}=-V_x+e\frac{\partial^2 N_\sigma}{\partial y^2}, \]

since for \(Q_x\) the positive direction is taken downward, while the reaction \(V_x\) from the load on the beam is directed upward, it should be considered negative.

Bearing in mind (9), equation (15) may be given the following form:

\[ EJ_z \frac{\partial^4 w}{\partial y^4} = \pm D\left[ \frac{\partial^3 w}{\partial x^3} +(2-\mu)\frac{\partial^3 w}{\partial x\,\partial y^2} \right] \mp e\left( \frac{\partial^3 \varphi}{\partial x\,\partial y^2} \right). \tag{16} \]

1. This boundary condition we shall find from the following considerations. The relative deformation of the plate and the beam in the direction \(Y\) at the joint must be the same. The deformation of the middle plane in the direction of the \(Y\)-axis is equal to:

\[ (\varepsilon_{yy})_s=\frac{N_y}{Eh} =\frac{1}{Eh}\frac{\partial^2\varphi}{\partial x^2}. \tag{17} \]

The deformation \((\varepsilon_{yy})_s\) of the beam fiber at a distance \(e\) from the axis can obviously be determined from (2):

\[ (\varepsilon_{yy})_e=(e_{yy})+e\frac{\partial^2 w}{\partial y^2}, \tag{18} \]

where \(\varepsilon_{yy}\) is the relative elongation of the elastic axis of the beam, or

\[ (\varepsilon_{yy})_e=\frac{N_\sigma}{EF}+e\frac{\partial^2 w}{\partial y^2}. \tag{19} \]

From the equality of deformations it follows that:

\[ (\varepsilon_{yy})_s=(\varepsilon_{yy})_e. \]

Taking (17) and (19) into account, we write the last condition in the following form:

\[ \frac{1}{Eh}\frac{\partial^2\varphi}{\partial x^2} = \frac{N_\sigma}{EF} + e\frac{\partial^2 w}{\partial y^2}. \tag{20} \]

To simplify the fourth boundary condition, expressed by equation (20), we differentiate the latter twice with respect to \(y\). Taking into account that, for

\[ x=\pm\frac{b}{2},\quad -N_x=\frac{\partial^2\varphi}{\partial y^2}=0 \]

and equation (12), we write the 4th boundary condition (20) in this form:

\[ \frac{1}{Eh}\frac{\partial^4\varphi}{\partial x^2\,\partial y^2} = \pm\left( \frac{\partial^3\varphi}{\partial x\,\partial y^2} \right)\frac{1}{Eh} + e\frac{\partial^4 w}{\partial y^4}. \tag{21} \]

We consider a plate (Fig. 5) of length \(l\), along whose edges \(\left(x=\pm \dfrac{b}{2}\right)\) the following conditions must be satisfied:

\[ \text{a) } w=0,\qquad \text{b) } M_y=0,\qquad \text{c) } N_y=0,\qquad \text{d) } N_x=0. \]

Fig. 5

The solution of the problem posed is thus reduced to finding the functions \(w\) and \(\varphi\), satisfying, respectively, equations (3) and (6) and the four boundary conditions stated above.

We shall seek the solution of equations (3) and (6) in the form [6]:

\[ w=\sum_{m=1}^{\infty} \left(A_m\cos \lambda_m x+B_m\lambda_m x\sin \lambda_m x+C_m\right)\sin \lambda_m y \tag{22} \]

and

\[ \varphi=\sum_{1}^{\infty} \left(D_m\cos \lambda_m x+K_m\lambda_m x\sin \lambda_m x\right)\sin \lambda_m y, \tag{23} \]

where

\[ \lambda_m=\frac{m\pi}{l}. \]

Substituting (22) and (23) into the boundary conditions (7), (8), (16), and (21), we obtain four equations for determining the constants

\[ A_m(1-\mu)\cos \frac{\lambda_m b}{2} + B_m\left[ \frac{\lambda_m b}{2}(1-\mu)\sin \frac{\lambda_m b}{2} + 2\cos \frac{\lambda_m b}{2} \right] = \mu C_m . \tag{24} \]

\[ D_m\cos \frac{\lambda_m b}{2} + K_m\frac{\lambda_m b}{2}\sin \frac{\lambda_m b}{2} =0 . \tag{25} \]

\[ \begin{aligned} &A_m\left[EJ_z\lambda_m\cos\frac{\lambda_m b}{2}+(1-\mu)\sin\frac{\lambda_m b}{2}\right]+\\ &\quad +B_m\left[EJ_z\lambda_m^2\frac{b}{2}\sin\frac{\lambda_m b}{2}-(1-\mu)D\frac{\lambda_m b}{2}\cos\frac{\lambda_m b}{2}+\right.\\ &\quad \left.+(1+\mu)\sin\frac{\lambda_m b}{2}\right]-eD_m\sin\frac{\lambda_m b}{2}-eK_m\left(\frac{\lambda_m b}{2}\cos\frac{\lambda_m b}{2}+\right.\\ &\quad \left.+\sin\frac{\lambda_m b}{2}\right)=-EJ_z\lambda_m C_m . \end{aligned} \tag{26} \]

\[ \begin{aligned} &A_m\cos\frac{\lambda_m b}{2} +B_m\left(\frac{\lambda_m b}{2}\sin\frac{\lambda_m b}{2}+2\cos\frac{\lambda_m b}{2}\right)+\\ &\quad +D_m\left[\frac{\mu}{EFe}\cos\frac{\lambda_m b}{2} +\mu\frac{\sin\frac{\lambda_m b}{2}}{EF\lambda_m e}\right]+\\ &\quad +K_m\left[\frac{\mu}{Ehe}\left(\frac{\lambda_m b}{2}\sin\frac{\lambda_m b}{2} +2\cos\frac{\lambda_m b}{2}\right)+\right.\\ &\quad \left.+\frac{\mu}{EFe\lambda_m}\left(\frac{\lambda_m b}{2}\cos\frac{\lambda_m b}{2} +\sin\frac{\lambda_m b}{2}\right)\right]=0 . \end{aligned} \tag{27} \]

Substituting (22) into (3), we obtain

\[ \sum \lambda_m^4 C_m\sin\lambda_m y=\frac{q}{D}. \tag{28} \]

To determine \(C_m\), we multiply (28) by \(\sin \dfrac{\kappa\pi y}{l}\,dy\) and integrate the right- and left-hand sides:

\[ \int_{-l/2}^{+l/2}\sum\left(\frac{m\pi}{l}\right)^4 C_m \sin\frac{m\pi y}{l}\sin\frac{\kappa\pi y}{l}\,dy = \int_{-l/2}^{+l/2}\frac{q}{D}\sin\frac{\kappa\pi y}{l}\,dy . \]

Hence

\[ C_m=\frac{4ql^4}{m^5\pi^5D}=\frac{ql^4}{78.5\,m^5D}. \tag{29} \]

Solving equations (24), (25), (26), and (27), we determine the constants \(A_m, B_m, D_m,\) and \(K_m\), which, owing to the cumbersomeness of the expressions, we do not present.

Let us write the expressions for the bending moments \(M_x\) and \(M_y\) [2]:

\[ M_x=D\sum_{1,2,3\ldots}^{\infty}\lambda_m^2 \left[(1+\mu)A_m\cos\lambda_m x +(1+\mu)B_m\lambda_m x\sin\lambda_m x -2B_m\cos\lambda_m x+\mu C_m\right]\cdot\sin\lambda_m y . \tag{30} \]

\[ M_y=D\sum_{1,2,3\ldots}^{\infty}\lambda_m^2 \left[(1+\mu)A_m\cos\lambda_m x +(1+\mu)B_m\lambda_m\sin\lambda_m x -2\mu B_m\cos\lambda_m x+C_m\right]\cdot\sin\lambda_m y \tag{31} \]

Fig. 6

Fig. 7

and for the longitudinal forces [2]

\[ N_x=-\sum_{1}^{\infty}\lambda_m^2\left(D_m\cos\lambda_m x+K_m\lambda_m x\sin\lambda_m x\right)\sin\lambda_m y. \tag{32} \]

\[ N_y=-\sum_{1}^{\infty}\lambda_m^2\cos\lambda_m x\left(D_m-K_m\lambda_m x\tan\lambda_m x\right)\sin\lambda_m y. \tag{33} \]

\[ N_{\sigma}=-\left(\frac{\partial \varphi}{\partial x}\right) =\sum_{1}^{\infty}\left( -\lambda_m D_m\sin\frac{\lambda_m b}{2} +K_m\lambda_m\sin\frac{\lambda_m b}{2} -K_m\lambda_m^2\frac{b}{2}\cos\frac{\lambda_m b}{2} \right)\sin\lambda_m y. \tag{34} \]

Owing to the very cumbersome final expressions for the bending moments and longitudinal forces, in this form they cannot be used for practical calculations.

We have carried the solution through to numerical results. The greatest stresses occur along the \(X\)-axis.

Graphs of the dependences \(W_{\max}, M_x, M_y\) as functions of \(\alpha=\dfrac{l}{b}\), \(\lambda=\dfrac{EJ_z}{bD}\), and the eccentricity \(e\) are given in Figs. 5, 6, 7, 8, and 9.

The graphs are constructed for steel \((\mu=0.3)\). Having \(M_y\), the strength condition may be written in the form:

\[ \sigma_{\max}=\frac{6M_y}{h^2}\le(\cdot). \]

It should be noted that comparison of our results with the results of the work of Academician B. G. Galerkin [3, p. 19, Table 3], where the special case of a square plate \(\left(\dfrac{l}{b}=1\right)\) on elastic supports was investigated, which corresponds to the case shown in Fig. 1a, gives exact agreement for the values of \(M_y\) and \(M_x\) at the center of the plate (see the circles with “T” in Figs. 6, 7).

Comparison of the results with those of the work of S. P. Timoshenko [2, p. 215] and [9] also gives agreement in the special case of a square plate for \(w_{\max}, M_y\), and \(M_x\). This agreement of the results indicates the correctness of the results obtained for the more general case considered here of bending of a rectangular plate supported along opposite edges by elastic I-beams.

It should be noted that for \(\dfrac{l}{b}\to 0\) and \(\dfrac{l}{b}\to\infty\), \(n=0\) (beams), the bending moment at the center is

\[ M_y=\frac{ql^2}{8}, \]

which agrees with the result obtained in strength of materials.

Comparing the arrangements of the plate relative to the beams, we see that the most unfavorable is the arrangement shown in Fig. 1в \((e=0)\), and the most advantageous is that in Fig. 1a \((e)\).

Let us note that the Tacoma Bridge (USA), which suffered collapse in 1940, had the arrangement of the slab relative to the beam as shown in Fig. 1b, i.e., the most unfavorable one [7].

Analysis of the curves (Figs. 5, 6, 7) shows that, for the ratios

\[ \frac{EJ}{bD}=4 \div 5, \]

usually encountered in practice, the influence of beam bending on the deformation and stresses of the slab is small (10–15%) and may be disregarded in practical calculations. For smaller ratios this influence proves to be very substantial (80–90%).

Consequently, in order to increase the rigidity and strength of a slab supported by elastic I-beams, it is sufficient to take

\[ \frac{EJ}{Db}=4; \]

a further increase of this ratio will not produce a noticeable effect.

We now add the temperature factor to the external load \(q\).

Let the temperature of the lower \(\left(z=+\frac{h}{2}\right)\) and upper \(\left(z=-\frac{h}{2}\right)\) surfaces of the plate be, respectively, \(t_1\) and \(t_2\), and let the difference be \(t_1-t_2=t\).

Since, by assumption, the plate is thin, the temperature may be considered to vary through the thickness of the plate according to a linear law.

As was noted, for

\[ \frac{EJ}{Db}=4 \div 5 \]

a slab resting on elastic beams may practically be regarded as simply supported along the entire contour.

Taking this important circumstance into account, we shall determine the thermal deflection following V. M. Maizel.

In the case of a linear temperature field it proves convenient, for a simply supported plate, to use the relations

\[ M_t=\frac{E\alpha}{1-\mu}\,\frac{h^2}{12}\,t, \tag{a} \]

\[ w_t=W_{M_t}, \tag{b} \]

according to which the temperature displacement in the supported slab coincides with the deflection in an unheated slab caused by a bending moment \(M_t\), uniformly distributed along the contour.

According to a remark of S. P. Timoshenko [6], for an unheated rectangular plate simply supported along the contour, the deflection \(w_{M_t}(x,y)\) under the action of a bending moment \(M_t\) uniformly distributed along the contour coincides with the torsional stress function,

\[ (\Delta\varphi=-2G\omega_0) \quad \text{when} \quad -2G\omega_0=-\frac{M_t}{D}. \]

Therefore, using the corresponding solution of the theory of torsion, one obtains the value of the deflection \(W_{M_t}\) for the unheated plate in the form:

\[ W_{M_t}=\frac{8}{\pi^3}\,\frac{M_t b^2}{D} \sum_{n=1,\,3,\,5}^{\infty} \frac{1}{n^3}(-1)^{\frac{n-1}{2}} \left( 1-\frac{\operatorname{ch}\frac{n\pi y}{b}} {\operatorname{ch}\frac{n\pi l}{2b}} \right) \cos\frac{n\pi x}{b}. \]

Using now relations (a) and (b), we obtain the required temperature deflection for a rectangular plate situated in a linear temperature field

\[ W_t(x,y)= \]

\[ = \frac{8(1+\mu)}{\pi^3 h}\,b^2\alpha t \sum_{n=1,3,5\ldots}^{\infty} \frac{1}{n^3}(-1)^{\frac{n-1}{2}} \left( 1- \frac{\operatorname{ch}\dfrac{n\pi y}{b}} {\operatorname{ch}\dfrac{n\pi l}{2b}} \right) \cos \frac{n\pi x}{b}. \tag{35} \]

It is known, however, that

\[ \sum_{n=1,3,5\ldots}^{\infty} \frac{(-1)^{\frac{n-1}{2}}}{n^3} \cos \frac{n\pi x}{b} = \frac{\pi^3}{32} \left[ 1-4\left(\frac{x}{b}\right)^2 \right] \tag{c} \]

as may be verified by expanding the right-hand side in a Fourier series. Substituting (c) into (35), we obtain

\[ W_t(x,y)= \frac{(1+\mu)\alpha t}{h} \left[ \left(\frac{b}{2}\right)^2-x^2 \right] \sum_{n=1,3,5\ldots}^{\infty} \left( 1- \frac{\operatorname{ch}\dfrac{n\pi y}{b}} {\operatorname{ch}\dfrac{n\pi l}{2b}} \right). \tag{36} \]

At the center of the plate the temperature deflection is

\[ W_t= \frac{(1+\mu)\alpha t b^2}{4h} \sum_{n=1,3,5\ldots}^{\infty} \left( 1- \frac{1} {\operatorname{ch}\dfrac{n\pi l}{2b}} \right). \tag{37} \]

If \(\dfrac{l}{b}\) is large, then

\[ W_t= \frac{(1+\mu)\alpha t b^2}{4h}. \tag{38} \]

Having determined the temperature deflection, we compute the bending and twisting moments

\[ M_{yt} = -D\left[ \left( \frac{\partial^2 w_t}{\partial x^2} + \mu\frac{\partial^2 w_t}{\partial y^2} \right) + \alpha t\,\frac{1+\mu}{h} \right], \]

\[ M_{xt} = -D\left[ \left( \frac{\partial^2 w_t}{\partial y^2} + \mu\frac{\partial^2 w_t}{\partial x^2} \right) + \alpha t\,\frac{(1+\mu)}{h} \right], \]

\[ M_{(xy)t} = -M_{(yx)t} = -D(1-\mu)\frac{\partial^2 w_t}{\partial x\,\partial y}. \]

Summing the temperature deflections, the bending and twisting moments with the corresponding values of these quantities due to the load \(q\), we obtain the total values of the plate deflection and of the bending and twisting moments, by which the plate is calculated. The work was carried out by order of the Manometer plant.

References

1. K. K. Chalyshev — On the problem of calculating plates resting on an elastic contour. Collection of the Institute of Railway Engineers, issue 87, 1914.

2. S. P. Timoshenko — Plates and Shells. Gostekhizdat, 1948.

3. B. G. Galerkin — Collected Works, vol. II. Academy of Sciences of the USSR, 1953.

4. E. Müller — Ingenieur-Archiv, Bd. 2, 1932.

5. S. W. Fuchs — Plates with boundary conditions of elastic support. Proceedings ASME, Vol. 79, No. 199, 1953.

6. S. P. Timoshenko — Theory of Elasticity. Gostekhizdat, 1934.

7. F. D. Dmitriev — Failures of Engineering Structures. Moscow, 1953.

8. B. Gilg — Schweizerische Bauzeitung, No. 48, 1953.

9. A. S. Kalmanok — Structural Mechanics of Plates. Mashstroiizdat, pp. 171–173, 1950.