Pergamon
PII: SOO45-7949(%)00029-6
NUMERICAL
SOLUTION
Compurrrs& Smcrures Vol. 61, No. I . pp. 21-29. 1996 Copyright0 Copyright0 19% Elsevier cienceLtd cienceLtd Printed n Great Britain.All Britain.All rightsreserved rightsreserved 0045-7949196$I 0045-7949196$I 5.00 + 0. 00
OF BOTH ENDS FIXED DEEP BEAMS
S. Reaz Ahmed, A. B. M. Idris and Md. Wahhaj Uddin Department
of Mechanical Mechanic al
Engineering, Engine ering, Bangladesh Banglade sh University Dhaka-1000, Bangladesh
of Engineering
& Technology
(Received 7 Ma rch 1995)
Abstract-This paper presents the exact solution of a two-dimensional elastic problem with mixed boundary bounda ry conditions. It gives the complete solution in terms of displacement and stress components of both ends fixed deep beams, subjected to uniformly distributed compressive loading on the top edge. new mathematical mathemat ical model has been used to formulate the problem. In this analysis both the governing equations and boundary bounda ry conditions c onditions are expressed in terms of a new potential function ‘Ywhich ‘Ywhich is defined the discretization of the in terms of displacement components. Finite-difference Finite-difference technique is used for the bi-harmonic equation and the associated boundary bounda ry conditions. All the parameters of interest in the solution are presented in the form of graphs. Finally, a comparison is made to ascertain how the elementary solutions differ with those of the present numerical numeri cal solutions. so lutions. The T he computed solutions are found to be highly accurate and rational, and thus establish establish the reliability and practicability of this numerical approach. Copyright 0 1996 Elsevier Science Ltd.
NOTATION
the stress distributi distribution on in bodies, all dimensions dimensions of which are of the same order, has to be investigated. They are only only approximately approximately correct in some cases,
rectangular co-ordinates elastic modulus of the material Poisson’s ratio stress X-component of displacement Y-component of displacement X-component of normal stress Y-component of normal stress shearing stress component in the xy plane length of the plate or beam in the y-direction depth of the plate or beam in the x-direction Airy’s stress function potential function defined in terms of displacements
to light by the more refined investigations investigations of the theory of elasticity. The theory of elasticity deals with general equations which must be satisfied by an elastic elasti c body in in equilibrium under any external forc system. Simplifying Simplifying assumptions are also used in elasticity, but usually usually with a better knowledge about the approximation involved. involved. Earlier, Earlier , two-dimensional mixed boundary value stress problems were solved either by a two-displacement function approach [l], or by by Airy’s stress function function approach [3]. Although the theories of elasticity had long been established, the solution of practical problems started mainly mainly after af ter the introduction of the stress function function by Airy. This stress function function approach can be used with ease only when the boundary conditions are known in terms of loading. As most of the problems of elasticity are of mixed boundary conditions, conditions, this approach fails to provide any explicit explicit understanding understanding of the stress distribution distribution in the critical regions of o f restrained boundaries. When the problems of elasticity are formulated in terms of two displacement functions, functions, the problem of the conditions conditions at the restrained boundaries is removed, but the simultaneous simultaneous evaluation evaluation of two functions functions satisfying two simultaneous simultaneous differential equations, is extremely difficult and this problem becomes more serious when the boundary conditions conditions are mixed [4].
INTRODUCTION
This paper is on the analysis of a two-dimension two-dimensional al mixed boundary boundary value stress problem [l], fixed at two opposing lateral edges, free at the bottom edge and subjected to uniformly uniformly distributed compressive loading at the top. The problem may be considered as a both ends fixed deep beam. The solutions are obtained using the present Y-formulation Y-formulation and are compared with the known solution of an elementary one-dimensio one-dimensional nal approach, as dealt with in undergraduate texts on mechanics of materials. In many cases, the elementary elementary theories of strength of materials are not sufficient to describe the stress distribution in engineering structures struc tures [2]. The elementary theory is inadequate to give information information regarding local stresses near the loads and near the supports of the beam. It fails fails also in in the cases when 21
22
S.
Reaz Ahmed
Moreover, it is not practical to solve this type of problem analytically because the assumption of trial solution satisfying all modes of mixed as well as changeable boundary conditions is an extremel difficult task. Considering all these limitations associated with the different approaches, the authors decided to solve the problems numerically based on the present formulation. In this computational approach, the elastic problem has been formulated in terms of a single potential function, Y, defined in terms of displacement components and is considered as parallel to Airy’s stress function @ [3], since both of them have to satisfy the same bi-harmonic equation. FORMULATION
OF THE PROBLEM
With reference to a rectangular co-ordinate system, the differential equations of equilibrium for plane stress and plane strain problems in terms of displacement components are [5]
g+(+)g+(yj&=o
et al.
Therefore, the whole problem has now been formulated in such a fashion that a single function of x and y, as in the case of stress function approach, has to be evaluated from the same bi-harmoni partial differential eqn (3) associated with the boundary conditions that are prescribed at the edges of the elastic body.
BOUNDARY
CONDITIONS
Normally, the boundary conditions in practical problems are prescribed in terms of restraints and loadings, that is, known values of displacements and stresses at the boundary. Both the displacement and stress components are defined by their respective normal and tangential components Referring to Fig. 1, for this particular problem, both the opposing lateral edges are fixed, the bottom edge is free of loading and the top edge is subjected to compressive loading. For the opposing lateral edges, AC and BD,
(1)
u(.u, y) = u(x,
Y) = 0
for e+(‘,‘)g+(+&=O,
(2) O
where u and u are the displacement components of point in the ?I- and y-directions, respectively. These two homogeneous elliptic partial differential equations, with appropriate boundary conditions, have to be solved for the case of the two-dimensional problem when the body forces are assumed to be absent. An attempt is made here to reduce the problem to the determination of a single variable, instead of evaluating two functions u and v simultaneously from the above equilibrium eqns (1) and (2). In this formulation, as in the case of Airy’s stress function a potential function Y of space variables is defined in terms of displacement components as
L’=
1, y/a=0
and 1.
For the bottom edge, CD, 0, (n, y) = U,, (x, y) = 0 for O
-[(+++(g$$J
When the displacement components in eqns (1) and (2) are replaced by the definition of Y (x, y), the first equilibrium eqn (1) is automatically satisfied. Therefore, Y has only to satisfy the second equilibrium eqn (2). Expressing this equation in terms of Y, the condition that Y has to satisfy becomes [5]
g+g&
+a’J/=,.
(3)
a------A Fig. 1. Both ends fixed deep beam subjected distributed
compressive
loading
to uniformly at the top edge.
Both ends fixed deep beams
and the corresponding edge, AB, are
boundary conditions
for top
6, (x, y)/E = 3 x lo-4 6,J (x, y) = 0
23
T-
for 0 ,< y/a < 1, x/b = 0.
In order to solve this problem by solving the function Y (x,~) of the bi-harmonic equation, the boundary conditions above are also needed to be expressed in terms of Y. The relations between known functions on the boundary and the function !P are a’* U=Z&
v=
(4)
(5)
-[(++$+(&)%I.
=&&g+]
2-z
-~
* (1 + v)Z 8-v’
0,)
=(1
(6)
(2 + v)
au
.I
(7)
dv
ay+a,
=2(1
ay
$q2--
ay
(8)
As far as numerical solution is concerned, it is evident from the expression of boundary conditions that all the boundary conditions of interest can easily be discretized in terms of displacement function Y, by the finite-difference method. METHOD OF SOLUTIONS
The essential feature of this numerical approach is that the original boundary value problem is replaced by a finite set of simultaneous algebraic equations,
Fig. 2. Discretization of the domain of interest to the co-ordinate system.
in relation
and the solution of this set of simultaneous equations provides us with an approximation for the displacement and stress within the solid body. The finite-difference technique is used here to the bi-harmonic partial differential discretize equation and also the differential equations associated with the prescribed boundary conditions. For this present problem, considering the rectangular shape of the boundary and also the nature of the differential equations involved, rectangular grid points are used all over the region concerned for numerical computation. The discrete values of the displacement function Y(.u, y) at mesh points of the domain concerned (Fig. 2) are solved from the system of linear algebraic equations resulting from the discretization of the bi-harmonic equation and the associated boundary conditions. There are numerous existing methods of solving a system of algebraic equations. The iteration method is advocated for this kind of large sparse system of linear algebraic equations, but considering the most unfortunate part associated with the iteration method of not always converging to a solution and sometimes converging, but very slowly, the present problem is solved by the use of the direct method of solution which ensures better reliability, as well as possibility of getting more accurate solution in a shorter period of time. Finally, the same difference equations are organized for the evaluation of displacement and stress components from the Y values at different sections of the body, as all the components are expressed as summation of different derivatives of the function Y. RESULTS AND DISCUSSIONS
In obtaining numerical values for the presen problem, the beam as the elastic body is assumed to be made of ordinary steel (v = 0.3, E 2 x 10” m-‘). Graphs are plotted both at different constant values of x for varying )’ and at constant y for varying x for each of the five quantities of interest, namely u, c, 0,, u, and cr,, Moreover, the effect of the ratio a/b on the relevant displacement
S. Reaz Ahmed et al.
24
6x1 O4
L-l xko.0
x/b=o.23
x/b=054
x/ b=
2
.4
.6
.8
1.0
YJa Fig. 3. Distribution
of displacement component u at different sections of the beam for a/b = 2.
and stress components is illustrated in this paper. In order to make the results nondimensional, the displacements are expressed as the ratio of actual displacement to the size of the plate, and the stresses are expressed as the ratio of the actual stress to the elastic modulus of the beam material. The soundness and accuracy of the solutions obtained by the present formulation are judged from different angles. For example, the solution can be intuitively guessed based on the basic principle of mechanics. Again, the physical symmetry or antisymmetry of the model and that of loading are always reflected in symmetric or antisymmetric distribution
of the parameters in the body. The famous Saint Venant’s principle must also be found true in the distribution of values of every parameter within the body of the structure. Figure 3 shows the distribution of displacement component u at different sections of the beam, which is observed to be parabolic in nature. It is seen from the graph that the displacement in the direction of loading is maximum at the top and minimum at the bottom edge, while the magnitude is zero at both ends of the beam, which is in conformity with the loading pattern as well as the end conditions. The effect of a/b ratio on the distribution of displacement componen
SKI25
.0020
Xl015
.OOlO
BOO5
-2
.4
.6
.8
Fig. 4. Distribution of displacement component u at section x/b = 0 for different a/
1.0
ratio.
25
Both ends fixed deep beams
2.5x1 04
.n
yfa Fig. 5. Distribution of displacement component 1’at section x/b = 0 for different
u at the top boundary is illustrated in Fig. 4. As appears from the graph, the displacement is increasing with higher values of a/b ratio. It conforms to the fact that, at lower a/b ratio, the end effects become very prominent and provide restriction to the deflection of the beam. Here the displacements are zero at both ends and maximum at the mid-section of the beam for each a/b ratio, which is in full agreement with the physical model of the elastic beam. From the distribution of the displacement component ~1with respect to y (Fig. 5) it is seen that the variation at the top boundary is sinusoidal, which is
.2
a/b ratio.
also in conformity with the physical model of the problem. The maximum values of displacement v are increasing with increasing values of a/b ratio. It should be noted that the displacement at both ends is not exactly zero. This is true only for the top fibre of the beam, because it is the fibre where the load is directly applied and the corner points of the top boundary are the points of singularity. At this point of singularity, the change in the variables is abrupt and the finite-difference method of solution always ends up in small residues, instead of zero values. As appears from the distribution of normal stress component cr, with respect to y (Fig. 6), the a/b ratio
.4
.8
.6
a/b=1
a/b=2 a/b=3
-2.6~10~
Fig. 6. Distribution of normal stress component C, at section
x/b
=
0.08 for different a/
ratio.
S. Reaz Ahmed er
26
al.
Fig. 7. Distribution of normal stress component uJ at section x/b = 0 for different
fails to produce any significant effect on the distribution. Here the distribution is shown for the section at x = O.O8b, as the distribution for all the beams is similar and has the same magnitude of 4 x 1O-4 at the top boundary. On the other hand, the ratio a/b produces a distinguishable effect on the distribution of normal stress a, with respect to (Fig. 7) at the top boundary. Stresses are maximum with higher values of a/b ratio and the maximum stresses are developed at the top boundary where the load is applied. The top fibre of the beam is in compression around the mid-point, but in tension near the fixed ends. The range of the tension zone
a/
ratio.
increases with increasing a/b ratio, but the magnitude of the tensile stresses in these zones decreases with increasing a/b ratio. Figure 8 shows the variation of the normal stress component a, at the restrained boundaries, showing the effect of a/b ratio on the distribution. Stresses are maximum at both the top and bottom fibres with zero value at the mid-section, which make the distribution symmetric about the longitudinal mid-section of the beam. This variation of normal stress component in the direction of y with respect to x, is analysed for a particular beam, mainly to compare how the elementary solutions differ with that of exact
.0030
.0015
-.OOlS
-.0030 ’ 0
.2
.4
.6
.8
1.0
xlb
Fig. 8. Distribution of normal stress component o, at section
y/ a
=
for different
a/
ratio.
Both ends fixed deep beams
a = 0. 23 & 0. 77 a 0. 15 8 0. 85 y/a = 0 . 0 8 & 0 9 2
a
- . 0 01 5
0. 46 & 0 54
- . 0 03 0 0
Fig. 9. Distribution
. 4
of normal
stress
component
obtained through this numerical approach. In the elementary solution it is assumed that th distribution of normal stress component varies linearly with depth everywhere and the magnitude is maximum at the top and bottom fibres. As can be seen from Fig. 9, the solutions obtained through the present formulation differ from that of elementary solution in the sense that the distribution is far from linear, rather it approaches towards linear only at the mid-section of the beam, which of course conforms to the famous Saint Venant’s principle that the end-effect decreases as we move away from the restrained boundary. Moreover, the magnitude of CT, solutions
. 6
0, at different
sections
of the beam
for
a/b =
at the top fibre is higher than that of the bottom fibre. However, in case of elementary solution this magnitude is the same for both the top and bottom fibres of the deep beam. Distribution of shearing stress (Fig. 10) reveals that shearing stress is zero at both the top and bottom edges, which conforms to the obvious fact that both the top and bottom edges of the physical model are free from shearing stresses. At the longitudinal sections, the shear stresses increase with increasing distance from mid-point and also with increasing a/b ratio. Figure 10 shows the non-linear distribution of shearing stress at the mid-section of the beams
yfa Fig. 10. Distribution
of shear stress component
CJ\, at section
x/b
=
0.54 for different
a/b
ratio.
S. Reaz Ahmed
28
et al.
.0020
0
.2
.4
.6
.8
1.0
x/b Fig. 11. Distribution
of shear stress component u,, at section
describing the effect of a/b ratio. Again Fig. 11 describes the variation of shearing stress component along the restrained boundary at y = 0, which varies completely in a non-linear way at the fixed boundaries and it becomes more critical when the length of the beam is increased, while the loading remains constant for each case. Finally, the variation of shearing stress with respect to x at various transverse sections of the beam is investigated mainly to compare its characteristic behaviour with the elementary solutions. From the distribution in Fig. 12, it is observed that the variation of this stress component over the depth is
-2
y/ a
=
for different a/
ratio.
almost similar to that of elementary solutions except at the fixed edges. Away from the boundary the distributions are parabolic in nature, and are identical in nature and magnitude. From the elementary solution it is observed that the magnitud of shearing stresses is maximum at the mid-section of the beam. This is not agreed by our numerical solutions and it differs mainly at the fixed ends. The distribution is not parabolic at the fixed ends, rather it is maximum at about x = 0.0%. Since in the elementary formulas of the strength of materials, the boundary conditions are satisfied in an approximate way, it fails to provide the actual distribution of
.4
.6
.8
1.0
x/b 12. Distribution
of shear stress component CT,, t different sections of the beam for
a/
= 3.
Both ends fixed deep beams
stresses at the boundaries, especially, at the restrained boundaries. The present Y-formulation is free from this type of shortcoming and is thus capable of providing the actual stress distribution at any critical sections either at, or far from, the restrained edges. CONCLUSION
In this paper, an attempt is made to obtain the numerical solutions of elastic deep beams through new mathematical formulation which has a very bright prospect in handling the two-dimensional mixed boundary value problems of elasticity.
suitability
29 of the method.
The
comparative
study
verifies that the elementary solutions are based on assumptions which fail to provide the solutions in the neighbourhood of the critical regions of restrained boundaries. Through the solutions of this exemplary problem, the rationality and appropriateness of the present approach are established. The solutions of the practical problems, like stresses in gear teeth and screw threads, can now be obtained exactly.
REFERENCES
1. M. W. Uddin, Finite difference solution of two-dimen-
The special feature of the present displacement potential approach over the existing approaches is that, here all modes of the boundary conditions can be satisfied exactly, whether they are specified in terms of loadings or restraints or any combination of them and thus, the solutions obtained are promising
2.
and satisfactory for the entire region of interest. Using the present Y-formulation, the complete
4.
solutions are obtained for deep beams and are presented in the form of graphs in such a way that provides a better compreh%on of the nature bf the solutions. Both the qualitative and quantitative results and, moreover, the comparison with those of the elementary solutions establishes the reliability and
5.
3.
sional Elastic problems with mixed boundary conditions. M.Sc. thesis, Carleton University, Canada (1966). S. P. Timoshenko, The approximate solution of two-dimensional problems in elasticity. Phil. Mag. 47, 1095 (1924). S. P. Timoshenko and J. N. Goodier, Theory of u s t i c i t _ vrd Edn. McGraw-Hill, New York (1970). A. B. M., Idris, A new approach to the solution of mixed boundary value elastic problems, M.Sc. thesis, Department of Mechanical Engineering, Bangladesh University of Engineering & Technology, Dhaka, Bangladesh, (1993). S. R. Ahmed, Numerical solution of mixed boundaryvalue elastic problems. M.Sc. thesis, Department of Mechanical Enaineerine. Bangladesh Universitv of Engineering & T&hnology, Dhika, Bangladesh (1493).
