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Elastic design of slabs for uniformly distributed loads
Compurcrs d Structures, Vol. 2, pp. 893-895.
ELASTIC
Pour1972. Printed io Chat
Per~amon
Britain
DESIGN OF SLABS FOR UNIFORMLY DISTRIBUTED LOADS M. A. MUSPRATT
Department
of Civil Engineering,
McGill University, Montreal, P.Q., Canada
descriptionsare obtained for the complete distribution of principal bending moment contours and trajectories, and deflections of rectangular slabs with four edges either simply supported or clamped, and subject to uniformly distributedload. Simple linear response is assumed.
Abstract-Graphical
GRAPHICAL
REPRESENTATION
THEORETICAL solutions deflections
in the form of infinite series are given in Ref. [l] for moments and of slabs. However, it can be seen that it is virtually impossible to obtain an
expression
for say the x-co-ordinate
purpose
of plotting,
without
given
the moment
and the y-co-ordinate,
some very indirect iterative process.
The approach
for the used
was therefore to calculate principal bending moments and their directions on a grid of points, statistically fit surfaces to these, and then plot contours from the least-squares surface expressions. Trajectories were plotted in a similar manner, but the handling of vectors instead of scalars required numerical solution of differential equations. The discrete vector plot containing information from both the contour and trajectory plots was obtained by dual surface fitting [2]. Principal moments and deflections were calculated for Poisson’s ratio of 0.0 and 0.3 and for aspect ratios of 1.0 and 1.5. Samples plots of these are shown in Figs. l-7. Figures 1 and 2 show maximum and minimum principal moment contours for the encastre slab. These may be compared with the plots presented by Wood [3] which were obtained by finite difference methods. The important discrepancy involves the singularities occurring along the diagonals. Ritz’s variational method also yielded these singularities [4]. When, however, a realistic value of Poisson’s ratio is used (e.g. 0.3), or when the aspect ratio is 1.5, only one central singularity occurs. In the encastre slab, there is a heavy concentration of negative bending action at the central region of the sides, while the centre of the slab behaves like a circular slab. (This is analogous to Prager’s lower bound solution for a simply-supported square slab which is that given by the circumscribed encastre axisymmetric slab [3].) There is an absence of moment in the comers, while a ‘corner lever’ effect is noticeable across the extremities of the diagonals. Minimum principal moment contours for the simply-supported slab are shown in Fig. 3. These may be compared with Morley’s lower bound optimal solution [5l in which moment contraflexure also occurs along diagonal lines joining the centre of the sides. When, however, Poisson’s ratio is increased to 0.5 corresponding to the normal incompressibility assumption in plasticity, the analogy becomes more indistinct. It may be shown [5] that the moment volume for the elastic (v=O.O) and Morley’s solutions are the same. 893
M. A. MUSPRATT
894
In Figs. 1-3, Moment =
q.a2 . CL
lo4
where CL = contour level . Other notation is as in Ref. [l]. Figure 4 shows deflection contours which are independent of Poisson’s ratio q.a?CL Deflection = D.10’ Figure 5 shows typical maximum bending moment trajectories. The saddle-type singularity at the centre of the slab did present some difficulty in numerical solution. Figures 6 and 7 are complete graphical representations of moments in magnitude and direction for the simply-supported slab. moment =q.a’. coeK coeff = coefficient given under the graph SUMMATION OF WFlNlTE
SERIES
The computational necessity of considering a finite subset of terms in the infinite summations previously mentioned necessitates convergence and remainder tests. Timoshenko [l] shows that Fourier series convergence is invariably rapid but the remainder is more difficult to ascertain. Euler’s summation technique [6] was used after previous conversion of the positive series into an alternating series by the Van Wijngaarden transformation The approach may be rendered feasible for series of only odd terms by replacing the summation index ‘m’ by ‘m-n+ 1’ where ‘n’ is the number of the term. Similar secondary transformations are required in other transcendental expressions. Using the above methods, high summation accuracy was achieved without the cumbersome evaluation of many series terms-e.g. evaluation of ‘n’ terms for deflection of the simply supported slab and application of the Euler-Van Wijngaarden summation procedures resulted in accuracy attainable normally by the evaluation of ‘1On’ terms. It should be noted also that the ergodic accuracy of summation of many terms of decreasing magnitude is subject to Cauchy’s Lemma of diminishing return. As magnitudes of the terms decrease, the effect of machine rounding or overflow becomes proportionately larger so that accuracy may actually decrease or oscillate if an ‘excessive’ number of terms are evaluated. A partly recursive evaluation method was used to minimise this latter effect. REFERENCES [l] S. Tmomxmo and 5. W~~NOW~KY-K~~W~, lbory o/Plates und Shek McGraw-Hill, New York (19S9). [2] M. M-IT, Graphic display. Inr. J. Camp. Math. 2,25Q-268 (1970) [3] R. WOOD. Pka& and Ehwtic De&n of Plates and Slabs. Thamea & Hudson, London (1967). [4] M. Mumurr, Behavior of concrete allab, Ph.D. Tbir, Dept. of Civil Engineering, Monash Univ., Australia (196Q).
40
8 .
2.00 I
r.00 I
6.00 I
6.00 1. A-
FIG. 1. Maximum principal moment contours for encastre slab.
to.00 I \'"I 2
---
&’
2s
_ 6.00
FIG. 2. Minimum principal moment contours
8.00
for encastre slab.
1o.oOi
2.00
FIG.3. Minimum principal moment contours for simply-supported slab.
2.00 i
I
2.00
9.00 I
6.00
I
Y.00
I
6.00
LO
I
I
6.00
I
6.00
FIG. 4. Deflection contours for encastre slab.
101
Era. 5. M~irn~ principalmoment trajectoriesfor encastreslab.
FIG. 6. Maximum principal moments in magnitude and direction for encastre slab.
FIG. 7. Minimum principal moments in magnitude and direction for encastre slab.
Elastic Design of Slabs for Uniformly Distributed
Loads
x95
[S] M. MUSPRAIT, Behavior of simply supported slabs, A.S.C.E., V.95, NSTl2, Dec., 1969, pp. 2703-272 I. [61 H.M.S.O., Modern Computing Methods, National Physical Laboratory, Notes on Applied Science, N.16 (1962). (Received 20 September 1971)
ELASTIC
Pour1972. Printed io Chat
Per~amon
Britain
DESIGN OF SLABS FOR UNIFORMLY DISTRIBUTED LOADS M. A. MUSPRATT
Department
of Civil Engineering,
McGill University, Montreal, P.Q., Canada
descriptionsare obtained for the complete distribution of principal bending moment contours and trajectories, and deflections of rectangular slabs with four edges either simply supported or clamped, and subject to uniformly distributedload. Simple linear response is assumed.
Abstract-Graphical
GRAPHICAL
REPRESENTATION
THEORETICAL solutions deflections
in the form of infinite series are given in Ref. [l] for moments and of slabs. However, it can be seen that it is virtually impossible to obtain an
expression
for say the x-co-ordinate
purpose
of plotting,
without
given
the moment
and the y-co-ordinate,
some very indirect iterative process.
The approach
for the used
was therefore to calculate principal bending moments and their directions on a grid of points, statistically fit surfaces to these, and then plot contours from the least-squares surface expressions. Trajectories were plotted in a similar manner, but the handling of vectors instead of scalars required numerical solution of differential equations. The discrete vector plot containing information from both the contour and trajectory plots was obtained by dual surface fitting [2]. Principal moments and deflections were calculated for Poisson’s ratio of 0.0 and 0.3 and for aspect ratios of 1.0 and 1.5. Samples plots of these are shown in Figs. l-7. Figures 1 and 2 show maximum and minimum principal moment contours for the encastre slab. These may be compared with the plots presented by Wood [3] which were obtained by finite difference methods. The important discrepancy involves the singularities occurring along the diagonals. Ritz’s variational method also yielded these singularities [4]. When, however, a realistic value of Poisson’s ratio is used (e.g. 0.3), or when the aspect ratio is 1.5, only one central singularity occurs. In the encastre slab, there is a heavy concentration of negative bending action at the central region of the sides, while the centre of the slab behaves like a circular slab. (This is analogous to Prager’s lower bound solution for a simply-supported square slab which is that given by the circumscribed encastre axisymmetric slab [3].) There is an absence of moment in the comers, while a ‘corner lever’ effect is noticeable across the extremities of the diagonals. Minimum principal moment contours for the simply-supported slab are shown in Fig. 3. These may be compared with Morley’s lower bound optimal solution [5l in which moment contraflexure also occurs along diagonal lines joining the centre of the sides. When, however, Poisson’s ratio is increased to 0.5 corresponding to the normal incompressibility assumption in plasticity, the analogy becomes more indistinct. It may be shown [5] that the moment volume for the elastic (v=O.O) and Morley’s solutions are the same. 893
M. A. MUSPRATT
894
In Figs. 1-3, Moment =
q.a2 . CL
lo4
where CL = contour level . Other notation is as in Ref. [l]. Figure 4 shows deflection contours which are independent of Poisson’s ratio q.a?CL Deflection = D.10’ Figure 5 shows typical maximum bending moment trajectories. The saddle-type singularity at the centre of the slab did present some difficulty in numerical solution. Figures 6 and 7 are complete graphical representations of moments in magnitude and direction for the simply-supported slab. moment =q.a’. coeK coeff = coefficient given under the graph SUMMATION OF WFlNlTE
SERIES
The computational necessity of considering a finite subset of terms in the infinite summations previously mentioned necessitates convergence and remainder tests. Timoshenko [l] shows that Fourier series convergence is invariably rapid but the remainder is more difficult to ascertain. Euler’s summation technique [6] was used after previous conversion of the positive series into an alternating series by the Van Wijngaarden transformation The approach may be rendered feasible for series of only odd terms by replacing the summation index ‘m’ by ‘m-n+ 1’ where ‘n’ is the number of the term. Similar secondary transformations are required in other transcendental expressions. Using the above methods, high summation accuracy was achieved without the cumbersome evaluation of many series terms-e.g. evaluation of ‘n’ terms for deflection of the simply supported slab and application of the Euler-Van Wijngaarden summation procedures resulted in accuracy attainable normally by the evaluation of ‘1On’ terms. It should be noted also that the ergodic accuracy of summation of many terms of decreasing magnitude is subject to Cauchy’s Lemma of diminishing return. As magnitudes of the terms decrease, the effect of machine rounding or overflow becomes proportionately larger so that accuracy may actually decrease or oscillate if an ‘excessive’ number of terms are evaluated. A partly recursive evaluation method was used to minimise this latter effect. REFERENCES [l] S. Tmomxmo and 5. W~~NOW~KY-K~~W~, lbory o/Plates und Shek McGraw-Hill, New York (19S9). [2] M. M-IT, Graphic display. Inr. J. Camp. Math. 2,25Q-268 (1970) [3] R. WOOD. Pka& and Ehwtic De&n of Plates and Slabs. Thamea & Hudson, London (1967). [4] M. Mumurr, Behavior of concrete allab, Ph.D. Tbir, Dept. of Civil Engineering, Monash Univ., Australia (196Q).
40
8 .
2.00 I
r.00 I
6.00 I
6.00 1. A-
FIG. 1. Maximum principal moment contours for encastre slab.
to.00 I \'"I 2
---
&’
2s
_ 6.00
FIG. 2. Minimum principal moment contours
8.00
for encastre slab.
1o.oOi
2.00
FIG.3. Minimum principal moment contours for simply-supported slab.
2.00 i
I
2.00
9.00 I
6.00
I
Y.00
I
6.00
LO
I
I
6.00
I
6.00
FIG. 4. Deflection contours for encastre slab.
101
Era. 5. M~irn~ principalmoment trajectoriesfor encastreslab.
FIG. 6. Maximum principal moments in magnitude and direction for encastre slab.
FIG. 7. Minimum principal moments in magnitude and direction for encastre slab.
Elastic Design of Slabs for Uniformly Distributed
Loads
x95
[S] M. MUSPRAIT, Behavior of simply supported slabs, A.S.C.E., V.95, NSTl2, Dec., 1969, pp. 2703-272 I. [61 H.M.S.O., Modern Computing Methods, National Physical Laboratory, Notes on Applied Science, N.16 (1962). (Received 20 September 1971)