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Comment on: A lower-bound solution to the collapse of uniform rectangular grids on simple supports
Int. J. mech. 8el. Pergamon Press. 1972. Vol. 14, pp. 795-796. Printed in Great Britain
LETTERS TO THE E D I T O R C o m m e n t on: A l o w e r - b o u n d solution to the collapse o f uniform rectangular grids on simple supports b y M. GRIGORIAN, / n t . J. mech. Sci. 13, 755 (1971). (Received 20 M a r c h 1972)
DR. GRIGORIAN deserves appreciation for obtaining a general lower-bound solution to the limit load of simple, s y m m e t r i c orthogonal grids subject to nodal loads. R e c e n t l y t h e limit analysis of grids has been receiving considerable a t t e n t i o n . Ref. 2 is p a r t i c u l a r l y n o t e w o r t h y a n d deals w i t h t h e general elasto-plastic analysis of grids w i t h c o m p l e x loading and b o u n d a r y conditions. The speciality in Dr. Grigorian's p a p e r is t h a t it deals w i t h a general grid w i t h ( m - 1 ) and ( n - 1 ) b e a m s respectively in the two orthogonal directions. The " a n a l o g y " b e t w e e n a grid a n d a plate is well k n o w n u n d e r elastic conditions. 5 The writer 4 believes t h a t e v e n u n d e r plastic conditions t h e grid analogue can be used. F o r e x a m p l e consider a s i m p l y s u p p o r t e d isotropic square grid w i t h a n e v e n n u m b e r of syrmnetric b e a m s s u b j e c t e d to n o d a l loads. The limit load is given b y e q u a t i o n (20) 16M m P = T m s --------~'
(a)
where L is t h e l e n g t h of the b o u n d a r y . I f a square slab w i t h a u n i f o r m l y distributed 10ad q is idealized as a grid t h e relation b e t w e e n the grid loads and the slab load is qL ~ = P(m-
1) ~.
(b)
I f we denote t h e m o m e n t per u n i t w i d t h of t h e isotropic slab as M 0, the relation b e t w e e n t h e grid m o m e n t M a n d t h e slab m o m e n t M 0 is M 0L m-- 1 = M.
(c)
U s i n g (b) and (c) in (a) t h e limit load of the idealized grid is obtained as q =
16M 0 m ( m - 1) L2 m s- 1 "
(d)
T a k i n g t h e limit m -~ ~ in (d), we o b t a i n t h e limit load of an isotropic simply supported square plate 16M0 qL = L 2 • (e) This value coincides w i t h t h e lower-bound solution o b t a i n e d b y P r a g e r a for a n isotropie, rigid-plastic, simply s u p p o r t e d square p l a t e assuming parabolic n o r m a l m o m e n t field (M~, M , ) a n d zero t w i s t i n g m o m e n t field (Mxy). I t is seen t h a t Dr. Grigorlan's expressions as well as t h e n u m e r o u s tables and c o m p u t e r p r o g r a m s available in the literature on grids can be used to o b t a i n a safe lower-bound solution for t h e design of m e t a l plates a n d reinforced concrete slabs. H o w e v e r , t h e converse process, viz. finding a v a l u e of t h e limit load of a grid f r o m a n e q u i v a l e n t p l a t e analysis, is dangerous as it gives an u p p e r - b o u n d solution for t h e grid a n d is hence unsafe. T h e writer a has also dealt w i t h t h e analysis of plates u n d e r a finite n u m b e r of conc e n t r a t e d loads using t h e grid m e t h o d . F o r example, w i t h an isotropic slab of dimensions 8a x 4a, t h e limit load b y t h e grid m e t h o d is o b t a i n e d as 0.74M 0 whereas t h e yield-line t h e o r y (which is k n o w n to give u p p e r bounds for slabs) gives 0-92M a. H e r e t h e p l a t e is s u b j e c t e d to 21 c o n c e n t r a t e d loads. I t is seen t h a t t h e grid m e t h o d yields a fairly safe lower bound. T h e solution can be i m p r o v e d b y refining t h e analysis a n d b y including torsion b y inserting spring elements in t h e grid. 1 795
796
Letters to the Editor
I n conchlsion the writer wishes to point out two typographical errors, viz. (i) in the first line under Example 1, the parenthetical term must read (m = 8, n = 4), and (ii) the expression at the top of p. 761 must read
Pa = --~ REFERENCES 1. A. H. S. A~o and L. A. LOPEZ, Prec. Am. Soc. ~iv. Engrs, Engng Mech. Div. 94, 271 (1968). 2. T. HOI~OLANDA:aOMI',E. C. Rossow and S. L. LEE, Prec. A m . Soc. civ. Engrs, Engng Mech. Div. 94, 241 (1968). 3. W. PRAOE~, A n Introduction to Plasticity, p. 65. Addison-Wesley, Reading, Mass. (1959). 4. K. RAJAGOPALA~¢, Limit analysis of two-way pre-stressed concrete plates. Thesis submitted to the Madras University (in partial fulfilment) for the degree of Master of Science in Structural Engineering (1972). 5. S. P. TIMOSH~.NKOand W. K~IEOER, Theory of Plates and Shells. McGraw-Hill, New York (1959).
Structural Engineering Research (Regional) Centre Adyar Madras-20 India
K. RAJ~.OOPALA~
Reply to discussion: (Received 13 J u l y 1972) THE AUTHORwould like to t h a n k Mr. Rajagopalan for his interest and comments on his paper, 1 and especially for pointing out the typographical errors. Mr. Rajagopalan may be interested to learn that since the appearance of this work similar cases of practical interest b u t with more complex boundary conditions under uniformly distributed loading have also been studied. 2, 3 Regarding the plate analogy for grids under plastic conditions, considerable progress has been made b y the author and his colleagues at Arya-Mehr University of Technology, Tehran, Iran, who have shown that, subject to restrictions, the yield-line analogy can be used to obtain valid, unique, generalized solutions to the collapse of twistless, orthotropic griUages of regular construction, with any combination of boundary support conditions along the sides of a parallelogram and carrying a continuous distribution of normal nodal forces applied over the entire surface of the grillage. REFERENCES M. GRIGORIAlq,Int. J. mech. Sci. 13, 755 (1971). M. GRIGORI~-W,lnt. J. mech. Sci. 14, 197 (1972). M. G~IOORIAN,J. Strain Anal. (in press) (1972). M. MEHR~IN and M. GRIOORIAN, Res. Report No. 1971-02, Struet. Eng. Dept., A.M.U.T. 5. M. GRIOORIAN and V. Sc~Rr~K~.R, Res. Report No. 1972-02, Struct. Eng. Dept., A.M.U.T. 1. 2. 3. 4.
Arya-Mehr University of Technology Tehran Iran
1~I. GRIGORIAN
LETTERS TO THE E D I T O R C o m m e n t on: A l o w e r - b o u n d solution to the collapse o f uniform rectangular grids on simple supports b y M. GRIGORIAN, / n t . J. mech. Sci. 13, 755 (1971). (Received 20 M a r c h 1972)
DR. GRIGORIAN deserves appreciation for obtaining a general lower-bound solution to the limit load of simple, s y m m e t r i c orthogonal grids subject to nodal loads. R e c e n t l y t h e limit analysis of grids has been receiving considerable a t t e n t i o n . Ref. 2 is p a r t i c u l a r l y n o t e w o r t h y a n d deals w i t h t h e general elasto-plastic analysis of grids w i t h c o m p l e x loading and b o u n d a r y conditions. The speciality in Dr. Grigorian's p a p e r is t h a t it deals w i t h a general grid w i t h ( m - 1 ) and ( n - 1 ) b e a m s respectively in the two orthogonal directions. The " a n a l o g y " b e t w e e n a grid a n d a plate is well k n o w n u n d e r elastic conditions. 5 The writer 4 believes t h a t e v e n u n d e r plastic conditions t h e grid analogue can be used. F o r e x a m p l e consider a s i m p l y s u p p o r t e d isotropic square grid w i t h a n e v e n n u m b e r of syrmnetric b e a m s s u b j e c t e d to n o d a l loads. The limit load is given b y e q u a t i o n (20) 16M m P = T m s --------~'
(a)
where L is t h e l e n g t h of the b o u n d a r y . I f a square slab w i t h a u n i f o r m l y distributed 10ad q is idealized as a grid t h e relation b e t w e e n the grid loads and the slab load is qL ~ = P(m-
1) ~.
(b)
I f we denote t h e m o m e n t per u n i t w i d t h of t h e isotropic slab as M 0, the relation b e t w e e n t h e grid m o m e n t M a n d t h e slab m o m e n t M 0 is M 0L m-- 1 = M.
(c)
U s i n g (b) and (c) in (a) t h e limit load of the idealized grid is obtained as q =
16M 0 m ( m - 1) L2 m s- 1 "
(d)
T a k i n g t h e limit m -~ ~ in (d), we o b t a i n t h e limit load of an isotropic simply supported square plate 16M0 qL = L 2 • (e) This value coincides w i t h t h e lower-bound solution o b t a i n e d b y P r a g e r a for a n isotropie, rigid-plastic, simply s u p p o r t e d square p l a t e assuming parabolic n o r m a l m o m e n t field (M~, M , ) a n d zero t w i s t i n g m o m e n t field (Mxy). I t is seen t h a t Dr. Grigorlan's expressions as well as t h e n u m e r o u s tables and c o m p u t e r p r o g r a m s available in the literature on grids can be used to o b t a i n a safe lower-bound solution for t h e design of m e t a l plates a n d reinforced concrete slabs. H o w e v e r , t h e converse process, viz. finding a v a l u e of t h e limit load of a grid f r o m a n e q u i v a l e n t p l a t e analysis, is dangerous as it gives an u p p e r - b o u n d solution for t h e grid a n d is hence unsafe. T h e writer a has also dealt w i t h t h e analysis of plates u n d e r a finite n u m b e r of conc e n t r a t e d loads using t h e grid m e t h o d . F o r example, w i t h an isotropic slab of dimensions 8a x 4a, t h e limit load b y t h e grid m e t h o d is o b t a i n e d as 0.74M 0 whereas t h e yield-line t h e o r y (which is k n o w n to give u p p e r bounds for slabs) gives 0-92M a. H e r e t h e p l a t e is s u b j e c t e d to 21 c o n c e n t r a t e d loads. I t is seen t h a t t h e grid m e t h o d yields a fairly safe lower bound. T h e solution can be i m p r o v e d b y refining t h e analysis a n d b y including torsion b y inserting spring elements in t h e grid. 1 795
796
Letters to the Editor
I n conchlsion the writer wishes to point out two typographical errors, viz. (i) in the first line under Example 1, the parenthetical term must read (m = 8, n = 4), and (ii) the expression at the top of p. 761 must read
Pa = --~ REFERENCES 1. A. H. S. A~o and L. A. LOPEZ, Prec. Am. Soc. ~iv. Engrs, Engng Mech. Div. 94, 271 (1968). 2. T. HOI~OLANDA:aOMI',E. C. Rossow and S. L. LEE, Prec. A m . Soc. civ. Engrs, Engng Mech. Div. 94, 241 (1968). 3. W. PRAOE~, A n Introduction to Plasticity, p. 65. Addison-Wesley, Reading, Mass. (1959). 4. K. RAJAGOPALA~¢, Limit analysis of two-way pre-stressed concrete plates. Thesis submitted to the Madras University (in partial fulfilment) for the degree of Master of Science in Structural Engineering (1972). 5. S. P. TIMOSH~.NKOand W. K~IEOER, Theory of Plates and Shells. McGraw-Hill, New York (1959).
Structural Engineering Research (Regional) Centre Adyar Madras-20 India
K. RAJ~.OOPALA~
Reply to discussion: (Received 13 J u l y 1972) THE AUTHORwould like to t h a n k Mr. Rajagopalan for his interest and comments on his paper, 1 and especially for pointing out the typographical errors. Mr. Rajagopalan may be interested to learn that since the appearance of this work similar cases of practical interest b u t with more complex boundary conditions under uniformly distributed loading have also been studied. 2, 3 Regarding the plate analogy for grids under plastic conditions, considerable progress has been made b y the author and his colleagues at Arya-Mehr University of Technology, Tehran, Iran, who have shown that, subject to restrictions, the yield-line analogy can be used to obtain valid, unique, generalized solutions to the collapse of twistless, orthotropic griUages of regular construction, with any combination of boundary support conditions along the sides of a parallelogram and carrying a continuous distribution of normal nodal forces applied over the entire surface of the grillage. REFERENCES M. GRIGORIAlq,Int. J. mech. Sci. 13, 755 (1971). M. GRIGORI~-W,lnt. J. mech. Sci. 14, 197 (1972). M. G~IOORIAN,J. Strain Anal. (in press) (1972). M. MEHR~IN and M. GRIOORIAN, Res. Report No. 1971-02, Struet. Eng. Dept., A.M.U.T. 5. M. GRIOORIAN and V. Sc~Rr~K~.R, Res. Report No. 1972-02, Struct. Eng. Dept., A.M.U.T. 1. 2. 3. 4.
Arya-Mehr University of Technology Tehran Iran
1~I. GRIGORIAN