Gravitational stresses in single span deep beams

BuiM. Sci. Vol. 2, pp. 173-179. Pergamon Press 1967. Printed in Great Britain

SfB

Ab 4

UDC

624.072

Gravitational Stresses in Single Span Deep Beams K. T. SUNDARA RAJA IYENGAR* K. CHANDRASHEKHARA*

A theoretical solution for the gravitational stresses in single span deep beams using Fourier series has been given. Numerical results for different span to depth ratios are given and these have been compared with the photoelastic results given by Saad and Hendry[1], and the finite difference results of Chow et al. [2,3].

1. INTRODUCTION BEAMS whose depths are comparable to their span are known as deep beams and the simple flexural theory for such cases yields erroneous results for the bending stress distribution since this theory does not take into accout the effect of normal stresses at top and bottom edges caused by the loads and support reactions. These normal stresses in fact make the bending stress distribution non-linear over vertical sections and shear stress distribution nonparabolic. Hence the assumptions made in the simple bending theory that transverse planes remain plane after bending and the neutral axis is in the middle plane are not valid for such cases. The loading coming on the deep beam may be classified under two categories; (i) surface loading, (ii) gravity loading. The analysis of deep beams under surface loading is fairly extensive and sufficient numerical results are available for design. However, the problem of stress distribution in deep beams under gravity loading has so far not been satisfactorily treated analytically. The importance of the analysis of deep beams under gravity loading can be realised from the fact that many and varied types of such problems arise in civil engineering construction especially reinforced concrete structures, where the self weight of the deep beam forms a major portion of the loading which cannot be neglected in the analysis. Hence it is necessary to analyse the deep beam under gravity loading to ensure an efficient design. The problem of gravitational stresses in deep beams can be effectively solved by converting it into surface traction problem. Several investigators have obtained approximate solutions to the problem of deep beam subjected to surface loading. Li Chow, Conway and Morgan[2] have obtained * Civil and Hydraulic Engineering Department, Indian Institute of Science,Bangalore 12, India.

the solution for single span deep beams using Fourier series and principle of least work, while Li Chow, Conway and Winter[3] have given the solution for the same problem using the finitedifference method. Archer and Kitchen[4], Guzman and Luisoni[5] have obtained the solution to this problem using the strain energy method. An exact solution for single span deep beams using Fourier series has been given by Sundara Raja Iyengar[6]. An exact solution has also been given by Varadarajan[7], for the same problem using orthogonal functions. Recently Sundara Raja Iyengar[8] has given an exact solution for the stress distribution in very deep beams using Fourier series and integrals. However, it appears that the problem of gravitational stresses in deep beams has not been completely solved yet. Only Durant and Garwood[9] have given an approximate solution to this problem and their results do not appear to be accurate. The only experimental investigation on this problem has been made by Saad and Hendry[1] using the photoelastic method. Since the experimental results have been obtained for certain span to depth ratios only, it has only a restricted application in practical design. The object of the present investigation is to give a theoretical solution to the same problem and to verify the available experimental results. The problem of gravitational stresses in deep beams has been converted to a surface traction problem using the concept given by Timoshenko and Goodier[10]. In the case of a single span deep beam shown in figure 1 the problem is solved by superimposing two stress systems (i) beam subjected to its own weight and supported continuously along its entire lower edge [figure l(a)] and (ii) the same beam resting on the supports and loaded at its lower edge by an uniformly distributed tensile load equa ! to the weight of thegirder [figure l(b)]. The 173

K. T. Sundara Raja lyengar am/K. Chandraskekkara

174 Y

Y

Y

I

I

llll Jill lilllilll

lIlIIIJII

=

14 lilill IIlllllll

lil

r¢oclion 2 ° ~ - - ~ i~- Conccntrot¢d reaction

@

llilli~41 lilli~ill llllillll

-'a+

~X

illlll

'llllllllllltllllllllllll q

lllllllllllllllllllllllll q,

reaction

~o

4 t--

O)

2

Cb)

Iq

,Co

Fig. 1. Beam under self-weight. stresses for the former case can easily be written as 6' = 0

)

X

The boundary conditions are, (i) along x = + a ~r.~ = 0, r.,.y (ii) along y = + b a y 0, r,.y = 0 (6) (iii) a l o n g y = - b aj; = J ( x ) , U, = 0 The loading on the edge y = - b can be expressed in the form of a Fourier series as --

~) = - p g ( b - y )

(1)

t

"Cxy -~ 0

where pg is the density of the material. The stress components in equation (1) satisfy the equations of equilibrium and compatability. This stress system gives rise to a uniform compressive stress ay = - 2 p g b on the bottom edge. To remove the boundary stress and to take the effect of support condition, a loading is to be applied as shown in figure l(b). Hence the body force problem is now reduced to one of surface traction problem and the stresses in the deep beam are now to be determined due to edge loads,

ql = 2pgb - 2pgb q = --tt

m=l

O"x

O'x

a, = a; + ay . (3) = ' + ~ ~ t Txy "Cxy -~- T.r v Hence the problem is now to find the stresses O'x, try and zxr due to the edge loads q and ql [figure 1(b)]. tt

V4cp" --- 0.

(4)

The stress components are determined from O2q)tt O"x = - 6 3 y 2 -

,, , O'y-

02q)tt (~X2 ,

"~xy --

a -,

(5)

(7)

mnx dx. a

(8)

For reaction distributed over a width 'c
K,, = 4pgbsin mn (1 -c0. c
(9)

For a point support, the concentrated reaction is first replaced by a uniformly distributed reaction on a small width equal t o ' ~a ' and then the expression for the Fourier coefficient Km be evaluated by taking the limit as oral2 -~ O. Hence for the concentrated support the Fourier coefficient Km may be obtained as K m

4 p g b ( - l) m+ 1 cos mn~

=

(10)

3. S O L U T I O N The expression for 4)" satisfying the differential equation (4) can be written as mTrx m m COS - -

q ° " = ? xt

( ~o)

2cosh-mnb n aY [m2Y = smh" a •

O2(P~t ~x~y"

O

(7/

Km = - 1 +" j" o v,, l y = - b c o s

t!

2. GENERAL EQUATIONS AND BOUNDARY CONDITIONS Formulation of the problem in terms of Airy's stress function leads one to the determination of a function, q¢' satisfying the biharmonic equation,

~

tl

where K m is determined from

(2)

n

n

~

f(x) = ~ K m cos mn_x

O~

If a~, ay and zxy are the stresses developed in the deep beam due to these edge loads, then the final stresses due to gravity loading are given by

n

t!

a

Gravitational Stresses in Single Span Deep Beams +~

~o

m=l

Bmcos mT~x -a [r~YeoshmnY ( ~ ) 2 sinh mnb.. a

+ ~(_l)m+n(~)(a)m2ntanhnb a n=x

n=l

2

[m2+ ( 7 ) 2] 2

a

-(l+m abtanhT) sinai: ] (3O

175

(14) and

I

C. COSnn_y (7) 2 b [~n~ sinhnn---f-x -cosh nn____aa b

1

2mnb 1 Bm sinh 2manb a____

b

-

l + ~n-nca o t h .- fnfn-a)\ cosn-ff-]' nnx] s=1.3

[mE+\2b]

12

Ds

D s sin shy

.-.. s=1,3 \ 2b]

, sna [-2ff sinh

Km (15)

2b

cosn-~

sna ,, sna~ _(l+_s_com g ) cosn -S-J" . s

xl

(11) It may be verified from the above expression for

~p"that the boundary condition for z~y is automatically satisfied. Applying the boundary condition a~ Ix= ±~ = 0, and after finite Fourier transformation and simplification, one obtains 1+

2nn__._aa 1

mn 2 + ~ ( - - 1 ) m+n

tanh

mnb

2 2 a Am .~_ 0

(12) and

smnT/ -b~(--1)m+~-'~2x.=l

m~:ithm~b

r~

2

[ mz+

Bm=

0.

1,2b] J (13)

Similarly by applying the boundary condition 0 and aj; ]y=-b = f i x ) and after finite Fourier transformation and simplification one obtains

a'; ]y=b=

1-F •

2 Thus it can be seen that equations (12) to (15) are the necessary relations to be satisfied by the four unknowns namely, Am, B,,, C. and Ds in order to satisfy all the boundary conditions. However, equations (12) to (15) from which the unknowns are determined lead to an infinite set of simultaneous equations. But, in practice, only a finite number of terms are chosen in each series and the resulting system of simultaneous equations are solved for the unknowns. Hence it follows that the solution given here satisfies the governing differential equation exactly and the boundary conditions only approximately since only a finite number of terms in the series are considered. However, the extent of nonsatisfaction of the boundary conditions can easily be investigated by comparing the numerical results obtained by considering different number of terms in the series.

a

sinh 2rnnb a

Am

4. NUMERICAL RESULTS Numerical results have been given for three different a/b ratios (namely 0.5, 1 and 2) and support conditions. For each a/b ratio equations (12) and (15) were solved by taking eight terms in the series. In order to study the convergence of the series, calculations were also done by taking four, six and twelve terms in the series, for a particular case. The horizontal stress distribution for different number of terms, for the distributed and concentrated reaction, are shown in figures 2(a) and (b). It may be seen from these figures that the convergence of the series is good by considering only eight terms in the series. It should be noted however, that if the distribution of stresses are required very near or along the boundaries, large number of terms in the series are to be considered for getting accurate values of stresses. The numerical results have been obtained for the following cases. (i)

a/b = 0.5, ~ = 0.224, 0.2(distributed reaction) (2) = 0"112 (concentrated reaction)

176

K. T. Sundara Raja lyengar and K. Chandrashekhara

(ii) a/b = 1, c~ = 0"216 0.2 (distributed reaction)

(;!=0108

(iii) a/b = 2, ~ = 0.15

5. DISCUSSION OF RESULTS The theoretical results obtained here for the various a/b ratios and support conditions have been compared with the results of Chow et al.[3] obtained by finite difference procedure and the photoelastic results of Saad and Hendry[1] Chow et al. have given the bending and shear stress distributions in deep beams having different span to depth ratios under surface loading at top and bottom edges of the beam. They have not solved the problem of gravitational stresses in deep beams

(concentrated reaction) 0.2 (distributed reaction)

( ; ) =0.075

(concentrated reaction)

The distribution of horizontal stress (a~) along the centre section of the beam for the above a/b ratios and support conditions are shown in figures 3-8.

"°1 }11 0.6

~-

x

O ~ - - - - - ~ r ~ l - - - 4. t e r m s I [ I -'-- 6 terms I --0-2~--~ ~ 8 terms m ; ~ I . . . . 12 t e r m s i I \ I o/b=l.O

'xl

,

-o.81--~+

-,.ol

T o.T o

I / [ /

-T

;'~.

!

i

1

I i

--t3

L

I

"--

"-°'1

I-0(,

I

1

-0.4

1

-0.8

i

-I.o

-0.2 -0.6

xi

- - - 4 terms x - - . - - 6 terms • -' ' 8 terms o . . . . 12 ¢¢rm5 a/b=l.O

~-T:

-fl

l

T

-~.0 0 ~.0 20 3.o 4-of go 6.o " - - - C~x/fgb) =-(b) tO) Fig. 2(a) and (b). Distribution of horizontal stress (a~) along the centre (x = 0). -I.O

O

I-O 2 . 0 3-0

4-0 5.0

•,,~-- C~x/t g b)

1.0

0"8 ~ T

1.0

ii

I

0-4

0.2

02

J

O

-0.4

"

I ' - - Chow ¢tol I a Authors _t K-T- S "IY~9O~

o/b=O.5

i

-0"8

-0-8 -,o

~0

3 - 0 4.'0

k

0'6

~-~x

-

0 - 4 -

-

0.2 - - ~---

~0-~--~ ~ ~

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0 . . . . Chow ctol / - O "2~ --i-~-- - - AuthorSa/b=l

°

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--0"6

--I-0 0 1.0 -.--- CoE//'gb)

- o.

I

0 ~! / -O'2

-0-6

--I " 0

;

i!

0-6

T

0.4

-0.2

,o

~Y

0-8

iy

0-6 . . . .

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1

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i

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,4,

-~-0 0 .

o/~O.S

i. ~

.

.

i .

.

.

.

.

~.0 2-0 3.0 4-0 s.o

(o~/tgb)

Fig. 3. Distribution of horizontal stress Fig. 4. Distribution of horizontal stress (a~) along the centre (x = 0). (ox) along the centre (x = 0).

The distribution of vertical stresses (ay) along horizontal sections and the central vertical section of the beam for a/b = 0 . 5 and 1 are shown in figures 9-11. The distribution of shear stress for different a/b ratios are shown in figures 12-14.

-~'0

0

~'0 2"0 3.0 4"0

Fig. 5. Distribution of horizontal stress

(¢rx) along the centre (x = 0).

directly. However, the results given by them for deep beams under surface loading acting along the bottom edge of the beam can be compared with the second solution given here. The transverse stress distribution obtained by them for different a/b

Gravitational Stresses in Single Span Deep Beams 1.0

I

X!

o.+

+I

"

177 y

f

o.+

/

o

[

Io.~ Fig. 6. Distribution of horizontal stress (o+)along the centre ~

,x:0,

0

t

~i j - - A u t h ° r ~ r ~ ° ~ l ~ r i

+ \

L-°'

....

?i',

Hcndry9

\',=+PEt, m+

_~.o! - 2 . 0 - 1 ' 0

0 I.O 2 " 0 3 - 0 4 " 0 S'O 6 " 0 •- - - - - - (o'x t?gb) ---

,

o~

"~

0"~ l

,

,o

'

,b

0.6

ol~==

%,,

0"4

a

L+L~

~ ' l

'"

'

'

. ",..,

e

~ >+-

0":~ ,

1

.... --

~

C h o w c t OI Authors

,

-.- Slmpt~ bending

-0"2

""+

'"'-""~+

--

-0'4

[

O,

t

j

-0'~

L

[ \+.~ ,~.

+

+

I\%/--AuthorS(Crag +

] --Authogs Qdtstlfil r¢Octlon-)

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t

I

~]---s+o+s+

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-0.8

II

r"%4"~+"+

).8--

-#'C -s.o-+o-~o-~.o-~o

o

~o ~.o s.o +.o s o 6.0

Fig. 7. Distribution of horizontal stress ( ~rx) along the centre (x = 0).

Fig. 8. Distribution of horizontal stress (ax) along the centre (x = 0).

o

?o~oo

i

-

I

66666 l

2b ~t j

8

I

+

O -"=

O -"=

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=

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,

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i

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.

+d,+G

1

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~~'

+ T

~ g°+°° OIb=O.S

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~

,,~

-~=os "~<~...

-08

~ _ "">f

-,.o-.~s-.s-.2s

•.--

+ m 6 =

--0"2 ~ i --0"4

-,>6

0=

'

11I.224a--III --

!

0 / I

l

;

;'/

0"2

'-

( ,----~4._.._L.~

q --

0"4.

++++++ +~ r

I

0

!

~o~ T T 3" '

•

++i

.., ~. ~

T

I

~

L+ K)

Y

0"8

0"6

~ 6I oI 6I 1

o o ^;~,+ ,

o .2s "~.o-v / e g ~ )

.s

.Ts

,.o

.

Fig. I0. Distribution of vertical stress (or) along the centre (x = 0).

I

~

o D.o 2.0 z.o 4.o s-o 6.0 (~/rgu) =

1,0

I t ° ;o;go

£88£ ,#oo?o

-s.o-4o-so-=.o-vo -

I;

~

,

1

+'9"

O/b= I

Distribution of vertical stress (oy) along different horizontal sections.

178

K. T. Sundara Raja lyengar and K. Chamh'ashekhara

ratios have been compared with those of the theoretical results obtained here in figures 3, 5 and 7. It may be observed from these figures that except for a/b = 0.5, the results are in good agreement with those obtained here. Also the distribution of shear stress for two a/b ratios namely 0.5 and 1 at

'°r~ } !

~.O

,o

O-8 -

y

. . . .

0 ~___~---Chow L~,\ I

-/

ct ol

~1- A u t h ° r s

,., ~ [ '~_~J~J~:~Simpl¢ bcndir I- v ' ' ' r " , L \ / i Ith~ory I I\,/~ ' - . . - K-T. S.

-0-8

-

_,.o~-T

!

i

o o.2 0.4 0.6 o-8 i.o 1.2 ('e~,y/pgb)"

-o.4

--~--*

-0-8~: _l.O--i

-

Fig. 13. Distribution of shear stress (r,-,,) along x/a = 0.5.

+----

i - 0 . 4 . - o . 2 o 0.2 0 . 4 o.s (:o-y/j,g b) -

-

Fig. 1l. Distribution of vertical stress (~y) along the centre (x = 0).

It may be observed from these figures that the agreement between the theoretical and experimental results is good for the case of distributed reaction. However, instead of distributed reaction at the bottom if a concentrated reaction is assumed then it may be observed from figures 4, 6 and 8 that it 1.0~

x/a = 0.5 has been compared with Chow et al. results in figures 13 and 14. It may be observed from these figures that the agreement between the two results is not good. This difference in results can be attributed to the approximations in the finite difference analysis.

o-8

\ .\

o.~

X \

0"4

1 ~

0"8 0.6 ~, I 0"4 . . . . .

:v

"~5~ \

!

'~

0

~-

•4 - -

-

-0-6

7" /

:

~, 0

- 0"2

!

Cho* , t ol

Authors Simple bending theory a/b=l.O

4

- - A u t h o r s ~distrib~t¢~ [ ', r11:oc,lon) I %\ l----Authors (Cone. [

"~k

reaction.) I

"',,L 1-5°°d ~

--I.O

0 0'25 O-5 0"75 I ~ 1"25 --'(:Cx

~/~.

-

\

" "~

I

',,~- ~ ---

_~0. 2

x

X

~

y/rgb)

Fig. 12. Distribution of shear stress ('rx~) along x/a = 0'6.

Saad and Hendry have obtained the photoelastic results for the gravitational stresses in deep beams having different a/b ratios. The distribution of transverse stress obtained by them for different a/b ratios have been compared with those obtained here in figures 4, 6 and 8. The distribution of shear stress for a/b = 1 has been compared in figure 12.

/

-o.s

2, o o

'~/+

~/~

0.2 0 . 4 - 0 . 6 0.8 I-o

( ~ x ylrgb) Fig. 14. Distribution of shear stress (rxy) along x/a = 0.5.

affects only the bottom fibre stress whose magnitude will be greater than that for the distributed reaction. But Saad and Hendry have determined the transverse stress distribution for a/b = 1 using the finite difference procedure for the support conditions namely distributed and concentrated reactions and show that the distributed reaction results in a greater bottom fibre stress than the concentrated reaction. According to the results obtained here this does not appear to be correct. The experimental results of Saad and Hendry indicate that the magnitude of transverse stress at the bottom, for different a/b ratios, does not vary much as one moves towards the support, the maxi-

Gravitational Stresses in Single Span Deep Beams m u m values always being at the centre section. F r o m the results obtained here, it was found that the maximum fibre stress does not always occur at the centre but at some other point away from the centre in some cases. In figures 10 and 11, the distribution of vertical stress for two a/b ratios namely 0.5 and 1 have been compared with the results of Saad and Hendry. It may be seen from these figures that the agreement between the two results is not good. In figure 9, the distribution of vertical stress along horizontal sections for two a/b ratios namely 0.5 and 1 is shown. It may be seen from this figure, that for a/b = 0.5, at distances equal to or greater than the span of beam, measured from the bottom, the

179

vertical stress distribution is almost uniform. Also the transverse stress is considerably small at such sections. However, the results of Saad and Hendry indicate that the vertical stress distribution at such sections is not uniform and the transverse stress has considerable magnitude even at sections very near the top of the beam. Further Saad and Hendry themselves state for this particular case, in their comparison of results, that ' a beam of the proportions just referred to acts roughly as a beam of depth equal to span with the top part merely resting on it. This top portion hardly contributes any resistance to the bending moment due to dead weight of beam '

REFERENCES 1. S. SAADand A. W. HENDRY,Gravitational stresses in deep beams, Struct. Engr 39 (6), 185 (1961). 2. LI CHow, H. D. CONWAYand G. W. MORGAN,Analysis of deep beams, J. appI. Mech. 18 (2), 163 (1951). 3. LI CHow, H. D. CONWAY and G. WINTER, Stresses in deep beams. Trans. Am. Soc. Cir. Engrs 118, 686 (1953). 4. F.E. ARCHERand F. M. KnrcEJEN,Stresses in single span deep beams. Aust. J. Appl. Sci. 7 (4), 314 (1956). 5. A. M. GUZMANand C. J. LtnsoNI, Solution variacional del problema de la viga rectangular simplements apoyada de gran altura. Cienciay Technica, Buenos Aires 111, 119 (1948). 6. K. T. SUNDARARAJA IYENGAR,Analysis of deep beams. Proc. 2nd. Cong. Theoret. appl. Mechanics, New Delhi, p. 87 (1956). 7. K. N. VARADARAJAN,Analysis of single span deep beams by basic functions method, M.Sc. Thesis, Indian Institute of Science (1965). 8. K.T. SUNDARARAJA IYENGAR,Contribution to the analysis of deep beams, Publications IABSE, 25 (1965). 9. N. J. DURANT and F. GARWOOD, Ministry of Works technical note No. 76, London (Oct. 1947). 10. S. TIMOSnENKOand J. N. GOODmR,Theory of Elasticity, McGraw-Hill, New York (1951). Une r6solution th6orique, se servant de s6rie de Fourier, a 6t6 donn6e traitant les tensions due ~ la pesanteur, dans les poutres hautes avec un seul espacement. Les r6sultats num6riques pour les rapports divers--l'espacement/hauteur ont 6t6 pr6sent6s et compar6s avec les r6sultats photom6triques de l'elasticit6 donn6s par Saad et Hendry[1], ainsi qu'avec les r6sultats de la diff6rence finale de Chow et d'autres[2, 3]. Eine theoretische L6sung ftir Schwerkraftbeanspruchungen, die in hohen Balken mit Einzelspannweite entstehen, mit Anwendung der Fourierschen Reihe ist angegeben. Die numerische Ergebnisse der Verh/iltnisse der verschiedenen Spannweiten zur H6he sind angegeben; sie wurden mit den von Saad und Hendry[1], angegebenen photoelastischen Ergebnissen und auch mit den endgiiltigen Differenz-Ergebnissen yon Chow und anderen[2, 3] verglichen.