FOUNDATION ENGINEERING

CALCULATION OF REINFORCED-CONCRETE

BEAMS

V. F. Tutynin and V. I. Solomin

FOUNDATION

UDC 624.072.233.5 : 624.15

The existing method of calculating foundation beams is based on the hypothesis of linear deformabil i - ty (ideal elasticity) of the foondation s t ruc ture and of the base.

Designing experience shows that the calculations per formed on the basis of the l inear model often r e - sult in overes t imated forces and overconsumption of re inforcement and concrete . It is usually considered that the cause of this is the imperfect ion of the model base, whereas a lmost no attention is given to the shor tcomings of the model s t ruc ture . Actually, many models of bases have been suggested, but only spo- rad ic at tempts have been made to refine the behavior of the s t ruc ture itself.

A model taking into account physically the nonlinear nature of the behavior of the s t ruc ture and soil would most correspond to real i ty . But since there is present ly no sufficiently substantiated nonlinear model of a base, only the nonlineari ty of the behavior of the foundation beam is taken into considerat ion in the present ar t ic le .

The hypothesis of l inear deformation of a beam essential ly determined the requi rement of its calcu- lation with respec t to the f i r s t limiting state (bearing capacity), which led to a number of contradict ions.

Firs t , in accordance with the existing method the forces in s t ruc tures are determined as in an ideal e last ic sys tem and the re inforcement is selected with respec t to the stage of failure, i .e. , the stage of de- veloped cracks and inelast ic deformations. But it is lmownthat in the f i r s t stage the st iffness of the s t ruc - ture changes markedly and a redistr ibution of fo rces occurs . Thus the f i r s t and second stages of calcula- tion are based on different p remises .

Second, since large inelastic deformations can develop in a r e in fo rced-concre te beam [1], the f i r s t limiting state occurs only as a consequence of an unlimited increase of displacement at a constant load. However, such a state is precluded, since we have accepted (at p r e s s u r e s on the soil not exceeding the nor - mal bed res i s tances , this is valid) that the beam res t s on a l inearly deformable base.

Consequently, the cr i ter ion of the f i r s t l imiting state cannot be determining in the calculation of foun- dation s t ruc tures .

At the same time, staying within the f ramework of the linear model of the s t ruc ture , we cannot move on to a calculation with respec t to the second (settlements) and third (crack formation) limiting s tates , since a l inear calculation, general ly giving overs ta ted internal fo rces , will inevitably relegate to the back- ground the requi rements of the second and third l imiting s tates .

We will set up the problem of calculating a foundation beam in the following way. The re in forced- concre te beam lies f ree ly on the surface of a l inear semi-infini te e last ic mass and is loaded by a rb i t r a ry ver t ica l loads, The fr ict ion between the beam and base is d is regarded. The following deformation depen- dences are valid for the beam [2]:

1 M = B (M} - - .

P

B----'0.85E hip, (M<Mc)

Chelyabinsk. Transla ted f rom Osnovaniya, Fundamenty i Mekhanika Gruntov, No. 2, pp. 16--18, March-Apri l , 1971.

© 1971 Consultants Bureau,. a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. IO011, All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15,00,

1 0 1

M, 1

1 !

2

Fig. 1. For determination of the st iffness B of the c ross section of a re inforced-concre te beam.

~P ...... _ _ ~

Cross section

5O0

a'=35 L ~

~ Fr ~

o--+ r ooo ;

Fig. 2. Loading diagram and c ross section of beam.

B = - - ~0 Z!

, ( M c < M < M I ) ~r ~;b Er Fr 4- (~, + ~) b h0 gb ~

l i p ' > P1

p l - h o z l -~ ( ¥ ' - ~ ) b h o E b-+ "

The formulas contained in [2, 3] were used for calculating M c and M 1.

We assumed that the second of the formulas given above for B is valid until the bending moment reaches the value M1, after which the moment in the given section remains constant, but the curvature con- tinues to increase (Fig. 1}. This made it possible to examine not only the "service" but also the "limiting" stage of behavior of the beam.

In such a statement of the problem the prepara tory stage of the calculation will include the following. As usual we assign the cross-sect ional dimensions. Then, before performing the calculation, we assign the re inforcement on the basis of the optimal percent of reinforcing.

The authors worked out an algorithm and compiled a p rog ram of the Minsk-22 computer which is based on Ya. B. L 'v in ' s method [4] (the beam is divided into 81 sections).

The computer per forms the following sys tem of calculations. In the f i rs t approximation the stiffness of the beam is considered constant and equal to B = 0.85 EbI p, the bending moments are calculated, and then the ver t ical displacements are found and f rom them the curvatures . Further , by means of the depen- dences given above the values of the bending moments are determined in 79 sections of the beam in the sec- ond approximation, etc.

Let us consider a beam whose pa ramete r s are borrowed f rom A. IV[. Ovechkin's book [5] with unessen- tial changes. A T-shaped re inforced-concre te beam (Fig. 2) res t s on a semi-infini te elast ic mass with charac te r i s t i c s : E 0 = 150 kg /cm 2,/~0 = 0.35, beam re inforcement of class A-II , F r = 42.47 cm 2, F~ = 21.24 cm 2, concrete grade 150.

For positive moments (extending the lower reinforcement) we find M c = 89.11 tons-m, M l = 145.22 tons-m, ~ = 0.72, In' = 0.36, 1/p l = 0.24.10 -2 l / m ; for negative, M c = 40.71 tons-m, M l = 76.69 tons-m, # = 0.35, ~' = 0.72, 1/p l = 0.16 -10 -2 1 /m.

Let us t race the kinetics of beam deformation at three loading stages covering the stages of behavior f rom l inear to the development of "plastic" zones (Fig. 3).

As we see f rom the bending-moment d iagrams, the l inear calculation by the conventional method shows that the load P = 80 tons is the design load, since in this case Mma x (the moment in the middle of

102

-3O0 (tons m)

-200

-100

~o,o

I00

200

0

tons

% \

\

°:zoo tons _ t_ _ ~

. . . . .

ton i - ~

1 .....

0

Ccm

2

- M 1

-Mc 4

+Mc

_ __ +M 1

t o n s •

tTl 2 )

0,1-Id

i

r p=2oo tons

I I.

Fig. 3. Diagrams obtained as a resul t of calculating a re in forced-concre te beam on a semi-infini te elast ic mass . a) Moments; b) set t lements ; c) reaction p re s - su res ; d) s t i f fnesses.

the span) reaches the value M 1. The nonlinear calculation, which takes into account the rea l law of defor- mation of a re in forced-concre te beam and redistr ibution of fo rces , shows that at P = 80 tons the value of !Vima x is much less than the value of M 1 and only reaches the moment of cracking M c.

The set t lements and react ion p r e s su re s obtained at this load with and without considerat ion of r ed i s - tribution of forces pract ica l ly coincide.

We increase the load to 200 tons (which is impermiss ib le f r o m the viewpoint of the l inear calcula- tion). With considerat ion of redistr ibution of fo rces , we obtain that in the middle of the span the moment reaches MI, and under the load the moment slightly exceeds the value of M c.

The set t lements and react ion p re s su res obtained f rom the l inear and nonlinear calculations differ in- significantly and can be admissible with respec t to magnitude.

At a load of 270 tons , azone fo rms in the middle of the span which amounts to t/8 of the span where the moments are equal to M 1.

The set t lements and react ion p r e s su re s found f rom the l inear calculation and with considerat ion of redis tr ibut ion of forces differ more appreciably and their nonuniformity increases markedly .

Also of in teres t is the stiffness diagram, which in Fig. 3 is shown for P = 270 tons. With the appear- ance of a c rack the stiffness of the section changes abruptly, then decreases gradually as the curvature in- c reases .

Thus we see that according to the present ly adopted method (linear calculation) a toad P = 80 tons is the design load for the beam considered, whereas the nonlinear calculation (with considerat ion of the red i s - tribution of forces) gives a value of the design load P = 200 tons, i .e. , 2.5 t imes g rea te r , ff we proceed f rom the requi rement of the second limiting state (the presence of c racks in the absence of an aggress ive

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environment is permissible), i.e., we take settlements as the criterion of service suitability, then the load

can be increased even more. An analysis of many beams (some results are given in [6]) with different sec- tion characteristics, loads, deformative properties, and models of bases showed that there is a substantial difference between the results of linear and nonlinear calculations. In all cases calculation with considera- tion of the real properties of reinforced concrete led to much more economical solutions than the linear

calculation.

The dependences B = B(M) used in the present work (excluding the stage in which the bending mo- ments are close to MI) have been confirmed by many theoretical and experimental investigations and have become popular in the designing of reinforced-concrete structures resting on rigid supports [I].

They are indisputably more substantiated than the hypothesis that a reinforced-concrete foundation

beam is ideally elastic and homogeneous.

The introduction of these dependences into the practice of calculating foundation beams will have a great economic effect and will make it possible to evaluate foundation beams from the viewpoint of the cri-

teria to which they should first of all correspond - the second arid third limiting states.

1.

2.

3.

4o

5.

6.

L I T E R A T U R E C I T E D

S° M. Krylov, Redistribution of Forces in Statically Indeterminate Reinforced-Concre te Structures [in Russian], Stroiizdat (1964). V. L Murashov, t~. E. Sigalov, and V. N. Baikov, Reinforced-Concre te Structures [in Russian], Gosstroi izdat (1962). B. F. Vasi l 'ev, I. L. Bogatkin, A. S. Zalesov, and L. A. Pan'shin, Calculation of Reinforced-Con- c re te St ructures for Strength, Deformations, and Formation and Opening of Cracks [in Russian], Stroiizdat (1965). Ya. B. L'vin, "Calculation of beams on a semi- inf ini te e las t ic mass and half-plane by the work meth- od," in: Investigations of the Theory of S t ruc tures [in Russian], NO. 5, Stroi izdat (1951). A, M. Ovechkin, Examples of Calculating Reinforced-Concre te Structures [in Russian], Vysshaya Shkola (1968). V. I. Solomin, V. P. Chirkov, and V. F. Tutynin, "Behavior of r e in fo rced -conc re t e beams on an e las - tic base with consideration of the specific p roper t ies of re inforced concre te ," Collection of Works No. 73 of the Chelyabinsk Polytechnic Institute, "investigations of Concrete and Reinforced Concrete" (1969).

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