X3. K. Smw RoY,~ J. N. B~~YoP~~~Y~ zmd D. P. IW.# ~~~~e~t of Civil Engiaeering, Regional Eking CoiIege, Durgapur-713 209, Mia

$Departmnt of Ciyil Enginemi~~g, Indian Institute of Technology, Kharagpur-7213Q2, India

Ahscraet-ms paper endeavmm to investigate the &I&t of Banges iu d#ep beams and to eampme the characteristic behaviour with thase af rectangular ones. Three flanged deep beam problems are analysed with depth-to-span ratios (H/L) of 1,2/3 and l/2 employing the finite element formulation with 2O-no~W isoparametrc &meats in a ~~-d~~~al state of stress. T&e authors gene&~I formulation has been mod&& to ignore the efR& of ~~0~~~ with B v&z9 to ana&ing pW eoate beams as a hom~~eo~ and is&o* materi& For ~~~~g the &fW of rift ts. The v&es of stress cxxwwxws e-x, us, a,, ‘cxy* t,q, an$ L, snd principal stresses q and ez or a, WQ ccquted in ~ondi~e~~~~~ form as eptined in the text. T%e pfinci@ stress contours am also ptotted for a spcc%c case of depth to span ratio of 1. A comparative study of the results reveals that a signifiican kmease in strength of Banged deep beams is noted over the m&mgular ones. Fkther, the increase in strength of both these types is also worth noting with the increase ia the H/L ratio.

I, INTRODUCITON

Deep beams in which depths are comparable to spans are widely ilsed in reinforced concrete ~~st~c~o~~ such as verticat ~ap~a~s in bridges* lotion waBs in buildings9 any other ~oad-~~~ walls, etc. It may be mentioned that the behaviour of flanged and mctanguiar deep beams is very different from that of ordinary flanged and rectangular shallow beams. A nmnber of jnv~ti~tors presented the c~~~s~ behaviour of alar deep beams as reviewed by the authors fZ$ Howems the smiks on the eilizt of the: flange of deep beams over the characteristic bebaviour of the re&sngular deep beams are very limited. Ray[2] analysed flanged deep beams based on the &s&al method and discussed the character- istic behaviour of f!anged deep beams (T-beam@ based on the results of tests booed by him. His analysis cotid hardly re&ct the effect of flange over the characteristic behaviour of rectangular deep ba&ms due to the adoption of the classicat method of analysis.

The objective of this paper is to study the charac- teristic behaviour of flanged deep beams and the e&t of flange on the performance of such a deep beam over that of the %Wtanguk%r deep beam fOF three

d%fema ~-~~-~~~ z%?h @f/L) of 1,2/S and t&z. A~~~y~ three flanged deep beam problems (called Problems 1, 2 and 3) are analysed employing the; 2%noded isoparametric element with grids of 4X4intheweba~d4x5j~~efl~~e~~~~ and without reinforcements. The symmetry of

dimensions and to&dings is considemd in the analysis” For the reinforced concrete flanged deep beams the tensile steel aeon ts consist of six 8 mm diameter bars io the bottom zone and vertk& stirrups of 6 mm diameter mild steel bars asps at 1sOmm centre-to-centre for ail the three beams, Furthermore, 6 mm diameter mild steel bars numbers i.ng IO, 6, and 4 for all the three types of deep beams of H/L 5= 1,2/3 and l/2, respectively~ are used as horizontal side-face ~~~o~rne~~~ fn addition, the %uzge is p~ovkkd with 8 mm diameter deformed bars in the form of stirrups at 150 mm centre-to- cantre distance across the span along with four straight deformed bars (8 mm diameter) in the longi- tudinal direction. Four 135” bends at either end ~~s~g of 8 mm diameter deformed bars are also provided to staff the jIm&on of the ftange and the web. The physical vision of the problems and two-point loadings piaced at one-third points are given in Fig. 1. The values of moduli of steel and concrete, E, and E,., and Poisson’s ratio, v, are takem as 2 x 10’ kg/c&, t53.85 kg&m2 and O,H, respects i&y.

For d the prop the results are presented using the foIlowing ~0n~rn~oR~ Parameters

e _ @X3 - 5’ % = {@% or ffp or ~~~~~~~~~

wlme &,, Hand L are the thickness of the web, depth and overall span, raspective~y, of the deep beams, and P fs the rn~~~ of the point load taken as unity for the analysis.

623

624 D. K. SINGHA ROY et al.

bf .WlOTH OF FLANGE

bw = WIDTH OF RIB

la) Front view (b) Cross-section at A-A

Fig. 1. Details of flanged deep beam problems.

3. THEORETICAL FORMULATION

Deep beams are analysed considering a three- dimensional state of stress and employing the 20- noded isoparametric element (Fig. 2). The stresses {a} at specified points on a given section of the beam

EC [DC1 = (1 + v)(l - 2v)

and

[RI = m4

l-v

V

V

I 0

0

0

Fig. 2. Parent element in 5-q-r; coordinates with global x-y-z axes.

(1) (E,/E,). It is assumed that the steel reinforcements are smeared in the concrete in an element.

The strains are obtained from

(2) (61 = m4, (4)

(3) where {d} = the displacement at nodes which is obtained from the standard relation knowing the stiffness matrix

PX 0 0000

0 PY 0000

0 0 p* 0 0 0

0 0 0000

0 0 0 000

0 0 0000

0 (1 - 2v)/2 0 0

0 0 (1 -2v)/2 0

0 0 0 (1 - 2v)/2

[~lTPIPII JI d5 dv 4 (5)

in which [B] is the strain-displacement relation and 1 J ) represents the determinant of the Jacobian matrix [J] with the following shape functions: Let

5i9 flir Ci= f1

NA = l/4( 1 - [*)(I + Wi)(l + CC0

where px, p,, and pz are the ratios of the reinforce- for nodes on < =0 and A = 17, 18, 19, 20

ments at the elemental level in x, y and z directions (Fig. l), respectively, and m is the modular ratio NB = l/4(1 + tl;i)(l - ~*)(l + C(i)

Comparative study of Banged and rectangular deep beams

for nodes on q = 0 and B = 10, 12, 14, 16

Nc = VW + t&)(1+ rMt)(l - C2)

for nodes on { = 0 and C = 9, 11, 13, 15

N, = l/411/2(1 + er,)cl + VI,)

x (1 + Kr) - 2(N, + NB + NJ

for comer nodes and i= 1,2,3,. ..,8, where (N,, + NB + NJ2 refers to only three nodes adjacent to node i.

4. RESULTS AND DJSCUSSION

The results are presented in two forms for the discussion. First, the principal stress trajectories are shown in Figs 3-8 at three different sections for a particular value of H/L = 1. The three speci&d sections are the longitudinal section through the middle plane marked as section (i), the section with the plane passing through the face of the flange and web of the beam along the span designated as section (ii) and the cross-section at mid-span referred to as section (iii). Moreover, the ~s~bution of stresses rS,, Csy and f_. are also given in Figs 9-l 1 in two specSed sections as indicated in the respective figures for the same value of H/L = 1. Secondly, the magnitudes of

Fig. 3. Compressive stress trajectories of section (i) (without . .

Fig. 4,

- P12 0.5.4

625

Compressive stress trajectories of section (ii) (with- out steel).

.O P bf = WIDTH OF FLANGE

b,=WIOTH OF RIB

tf =THICKNESS OF FLANGE

Fig. 5. Compressive atress trajectories of section (iii) (with- StUU). out steel).

626 D. K. SINGHA ROY et al.

Fig. 6.

P/2

bf =WIDTH OF FLANGE

b, = WIDTH OF RIB

t f = THICKNESS OF FLANGE

Tensile stress trajectories of section (i) (without Fig. 8. Tensile stress trajectories of section (iii) (without

L .l

o.y-\ ,I \y p ,#.pl :’ 1 , ‘,

/

P/2 0.5

I 1 -I- i

steel).

the peak values along with their locations are given for three stresses 6,, 6,, and fxY in Tables l-3 for three different values of H/L ratios of 1,2/3 and l/2. These tables also provide the respective results of the rec- tangular deep beams taken from an earlier paper [ 11.

The compressive isostatic stresses of intensity rang- ing from 0.1 to 0.5 (Figs 3 and 4) are concentrated under the load within a short depth (2 = 0.30-0.45 and jj = 0.90-l .O) from the top edge of the deep beam for both sections (i) and (ii) and thereafter these stresses are found to be fairly distributed throughout the depth of the beam. It is further noticed that the compressive isostatic lines of very high intensity

Fig. 7. Tensile stress trajectories of section (ii) (without - 6x steel). Fig. 9. Variation of b, at (a) Z = 0.3472 and (b) f = 0.50.

Comparative study of tIanged and rectangulsr deep beams 627

Variation of a’, at(a)2 = 0.3472 and (I$ f =O.50.

- hy

Fig. 11. Variation of Zxy at (a) 1 = 0.0833 and (b) 2 = 0.3125.

(rangmg from 1.0 to 3-O) radiate from the edge of the support locations and confmed within a depth expected to be widened graduz& tcr form a continu- 9 = 0.073-0.591 (_% = 0.0~=50~= In some situatioris ous band and thereby the arching a&iou may be these compressive isostatic lines are fotmd in the form observed. On the other hand, in section (iii) of of either open or dosed loops. Few of these are Fig. 5 wider pockets of compressive isostatic lines located in a diagonal fashion discretely. Under the (0.1-3.0) are formed throughout the depth of the imposition of increasing loads these loops are flange width and pockets of compressive isostatic

Table 1. Peak values of horizontal bending stress (ex) and their locations Horizontal bending stresses (a’,)

2 = 0.3472 R = 0.50

Flancmeep I2ectanguIar deep beams Plan?z!l?ep

Rectangular deep beams

Depthtp

s~~~~~ With

w~~ut With Without Without With Without

steel St4 steef zz steei sted steel

Negative 0.63 0.57 4.33 3.47 0.57 0.53 1.10 1.00

$LJ$ 1.00 1.00 1.00 1.00 0.86 0.86 0.90 0.90 1

positive 1.07 0.77 1.083 0.80 0.813 0.733 1.27 1 .oo value (j-1

iizzi$” 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Negative 1.00 0.90 7.80 5.60 0.85 0.78 2.10 1.50 value (-1

: 10cauon Peak. (u’)

l.O” 1.00 1.00 1.90 0.86 0.86 0.905 0.89

Positive 1.40 1.15 1.65 1.20 1.20 1.00 1.92 1.50 value (,- )

E%i$’ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Negative 1.20 1.10 8.00 5.80 1.05 0.95 3.40 F$; ($5’

2.20

1 location 1.00 1.00 1AM 2 1.00 0.86 0.86 0.91 0.875

Positive 1.80 1.45 2.25 1.80 1.80 1.70

~~~~~

2.48 2.10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Negative value indicates compression; positive value indiites tension.

628 D. K. SINGHA ROY et al.

Depth to span ratio

(H/L)

Table 2. Peak values of vertical stress (~7~) and their locations

Vertical stress (eY)

2 = 0.3472 f = 0.50

Flancateep Rectangular deep beams

Flanra;;eep Rectangular deep beams

With Without With Without With Without With Without steel steel steel steel steel steel steel steel

Negative 1.70 1.73 4.53 5.0 1.53 1.57 1.07 1.40 value (-)

;+=l:i.) 0.83 0.83 1 .oo 1 .oo 0.83 0.83 0.75 0.75 1

Positive 0.020 0.057 NA NA 0.47 0.57 NA NA value (+)

;zl$) 0.03 0.03 NA NA 1.00 1 .oo NA NA

Negative 1.65 1.70 4.60 5.30 1.35 1.40 0.875 1.10 value ( - ) Peak (jj)

2 location 0.83 0.83 1.00 1 .oo 0.83 0.83 0.70 0.65 5

Positive 0.019 0.050 NA NA 0.40 0.50 NA NA value (-)

Fati.) 0.03 0.03 NA NA 1 .oo 1 .oo NA NA

Negative 1.52 1.55 4.70 5.50 1.05 1.10 0.70 0.90 value ( - ) Peak (9)

1 location 0.83 0.83 1.00 1 .oo 0.83 0.83 0.65 0.63 z L

Positive 0.018 0.025 NA 0.35 0.40 value (+)

;z;J;) 0.03 0.03 NA NA 1.00 1 .oo NA NA

Negative value indicates compression; positive value indicates tension. NA: Not applicable, i.e., the type of stress is absent.

lines of comparatively lower intensity (0.1-1.0) are seen in the web. The tensile isostatic lines of intensity 0.1-0.7 are concentrated under the loaded area and are fairly distributed in the depth of the flanged deep beams in the form of loops within jj = 0.0-0.75 for both sections (i) and (ii) (Figs 6 and 7). Section (iii) (Fig. 8) exhibits concentration of the tensile isostatic lines (0.01-0.7) at discrete pockets scattered through- out the depth of web and flange width. In general, the tensile isostatic lines are found either in a closed or open loop form and are placed in different zones, namely under load points, two-thirds to one quarter depth, and the bottom zone of the beam. Some of these are also oriented in a diagonal fashion. The shape and pattern of both tensile and compressive isostatic lines of flanged deep beam with steel reinforcements are found to be almost similar but a little distorted as compared to those of the flanged deep beam without steel reinforcements in all the three sections. This may be primarily due to the influence of steel reinforcements showing higher load- carrying capacity of flanged deep beam with steel reinforcements.

In an earlier paper [l] rectangular deep beams were studied both with and without steel reinforcements. It has been observed that the isostatic compressive lines of high intensity are radiating from the edge of the support location as in the case of flanged deep beam. However, the pressure bulbs created under the load- ing points have also been observed in the case of a rectangular beam which is not so in the case of a flanged beam. With regard to the tensile isostatic lines (O.l-0.7), the rectangular deep beam has shown fairly large open loops while flanged deep beam exhibits small as well as large loops either in open or closed forms.

It is well known that the tensile and compressive isostatic lines are of equal intensities and are dis- tributed symmetrically with respect to the neutral axis of the normally proportioned flanged and rectangular beams (H/L = l/10-1/12) both with and without steel reinforcements. Accordingly, the magnitude of the total compressive force is same as that of the total tensile force and the bending action is predominant. Thus, the stresses in these beams are developed primarily due to the flexure associated with shear

Comparative study of flanged and rectangular deep beams 629

Table 3. Peak values of shear S- (r,) and their kMX~OllS

Shear stress (f,)

R = 0.0833 R = 0.3125

magz;=P Rectangular flanf&&ep Rectangular deepbeams deepbeams

Depth to span ratio With Without With Without With Without With Without

(H/L) steel Steel steel steel steel steel Steel Steel

Negative 0.15 0.142 NA NA NA NA NA NA valie ( - ) p”ocJi)

0.98 0.98 NA NA NA NA NA NA 1

Positive 0.875 0.90 1.63 1.933 0.57 0.60 0.95 0.967 value ( + )

Z!Z? 0.03 0.03 0.03 0.03 0.19 0.19 0.90 0.90

Negative 0.138 0.125 NA NA 0.025 0.05 NA NA valiie (-) Pd. (1)

$ location 0.98 0.98 NA NA 0.02 0.02 NA NA J

Positive 1.20 1.23 1.95 2.50 0.70 0.725 1.20 1.20 value (+)

;=&..z) 0.03 0.03 0.03 0.03 0.40 0.40 0.88 0.88

Negative 0.125 0.113 NA NA 0.038 0.065 NA NA valk (-) Peak (7)

1 location 0.98 0.98 NA NA 0.02 K L

Positive 1.59 1.63 2.425 3.375 0.95 value (_+)

Fg\J;) 0.03 0.03 0.03 0.03 0.50

NA: Not applicable, i.e., the particular type of stress is absent.

0.02 NA NA

1.00 1.50 1.50

0.50 0.85 0.85

while the vertical stress component (cQ) is absent. On the other hand, for plain and reinforced concrete flanged and rectangular deep beams the tensile and compressive isostatic lines are not symmetricany dis- tributed nor are the intensities equal in magnitude. Moreover, there may be arching action present in the deep beams as mentioned earlier. This difference in the behaviour of deep beams may be due to the presence of the vertical stress component (c?~,) causing direct transfer of the load from its points of appli- cation to the support through its compressive action associated with high magnitude of shear. Thus the simple beam theory (with equal tension and com- pression) cannot be applied in the case of these deep beams due to the different patterns and mechanisms of load transfer.

Figures 9 and 10 present the variation of the horizontal bending stress (5,) and vertical stress (C,), respectively, along 2 = 0.3472 and 0.50 both with and without steel reinforcements for a ilanged deep beam having H/L = 1. Moreover, Fig. 11 shows shearing stresses (T,) for a tlanged deep beam with H/L = 1 along 2 = 0.0833 and 0.3125 both with and without

steel reinforcements, respectively. It is quite evident from these figures and those of the rectangular deep beam of [l] that the shape and pattern of distribution of I?~, cF,, and ?_, are somewhat similar throughout the depth of both the flanged and rectangular deep beams with and without the presence of steel reinforcements. However, the magnitude of these stresses is seen to vary for both the types of deep beams as indicated below. The horizontal bending stress (rTX) at any depth of both the plain and reinforced concrete flanged deep beams is lower than that of the rectangu- lar deep beams both with and without steel reinforce- ments. Similar results are also found for shear stresses. On the other hand, the vertical stresses (eY) both with and without the reinforcements of flanged deep beam show much lower values for y = 0.90-l .O and remaining depth (1 = 0.0-0.90) exhibits fairly close values in comparison to those of the rectangular deep beam for both with and without steel reinforce- ments. Moreover, the incorporation of steel into the concrete for all cases of deep beams shows lower values of stresses (ZX, eY and fV) in comparison to the plain concrete deep beams. It is, therefore+

630 2>. K. SINGHA ROY et al.

evident from the above discussion that the load- carrying capacity is comparativeiy higher for Banged deep beam than the same for the reEt~~~nlar deep beams

Tables I-3 present the peak values of the horizon- tal bending stress feX)$ vertical stress (ir?,) and shear stress <?l,)* res~ctiveIy, afang with their locations at different sections as detailed befow For both the Banged and rectangular deep beams with and without steel reinforcements having three different depth to span ratios, namely, H/L = I, 213 and K/2. The peak values and their locations of rectangular deep beams have also been furnished in these tables as a ready reference [I]. The abbreviation ‘NA’ (not applicable) is used when the value does not exist for a particular stress.

The peak values of 6, (both positive and negative) are seen to occur almost at the same locations for both the Banged and rectangular deep beams with and without steel ~info~ents of a particular value of the H/L ratio as furnished in Table 1 at two different sections [Z = 0.3472 and @.t.sO), While the peak values of gY (both positive and negative) at the same sections are found to occur at different locations for both with and without steel reinforcements of flanged and rectangular deep beams for the same value of Ii/L as seen from Table 2. On the other hand, the peak values of TX,” (both positive and negative) are seen to occur at the same locations in section 2 =0.0833 and at different locations in section 9 = 0.3 12.5 for both types of deep beams with and without steel re~nfor~~ents For a particular value of H/L. The presence of steel ~infor~ments appears to have marginal influence in shifting the Iocation of the above peak vafue for the three types of stress sX, gY and fXY in both flanged and rectangular deep beams for all the three different H/L ratios. These values of stresses fgX,, o’,, and 2;;,) are taken up to study the effects of the H/L ratios for these beams.

A critical study of Table 1 shows that the magni- tudes of the positive and negative horizontal bending stress (ai,) are decreasmg at P = 0.3472 and 0.50 with increasing depth to span ratios <H./L) for both with and without steel r~nfo~~me~ts of flanged and reetanguhn deep beams. The same trend is observed from Table 2 for the negative vertical stress (sY) at f 5 0.3472 for both with and without steeE reinforce- ments of rectangular deep beams. However, a reverse trend is observed in the flanged deep beam at 2 = 0.3472 and 0.50 and in rectangular deep beam at p = 0.50 for both with and withaut steel reinforce- ments, i.e. the positive or negative verd~al stresses (gY) are seen to increase with the increase in depth to span ratios (H/L). On the other hand, the magnitude of the positive shear stress ii,> as seen from Table 3 decreases with increasing Hjt ratios for both with and without steeI ~i~o~ments of Banged and rectanguIar deep beams at ah se&on% name& f = 0.0833 and 0.3125 (same as that of ci,). More-

over, the negative shear (a) for both with and without steel reinf~~ments of flanged deep beams increases with increasing H/L ratios at .i = 0.0833 and at section W =: 0.3125 the reverse trend is observed for flanged deep beam with and without steef reinforcements for aII &f/L ratios except when H/L = 1.

A comparative study of the three stresses (gX, CT,, and QY) for pgain and reinforced concrete ganged and rectangular deep beams shows the fotfowing, The peak vafues of ~o~zontai bending stress {ii;), either positive or negative, are greater for a rectangular deep beam than for those of the corresponding Banged deep beam at .E = 0.3472 and 0.50 for all the H/L ratios. A similar trend is observed for the vertical stress (#),) at f = 0.3472 and shear stress (f,) at X = 0.0833 and 0,3125 for all the H/L ratios, On the other hand, the vertical stress at 2 = 0.50 is higher for a flanged deep beam than those of the rectangular deep beam both with and without steef reinforce- ments for ah values of H/L ratios. Furthe~ore~ the horizontal bending stress (CT,) at these sections are more for Aanged and rectangular deep beams with steel reinforcements for ah vafues of H/L. The reverse trand is observed for positive or negative vertical stress (ir,) at .li- = 0.3472 and 0.50, and for positive shear stress (g) at .< = 0.0833 and 0.3125 for both the flanged and rectangular reinforced concrete deep beams for all H/L ratios. While negative shear stress at all sections are more for reinforced concrete deep beam compared to those of the pfain concrete flanged deep beam for the same H/L do.

The above discussion supports the normally expected su~e~o~ty of Ranged deep beam to the rectangular deep beam for the same H/L ratio with respect to the load-carrying capacity. Again it is also seen that the characteristic strength of both flanged and rectangular deep beams increases with the increase of the H/X, ratio.

5. CONCLUSONS

The F&.X& of tensile and compressive isostatic Iines and the stresses c?,, c?,# and TV of both plain and reinforced concrete deep beams with and without flange reveaIs the characteristic behaviour and intrin- sic features of these deep beams for a particufar case of H/L =: 1. A detailed study of the magnitudes and location of peak vaXnes of these deep beams shows the expected increase of strength of the reinforced concrete deep beams when compared to those of plain concrete deep beams for a specific value of H]L. Such expwted increase of strength is also found with the presence of flange when compared to the WtanguIar ones for both plain and reinforced con- crete deep beams. Moreover, the strength of both p&n and reinforced concrete deep beams increases with the increase of H/L ratio for both the rectangu- Iar and Banged cases. The validity of the simpIe beam

Comparative study of flanged and rectangular deep beams 631

theory is not applicable for the flanged deep heam CYBER-180/84OA sy&m belonging to the Computer

as was found for the rectangular deep heam in Centre, IIT, Kharagpur,

the earlier paper [l] due to the difference in the hehaviour and mechanism of load transfer in the deep beams. 1.

Ack~o~le~g~nts-~~ paper is based on a part of the doctoral work carried out by the first author under the supervision of the second and third authors at IIT, Kharagpur. The ~mpu~tion~ work was done using the

2.

REFERFJNCES

D. K. Singha Roy, J. N. Bandyopadhyay and D. P. Ray, Comparative analysis of deep beams. Cornput. Struct. q459-467 (1992). D. P. Ray, An investigation on ultimate strength of reinforced concrete deep beams of rectangular and tee-sections. Thesis submitted for Ph.D. degree, IIT, Kharagpur (1962).