INSTABILITY OF SLENDER CONCRETE DEEP BEAM-PANELS - COMPARISON OF TEST RESULTS WITH EXISTING DESIGN AIDS –

M. Chemrouk*, University of Science & Technology (USTHB), Algeria F. K. Kong, University of Newcastle UponTyne, United Kingdom

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28th Conference on OUR WORLD IN CONCRETE & STRUCTURES: 28 - 29 August 2003, Singapore

INSTABILITY OF SLENDER CONCRETE DEEP BEAM-PANELS - COMPARISON OF TEST RESULTS WITH EXISTING DESIGN AIDS -

M. Chemrouk*, University of Science & Technology (USTHB), Algeria F. K. Kong, University of Newcastle UponTyne, United Kingdom

Abstract

Tests conducted by the authors have revealed that buckling is a possible design criterion for deep beams particularly for the more slender sections. Of the major deep beam design documents currently in use, the CIRIA guide is the only one that gives guideline for the buckling strength of deep beam­panels. It is stated, however, that in the absence of experimental evidence, such recommendations were based on theoretical studies and engineering judgement. The Portland Cement Association Design Aid for the buckling of the tilt-up load-bearing walls is the only other design document which could be adapted for the buckling of beam-panels, though its use is limited by a number of parameters. The present experimental programme, which consists of 7 slender concrete deep beams having height/thickness (h/b) ratios in the range of 20 to 70, models as closely as possible the PCA Design Aid and aims at comparing for the first time the two buckling design procedures. The tests show that buckling failure occurs at a significantly reduced strength capacity, particularly for higher slenderness (h/b ratios). Deep beams having h/b ratios of around 70 should be avoided in practice as they might buckle at less then a 1/3 of their strength capacity. The test results also show that, while both design methods are safe against buckling, the PCA method, when applicable, is easier to use, less conservative and hence more accurate. In contrast, the CIRIA buckling provisions have a broader use but are more conservative particularly for the very slender sections and have a more complicated use. Indeed, first time users may find them difficult to follow.

Keywords: Instability, Slender Concrete Deep Beam, Slenderness Ratio, Buckling Strength, Tilt-up Load-bearing Walls, Diagonal Cracks, Horizontal Cracks, Local Crushing, Sheer Strength.

1. INTRODUCTION In the past, researchers and designers alike have always avoided the problem of buckling in concrete deep beams by opting for stocky sections. With the advance in materials technology, higher-strength concrete has become commercially available and, as a result of its expanding use, more slender members will be used in the construction industry such as slender concrete deep beams in high rise buildings or in offshore structures. Consequently, stability rather than strength requirements might be the main criterion for design in order to avoid premature buckling failures. Of the major deep beam design documents currently in use throughout the world, namely the American building code ACI (318-89) (1), the Canadian building code CAN3-A23.3-M84 (2) the model code CEB-FIP (3) and the CIRIA Guide N° 2 (4), the CIRIA Guide is the only one that gives recommendations on the buckling strength of deep beams. The Guide states explicitly that buckling should not be disregarded as a possible failure criterion. However, in the absence of experimental data, the CIRIA buckling provisions were based on theoretical studies and engineering judgement; to quote from the CIRIA Guide (4) .. there is no experimental data to substantiate these procedures n. In the late seventies, the development of the tilt-up method of construction, particularly in north America, highlighted the need for a buckling design procedure to deal with slender load-bearing panels. The Portland Cement Association responded to such need by publishing a buckling design aid for tilt-up load bearing walls (5) based on numerical analYSis on column models. As for the CIRIA Guide, the

253

PCA Design Aid is not founded on experimental data on the buckling of wall-like panels or deep beam-panels. The scarcity of experiments on such structural members is believed to be due to the complexity of the buckling tests which are difficult to carry out and require attention to details in order to achieve reliable results and to prevent any risks such as injury to personnel or damage to equipment.

In the past, the authors, jointly with others, have reported for the first time buckling tests on 38 slender concrete deep beams (6) where the CIRIA buckling provisions were assessed. The PCA buckling design method could not be assessed since the tests reported did not fulfil the requirements of the PCA Design Aid which, strictly speaking, has a more specific use. In the present tests, the 7 deep beam-panels with h/b ratios ranging from 20 to 70 were conceived so as to meet the requirements of the PCA Design Aid and hence a comparison of the two buckling design manuals could be made for the first time. It is believed that the 7 deeper beam-panels (span/height Uh=1.0) together with the 38 slender deep beams (Uh=1.4) reported earlier by the authors jointly with others (6) represent most of the experimental data available in the literature and that deep beams with slenderness ratio as high as 70 such as reported in the present tests are among the highest ever tested though structural concrete members with such a high slenderness ratio are very few in practice.

2. TESTING PROGRAMME 2.1 Description of the specimens

The test specimens (Fig.1) consisted of 7 slender reinforced concrete deep beams of height h 1400 mm, overall length 1700 mm and simple span L 1400mm giving a span/height ratio of 1.0. The thickness b varied from 70 mm to 20 mm giving height/thickness ratios h/b (slenderness ratios) ranging from 20 to 70. The PCA Design Aid, based mainly on column analysis, does not recommend the use of h/b ratio higher than 50" because of luck of experimental data" to quote from it. The main reinforcement was so designed as to avoid flexural failure and consisted of high yield deformed bars of either 10 mm or 12 mm size used in numbers of 3 or 6 depending on the thickness of the beam (table 1). The web reinforcement used followed one of the patterns adopted in the PCA Design Aid, namely a central layer of vertical bars restrained by horizontal ones (Fig.2). Plain round bars of 6 mm diameter were used at various spacing to achieve a steel ratio of 0.5% both vertically and horizontally (Fig.2, table 1) to model one of the design tables of the PCA Design Aid, namely table A 1. In addition, reinforcement cages were used at the loading and support regions to confine concrete and avoid premature local crushing. Due to the concrete strength limitation of the PCA Design Aid (f c less than 4000 psi), a sand based concrete mix, with a water/cement ratio of 0.7 and an aggregate/cement ratio of 5.0 was used; this concrete mix resulted in a cube strength of around 40 N/mm2 (table 1). The cylinder compressive strength was deduced from the cube strength as f c= 0.8 feu , a relation widely used and confirmed experimentally by the authors (7). For the beam notation, the series of beams is indicated by the letter B followed by the slenderness ratio after the first hyphen and the e/b ratio after the second hyphen.

2.2 Testing Procedure Strictly speaking, the design tables given in the PCA Design Aid are meant for panels of practical scale with thickness ranging between 140 mm and 241 mm and load-eccentricities between 25 mm and 210 mm. It is obviously rather difficult to handle the testing of such specimens in laboratory. However, following recent tests by the authors jointly with other (6), the buckling strength was found to depend more on the eccentricity/thickness ratio e/b than on the two parameters taken individually. In addition, the e/b ratio, being dimensionless, is more convenient to describe the buckling behaviour of a deep beam-panel. Consequently, e/b ratio of 0.182 corresponding to an eccentricity e of 1.0 inch and a thickness b of 5.5 inches (1.0/5.5=0.182 ) in the PCA document (table A1) was used in the present tests. The beams were tested under two point eccentric loads (e = 0.182 b) and rested on two simple supports with concentric reactions as shown in (Fig.1-b) to give a total shear-span/height ratio and a clear shear-span/height ratio of 0.29 and 0.12 respectively (Fig 1-a; table 1). Lateral displacements were measured at discrete positions with 15 LVDT transducers, placed in 3 columns of 5 each, one column above each support and one at mid-span. In addition, due to the sudden nature of buckling (6), two dial gauges were placed on the back to signal an impending failure. This safety precaution proved in the past (6) very useful in giving warning of buckling failure. Strains were monitored with demountable strain transducers in various locations on the back face of the beams. On the front face, strains were measured with demec gauges at mid-span across the depth to assess the longitudinal strain distribution in deeper beams. To facilitate crack observation and

254

locations, a 100 mm square grid was marked on the white-washed faces of the beams. The measurements were taken at each load increment and the hand ones were stopped when buckling danger was felt (lateral deflection exceeding 4.0 mm ) .

3. TEST RESULTS AND DISCUSSION 3.1 Crack Patterns, Failure Modes and Ultimate Loads

Figure 3 shows the crack patterns up to failure of the beams. The load at which each crack was first observed is indicated together with the extent of the crack at that load. The general trend of the crack development is similar to that described elsewhere (8 ; 13) for slender deep beams with the flexural cracks being the first to form in the maximum bending moment region. The maximum flexural crack width reached the 0.3 mm serviceability in the less slender beams (0.3 mm in 8-20-0.182; 0.36mm in 8-25-0.182; 0.40mm in 8-30-0.182) before closing up after full development of diagonal cracking. One common feature with flexural cracking in these deeper beams ( Uh = 1.0 ) is that it extended upward to as high as 0.75h within the maximum bending moment region and was maximum in width above the tension steel, in some cases, at mid-height region (8-25-0.182; 8-30-0.182). This suggests that in the deeper beams the tensile steel should not be contained in a narrow region at the bottom but distributed at least over half the depth from the bottom. Indeed strain measurements have revealed an important tension zone above mid-height (Fig 4) in addition to the one at the soffit. These two tension zones are linked with an almost unstressed zone which becomes tensile after full development of flexural cracking. Diagonal cracks formed within the shear-spans from support to loading points and were very long even when first observed. Their maximum width exceeded by far the 0.3 mm limit particularly in the less slender beams and caused splitting failure in beam 8-25-0.182. With the beams having the same load-eccentricity/thickness ratio e/b, the failure mode depended mainly on the slenderness ratio h/b :

-For h/b ratios greater than 25, failure was by buckling. The beam split horizontally approximately along mid-height (Fig.3). Horizontal cracks appeared across the full length of the beam at failure and hence could not be used as a warning sign for collapse. In some cases, the beam may appear to be stable after the final load increment when a sudden buckling failure occurs. With dial gauges placed at the theoretical critical section, the prominent failure could be detected by the continuous creeping of these dial gauges. From the cracking patterns of Fig.3 and from the failure loads given in table 2 it can be deduced that the buckling strength was about 30% greater than the diagonal cracking load for beams with h/b ratio of 30 and 35 (8-30-0.182; 8-35-0.182), 20 % greater for those with h/b ratio of 40 and 50 (8-40-0.182; 8-50-0.182 ) and that with htb ratio of 70 buckled at the same load at which the diagonal crack formed ( 8-70-0.182) and hence diagonal cracking evolution could not be possibly used as a warning sign of failure as suggested earlier by Kong et al. (6).

- For htb ratios smaller than or equal to 25, the beams failed either in shear (8-25-0.182) or at the bearing (8-20-0.182). However, the measured lateral defections indicatey that even beam (8-25-0.182) which failed in shear was on the verge of buckling; its maximum lateral deflection just prior to failure was 6.0 mm as recorded by both a dial gauge and displacement transducers above one of the supports, an order of magnitude similar to those recorded for the beams that buckled. It should also be pointed out that 'buckling' in this work is regarded as failure with pronounced lateral deflection. Following this argument, beam 8-25-0.182 was assumed to have reached its ultimate shear as well as buckling strength capacities. In contrast, beam 8-20-0.182 being the least slender, did not reach its buckling nor its shear capacities when it failed prematurely by crushing at the loading point despite additional confining reinforcement. In practice however, such failure could be avoided by the use of high strength concrete. A close examination of the lateral displacement profiles (Fig.5) reveals that for such beam the maximum deflection was 3.5 mm despite that the ultimate load was highest. 8y experience gained from these tests and from others (6), buckling failure is likely to occur when the lateral deflection exceeds 4.0 mm.

It would seem that buckling failure is most likely to occur with slenderness ratio htb of 25 and up when the load is eccentrically applied with a significant reduction in strength for h/b ratios above 35 (table2). The measured ultimate loads of the 7 beams are given in the table 2. The loads based on the shear strength capacity as estimated by the modified Kong-CIRIA formulae for slender deep beams (8) are also given in the same table. It can be seen that the beams buckled at lower loads than their shear capacities, particularly for the higher h/b ratios. Figure 6 below shows clearly that, for a constant etb ratio, the buckling strength is very much dependent on the h/b ratio and decreases sharply as this ratio increased .

255

3.2 Lateral Displacements Figure 5 shows the lateral displacement profiles over each support and at mid-span section. It can be seen that the maximum lateral deflection often occurred at mid-height over one of the two supports. In general, these deflections were small until just before failure when the beam would creep rapidly towards buckling collapse. The maximum recorded deflection was 7.6 mm (8-40-0.182) just prior to failure. To a naked eye, such displacement may not be evident and the beam may appear to be straight. Such is the imminent buckling danger without visible warning. Figure 7 shows the load against the maximum lateral deflection at mid-height for all the beams. It can be seen that, with the exception of the least slender beams (8-20-0.182), all the others exhibited clearly a 3-stage behaviour: - An initial stage where lateral deflections are relatively small. - An second stage with an increased rate of change of deflection with load, flattening the curves from their initial slope. - A third stage where, following a small increase in load, the deflection would increase continuously until failure, reflecting the continuous creeping towards failure as observed through the dial gauges. This last stage was most evident in beams with high h/b ratio (8-40-0.182; 8-50-0.182; 8-70-0.182). 8eam 8-20-0.182 exhibited only the first two stages of behaviour and hence did not fail by buckling.

4. PREDICTION OF THE BUCKLING STRENGTH 4.1 The CIRIA Guide Prediction

Strictly speaking, the CIRIA guidelines are intended for designing the reinforcement against buckling and are not directly applicable for predicting ultimate buckling strengths. However, the authors, jOintly with others, have presented a way of adapting these guidelines to predict the buckling loads for deep beams. Interested readers can consult references (9 ; 10) where the procedure is clearly explained with illustrative examples. As a brief explanation, the CIRIA buckling guidelines recommend that the slenderness effect should be taken into account when a deep beam can not be defined as a short braced wall (clause 1.2.4 of 8S 8110). For this, the guide assumes that a deep beam is made up of an assembly of unit width column strips spanning vertically and horizontally and emphasises on the determination of the effective height and length of the deep beam panel. The 'additional moment concept' approach is then used as for slender columns. The CIRIA guide gives three methods for the determination of the effective height and length, namely: -The supplementary rules -The single-panel method -The two-panel method. The CIRIA guide specifies that the three methods are to be used in conjunction with CP110 (12). However, since this was superseded by 8S 8110 (11), it is felt more appropriate to use them in conjunction with 8S 8110. In table 2, the factors of safety, defined as the measured buckling load/predicated buckling load, provided by the three methods are indicated by Rsr, Rsp, Rip where subscript sr is for supplementary rules, subscript sp is for single panel method and tp that for the two panel method. The following comments can de made: - The CIRIA methods are very conservative with mean safety factors of Rsr = 41.1 ; Rsp = 19.1; Rip = 7.4. Comparatively, the two panel method gave the most realistic results of the three. Table 2 shows that both the supplementary rules and the single panel method are unduly conservative. For the three methods, the degree of conservatism increases as the slenderness ratio h/b increases; such trend is inhibited in the 'additional moment concept'. -For hlb ratio in excess of 40, the two panel method is expected to give safety factors of around 10 or slightly more. It can be argued, however, that due to the nature of buckling often sudden and unpredictable and the rare occurrence in practice of very slender deep beams (h/b >40), the two -panel method results can be considered as acceptable. -The average safety factor for h/b ratios between 25 and 40, representing the most likely range to be used in practice, is 4.1 which should not be considered as high for buckling failure. 80th, the supplementary rules and the single panel method, need more refinement.

4.2 The PCA Prediction To a certain extent, the test specimens were designed to fulfil the requirements of the PCA Design Aid. However, the section properties, specifically the thickness and the load eccentricity were scaled down for ease of laboratory handling. The scaling down has followed the argument (6) that the slenderness ratio h/b and the eccentricity thickness ratio elb are the most important parameters affecting the buckling behaviour. Indeed, a close examination of the PCA tables (5) reveals that for

256

the same e/b ratio and the same steel ratio the load capacity factors are independent of the panel thickness b and the eccentricity e taken as separate factors and depend only on h/b ratio. An example for e/b =0.5 and a steel ratio of 0.25 is reproduced here:

b (mm) e( mm) hlb 20 30 40 50

140.0 70.0 0.110 0.051 0.030 0.017 (table A 1 of PCA) 165.0 82.5 0.110 0.050 0.026 0.018 (table AS of PCA) 190.0 95.0 0.110 0.054 0.030 0.010 (table A9 of PCA)

Following this, the use of the PCA method to the present beam-panels seems to be justified. The e/b ratio of 0.182 adopted in the present tests corresponds to table A 1 with a steel ratio of 0.5 % in the PCA Design Aid. Table 2 shows the buckling loads for the tested beams predicted by the PCA method together with the safety factors indicated by a subscript pca to R. It can be seen from table 2 that, compared to the two-panel method of the CIRIA guide, the PCA method gives a safe and more accurate prediction of the buckling strength with an average safety factor of 2.76 for the beams for which the method is applicable. For these beams (h/b between 25 and 50 ), the two-panel method gave a safety factor of 6.68 and hence appears to be twice more conservative than the PCA method. For h/b ratios between 25 and 40, the PCA method gives a safety factor of 2.33 as compared to almost twice this value ( 4.10) given by the two-panel-method. Moreover, the PCA method is easier to use and hence more practical compared to the two-panel method given in the CIRIA guide. However, the application of the PCA method is limited by a number of parameters such as the material properties ( concrete f c ::; 4000 Psi; reinforcement fy ::; 60000 Psi ), the steel ratio (between 0.25% and 0.75% ), the loading arrangement ( eccentricity at the load only) and a slenderness ratio not exceeding 50. The relatively more conservative two-panel method of the CIRIA guide, though more complicated in use, copes with wider ranges of these parameters. Furthermore, for slenderness ratio hlb between 20 and 25, the PCA may not be safe enough, though bearing capacity or shear may govern the design for such cases. Beam B-20-0.182 had 3.5 mm maximum lateral deflection when it crushed at the bearing. From experience, if such beam did not fail at the bearing it could not have sustained more than 15 % of the 800 kN bearing failure load and the safety factor would have been 1.1 according to the PCA method, hence, insufficient against buckling.

5. CONCLUSIONS a- The failure mode of slender concrete deep beams is strongly dependent on the

height/thickness ratio h/b and the eccentricitylthickness ratio e/b. Beams with h/b ratios as high as 50 failed in shear when the e/b ratio was zero in previous tests (6; 8). In the present tests, deep beams of slenderness ratio as low as 25 buckled when the load was applied eccentrically (e/b = 0.182).

b- Buckling failure is accompanied by a significant reduction in strength which could be as high as high 50 % for the upper range of slenderness ratio h/b. Deep beams having slenderness ratio of around 70 should be avoided in practice as they might buckle at less than a 1/3 of their strength capacity .

c- The buckling design recommendations in both the CIRIA guide and the PCA Design Aid were in general found to be safe for use. Of the three methods given in the CIRIA guide, the two­panel method is the most realistic, though still conservative for higher h/b ratios.The supplementary rules and the singles-panel method are excessively conservative and need more refinement. The designer using the CIRIA guide is advised to work with the two-panel method despite its complicated use.

d- Between the two-panel method of the CIRIA guide and the PCA method, the latter is more accurate and easier to use for designers with the two-panel method giving safety factors twice as much. However, while the PCA method has its use restricted by many parameters, the two­panel method covers a wider range of deep beam-panels. Moreover, the PCA method may not be safe enough for the buckling of deep beams with smaller h/b ratios, though in this range of slenderness, other failure criterion may govern the design.

257

REFERENCES

[1] ACI Committee 318 "Building Code Requirements for Reinforced Concrete", ACI(318-89) and the Commentary (ACI 318R-89); American Concrete Institute, Detroit, July 1990.

[2] Canadian Standards Association: "Design of Concrete Structures for Buildings (CAN-A23.3-M84),,; CSA, Rexdale, Ontario; December 1984.

[3] CEB-FIP Model Code for Concrete Structures; Cement and Concrete Association(British Cement Association), London 1978.

[4] CIRIA Guide No.2 "The Design of Deep Beams in Reinforced Concrete"; Construction Industry Research and Information Association; London 1977 (Reprinted 84).

[5] Portland Cement Association (PCA) ; "Tilt-Up load bearing walls: A Design Aid"; Publication No.EB074.02D; Illinois, 1977,27 pages.

[6] Kong, F.K.; Garcia, R.C.; Paine, J.P.; Wong, H.H.A.; Tang, C.W.J.; and Chemrouk, M.; "Strength and Stability of Slender Concrete Deep Beams"; The Structural Engineer; vo1.64B; No.3, September 1986, pp.44-56.

[7] Chemrouk, M. "Slender Concrete deep beams: Behaviour, Serviceability and Strength"; Ph.D Thesis; University of Newcastle Upon Tyne, England, 1988.

[8] Chemrouk, M. and Kong, F.K; "Diagonal Cracking and Ultimate Shear Strength of slender Concrete Deep Beams"; paper accepted for publication in "Advances in Structural Engineering - an International Journal". Hong Kong Polytechnic University, Hong Kong.

[9] Kong, F.K.; Wong, H.H.A.; Tang, C.W.J.; and Chemrouk, M.; "Worked Examples on the use of the CIRIA Guide No.2 to calculate the buckling strengths of slender Deep Beams"; Technical Report STRUCT 7/3A; Department of Civil Engineering, University of Newcastle Upon Tyne, 1987, 17pp.

[10] Kong, F.K., Chemrouk, M, Tang, C.W.J. and Wong, H.H.A.; "Worked Examples on the buckling design of slender concrete deep beams"; Technical Report SRUCT6/4A; Department of Civil Engineering, University of Newcastle upon-Tyne; 1986; 22pp.

[11] BS 8110; "The Structural Use of Concrete"; Part 1; British Standards Institution London; 1985.

[12] CP11 0; "The Structural Use of Concrete" ; Part 1; British Standards Institution London; 1972.

[13] Kong, F.K., TENG, S., MAIMBA, P.P., TAN, K.H. and GOAN, L.W.; "Single Span, Continuous and Slender Deep Beams Made of High-Strength Concrete"; ACI Special Publication SP149 on 'High Performance Concrete'; American Concrete Institute; Detroit, 1994; pp.413-432.

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Table 1 : Details of the Test Specimens

Thick. b Span/height Total shear Clear-shear- Cube stren~th Splitting Main Steel Web Steel 8eam (mm) Llh Span/height span/height feu (N/mm ) strength ft horizontal Vertical a/h x/h (N/mm2

) Size % Size Size (mm) (mm) % (mm) %

8-70-182 20 1.0 0.29 0.12 41.2 3.34 10 0.84 6 0.50 6 0.50 8-50-182 28 1.0 0.29 0.12 41.0 2.96 10 0.60 6 0.50 6 0.50 8-40-182 35 1.0 0.29 0.12 42.2 2.92 12 0.69 6 0.50 6 0.50 8-35-182 40 1.0 0.29 0.12 36.9 2.70 10 0.84 6 0.50 6 0.50 8-30-182 47 1.0 0.29 0.12 41.7 2.97 10 0.72 6 0.50 6 0.50 8-25-182 56 1.0 0.29 0.12 40.4 2.98 12 0.87 6 0.50 6 0.50 8-20-182 70 1.0 0.29 0.12 44.4 3.69 12 0.69 6 0.50 6 0.50

Notation: The series of beams is indicated by the letter 8 followed by the slenderness ratio after the first hyphen and the e/b ratio after the second hyphen.

Table: 2 Test results and Prediction of Ultimate Loads

Modified Supp.Rules Single-Panel Tow-Panel PCA

8eams Measured Kong-CIRIA Load (kN) Load Rsr Load Rsp Load Load

Equation(8) (kN) (kN) Rtp (kN) Rpca (kN)

8-20-0.182 800 1045.7 / / / / / / 835.5 0.96 8-25-0.182 750 885.4 44.0 16.93 99.0 07.57 250.0 03.0 453.5 1.65 8-30-0.182 620 691.2 22.6 27.47 56.4 11.0 133.2 04.65 284.6 2.18 8-35-0.182 560 611.3 13.6 41.2 30.6 18.31 075.2 07.45 168.0 3.33 8-40-0.182 280 516.4 10.2 27.4 20.4 13.7 050.2 05.57 129.6 2.16 8-50-0.182 280 387.0 04.0 0.56 08.6 32.57 022.0 2.75 062.8 4.46 8-70-0.182 90 315.6 01.4 63.24 09.9 31.6 008.8 10.28 / /

Mean 41.13 19.12 07.28 02.76

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260

8-ll-0.192

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261

1400 B - 20 -0.192 1400

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"0 400 'is

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34567 -1012345678 lateral deflection (mm)

-1 0 1234567S -1 0 1234567 -1 0 123456 lateral deflection (mm)

-2 -1 ° 1

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700

600

500

400

300

200

100

0

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600

500

400

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O~~~~~~~~_ 20 30 40 50 60 70 h/b

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B- 20- O.1S2

B- 30- 0.192

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B -50-0.1S2 6-40-0.182

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2 3 4 5 6 7 lateral deflection (mml

Figure 7 : Buckling Load against Maximum Lateral Deflection

263