T A 1
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UC-NRLF
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EXCHANGE
OCT 21 1913
UNIVERSITY OF ILLINOIS BULLETIN
ISSUED WEEKLY Vol. XI SEPTEMBER 1, 1013 No. 1
[Entered as second-class matter Dec. 11, 1912, at the Post Office at Urbana. 111., under the Act of Aag. 24, 1912.]
BULLETIN NO. 68
THE STRENGTH OF I-BEAMS IN FLEXURE
BY
HERBERT F. MOORE
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
PUBLISHED BV T^B; 'IJiwinBtolTY OF JLJ;K?OIS, URBANA
PRICE: TWENTY CENTS
EUROPEAN AGENT CHAPMAN AKD HALL, LTD., LONDON
THE Engineering Experiment Station was established by act of the Board of Trustees, December 8, 1903. It is the purpose of the Sta- tion to carry on investigations along various lines of engineering and to study problems of importance to professional engineers and to the manufacturing, railway, mining, constructional, and industrial interests of the State.
The control of the Engineering Experiment Station is vested in the heads of the several departments of the College of Engineering. These ^constitute the Station Staff, and with the Director, determine the char- acter of the investigations to be undertaken. The work is carried on under the supervision of the Staff, sometimes by research fellows as graduate work, sometimes by members of the instructional staff of the College of Engineering, but more frequently by investigators belonging to the Station corps.
The results of these investigations are published in the form of bulletins, which record mostly the experiments of the Station's own staff of investigators. There will also be issued from time to time in the form of circulars, compilations giving the results of the experiments of engineers, industrial works, technical institutions, and governmental testing departments.
The volume and number .at the top of the title page of the cover are merely arbitrary numbers and refer to the general publications of the University of Illinois; above the title is given the number of the Engineering Experiment Station bulletin, or circular, which should be used in referring to these publications.
For copies of bulletins, circulars or other information address the Engineering Experiment Station, Urbana, Illinois.
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
BULLETIN No. 68 SEPTEMBER, 1913
THE STRENGTH OF I-BEAMS IN FLEXURE
By Herbert F. Moore, Assistant Professor in Theoretical and Applied
Mechanics
CONTENTS
Page
1. Introduction 3
2. Acknowledgment 3
3. Phenomena of Flexural Failure 4
4. Earlier Tests of I-Beams 5
5. I-Beam Tests at the University of Illinois 6
6. Yield Point of Structural Steel in Tension and in Compression 7
7. Failure of I-Beams by Direct Flexure 9
8. Inelastic Action of I-Beams Under Low Stress 10
9. Buckling of Compression Flanges of I-Beams; Equivalent
Column Length 16
10. Buckling of Compression Flanges of I-Beams; Tests 20
11. Tests to Failure of Beams Restrained from Twisting of Ends
and Beams Restrained from Sidewise Buckling 24
12. Effectiveness of Sidewise Restraint of I-Beams 26
13. Web Failure of I-Beams 28
14. Web Failure of I-Beams; Tests 30
15. Stiffness of I-Beams 31
16. Summary 32
LIST OF FIGURES.
1. Apparatus for Compression Tests of Steel 8
2. Deflection of I-Beam Under Repetitive Loading 12
3. Diagram of Stress in Compression Flange 16
4. Apparatus for Testing I-Beams without Restraint of Ends or of
Compression Flange 22
5. Results of Tests for Sidewise Buckling of I-Beams 23
6. Apparatus for Testing I-Beams with Restraint Against End
Twisting 25
268748
Page
7. Apparatus for Testing I-Beams with Eestraint Against Side-
wise Buckling 27
8. Shapes Assumed by I-Beams after Web Failure 31
9. Diagram of Compressive Stress in Web; of I-Beams over Bear-
ing Block 31
10. Eesults of Tests 1-6 35
11. Eesults of Tests 7-11 36
12. Eesults of Tests 12-17 37
13. Eesults of Tests 18-23 38
14. Eesults of Tests 24-29 39
15. Eesults of Tests 30-33 40
LIST OF TABLES.
1. Yield Point of Structural Steel in Tension and in Compres-
sion 11
2. Tests of I-Beams; Primary Failure by Direct Flexure 13
3. Tests of I-Beams at the University of Illinois; Primary Fail-
ure by Direct Flexure 14
4. Sidewise Buckling of I-Beams; Values of the Coefficient m
for Various Loadings of Beams 19
5. Tests of I-Beams; Primary Failure by Sidewise Buckling.... 21
6. Effect on the Elastic Limit of I-Beams of Eestraint against
Twisting of Ends and Against Sidewise Buckling 26
7. Sidewise Deflection of I-Beams at a Computed Fiber Stress of
16,000 Ib. per sq. in. . 29
8. Web Failure of I-Beams 33
9. Modulus of Elasticity of I-Beams and of I-Beam Material .... 34
THE STRENGTH OF I-BEAMS IX FLEXURE.
1. Introduction. — The mathematical theory of the resistance of materials in flexure has been extensively developed, but much less has been done in the experimental study of the phenomena of flexural stress. A striking fact brought out in the tests which have been made is the tendency of metal beams to fail by reason of column action in fibers which are under compressive stress. This tendency is especially strong in I-beams, channel-beams, and other forms of beams having tension and compression flanges connected by a comparatively thin web. The tests of Marburg*, Christief, and Burr and ElmoreJ show that this column action in I-beams may cause failure of test pieces by sidewise buckling or on account of excessive stresses in the web. These tests emphasize the importance of taking into account of stresses other than the direct flexure stresses in the flanges.
The wide-spread use of I-beams as flexural members makes the sub- ject of their flexural strength a matter of general engineering interest. To obtain experimental data on the action of I-beams under load, sev- eral series of tests of I-beams were carried out in the Laboratory of Applied Mechanics of the University of Illinois. This bulletin 'records and discusses the results of these tests and of others of similar kind. A formula is deduced for -the flexural strength of I-beams which are not restrained against sidewise buckling. There also is given a dis- cussion of the stiffness of I-beams, a discussion of the action of I-beams restrained against sidewise buckling and restrained against twisting at the ends of the beams, and a discussion of web failure of I-beams.
2. Acknowledgment. — The experimental work was a part of the research work of the department of Theoretical and Applied Mechanics, and was done under the general direction of the head of that depart- ment, Professor A. N. Talbot. The tests were made under the direct supervision of the writer. Acknowledgment is hereby made to the following students in civil engineering who assisted in making tests: F. J. Weston and W. E. Deuchler of the class of 1910, and M. H. Froelich and F. C. Lohman of the class of 1911. Analytical methods devised by various investigators have been use'd in this bulletin, and experimental data from various sources have been quoted; in all cases an attempt has been made to give due credit for methods and data.
*Proceedings of the American Society for Testing Materials, Vol. 'IX (1909), p. 378.
tPencoyd Steel Handbook (1898 Edition), p. 23.
^Selected Papers of the Rensselaer Society of Engineers, Vol. 1, No. 1. An abstract of the results of these tests is given in Burr's "Elasticity and Resistance of the Materials of Engineering," p. 694.
ILLINOIS ENGINEERING EXPERIMENT STATION
3. Phenomena of Flexural Failure. — The formulas commonly used for computing the stresses and deflections in beams are based on the assumptions (1) that a plane cross-section of a beam remains plane during flexure and (2) that the moduli of elasticity of the beam material for tension and for compression are equal and constant. For low stresses and for beams of medium or long spans these assumptions are very nearly exact. They become rough approximations when a beam is loaded to a point near failure.
The failure of beams of brittle material usually occurs by snapping of the extreme tension fibers at a computed fiber stress higher than the tensile strength of the material as determined by tension tests of specimens. Brittle material is nearly always much weaker in tension than in compression. As the fiber stress in beams of such material increases with increasing load, by reason of the change which takes place in the values of the moduli of elasticity, the tension side of the beam stretches more readily than the compression side shortens. The effect is to shift the neutral axis of the beam toward the compression side. This, together with the difference in strength, causes the actual tensile fiber stress to be less than the computed tensile fiber stress.
The failure of beams of ductile material may take place in one of a number of ways :
(1) The beam may fail by direct flexure. Under increasing load the usual flexure formulas are very nearly exact up to a load which stresses the extreme fibers of the beam to the yield-point strength of the material. When the yield point is reached in the extreme fibers, the deflection of the beam increases more rapidly with respect to an increase of load; and if the beam is of a thick, stocky section or is firmly held so that it can not twist or buckle, failure takes place by a gradual sagging which finally becomes so great that the usefulness of the beam as a supporting member is destroyed. ,
(2) In a beam of long span, the compression fibers act somewhat as do the compression fibers of a column, and failure may take place by buckling. Buckling failure, in general occurs in a sidewise direc- tion. Sidewise buckling may be either the primary or the secondary cause of failure. In a beam in which excessive flexural stress is the primary cause of failure and in which the beam is not firmly held against sidewise buckling, the primary overstress may be quickly followed by the collapse of the beam due to sidewise buckling. The sidewise re- sisting strength of a beam is greatly lessened if its extreme fibers are stressed to the yield point. Sidewise buckling may in some cases be a primary cause of beam failure, in which cases the computed fiber
MOORE STRENGTH OF I-BEAMS 5
stress, in general, does not reach the yield-point strength of the mate- rial. Sidewise buckling not infrequently limits the strength of nar- row, deep beams, especially beams of I-section or channel-section with tension and compression flanges connected by a thin web. Whether it is a primary cause of failure or a final manner of failure, sidewise buckling results in a clearly marked and generally quite sudden failure of a beam.
(3) Failure in an I-beam or a channel-beam may occur by ex- cessive shearing stress in the web, or by buckling of the web under the compressive stresses which always accompany shearing stress. If the shearing fiber stress in the web reaches a value as great as the 3'ield-point strength of the material in shear, beam failure may be expected and the manner of failure will probably be by some secondary buckling or twisting action. The inclined compressive stress always accompaning shear may reach so high a value that the buckling of web of the beam is a primary cause of failure. Danger of web failure as a primary cause of beam failure exists, in general, only for short beams with thin webs.
(4) In the parts of beams adjacent to bearing blocks which trans- mit concentrated loads or reactions to beams, high compressive stresses may be set up, and in I-beams or channel-beams the local stress in that part of the web nearest a bearing block may become excessive. If this local stress exceeds the yield-point strength of the material at the junction of web and flange, the beam may fail primarily on account of the yielding of the overstressed part and finally by a resulting twisting action of the beam.
4. Earlier Tests of I-beams. — Data of a considerable number of tests of I-beams have been published. A list of some important tests with references follows:
(1) Tests of twenty 6-in. wrought-iron I-beams loaded at the mid- point of the span and so held in the testing machine as to be free to buckle sidewise. The spans varied from 4 ft. to 20 ft. These tests were made by Burr and Elmore at the Rensselaer Polytechnic Institute and are reported in "Selected Papers of the Rensselaer Society of En- gineers," Vol. 1, No. 1. An abstract of the results is given in Burros "Elasticity and Resistance of the Materials of Engineering," 1905 edi- tion, p. 694.
(2) A series of tests on wrought-iron I-beams made by Tetmajer at the Materialpruefungsanstalt at Zurich, Switzerland. The results of the tests are given in Heft IY. of the Mitteilungen of that laboratory. The results of nine of the tests are given in Lanza's "Applied Mechan- ics/7 1905 edition, p. 443.
6 ILLINOIS ENGINEERING EXPERIMENT STATION
(3) A series of tests of twenty-one steel I-beams made by Christie at the Pencoyd Iron Works. In these tests the beams were somewhat restrained from sidewise buckling by the friction of the bearing blocks which were directly attached to heads of the testing machine. These tests were reported by Mr. Christie in the Transactions of the American Society of Civil Engineers for 1884. A summary of results is given in Burr's "Elasticity and Resistance of Materials of Engineering," 1905 edition, p. 689.
(4) Tests of wrought-iron and of steel I-beams made at the Massa- chusetts Institute of Technology. Eesults of twenty-nine such tests are reported in Lanza's "Applied Mechanics/' 1905 edition, p. 444 and p. 497.
(5) Tests of thirty-one steel I-beams and rolled girders made by Marburg at the University of Pennsylvania. These tests included I- beams of standard cross-section, I-beams with specially wide flanges, rolled by the Bethlehem Steel Company, and broad-flanged girder beams rolled by the same compan}r. In Marburg's tests great care was taken to secure the greatest freedom of motion possible for the beam in the testing machine. The tests are reported in the Proceedings of the American Society for Testing Materials for 1909, p. 378.
(6) A test of a built-up plate girder made by Turneaure at the University of Wisconsin. This girder was so designed and tested that failure occurred by buckling of the web.
The results of the first five series of tests are summarized in Table 2. The last named test is reported in the Journal of the Western So- ciety of Engineers for 1907, p. 788, and a summary of results is given in Table 8.
5. I-l)eam Tests at the University of Illinois. — The tests made in the Laboratory of Applied Mechanics of the University of Illinois and described in this bulletin include forty steel I-beams. These tests are summarized in Tables 3, 5 and 8. The general features of the tests may be indicated as follows :
(1) Ten 8-in., 18-lb. I-beams (Tests 1, 2, 7, 8, 12, 13, 18, 19, 32 and 33) were tested under loads applied at the one-third points of the span with spans varying from 5 ft. to 20 ft. These beams were so held in the testing machine as to afford the maximum possible free- dom of motion for the beam (see Fig. 4).
(2) Eight 8-in., 18-lb. I-beams (Tests 3, 4, 9, 10, 14, 15, 20, and 21) were tested under conditions similar to those described in (1) ex- cept that the ends of the beams were firmly held so that they could not twist. (See Fig. 6.)
MOORE STRENGTH OF I-BEAMS
(3) Seven pairs of 8-in., 18-lb. I-beams (Tests 5, 6, 11, 16, IT, 22, and 23) were tested under conditions similar to those described in (1) except that the beams were firmly restrained against sidewise buckling. (See Fig. 7.)
(4) Four 8-in., 18-lb. I-beams (Tests 24, 25, 26, and 27) were tested with 10-ft. span. Two beams were loaded at the mid-point of the span, and two were loaded at the one-sixth points of the span. The beams were not restrained against sidewise buckling or against end twisting action. (See Fig. 4.)
(5) Two 8-in., 25.25-lb. I-beams (Tests 28 and 29) were tested with loads at the one-third points of a 10-ft. span. The beams were without sidewise or end restraint. (See Fig. 4.)
(6) Two 8-in., 18-lb. I-beams (Test 31) were tested under a load continued on the beams for 107 days. The computed stresses under the applied load were about equal to the yield-point strength of the material. The beams were symmetrically loaded at two points in a span of 8 ft. 9f inches.
(7) One pair of 8-in., 18-lb. I-beams (Test 30) with their webs fastened together by separators were tested with loads at the one-third points of a 10-ft. span.
(8) Six 12-in., 31.5-lb. I-beams (Tests 34-39) were tested over a span of 3 feet, with two symmetrical loads near mid-span. Different web conditions were obtained by varying the thickness of web by planing down the webs of some beams. These beams all failed in the web.
Nearly all tests were made on a four-screw 200,000-lb. Olsen testing machine with long table for beam tests. The instrument used for measuring deflections consisted of a framework supported entirely on the test beam, fitted with an extensometer dial by means of which de- flection could be measured to one one-thousandth inch. For measuring longitudinal fiber deformation in the beams various forms of exten- someters were used. A strain gauge of the Berry type proved the most satisfactory extensometer for this purpose.
The I-beams tested were bought in the open market at various times, and the beams may be expected not to differ far from the range of material found in practice.
6. Yield Point of Structural Steel in Tension and in Compres- sion.—For I-beams the yield-point strength of the steel is, perhaps, the most important physical characteristic. The tension test is the most usual test for determining the physical properties of structural steel, and the yield point reported (frequently but incorrectly called the "elastic limit") is the yield point in tension. As flexural members of
ILLINOIS ENGINEERING EXPERIMENT STATION
structural steel may fail through the yielding of the compression fibers, the yield point of structural steel in compression seemed worthy of in- vestigation. A considerable amount of test data are available on this subject, and the conclusion seems fairly well established that for the softer grades of steel the yield-point determined for tension is about the same as the yield-point in compression.* This conclusion is cor- roborated by the results of tests made in connection with the investiga- tion of I-beams made at the University of Illinois. Table 1 gives the values of the stress at yield-point in tension and in compression for specimens cut from the flanges of I-beams and from -flat bars. Both in the compression tests and in the tension tests the specimens were held in wedge-shaped grips, a special head being used in the compression tests, as shown in Fig. 1. The specimens were of such length that column action was not noticeable below the yield-point in the compression specimens.
Moving Crosshsod of Tesrino Machine
Weighing fable of Testing Machine. FIG. 1. APPARATUS FOR COMPRESSION TESTS OF STEEL.
The yield-point reported is the average of the values obtained by three methods: (1) the drop of the beam of the testing machine, which was well marked, as the speed used in testing below the yield-point was slow (0.1 in. per min.) and as the poise was easily kept in balance; (2) the "knee" of an autographic diagram of load and deformation, in which diagram the stretch or compression was magnified five times; and (3) the scaling of the specimens as shown by the flaking of the plaster of paris with which the specimen had been coated.
*J. B. Johnson, "The Materials of Construction," p. 502. This is a summary of tests by Chas. A. Marshall, and of tests at the Watertown Arsenal. Recent tests in British laboratories also show practically the same values for the yield-point strength of mild steel in tension and in compression.
MOORE — STRENGTH OF I-BEAMS . 9
7. Failure of I-beams by Direct Flexure. — In studying the failure of I-beams care must be exercised to distinguish between the primary failure and the final failure as judged by the shape of the beam after the test. Beams in which the primary cause of failure is excessive flex- ural stress not infrequently buckle side wise after this excessive stress has weakened the flanges of the beam; in other cases the yielding of the flanges allows stress to be transferred to the web which then may twist or buckle. If a beam is held firmly against sidewise bending and has a thick web, the final failure under a load as applied in a testing machine will be by gradual sagging and the exact instant of failure will not be very clearly marked. Under excessive flexural fiber stress the time of application of load affects the deformation under load, and a beam may carry momentarily a load applied in a testing machine larger than that which if continued for several days would cause col- lapse of the beam.
In Table 2 are given test results obtained by various investigators in I-beam tests in which excessive flexural stress seems to have been the primary cause of failure. While it can not be certain that in every test the primary cause of failure was excessive flexural stress, yet, since every beam tabulated in Table 2 developed before failure com- puted stresses as high as might be expected for the yield-point strength of the material, and since in most cases friction of testing machine heads acted to prevent sidewise buckling, these tests appear to furnish a fairly satisfactory basis for the study of failure by direct flexure.
In Table 3 are given the test results for those I-beams tested at the University of Illinois for which excessive flexural fiber stress ap- peared to be the primary cause of failure. Figs. 10 to 15 inclusive (at the end of the text) give stress-deformation curves and stress-deflection curves for most of the beams tested at the University of Illinois.
A study of Table 2 and Table 3 shows that some beams which failed by direct flexure developed computed fiber stresses but little in excess of the usual values of yield-point strength of the material, while other beams developed momentarily stresses considerably higher. In Table 3 it will be seen that for nearly all the beams tested at the University of Illinois excessive deflection, large permanent set, or other sign of structural damage was observed at computed fiber stresses not much higher, if any, than the yield-point strength of the material. The load temporarily carried in a laboratory test depends in part on the speed of testing and the nature of the support of the beam in the testing machine. However, since a long-continued dead load or an oft-repeated live load would be more liable to injure a beam than an equal load ap-
10 ILLINOIS ENGINEERING EXPERIMENT STATION
plied by a testing machine, it is apparent that even under circum- stances most favorable to the development of high fiber stresses, it is unsafe practice to regard as the ultimate fiber stress in flexure any value higher than the yield-point strength of the material of the beam.
Under long-continued static load the deformation of beams having their extreme fibers stressed to the yield-point of the material, increases for some time, frequently for several days, but the member does not necessarily fail. An illustration is furnished by a test made on two 8-in., 18-lb. steel I-beams loaded at two points each 12 inches to one side of the mid-point of an 8-ft. span. (Test No. 31, Fig. 15.) The load was applied to both beams until, as was shown by extensometers at- tached to the flanges, some fibers were stressed to the yield-point. No- ticeable sidewise buckling of the beams had begun, and apparently fail- ure was imminent. The load was kept constant for 107 days, and the extensometer on the most stressed flange was read from time to time. After a few days the fiber deformation reached a value practically con- stant, and the beam did not collapse during the test period. Thurston* reports tests showing similar results on transverse tests of steel of square section.
The excessive deflection of beams found when fibers are stressed be- yond the yield-point, and the possibility of collapse emphasize the con- clusion that for I-beams the yield-point strength of the material in the flanges should be regarded as the ultimate fiber stress in flexure. It is especially absurd to regard the ultimate tensile strength of steel as the ultimate fiber stress, as even under the most favorable conditions of service this fiber stress can not be developed before the I-beam collapses.
The results of those tests of I-beams in which failure occurred at stresses lower than the yield-point strength of the flange material due to sidewise buckling or to web failure, are tabulated and discussed in subsequent paragraphs.
8. Inelastic Action of I-beams under Low Stress. — Many recent writers on structural design have pointed out that in practically all steel structures the bending of beams and rods incidental to the erection of the structure and the use of drift pins in aligning rivet holes cause local stresses in excess of the yield-point strength of structural steel. These local stresses do not, however, cause the failure of the whole mem- ber. In Marburg's tests of I-beams and in those made at the University of Illinois frequent evidences of slight inelastic action at low stresses were observed. It is believed, however, that this inelastic action is the result of local stress and that it does not indicate the limit of load
'Thurston, "Text Book of the Materials of Construction," p. 516.
MOORE — STRENGTH OF I-BEAMS 11
TABLE 1.
YIELD POINT OF STRUCTURAL STEEL IN TENSION AND IN COMPRESSION.
Specimens cut from flanges of I-beams and from flat bars. The length of specimens was about 9 in., 4 in. between grips, the width was from 1 in. to 1.5 in., and the thickness 0.25 in. to 0.50 in. The specimens from flanges of I-beams were planed down till the cross-section was rectangular.
|
Number of Specimens |
Fiber Stress at Yield Point, Ib. per sq. in. |
Ratio of Yield Point in |
|||
|
Specimen |
Tension to |
||||
|
from |
Ten- sion |
Compres- sion |
Tension |
Compression |
Yield Point in Compression |
|
I-beam 16a |
1 |
2 |
34,700 |
34,500 |
1.00 |
|
I-beam 16b |
2 |
2 |
34,100 |
34,800 |
0.98 |
|
I-beam 26 |
2 |
2 |
32,500 |
34,600 |
0.94 |
|
I-beam 27 |
2 |
2 |
34,200 |
34,400 |
0.99 |
|
I-beam 24 |
2 |
2 |
34,700 |
35,000 |
0.99 |
|
I-beam 17a |
2 |
2 |
36,300 |
35,100 |
1.03 |
|
I-beam 17b |
2 |
2 |
35,400 |
36,100 |
0.98 |
|
I-beam 25 |
2 |
1 |
31,100 |
35,400 |
0.88 |
|
I-beam 21 |
2 |
2 |
33,800 |
32,600 |
1.03 |
|
I-beam 23 |
2 |
2 |
37,900 |
40,300 0.94 |
|
|
I-beam 14 |
2 |
2 |
36,400 |
32,000 |
1.14 |
|
I-beam 15 |
2 |
2 |
33,800 |
33,500 |
1.01 |
|
Flat 1 |
1 |
1 |
41,700 |
46,000 |
0.91 |
|
Flat 2 |
1 |
1 |
42,500 |
42,900 |
0.99 |
|
Flat 3. |
1 |
1 |
41,400 |
40,000 |
1.03 |
|
Flat 4 |
1 |
1 |
39,700 |
39,500 |
1.00 |
|
Flat 5 |
1 |
1 |
40,700 |
39,000 |
1.04 |
|
Flat 6 |
1 |
1 |
38,500 |
37,500 1.02 |
|
|
Flat 7 |
1 |
' 1 |
39,600 |
44,800 |
0.88 |
|
Flat X |
1 |
1 |
37,900 |
35,000 |
1.08 |
|
31 |
31 |
Av. 0.993 |
carrying capacity for the beam as a whole. In support of this belief the following facts are presented :
(1) In any test of material if apparatus of sufficient delicacy is used inelastic action can be detected at comparatively low stresses.*
(2) The physical properties of the material in various places in an I-beam may vary considerably ;t the material at the root of the flange is usually weaker than the material in the flange or than the material in the web, and inelastic action of the beam under low stresses may be due in part to yielding of this weaker material while the flange material can develop further strength.
(3) If the load which at first application causes inelastic action be removed and reapplied, this second cycle of loading and removal of load will usually show much less inelastic action than is shown during the first cycle of loading and unloading, and successive cycles will show
*Moore, "The Physical Significance of the Elastic Limit," Proceedings of the Sixth Congress (1912) of the International Association for Testing Materials.
tMarburg, Proceedings of the American Society for Testing Materials, Vol. IX (1909),
Moore, Proceedings of the American Society for Testing Materials, Vol. X (1910),
Hancock, Proceedings of the American Society for Testing Materials, Vol. X (1910), p. 248; Vol. XI (1911), p. 477.
12
ILLINOIS ENGINEERING EXPERIMENT STATION
gradual improvement of the elastic qualities of the beam. Fig. 2 shows the computed fiber stresses and the deflections observed with successive applications of load in a test of an 8-in. I-beam. (Test No. 13, Table 5.) The energy lost in inelastic action for a cycle of loading and un- loading is shown by the shaded area. It will be seen that during the third cycle the action of the beam was almost perfectly elastic. Only
One Division = O.I in. Deflection FIG. 2. DEFLECTION OF I-BEAM UNDER REPETITIVE LOADING.
a few moments elapsed between successive loadings so that rest of mate- rial could not have played an important part in the result. Similar re- sults were obtained in tests of seven other beams.
For static load or for loads repeated at infrequent intervals and al- ways acting in the same direction, local inelastic action does not seem important. The writer has not been able to discover any record of the failure by fatigue of metal originally sound in a bridge, building, or other structure designed to carry static load, but in structures under loads applied successively in opposite directions or loads rapidly re- peated many millions of times it might be expected that local inelastic action may cause minute cracks or miscroscopic flaws which, spreading, would eventually cause the failure of the structure by fatigue of metal.
MOORE STRENGTH OF I-BEAMS
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14
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 3.
TESTS OF I-BEAMS AT THE UNIVERSITY OF
In all tests included in this table the beams developed computed fiber stresses equal to
Test No.*
Beam
Span, feet
Loading
Yield-point
Strength of
Material in
Flanges,0
Ib. per sq. in.
i> Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
4 8-in., 18-lb. I-beam
Restrained from end twisting.
1 8-in., 18-lb. I-beam
No restraint.
8-in., 18-lb. I-beam
No restraint.
11 Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
10 8-in., 18-lb. I-beam
Restrained from end twisting.
16 Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
17 Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
8-in., 18-lb. I-beam
Restrained from end twisting. 8-in., 18-lb. I-beam
No restraint
30 Pair of 8-in., 18-lb. I-beams
With separators.
28 8-in., 25.25-lb. I-beam
No restraint.
26 -in., 18-lb. I-beam
No restraint.
27 8-in., 18-lb. I-beam
No restraint.
22 Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
23 Pair of 8-in., 18-lb. I-beams
Restrained from sidewise buckling.
7.50 7.92
10
10
10 10
10
10 10
10 15
20
l/3 points l/3 points l/3 points
l/3 points l/3 points l/3 points */3 points l/3 points Y$ points
l/3 points l/3 points
l/3 points
l/3 points Mid-point
Mid-point l/3 points
l/3 points
35,300 35,300 33,800
37,000 37,000 35,200 33,900 34,300 35,900
33,800 33,800
32,400
34,100 32,500
84,200 34,200
35,800
*See Fig. 10-15 at the end of the bulletin. "Determined from tests of specimens cut from flanges.
MOORE STRENGTH OF I-BEAMS
15
TABLE 3 (Continued)
ILLINOIS — PRIMARY FAILURE BY DIRECT FLEXURE.
or greater than the yield-point strength of the material.
|
Computed Fiber Stress at Maximum Applied Load, Ib. per sq. in. |
Deflection under Maxi- mum Load, inches |
Modulus of Elasticity, Ib. per sq. in.00 |
Remarks |
|
41,500 |
0.30 |
26,500,000 |
0.28 in. sett after fiber stress of 88,400 |
|
41,500 |
0.33 |
23,000,000 |
Ib. per. sq. in. Beams sagged gradually. 0.18 in. set after fiber stress of 33,400 Ib. |
|
35,300 |
per sq. in. Beam sagged gradually. Final failure by sidewise buckliniz after |
||
|
yield-point strength of material was |
|||
|
developed. |
|||
|
38,400 |
0.20 |
Final failure by sidewise bucklini? after |
|
|
yield-point strength of material was |
|||
|
developed. |
|||
|
37,800 |
0.13 |
30,800,000 |
Final failure by sidewise buckling after |
|
yield-point strength of material was |
|||
|
developed. |
|||
|
37,300 |
„_ |
Beam sagged gradually excessive fiber |
|
|
deformation at 84,300 Ib. per sq. in. |
|||
|
fiber stress. |
|||
|
34,300 |
|
Final failure by very gradual sidewise buckling after yield-point strength of |
|
|
material was developed. |
|||
|
35,200 |
0.76 |
27,800,000 |
0.12 in. set after fiber stress of 88,700 |
|
Ib. per sq. in. Beams sagged grad- ually. |
|||
|
36,600 |
0.70 |
26,300,000 |
0.08 in. set after fiber stress of 81,000 |
|
Ib. per sq. in. Beams sagged grad- ually. |
|||
|
35,400 36,000 |
0.76 0.58 |
25,800,000 30,000,000 |
Ultimate strength apparently reached. Beam sagged gradually. Final failure by sidewise buckling after |
|
yield-point strength of material was |
|||
|
88,200 |
0.55 |
30,400,000 |
developed. 0.11 in. set after fiber stress of 84,800 |
|
Ib. per sq. in. Final failure by side- |
|||
|
wise buckling after yield-point strength of material was developed. |
|||
|
39,800 |
0.78 |
27,800,000 |
Flanges showed signs of crippling at |
|
fiber stress of 33,400 Ib. per. sq. in. |
|||
|
36,200 |
0.63 |
28,400,000 |
Final failure by sidewise buckling after |
|
yield-point strength of material was |
|||
|
37,600 |
0.45 |
28,000,000 |
developed. Final failure by sidewise buckling after yield-point strength of material was |
|
35,300 |
|
developed. Restraining batten plates clamped to |
|
|
beams. Hold of clamps loosened and |
|||
|
final failure of beams was by sidewise |
|||
|
buckling after yield-point strength of |
|||
|
material was developed. |
|||
|
36,500 |
2.10 |
24,000,000 |
0.54 in. set after fiber stress of 81,600 |
|
Ib. per sq. in. Beams sagged grad- |
|||
|
ually. |
00 Determined from deflection of beams. tSet denotes deflection remaining after the
removal of load from the beam.
16 ILLINOIS ENGINEERING EXPERIMENT STATION
9. Buckling of Compression Flanges of I-beams: Equivalent Col~ umn Length. — In an I-beam under a load which acts in the plane of the web there is a tendency for the compression flange to buckle side- wise due to column action. On account of this column action the ul- timate available fiber stress for a beam with no restraint against side- wise buckling, as calculated by the flexure formula, may be less than the yield-point strength of the material in the flanges. Column action in a beam is more complex than the action of a strut under direct
t=W
i '
A -->—-- *U
L 7
f.— — / - -*j
FIG. 3. DIAGRAM OF STRESS IN COMPRESSION FLANGE.
compression. In a strut a compressive load P is applied at one end and the average stress over the cross section due to the direct compression
is the same all along the strut. For a strut the average stress over any
•p cross section is given by the equation fc = — , where P is the total
A
compressive load, A the cross sectional area of the strut, and /c the average stress due to direct compression. In a beam the compressive stress in the extreme fibers of the compression flange due to flexural action varies according to the location of the section, increasing from zero at an end support (or point of inflection for a fixed-ended beam) to a maximum at some point in the span. The compression flange of the I-beam then may be regarded as a strut loaded with a large number of elementary loads dP applied at successive points along the span. Each elementary load or increment of load may be considered to pro- duce an increment in the compressive stress of the flange. The value of the compressive stress at any section will then be the summation of the increments of compressive stress from the point of zero stress, or f = f df. This stress / for the remotest fiber of any section will be very nearly equal to the flexural stress produced by the flexure of the beam, which stress is computed by the usual flexure formula / = _•, in
which M = the bending moment at the point of span considered, I = the moment of inertia of cross-section of beam, and c = distance from the neutral surface of the beam to the extreme compression fiber. To
MOORE - STRENGTH OF I-BEAMS 17
find the effect of this kind of column loading it will be necessary to de- duce an expression for thd value of the compressive stress due to column action which will be developed with a column loaded with increments of load as here considered.
For a strut under direct compression the average compressive stress developed at failure fc may be expressed fairly well by the formula
in which
fe = o, value of stress about equal to the yield-point strength of the material,
I = length of strut,
r = minimum radius of gyration of cross-section of strut,
k = an experimentally determined constant.
We may regard the term&— in either of two ways, (1) as represent-
ing a reduction from /e of available ultimate fiber stress in the strut, due to the effect of column action, or (2) as a fiber stress which is due to a bending action in the column, produced by the same load as pro- duces /c, and which added to /c brings the extreme fiber stress up to /e. By the second conception the stress /e is the sum of the direct com- pressive stress fc and the stress due to column bending.
For the purposes of this discussion it will be convenient to use the second conception; i. e., to consider the last term as a stress produced by a bending action in the column.t In the case of the I-bearn, the stress in the remotest fiber of the compression flange on the edge having the highest compression may be considered to be made up of the sum of the flexural fiber stress f\ (computed by the usual flexure formula,
/! = ~^) and the column bending stress /'. This stress will be a maxi-
mum at the section where the bending moment will be a maximum, and at failure by side buckling of the flange we may consider that fe = fi + f, where f\ is the computed flexural fiber stress at the dangerous sec- tion, and /Q has a value not greatly different from the yield point of the material.
'While this "straight line" formula for columns is based directly on experiment rather than on mathematical reasoning, it is generally accepted as expressing with a good degree
of accuracy the law of failure for columns whose— — is not greater than about 150 and
which are of sufficiently stocky construction to avoid danger of failure by "wrinkling" of parts or local collapse.
tOther methods of analytical treatment of the sidewise buckling of I-beams have been proposed. Some of them are based on the Rankine-Gordon-Schwartz column formula, others on reasoning analagous to that used in developing the Euler column formula.
See Michell, in the Philosophical Magazine for 1899, p. 298; Reissner, in the American Machinist for March 10, 1906; H. D. Hess, in the Proceedings of the Engineers' Club of Philadelphia for April, 1909; Boyd, "Strength of Materials," p. 219; Cambria and Carnegie Steel Handbooks.
18 ILLINOIS ENGINEERING EXPERIMENT STATION
To determine the column bending stress /', it will be necessary to take into account the manner of application or of distribution of the assumed compressive loading along the length of the compression flange. At any point along the compression flange the column load may be taken to be the flexural fiber stress at that point, since by this con- ception the amount of the column load per unit of area of section is the flexural fiber stress. This flexural fiber stress increases from the point of zero bending moment in the beam to the dangerous section. To determine the effect of column action, the increment or differential of flexural fiber stress df applied along a differential of length of flange d\ (which is the only column load which acts throughout the column length A) will be considered as producing column bending stress at the dangerous section. This increment of load is applied at any two points A and A (Fig. 3) and acts upon a column of the length A (the cal- culated flexural fiber stress being the same at A and A). The sum of all the loads on the whole length of column ( C df) will be the flexural fiber stress at the dangerous section (/i). To determine the total column bending stress /' at the dangerous section A it will be necessary to make a summation of the effects of the increments of column load df over the length of flange to be considered. By analogy with the straight-line column formula, adopting q as the coefficient for the formula as applied to sidewise buckling of the flange, the term expressing the column bend-
ing stress will be of the form q — ^, where r is the radius of gyration
of a cross-section of the compression flange about a gravity axis parallel to the depth of the web. As the elements of column load df vary along the flange and as the proportional effect of each elementary load as compared with the sum of all the loads must be used in the summation,
it is necessary to introduce the ratio —r- into the term. Then
This is the column bending stress at the dangerous section.
It will be found convenient to consider this stress as equal to the column bending stress in an ordinary strut loaded with a load f\, and having a length of ml, where m is a coefficient depending upon the method of loading and conditions of continuity and I is the length of the beam, ml may be called the equivalent column length. Equation (2) may then be written
J* ................. (3)
MOORE — STRENGTH OF I-BEAMS 19
The equation- for the computed flexural fiber stress at failure due primarily to sidewisc but-kling will be
/,=/e-9^ ................... (4)
where /e = a value not greatly different from the yield-point strength of the material in the 'flange.
q = a coefficient of column action.
ml — the equivalent column length of the flange of the beam, the coefficient m being found by equation (3) for different loadings and different conditions of continuity.
r = the radius of gyration of the compression flange about a gravity axis parallel to the web. For practical purposes / may be taken as the radius of gyration of the I-section about a gravity axis parallel to the web.
From equation (3) an expression for ml may be written ml — \ -I-.
J fi
It may help in integration to consider that f \df is the same as the area under the curve of flexural stress for the full length of the beam in simple supported beams, as is indicated in Fig. 3. Then, if /a is the mean ordinate of the curve of flexural stress, f \df = fj.
rf/ fj
That is, the equivalent column length of the compression -flange of an I-beam is equal to the span multiplied by the ratio of the mean flexural fiber stress in the compression flange to the flexural fiber stress in the compression flange at the dangerous section. For a uniformly loaded beam of constant cross section m is found to be f . For a beam with a single load at any point between supports m is \. Table 4: gives values of m for various beam loadings. For beams fixed at the ends it will be
TABLE 4.
SIDEWISE BUCKLING OF I-BEAMS : VALUES OF THE COEFFICIENT m FOR VARIOUS LOADINGS OF BEAMS.
|
Loading |
Value of m |
|
0 667 |
|
|
0.500 |
|
|
Simple beam, single concentrated load at any point of span |
0.500 |
|
0 667 |
|
|
Simple beam one-quarter point loads |
0.750 |
|
0 888 |
|
|
0 667 |
|
|
1 000 |
|
|
0 281 |
|
|
Fixed-ended beam, mid-point load |
0 250 |
20 ILLINOIS ENGINEERING EXPERIMENT STATION
seen that one limit for A will be the distance between points of inflexion of the elastic curve.
10. Buckling of Compression Flanges of I-beams; -Tests. — The value of the coefficient q of equation (4) is to be determined from the re- sults of flexure tests of I-beams. All available data of tests of I-beams were studied and the test results given in Table 5 were chosen as fur- nishing the best basis for the determination of q. In selecting data suitable for the study of resistance to sidewise buckling, only those tests were considered in which the primary cause of failure was evidently sidewise buckling. Beams which were wholly or partially restrained laterally -by the method used in supporting them in the testing machine were not considered. Fig. 4 shows the method used at the University of Illinois for supporting beams and preserving freedom in respect to sidewise buckling. Beams in which the yield-point strength of the material in the flanges was developed before failure were not considered, as the mere presence of such a fiber stress would explain the failure of a beam and might readily be the cause of sidewise buckling, which would then be a secondary and not a primary failure. The non-con- sideration of beams free to buckle sidewise which develop fiber stresses as great as the yield-point strength of the material affected only beams of short span or beams of medium span in which the flange material was unusually weak. As a matter of fact the results obtained for re- sistance to sidewise buckling would not be materially affected whether such beams were considered or not. In making up Table 5 tests were not considered in which web failure seemed to be the primary failure. Fig. 5 shows graphically the results of the tests given in Table 5. The computed fiber stress at failure (generally called the modulus of rupture) was chosen as a criterion of the strength of an I-beam, rather than the "elastic limit" of the beam, for the following reasons: (1) The failure of beams which buckle sidewise is sharply marked, and the personal equation of the observer will affect the determination of the point of failure but slightly. On the other hand any determination of the elastic limit is dependent upon the sensitiveness of apparatus used in obtaining readings of deformation and upon the interpretation of a plotted curve, and it is much more subject to variations due to personal equation than is the computed fiber stress at failure. (2) The load at failure is more dependent upon the average physical properties of the beam material and less on local stresses and individual peculiarities than is the elastic limit. As the yield point of the material was not exceeded, the computed fiber stress at failure may be considered to vary but little from the actual fiber stress.
MOORE STRENGTH OF I-BEAMS
21
II
m .§
3
&^ 3
* li s -s ! S -^ s -I
a
'5 .2
0 -S
OQ •« M
I a
*
ssss.ss g s § g § §
«
£* o« oo oo ob ob ob ob oo ob co ob«o
t-SrtrtS s --"•••- -a
•7'TCCCC C n C C C CC •O^'r'7-7'7 '7 '7 '7 '7 '7 '7'7
22
ILLINOIS ENGINEERING EXPERIMENT STATION
The advisability of adjusting the fiber stresses developed in tests of I-beams to compensate for variation in strength of material in the flanges of different test beams was considered, but it was decided to base conclusions on the stresses computed for the tests. Two reasons led to this decision : ( 1 ) Due to cold-straightening and other bending which a beam receives there is considerable variation in strength in different parts of the same beam,, and the strength of test pieces from one part of the beam would not be wholly representative of the strength of other parts. (2) For beams of long span the resistance to sidewise buckling is dependent not so much on the strength of material as on its stiffness (of which the modulus of elasticity is an index) ; for beams of medium span the resistance to sidewise buckling is dependent partly on the strength of material and partly on its stiffness ; hence the proper adjustment of stresses to compensate for variation of material would be a matter of no small difficulty.
From Fig. 5 it may be seen that the equation
/, = 40,000 — 60 *p
(5)
represents the results with a fair degree of accuracy. The extreme values observed fall .within 2,500 Ib. per sq. in. of the values given by the above equation.
Sphere and Plate Searing
Roller Bearing
Moving Crosshead of resting Machine
I* I
""'
TesrBeam
and Plate Bearing
/ V^tacter Bearing
wm.Spnene and Plate Bearing
'Jw////?/////////^^^
Weighing Table of Testing Machine
FIG. 4. APPARATUS FOR TESTING I-BEAM WITHOUT RESTRAINT OF ENDS OR OF
COMPRESSION FLANGE.
A comparison of equation (4) with the results of tests of columns is of interest. Tests made by J. E. Howard at the Watertown Arsenal* on H-section steel columns with pin ends have been chosen as tests which furnish an excellent basis of comparison of column test results and I-beam test results. The results of Howard's tests of H-section columns with pin ends may be expressed by the equation
P/A = 36,000 -- 100 ~
(6)
'Tests of Metals for 1909, p. 754; Proceedings of the American S.ociety for Testing Materials, Vol. IX (1909), p. 413.
MOOKE STRENGTH OF I-BEAMS
23
24 ILLINOIS ENGINEERING EXPERIMENT STATION
in which P/A is the average intensity of compressive stress at failure, Z is the length of the column and r is the least radius of gyration of the column section.
Comparing equation (5) with equation (6) it is seen that, as might be expected, the coefficient of equivalent slenderness ratio for the beam formula is somewhat less than the coefficient of slenderness ratio for the column formula. The smaller value of coefficient in the beam formula is doubtless due to the fact that in a beam there is end restraint against sidewise buckling and a restraining action of the web and the tension flange. The first term in the beam equation (5) is larger than the first term in the column equation (6). The flanges and webs of the I-beams were rolled thinner than the flanges and webs of the H-sections, and the additional work of rolling done on the I-sections may explain the in- crease in the yield-point strength of the material over that of the H-sec- tions.
It should be noted that all but one of the beams given in Table 5 are "light" sections. The web and the tension flange of "heavy" I-beams would offer more restraint against sidewise buckling than do the web and the tension flange of "light" I-beams, and the fiber stresses developed at failure may reasonably be expected to be higher. Such a result is indi- cated by the tests made by Burr and Elmore at Rensselaer Polytechnic Institute, to which reference has already been made. The results of these tests are shown in Fig. 5 by small black squares. The tests were made on medium-weight wrought-iron I-beams, 6 in. deep, and it is seen that the greater strength and stiffness of steel I-beams was about offset by the greater stockiness of section of the Burr and Elmore wrought-iron test beams.
As test data are lacking for "heavy" steel I-beams, as the formula (equation 5) derived from tests of "light" I-beams gives results which err on the side of safety when applied to "heavy" I-beams, and as the reduction below yield-point strength of material of fiber stress at failure is not large for ordinary spans, no attempt will be made to derive a separate formula for I-Beams of medium-weight or heavy-weight sections.
Attention is called to the fact that in no case should the ultimate flexural stress be taken as higher than the yield-point strength of the material in the flanges. In the absence of special tests of material 35,000 Ib. per sq. in. may be used as an average value for the yield point of structural steel. Especial attention is called to the fact that equation. (5) gives ultimate values of fiber stress and not working values, which should, of course, be much lower.
11. Tests to Failure of Beams Restrained from Twisting of Ends and Beams Restrained from Sidewise Buckling. — Two series of tests were
MOORE STRENGTH OF I-BEAMS 25
carried out for the purpose of investigating the action of I-beams re- strained against end twisting and oi' beams restrained against sidewise buckling. Fig. G shows the arrangement of apparatus used in testing beams restrained against end twisting. To each end of the web of the test beam heavy angles were bolted by one leg and the other leg of each angle was bolted to an end piece which rested on a roller. When the test beam was placed in the testing machine evenness of bearing under rollers was secured by the use of thin metal shims. This method of supporting the test beams proved effective in preventing end twisting and did not affect the tendency of the beam to buckle sidewise.
FIG. 6. APPARATUS FOR TESTING I-BEAM WITH RESTRAINT AGAINST END TWISTING.
The fiber stresses developed at failure in those beams which were restrained against end twisting are given in Tables 3 and 5. Short-span beams so restrained did not develop quite so high stresses at failure as did similar beams tested without restraint against end twisting. This was probably due to imperfect bearings at the ends. For all except the short-span I-beams the fiber stresses developed at failure by beams re- strained from end twisting did not differ appreciably from the fiber stresses developed at failure by the beams not so restrained. Failure for both kinds of beams occurred by sidewise buckling, and it would seem that restraint of ends of I-beams against twisting does not appre- ciably increase their resistance to sidewise buckling.
The method of testing beams restrained against sidewise buckling is shown in Fig. 7. For each test two beams were fastened together along their compression flanges by means of batten plates spaced about ten inches apart. Each batten plate was fastened to the flanges of the beams by four studs of cold-rolled steel fitting snugly in drilled holes. This
ILLINOIS ENGINEERING EXPERIMENT STATION
device prevented appreciable sidewise buckling and all beams thus re- strained failed very gradually by vertical sagging with the exception of the beams with lo-l't. span in which the batten plates were merely clamped to the flanges of the beams, and in which, though the full yield- point strength of the material Avas developed, the beams finally buckled sidewise. In all beams tested with restraint against sidewise buckling the maximum computed fiber stress developed in the test was equal to or slightly greater than the yield-point strength of the material in the flanges. It would seem that for beams effectively restrained against side- wise buckling the fiber stress developed before failure in flexure will be as great as the yield-point strength of the material, regardless of the length of the span. What constitutes effective restraint is discussed in the next paragraph.
TABLE 6.
EFFECT ON THE ELASTIC LIMIT OF I-BEAMS OF EESTRAINT AGAINST TWISTING OF ENDS AND AGAINST SIDEWISE BUCKLING.
All tests made on 8-in., 18-lb. I-beams loaded at the one-third points of the span.
|
Span ft. |
Number of Beams Tested for Each Item |
Computed Fiber Stress at the First Observed Elastic Limit Ib. per sq. in. |
||
|
No Restraint |
Restrained against Twisting of Ends |
Restrained against Sidewise Buckling |
||
|
5 7.5 10 15 20 |
2 2 2 1 1 |
27,300 27,900 23,000 25,200 21,000 |
23,000 23,300 19,300 22,000 23,400 |
22,300 26,300 26,600 21,200 24,000 |
12. Effectiveness of Sidewise Restraint of I-beams. — In the tests of 8-in. I-beams, measurements of the extreme fiber deformation in the flange (stretch and shortening) at mid-span were made. By plotting the observed fiber deformations against the fiber stress computed by the usual flexure formulas, curves were obtained showing local action of the beams under load. These curves (Fig. 10-15) are given at the end of the bulletin. From these curves fiber stress at the elastic limit first ob- served at any part of the beam was determined,* and these stresses have been tabulated in Table 6. An examination of this table shows that in- elastic action was detected in some restrained beams at computed fiber stresses lower than was the case for the corresponding unrestrained beams, and that, in general, the effect of restraint on elastic limit is not great. A reasonable explanation of this would seem to be that the re-
*The elastic limit was located by the method proposed by the late Prof. J. B. Johnson. His method consists in finding the point on a stress-deformation curve at which the deforma- tion is increasing fifty per cent more rapidly than its initial rate of increase.
See Johnson, "The Materials of Construction," pp. 18-20.
MOORE — STRENGTH OF I-BEAMS 27
FIG. 7. APPARATUS FOR TESTING I-BEAM WITH RESTRAINT AGAINST SIDEWISE
BUCKLING.
straining devices sometimes introduce additional stresses into the beam. As noted in "11. Tests to Failure of Beams Kestrained from Sidewise Buckling and Beams Eestrained Against Twisting of Ends," restraint against twisting of ends produced no marked effect on the ultimate strength of medium-span and long-span I-beams, while with restraint against sidewise buckling the fiber stresses developed before failure, even for the longest-span beams, were as high as the yield-point strength of the material.
.The elastic limit observations in connection with the observations of ultimate fiber stress seem to indicate that the resisting effect exerted by restraint is not noticeable until failure is imminent. Observations on sidewise deflection tend to confirm this conclusion. In the tests of certain of the unrestrained I-beams the sidewise deflection was measured. Table 7 records the sidewise deflection observed at loads which give a computed flexural fiber stress of 16,000 Ib. per sq. in. (an ordinary work- ing stress). It will be seen that the sidewise deflection is small, and gen- erally in the tests it continued to be small until failure of the beam became imminent. So small is this sidewise deflection for working stresses that it is questionable whether such restraining members as usually would be attached to beams in actual structures will be stiff enough to prevent it. In structures the usefulness of restraining I-beams against sidewise buckling lies mainly in the fact that such restraint ren- ders available the full yield-point strength of the material in the flanges of the beams, and that should failure occur, with effective sidewise and
28 ILLINOIS ENGINEERING EXPERIMENT STATION
end restraint, the beam will fail by gradual sagging rather than by sudden collapse.
Any restraining devices to prevent sidewise buckling of I-beams should provide resistance to sidewise bending moments. Separators be- tween the ribs of a pair of beams do not provide such resistance. A pair of beams held together merely by bolts and separators was tested (Test No. 30), and though the yield-point strength of the material was de- veloped, final failure occurred rather suddenly by sidewise buckling. While comparatively slight sidewise restraint may enable a beam to develop the full yield-point strength of the material, a strong, stiff re- straining system is needed to prevent sudden collapse when final failure does occur. Another illustration of this was furnished by testing a pair of 8-in., 18-lb. I-beams having a span of 15 ft. (Test No. 22) and re- strained against sidewise buckling. In this test the batten plates hold- ing the beams together were merely clamped to the flanges and not bolted. In the test the full yield-point strength of the flange material was developed, but soon afterward the beams failed quite suddenly by sidewise buckling. The clamps were not strong enough to hold the batten plates to the beam flanges under the large sidewise force developed when the yield point of the material in the beam was reached.
13. Web Failure of I-beams. — I-beams and built-up girders of short span are sometimes in danger of failure by crippling of the web. Web failure may be caused in several ways: (1) The fiber stress in shear at the middle of the web may exceed the yield-point strength in shear of the web material. (2) Accompanying the shearing fiber stress at any point of the web is a compressive stress of equal intensity acting in a direction inclined at 45 degrees .with the direction of the shearing stress, and this compressive stress may become so great as to cause buckling. (3) There may be an excessive compressive stress near the junction of web and flange and adjacent to a concentrated load or reaction. The shapes assumed by a cross-section of an I-beam after web failure are shown in Fig. 8. The shape and position at (a) is that due to torsion of the beam as a whole; that at (b) to buckling of the web; and that at (c) to local compressive stress at root of flange. What has been re- ferred to previously as failure by twisting of ends of I-beams is in most cases primarily caused by excessive local compression at the root of the flange.
An approximate method of computing the compressive stress at the root of the flange adjacent to a concentrated load or an end reaction, has been given by C. W. Hudson* as follows: Imagine a small piece cut
* Engineering News, December 9, 1909.
MOORE — STRENGTH OF I-BEAMS 29
TABLE 7.
SIDEWISE DEFLECTION OF I-BEAMS AT A COMPUTED FIBER STRESS OF 16,000 LB. PER SQ. IN. — BEAMS FREE TO MOVE LATERALLY.
|
Beam |
Material |
Span ft. |
Loading |
Deflection in. |
|
8-in., 18-lb. I-beam 8-in., 18-lb. I-beam 8-in., 25.25-lb. I-beam... 8-in., 25.25-lb. I-beam... 17.5-in. built-up beam... 24-in built-up beam |
Steel Steel Steel Steel Wrought iron Wrought iron |
10 20 10 10 12.9 14 2 |
One-third points ... . One-third points... . One- third points One-third points. .. .. One-third points... . Load 12 in. each side |
0.046 0.036 0.026 0.019 0.080 |
|
of center |
0.067 |
from the flange and web of an I-beam immediately over a bearing block (as shown in Fig. 9)', and imagine this piece to be held in equilibrium by the elastic forces which act on it while it is in its place in the beam. The forces are (1) the pressure of the reaction at the bearing block P; (2) the compression in the web which equals f^ib, when /w = the aver- age intensity of compressive stress, t = the thickness of web, and & = the length of bearing block; (3) a horizontal shearing force S^', and (4) a vertical shearing force Sv. Very little of the total shear would be bal- anced by the small internal shearing stress in the flange of an I-beam, and if the section considered be taken at the root of the flange we may write without serious error
&=&==<)
Then the compressive stress on the web is balanced by the reaction on the bearing block. The compressive stress may be regarded as uniformly dis- tributed, and we may write
In the above discussion the case considered is for the compressive stress adjacent to an end reaction. The reasoning for the compressive stress in the web adjacent to a concentrated load would be similar.
The compressive stress in the web of an I-beam necessary to cause buckling of the web is computed in most text books on strength o£, ma- terials on the assumption that the web of the I-beam is in the same con- dition of stress as a fixed-ended column whose length is equal to the vertical distance between flanges multiplied by the secant of 45 degrees, and whose radius of gyration is equal to the thickness of the web divided by VI 2, and in which the average intensity of compressive stress is equal to the maximum intensity of shearing stress in the web of the I-beam. This shearing stress is very nearly equal to the total shear divided by the area of the web. The assumption of fixed-ended conditions and the
30 ILLINOIS ENGINEERING EXPERIMENT STATION
neglect of the restraint against the buckling of the web by tensile stress in the lower part of the beam render the accuracy of this method some- what uncertain.
14. Web Failure of I-beams; Tests. — After gathering the data of tests it is realized that the whole amount of data on web failure of I- beams is small. The drawing of conclusions from these data is further complicated by the fact that several web failures of test beams seemed to be due partly to shearing stress in the web and partly to compressive stress in the web adjacent to bearing blocks.
In selecting test data for the study of web failure of I-beams, only those tests were taken in which at failure the fiber stress in the flanges was less than the yield-point strength of the material and in which the failure took place by crippling of the web.
Six of the tests selected were made at the University of Illinois. As noted on p. 33, variation in web dimension was obtained by planing down the webs of some of the beams.
The results of the tests selected for the study of web failure are given in Table 8. In the tenth line of the table is given the slenderness ratio of the web computed on the assumptions usually made in text books on mechanics of materials and named in the preceding paragraph. In the thirteenth line of the table is given the computed fiber stress at fail- ure of the web by buckling as determined by Euler's formula for fixed- ended columns. Euler's formula was chosen on account of the high values of slenderness ratio. It will be seen that the calculated compres- sive stresses corresponding to the loads carried (tabulated in the twelfth line of the table) were in three cases very much in excess of the value given by Euler's formula. This excess is so marked that even these few tests may be taken to indicate that for computing the safety of I-beam webs against buckling the method common in texts on mechanics of ma- terials gives results which are on the side of safety.
From the twelfth line of Table 8 it will be seen that in all beams but one the fiber stress in shear at mid-web was not much below the yield- point in shear for structural steel, which averages from 25,000 to 35,000 Ib. per sq. in. Of course, even under the most favorable circumstances the web of an I-beam may not be counted on to develop without failure a stress in excess of the yield-point in shear of the web material.
In the fifteenth line of Table 8 is given the computed fiber stress in compression developed at the roots of the flange. This fiber stress is com- puted from equation (7). In the sixteenth line of the table is given the yield-point strength of the material of the beams at the root of the flange.
MOORE STRENGTH OF I-BEAMS
31
FIG. 8. SHAPES ASSUMED BY I-BEAMS AFTER WEB FAILURE.
This was determined by means of tests of specimens cut from the beams. The compressive fiber stress developed was in all cases not much greater than this" yield-point strength of material. However, in all the tests for web failure made at the University of Illinois, before final failure occurred evident signs of structural injury, scaling, etc., had appeared. It is un- wise to regard the ultimate compressive fiber stress in the web adjacent to a bearing block as higher than the yield-point strength of the material at the root of the flange. Moreover, the fact should be borne in mind that the material at the root of the flange of an I-beam has a yield-point strength somewhat lower than the material in the flange or in the web. In the absence of special tests the yield-point strength of the structural steel at the root of the flange of an I-beam may be taken as about 30,000
FIG. 9.
DIAGRAM OF COMPRESSIVE STRESS IN WEB OF I-BEAM OVER A BEARING BLOCK.
Ib. per sq. in., which is near the value obtained in various tests at Illinois and by Marburg and by Hancock. In view of the small amount of data of failure of I-beams by buckling of web, the conclusions given should be regarded as tentative.
15. Stiffness of I-beams. — In most of the tests of I-beams referred
32 ILLINOIS ENGINEERING EXPERIMENT STATION
to in this bulletin the value of the modulus of elasticity based on the beam deflections and the common theory of flexure is reported. Tables 2, 4 and 5 give values of the modulus of elasticity so computed. These values of the modulus of elasticity seem in general to be lower than the values usually obtained from tension tests of structural steel or of wrought iron. This difference in the values obtained in the two ways is confirmed by Marburg's tests. By means of extensometer tests of samples of material cut from I-beams, Marburg determined the modulus of elas- ticity of the beam material, and also computed the modulus of elasticity of the beam from load-deflection curves. Table 9 summarizes the aver- age values obtained from the published data of Marburg's tests. It will be seen that the modulus of elasticity obtained from beam deflections is about 10 per cent less than the modulus of elasticity obtained from tension tests of samples of beam material ; or in other words, the stiffness of the I-beams was about 10 per cent less than that indicated by tension tests of material.
16. Summary. — The following summary is given :
1. The yield-point strength, not the ultimate tensile strength, should be regarded as the ultimate fiber stress for structural steel in flexure.
2. The yield-point strength of structural steel in compression is about the same as the yield-point strength in tension.
3. The slight inelastic action which may be observed in steel I-beams under stresses as low as those used in practice is in general local in its effects and does not indicate the load-carrying capacity of the I-beam, if the load is not reversed in direction.
4. The computed ultimate fiber stress for steel I-beams not re- strained against sidewise buckling of the compression flange is given by the formula
7
A = 40,000 — 60 ^
in which /i is the extreme fiber stress, in pounds per square inch, computed by the usual flexure formula, I is the length of span of beam in inches, r' is the radius of gyration of the I-section about a gravity axis parallel to the web, and m is a coefficient dependent on the method of loading, ml being a so-called equivalent column length. Values of m for various loadings are given in Table 4. In no case should /i be taken greater than the yield-point strength of the material in the flanges. It should be borne in mind that f\ of this formula is an ultimate, not a working value.
5. A light system of sidewise bracing may so strengthen an I-beam that the full yield-point strength of the material will be developed before
MOORE — STRENGTH OF I-BEAMS
33
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34 ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 9.
MODULUS OF ELASTICITY OF I-BEAMS AND OF I-BEAM MATERIAL.
Values from results of tests by Marburg at the University of Pennsylvania.
|
Item |
Standard I-beams |
Bethlehem I-beams |
Bethlehem Girder Beams |
|
Average modulus of elasticity of tension test §ieces cut from web, flange and root of ange of I-beam. Ib per sq. in |
29,500,000 |
28,810,000 |
29,660,000 |
|
(A) Average modulus of elasticity of beams de- termined from deflections, Ib. per sq. in (B) (B) • (A) |
26,300,000 0 892 |
26,570,000 0 987 |
26,120,000 0.882 |
failure occurs, but a stiff bracing capable of resisting sidewise bending moment is necessary to prevent sudden failure by sidewise buckling, once the yield-point of the beam flanges is reached. Separators between the webs of I-beams do not furnish a stiff bracing against sidewise buckling. 6. In investigating the safety of an I-beam as regards web failure three possible causes of failure should be considered :
(a) Failure by shearing stress in the web. The yield-point strength of structural steel in shear should be regarded as the ultimate fiber stress for the web.
(b) Failure by buckling of web. The buckling strength of a strip of web inclined 45 degrees to the flanges as computed by Euler's formula for fixed-ended columns was developed in several tests without collapse of the web.
(c) Failure by compressive stress in the part of the web adjacent to a bearing block. The value of this stress may be roughly estimated from the formula given by Hudson,
f _L
Tw ~ u
in which /w is the fiber stress in compression in pounds per square inch, 6 is the length of bearing block in inches, t is the thickness of web in inches, and P is the concentrated load or the reaction in pounds. The yield-point strength of the material at the root of the flange of the I-beam should be regarded as the ultimate value for /w.
MOORE STRENGTH OF I-BEAMS
35
40000
30000
10000
One Division = O-OS in. Deflection 'OOio in. CxtensometerMOi
FIG. 10. RESULTS OF TESTS 1-6.
36
ILLINOIS ENGINEERING EXPERIMENT STATION
40000
30000
40000
40000
loooo hM
One Division -=o.lin. Deflection = 0.010 in. Ettensometer Movement
FIG 11. RESULTS OF TESTS 7-11.
MOORE STRENGTH OF I-BEAMS
37
4OOOO
30000
One Division = 0.10 in. Deflecf ion *O£ioin. Extensomerer Movement FIG. 12. RESULTS OF TESTS 12-17.
38
ILLINOIS ENGINEERING EXPERIMENT STATION
40000
One Division - o.4in. Deflection -O.oioin. Extensomefer Movement FIG. 13. RESULTS OF TESTS 18-23.
MOORE— STRENGTH OF I-BEAMS
39
soooo
4OOOQ
30OOO
30000
aoooc
One Division *o 10 Def lection 'O.OSO in. F xfensomerer Movement FIG 14. RESULTS OF TESTS 24-29.
40 ILLINOIS ENGINEERING EXPERIMENT STATION
4OOOO
30000
eoooo
IOOOO
,40000
30000
20000
IOOOO
-+- .- + -.- + 4- + - 1 pefh ttion ifdeim
One Division =aio in. Def/ection =o.o/oin. Exflensometer Movement
FIG. 15. RESULTS OF TESTS 30-33.
PUBLICATIONS OF THE ENGINEERING EXPERIMENT STATION
Bulletin No. I. Tests of Reinforced Concrete Beams, by Arthur N. Talbot. 1904. None available.
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THE UNIVERSITY OF ILLINOIS
THE STATE UNIVERSITY
Urbana
EDMUND J. JAMES, Ph. D., LL. D., President
The University includes the following departments:
The Graduate School
The College of Liberal Arts and Sciences (Ancient and Modern Languages and Literatures; History, Economics and Account- ancy, Political Science, Sociology; Philosophy, Psychology, Edu- cation; Mathematics; Astronomy; Geology; Physics; Chemistry; Botany, Zoology, Entomology; Physiology; Art and Design; Ceramics)
The College of Engineering (Architecture; Architectural, Civil, Elec- trical, Mechanical, Mining, Municipal and Sanitary, and Railway Engineering)
The College of Agriculture (Agronomy; Animal Husbandry; Dairy Husbandry; Horticulture and Landscape Gardening; Veterinary Science; Agricultural Extension; Teachers' Course; Household Science)
The College of Law (Three years' course)
The School of Education
The Courses in Business (General Business; Banking; Accountancy; Railway Administration; Insurance; Consular Service)
The Course in Journalism
The Courses in Chemistry and Chemical Engineering
The Courses in Ceramics and Ceramic Engineering
The School of Railway Engineering and Administration
The School of Music (Voice, Piano, Violin; four years' course)
The School of Library Science (two years' course)
The College of Medicine (in Chicago)
The School of Pharmacy (in Chicago; Ph. G. and Ph. C. courses)
The Summer Session (eight weeks)
Experiment Stations: U. S. Agricultural Experiment Station; En- gineering Experiment Station; State Laboratory of Natural His- tory; State Entomologist's Office; Biological Experiment Station on Illinois River; State Water Survey; State Geological Survey; Mine Rescue Station
The library collections contain (May 30, 1913) 259,856 volumes, includ- ing the library of the State Laboratory of Natural History (8,100 volumes), the Quine Medical Library (14,000 volumes), and the library of the School of Pharmacy (2,000 volumes).
For catalogs and information address
THE REGISTRAR
Urbana, Illinois
UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY
Return to desk from which borrowed. This book is DUE on the last date stamped below.
CALIF. HALL
LD 21-100m-9/48(B399sl6)476
48
UNIVERSITY OF CALIFORNIA LIBRARY