THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING COMMENTS ON THE USE OF SEMI-RIGID CONNECTIONS IN STEEL FRAMES Lawrence C. Maugh March, 1960

ACKNOWLEDGMENTS The author wishes to express his appreciation to Messrs. D. N. Frayne and Nabil Nassar, former graduate students at The University of Michigan, for use of the experimental data in this paper. ii

LIST OF FIGURES Figure Page 1 Arrangement of Specimens in Testing Maching..........e 2 Location of Angle cp and Displacement A.................. 4 3 M-cp Diagram Determined from Displacement A............. 6 4 M-c Diagram for Specimen No. 3........... 0............ 7 5 Connection Details of Specimen No. 1....,..,.......... 9 6 Connection Details of Specimen No. 2................... 10 7 Connection Details of Specimen No. 3................... 11 8 Load-Deflection Diagram for Specimen No. 2............. 12 9 Values of C1, C2 and MF........................... 15 iii

INTRODUCTI ON For almost thirty years the effect of the deformations in ihe connections of structural members, particularly beams to coldinS upon- the stresses in steel frames has been investigated both experimentally and analytically. Although considerable progress has been indicated by t'he laboratory and research reports (see references) on methods for deterraini-~ the physical characteristics of various riveted and welded connections, th.e uncertainties that are involved when variations occur in both beams, col.9its: and connecting elements still plague the structural designer~ Now with the increased use of plastic or ultimate methods of design, it becomes even more essential. to consi der the influence of the connecting elements in the structural behavior of -the frame. If the design assumes elastic deformatio-ns in both members and connections,9 then only certain types of conrections w ill qualify. It is therefore important that the engineer should tes-t t;he particular connections that are likely to be used and that he should be able to make such tests quickly and with reasonable accuracy. In this paper a relatively simple laboratory procedure that requires no sensitive eq7it pment or experienced personnel is described and the use of the data in obtt anrding the necessary beam coefficients is discussed. Once these coefficients are determined, the analysis can proceed as for any frame with variable momeni; of inertia. Test Procedure For Measuring Beam Connectioon Properties. The following laboratory procedures and interpxretation ef the data has been founnd to be convenient and sufficien.tly accurate for measr rinfg thhe

rotational restraint of beam connections,. The elastic portion a of a tlypical test specimen. as shown in Figure 1 is assuLmed to extend within three inches of the edge of tbhe connelct ion, Strain gage measurements have shown that between. ' this sectior and tahe face of the column, the strain distribution across any raansverse section is nonlinear.- This region is called inelastic althloulgh sulch a description is open to question, 2o The resultanit angle changes cp. Figure 2, for the inelast;ic ranwgne is assured to occur at each face of the co lunri for p-urposes of r 'e f.erenc,' O 3, A laborat. ory t+est specimen as shown in Fig 're I is therefore divided into an elas+;ic zone a and an inelastic zone L-a which contains t;he conn.e.cting eleme.nts and colromn section. 4, The specimen is loaded as shown in Figuie and the only measunrme<nt needed besides +the central load P is the displacement A at, +t,.'e e-er of the span. This displacement" can 'be mreasured with ordinary dial gages. 5. The n(mrerical1 valiue of the rotation. cp for the in oi asi z7orne "can,nolow be det;ermined by subtracting the calculated dlspl.acenerwe nt at 'C cet;er d-ue to the strain in the el.astic portions a from -the toali mieas.i.-ed deflection A. Thus, from Figur-e 2, the following relations can; be estab lshed Pa3 p (L d)= A 4 Pa3 6E 1 ~'~ S PL~d

LOAD P THROUGH SPHERICAL HEAD i I2 - Elastic a Inelastic 2 L- -'-Elastic a F RA1N GAG ES I \|o GAGES TO MEASURE DEFLECTION A BASE OF MACHINE ARM ARM Figure 1. Arrangement of Specimens i Testing Machine.

P I I ____ I I I 2 2LH LOCATION AND VALUE OF Pa Po I I 2 ELASTIC I INELASTIC ELASTIC REGION REGION REGION MOMENT DIAGRAM P12 P/2 DEFLECTION DIAGRAM Figure 2. Location of Angle cp and Displacement A.

where P = total load on the specimen E modulus of elasticity I = moment of inertia a = length of elastic portion L = 1/2 span of specimen d - 1/2 width of column 6. The values of cp that are determined from the measured valuhe of A by means of Equation (1) when plotted as abscissae against the morr-ment. at. the face of the column as ordinate provide typical moment-rotation (M, cp) curves as shown in Figures 3 and 4. In the diagrams are also shown corres= ponding rotations cp (see broken lines) which are obtained from the horizontal movement between two reference points that were established in each flange at the edges of the inelastic zone. These gage distances are shown by point:s 1, 29 3. and 4 in Figure i. The sum of the horizontal displacements between points 1 and 2 and between 3 and 4 divided by the vertical distance between the points was used to check the value of cp. Results of Typical Tests, The test procedure as described above was used on the three specimens whose details are shown in Figures 5, 6, and 7. An initial load was applied through a movable head with a spherical beacing and removed several times before the final measurements were made. The load-deflection curve for specimen number 2 is shown in Figaure 8. From such diagrams the angle change cp in the inelastic zone was calculated for each specimen by means of Equation (1). The values of cp are shown in

700 1' I........... 600 t/) 500 /_/ z_ 400,..... 300..... ~~~~~~~~~~~~~~~~~~~..2 IaIL 4 iN 200... '... ALUE OF FROM gqu. I.... ~mDETERMINED DIRECTLY FROM HORIZONTAL STRAINS 300 OF 0 -0 O. 001 0.002 0. 003 0.004 O. 005 0.006 0. 007 O. 008 0.009 00 F VALUE OF IN RADIANS Figure.. DET Diagram DetermineE frM D DisplacematT

1400 (I) 0. 1200 - 1000 -:|*- r SPECIMEN NO, 3 400 200 0 VALUE OF IN RADIANS Figure 4. M-~cp Diagram for Specimen No. 3.

Figures 3 and 4o The magnitudes of the inelastic portions a are 46, 43, and 46 inches for specimens 1, 2, and 3, respectively. L is equal to 6o inches and d is 5 inches for all specimens. Use of M, Lp Curves in Design. If the actual M, cp curves in Figure 3 are approximated by a straight line the moment M can be expressed in terms of the rotation cp bcy the relation M =r cp (2) where 4 is the slope of the M, cp diagram. The quantity -- is therefore Mdx equivalent to E — in a beam and can be treated as such in the calculations. In any frame where the steel girders support a reinforced concrete floor, the actual EI value of the beam is uncertain. However, when only the steel members are considered in determining the beam coefficients, t..he following assumptions are recommended for design calculationso (a) Consider the beam as a member with constant EI except at, the ends where a concentrated angle change of occurs, (b) When the connection stiffness ' is the same for both ends of the beam, the coefficients 4 and 2, and the fixed-end moments MFab and lFba in the slope-deflection equations ab = EI ( 48a + 28b) + MFab Mb (28 + 4 (3b5' Mba L (2a + 4b) + MFba a.r. be replaced by

SPECIMEN NO. I RIVETED 3 RIVETS | 4~> 4 a2>l C tIOH49x2'-6+ _ 2- f -2 ILb:; T at, eL6xx ox 6-3 Lb,:: 12 VF 27 x 5'-0 I12 W' 27x 5-O - N Zi I Figure 5. Connection Details of Specimen No. 1.

SPECIMEN NO.2 RIGID JOINT c _j 4~ r ~1 --- —-- 10* ~F1w2 -TYP / TYP \~~~~~~~~~~II- ' 'IItp -t~u -- Q.64 HT BOL BL ii S2T 8 4F25 14 F 3 x 04 nl 9 14 VIF 30 x 5'- 0 mlT~ 7 BOLTS 4 REQD6 HT Figure 6. Connection Details of Si. 0 ~ ~ ~ ~ ~ ~ ~ ~~~~~ ST 8 V' 25 CL~~~~~~~~~~' -J ~BAR 4 x.x0-8 4 8 4 4 REQ'D "J Figure 6. Connection Details of- Specimen No. 2.

SPECIMEN NO. 3 WELDED STD. CONN. TYP 3 I.1 _.1 45* It 6oxx '-6 12 %F 36 x 5' -0 1 LQ 12 %F 36 X 5'_0 ST 8! — 25 BAR 4- L x 3 x'-.4 4 REO'D 4 Figure 7. Connection Details of Specimen No. 3

28,000 24,000. 20,000 u~vvv -... I. 12,000 L I 8,000 __._.... 4,000 0.1 0.2 0.3 0.4 0.5 DEFLECTION -,Inches Figure 8. Load-Deflection Diagram for Specimen No. 2.

-13 -Mab - (Ca + C2b) + M 'Fab(4a EI Mba (C20 + C0b) + M'Fba (4b in which. assaming that 4*a 1=b = V 12A (5a) 4A2 - 1 6(5) C=4A2 - 1 where 3EI 3K A 1-+ -- (52) K = EI L in which L= distance center to center of columns. Also, MlFab = ~ [MFab (2C1 - C2) + MFba (2C2 - C) ] (6a) M Fba = ~ [MFab (2C2 C1) + MFba (2C1 - C2)] (6b) where MFab and MFba are the usual fixed-end moments in Equations (3a) and (3b) that is for V equals infinity and A equal to one. For a symnnmetrical loading such that

14 -~Cab =-MThba t;hen MFab = [MFab (2C1 C 2C + C) ] Fab ~ 1 2 2 9] or M Fab= 2 (C1 C2) MFab Important Features of Semi-Rigid Connections, As variations in the coefficients C1, C2, M Fab9 and M'Fba are important factors in a structural design, it is interesting to note that a particular end connection may provide considerable restraint for a beam with small K ut - o a sma.i value, but relatively little if the beam has a large K value0 The variation of the coefficients C1 and C2 with respect to K are shown in Figure 9. A disturbing feature of these diagrams is the rapid change that may occur in the values of C1i C2' and M' for small changes in K. The changes in the fixed-end moments M'F are indicated in Figure 8 for a u:niform load over the entire span.. It is apparent that the fixed-end moments may change rapidly for even small changes in the - values. If a semi-rigid connection such as in specimen 3 is used instead of a rigid connection than a constant value of l of 385 x 106 in=lhs is obtained from the slope of the M, cp curve in Figure 3. When this connection is used an a 12 WF 36 beam of 17 feet length the value of is K El 29 x 106 x 280.8 L 17 x 12 x 385 x 106 =.104 Froml Figlure 8 or from Equations (5a, 5b, 5c) we obtain,

. 500 2.... 25 4.0 0 c 2.0 30 10 0 0.5 1.0 1.5 2.0 5 3.0 35 4.0.5 Figure 9. Values of C, C2 and M..

-16 -= 2.68 C2 = 1.02 (268 - 12 = (.8 ) = 69 F=~~~~~ 2w2 = 0o69 wL~

SUMAMARY In this paper the importance of considering the deformation of beam connections in the design of steel frames has again been emphasized. A laboratory procedure for determining the M, cp diagram for any type of beam connection has been discussed. This method involves measuring only a single vertical displacement by means of dial gages. Analytical methods for incorporating the properties of the connections into the slope-deflection equations are presented. It has been showmn that the stiffness of the beam and the magnitude of the end couples may be modified considerably by the rotational restraint factor * of the connections. Therefore the actual beam coefficients and fixedAend moments, for the particular beam and connection,should be determined from test results and used in the structural analysis.

REFERENCES 1. Rathbun, J. Charles, "Elastic Properties of Riveted Connections," Trans. Arnom. Soc. C E., 101, 1936. 2, Johnston, Bruce and Mount,, E. H., "Analysis of Building Frames with Semi-Rigid Connectionss," Trans. Am. Soc. CE., 107, 1942. 3. Johnston, Bruce and Hechtman, Robert, "Design Economy by Connection Restraint -,' Eng. News-Record, Oct. 10, 1940. 4, First, Second and Final Reports of the Steel Structures Research Committee, Department of Scientific and Industrial Research, Great Britain, 1931-1936o 5. Lyse, Inge and Gibson, G. J. "Welded Beam-Column Connections," Am. Welding Soco, 15, 34-50 (1936), and 16, 2-9 (1937) 6, Hechtman, R. A. and Johnston, B. G., "Riveted Semi-Rigid Beam.toColumn Building Connections," A.I.SC., Progress Report No. 1. 70 Lothers, J. E., "'Elastic Restraint Equations for Semi-Rigid Connec~ tions," Trans. Am, Soc. C oE, 116, 1951. 8. Maugh, L. C.,, Statically Indeterminate Structures, 292=300, John Wiley & Sons, New York. 9. Pippard, A, J. S. and Baker, J, Fo,, "The Analysis of Engineering Structures, "196-216, Longmans, Green and Co. ~ ~.d8P