THlE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THlE COLLEGE OF~ -ENGINEERING A STUDY TO DETERMINE THE INELASTIC MOMENT-CURVATURE RELATION OF STRUCTURAL MEMBERS M.. J. Kalajilan October, 1966 IP- 753

TABLE OF CONTENTS Page LIST OF TABLES................................................. iii LIST OF FIGURES......................................... iv INTRODUCTION..............*...... o o o.... o..o................. 1 ASSUMPTIONS...................................................... 4 RESULTANT FORCE AND MOMENT...................................... 7 STRESS CENTER (CENTER OF PRESSURE)............................... 9 WIDE-FLANGE SECTIONS............................ 10 NUMERICAL WORK................................................... 16 SIMPLIFIED EXPRESSIONS........................................... 16 ALTERNATE EXPRESSION............................................o 22 DISCUSSION OF RESULTS,........................................... 23 CONCLUSIONS...................................................... 23 REFERENCES...................................................... 25 ACKNOWLEDGEMENTS................................................ 25 APPENDIX - NOTATIONS........... o................. 26 ii

LIST OF TABLES Table Page. I APPROXIMATED SECTION PROPERTIES.....sopp000-s 10 II MOMENT-CUJRVATURE PARAMETERS COMPARED FOR r = 10, G = 36 k s i, EY.=.0012 in/in, [i = 10................. 17 III MOMENT-CURVATURE PARAMETERS COMPARED FOR r = 10, aY= 36 ksi, E Y =.0012 in/in, ji.....5.............. 18 IV MOMENT-CURVATURE PARAMETERS COMPARED FOR r=10,~ C~= 36 ksi, C y =.0012 in/in, ~L 20................... 19 V MOMENT-CURVATURE PARAMETERS COMPARED FOR r = 5, ay= 36 ksi, E =.0012 in/in, p. = 10............20 iii

LIST OF FIGURES Figure Page 1. Ramberg-Osgood Functions................................. 2 2. Ramberg-Osgood Load-displacement Relations,............... 2 3. Experimental Hysteresis Loops............................. 3 4. Stress-strain Relation.................................... 6 5. Stress and Strain Distribution Across Section............. 6 6. Wide-Flange Section Approximation......................... 6 7. Moment-curvature Relation For 21WF62 (ay = 36, Ey =.0012 and = 10)................................... 12 8. Simplified Moment-curvature Expressions Compared For 21WF62 (ay = 36, Ey =.0012 and l = 10)................... 13 9. Moment-curvature Relation For 21WF62 (ay = 36, e =.0012 and =- 10)................. 14 10. Simplified Moment-curvature Expressions Compared for 21WF62 (ay = 36, y =.0012 and = 10)..............15 iv

INTRODUCTION Experimental work has shown that the load-displacement (or moment-curvature) relationship for structural members, structural steel in particular, is not an elasto-plastic curve. The actual load displacement curve has an elastic branch followed by a transition curve that leads to a plastic branch(l). When the displacement is reversed, due to Bauschinger effect, the transition becomes more gradual. Such a relationship can be expressed quite closely by a Ramberg-Osgood function, (2)(3)(4) y y y ( where x = the displacement (or curvature) Xy = a characteristic displacement q = the load (or moment) q= a characteristic load r = an exponent xy, qy and r are the Ramberg-Osgood parameters. Plots of Equation (1) are shown in Figure 1 for various values of ro It also includes as limiting cases the elastic (r=l) and the elastoplastic (r = Io) relations. A complete description of a Ramberg-Osgood equation is found in Figure 2. Ramberg-Osgood parameters so far have been determined from test results onlyo Data for a given section are obtained in the form of loaddisplacement (moment-curvature) hysteresis loops and the parameters qy xy and r are chosen to give the best fit in the sense of least squares. A typical example of this is shown in Figure 3 (2) -1 -

-2 - 1.0.5 y = Ly ( + q r-i x q I ly 0 I 2 3 4 5 Figure 1. Ramberg-Osgood Functions. x7, qy q 7, - - I X 2q (1+qr-' 7 qy 2qy Xy q-q r2 - 2q 1+ 2q I 2xy 2qyy (! xt ql ) SXy qy - -.- -I Figure 2. Ramberg-Osgood Load-displacement Relations.

:R3> a,) 0 -J 8WF20 TOP FIBER STRAIN a_. I 1.5 %/ Figure 3. Experimenri-tal Hysteresis Loops.

-4 - The purpose of the present study is to find an analytical basis to determine these parameters for various sections a) from a stress-strain consideration, when the stress-strain relation across a section in bending can be expressed by a Ramberg-Osgood function(5)o b) when a set of parameters for a section is available, say from experimental resultso Finding the Ramberg-Osgood parameter is essential in seismic design. Dynamic analysis of a multi-degree-of-freedom structural frame can be performed only after the parameters x, q and r are known for all the members in the frame. ASSUMPTIONS are being To develop moment-curvature relations, the following assumptions made: a. Beams are prismatic and straight, and have a cross-sectional area of symmetry about the plane of bending. b. Planes normal to the axis of the beam remain plane after deformation, i.e., strains vary linearly from the neutral axis. co The stress-strain relation is of the Ramberg-Osgood type and is applicable to the individual fibers in tension as well as in compression. do A wide-flange section can be closely approximated by the difference of two rectangleso

-5 - From assumption (c), the stress-strain relation is given as a Cr (1+r-l e - + I (2).y y y where C = the strain a = the stress ey a characteristic strain ca = a characteristic stress r = an exponent ~, and r are the Ramberg-Osgood stress-strain parameters, and y,y can be determined from test results. For a given bending moment M at a section along a beam let the maximum stress and the maximum strain in the extreme fibers be represented by am and em respectively, see Figure 4. The stress and the strain distribution across the section is shown in Figure 5. The stress at any point y from the neutral axis (Figure 5) can be expressed as a =C m - xa where xa is the difference between the stress at the extreme fiber and the stress at point y. The corresponding expression for strain from Equation (2) becomes E =E ( (1 + am. I|r (3a) [\ ^ [ ^

c (1+ - r-i Figure 4. Stress-strain Relation. el,. b s c 00, F A c (a) Cross-sectional area (b) Strain Distribution (c) Stress Distribution Figure 5. Stress and Strain Distribution Across Section. C J ~ i, t! c c H 1-100" gm b 0 L b=b-w Figure 6. Wide-Flange Section Approximation.

-7-.A~t the extreme fiber the strain = E[ (' 1)l~ + l](3b) RESULTANT FORCE AND MO MENT For a rectangular cross-section of width b and depth 2c the differential element of force dF and moment dM from Figure5 are dF = b (c - y) dx~ dM -c+ ' _ F=l 2)dc 2 2y)dx (4) (5) and also from Figure 3 c (6) Substituting Equations (3a)., (3b) and (6) into Equations (4) and (5) and simplifying, results in dF =bc) 1 j L m X +(~ x) jdx ~ (7) and bc2 dM -.; 2 L dxc (8) where - c m = Cm + =r y y Note that the absolute value notations left out -because ( cTm - x.c)/ lr in thi s the ductility ratio. in Equations (7) and (8) have been analysis is always positive.

-8 - The resultant force F over half the section, i.e., to one side of the neutral axis, is obtained by integrating Equation (7) from xQ = 0 to xm= am F = fIm bc ( - 1 a + a 1 dx Performing the integration over the limits, the resultant force becomes.__ +..- 2. (9) F= abc 1 r+1 a To obtain the total moment M, Equation (8) is integrated over the whole section, thus, 2am. 1 a - X a - X. r- 2 bM bc2f {Ma (m- r } a ___ dxr b m i -a 2 m I 0O { [ [ ay a + a_ ] which, upon integrating and simplifying, reduces to M2 r2 ( r+l M C2 1 1 |m + | 2r+ 1 ( ) 2r 2r + 1 The curvature 0 corresponding to the moment above is obtained by dividing the extreme fiber strain Em by its distance from the neutral axis. In equation form the curvature E0= m = ( [ r (11) c c ay

STRES$ CENTER (CENTER OF PRESSURE) The stress center yR for a rectangular section is found by R taking half the total moment and dividing it by the resultant force, i.e., yM R 2F Substituting Equations (9) and (10) into above and simplifying, yields the desired expression for the stress center 1 1 2 + 6 zr+l + 3 z2r c 3 r + 2 m 2r + 1 m (12) 1 - 2 Z-+ zr+ r +1 where z = m m y When the section approaches the fully plastic condition of stress, the ductility ratio p. tends to infinity making the stress center YR in Equation (12) approach c/2. On the other hand, a fully elastic dis2 tribution of stress in the section reduces YR to 2 c. These are to be expected for rectangular sections as limiting values for y. The R stress center for any section must therefore lie between y yR _ P- YR - YE where y = the stress center for the fully plastic case and YE = the stress center for the fully elastic case. YE=tesrsetrfor th fll easi — cse -9 -

-10-, TABLE I APPROXIMATED SECTION PROPERTIES IfSec. Mod.I Plas. Mod.. Section Are a sYp 4 LC 13 3WF 17 8 WF 20 10 WF 25 12 WF 36 16 WF 5 0 21 w~p 62 24 WF 84 30 WF 108 8 x 4 _a 3.765 (3.82) 4.932 (5.00) 5..14 (5.88) 7.279 (7.35) 10. 494 (10 p59) 14.581 (14.70) 18.039 (18.23) 24.516 (24.71) 31,444 (31.77) 32.00 5.225 (5.2) 13.895 (14.1) 16.795.(17.0) 26.170 (26.4) 45.448 (45,9) 79.883 (80.7) 124.76 (126.4) 194.41 (196.3) 295.12 (299.2) 42.667 6.012 (6.1) 15.573 (is.8) 18.837 (19.1) 29.265 (29.5) 50.974 (51.4) 90.748 (92.7) 142.30 (144.1) 222.01 (224.0) 341.27 (345.-5) 64.oo 1.597 3.157 (3.16) 3.240 (3.24) 4,0021 (4.02) 4.857 (4.86) 6.224 (6.24) 7.888 (7.90) 9,0o56 (9.07) 10.853 (lo,88) 2.00 1.738 3.569 (3.566)i 3.629 (3.626)1 5.457 (5.455) 7.152 (7.16) 9.202 (9.20) 10, 547 (10.ss) 12.894 (12.90) 2.667 NOTE: Bracketed numbers are from acul s tin fo copr on actual sections for comparison.

WIDE-FLANQE SECTIONS A wide-flange section can be closely approximated by the difference of two rectangles as shown in Figure 6. The moment MWF in a wide-flange section can be given as MwF MA MB (13) The subscripts A and B refer respectively to rectangles A and B in Figure 6. For a given am in the extreme fiber of wide flange MA, the moment in rectangle A, is found explicitely from Equation (10). There is, however, no direct way of obtaining MB. To find the latter, the fiber strain at c' (i.e., -c e) in Figure 6 is first calculated and the corresponding stress is found from Equation (2) by a numerical method. This is the extreme fiber stress in rectangle B and is used in Equation (10) to obtain the value of MB 0 With MA and MB known the moment MWF in the wide-flange section is determined from Equation (13), and the curvature 0, corresponding to MWF, is obtained from Equation (11). MOMENT- CURVATURE PARAMETERS Points along the moment-curvature plot for a wide-flange section are calculated from Equations (10), (11) and (13) by varying the extreme fiber stress am, and the moment-curvature parameters My (a characteristic moment), 0y (a characteristic curvature) and R (an exponent) are chosen to give the best fit in the sense of least squares. Fitting the curve through the points is done with the aid of a computer.

8000i 6000 - _J r= 5 Z 4000 -E / 0 2 2000 -C0 STRESS- STRAIN RESULTS EQ. 10 CURVE FITTING RESULTS EQ.14 0 I I I I Ii I I I 0 2 4 6 8 10 12 CURVATURE (RADIANS x10 ) Figure 7. Moment-curvature Relation For 21WF62 (ay = 36, Cy =.0012 and iL = 10).

8000 6000 - /) am 4000 - Iz 0 2000 - 0 0 r=5 0 STRESS-STRAIN RESULTS EQ. 10 --- El CONSTANT RESULTS EQ. 15, 16 8 2! - MODIFIED El RESULTS EQ. 15, 16 8 17 I H UJ 2 4 6 8 CURVATURE (RADIANS x 10-4) 10 12 Figure 8. Simplified Moment-curvature Expressions Compared For 21WF62 (ay = 36, c =.0012 and pt = 10). y

6000 6-1 0. 4000 Z Iz LJ 2000 0 0 0 1 r=10 EQ.10 RESULTS EQ. 14 RESULTS I 4-u o STRESS-STRAIN CURVE FITTING 0 2 4 CURVATURE 6 (RADIANS x 10 ) 8 10 12 Figure 9. Moment-curvature Relation For 21WF62 (oy = 36, cy =.0012 and p = 10).

6000 n a' 4000 z 1 -z L 2000 0 2 r=10 0 STRESS- STRAIN EQ. 10 --— El CONSTANT EQ. 15,16,8 21 - MODIFIED El EQ. 15,16 8 17 I k-J 0 0 2 4 6 (RADIANS x 10-4) 8 10 12 CURVATURE Figure 10. Simplified Moment-curvature Expressions Compared for 21WF62 (ay = 36, ey =.0012 and =- 10).

Thus, a single relation between moment and curvature is established: M l + I M (14) Moment-curvature plots from stress consideration as well as from curve fitting Equation (14) are shown plotted in Figures 7 and 9 for 21 WF 62. The results are typical for the wide-flange sections considered. NUMERICAL WORK To test the preceding presentation numerically, nine structural steel sections are chosen ranging from 4 LC 13 to 30 WF 108. Also a rectangular section 8" x 4" is considered for comparison. To obtain the moment-curvature parameters, a stress-strain curve of the Ramberg-Osgood type and a ductility ratio are assumed. Points along the moment-curvature plot are calculated for each section from Equations (10), (11) and (13), and the parameters My, 0y and R for Equation (14) are chosen. Moment-curvature parameters are determined for various stressstrain parameters and ductility ratios, and are shown tabulated in Tables II through V. SIMPLIFIED EXPRESSIONS To obtain the moment-curvature parameters the need for simplicity is apparent. The following simplified relations are being proposed for wide-flange sections: a) When the stress-strain parameters (ey, ay and r) are known,

TABLE II MOME~NT-CURVATURE PARAMtETERS COMAPARED FOR r = 10,9 cT = 36 ksi, Ey =.0012 in/in 11= 10 M x 104 EI x10 - Section R -Z x 10 - (in-kips) (radians) vy (kip-in2) 4 LC 13 9.845 216.11 7.192 30.05 31.35 (10.0) (216.43) (7.196) (30.08) — 9.715 215.82 7.205 29.95 8 WF 17 9.779 559.49 3.570 156.72 166.74 (10.0) (560.63~) (3.568) (157.12)9.715. 559.04 3.572 156.50 8 WF 20 9.796 676.85 3.494 193.71 205.-07 (io.o) (678.13) (3.494) (194.08).9.715 676.21 3.498 -193.31 -10 WF 25 9.795 1051.57 2.815 373.55 395.69 (10.0) (10535.5) (2.814) (374.37) 9.715 1o50.56_. 2.818 372.80 12 WF 36 9.790 1831.5 2.327 787.06 834.43 (io.o) (1835.1) (2.327) (788.61) 9.715 1829.9 2.330 785.36 16 WF 50 9.744 3258.8 1.795 1815.48 1947.15 (10.0) (3266.9) (1.794) (1821.01) -9.715 3257.7 1.796 1813.86 21 WF 62 9.715 5108.3 1.4o6 3633.21 3928.07 (1o.0) (5122.7) (1.404) (3648.64) 9.715 5108.3 1.406 3633.21 24 WF 84 9.717 7969.9 1.225 6506.04 7025.0 (io.o0) (7992.4) (1.224) (6529.73) — 9.715 7969.7- 1.226 6500.57 -30 wF' 108 9.674 12244 1.012 12098.8 13245.0 (10.o) (12286) (1.011) (12152.8) -9.715- 12251 1.012 12105.7 -8 x 4 9.210 2272.4 5.032 451.59 512.00 (10.0) (2304.0) (5.142) (448.07) 9.715 2297.5 5.149 446.20 NOTE: Bracketed numbers are derived from Equations 15., 16 and 17. Underlined numbers are from Equations 18, 19 and 20 based on Section 21 WF 62 values.

-18,.TABLE III MOMENT-.CURVATURE PARAMETERS COMPARED FOR =10, a 36 ksi>, E.0012 in/in kL 5 x 10 MEJIxlO4.Section R M lO 2 (in-kips) (radians) Oy (kip-in) 4 L.C 1 3 9.577 214.51 6.948 30.87 31.33 (1.) (216.43) (90)(31.35) 9.231) 212.82 6.983 30.48 8WF 20 9.442 669.98 3.333 20.1205.07 (io-o) (678.13) (3.307) (205.06) 9.231 666.81 3.345 199.35.21 WF 62 9.23L- 5037.3 1.319 3819.0 3928.07 (10.0) 3122.7) (1.304) (3928.5) I.9.231 5037.3- -139 3819,0 30 WE 108 9.130 120533.943 12782 13245.0 (10.0) (12286) (.931) (13197) 9.231 12081.941 12838 8x4813 2215. 4.526 489.5 512.00 (10.0) (2304.0) (4.500) (512."0) 921 2266 4.552 49.8 NOTE: Bracketed numbers are derived from -EquatiQns 15, 16 and 17a. Underlined numbers are from.Equations 18, 19 arid 20a based on Section. 21 WE 62 values.

-19 - TABLE IV M01~gNT-CUEVATURF, PARAN~ETERS C0MKPARED FOE~.r =10, 3 6 ksi,. Ey =.0012 in/in ~t=20 Sectioni R MY 10 4 El x 10~ (in-kips) (radians) o 0 i-n 4 LC 13 9.967 217.39 7.422 29.29 31.35 (10.0) (216.43) (7.52) (28.78) 9.937 218910 7.340 29.71 8 WF 20.9.956 682.14 3.644 187.20 205.07 (1o.0) (678.13) (3,71) (182.8 9.937( 683.35 3.618 188.88 21. WF 62 9.937 51-62.2 1.486 3473.9 3928.07 (10.0) (5122,7) (1.522) (3365.8) 996,93 5162.2 1.486 3473.9 30 WF 108 9.926 1,2388 1.076 11,513 13245.0 (1o.o) (12286) (i-ios) (11119) 9.937 1,2380 lo8o 11463 8 x 4 9.771 2321,4 5.570 416.8 512,00 (10.0) (2304.0) (6.oo) (384.0) 9.937 2321,7 5.861 396.1 NOTE: Bracketed numbers are derived from Equations 15, 16 and l7b. Underlined numbers are from Equation. 18, '19 and 20b based on Section 21 WF 62 values.

TABLE V MOMENT~- CURVATURE -PARAM~ETERS COM~PARED FOR r = 5, c = 36 ksil E~ =..0012 in/in ji= 10 Section R MY10 E0l4 2 (in-.kips) (radians) (kip-in) y 4LC 13 4.966 215.45 6.992 3.81315 (5.0) (216.43) (7.196) 30.08 -4.937 214.61 -6.841 -31.37 8 WF 20 4.955 674.05 3.363 ~ 200.43 205.07 (~.O) (678.13) (3.494) (194.08) 4.937 672.41 3.321 202.47 21 WF 62 4.937 5079.6 1.335 3805.0 3928.07 (5.0) (5122.7) (1.4o4) (3648.6) 4.937 5079.135 3805.0 30 WF 108 4.928 12167.956 12727 13245.0 (5.0) (12286) (1.011) (12152) 4.937 1,2182.961 12676 8 x 4 4.828 2249.8.4.664 482.4 512.00 (5.0) (2304.0) (5.142) (448.1) 4,937 2284.6 4.889 467.3 NOTE: Bracketed numbers are derived from Equations 15, 16, and 17, Underlined numbers are from-Equations 18, 19 Section 21 WF 62 values. and 20 based on

My = Zx Cy (15) R =r (16) 0; 2 () p YE y (17) where Zx = the plastic section modulus. Note that when pi is small the characteristic curvature approaches the lower limiting value of OY = Y YE (17 a) and for large values of pi it approaches the upper limiting value of y yp Yp (l7b) b) When the moment-curvature parameters are known for a particular section, (B) z (B) (A) (18) x(A) R(B) - R(A) (19) (B) (Y + YE) (A) (20) Y P + yE)(B) y (20) Superscript (A) denotes the particular section for which the momentcurvature parameters are known, and superscript (B) denotes the section for which the parameters are being found. Similar to (a) above, when [ is small the yield curvature approaches

,22 - 0(B)= YE(A) (A) (20a) and for large values of p it approaches 0 (B) yp(A) (A) (2 yp(B) Moment-curvature parameters obtained from Equations (15) to (17) are shown in Tables II through V for comparison. These Tables also show the results of Equations (18) to (20) based on 21 WF 62 values. ALTERNATE EXPRESSION For small values of M (compared to My ) the moment-curvature relation of a section is elastic. The slope of Equation (14-) at the origin dM M d0 0y therefore must equal the elastic slope El (Young's Modulus times moment of inertia) of the section. This gives the relation 0 1 1 (21) y EI Equation (17) may be replaced by Equation (21) if desired. These are shown compared in Figures 8 and 10. For large values of r and i ~ Equation (17) results are closer to the actual momentcurvature plot.

DISCUSSION OF RESULTS The variation in yp or YE from the average value (yp + yE)/2 is found to be between 4.2 and 8.4% for the sections used. See Table 1o Since the actual center of stress YR is between yp and YE the above variation becomes an upper limit for YR in Equation (12). Moment-curvature parameters are found somewhat dependent on the curve length (or the ductility ratio At) employed in curve fittingO This is found to be the case with experimental data as wello The exponent R approximately equals r of the corresponding stress-strain curve. The difference between them becomes negligible as the ductility ratio increases. Varying the ductility ratio or the exponent r effects the characteristic moment My very little. The latter remains practically unchanged. Reducing the exponent r or the ductility ratio |j has similar effects on the characteristic curvature parameter 0y o CONCLUSIONS A stress-strain relation of the Ramberg-Osgood type across a beam section has been shown to result in a Ramberg-Osgood moment-curvature function for the section. The plastic section modulus of a cross-sectional area appears to be an important parameter for determining the characteristic moment parameter My o -23 -

-24 -Simple expressions are presented for finding the Ramberg-Osgood moment-curvature parameters for wide-flange sections a) from the stress-strain parameters of the material. b) from the moment-curvature parameters of a section. Remarkable agreement exists between these simple expressions and the actual results. It should be noted in closing that the data available in the literature were not sufficient to allow the experimental verification of the results presented in this paper.

REFERENCES 1. Popov, E. P., and Franklin, H. A., "Steel Beam-to-Column Connections Subjected to Cyclically Reversed Loading," Proceedings, Structural Engineers Association of California, October, 1965 2. Berg, G. VO, "A study of the Earthquake Response of Inelastic Systems," AISI Project 119, Proceedings, Structural Engineers Association of California, October, 1965 3. Jennings, P. Co, "Response of Simple Yielding Structures to Earthquake Excitation, " Ph.Do Thesis, California Institute of Technology, June 1963. 4- Hanson, R. D., "Post-Elastic Dynamic Response of Mild Steel Structures," Ph.D. Thesis, California Institute of Technology, June 1965. 5. Morrow, J., "Cyclic Plastic Strain Energy and Fatigue of Metals," ASTM Special Technical Publication No. 378, 1965. ACKNOWLEDGEMENTS This paper was developed in connection with earthquake response studies AISI Project 119, Department of Civil Engineering, University of Michigan, Ann Arbor, Michigan. -25 -

APPENDIX - NOTATIONS The following symbols are used in this paper: b, bY = C, ct q = w = W = X = y = YE ' YP YR = z = am m - y EI = F = M = y MA, MB, F = R = Sx = Zx = width of rectangle half depth of rectangle force characteristic force an exponent flange thickness web thickness displacement characteristic displacement stress distance from the neutral axis stress center elastic, plastic and Ramberg-Osgood stress at the extreme fiber of yield stress Young's modulus times moment of inertia force moment characteristic moment moment an exponent section modulus plastic section modulus -26 -

-27 - ~j= = /E ay= strain characteristic strain ductility ratio stress characteristic stress curvature characteri stic curvature

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