Influence of residual stress on the carrying-capacity of steel framed structures. Numerical investigation

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1 Stability and Ductility of Steel Structures (SDSS'99) D.Dubina & M. Ivany (editors) Elsevier Science Pub. Pages Influence of residual stress on the carrying-capacity of steel framed structures. Numerical investigation G. M. Barsan and C. G. Chiorean Faculty of Civil Engineering, Technical University of Cluj, 34 Cluj-Napoca, Romania ABSTRACT In order to assess explicitly the influence of residual stresses on the inelastic behavior of the steel frames a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors. The states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses are accurately included in the analysis. Three residual stress patterns, constant, linear and parabolic distribution has been considered for three types of calibration frames: a portal frame, a six-story two-bay frame and a two-story frame. The numerical tests and the comparisons made proved the effectiveness of the proposed analysis method. KEYWORDS Residual stress, material imperfection, advanced analysis, steel frames, inelastic analysis, semi-rigid frames, large-deflection analysis. INTRODUCTION An upcoming technology for limit-states design, termed advanced analysis, is defined as any method of analysis that sufficiently represents the strength and stability behavior, such that specification member capacity is not required ( Chen & Toma, 1994). The reason for this is that, since advanced analysis can directly assess the strength and stability of the overall structural system as well as interdependence of member and system strength and stability, separate member capacity checks are not required. A plasticzone analysis that includes the spread of plasticity, residual stresses, initial geometric imperfections, and any other second-order behavioral effects, would certainly be suitable to be classified as advanced inelastic analysis in which the checking of beam-column interaction is not required. An efficient object-oriented Turbo-Pascal computer program NEFCAD for large deflection elastoplastic analysis of semi-rigid steel frames, in the plastic-zone approach has been developed recently, by the authors (Barsan & Chiorean, 1999). In this approach, the gradual plastification of the cross-section of each member are accounted for by smooth moment-rotation curves of Ramberg-Osgood type, experimentally calibrated, and the residual stresses are only implicitly considered. In order to assess explicitly the influence of residual stresses on the behavior of the structure, and in particular, on the carrying-capacity of the frames, and to compare this to results obtained by NEFCAD program, a new

2 computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors. This time the states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses can be accurately included in the analysis. Three residual stress patterns, constant, linear and parabolic distribution has been considered for calibration frames, as it will be shown in the following. In the same purpose three types of frames were selected as well: a portal frame first analyzed by El-Zanaty (198) followed by others researchers, a six-story twobay frame (Vogel, 1985) and a two-story frame that has been proposed by Ziemian (1992). MATHEMATICAL FORMULATION The magnitude and distribution of residual stresses in hot-rolled members depend on the type of cross section and manufacturing processes and different patterns are proposed. In the US, the residual stress is considered constant in the web although, when the depth of a wide flange section is large, it varies more or less parabolically. Another possible residual stress pattern in the web is the one simplified by a linear variation as used in European calibration frames (Vogel, 1985).These three residual stress patterns for the web, i.e., linear, constant, or parabolic distribution, were chosen for calibration frames (Fig. 1.a, b, and c). a. b. c. Figure 1: Residual stress patterns Figure 2: Stress-strain relation Consider the cross-section of a beam-column subjected to the action of the bending moment M e axial force N e and residual stresses σ r, as shown in Figure 3. The strain ε ij in an arbitrary point of the section can be expressed as follows: Y X yi εrij φ yi u X M e e N φ Y

3 Figure 3: Cross-section characteristics ε ij = u + φ. y i + ε r,ij (1) in which u is the strain produced be axial force, φ is the curvature in the section and ε r,ij is the strain produced by residual stresses. The equations of static equilibrium in the section are: N e - N int (u, φ) = M e - M int (u, φ) = (2) where N int (u, φ) is the internal axial force and M int (u, φ) is the internal bending moment in the section. They can be expressed in the following form: N ( u, φ) = σ( u, φ) da int A M ( u, φ) = σ( u, φ) yda int A The curvature φ and the axial strain u are the solutions of Eqs. (2) that may be rewritten as follows: e Nu (, φ) = Nint ( u, φ) N = (4) e Mu (, φ) = Mint ( u, φ) M = The Eqs. 4 are solved numerically using the modified Newton-Raphson method, and results in two recurrence relationships to obtain the unknowns u and φ. The stress-strain relation of the material is assumed as elastic-perfectly plastic, and a tri-linear type of curve such as shown in Figure 2 is used in the analysis. The elastic modulus E = 29 ksi and f y = 32.4 ksi. Two types of residual stress pattern was considered: type 1, European linear distribution (Fig.1.a) and type 2, US constant distribution (Fig. 1.b). Using these data and the recurrence relationships obtained from Eqs.4 the M-N-Φ curves of a W8x31 section were computed for different N/N p ratio considering, both compression and tension, with and without residual stresses, as they are shown in Figure 4.a and b. A detailed layout of the procedure used to consider the development of plastification, second-order effects and non-linear behavior of the connections, is presented by Barsan and Chiorean (1999). These curves compare favorably to those computed by Kanchanalai (1977) presented by Chen and Toma (1994). (3) Bending Moment(kNm) a. Tension Figure 4: M-N- φ re N/Np=.2 stresses, type 2 stresses, type 1 Without residual stresses Bending Moment (knm) b. Compression N/Np=.2 stresses, type 2 stresses, type 1 Without residual stresses Curvature Curvature Figure 4: M-N-Φ relationship of section W8x31

4 NUMERICAL EXAMPLES Based on the mathematical formulation above described an object-oriented Turbo-Pascal program, NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) has been developed to study the influence of residual stresses on the behavior of steel-framed structures. It combines the structural analysis routine with a graphic routine to display the results, as deformed shape of the structure, M, T, N diagrams, percentages of section areas yielded at ultimate stage, in compression or tension, loaddeflection response curves and the development of plastic zones in the cross-sections and along the members of the structure. This program was used to study the influence of residual stresses on the behavior of steel frames. As a standard for comparison three types of frames were selected: a portal frame first analyzed by El-Zanaty (198), a six-story two-bay frame (Vogel, 1985) and a two-story frame that has been proposed by Ziemian (1992). Three residual stress patterns, constant, linear and parabolic distribution in the web has been considered for calibration frames. The differences between the linear and parabolic distribution was irrelevant so only the linear (type 1) and constant (type 2) distribution have been represented. Example 1: Portal frame The El-Zanaty (198) portal frame, member and material properties, and applied loading are shown in Figure 5. The plastic zone developments are represented in the Figure 6. In Figure 7 the lateral-load displacement curves are represented for type 1, type 2 distributions of the residual stress, and without residual stress. Example 2: Six-story frame The Vogel s six-story frame, member and material properties, and the applied loading are shown in Figure 8. In this example the influence of residual stress on the inelastic behavior of the structure and a comparison of computer time necessary to carry out the analysis using NEFCAD and NILDFA programs are presented. Comparable lateral load-displacement curves for the sixth-stories are shown in Figure 9. The NEFCAD program (a=1 and n=35) results are in close agreement with NILDFA program where the effects of residual stresses are explicitly taken. Computing time for NEFCAD program was 1 times shorter than the time taken to complete the same analysis with NILDFA program. Percentages of section-areas yielded (Fig. 1.a) and spread of plastic zones in the characteristic cross-sections (Fig. 1.b and Fig. 11.a,b,c) are represented. Example 3: Two-story frame The Ziemian s two-story frame, member and material properties, and applied loading are shown in Figure 12. The percentages of section-areas yielded are represented in Figure 13 for all three cases considered and the lateral load-displacement curves, for the same three cases, are shown in Figure 14. CONCLUSIONS In order to assess explicitly the influence of residual stresses on the inelastic behavior of the structures, and in particular, on the carrying-capacity of the steel frames, and to compare this to results obtained

by NEFCAD program, a new computer program NILDFA (Non-linear Inelastic Large

2 stress, type 2, Plim=1.3 stress, type 1, Plim=1.3 Without residual stress, Plim=1.6.

Portal frame description Figure 7: Lateral load-displacement curves for portal frame

Percentages of section areas yielded at ultimate for the portal frame Type 1

5 by NEFCAD program, a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors Applied load ratio, p.2 stress, type 2, Plim=1.3 stress, type 1, Plim=1.3 Without residual stress, Plim= D(m) Figure 5: Example 1. Portal frame description Figure 7: Lateral load-displacement curves for portal frame 45.6 (38.3) [.] 52. (45.2) [41.1] 51.2 (48.8) [45.6] 62.6 (57.6) [57.5] a. Percentages of section areas yielded at ultimate for the portal frame Type 1 distribution. (Type 2 distribution.) [Without residual stress] b. Type 1 c. Type 2 d. Without residual stress Figure 6: Spread of plastic zones for portal frame

1.2 1 Applied load ratio, p.2 stresses, type 1, Plim=1.18 Without residual stresses, Plim=1.2 Ramberg-Osgood, n=35, Plim=1.18 Ramberg-Osgood, n=3, Plim=1.14.5.1.15.2.25 D6(m) Figure 8: Example 2.

6 1.2 1 Applied load ratio, p.2 stresses, type 1, Plim=1.18 Without residual stresses, Plim=1.2 Ramberg-Osgood, n=35, Plim=1.18 Ramberg-Osgood, n=3, Plim= D6(m) Figure 8: Example 2. Six-story frame description Figure 9: Lateral load-displacement curves for sixstory frame 89.6 (9.2) [9.2] 96.6 (96.6) [96.6] 55. (53.6) [57.2] 75.7 (74.8) [77.] 42.7 (39.8) [4.6] 86.2 (86.6) [86.6] Value indicates total % of section area yielded, Type 1 distribution (Type 2 distribution) [Without residual stress] 63.1 (61.4) [68.9] 61. (59.3) [66.9] Compression Tension 14.4 (9.8) [.] 17. (1) [.] Compression Tension 95.8 (95.8) [95.6] 95.9 (95.9) [95.9] Cross section 1-1 Cross section (88.6) [88.6] 24.1 (15.1) [.] Compression 2 (23.3) [.] 58.8 (58.2) [58.9] 58.1 (57.2) [57.5] Figure 1.a: Percentages of section areas yielded Cross section 3-3 Figure 1.b: Spread of plastic-zones

α β γ a. Type 1 distribution b. Type 2 distribution c. Without residual stress Figure 11: The spread of plastic zones of the AB beam α-top flange; β-lateral view of the beam; γ-bottom flange.

7 α β γ a. Type 1 distribution b. Type 2 distribution c. Without residual stress Figure 11: The spread of plastic zones of the AB beam α-top flange; β-lateral view of the beam; γ-bottom flange. Figure 12: Example 3. Two-story frame description 95.3 (95.7) [95.3] 83.3 (83.3) [83.3] 84.9 (82.) [78.5] 55.5 (56.3) [43.2] 97.1 (97.1) [97.] 96.5 (95.1) [96.2] 58.7 (55.8) [6.2] 96.9 (97.) [96.8] 16.6 (21.5) [.] 58.8 (57.3) [57.8] Type 1 distribution (Type 2 distribution) [Without residual stress] 88.1 (87.9) [87.9] (36.8) [11.5] Figure 13: Percentages of section areas yielded at ultimate for the two-story frame

8 stresses, type 2, Plim=1.1 stresses, type 1, Plim=1.1 Without residual stresses, Plim= Applied load ratio, p stresses, type 2, Plim=1.1 stresses, type 1, Plim=1.1 Without resisual stresses, Plim= Applied load ratio, p D1(m) Figure 14: Lateral load-displacement curves for two-story frame The states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses are accurately included in the analysis. Three residual stress patterns, constant, linear and parabolic distribution has been considered for calibration frames. Three frame types were selected: a portal frame, a six-story two-bay frame and a two-story frame. From the numerical tests it may be concluded that the influence of the residual stress on the carryingcapacity of the steel frames analysed vary within 1-3 % but the influence on inelastic behavior during the loading process is more important and must be considered in the analysis. Also, the NEFCAD program (a=1 and n=35) results are in close agreement with NILDFA program where the effects of residual stresses are explicitly taken. That means that NEFCAD program correctly calibrated can implicitly capture accurately the influence of residual stress. Computing time for NEFCAD program was 1 times shorter than the time taken to complete the same analysis with NILDFA program. REFERENCES Barsan G.M. and Chiorean C.G. (1999). Computer program for large deflection elasto-plastic analysis of semi-rigid steel frameworks. Computers & Structures, 72 (1999) Chen W.F. and Toma S. (1994). Advanced Analysis of Steel Frames, CRC Press, U.S. El-Zanaty M.H., Murray D.W. and Bjorhodve R. (198). Inelastic Behavior of Multystory Steel Frames, Structural Engineering Report, No. 83, The University of Alberta, Edmonton, Canada, April, 248 pp. Kanchanalai T. (1977). The Design and Behavior of Beam-Column in Unbraced Steel Frames. AISI Project No. 189, Report No. 2, Civil Engineering/Structures Research Laboratory, University of Texas, Austin, 3pp. Vogel U. (1985). Calibration frames. Stahlbau, 1, 1-7. Ziemian R.D. (1992). A Verification Study for Methods of Second-Order Inelastic Analysis, Proceedings, Annul Technical Session, Structural Stability Research Council, Pittsburgh, PA, D2(m)

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