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Effects of panel zone deformations on seismic response
J. Construct. Steel Research 8 (1987) 233-250
Effects of Panel Zone Deformations on Seismic Response
H . K r a w i n k l e r and S. M o h a s s e b Department of Civil Engineering,StanfordUniversity,Stanford,California94022, USA
SYNOPSIS This paper shows that joint panel zone shear deformations may have a significant effect on the structure strength and stiffness and on the distribution o f inelastic deformations in a frame structure subjected to earthquake excitations. If panel zones are strong in shear, frame behavior is as predicted from conventional rigid frame analysis with inelastic deformations concentrated in plastic hinge regions in beams or columns, if panel zones are weak in shear, the joints are weak links in which inelastic deformations are concentrated and the beams and columns will not develop their bending strength under lateral loading. As a consequence the structure strength and stiffness may be reduced considerably. Since designs with weak joints are becoming the rule rather than the exception, due consideration should be given to the effects of joint shear deformations on frame behavior.
1 INTRODUCTION In seismic design of steel frame structures the question of adequate shear strength and stiffness of joint panel zones has been a much discussed issue over the last two decades. Seismic codes have fluctuated between very severe and very relaxed requirements for shear strength. During the last ten years it was customary practice in California to provide panel zones with sufficient shear strength to permit the development of the plastic moment capacity in both beams framing into a joint. This practice was based on a recommendation made in the commentary to the SEAOC R e c o m m e n d e d Lateral Force Requirements 1 and has led to joint panel zones that often had to be reinforced with very thick doubler plates. The need for these thick 233 J. Construct. Steel Research 0143-974X/87/$03-50© ElsevierAppliedSciencePublishersLtd, England, 1987. Printed in Great Britain
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doubler plates, which are a considerable cost item, is in question because of the proven ductility of panel zones in shear. 2-5Also, the restrained welding n e e d e d to connect thick plates to the joints will cause large residual stresses and may lead to column flange distortions. Thus, there appear to be good reasons to re-evaluate the seismic shear strength requirements for joint panel zones. In fact, the 1985 draft of the S E A O C R e c o m m e n d e d Lateral Force Requirements has considerably reduced these strength requirements as will be discussed later. It must be recognized, however, that joints, if they are weaker than the elements framing into them, will have important effects on the seismic response of steel frame structures. These effects include a decrease in structure strength and stiffness in the elastic and inelastic range, changes in the dynamic characteristics, and changes in the inelastic deformation demands in the elements of the structure. It is the purpose of this paper to discuss simple mathematical models that permit an evaluation of these effects and to present a few numerical results obtained from a pilot study on two frame structures.
2 JOINT P A N E L Z O N E B E H A V I O R The shear behavior of joint panel zones has been investigated in a number of experimental studies, e.g. References 2 to 7. In general, the conclusions drawn from these studies are similar and can be summarized as follows. (i) Joints respond in a ductile mode of deformation when subjected to an unbalance of beam moments that causes high shear forces across the panel zone. Under load reversals, stable hysteresis loops and considerable cyclic work-hardening are observed (see Fig. 1). (ii) Joints with thin panel zones may buckle in shear but the post-buckling tension field causes stable hysteresis loops unless the width-to-thickness ratio of the panel zone is very large. (iii) Resistance to shear forces in a beam-column joint is provided by the joint panel zone but also by the elements surrounding the panel zone, particularly the column flanges. (iv) The shear force-shear distortion behavior of a joint is characterized by an elastic range, a range of gradually decreasing stiffness, and a range of almost constant strain-hardening stiffness. The thicker the column flange, the slower is the decrease in post-elastic stiffness and the higher is the shear strength of the joint. In addition it was observed, in the studies summarized in References 3 and 5, that severe kinking occurs at the four corners of joints with thin
235
Effects of pand zone deformations on seismic response 21
-0.1
Fig. 1. Hysteretic behavior of a beam--column joint.
c o l u m n flanges when these joints are subjected to very large inelastic shear distortions. This kinking leads to high curvatures at the welded beam flange connections and in turn may cause crack propagation and fracture at the b e a m flange welds. Based on these observations it can be concluded that, because of the very ductile shear response of joints, it is desirable to have joints participate in energy dissipation if a steel frame structure is subjected to a severe earthquake. This can be achieved by designing the joints such that they will yield before the strength of the elements framing into them is developed. However, the observations from past studies also indicate that it is not advisable to m a k e joints too weak in shear. If the joints are too weak, the strength of a frame may be reduced more than desirable and most or all of the inelastic d e f o r m a t i o n s have to take place in the joints, which may cause weld fracture p r o b l e m s as observed in the studies mentioned above.
2.1 Shear strength and stiffness of joints Several models have been proposed in the literature for the evaluation of the shear force versus shear deformation behavior of joints, ranging from
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simple strength estimates to complex models that account for the effects of axial load and bending moment on the joint shear strength and stiffness. 8The simplest strength prediction is that contained in Part 2 of the AISC Specifications, 9 i.e. Vy = 0.55Fydct
(1)
in which Fy is the yield strength of the material, de is the depth of the column, and t is the thickness of the panel zone. Based on experimental evidence, this prediction usually coincides well with the onset of gross yielding in the panel zone and may be referred to as panel yield strength, Vy. Assuming uniform shear stresses in the panel zone and neglecting the effects of normal stresses on joint yielding, the angle of shear distortion at general yielding can be expressed as Ty = F y / ( X / 3 G )
according to von Mises' yield criterion. With these simplifications, the elastic shear stiffness of the joint is given as Ke = V/3, = 0.95dctG
(2)
Figure 2 shows, in solid lines, several experimentally obtained shear force-distortion diagrams1° normalized with respect to Vy and yy. The figure indicates that the strength prediction given by eqn (1) does not capture the reserve strength beyond panel zone yielding, which is caused primarily by the column flanges bounding the panel zone. Based on a spring model presented in Reference 11, the post-yield shear stiffness of a joint can be expressed by the equation Kp = AV/AT = l'095bct2cfG/db
(3)
in which b c and tcf are the width and thickness of the column flange, respectively, and db is the depth of the beam framing into the column. In Reference 11 it is assumed that this post-yield stiffness is valid for a range of A~ = 3Ty and that for larger distortions a constant strain-hardening stiffness will describe the joint behavior. With these assumptions, the shear strength of a joint, at a total distortion of 4"yy, can be computed as Vp = 0.55Fydct(1 + 3.45bct2cf/dbdct)
(4)
The second term in parenthesis represents the increase in strength beyond
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Vy. This increase is small for joints in columns with thin flanges but may be substantial if the column flanges are thick. The shear force in the joint can be related approximately to the applied beam moments by the equation V = ( A M / O ' 9 5 d b - Vcol)
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3 ANALYTICAL STUDY In order to assess the effects of joint strength and deformations on the seismic response of frame structures, a pilot study was performed on two different frame configurations. The joints in these frame structures were designed according to different design philosophies so that a wide spectrum of joint effects could be investigated. A series of non-linear dynamic
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analyses was performed on the frame structures utilizing a recorded seismic g r o u n d motion as input. From the results of these analyses preliminary conclusions are drawn on the effects of joint strength and stiffness on the structure strength and stiffness and on the inelastic deformation demands in the frame elements.
3.1 Frame designs A seven-story and a ten-story frame structure were used in this study. Both structures were designed in accordance with the 1982 Uniform Building Code, considering all requirements for ductile m o m e n t frames located in Seismic Z o n e 4. The frames are made of A36 structural steel. The strong c o l u m n - w e a k girder concept was followed except in the upper stories where the axial load in the columns was less than 0.2 Py. Many of the elements were governed by drift limitations resulting in elements that were larger than required by strength considerations. The design of the two structures is discussed in detail in Reference 12. Seven-story frame structure: This structure has two exterior four-bay moment-resisting frames in the E - W direction and is braced in the N-S direction as shown in Fig. 3. The structure is designed for a typical floor live load of 50 psf and a dead load of 85 psf. All columns in the two momentresisting frames are subjected to strong axis bending only. The elements used for beams and columns are summarized in Table 1. Ten-story frame structure: This structure consists of five two-bay momentresisting frames in the N-S direction and has two braced frames in the E - W direction as shown in Fig. 4. The structure is designed for a floor live load of 80 psf and a dead load of 95 psf. Again, all columns in the moment-resisting frames are subjected to strong axis bending only. Only a typical interior frame is considered in this study. The elements used for beams and columns are shown in Table 2. The main difference between the two structures is that the two E - W perimeter frames in the seven-story building must resist the full seismic loading whereas in the ten-story building all moment-resisting frames resist lateral loads. Thus, in the seven-story frames the ratio of seismic moments to gravity load moments is much larger than in the ten-story frames.
3.2 Joint designs T h e joints were designed according to four different design criteria, leading to panel zones of different thicknesses. The following criteria were employed.
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(1) Allowable stress design. In this design the panel zone thicknesses are determined from the criterion that the joint shear force due to gravity plus seismic moments cannot exceed the value 0.53 Fydct. The value 0.53 Fy corresponds to the allowable stress value in shear for seismic loading (1.33 × 0.4Fy). (2) 1985 SEAOC design. In this design the criterion given in the 1985 draft of the SEAOC Recommended Lateral Force Requirements is employed. This draft states that the shear force due to gravity load moments plus 1.7 times the seismic design moments shall not exceed the shear strength of the joint. The joint shear strength is determined from eqn (4) with the simplification that the constant 3.45 in this equation is replaced by 3.0. (3) Gravity plus 3E design. Here the criterion employed is that the shear force due to gravity load moments plus three times the seismic design moments shall not exceed the joint shear strength given by eqn (4).
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This criterion was under discussion in older drafts of the S E A O C Recommended Lateral Force Requirements and would have resulted in stronger joints than the criterion in the 1985 draft. (4) 1980 SEAOC design. This design, which was customary practice in California over the last ten years, is based on the criterion that the joint shear 'strength' determined from eqn (1) shall be sufficient to permit the development of the plastic m o m e n t capacity in the beams framing into the joint. Thus, the sum of the plastic m o m e n t capacities of the beams is to be used for AM in eqn (5). The panel zone thicknesses resulting from the four designs are summarized in Tables 1 and 2 for the joints in the exterior and interior columns of the two frame structures. As can be seen from these tables, the thicknesses vary considerably between the four designs with the smallest thickness required in the Allowable stress design and very large thicknesses in the 1980 SEAOC design. In general, the differences between the
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Allowable stress design and the 1985 SEAOC design are small whereas the differences between the 1980 S E A O C design and the 1985 S E A O C design are very large. This indicates a considerable change in design philosophy between the present and proposed SEAOC provisions. In the analyses performed on the two structures, the panel zone thicknesses listed in Tables 1 and 2 were used unless the required thickness was smaller than the column web thickness, in which case the latter was used. This happened primarily in the Allowable stress design and the 1985 S E A O C design. Because in these two designs very often the unreinforced web thickness governed and the joint designs were similar, the seismic responses of these two designs were similar as well. Therefore, in the subsequent discussion, only results for the Allowable stress design will be presented. The conclusions drawn on this design can be applied also to the 1985 S E A O C design.
3.3 Static and dynamic analyses The two frame structures with the different joint designs were analyzed statically and dynamically using the computer program STANON.13 This non-linear analysis program considers inelastic deformations in joints, beams, and beam-column connections. Incremental static loading and unidirectional seismic ground motions can be used as load input. The program performs only two-dimensional analysis, thus, it is assumed that all frame units are coupled by rigid diaphragms and that torsional vibrations are negligible. Beam-column joints are modeled in this program as a pair of scissor-type elements connected by a hinge and a rotational spring that allows relative rotation between the elements. This rotation represents the shear distortion of the joint (see Fig. 5). The lengths of the scissors are equal to the beam and column depths. The properties of the rotational spring are defined by the joint shear force--distortion characteristics. In this study the joints were ¥
"---.--2/ Fig. 5. Computermodelfor joint.
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modeled as bi-linear elements with the strength and stiffness parameters Vy, Ke, and Kp. In the beams, inelastic deformations are considered to be concentrated in plastic hinges at the beam ends. These plastic hinges are modeled by rotational springs between the elastic beams and the scissors representing the joints. The springs are of infinite stiffness until a plastic hinge forms and thereafter represent the rotational strain-hardening stiffness at the plastic hinge. Based on a parameter study of wide-flange sections it was found that a spring that is activated at 1.05Mp and has a rotational stiffness of 0.1EI/L represents the plastic hinging in mild steel beams subjected to lateral loading rather accurately. Two kinds of analyses were performed on the two structures; static analysis to evaluate the effects of joint strength and stiffness on the structure strength and stiffness, and dynamic analysis to evaluate the effects on member deformation demands. In the static analysis the gravity loads are applied first and then the seismic design loads, E, which are distributed triangularly over the height of the structure, are applied and incremented until a mechanism is formed in the structure. Equilibrium in the non-linear range is maintained through a Newton-Raphson iteration procedure. In the dynamic analysis, an acceleration-time history is used as input. The dynamic response is computed using step-by-step integration assuming constant input acceleration during each time step. Masses are lumped at the floor levels and the mass moment of inertia is neglected. Stiffnessdependent damping of 5% of critical damping is assumed in the first mode. The N21E component of the Taft Lincoln School Record of the 1952 Kern County earthquake was used as input ground motion. Even though the record was scaled to a peak ground acceleration of 0.6g, the inelastic deformations in the two structures were relatively small. In part this can be attributed to the inherent reserve strength of ductile moment resisting frames in which gravity load moments add to the member strength requirement and drift requirements govern often over strength requirements. In part this must be attributed also to period problems, since the fundamental periods of the bare frame structures used in the dynamic analysis were about twice the periods used in design (T = 0.1 N). Thus, the relatively small inelastic deformation demands obtained in the analyses may not be fully representative for real structures. However, the conclusions drawn on the relative deformation demands should not be greatly affected by this consideration. 3.4 Effects of joint deformations on structure strength and stiffness
Figure 6 shows the static lateral load versus roof displacement diagrams for the two structures. The differences in the diagrams in each figure are caused
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diagrams is that, if the need exists to predict accurately the strength and stiffness of steel frames, the effects of joint shear strength and stiffness should be included in the mathematical model. This holds true particularly for frames with weak joints which appear to become common practice even in regions of high seismicity. In these cases, strength and stiffness predictions based on centerline dimensions may lead to poor results.
3.5 Effects of joint strength and stifihem on dynamic response The time history results of the non-linear dynamic analyses are discussed in detail in Reference 12. There are considerable differences in the story shear histories between the four designs because of the variations in joint strengths. However, the overall dynamic characteristics represented by maximum interstory and roof displacements were similar for all designs. In fact, the maximum roof displacements for the two Allowable Stress Designs were slightly smaller than those for the other designs. The preliminary conclusion to be drawn from this pilot study is that the global dynamic response of frame structures is not affected greatly by variations in joint strengths and stiffnesses. However, the contrary is observed in the localized inelastic deformation responses of the individual members. Here the joint shear strength has a considerable effect on the relative deformation demands in beams and joints. This is evident from Figs 7 and 8 which show ductility demands in beams and joints. The figures display plots of the maximum ductility ratios in each story and for each design. The ductility ratio for plastic hinge rotation at the beam end is defined as /~0 = 1 +10p]/0y, where 0p is the plastic hinge rotation and Or is defined as MyL/(6EI). The ductility ratio for joint distortion is given a s / ~ = I~/I/Yy, with yy = Fy/(V'3G) = 0.0019 for A36 structural steel. In the Allowable Stress Designs no plastic hinge rotations occur in any of the beams and, because of the weak panel zones, all inelastic deformations are concentrated in the joints. In both the seven- and ten-story frames the maximum joint distortion ductility ratios are less than 8-3 which are values that should be tolerable in well designed joints. However, it must be pointed out again that the two structures did not undergo very large inelastic deformations in the postulated earthquake. This is evident from the small plastic hinge rotation ductilities in the frames analyzed with centerline dimensions and rigid joints. In the Gravity plus 3E Designs the deformation demands in beams and joints appear to be well balanced, with both the plastic hinge rotations and joint distortions being relatively small. In the 1980 SEAOC Design the
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demands on plastic hinge rotations are largest. With the mathematical models used in the frame analysis, the inelastic joint distortions are not negligible in this design because the bending strength of the beams is taken as 1.05Mp while the joints are designed to resist 1.0Mp. The results presented for Centerline Dimensions are a measure of the error that is encountered when conventional frame analysis with rigid joints is performed. This analysis gives reasonable results for plastic hinge rotations in frames with strong joints (1980 SEAOC) but gives completely misleading results for frames with weak joints (Allowable Stress Design) where the inelastic deformations are concentrated in the joints.
4 CONCLUSIONS Because well designed joints are ductile elements when subjected to an unbalance of beam moments that causes inelastic joint distortions, it may be advisable to design the joint panel zones so that they yield in shear and participate in energy dissipation in severe earthquakes. This is done inadvertently in allowable stress design and intentionally with a provision given in the 1985 SEAOC draft. The best design would be one in which the inelastic deformation demands in beams and joints are well balanced. This appears not to be the case in the two aforementioned design approaches which often lead to very weak joints that have to undergo large inelastic distortions whereas the beams remain elastic. When the joints are weaker than the connected elements, they will greatly affect the structure strength and stiffness and the distribution of inelastic deformations in structural elements. Thus, if attempts are made to predict the inelastic response of frame structures with weak joints, mathematical models should be used that consider the strength and stiffness characteristics of the beam-column joints.
REFERENCES 1. Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, San Francisco, SEAOC, 1980. 2. Fielding, D. J. and Huang, J. S., Shear in steel beam-to-column connections, Welding Journal (July 1971). 3. Krawinkler, H., Bertero, V. V. and Popov, E. P., Shear behavior of steel frame joints, J. Struct. Div., ASCE, 101, No. STll (November 1975). 4. Slutter, R. G., Tests of Panel Zone Behavior in Beam--column Connections, Fritz Engineering Laboratory Report No. 200.81.403.1, Lehigh University, Bethlehem, USA, 1981.
250
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5. Popov, E. P., et al., Cyclic behavior of large beam-column assemblies, Earthquake Spectra, EERI, 1, No. 2 (February 1985). 6. Becker, R., Panel zone effect on the strength and stiffness of steel rigid frames, Engineering Journal, AISC, 12, No. 1 (1975). 7. Naka, T., et al., 'Research on the behavior of steel beam-to-column connections in the seismic-resistant structure', Proceedings of Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1969. 8. Hjelmstad, K. D. and Popov, E. P., Seismic behavior of active link beams in eccentrically braced frames, EERC Report No. 83/15, University of California, Berkeley, USA, July 1983. 9. American Institute of Steel Construction, Manual of Steel Construction, 8th edn, New York, AISC, 1980. 10. Krawinkler, H., Bertero, V. V. and Popov, E. P., Inelastic Behavior of Steel Beam-to-Column Subassemblages, EERC Report No. 71-7, University of California, Berkeley, USA, October 1971. 11. Krawinkler, H., Shear in beam--column joints in seismic design of steel frames, Engineering Journal, AISC, 15, No. 3 (1978). 12. Mohasseb, S., Effects of beam-column joints on the seismic behavior of steel frames, Engineer Thesis, Department of Civil Engineering, Stanford University, USA, December 1982. 13. Lee, D. G., Accurate and simplified models for seismic response prediction of steel frame structures, Ph.D. Dissertation, Department of Civil Engineering, Stanford University, USA, February 1984.
Effects of Panel Zone Deformations on Seismic Response
H . K r a w i n k l e r and S. M o h a s s e b Department of Civil Engineering,StanfordUniversity,Stanford,California94022, USA
SYNOPSIS This paper shows that joint panel zone shear deformations may have a significant effect on the structure strength and stiffness and on the distribution o f inelastic deformations in a frame structure subjected to earthquake excitations. If panel zones are strong in shear, frame behavior is as predicted from conventional rigid frame analysis with inelastic deformations concentrated in plastic hinge regions in beams or columns, if panel zones are weak in shear, the joints are weak links in which inelastic deformations are concentrated and the beams and columns will not develop their bending strength under lateral loading. As a consequence the structure strength and stiffness may be reduced considerably. Since designs with weak joints are becoming the rule rather than the exception, due consideration should be given to the effects of joint shear deformations on frame behavior.
1 INTRODUCTION In seismic design of steel frame structures the question of adequate shear strength and stiffness of joint panel zones has been a much discussed issue over the last two decades. Seismic codes have fluctuated between very severe and very relaxed requirements for shear strength. During the last ten years it was customary practice in California to provide panel zones with sufficient shear strength to permit the development of the plastic moment capacity in both beams framing into a joint. This practice was based on a recommendation made in the commentary to the SEAOC R e c o m m e n d e d Lateral Force Requirements 1 and has led to joint panel zones that often had to be reinforced with very thick doubler plates. The need for these thick 233 J. Construct. Steel Research 0143-974X/87/$03-50© ElsevierAppliedSciencePublishersLtd, England, 1987. Printed in Great Britain
234
H. Krawinkler, S. Mohasseb
doubler plates, which are a considerable cost item, is in question because of the proven ductility of panel zones in shear. 2-5Also, the restrained welding n e e d e d to connect thick plates to the joints will cause large residual stresses and may lead to column flange distortions. Thus, there appear to be good reasons to re-evaluate the seismic shear strength requirements for joint panel zones. In fact, the 1985 draft of the S E A O C R e c o m m e n d e d Lateral Force Requirements has considerably reduced these strength requirements as will be discussed later. It must be recognized, however, that joints, if they are weaker than the elements framing into them, will have important effects on the seismic response of steel frame structures. These effects include a decrease in structure strength and stiffness in the elastic and inelastic range, changes in the dynamic characteristics, and changes in the inelastic deformation demands in the elements of the structure. It is the purpose of this paper to discuss simple mathematical models that permit an evaluation of these effects and to present a few numerical results obtained from a pilot study on two frame structures.
2 JOINT P A N E L Z O N E B E H A V I O R The shear behavior of joint panel zones has been investigated in a number of experimental studies, e.g. References 2 to 7. In general, the conclusions drawn from these studies are similar and can be summarized as follows. (i) Joints respond in a ductile mode of deformation when subjected to an unbalance of beam moments that causes high shear forces across the panel zone. Under load reversals, stable hysteresis loops and considerable cyclic work-hardening are observed (see Fig. 1). (ii) Joints with thin panel zones may buckle in shear but the post-buckling tension field causes stable hysteresis loops unless the width-to-thickness ratio of the panel zone is very large. (iii) Resistance to shear forces in a beam-column joint is provided by the joint panel zone but also by the elements surrounding the panel zone, particularly the column flanges. (iv) The shear force-shear distortion behavior of a joint is characterized by an elastic range, a range of gradually decreasing stiffness, and a range of almost constant strain-hardening stiffness. The thicker the column flange, the slower is the decrease in post-elastic stiffness and the higher is the shear strength of the joint. In addition it was observed, in the studies summarized in References 3 and 5, that severe kinking occurs at the four corners of joints with thin
235
Effects of pand zone deformations on seismic response 21
-0.1
Fig. 1. Hysteretic behavior of a beam--column joint.
c o l u m n flanges when these joints are subjected to very large inelastic shear distortions. This kinking leads to high curvatures at the welded beam flange connections and in turn may cause crack propagation and fracture at the b e a m flange welds. Based on these observations it can be concluded that, because of the very ductile shear response of joints, it is desirable to have joints participate in energy dissipation if a steel frame structure is subjected to a severe earthquake. This can be achieved by designing the joints such that they will yield before the strength of the elements framing into them is developed. However, the observations from past studies also indicate that it is not advisable to m a k e joints too weak in shear. If the joints are too weak, the strength of a frame may be reduced more than desirable and most or all of the inelastic d e f o r m a t i o n s have to take place in the joints, which may cause weld fracture p r o b l e m s as observed in the studies mentioned above.
2.1 Shear strength and stiffness of joints Several models have been proposed in the literature for the evaluation of the shear force versus shear deformation behavior of joints, ranging from
236
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simple strength estimates to complex models that account for the effects of axial load and bending moment on the joint shear strength and stiffness. 8The simplest strength prediction is that contained in Part 2 of the AISC Specifications, 9 i.e. Vy = 0.55Fydct
(1)
in which Fy is the yield strength of the material, de is the depth of the column, and t is the thickness of the panel zone. Based on experimental evidence, this prediction usually coincides well with the onset of gross yielding in the panel zone and may be referred to as panel yield strength, Vy. Assuming uniform shear stresses in the panel zone and neglecting the effects of normal stresses on joint yielding, the angle of shear distortion at general yielding can be expressed as Ty = F y / ( X / 3 G )
according to von Mises' yield criterion. With these simplifications, the elastic shear stiffness of the joint is given as Ke = V/3, = 0.95dctG
(2)
Figure 2 shows, in solid lines, several experimentally obtained shear force-distortion diagrams1° normalized with respect to Vy and yy. The figure indicates that the strength prediction given by eqn (1) does not capture the reserve strength beyond panel zone yielding, which is caused primarily by the column flanges bounding the panel zone. Based on a spring model presented in Reference 11, the post-yield shear stiffness of a joint can be expressed by the equation Kp = AV/AT = l'095bct2cfG/db
(3)
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(4)
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Vy. This increase is small for joints in columns with thin flanges but may be substantial if the column flanges are thick. The shear force in the joint can be related approximately to the applied beam moments by the equation V = ( A M / O ' 9 5 d b - Vcol)
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3 ANALYTICAL STUDY In order to assess the effects of joint strength and deformations on the seismic response of frame structures, a pilot study was performed on two different frame configurations. The joints in these frame structures were designed according to different design philosophies so that a wide spectrum of joint effects could be investigated. A series of non-linear dynamic
238
H. Krawinkler, S. Mohasseb
analyses was performed on the frame structures utilizing a recorded seismic g r o u n d motion as input. From the results of these analyses preliminary conclusions are drawn on the effects of joint strength and stiffness on the structure strength and stiffness and on the inelastic deformation demands in the frame elements.
3.1 Frame designs A seven-story and a ten-story frame structure were used in this study. Both structures were designed in accordance with the 1982 Uniform Building Code, considering all requirements for ductile m o m e n t frames located in Seismic Z o n e 4. The frames are made of A36 structural steel. The strong c o l u m n - w e a k girder concept was followed except in the upper stories where the axial load in the columns was less than 0.2 Py. Many of the elements were governed by drift limitations resulting in elements that were larger than required by strength considerations. The design of the two structures is discussed in detail in Reference 12. Seven-story frame structure: This structure has two exterior four-bay moment-resisting frames in the E - W direction and is braced in the N-S direction as shown in Fig. 3. The structure is designed for a typical floor live load of 50 psf and a dead load of 85 psf. All columns in the two momentresisting frames are subjected to strong axis bending only. The elements used for beams and columns are summarized in Table 1. Ten-story frame structure: This structure consists of five two-bay momentresisting frames in the N-S direction and has two braced frames in the E - W direction as shown in Fig. 4. The structure is designed for a floor live load of 80 psf and a dead load of 95 psf. Again, all columns in the moment-resisting frames are subjected to strong axis bending only. Only a typical interior frame is considered in this study. The elements used for beams and columns are shown in Table 2. The main difference between the two structures is that the two E - W perimeter frames in the seven-story building must resist the full seismic loading whereas in the ten-story building all moment-resisting frames resist lateral loads. Thus, in the seven-story frames the ratio of seismic moments to gravity load moments is much larger than in the ten-story frames.
3.2 Joint designs T h e joints were designed according to four different design criteria, leading to panel zones of different thicknesses. The following criteria were employed.
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(1) Allowable stress design. In this design the panel zone thicknesses are determined from the criterion that the joint shear force due to gravity plus seismic moments cannot exceed the value 0.53 Fydct. The value 0.53 Fy corresponds to the allowable stress value in shear for seismic loading (1.33 × 0.4Fy). (2) 1985 SEAOC design. In this design the criterion given in the 1985 draft of the SEAOC Recommended Lateral Force Requirements is employed. This draft states that the shear force due to gravity load moments plus 1.7 times the seismic design moments shall not exceed the shear strength of the joint. The joint shear strength is determined from eqn (4) with the simplification that the constant 3.45 in this equation is replaced by 3.0. (3) Gravity plus 3E design. Here the criterion employed is that the shear force due to gravity load moments plus three times the seismic design moments shall not exceed the joint shear strength given by eqn (4).
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This criterion was under discussion in older drafts of the S E A O C Recommended Lateral Force Requirements and would have resulted in stronger joints than the criterion in the 1985 draft. (4) 1980 SEAOC design. This design, which was customary practice in California over the last ten years, is based on the criterion that the joint shear 'strength' determined from eqn (1) shall be sufficient to permit the development of the plastic m o m e n t capacity in the beams framing into the joint. Thus, the sum of the plastic m o m e n t capacities of the beams is to be used for AM in eqn (5). The panel zone thicknesses resulting from the four designs are summarized in Tables 1 and 2 for the joints in the exterior and interior columns of the two frame structures. As can be seen from these tables, the thicknesses vary considerably between the four designs with the smallest thickness required in the Allowable stress design and very large thicknesses in the 1980 SEAOC design. In general, the differences between the
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Allowable stress design and the 1985 SEAOC design are small whereas the differences between the 1980 S E A O C design and the 1985 S E A O C design are very large. This indicates a considerable change in design philosophy between the present and proposed SEAOC provisions. In the analyses performed on the two structures, the panel zone thicknesses listed in Tables 1 and 2 were used unless the required thickness was smaller than the column web thickness, in which case the latter was used. This happened primarily in the Allowable stress design and the 1985 S E A O C design. Because in these two designs very often the unreinforced web thickness governed and the joint designs were similar, the seismic responses of these two designs were similar as well. Therefore, in the subsequent discussion, only results for the Allowable stress design will be presented. The conclusions drawn on this design can be applied also to the 1985 S E A O C design.
3.3 Static and dynamic analyses The two frame structures with the different joint designs were analyzed statically and dynamically using the computer program STANON.13 This non-linear analysis program considers inelastic deformations in joints, beams, and beam-column connections. Incremental static loading and unidirectional seismic ground motions can be used as load input. The program performs only two-dimensional analysis, thus, it is assumed that all frame units are coupled by rigid diaphragms and that torsional vibrations are negligible. Beam-column joints are modeled in this program as a pair of scissor-type elements connected by a hinge and a rotational spring that allows relative rotation between the elements. This rotation represents the shear distortion of the joint (see Fig. 5). The lengths of the scissors are equal to the beam and column depths. The properties of the rotational spring are defined by the joint shear force--distortion characteristics. In this study the joints were ¥
"---.--2/ Fig. 5. Computermodelfor joint.
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modeled as bi-linear elements with the strength and stiffness parameters Vy, Ke, and Kp. In the beams, inelastic deformations are considered to be concentrated in plastic hinges at the beam ends. These plastic hinges are modeled by rotational springs between the elastic beams and the scissors representing the joints. The springs are of infinite stiffness until a plastic hinge forms and thereafter represent the rotational strain-hardening stiffness at the plastic hinge. Based on a parameter study of wide-flange sections it was found that a spring that is activated at 1.05Mp and has a rotational stiffness of 0.1EI/L represents the plastic hinging in mild steel beams subjected to lateral loading rather accurately. Two kinds of analyses were performed on the two structures; static analysis to evaluate the effects of joint strength and stiffness on the structure strength and stiffness, and dynamic analysis to evaluate the effects on member deformation demands. In the static analysis the gravity loads are applied first and then the seismic design loads, E, which are distributed triangularly over the height of the structure, are applied and incremented until a mechanism is formed in the structure. Equilibrium in the non-linear range is maintained through a Newton-Raphson iteration procedure. In the dynamic analysis, an acceleration-time history is used as input. The dynamic response is computed using step-by-step integration assuming constant input acceleration during each time step. Masses are lumped at the floor levels and the mass moment of inertia is neglected. Stiffnessdependent damping of 5% of critical damping is assumed in the first mode. The N21E component of the Taft Lincoln School Record of the 1952 Kern County earthquake was used as input ground motion. Even though the record was scaled to a peak ground acceleration of 0.6g, the inelastic deformations in the two structures were relatively small. In part this can be attributed to the inherent reserve strength of ductile moment resisting frames in which gravity load moments add to the member strength requirement and drift requirements govern often over strength requirements. In part this must be attributed also to period problems, since the fundamental periods of the bare frame structures used in the dynamic analysis were about twice the periods used in design (T = 0.1 N). Thus, the relatively small inelastic deformation demands obtained in the analyses may not be fully representative for real structures. However, the conclusions drawn on the relative deformation demands should not be greatly affected by this consideration. 3.4 Effects of joint deformations on structure strength and stiffness
Figure 6 shows the static lateral load versus roof displacement diagrams for the two structures. The differences in the diagrams in each figure are caused
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diagrams is that, if the need exists to predict accurately the strength and stiffness of steel frames, the effects of joint shear strength and stiffness should be included in the mathematical model. This holds true particularly for frames with weak joints which appear to become common practice even in regions of high seismicity. In these cases, strength and stiffness predictions based on centerline dimensions may lead to poor results.
3.5 Effects of joint strength and stifihem on dynamic response The time history results of the non-linear dynamic analyses are discussed in detail in Reference 12. There are considerable differences in the story shear histories between the four designs because of the variations in joint strengths. However, the overall dynamic characteristics represented by maximum interstory and roof displacements were similar for all designs. In fact, the maximum roof displacements for the two Allowable Stress Designs were slightly smaller than those for the other designs. The preliminary conclusion to be drawn from this pilot study is that the global dynamic response of frame structures is not affected greatly by variations in joint strengths and stiffnesses. However, the contrary is observed in the localized inelastic deformation responses of the individual members. Here the joint shear strength has a considerable effect on the relative deformation demands in beams and joints. This is evident from Figs 7 and 8 which show ductility demands in beams and joints. The figures display plots of the maximum ductility ratios in each story and for each design. The ductility ratio for plastic hinge rotation at the beam end is defined as /~0 = 1 +10p]/0y, where 0p is the plastic hinge rotation and Or is defined as MyL/(6EI). The ductility ratio for joint distortion is given a s / ~ = I~/I/Yy, with yy = Fy/(V'3G) = 0.0019 for A36 structural steel. In the Allowable Stress Designs no plastic hinge rotations occur in any of the beams and, because of the weak panel zones, all inelastic deformations are concentrated in the joints. In both the seven- and ten-story frames the maximum joint distortion ductility ratios are less than 8-3 which are values that should be tolerable in well designed joints. However, it must be pointed out again that the two structures did not undergo very large inelastic deformations in the postulated earthquake. This is evident from the small plastic hinge rotation ductilities in the frames analyzed with centerline dimensions and rigid joints. In the Gravity plus 3E Designs the deformation demands in beams and joints appear to be well balanced, with both the plastic hinge rotations and joint distortions being relatively small. In the 1980 SEAOC Design the
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demands on plastic hinge rotations are largest. With the mathematical models used in the frame analysis, the inelastic joint distortions are not negligible in this design because the bending strength of the beams is taken as 1.05Mp while the joints are designed to resist 1.0Mp. The results presented for Centerline Dimensions are a measure of the error that is encountered when conventional frame analysis with rigid joints is performed. This analysis gives reasonable results for plastic hinge rotations in frames with strong joints (1980 SEAOC) but gives completely misleading results for frames with weak joints (Allowable Stress Design) where the inelastic deformations are concentrated in the joints.
4 CONCLUSIONS Because well designed joints are ductile elements when subjected to an unbalance of beam moments that causes inelastic joint distortions, it may be advisable to design the joint panel zones so that they yield in shear and participate in energy dissipation in severe earthquakes. This is done inadvertently in allowable stress design and intentionally with a provision given in the 1985 SEAOC draft. The best design would be one in which the inelastic deformation demands in beams and joints are well balanced. This appears not to be the case in the two aforementioned design approaches which often lead to very weak joints that have to undergo large inelastic distortions whereas the beams remain elastic. When the joints are weaker than the connected elements, they will greatly affect the structure strength and stiffness and the distribution of inelastic deformations in structural elements. Thus, if attempts are made to predict the inelastic response of frame structures with weak joints, mathematical models should be used that consider the strength and stiffness characteristics of the beam-column joints.
REFERENCES 1. Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, San Francisco, SEAOC, 1980. 2. Fielding, D. J. and Huang, J. S., Shear in steel beam-to-column connections, Welding Journal (July 1971). 3. Krawinkler, H., Bertero, V. V. and Popov, E. P., Shear behavior of steel frame joints, J. Struct. Div., ASCE, 101, No. STll (November 1975). 4. Slutter, R. G., Tests of Panel Zone Behavior in Beam--column Connections, Fritz Engineering Laboratory Report No. 200.81.403.1, Lehigh University, Bethlehem, USA, 1981.
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5. Popov, E. P., et al., Cyclic behavior of large beam-column assemblies, Earthquake Spectra, EERI, 1, No. 2 (February 1985). 6. Becker, R., Panel zone effect on the strength and stiffness of steel rigid frames, Engineering Journal, AISC, 12, No. 1 (1975). 7. Naka, T., et al., 'Research on the behavior of steel beam-to-column connections in the seismic-resistant structure', Proceedings of Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1969. 8. Hjelmstad, K. D. and Popov, E. P., Seismic behavior of active link beams in eccentrically braced frames, EERC Report No. 83/15, University of California, Berkeley, USA, July 1983. 9. American Institute of Steel Construction, Manual of Steel Construction, 8th edn, New York, AISC, 1980. 10. Krawinkler, H., Bertero, V. V. and Popov, E. P., Inelastic Behavior of Steel Beam-to-Column Subassemblages, EERC Report No. 71-7, University of California, Berkeley, USA, October 1971. 11. Krawinkler, H., Shear in beam--column joints in seismic design of steel frames, Engineering Journal, AISC, 15, No. 3 (1978). 12. Mohasseb, S., Effects of beam-column joints on the seismic behavior of steel frames, Engineer Thesis, Department of Civil Engineering, Stanford University, USA, December 1982. 13. Lee, D. G., Accurate and simplified models for seismic response prediction of steel frame structures, Ph.D. Dissertation, Department of Civil Engineering, Stanford University, USA, February 1984.