01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only affect the behaviour of the structure in nonlinear static and nonlinear direct-integration time-history analyses. Yielding and post-yielding behaviour can be modelled using discrete automatic or user-defined hinges at any location along that element. Plastic Hinges can only be introduced into frame elements (not into cables or Tendons). 2 Hinge Properties A hinge property is a named set of rigid-plastic properties that can be assigned to one or more Frame elements. You may de fine as many hinge properties as you need. For each force degree of freedom (axial and shears), you may specify the plastic force-displacement behaviour. For each moment degree of freedom (bending and torsion) you may specify the plastic moment-rotation behaviour. Each hinge property may have plastic properties specified for any number of the six degrees of freedom. The axial force and the two bending moments may be coupled through an interaction surface. Degrees of freedom that are not specified remain elastic. There are multiple possibilities to model the plastic hinge when this concept is used in structural analysis. Uncoupled moment, torsion, axial force and shear hinges are available. There is also a coupled P-M2-M3 hinge which yields based on the interaction of axial force and bi-axial bending moments at the hinge location. Subsets of this hinge include P-M2, P-M3, and M2-M3 behaviour (see Figure 1). Figure 1: Hinge Property Data More than one type of hinge can exist at the same location, for example, you might assign both an M3 (moment) and a V2 (shear) hinge to the same end of a frame element. Hinge properties can be computed automatically from the element material and section properties according to ASCE-41 criteria. Automatic hinge properties for steel members are based on Tables 9-6 of ASCE-41, and for concrete members, Tables 10-7 and 10-10 ASCE-41 [1]. Similar curve can be also obtained using Eurocode 8 part 3 (see Table 1). Figure 2 shows the generalized load-deformation curve with parameters a, b, and c as defined in Tables 9-6 and 9-7 (ASCE-41).
Figure 2: Generalised Force-Deformation relationship used for the plastic hinges according to ASCE-41. Table 1: Plastic rotation capacity at the end beams or columns with dimensionless axial load v not greater than 0.30 [Eurcode 8 part 3]. 2.1 Plastic Deformation Curve For each degree of freedom (DoF), you define a force-displacement (moment-rotation) curve that gives the yield value and the plastic deformation following yield. This is done in terms of a curve with values at five points, A- B-C-D-E, as shown in Figure 2. You may specify a symmetric curve, or one that differs in the positive and negative direction. The following points should be noted: Point A is always the origin. Point B represents yielding. No deformation occurs in the hinge up to point B, regardless of the deformation value specified for point B. The displacement (rotation) at point B will be subtracted from the deformations at points C, D, and E. Only the plastic deformation beyond point B will be exhibited by the hinge. Point C represents the ultimate capacity for push-over analysis. However, you may specify a positive slope from C to D for other purposes. Point D represents a residual strength for push over analysis. However, you may specify a positive slope from C to D or D to E for other purposes. Point E represent total failure. Beyond point E the hinge will drop load down to point F (not shown) directly below point E on the horizontal axis. If you do not want your hinge to fail this way, be sure to specify a large value for the de formation at point E. You may specify additional deformation measures at points IO (immediate occupancy), LS (life safety), and CP (collapse prevention), see Figure 3. Their values are available in ASCE-41 Table 9-4; Table 9-6; and they are informational measures that are reported in the analysis results and used for performance-based design. They do not have any effect on the behaviour of the structure. Prior to reaching point B, all de formation is linear and occurs in the Frame element itself, not the hinge. Plastic deformation beyond point B occurs in the hinge in addition to any elastic deformation that may occur in the element. When the hinge unloads elastically, it does so without any plastic deformation, i.e., parallel to slope A- B. 2
3 Hinge Definition 3.1 M3 Hinge Figure 3: The A-B-C-D-E curve for Force/Moment vs. Displacement/Rotation. For the automatic FEMA frame hinges, the yield rotation is specified in the code (ASCE 41-06), Table 5-6 as the chord rotation of the full member length due to plastic moment at one end. Specifically, the yield rotation is defined as: Beams: = ; Columns: = (1 ), where =, is the frame object (not element) length and is the expected yield stress 1. Figure 4: Definition of Chord Rotation If the beam element with plastic hinge does not span between two columns, or for a column between two floors, but it is only a portion of it, the yield rotation has to be entered manually using equal the span/length of the beam/column. 1 The expected yield ( ) stress is the product of a code-prescribed factor and the expected strength of the material. This factor is typically around 1.1, as with ASCE-41 Table 9-3. These effective stress values represent the material response which occurs approximately halfway along the x-axis of the force-deformation relationship. ASCE-41 recommends using effective strength for deformation-controlled actions. Minimum strength represents the lower bound of nonlinear material response, which is best for force-controlled actions. Expected stress values are used to automatically generate hinge properties for P-M2-M3 and P-M hinges [2]. If Eurocodes are used, may be taken equal to the nominal value multiplied by the over strength factor for the steel of the beam, determined in accordance with EN 1998-1: 2004 [Eurocode 8 part 3]. 3
3.1.1 Hinge Definition in SAP2000 In SAP2000 each plastic hinge is modelled as a discrete point hinge. All plastic deformation, whether it be displacement or rotation, occurs within the point hinge. The hinge can be of two types: - Force/Displacement (or Moment-Rotation) and no additional hinge length is required; - Force/Strain (or Moment-Curvature). This means a length for the hinge over which the plastic strain or plastic curvature is integrated must be assumed. There is no easy way to choose this length, although guide lines are given in FEMA-356. Typically it is a fraction of the element length, and is often on the order of the depth of the section, particularly for moment-rotation hinges. Values accordingly with ASCE-41 or EC8 Only for Moment-Curvature hinge type the Hinge length has to be assumed : to be defined manually Safety Factors assumed in accordance with the Standard (ASCE-41, EC8) : to be defined manually Figure 5: Definition of Chord Rotation 3.2 Axial Hinge The lower-bound strength,,, of steel columns under axial compression shall be the lowest value obtained for the limit states of column buckling, local flange buckling, or local web buckling. The expected axial strength of a column in tension shall be computed as:, =! ; while the Yield Displacement is defined as = # $, where is the length of the beam element. 3.3 Shear Hinge Shear hinges are modelled to account for inelastic shear behaviour. It is recommended to avoid placing shear hinges at either end of a member. Only one shear hinge is necessary. Once a shear hinge begins yielding, capacity begins decreasing, and load is redistributed. If a second hinge then fails, the member will no longer be able to transfer lateral load applied between hinge locations. Gravity load must be carried during nonlinear analysis. CSI Software will stop analysis if gravity load cannot be redistributed. When gravity load is present, users are recommended to replace distributed loading with equivalent loads concentrated at either element end. When using shear hinges, please keep in mind that once shear capacity is exceeded, the model will not provide any further useful information. There is no sense in continuing with analysis since shear connections must be modified. Once shear capacity is increased, analysis may then proceed toward more ductile response mechanisms. SAP2000 automatic shear hinge properties are defined following FEMA 365 and FEMA 273 [3]; in details: % = 0.55! ) and = 0.01. 4 Cross-section requirements for plastic global analysis For plastic hinge design the beam section should satisfy the requirements defined in the Eurocode 3-1-1, section 5.4, 5.5 and 5.6. Some of them are: 4
- At plastic hinge locations, the cross-section of the member which contains the plastic hinge should have a rotation capacity of not less than the required at the plastic hinge location. - In a uniform member sufficient rotation capacity may be assumed at a plastic hinge if both the following requirements are satisfied: a) the member has Class 1 cross-sections at the plastic hinge location; b) where a transverse force that exceeds 10 % of the shear resistance of the cross section, is applied to the web at the plastic hinge location, web stiffeners should be provided within a distance along the member of h/2 from the plastic hinge location, where h is the height of the cross section. 5 Define and Assign Frame Hinges 5.1 Define Hinge Properties Figure 6 shows the SAP2000 window to setup the hinge properties available at Define Section Properties Hinge Properties. Figure 6: Definition of Frame Hinge Properties In order to define a plastic hinge click on Add New Property; the window in Figure 7a will appear; choose the material (steel or concrete) and click OK to get into Frame Hinge Property Data Figure 7b. (a) (b) Figure 7: Add Hinge Properties SAP2000 windows - Define a Hinge Name and select the Hinge Type (example Moment M3) A new window Frame Hinge property data will pop up (See Figure 6) - Enter the Displacement Control Parameters; Yield Moment and Yield Rotation, Type and Acceptance Criteria OK. 5.2 Assign Hinge Properties In order to assign the Hinge properties go to Assign Frame Hinges. 5
In the window Assign Frame Hinges of Figure 8, select the Hinge Property previously created and introduce the Relative Distance value, which will locate the hinge along the beam Add Hinge. Multiple Hinges can be added into the same beam. Figure 8: Assign Frame Hinges - Then select the beam elements where you are planning to introduce the Hinges OK to complete the hinge assignment. 5.3 Recommendation Sudden strength loss is often unrealistic and can be very difficult to analyse. It is encouraged to consider strength loss only when strictly necessary, to use realistic negative slopes instead, and to carefully evaluate the results. 6 Run Nonlinear Analysis Time-history analysis may be initiated using the process which follows: - Create the model and assign support conditions to restrained joints. - Select Define Functions Time History to define a time-history function which characterizes load variation over time. - Assign load conditions to the model through Assign Joint Loads or Frame Loads. - Define either a modal or direct-integration time-history load case through Define Load Cases. More details are available in the report Fast Nonlinear Analysis_20160118_MD 7 Analysis Results For each output step in a nonlinear static or nonlinear direct-integration time-history Load Case, you may request analysis results for the hinges. These results include: The forces and/or moments carried by the hinge. Degrees of freedom not defined for the hinge will report zero values, even though non-zero values are carried rigidly through the hinge. The plastic displacements and/or rotations. The most extreme state experienced by the hinge in any degree of freedom. This state does not indicate whether it occurred for positive or negative deformation: A-B; B-C; C-D; D-E; >E The most extreme performance status experienced by the hinge in any degree of freedom. This status does not indicate whether it occurred for positive or negative deformation. 6
When you display the deflected shape in the graphical user interface for a nonlinear static or nonlinear directintegration time-history Load Case, the hinges are plotted as coloured dots indicating their most extreme state or status. The colours used for the different states are indicated on the plot. Hinges that have not experienced any plastic deformation (A to B) are not shown. These outputs are available under: Display In particular the plastic Hinges Moment-Rotation diagram can be seen in: Display Show Hinge Results (Figure 9). Figure 9: Hinge Results 8 Difference between hinge and hysteretic-link application The energy dissipation which occurs during time-history analysis may be modelled using hysteretic links. Links are useful for capturing dynamic loading and unloading because of their multi-axial response. Isotropic, kinematic, Takeda, and pivot hysteresis models are available for single DOF hinges. For isotropic hysteresis, hinges unload elastically, parallel to the initial stiffness tangent (A-B slope); while for other hysteresis types, unloading follows a more complex nonlinear relationship. For links, several additional hysteretic models are available. Hysteretic behaviour may be specified for multiple degrees-of-freedom using a single link. PM and PMM hinges always use isotropic dissipation, which dissipates more energy than the kinematic, Takeda, and Pivot rules available for the single DOF hinges. 8.1 Multilinear plastic Link to replace plastic Hinge Hinge can be modelled as a separate link element. This method is currently only available in the ETABS Ultimate level, and enables hinge behaviour to be considered in nonlinear model time-history (FNA) load cases. As a rule, 7
FNA analysis runs significantly faster than nonlinear direct-integration time-history analysis. Nonlinear static analysis and nonlinear direct-integration time history analysis are available for both types of analysis modelling [3]. Multilinear plastic links can be used to replace plastic hinges with exactly the same results if the plastic hinge is located at the mid-span of the element. A 3m long column is studied in the following as practical application of Nonlinear Links to replace Plastic Hinges. Only the M3 plastic hinge will be introduced, while any other hinge will follow the same procedure. The Column is an IPE270 (S355). IPE 270 Units [kn, m] Link Properties E = 210 GPa Mass = 0.00721 T v = 0.3 Weight = 0.07068 kn G = 80.769231 GPa gm = 7.849 T/m3 EA = 9.64E+05 KNm EI = 1.22E+04 KNm^2 A = 4590 mm2 I = 57900000 mm4 K_axial = 4.8195000E+06 kn/m As = 1782 mm2 K_shear = 1.8958154E+07 kn/m I_tors = 159000 mm4 K_bending = 6.0795000E+04 kn m/m L_link = 0.2 m K_torsion = 64.21153846 kn m/m Given the material and section properties the Yield Bending Moment and Yield rotation can be evaluated in accordance with Subsection 3.1. L_beam = 3 m 1.8900E+02 knm Z = 4.84E-04 knm 7.7721E-03 rad Fy = 390.5 MPa Given also the Moment-Rotation diagram for the section it is possible to build the corresponding Moment- Rotation diagram for the Link. A diagram with 5 points is considered in the following: Point B: (*) = and (*) = +,-./0.1 Point C: () = 1.25 and () = (*) + (4)5 (6) +,-./0.1 + (4 0) Difference between the rotation coefficients at point C and B Point D: (8) = 1.26 and (8) = () + (:)5 (4) +,-./0.1 + (8 4) Point E: (<) = 1.27 and () = (8) + (>)5 (:) +,-./0.1 + (12 8) The following table summarise the diagram values for the IPE270 (S355). Difference between the rotation coefficients at point D and C Hinge Link Moment Rotation Rotation Moment 1-9.721346E-02-2.400325E+02 2-6.609396E-02-2.381425E+02 3-3.497446E-02-2.362525E+02 4-3.108841E-03-1.890020E+02 A 0 0 5 0.000000E+00 0.000000E+00 B 1 0 6 3.108841E-03 1.890020E+02 C 1.25 4 7 3.497446E-02 2.3625250E+02 D 1.26 8 8 6.609396E-02 2.3814252E+02 E 1.27 12 9 9.721346E-02 2.4003254E+02 8
Same yield Moment The rotation of the link at Yield is not the plastic rotation but the sum of yield rotation and elastic rotation of the element Figure 10: Nonlinear Link assignment NOTE! Nonlinear link replaces a whole element including the plastic hinge and as consequence it has to include both the elastic deformation of the beam and the plastic deformation of the hinge. The Moment-Rotation Diagram for the Nonlinear Link will be different than that one of the Plastic Hinge as defined in Figure 10 (left). Figure 11 shows exactly the same lateral displacements for a column with plastic Hinge and an equivalent column with nonlinear link. Plastic Hinge Nonlinear Link Figure 11: Comparison between Plastic Hinges (beam) and Nonlinear Link (right) 9 References [1] https://wiki.csiamerica.com/display/kb/hinge+faq#hingefaq-defaulthingepropertiesarederivedfromwhichsource? [2] https://wiki.csiamerica.com/display/kb/yield,+ultimate,+and+expected+stress [3] SAP2000 Graphic User Interface Manual [3] Eurocode 3. Design of steel structures. General rules and rules for buildings [4] Eurocode 8 9