26/01/2016 Marco Donà Multi Linear Elastic and Plastic Link in SAP2000 1 General principles Link object connects two joints, i and j, separated by length L, such that specialized structural behaviour may be modelled. Linear, nonlinear, and frequency-dependent properties may be assigned to each of the six deformational degrees-of-freedom (DOF) which are internal to a link, including axial, shear, torsion, and pure bending. Internal deformation is then calculated from joint j displacement relative to joint i, where i may be grounded to simulate a support point. There are two types of Links: - Two-joint links - Single-joint links which are those grounded at support points. Six independent internal deformations are defined for the Link/Support element. These are calculated from the relative displacements of joint j with respect to: - Joint i for a two-joint element - The ground for a single-joint element For two-joint Link/Support elements the internal deformations are defined as: Axial (U1): d u1 = u 1j u 1i Shear in the 1-2 plane (U2): d u2 = u 2j u 2i d j2 r 3j (L d j2 ) r 3i Shear in the 1-3 plane (U3): d u3 = u 3j u 3i d j3 r 2j (L d j3 ) r 2i Torsion (R1): d r1 = r 1j r 1i Pure bending in the 1-3 plane (R2): d r2 = r 2j r 2j Pure bending in the 1-2 plane (R3): d r3 = r 3j r 3i Where u ki and r ki are the translations and rotations at joint i while d jn is the distance you specify from joint j to the location where the shear deformation d un is measured (the default is zero, meaning at joint j. Three of these internal deformations are illustrated in Figure 1. Figure 1: Internal Deformations for a two-joint Link Element. For one-joint grounded-spring elements the internal deformations are the same as above, except that the translations and rotations at joint i are taken to be zero.
2 Link/Support Properties A Link/Support Property is a set of structural properties that can be used to define the behaviour of one or more Link or Support elements. Each Link/Support Property specifies the force-deformation relationships for the six internal deformations. Mass and weight properties may also be specified. Each Link/Support Property is assumed to be composed of six internal springs or Hinges, one for each of six internal detonations. Each spring may actually consist of several components, including springs and dashpots. The force-deformation relationships of these springs may be coupled (available only for linear links) or independent of each other. Figure 2: Three of the Six Independent Spring Hinges in a Link/Support Element Figure 2 shows the springs for three of the deformations: axial, shear in the 1-2 plane, and pure-bending in the 1-2 plane. It is important to note that the shear spring is located a distance d j2 from joint j. All shear deformation is assumed to occur in this spring; the links connecting this spring to the joints (or ground) are rigid in shear. Deformation of the shear spring can be caused by rotations as well as translations at the joints. The force in this spring will produce a linearly-varying moment along the length. This moment is taken to be zero at the shear spring, which acts as a moment hinge. The moment due to shear is independent of, and additive to, the constant moment in the element due to the pure-bending spring [1]. 3 Types of Linear/Nonlinear Properties The primary Linear/Nonlinear Link/Support Properties may be of the following types: Coupled Linear Damper Gap Hook Multi-linear Elastic Multi-linear Plastic Plastic (Wen) Hysteretic (Rubber) Isolator Friction-Pendulum Isolator Tension/Compression Friction Pendulum Isolator The Coupled Linear may have fully coupled linear stiffness and damping coefficients. All other property types are considered nonlinear. However, for each nonlinear property type you also specify a set of uncoupled linear stiffness and damping coefficients that are used instead of the nonlinear properties for linear analyses. These substitute linear properties are called linear effective stiffness and linear effective damping properties. 2
4 Links to replace beam elements Links can be used to replace frame elements or part of them. The equivalent axial, shear, bending stiffness and torsional stiffness (k A, k S, k B and k T respectively) to include into the link properties can be obtained using the following relations: k A = E A ; k L S = G A s E I + 12 ; L L 3 k B = E I L, k T = G I T L, where E is the Young modulus of the material, G is the shear modulus, A and I are the area and second moment of inertia of the cross section; I T is the torsional constant and A s is the Shear area. The shear spring, k S, is located at d jn = L/2. The nonlinear behaviour of beam elements can also be included but it is recommended to use a short link, as long as the plastic hinge L=h. Because, in this small portion of the beam, the stress parameters are almost constant and then the link can still represent the behaviour of the hinge. For example if the link is short, the bending moment at mid span is a good approximation of the bending moment along the whole link. Only the relative displacement, or rotation, between the two nodes is considered in the calculation. Is then suggested to replace the beam element only in the portion where the plastic hinge is supposed to develop. As example in Figure 3 a clamped-simply supported beam has hinges at the clamp and at mid-span only. Figure 3: Clamped-pinned beam with links at the expected plastic hinge locations only Important! Beam elements replaced by the links have to be removed from the model 4.1 Limitations Nonlinear plastic links moment cannot take into the interaction between axial force and bending moment, typical of a plastic hinge. Only single moment-rotation diagram corresponding to a specific axial force can be introduced in the link properties. Figure 4: N-M interaction diagram for an I steel section 3
5 Multi Linear Plastic link There are six force-deformation relationships that govern the behaviour of the element, one for each of the internal springs (Axial, Shear, Torsional and pure bending). Each of these relationships may be zero, linear only, or linear/nonlinear for a given Link/Support Property. Figure 5 shows the SAP2000 window to setup the link properties for the Multi Linear Plastic Link, available at Define Section Properties Link/Support Properties. Mass and Weight of the beam replaced by the Link Fixed Link: If selected a relatively large stiffness will be assigned. Their deformation is zero. Nonlinear: To be selected in order to apply the nonlinear behaviour Directional properties: U1, U2, U3, R1, R2, R3 Figure 5: Link Property Data - SAP2000 In order to introduce the nonlinear Link properties for example of U1: - Tick the box next to U1 - Tick the Nonlinear box - Enter the link properties by clicking on Modify/Show for U1 A new window Link/Support Directional Properties will pop up (See Figure 6). Elastic Stiffness as defined in Section 4 4
Figure 6: Nonlinear Plastic Link Directional Properties - SAP2000 - The Multi-Linear Force Deformation Definition has to be defined in accordance with the plastic hinge properties of the beam. Important! When a link is used to replace a beam element in a 3D frame all the directional properties have to be activated otherwise the inactive stiffness will be lost. Important! In a first analysis it is suggested to use only nonlinear plastic hinges for bending moment (R2, R3) which are the most likely to occur. Only later, if strictly required, the user can introduce the Axial and Shear ones. If the Nonlinear box is unchecked, the link behaves as linear and the following window, for the link properties, will pop-up: Shear spring location at L_link/2 Figure 7: Linear Link Directional Properties - SAP2000 Important! The Shear directions require the definition of the Shear Deformation Location which has to be set as half the length of the link (in order to match with the beam element deformations). 6 Example of application of Multi-Linear Elastic link A 2D three-story frame model is studied in this example as practical application of the Multi linear Plastic Links. The frame has story height of 3m and bay width of 6m. Columns and beams are IPE270 (S355). The frame is clamped at the ground level. Two model configuration are studied: (i) beam element only; (ii) beam element and 2 links at the two ends of the columns at the ground floor (Figure 8). For the model with links the columns have been split at 0.2m from the node and replaced by the link. 5
Link (L_link=0.2m) Link (L_link=0.2m) Figure 8: 2D view of the frames: (left) without links; (right) with 4 links at ground floor Figure 9 and Figure 10 shows the section properties of the IPE270. Figure 9: IPE270 - Section dimensions Figure 10: IPE270 Property Data 6
In this example the length of the link (L_link) is assumed to be 0.2m (It usually should be taken equal to the length of the plastic hinge). The Link properties data are first evaluated in the following table and the introduced into the model (Figure 11, Figure 12) IPE 270 Units [kn, m] 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 A = 4590 mm2 EI = 1.22E+04 KNm^2 I = 57900000 mm4 ka 4.8195000E+06 kn/m As = 1782 mm2 Ks 1.8958154E+07 kn/m L_link = 0.2 m Kb 6.0795000E+04 kn m/m Figure 11: IPE270Link Property Data (U1) (U2) (R3) Figure 12: IPE270Link Directional properties 7
6.1 Loads Loads applied are the self weight dead load and a lateral unit point loads at each floor level (Figure 13). Figure 13: Lateral unit loads 6.2 Output Figure 14: Load Case definition The outputs show exactly the same results for bending moment, shear force and reactions. Figure 15: Lateral unit loads 8
Figure 16: Lateral unit loads Figure 17: Lateral unit loads 7 Summary Linear links can be used to exactly replace beam elements. This is not the same for nonlinear plastic link, where the internal stresses are evaluated at the shear spring location. A good approximation can still be achieved if short links are considered. Further detail for the nonlinear properties definition of the plastic link will be provided with the report on Plastic Hinges. 8 Reference [1] CSi SAP, Analysis Reference Manual, Computers and Structures, Inc., Berkeley, California, 2013 9