MASTER MMS
Applied mechanics of materials and structures
MECHANICS of MATERIALS & STRUCTURES
UdG - 3501MO2062
Academic year 2012-2013
MODELING OF A PLASTIC HINGE
INDEX 1.
PROBLEM DESCRIPTION ........................................................................................................................ 2
2. GEOMETRY OF THE STRUCTURE, LOADS AND INTERNAL FORCES FROM SEISMIC AND GRAVITACIONAL LOADS ................................................................................................................................ 5 3.
PLASTIC MOMENT AND PLASTIC HINGES ............................................................................................. 7 3.1
Plastic Moment ............................................................................................................................. 7
3.2
Moment‐Curvature & Moment‐Rotation relationships ............................................................... 9
4.
INTERNAL FORCES OF A BEAM BASED ON THE ASSUMPTION OF PLASTIC HINGE FORMATION ....... 12
5.
GEOMETRIC DESIGN OF A STEEL PLASTIC HINGE FOR I‐SECTIONS (FEMA‐350) ................................. 14
6.
QUESTIONS TO BE SOLVED ................................................................................................................. 16
References .................................................................................................................................................. 17
1. PROBLEM DESCRIPTION
Ground motion produced by earthquakes mean a huge amount of energy that can be transmitted to the buildings. The amount of energy introduced into a structure depends mainly on its fundamental period. This energy can be transformed in elastic deformation or dissipated by viscous damping and plastic deformation. A structure that can dissipate most of the energy input can be lower sized. The conventional design of buildings from the seventies to now has been based on seismic forces reduction by means of the yielding of determined parts of the structure. This design means more economical structures but also possible severe damage spread all over the building that could suppose the building demolition when the cost of repairing the damage is too high. Most of new buildings are still based on conventional design. Fortunately some of them are being designed with reparable or non-damaging special elements to concentrate the energy dissipation process (known as energy dissipators) or with base isolation systems to reduce most of the energy input.
Fig. 1 Plastic hinges distribution in a frame based on the “strong column – weak beam” concept.
The Special Moment Frame (SMF) is considered as a conventional seismic design system for steel structures. A SMF is designed to withstand significant inelastic deformations when subjected to the forces resulting from the motions of the design earthquakes. This inelastic deformation has to be located in specially designed plastic hinges in the beam endings, based on the concept of “strong column – weak beam” (Fig. 1). To withstand with large plastic deformations, the plastic hinges has to be located in a compact classified section and it has to be experimentally proved that their geometry and materials provide such plastic rotations.
Although it is common to visualize inelastic rotations in moment frames occurring at beam or column “hinges,” analyses and testing demonstrate that the inelastic rotations actually mainly combine flexural deformations at the hinges and shear deformations of the panel zones (Fig. 2). In this problem we will design the panel zone to behave elastically, by reinforcing it with additional welded plates (Fig. 3).
Fig. 2 Beam-column steel joint
Fig. 3 Panel reinforcement with additional welded plates The yielding elements have to sustain a non-linear hysteretic response that has to be proved with experimental analysis. As experimental analysis of buildings are very expensive (they need huge laboratories and dynamic actuators fed with enormous hydraulic pumps and sophisticated servocontrolled systems) experimental models use to be scaled and often they are tested by equivalent quasi-static loading methods (Fig 4). Frequently a substructure is tested instead of the all frame. Fig. 5b shows a substructure that was tested to obtain the non-linear behavior of a slab-to-column interior connection (Fig. 5a). It was considered that in the endings of the columns and frames of the model there was not bending moment, based on responses to lateral forces of frames (Fig. 7b) The main goal of this problem is obtain the non-linear response of beam-column joint by a simplified method and by a FEM analysis of a substructure, and compare their results.
Fig. 4 Hysteretic response of a shear panel [5]
Fig. 5 a) 2D steel frame model, b) experimental model of an interior joint, c) lateral imposed displacement, d) Displacement-force hysteretic response [6]
2. GEOMETRY OF THE STRUCTURE, LOADS AND INTERNAL FORCES FROM SEISMIC AND GRAVITACIONAL LOADS
Fig.6 Geometry of the problem’s frame (left) and the load gravity forces (q) and the lateral earthquake forces (E) obtained from modal spectral analysis Load Forces: A simplified load case has been considered, taking into account the same live load for all the floors. Earthquake forces has been obtained from spectral modal analysis, considering a ductility factor of 4, ground type A, ground acceleration 0.23g 1st G (kN/m) Q (kN/m) E (kN)
5.26
floor
2nd floor
10.5
3rd floor 18 12 15.6
4th floor
20.8
Where: G: permanent gravity loads, Q: live loads, E: earthquake horizontal loads Combination for persistent or transient design situations: 1.35* G + 1.5 Q Combination for seismic design situation G + 0,3 Q + E
5th floor
25.6
Fig.7 Bending moments of the frame with the gravity loads (left) and the lateral earthquake equivalent forces (right).
Fig.8 Internal Forces of the frame from the seismic combination
Fig.9 Lateral deformation from the seismic combination: deformation from equivalent forces (reduction factor of 4 because ductility) and linear elastic analysis (left) and expected deformations considering the ductility factor uniformly distributed over the all high of the frame
3. PLASTIC MOMENT AND PLASTIC HINGES 3.1
Plastic Moment
Fig. 10
In simple bending the neutral axis and the sectional moment can be obtained from the static equilibrium equations:
N .dA 0 A
[3.1]
M .z.dA
[3.2]
A
Where is the normal strain, A the cross-sectional area, z the distance from the neutral axis and N and M the axial internal force and the bending moment respectability. For an ideally section where all fibers are plasticized, the before equations become:
N .dA f y dA f y dA 0 A
Ac
At
and:
dA dA
Ac
At
Where Ac and At are compressed and tensed areas respectably.
Fig. 11 It means that, in this ideally yielded section, the neutral axis is located from the condition that the compressed and the tensed areas are equal. In symmetrical sections, like double T sections, the neutral axis and the symmetry axis are the same, but not in the case of non-symmetrical sections (Fig. 11). The ideal plastic moment Mpl can be obtained from equation 3.2:
M pl .z.dA f y zdA f y zdA A Ac At M pl f y S Ac f y S At
[3.3]
SAc and SAt are the first moment of the compressed and tensed areas, respectably, referred to the neutral axis. In the case of symmetrical sections fist moment values are the same so: M pl 2 f y S
If the plastic section modulus Zpl is defined as:
Z pl 2 S
Then the plastic moment can be rewritten as:
[3.4]
M pl f y . Z pl
[3.5]
3.2
Moment‐Curvature & Moment‐Rotation relationships
Fig. 12
Fig. 13 The section curvature k can be easily obtained from Euler‐Bernoulli theory of slender beams. The assumption that plane sections remain plane after flexural deformation supposes (Fig. 14):
k
1
z
[3.6]
Fig. 14
For a defined section where the constitutive stress-strain relationship is known (Fig. 15), it is possible to obtain the curvature – moment relationship. Elastic moment can be obtained:
M el f y
Iy z max
f yWel
Curvature el can be determined from equation 3.6, where the maximum deformation of the section is the yielding deformation y. From a set of incremental curvatures in the plastic range, the strain and stress diagram can be determined (Fig.13) taking into account the strain-stress relationship of the steel. Finally the moment correspondent to every curvature can be calculated by the equation 3.2. Finally the Moment – Curvature curve can be represented (figure 15.a) As curvature can be defined as:
d 2 z d dx 2 dx
[3.7]
The Moment – Rotation curve (Fig. 15.b) of a plastic joint can be determined by integrating the curvatures along the length of the plastic hinge:
H kd x LH
Fig. 15 Stress – strain relationships: a) realistic, b) bilinear
Fig. 16 a) Moment-Curvature relationship, b) Moment-Rotation relationship
4. INTERNAL FORCES OF A BEAM BASED ON THE ASSUMPTION OF PLASTIC HINGE FORMATION
Fig. 17
Fig. 18
Fig. 19
5. GEOMETRIC DESIGN OF A STEEL PLASTIC HINGE FOR I‐SECTIONS (FEMA‐350)
Fig. 20 Reduced Beam Section (RBS) connection [2]
6. QUESTIONS TO BE SOLVED a.
Determine the geometry of the plastic hinge as defined in figure 20 and taking into account the steps 1 and 2 of the design procedure in chapter 5.
b.
Determine the M-H relationship for the plastic hinge taking into the stress-strain relation-ship or a simplified one that you find suitable enough (bilinear, for example): Yielding point Ultimate point
Strain Stress (mm/mm) (MPa) 0 0 0,0022
285
0,0407 0,0881 0,1495
417 488 534
0,2045
557
Fig. 21 c.
Determine the response of the joint by a FEM analysis of next representative substructure (Fig. 22). Impose a lateral displacement =1.5 times the interstory drift of Fig. 8b. Reinforce the panel joint by increasing the panel thickness (if necessary) to avoid plastic yielding in the panel. Obtain the Moment – Rotation and define the hinge length and its location on the beam.
Fig. 22 Sketch of a substructure to model the joint behavior
d.
Compare the both obtained plastic hinge rotation curves.
References [1] ANSI/AISC 341‐05; ANSI/AISC 341s1‐05 Seismic Provisions for Structural Steel Buildings; American Institute of Steel Construction, INC., Chicago, 2005 [2] FEMA‐350. Recommended Seismic Design Criteria for New Steel Moment‐Frame Buildings. Federal Emergency Management Agency, June 2000 [3] CEN prEN 1998‐1 Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. 2004 [4] CEN prEN 1993‐1‐1:2005 Eurocode 3: Design of steel structures, Part 1.1: general rules for buildings, 2005 [5] D.D. Kim, M.D. Engelhardt (2002). “Monotonic and cyclic loading models for panel zones in steel moment frames” Journal of Constructional Steel Research, Vol. 58, Issues 5‐8, 605‐63 [6] A. Benavent‐Climent, X. Cahís and A. Catalan (2008) “Seismic behavior of interior connections in existing waffle‐flat‐plate structures”. Engineering Structures 30, 2510–2516
