4 PH Approach

Credits: 3 + 0
PG 2019
Spring 2020 Semester
Performance-based Seismic Design of Structures
Fawad A. Najam
Department of Structural Engineering
NUST Institute of Civil Engineering (NICE)
National University of Sciences and Technology (NUST)
H-12 Islamabad, Pakistan
Cell: 92-334-5192533, Email: fawad@nice.nust.edu.pk
Modeling for Structural Analysis
by Graham H. Powell
http://structurespro.info/nl-etabs/
Approaches for Nonlinear Modeling of Structures
Practical Approaches for Nonlinear Modeling of Structures
Defining Inelastic
Behavior at
Nonlinear Modeling
Material Level Fiber Modeling Approach
of Materials
Nonlinear Modeling
Cross-section Level
of Cross-sections Plastic Hinge Modeling
Nonlinear Modeling Approach
Member Level
of Members
Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 5

An Introduction to Lumped Plastic Hinge Approach for Nonlinear
Modeling of Structural Components

What is a Plastic Hinge?

RZ MZ
UZ VZ
UX RX VX MX
UY P
RY MY
The six degrees-of-freedom and corresponding actions for a node in 3D space.

Plastic Hinge Modeling Approach
Plastic Hinge Shear Hinges Force-Displacement Type Hinges

Elements in ETABS
Axial Hinges Stress-Strain Type Hinges
Torsion Hinges
Flexural Hinges
Interacting
P-M-M Hinges
Moment-Curvature Type Hinges
Moment-Rotation Type Hinges

Force Distributions on Buildings
Based on Dr. Pramin Norachan, AIT Solutions

Moment-Rotation Plastic Hinge Model for Columns
Illustration of modeling components for a reinforced concrete beam-column: (a) inelastic hinge model; (b) initial
(monotonic) backbone curve; and (c) cyclic response model (Haselton et al. 2008).

Moment-Rotation Plastic Hinge Model for RC Beams
Moment-rotation behaviour assigned

to a plastic hinge element
Moment-Rotation Plastic Hinge Model for RC Beams and Columns

Moment-Rotation Plastic Hinge Model for RC Beams and Columns
2-hinge beam
element

An Introduction to Plastic Hinges (and Inelastic Components) in
CSI ETABS and CSI PERFORM 3D

Nonlinear Modeling of Structures in ETABS

Uncoupled hinges – corresponding
to each degree of freedom
Coupled hinges – These hinges yield based on the

interaction of (any combination of) axial force and bi-axial
bending moments at the hinge location

ASCE 41 Approach for Nonlinear Modeling of Structural
Components

ASCE 41 Approach for Nonlinear Modelling of Structural Components
• ASCE/SEI 41 prescribes nonlinear Modeling Parameters(MP) and Acceptance Criteria(AC) for various
structural components.
• For Beams and Columns, MP and AC are given as limiting plastic rotations.
• MP are used to build analytical models of structures for seismic evaluation.
• AC provide deformation limits below which member performance is deemed acceptable.

ASCE 41-17 Generalized F-D Relation
Source: AIT Solutions

ASCE 41-17 Generalized F-D Relation

Basic Force-Deformation Relationship in perform 3d
Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 22 22

Hinge Properties
• Five points labeled A, B, C, D, and E are used to define the

force deflection behavior of the hinge.
The acceptance criteria (capacities) marked on the force-

deformation behavior of hinges.

Hinge Properties
• 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 Pushover analysis
• Point D represents a residual strength for Pushover analysis
• Point E represents total failure. Beyond point E the hinge will drop load down to point F (not shown)
directly below point E on the horizontal axis. To prevent this failure in the hinge, specify a large
value for the deformation at point E

RC Beams
Source: UCLA-SGEL Report 2009/06
RC Columns other
than Circular with
Spiral Reinforcement
27
RC Circular Columns
with Spiral
Reinforcement
28
RC Beam-Column Joints

Two-way Slabs and Slab-
Column Connections

Plastic Hinge Modeling of RC Beams using ETABS
Option 1: Manual Definition of Hinge Properties for Every Hinge Type
Option 2: Automatic Definition of Hinge Properties by the Program

ASCE 41-17 Modelling Parameters
and Acceptance Criteria - RC Beams
ASCE 41-17 Modelling Parameters and
Acceptance Criteria for RC Beams

Backbone Curve
Hysteresis
Acceptance Criteria

Seismic Design Shear (V) in Plastic Hinge Regions
• Ductile response requires that members yield in flexure, and that shear failure be avoided. Shear failure is avoided
through use of a capacity-design approach. The general approach is to identify flexural yielding regions, design those
regions for code-required moment strengths, and then calculate design shears based on equilibrium assuming the
flexural yielding regions develop probable moment strengths.
• Seismic design shear (V) in plastic hinge regions is associated with maximum inelastic moments that can develop at
the ends of members when the longitudinal tension reinforcement is in the strain hardening range (assumed to
develop 1.25 𝑓𝑦 ) This moment level is labeled as probable flexural strength, 𝑀𝑝𝑟 .
• Probable moment strength is calculated from conventional flexural theory considering the as- designed cross section,
using 𝜙 = 1.0, and assuming reinforcement yield strength equal to at least 1.25𝑓𝑦 .

Beam Shear Demand (based on Capacity Design Approach)
• Assuming beam is yielding in flexure, beam end moments are set

equal to probable moment strengths.
• Design shear is based on the probable moment to maintain the

moment equilibrium. This is worst case scenario for shear.
• ACI 318 defines probable moment strength as moment strength of a

member, determined using the properties of the member at the joint
faces assuming a tensile stress in longitudinal bars of “at least 1.25 𝑓𝑦 ”
and a strength reduction factor 𝜙 = 1.0.
• Controlling Load Combination 1.2𝐷 + 𝐸𝑣 + 1.0 𝑜𝑟 0.5 𝐿 + 0.2 𝑆

SUMMARY – Beam Shear Demand (based on Capacity Design Approach)
• A capacity design approach is used to guide the design of a special moment frame.
• The process begins by identifying where inelastic action is intended to occur. For a special moment frame, this is intended to
be predominantly in the form of flexural yielding of the beams.
• The building is analyzed under the design loads to determine the required flexural strengths at beam plastic hinges, which are
almost always located at the ends of the beams.
• Beam sections are designed so that the reliable flexural strength is at least equal to the factored design moment, that is,
𝜙𝑀𝑛 > 𝑀𝑢
• Once the beam is proportioned, the plastic moment strengths of the beam can be determined based on the expected material
properties and the selected cross section. ACI 318 uses the probable moment strength 𝑀𝑝𝑟 for this purpose.
• Probable moment strength is calculated from conventional flexural theory considering the as-designed cross section, using 𝜙 =
1.0, and assuming reinforcement yield strength equal to at least 1.25 𝑓𝑦 .

• The probable moment strength is used to establish requirements for beam shear strength, beam-column joint strength, and
column strength as part of the capacity-design process. Because the design of other frame elements depends on the amount
of beam flexural reinforcement, the designer should take care to optimize each beam and minimize excess capacity.

Probable Moment Strength, 𝑀𝑝𝑟
• The overstrength factor 1.25 is thought to be a low estimate of the actual overstrength that might occur for a beam.
Reinforcement commonly used in the U.S. has an average yield stress about 15 percent higher than the nominal value
(𝑓𝑦 ), and it is not unusual for the actual tensile strength to be 1.5 times the actual yield stress. Thus, if a reinforcing bar
is subjected to large strains during an earthquake, stresses well above 1.25 𝑓𝑦 are likely.
• The main reason for estimating beam flexural over-strength conservatively is to be certain there is sufficient strength
elsewhere in the structure to resist the forces that develop as the beams yield in flexure. The beam overstrength is
likely to be offset by overstrength throughout the rest of the building as well.
• The factor 1.25 in ACI 318 was established recognizing all these effects.

Beam Design Shear (in SMRFs)
• Figure illustrates this approach applied to a beam. A free body diagram of the beam is isolated
from the frame, and is loaded by factored gravity loads (using the appropriate load combinations
defined by ASCE 7) as well as the moments and shears acting at the ends of the beam.
• Assuming the beam is yielding in flexure, the beam end moments are set equal to the probable
moment strengths 𝑀𝑝𝑟 .
• The design shears are then calculated as the shears required to maintain moment equilibrium of
the free body (that is, summing moments about one end to obtain the shear at the opposite
end).
• This approach is intended to result in a conservatively high estimate of the design shears. For a
typical beam in a special moment frame, the resulting beam shears do not trend to zero near
mid-span, as they typically would in a gravity-only beam. Instead, most beams in a special
moment frame will have non-reversing shear demand along their length. If the shear does
• reverse along the span, it is likely that non-reversing beam plastic hinges will occur.
• Typical practice for gravity-load design of beams is to take the design shear at a distance d
away from the column face. For special moment frames, the shear gradient typically is low such
that the design shear at d is only marginally less than at the column face. Thus, for simplicity
Beam shears are calculated based on provided probable
the design shear value usually is evaluated at the column face. moment strengths combined with factored gravity loads.

SUMMARY – Beam Shear Demand (based on Capacity Design Approach)

Definition of M3 Plastic Hinges for Beams (Manual vs. Automatic)
Backbone Curve
All Inputs Required:
• Yield Moment (𝑀𝑦 )
• Yield Rotation (𝜃𝑦 )
• MPs: a, b, c
• AC: IO, LS, CP
𝜌−𝜌′
•
𝜌𝑏𝑎𝑙
• Shear reinforcement C/NC

Hysteresis
𝑉
•
𝑏𝑤 𝑑 𝑓′𝑐𝐸
• Hysteretic Model (and its

Acceptance Criteria
parameters)

Definition of M3 Plastic Hinges for Beams (Manual vs. Automatic)
OPTION 1
1) Define and assign new beam sections based on difference in reinforcement [to automatically determine 𝑀𝑦 for 𝑀-𝜃 curve and determine
(𝜌 − 𝜌′)/𝜌𝑏𝑎𝑙 ) factor for using in Table 10-7, ASCE 7-16].
2) Manually determine 𝑉 using Excel sheet. Run the gravity load analysis first to extract the maximum shear in all beams.
3) Select beams of same type and use Auto option to define PHs.
4) Give manually calculated 𝑉 (maximum value for one type of beams) from the Excel Sheet. [Same type = Beams having same cross-
section size, reinforcement and span (as the V value will depend on span also)].
5) Use "From Current Design" for (𝜌 − 𝜌′)/𝜌𝑏𝑎𝑙 factor.
6) Manually modify each generated PH to remove moment over-strength and assign suitable hysteretic model.
7) Repeat steps 3 to 6 for each type of beams.
OPTION 2
1) Define and assign new beam sections based on difference in reinforcement [to automatically determine 𝑀𝑦 for 𝑀-𝜃 curve].
2) For the same type of beams, manually define one PH using a, b, c, IO, LS and CP from the Excel sheet. Select “Use Yield Moment” for
the scale factor of moment. Select a suitable hysteretic model. [Same type = Beams governed by the same row in Table 10-7, ASCE 7-
16].
3) Manually select all beams which should be assigned the same 𝑀-𝜃 curve and IO, LS and CP [i.e. which belong to same row in Table 10-
7, ASCE 7-16] and assign that pre-defined PH. Software will generate PHs for all beams with same 𝑀-𝜃 curve, IO, LS and CP, and
hysteretic behaviour but the 𝑀𝑦 will be different for each single beam depending upon its reinforcement.
4) Repeat step 3 for all beam types (corresponding to same row of Table 10-7, ASCE 7-16).

Definition of M3 Plastic Hinges for Beams
OPTION 1 OPTION 2
Effort Required vs. Saved Effort Required vs. Saved
1) Define and assign new beam sections based on 1) Define and assign new beam sections based on
difference in reinforcement. difference in reinforcement.
2) Run the gravity load analysis to extract the maximum 2) Run the gravity load analysis to extract the maximum
shear in all beams. shear in all beams.
3) Fill the Excel sheet with all details of beams (cross- 3) Fill the Excel sheet with all details of beams (cross-
section, reinforcement, spans and V from gravity load section, reinforcement, spans and V from gravity load
combination) combination).
4) No need to manually define Master PHs 4) Define 1 master PH for each type of beams.
5) Selecting same type of beams (Same type = Beams 5) Selecting same type of beams [Same type = Beams
having same cross-section size, reinforcement and governed by the same row in Table 10-7, ASCE 7-
span (as the V value will depend on span also)]. 16].
6) More number of beam “types”. 6) Less number of beam “types”.
7) Give manually calculated 𝑉 while defining Auto PHs 7) No need to manually give 𝑉 while defining PHs.
8) Manually modify each generated PH to remove moment 8) No need to manually modify each generated PH to
over-strength and assign suitable hysteretic model. remove moment over-strength and assign suitable
hysteretic model.

ETABS Demonstration on
Moment-Rotation Type Plastic
Hinge Modeling of RC Beams

Plastic Hinge Modeling of RC Columns using ETABS
Option 1: Manual Definition of Hinge Properties for Every Hinge Type
Option 2: Automatic Definition of Hinge Properties by the Program

Analytical Plastic Hinge Response CASE A – [ P = 0 kN ]
600
400
Vertical Force (kN)

200
Constant Axial Force (P)
0
-200
-400
Reversed cyclic
displacement -600
-0.012 -0.008 -0.004 0 0.004 0.008 0.012
Tip Displacement (m)
CASE B – [ P = 100 kN ]
1000 mm
600
A
400
SECTION A-A
Vertical Force (kN)

(dimension in mm) 200
-200
-400
-600
-0.012 -0.008 -0.004 0 0.004 0.008 0.012
Tip Displacement (m)

Uncoupled hinges – corresponding
to each degree of freedom
Coupled hinges – These hinges yield based on the

interaction of (any combination of) axial force and bi-axial
bending moments at the hinge location

Interacting Plastic Hinges
• Interacting Hinge Types

• M2-M3 Hinge Rotation and Curvature Types
• P-M2 Hinge Rotation and Curvature Types
• P-M3 Hinge Rotation and Curvature Types
• P-M2-M3 Hinge Rotation and Curvature Type
• V2-V3 Shear Hinge (Not available in ETABS)
• Fiber P-M2-M3 Hinge
Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 50 50

Moment vs. Rotation
Behaviour for P-M2-M3
Interacting Hinges
51
Moment vs. Rotation
Behaviour
(Defined as a function of
Axial Force and Angle)
52
Hinge Interaction Surface
53
User-defined
Hinge Interaction
Surface
54
Types of P-M2-M3 Hinges
• Normally the hinge properties for each of the six degrees of freedom are uncoupled from each other. However, you
have the option to specify coupled axial-force/bi-axial-moment behavior.
• This is called a P-M2-M3 or PMM hinge. Three types are available. In summary:
a) Isotropic P-M2-M3 hinge: This hinge can handle complex and unsymmetrical PMM surfaces and can
interpolate between multiple moment-rotation curves. Two-dimensional subsets of the hinge are available. It is
limited to isotropic hysteresis, which may not be suitable for some structures.
b) Parametric P-M2-M3 hinge: This hinge is limited to doubly symmetric section properties and uses a simple
parametric definition of the PMM surface. Hysteretic energy degradation can be specified, making it more
suitable than the isotropic hinge for extensive cyclic loading.
c) Fiber P-M2-M3 hinge. This is the most realistic hinge, but may require the most computational resources in
terms of analysis time and memory usage. Various hysteresis models are available and they can be different
for each material in the hinge.

Isotropic P-M2-M3 Hinge
• This hinge can handle complex and unsymmetrical PMM surfaces and can interpolate be tween multiple moment-
rotation curves. It is limited to isotropic hysteresis, which may not be suitable for some structures.
• Three additional coupled hinges are avail able as sub sets of the PMM hinge: P-M2, P-M3, and M2-M3 hinges.

Isotropic P-M2-M3 Hinges - Interaction (Yield) Surface
• For the PMM hinge, you specify an interaction (yield) surface in three-dimensional P-M2-M3 space that represents where yielding first
occurs for different combinations of axial force P, minor moment M2, and major moment M3.
• The surface is specified as a set of P-M2-M3 curves, where P is the axial force (tension is positive), and M2 and M3 are the moments.
For a given curve, these moments may have a fixed ratio, but this is not necessary.
• The following rules apply:
• All curves must have the same number of points.
• For each curve, the points are ordered from most negative (compressive) value of P to the most positive (tensile).
• The three values P, M2 and M3 for the first point of all curves must be identical, and the same is true for the last point of all curves.
• When the M2-M3 plane is viewed from above (looking toward compression), the curves should be defined in a counter-clock wise direction.
• The sur face must be con vex. This means that the plane tangent to the surface at any point must be wholly outside the surface. If you define a surface
that is not convex, the program will automatically increase the radius of any points which are “pushed in” so that their tangent planes are outside the
surface. A warning will be issued during analysis that this has been done.
• You can explicitly define the interaction surface, or let the program calculate it using one of the following formulas:
• Steel, AISC-LRFD Equations H1-1a and H1-1b with phi = 1
• Steel, FEMA-356 Equation 5-4
• Concrete, ACI 318-02 with phi = 1
• You may look at the hinge properties for the generated hinge to see the specific surface that was calculated by the program.

Isotropic P-M2-M3 Hinges - Moment-Rotation Curves
• For PMM hinges you specify one or more moment/plastic-rotation curves corresponding to different values of P and
moment angle q. The moment angle is measured in the M2-M3 plane, where 0° is the positive M2 axis, and 90° is the
positive M3 axis. You may specify one or more axial loads P and one or more moment angles q.
• During analysis, once the hinge yields for the first time, i.e., once the values of P, M2 and M3 first reach the interaction
surface, a net moment-rotation curve is interpolated to the yield point from the given curves. This curve is used for the
rest of the analysis for that hinge.
• If the values of P, M2, and M3 change from the values used to interpolate the curve, the curve is adjusted to provide
an energy equivalent moment-rotation curve. This means that the area under the moment-rotation curve is held fixed,
so that if the resultant moment is smaller, the ductility is larger. This is consistent with the underlying stress strain
curves of axial “fibers” in the cross section.
• As plastic deformation occurs, the yield surface changes size according to the shape of the M-Rp curve, depending
upon the amount of plastic work that is done. You have the option to specify whether the surface should change in
size equally in the P, M2, and M3 directions, or only in the M2 and M3 directions. In the latter case, axial deformation
behaves as if it is perfectly plastic with no hardening or collapse.

Parametric P-M2-M3 Hinge
• This hinge is limited to doubly symmetric section properties and uses a simple parametric definition of the PMM
surface. Hysteretic energy degradation can be specified, making it more suitable than the isotropic hinge for extensive
cyclic loading.
• Two versions of the hinge are available, one for steel frame sections, and one for reinforced-concrete frame sections.
• Currently this hinge is only available in ETABS, and will be added to SAP2000 and CSI Bridge in subsequent versions.
• The description and theory for this hinge formulation are presented in the Technical Note “Parametric P-M2-M3 Hinge
Model”. This document can be found in the Manuals subfolder where the soft ware is installed on your computer. It
can be accessed from inside the software using the menu command Help > Documentation > Technical Notes.
• Detailed descriptions of the input values needed to define the properties for either the steel or concrete hinge are
available from the Help facility within the software.
• This can be accessed using the menu command Help > Product Help, or pressing the F1 key at any time.

Parametric Concrete P-M2-M3 Hinges



Automatic Definition of
Column Hinge Properties
63
ETABS Demonstration on
Moment-Rotation Type Plastic
Hinge Modeling of RC Columns

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Performance-based Seismic Design of Structures

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 5

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 6

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 7

The six degrees-of-freedom and corresponding actions for a node in 3D space.

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 8

Plastic Hinge Shear Hinges Force-Displacement Type Hinges

Moment-Rotation Type Hinges

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 9

Based on Dr. Pramin Norachan, AIT Solutions

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 11

Moment-rotation behaviour assigned

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 13

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 14

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 15

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 16

Coupled hinges – These hinges yield based on the

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 17

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 18

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 19

Source: AIT Solutions

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 20

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 21

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 22 22

• Five points labeled A, B, C, D, and E are used to define the

The acceptance criteria (capacities) marked on the force-

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 23

• Point A is always the origin

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 24

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 29

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 30

Option 1: Manual Definition of Hinge Properties for Every Hinge Type

Option 2: Automatic Definition of Hinge Properties by the Program

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 31

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 33

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 34

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 37

• Assuming beam is yielding in flexure, beam end moments are set

• Design shear is based on the probable moment to maintain the

• ACI 318 defines probable moment strength as moment strength of a

• Controlling Load Combination 1.2𝐷 + 𝐸𝑣 + 1.0 𝑜𝑟 0.5 𝐿 + 0.2 𝑆

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 38

1.0, and assuming reinforcement yield strength equal to at least 1.25 𝑓𝑦 .

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 39

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 40

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 41

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 42

• Shear reinforcement C/NC

• Hysteretic Model (and its

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 43

Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 44

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Performance-based Seismic Design of Buildings – Semester: Spring 2020 (Fawad A. Najam) 46

Option 1: Manual Definition of Hinge Properties for Every Hinge Type

Option 2: Automatic Definition of Hinge Properties by the Program

Performance-based Seismic Design of Buildings – Semester: Spring 2018 (Fawad A. Najam) 47

Vertical Force (kN)

Tip Displacement (m)