CE 5541 Advanced Structural Analysis Final
Term: Spring 2025
Instructor: Dr. Lomiento, Ph.D.
Submitted By:
Billy Jimenez (008534851)
Due date: May 16th at 5:00pm
Last Name
Jimenez Pacherrez
First Name
Billy
C
Assigned Parameters
B
A
D
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Table of Contents:
1. Nonlinear geometric analysis of a truss structure (10 points)
1.1. Introduction
1.2. Truss Model
1.3. Nonlinear analysis
1.4. Analysis Results
1.5. Discussion of results
2. Material and geometric nonlinear analysis of a Moment Resisting Frame (20 points)
2.1. Introduction
2.2. Design of the special MRF
2.2.1. Loads
2.2.2. Design of the MRF
2.4. Modes of vibration and structural design
2.5. Pushover analysis results
2.6. Energy assessment
2.7. Demand-capacity analysis
2.7.1. Description of the method
2.7.2. Application of the method
2.7.3 Discussion of the results
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Student name: Billy Joel Jimenez Pacherrez
1. Nonlinear geometric analysis of a truss structure
1.1. Introduction
A nonlinear geometric analysis accounts for the deformation of the structure as it deforms. In other words, as the
structure deforms, additional stresses are induced due to the deformation of the structure. Additional stresses can
reduce or increase the overall stress of the structure depending on the deformed shape.
Examples of real structures that would require nonlinear geometric analysis:
 Slender Piers: A bridge with tall and slender piers with substantial vertical load. During an earthwake the
vertical load will amplify the stresses in the pier due to the P-Δ effects. The P-Δ effects potentially can lead
instability or even failure.
(source: https://medium.com/@Parishith_Jayan/p-delta-effect-on-structures-2dc243fd96e7 )
 Electrical cables between electrical towers: The interaction between the deck and cables introduces
geometric nonlinearity, especially under asymmetric loading or thermal effects.
Source (https://www.linkedin.com/pulse/sagging-overhead-conductor-lines-iengineeringgroup/ )
The truss analysis is an application of nonlinear geometric analysis because each member in the truss is a
compression or tension member. When the truss deforms additional stresses develop in the compression
members, causing a p-delta effect that causes the members to fail quicker (with less load due to p-delta effect)


Explanation of how the analysis works for a truss:
In a typical truss analysis, each member of truss (diagonals, vertical members, and horizontal in between
verticals) are pinned connected at each of its joint ends. For this reason, each member in SAP 2000 model has
released from moments at each end of the member.
In addition to the releases, each of the members needs to be divided into two sub-members. This “break” the
element, creating a joint in between the members, which lets SAP 2000 know that the sub-members can deflect
(or buckle) at that joint.
It is important for the dead load in sequence prior to the superimposed load. The dead load provides the initial
deformation that the truss needs to start deformation, which causes the p-delta effect.
The truss analysis done for this report has 4 bays 10.5 feet wide each, height of truss of 3.5 feet. All top
cord/external diagonals section are HSS3x3x5/16 and the bottom cord/internal webbing sections are
HSS3x3x3/16. The steel specification for all members is ASTM A500 Grade B (Fy=46ksi)
Goals of this analysis:
The goal of this analysis is to capture non-linear behavior at members in compression and tension and double the
capacity of the original structure as a whole by strategically upsizing member failing (buckling). The non-linear
behavior of the truss will increase/decrease stresses depending on geometry of each member. For compression
members, p-delta effects increase the demand.
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1.2. Truss Numerical Model
The truss model utilizes numerical analysis iterative methods in order to analyze non-linear geometric behavior.
This truss model was analysis using the Iterative only method, Newton-Raphson method. The analysis inputs can
be seen in Figure 1.10. The number of steps in the iterative method is 100 steps with a iterative analysis tolerance
of 0.001, a maximum of 50 null steps and a final target displacement of 1 feet. Additional info:
GEOMETRIC PARAMETERS
Length of divisions
Height of truss
Top cord/external diagonals section
Bottom cord/internal webbing section
10’ 6”
3’ 6”
HSS3x3x5/16
HSS3x3x3/16
Structural layout: roof truss with 4 divisions
Dead load: only self-weight
Superimposed load: downward point loads on top joints
Controlled displacement: vertical midspan displacement
Steel: ASTM A500 Grade B
Sections: tubes as specified below
Dead load: only self-weight
Superimposed load: downward point loads on top joints
Controlled displacement: vertical midspan displacement
Steel: ASTM A500 Grade B
Sections: tubes as specified below
Fig. 1.1 Truss model with sections and releases
Fig. 1.2 Truss model with superimposed loads
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1.3. Nonlinear analysis
The truss nonlinear analysis has a total of 100 step. The analysis successfully captured 100 steps without
reaching the maximum amount of null steps. The maximum vertical displacement at the middle top chord of 1 feet
was successfully reached within the 100 iterative numerical analysis steps.
Dead Load
Fig. 1.3 Load case window
Fig. 1.4 Full load/number of steps window
Fig. 1.5 Force/Displacement control window
Fig. 1.6 Analysis parameters window
Superimposed Load
Fig. 1.7 Load case window
Fig. 1.8 Full load/number of steps window
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Fig. 1.9 Force/Displacement control window
Fig. 1.10 Analysis parameters window
1.4. Analysis results
The main results of the original analysis capture the buckling of two members, the outermost diagonals braces.
The result makes sense because the axial load distribution is the highest at the top cord and at the outermost
diagonals braces as shown on Figure 1.15. Since all the diagonals are of similar size and thickness, it makes
sense that the outer most diagonal braces buckle first.
Fig. 1.11 Deformed shape of the truss under the dead load (scale factor = 150)
Note from Billy: STEP 17 (based on element force step curve and pushover table, not really visible on scale factor= 1)
Fig. 1.12 Deformed shape of the truss that shows buckling under the superimposed load (scale factor = 1)
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Vertical, Kip
Horizontal, inches
Fig. 1.13 Static pushover curve for the truss: superimposed load on the vertical axis (kips), and vertical displacement of
the middle node on the horizontal axis (in)
Fig. 1.14 Pushover data table (circle the step at which buckling occurs)
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1.5. Discussion of results and improved design
The critical Euler buckling load is for the outer most diagonal brace HSS 3x3x5/16 is dependent of unbraced
length. The unbraced length is approximately 11 feet.
Lc/r = 11*12 / 1.08 = 122.2
= pi2*29000 ksi / 122.22 = 56.35 kips
The theoretical load of 56.35 is very similar to the results for buckling of those member from SAP2000 of 55.46.
The results are as expected since the analytical results are based on steps of iteration, which can underestimate
of overestimate the buckling load.
Fig. 1.15 Normal forces in the truss bars at buckling (kips)
Fig. 1.16 Truss model with modified sections and releases
Modified Truss
Original Truss
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Fig. 1.17 Plot of the two static pushover curves for the truss and the modified truss: superimposed load on the vertical
axis (kips), and vertical displacement of the middle node on the horizontal axis (in)
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2. Material and geometric nonlinear analysis of a Moment Resisting Frame
2.1. Introduction
Similar to section 1, nonlinear geometric analysis will increase or decrease the stresses in a structure/system as
the structure deforms while the load is applied. The nonlinear material analysis means the model captures effects
like material yielding that causes the member/structure to go through softening (loss of strength) or hardening
(gain in strength). Together, these nonlinearities provide a more realistic prediction of how a structure behaves
under large loads or displacements
Examples of real structures that require nonlinear material and geometric analysis:
 Car crash analysis: Energy absorption is greatly affected by the geometric nonlinear deformation and by the
material nonlinear hardening/softening of those members when they deform.
(Source: https://enteknograte.com/finite-element-simulation-fea-crash-test-crashworthiness/ )
 Steel RBS analysis in a moment frame: The study/testing of the energy absorption at the reduced beam
section shows how ductile this sections can be due to nonlinear material and geometric analysis.
(Source:https://www.ideastatica.com/support-center/reduced-beam-section-rbs-prequalified-connection-aisc )
The seismic analysis is a is a typical application of material nonlinear geometric analysis because it captures the
seismic load induces p-delta effects while gravity loads remain on the elements.
In a moment frame, the material and geometric nonlinearities capture p-delta effects due to gravity loads and
lateral loads. When analysis the mentioned nonlinearities, the actual behavior of the structure can be fully
captured by target earthquake/loads.
The moment frame analyzed has 3 stories, 2 bays at the middle roof bays, 4 bays at the bottom bays. The 1st
story height is 15 feet, 2nd and 3rd story are 12 feet each. Each bay is 28 feet wide. See the following data for
more info:
Steel: ASTM A992 Grade 50
Base supports: fixed ends
Dead loads (not including self-weight of the MRF): 100 psf
Roof Dead loads (not including self-weight of the MRF): 90 psf
Live loads: 60 psf
Roof Live loads: 20 psf
Tributary width for gravity loads: 24’
Tributary width for seismic masses: 48’
Column
Framing layout 3
28’
bay (girder
4 floor, 39’
span)
Location
Riverside
3 floor, 27’
Zip code
92508
th
rd
nd
2 floor, 15’
st
Soil Class
E
1 floor, 0’
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
Goals of this analysis:
The goal of this analysis is to design the frame with sizes while providing enough strength without oversizing each
member. The target DCR is from 0.5 to less than 1.0. The frame also verifies the structure first 3 modes and mass
participation ratios at each mode thru a modal analysis. A response spectrum analysis is also performed to show
the capacity of the structures at a design spectrum level. Finally, an energy comparison for the input energy and
inelastic energy is provided for reference to show the level of energy absorption in the structure.
2.2. Linear-elastic analysis and design of the special MRF

Description of the MRF numerical model showing cross sections and restraints:
The moment frame model utilizes numerical analysis iterative methods in order to analyze the linear-elastic and
also captures the non-linear geometric behavior. All the moment frame joints are moment connected, including
the foundation. This model was analysis using the Iterative only method, Newton-Raphson method. The analysis
inputs can be seen in the following sections. See the initial analysis cross section and the final cross section with
DCR can be seen in the following figures.
Fig. 2.1 SAP model with original cross sections (before design iterations)
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2.2.1. Loads
The dead loads, live loads at the floor and roofs were calculated based on the parameter shown on Section 2.1.
The results of the line loads are show on the following sections. Note that the structure has roofs at two levels,
level 3 and level 2.
Loads
2.2 Dead load assignment
2.3 Live load assignment
Note: all dead loads were combined in one load
combination purposely.
2.4 Roof Dead load assignment
2.5 Roof Live load assignment
2.6 Response spectrum function from SAP
2.2.2. Special MRF design
 Description of the design process, present all the seismic parameters, and discuss DCR values:
The design process was iterative process. The Beam members were design first to address the gravity loads and
moments. The columns were designed to address the lateral loads as a governing load compared to gravity
loads. Lastly, each member at the structure was optimized to conform to the AISC member ratio requirements
such as weak beam -strong column. The final members and ratios are shown on Figure 2.8.
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2.7 Response spectrum load case window
(with an appropriate scale factor)
2.8 DCR color map for final design
2.9 Screenshot of the SAP model with final cross sections
2.10 Screenshot of extruded view of the SAP model with
final cross sections
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2.4. Modes of vibration and structural design
A modal analysis is used in order to determine a structure’s natural vibration modes, including its natural
frequencies, mode shapes, mass participation ratios and periods of vibration. The mass of the structure tends to
increase the period (slower vibration), while increase of stiffness tends to decrease the period (faster vibration).
The period is proportional to the square root of mass over stiffness.
The modal analysis results are as expected since the 1 st mass participation of 84% and a period of 1.353
seconds.
See the following figure for mass assignments, shape, and period of the first three fundamental modes, and mass
participating factors for each mode.
2.11 Mass assignment
2.12 Shape Mode 1 – Period T = 1.353
2.13 Shape Mode 2 – Period T = 0.54211
2.14 Shape Mode 3 – Period T = 0.2196
2.15 Table with mass participating ratios
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2.5. Nonlinear frame model
Discussion of plastic hinge definition and assignment to the MRF:
A plastic hinge model with lumped plasticity concentrates all nonlinear deformations in a single hinge location.
Typically at the single hinge model is assigned at the member ends, while the rest of the member remains elastic.
In a good MRF structure, the sequence of hinges is: beam ends, then column beam connections, lastly the base
of the column is the least preferred hinge location since it compromises the stability of the structure.
The difference between the length of plastic hinge and location of hinge is that location can be placed anywhere
along a member, while the hinge length is solely related to the hinge length. Both are difference and cannot be
used interchangeably.
The relative/absolute location for the plastic hinges assigned to one girder and one column are shown on the
following pages and figures.
Table 2.1. Plastic hinge properties
Number Plastic hinge
Plastic Hinge
label
Relative
(SAP) Used
1
FH1
0.05
Plastic hinge length
(absolute, in inches)
0.05 * 28’ *12 = 16.8”
Element label which
the hinge is assigned
to
Beam element B1
2
FH2-Col
0.05
0.05* 15’ *12 = 9”
1st Story Column C1
3
FH3-Col
0.05
0.05* 12’ *12 = 7.2”
2st Story Column C2
4
FH4-Col
0.05
0.05* 12’ *12 = 7.2”
3rd Story Column C3
Plastic hinge location
(absolute, in inches)
6 inches from node i
6 inches from node j
0 inches from node j
7 inches from node j
7 inches from node j
6 inches from node j
7 inches from node j
6 inches from node j
2.16 Windows with plastic hinge definition for columns
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2.17 Windows with plastic hinge definition for girders
2.18 Windows for plastic hinge location for a girder and a column
2.19 SAP model with plastic hinges
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Dead Load
2.20 Load case window
2.21 Full load/number of steps window
2.22 Force/Displacement control window
2.23 Analysis parameters window
Lateral Load
2.24 Load case window
2.25 Full load/number of steps window
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2.26 Force/Displacement control window
2.27 Analysis parameters window
Discussion of pushover analysis results:
 Identify in the following figure at what level of displacement the first plastic hinge form, and if/when the MRF
tangent stiffness becomes null, if/when the structure undergoes softening:
As shown on Figure 2.28, structure experiences softening after the first hinge forms.

Show the plastic hinges in the structure at the peak (ultimate) strength, and discuss if the weak-beam strongcolumn behavior was achieved:
Figure 2.29 shows the plastic hinge in the structure peak strength at step 20. The weak-beam strong column
behavior was achieved. Although the hinge was developed in the beam-column interface, the column was
located at the roof level. A hinge failure at the base of the column is not developed, which is structurally
acceptable.

Determine what is the available displacement ductility of the structure as the ratio between ultimate-displacement
(at peak strength) and displacement at first yielding (first plastic hinge formation):
The displacement ductility is calculated by ratio of ratio between ultimate-displacement (at peak strength) and
displacement at first yielding (first plastic hinge formation). The displacement ductility is 43.5mm/12.7mm = 3.4.
Notes: The structure experiences softening after the first hinge.
2.28 Static pushover curve for the MRF: lateral load on the vertical axis (kips), and roof displacement on the horizontal
axis (in)
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STEP 20
2.29 Screenshot of the deformed shape of the model with plastic hinges at the ultimate (peak) strength
2.6. Energy assessment
2.30 Input energy and inelastic energy vs roof
displacement (absolute values in k-in)
2.31 Input energy and inelastic energy vs roof
displacement (relative values, normalized to input energy)
2.7. Demand-capacity analysis
2.7.1. Description of the method
Explanation of the ADRS method to find the performance point:
The ADRS (Acceleration Displacement Response Spectrum) method finds the performance point by plotting a
structure’s first-mode pushover curve (capacity) in spectral acceleration vs. displacement. Then the roof
displacement is converted to spectral displacement using modal participation factors. A constant damping
spectrum assumes a fixed damping ratio (say 5%), while a variable damping spectrum adjusts the spectrum
based on the structure’s ductility. The variable damping is more accurate demand and compared to the inelastic
response.
2.7.2. Application of the method:
The design spectrum analysis was performed. No amplification or reduction on the response spectrum. MCE and
the serviceability earthquake checks are not performed.
Seismic performance point info:
Step: 9 (see figure 2.33)
Roof displacement: 11.05 mm
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2.32 Screenshot of results window showing the seismic performance point
2.7.3 Discussion of the results:
See below for deformed shape of the model with plastic hinges at the performance point. The level of
performance achieved for the design spectrum is Life-Safety (LS), according to Vision 2000 performance levels.
This is based in the “Collapse Margin Ratio”. The CMR is the ratio between ultimate displacement (roof
displacement at the max strength ie; the null tangent stiffness point) and roof displacement at the performance
point.
CMR = 31”/11.065” = 2.8.
 The structure is life-safe if the Collapse Margin Ratio is ≥1.5.
 The structure is preventing collapse if the displacement safety factor is ≥1.25.
 The structure can be immediately occupied if the displacement safety factor is ≥3.
Deformation with Cd at Response Spectrum from Riverside: 2.18” *5.5 = 12” (see figure 2.33)
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2.33 Screenshot of the Quake Riverside Maximum Elastic Deformation at Roof
2.34 Screenshot of the deformed shape of the model with plastic hinges at the seismic performance point
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