SPRING 2016
LINEAR DYNAMIC ANALYSIS AND SEISMIC EVALUATION OF A FULL-SCALE RC MODEL Earthquake-Resistant Design
Abhinanda Dilip, Abhishek Salkar, Qudsia Wahab
University of California, Berkeley
1
Table of Contents ABSTRACT ................................................................................................................................................. 2
INTRODUCTION ....................................................................................................................................... 2
PAST RESEARCH ..................................................................................................................................... 3
PROBLEM STATEMENT AND OUTLINE ........................................................................................... 4
DETAILS OF FULL-SCALE MODEL TESTED ON SHAKE TABLE ............................................... 6
Materials Used .................................................................................................................................... 7
Setting of Specimen ............................................................................................................................ 7
MODELING IN SAP2000 .......................................................................................................................... 8
SMRF .................................................................................................................................................. 8
Shear Walls ......................................................................................................................................... 9
Loading ............................................................................................................................................... 9
RESPONSE-SPECTRUM ANALYSIS IN SAP2000 ............................................................................. 12
CALCULATIONS AND RESULTS........................................................................................................ 16
Girder 1 (2nd Floor) .......................................................................................................................... 16
Column 1 (1st Floor) ......................................................................................................................... 19
Exterior Joint Shear (1st Floor) ....................................................................................................... 22
Wall (1st Floor) ................................................................................................................................. 24
SERVICEABILITY CHECKS ................................................................................................................ 26
Special Moment Resisting Frames.................................................................................................. 28
Special Shear Walls.......................................................................................................................... 33
PERFORMANCE OF FULL-SCALE MODEL TESTED ON SHAKE TABLE ............................... 36
Performance of Columns on 1st floor ............................................................................................. 36
Performance of Girders and Joints on 2nd, 3rd and 5th floors ................................................... 36
Performance of Shear Wall on 1st Floor......................................................................................... 38
REFERENCES .......................................................................................................................................... 39
ACKNOWLEDGMENTS ........................................................................................................................ 39
APPENDIX ................................................................................................................................................ 40
2
Linear Dynamic Analysis and Seismic Evaluation
of a Full-Scale RC Model Abhinanda Dilip1, Abhishek Salkar1 and Qudsia Wahab1
1Graduate Student, Department of Civil and Environmental Engineering, University of California, Berkeley,
Berkeley, CA 94720
ABSTRACT
It is a well known fact that Japan is one of the most seismically active countries in the world.
There have been 11 earthquakes with a moment magnitude of over 7.0 since 2010 in Japan. As a
result, Japan invests significant amount of resources in advancing knowledge and technology in
the area of seismic design of structures. In this project, a ten-story structure was modeled in
SAP2000. The structure had SMRFs in the longer direction and RC shear walls in the shorter
direction as its lateral force resisting systems. The structure was modeled to represent the
behavior of a full scale model that was tested on the E-Defense Shake Table at the Hyogo
Earthquake Engineering Research Center, Japan. Capacities and demands were obtained at
critical members and joints using ACI 318-11 and SAP2000 respectively. Time period of the
structure and other important parameters were obtained using ASCE 7-10. Appropriate spot
checks were applied to determine if the structure conformed to the ASCE 7-10 requirements and
the shake table test results were compared to SAP2000 results.
Keywords: Seismic Design, ASCE 7-10, ACI 318-11, RC Building, SMRF, Special Shear Wall
INTRODUCTION
Earthquake engineering is an interdisciplinary branch of engineering that designs and analyzes
structures, such as buildings and bridges in order to make them more resistant to earthquakes.
Early stages of earthquake engineering and earthquake resistant design were shaped by major
earthquakes that occurred in the early 20th century such as the 1906 San Francisco earthquake
(Figure 1-a) in the United States, the 1908 Messina earthquake in Italy and the 1923 Kanto
earthquake (Figure 1-b) in Japan [1]. The immediate reaction of structural engineers to the idea
of earthquake resistant design was to approximate the seismic action by static horizontal forces
that will be resisted elastically. The first such report was prepared by the commission formed by
the Italian government after the 1908 Messina earthquake and recommended designing the
building to withstand a static horizontal force of approximately 10% of the total building weight.
This method was then adopted by building codes worldwide. The Los Angeles Building Code
adopted in 1943 addressed the importance of the height of the building for estimating the seismic
response for the first time. Thus, a "building flexibility" associated with the number of stories
was introduced. In 1952, 20 years after the development of the concept of response spectra, the
period of vibration of the building was introduced as a means of determining the base shear
coefficient. In 1957, a coefficient considering the inherent ductility and energy dissipation
characteristics of structures was introduced in the base shear equation. It was not before 1978
that seismic hazard was explicitly considered in the seismic design recommendations, namely by
3
Cornell who developed contour maps for effective peak acceleration (EPA) and peak velocity
(EPV). A period-dependent lateral force coefficient based on the ground motion spectrum was
proposed to be used in the structural design similar to the design spectrum that is used today.
Further, the introduction of a response modification factor permitted the use of an elastic force
design that is expected to respond inelastically.
Figure 1 (a) 1906 San Francisco Earthquake and (b) 1923 Kanto Earthquake
The US Geological Survey (USGS) estimates that several million earthquakes occur in the world
each year. Many earthquakes go undetected because they hit remote areas or have very small
magnitudes [2]. Experience has shown that reinforced concrete structures have great advantage
in such situations. The successful performance of a large number of reinforced concrete
buildings in earthquake zones in the U.S. and around the world has proved that it is possible to
design structures with the resilience to withstand earthquakes of relatively high magnitude. It is
uneconomical to design a structure to respond in the elastic range to the inertial forces caused by
the maximum considered earthquake. Accordingly, the design seismic lateral forces prescribed in
ASCE 7-10 are less than the elastic response inertial forces caused by the intended design
earthquake. The purpose of these detailing and proportioning requirements is to avoid all forms
of brittle failure and insures that the structure will have sufficient inelastic deformability. This is
to enable the structure to survive without collapse when subjected to several cycles of loading
within the inelastic range. ASCE 7-10 will be extensively used ahead in this project.
PAST RESEARCH
The ten story structure considered in this project consists of RC SMRF’s and RC Special Shear
Walls in the longer and shorter directions, respectively. Reinforced concrete special moment
frame concepts were introduced in the U.S. starting around 1960 [3]. It was not until 1973 that
the Uniform Building Code first required use of the special frame details in regions of highest
seismicity [4]. The earliest detailing requirements are remarkably similar to those in place today.
Studies related to modeling of reinforced concrete walls date back to the 1970s. Cervenka and
Gerstle (1971, 1972) tested a number of RC shear walls under monotonic and cyclic loading.
They also developed a model for reinforced concrete based on these experimental results.
4
PROBLEM STATEMENT AND OUTLINE
The main objective of this project is to model a ten-story structure to represent a full-scale model
which was tested on the E-Defense Shake Table in Japan and check if it conforms to the ASCE
7-10 and ACI 318-11 requirements. The procedure for this project has been briefly summarized
below (Figure 2).
Figure 2 – Procedure followed for this project
A report was obtained using the USGS website. The DBE and MCE response spectra (Figures 3
and 4) along with other important information from the report have been summarized below:
The structure was modeled using SAP2000 and non-linear geometry effects were taken into consideration.
The structure was assumed to be located at 1916 Shattuck Avenue at Berkeley in order to represent a seismic zone similar to Japan.
The response spectra, peak ground acceleration and other important parameters were obtained using the USGS website. The soil was assumed to be stiff (Site D).
The time period of the structure, importance factor, risk category, seismic design category and necessary information was obtained using the ASCE7-10 requirements.
Capacities of critical members and joints were obtained using ACI 318 code provisions and compared to demands obtained from SAP2000 analysis to determine
if the structure was adequate. ASCE 7-10 requirements were used.
The results from SAP2000 analysis were also compared to the results from the shake table test.
5
Table 1 – USGS Summary of Parameters
SMS 2.373g
SM1 1.480g
SDS 1.582g
SD1 0.987g
TL 8 seconds
PGA 0.914g
Figure 3 – Design Response Spectrum
Figure 4 – Maximum Considered Earthquake (MCER) Response Spectrum
6
DETAILS OF FULL-SCALE MODEL TESTED ON SHAKE TABLE
A full scale model of the ten story RC structure was tested on the E-Defense Shake Table at the
Hyogo Earthquake Engineering Research Center, Japan in 2010. The structure remained stable
after the test but sustained severe damage. The lateral force resisting system of the structure
consisted of SMRF’s in the longer (Y) direction and special shear walls up to the first 7 stories in
the shorter (X) direction. The total floor area is 1297m2. The plan and elevation of the structure
have been shown below (Figure 5) along with the loading on the structure (Table 2).
Figure 5 – Plan and Elevation of the Structure
Table 2 - Weight and Seismic Loads
7
Materials Used
The materials used for the structure have been shown below (Table 3 and Figure 3).
Table 3 (a) – Concrete and (b) - Steel
Figure 6 (a) – Concrete and (b) - Steel
Setting of Specimen
The first six floors and the rest of the floors (floor 7 to roof) were constructed separately as two
different specimens. These two specimens were taken to the testing site and connected together
on the shake table as illustrated below (Figure 7).
Figure 7 – Two separate specimens of the structure to be connected on shake table
8
MODELING IN SAP2000
A finite element model of the building was generated in SAP2000 and analyzed under
earthquake forces. The first floor of the building was not included in the model as it had no
contribution to the inertia forces. The supports were defined to be restrained against translation
and rotation in all three directions. All connections except gravity-only beams were modeled as
fixed connections as shown below (Figure 8).
Figure 8 – (a) Plan View and (b) Elevation (SMRF and Shear Wall Direction)
SMRF: The SMRF was modeled with frame elements and were meshed at intermediate joints
and at frame intersections. The local axes of the members were oriented such that axis 3-3 was
the major bending axis. Sections were defined from the structural drawings and material
properties were chosen according to the story levels (Figure 9 (a)). The insertion point of the
beams was set at the top-centre of slab to include the slab in the effective flange width (Figure 9
(b)). End length offsets were defined from connectivity with a rigid zone factor of 1.0 and panel
zones were defined at the joints. Flexural stiffness modifiers of 0.7 and 0.5 were assigned to the
columns and the beams respectively, to account for cracking.
9
Figure 9 – (a) Typical Column Section and (b) Frame Insertion Point
Shear Walls: The shear walls were modeled as thin shell elements with both membrane and
plate bending characteristics. The transition between the shear wall and moment frame at the 8th
story level was captured by modeling a frame element at the transition line. The shell elements
were meshed using automatic area meshing at edge points and edge constraints were generated.
Stiffness modifiers of 0.5 and 0.7 were assigned for the membrane and bending behaviors,
respectively. In order to constrain the rotation of the beams framing into the walls, rigid links
were defined at the boundary of vertically adjacent wall segments using frame elements with
flexural rigidity of 100 times that of the other beam elements (Figure 10 (a)).
Figure 10 – (a) Rigid Links at Shear Walls and (b) Mass Source Definition
Loading: The element self-weight was included in the dead load pattern. Additional joint loads
were defined under dead load pattern to include the weight of the cantilevered slab edges. Live
loads were added as surface loads on the slab elements. For modal analysis, all loads under the
10
dead load pattern and and 50% of loads under live load pattern were defined in the mass source
(Figure 10 (b)). The response spectrum for DBE intensity level was generated from USGS
website and the parameters including soil type were defined in the response spectrum function.
SAP2000 generates the response curve (Figure 11). Load cases were defined for modal analysis,
combined modal response spectrum analysis in the X( shear wall) and Y (SMRF) directions
(Figure 12). Nonlinear geometry effects were also included by using the stiffness matrix at the
end of P-Δ geometry under dead load, for all load cases (Figure 13).
Figure 11 – Response Spectrum Definition in SAP2000
Figure 12 – Definition of Load Cases Figure 13 – Inclusion of non-linear geometry effects
11
For the response spectrum cases, the spectrum was scaled using appropriate reduction factor and
scaling factor to match with the base shear obtained from equivalent lateral force procedure in
both directions (Figure 14).
Figure 14 – Scaling of Response Spectrum in X and Y Directions
Load combinations were defined as per ASCE 7-10 including effects of seismic forces
simultaneously occurring in orthogonal directions. Envelope case was defined to obtain the
critical maximum and minimum analysis results (Figure 15).
Figure 15 – Defining Load Combinations in X and Y Directions as per ASCE 7-10
12
RESPONSE-SPECTRUM ANALYSIS IN SAP2000
Response-spectrum analysis (RSA) is a linear-dynamic statistical analysis method which
measures the contribution from each natural mode of vibration to indicate the likely maximum
seismic response of an essentially elastic structure. It is the representation of the maximum
response of idealized single degree freedom system having certain time period and damping
during earthquake ground motions [5]. The maximum response plotted against un-damped
natural period for various damping values can be expressed in terms of maximum absolute
acceleration, maximum relative velocity or maximum relative displacement. Response-spectrum
analysis provides insight into dynamic behavior by measuring pseudo-spectral acceleration,
velocity, or displacement as a function of structural period for a given time history and level of
damping. It is practical to envelope response spectra such that a smooth curve represents the
peak response for each realization of structural period. Response-spectrum analysis is useful for
design decision-making because it relates structural type-selection to dynamic performance.
Structures of shorter period experience greater acceleration, whereas those of longer period
experience greater displacement. Structural performance objectives should be taken into account
during preliminary design and response-spectrum analysis. The main limitation of response
spectra is that they are only universally applicable for linear systems. Response spectra can be
generated for non-linear systems, but are only applicable to systems with the same non-linearity.
As it has been mentioned earlier, the ten story structure consists of SMRF’s in the Y (longer)
direction and Special RC Shear Walls up to the seventh story in the X (shorter) direction as its
lateral force resisting systems. In order to determine if the lateral force resisting system in the X
direction behaved like a dual system or not, earthquake loading was applied in the X direction.
The columns in the X direction were found to take 15% of the base shear. ASCE 7-10 [6]
Section 12.2.5.1 provisions state that the moment frames should be capable of resisting at least
25% of the design seismic forces in order for the system to be considered as a dual system.
Hence, the lateral force resisting system in the X direction was treated as a Bearing Wall System.
The Mass Source was defined to include 100% of the dead load and 50% of the live load. The
dead load only included the self weight of the structure while the live load was defined under
load patterns. Once the mass source was defined, modal analysis of the structure was performed
in SAP2000 in X and Y directions.
The fundamental period of the structure was found to be 0.538 seconds in the X direction (Figure
16 (a)) and 0.947 seconds in the Y direction (Figure 16 (b)). The mass participation of the first
mode was found to be 73.43% and 78.25% in the X and Y directions, respectively. As we see,
the mass participation in both directions was less than 90% as required by Section 12.9.1 ASCE
7-10. A total of 60 modes were considered for analysis such that the mass contribution added up
to 99% of the mass of the structure as required for this project.
13
Figure 16 - (a) Deformed Shape in the fundamental mode in X-direction and (b) Deformed
Shape in fundamental model in Y- direction
Section 12.8.2 ASCE 7-10 Code Provisions were used in order to determine the fundamental
periods and the upper limits of time periods that could be considered for the structure. Section
12.8.2.1 of ASCE 7-10 states that the fundamental period of the structure is given by:
Ta = Cthnx (1)
Where
hn = structural height of the structure
Ct and x = Period parameters obtained from Table 12.8.2 of ASCE 7-10 (shown below)
14
Using the equation (1), the fundamental periods in the two directions were found to be:
X-direction Ta = 0.04880*(27.45)0.75 = 0.585 sec
Y-direction Ta = 0.04660*(27.45)0.90 = 0.918 sec
Section 12.8.2.1 of ASCE 7-10 also states an upper limit on the fundamental periods that can be
used for a structure given by:
Tlimit = CuTa (2)
Where,
Ta = Fundamental Period of structure obtained from the above equation
Cu = Coefficient for upper limit on calculated period obtained from Table 12.8.1 of ASCE 7-10
(shown below)
Using equation (2), the upper limits for calculated periods in the two directions were found to be:
X-direction Tlimit = 1.4 * 0.585 = 0.819 sec
Y-direction Tlimit = 1.4 * 0.918 = 1.285 sec
All the values of fundamental periods obtained in SAP2000 and ASCE7-10 (including the upper
limits) have been summarized below (Table 4)
Table 4 – Comparison of Fundamental Periods from SAP2000, drawings and ASCE 7-10
Ta (ASCE 7-10) T (drawings) T (SAP2000) Tlimit (ASCE 7-10)
X-direction 0.585 sec 0.610 sec 0.538 sec 0.819 sec
Y-direction 0.918 sec 0.800 sec 0.947 sec 1.285 sec
It is observed that all the calculated fundamental periods are below the upper limits obtained
from ASCE 7-10. Hence, the time periods obtained from SAP2000 analysis were used.
The base shears were found to be 1784.43 kN and 804.11 kN in the X and Y directions,
respectively. Section 12.8 of ASCE 7-10 was used to calculate the seismic base shear using the
Equivalent Lateral Force Method. The seismic base shear is given by:
V = CsW
15
Where
V = Seismic base shear
W = Weight of the structure (7683.19 kN)
Cs = Base Shear Coefficient given by
Cs = (𝑆𝑑𝑠
𝑅
𝐼𝑒
)
The building falls under risk category II according to Table 1.5-1 of ASCE 7-10. The importance
factor, Ie of the building was determined to be 1.00 from Table 1.5-2 of ASCE 7-10. The
response modification factor, R was found to be 5 and 8 in the X and Y directions, respectively
from Table 12.2-1 of ASCE 7-10.
The upper and lower limits of Cs are given by:
Cs, max = 𝑆𝑑1
𝑇(𝑅
𝐼𝑒) for T < TL = 8sec
Cs, min1 = 0.5𝑆1
(𝑅
𝐼𝑒)
Cs, min2 = 0.044SDSIe > 0.01
The base shear values obtained from the above equations using ASCE 7-10 have been
summarized below (Table 5).
Table 5 – Summary of Base Shear Calculations using ASCE 7-10
T (sec) Cs Cs,max Cs, min V=CsW
SMRF 0.947 0.1978 0.1303 0.0690 1000.98 kN
Shear Wall 0.538 0.3164 0.3670 0.0987 2430.96 kN
The base shear values obtained using ASCE 7-10 were compared with the base shear values
obtained from SAP2000 analysis. The comparison has been summarized below (Table 6).
Table 6 – Comparison of Base Shear Values from SAP2000 and ASCE 7-10
V (ASCE 7-10) V (SAP2000)
SMRF 1000.98 kN 780.5850 kN
Shear Wall 2430.96 kN 1754.678 kN
As it can be seen, the values obtained from SAP2000 were found to be lower than those of the
code. Therefore, the values obtained from analysis were increased by a factor of 1.2448 and
1.3623 for SMRF and Shear Wall directions, respectively. The building was analyzed for these
forces and member forces were obtained for the critical load combinations. A few members were
picked and demand versus capacity ratios were checked to understand the strength level of the
building per ASCE and ACI 318 provisions.
16
CALCULATIONS AND RESULTS
Girder 1 (2nd Floor)
The girder spans the EW direction with a span of 4000 mm. SAP2000 is used to find the design
moment, Mu. In SAP2000, the girders of the moment frame are modeled with no moment
releases at column intersections. The girders are designed using a rectangular cross-section
(Figure 17 (a)) ignoring any reinforcement in the slab, which gives a conservative result. The
following calculations adhere to ACI 318-11 [7] and ASCE 7-10.
Moment:
Design at Support (Rectangular Cross-Section):
Given From SAP2000:
From load combinations, the maximum negative moment is 304 kN-m
From load combinations, the maximum positive moment is 290 kN-m
Reinforcement:
Longitudinal reinforcement – 22 mm dia. bars
Transverse reinforcement – 10 mm dia. stirrups
𝜑 = 0.9 (moment)
𝜑 = 0.75 (shear)
∈𝑠 ≥ ∈𝑦
Parameters:
𝑓𝑐′ = 42 𝑀𝑃𝑎 → 𝛽1 = 0.745 (Figure 17 (b))
𝑓𝑦 = 345 𝑀𝑃𝑎
𝑓𝑦𝑡 = 295 𝑀𝑃𝑎
𝐸𝑠 = 200000 𝑀𝑃𝑎
∈𝑐 = 0.003
𝑑 = 550 − 61 = 489 𝑚𝑚
ℎ = 550 𝑚𝑚
𝑏 = 350
Cover = 61 mm
Figure 17 – (a) Rectangular cross section, (b) 𝛽 values from concrete compressive strength
17
Rectangular Design:
∈𝑠
(𝑑−𝑐)=
∈𝑐
𝑐 → ∈𝑠= 0.003 (
𝑑
𝑐− 1)
∈𝑠′
(𝑐−𝑑′)=
∈𝑐
𝑐 → ∈𝑠 ′ = 0.003 (1 −
𝑑′
𝑐)
𝑇𝑠 = 𝐴𝑠𝑓𝑦
𝐶𝑐 = 0.85𝑓𝑐′𝑏𝛽1𝑐
𝐶𝑠′ = 𝐴𝑠′ 𝐸 ∈𝑠
o +↺ ∑ 𝐹 = 0 → 0.85𝑓𝑐′𝑏𝛽1𝑐 + 𝐴𝑠′𝐸 ∗ 0.003 (1 −
𝑑′
𝑐) − 𝐴𝑠𝑓𝑦 = 0
o 9314 c2 +615815 c – 80968267 =0
o c = 65.9 mm
Check:
Moment:
o + ↺ ∑ 𝑀 = 0
o 𝑀𝑛 = 0.85𝑓𝑐′𝑏𝛽1𝑐 ∗ (𝑑𝑠 − 0.425𝑐) + 𝐴𝑠′𝐸 ∈𝑐 (1 −
𝑐𝑜𝑣𝑒𝑟
𝑐) (𝑑𝑠 − 𝑐𝑜𝑣𝑒𝑟)
o 𝑀𝑛+= 246 𝑘𝑁 − 𝑚
o 𝑀𝑛−= 303 𝑘𝑁 − 𝑚
o 𝑀𝑢 > 𝜑 𝑀𝑛 Girder is NOT adequate for flexure
Phi Factor:
o ∈𝑠
(𝑑−𝑐)=
∈𝑐
𝑐 → ∈𝑠= 0.003 (
𝑑
𝑐− 1) = 0.0188
o ∴ ∈𝑠 > 0.005 → 𝜑 = 0.9
Shear:
Given From SAP2000:
Vu = 𝑀𝑝𝑟1+𝑀𝑝𝑟2
𝐿𝑛+
𝑤𝐿𝑛
2 = 228 kN.
Vu ≤ 𝜑(𝑉𝑠 + 𝑉𝑐)
o 𝑉𝑐 = 2√𝑓𝑐′𝑏𝑤𝑑
o 𝑉𝑠 = 𝐴𝑣 𝑓𝑦 𝑑𝑠
𝑠
o 𝑉𝑛 = 𝑉𝑠 + 𝑉𝑐
o 𝜑𝑉𝑛 = 347 𝑘𝑁
o 𝑉𝑢 ≤ 𝜑 𝑉𝑛 Girder is adequate for shear
The calculations were done in excel. The rest of these calculations have been provided in the
Appendix. The results from SAP2000 analysis for Girder 1 have been shown below (Figure 18).
The demands capacity ratios were then obtained and have been summarized in Table 7.
18
Figure 18 – SAP2000 results for girder G1
Table 7 – Demand Capacity Ratios for Flexure and Shear
D/C FOR MOMENT AND SHEAR
Mu/ΦMn (+) Mu/ΦMn (-) V
2nd Floor
G1 1.31 1.12 0.66
G2 0.99 1.04 0.66
3rd Floor
G1 1.31 1.12 0.76
G2 1.02 1.07 0.76
5th Floor
G1 1.24 1.33 0.86
G2 1.21 1.03 0.77
As it can be seen from the above table, most of the D/C in flexure are greater than 1, which
means that they don’t meet the demand requirements as per ACI 318. D/C ratios for shear are all
lower than 1, and therefore the girders are sufficient in shear.
19
Column 1 (1st Floor)
The column considered is C1 on grid A1 and spans from level 1 to 2. The size of the column is
550 mm x 550 mm and its height is 2800 mm. SAP2000 is used to find the design moment, Mu
and axial force, Pu. SAP2000 is also used to generate a P-M interaction diagram (Figure 19) for
the cross section. The following calculations adhere to ACI 318-11 and ASCE 7.
Reinforcement:
Longitudinal reinforcement – (20) 22 mm dia. bars
Transverse reinforcement – 10 mm dia. Hoops @ 100 mm o.c.
Parameters:
𝑓𝑐′ = 42 𝑀𝑃𝑎
𝑓𝑦 = 345 𝑀𝑃𝑎
𝑓𝑦𝑡 = 295 𝑀𝑃𝑎
𝐸𝑠 = 200000 𝑀𝑃𝑎
Figure 19 - P-M Interaction Diagram of the Column
As it can be seen in the graph, some of the demand points lie outside of the P-M interaction
diagram, suggesting that the column does not have adequate capacity to resist applied loads.
(Pu, Mu) > Φ (Pn, Mn)
-4000
-2000
0
2000
4000
6000
8000
10000
12000
0 200 400 600 800 1000 1200
Axi
al F
orc
e, P
(kN
)
Moment, M (kN-m)
C1: P-M Interaction Diagram
Pu,Mu
ΦMn, ΦPn
Pn,Mn
20
Column Shear Calculations
Vu ≤ Φ Vn
- Φ = 0.75
- Vn = Vc + Vs
𝑉𝑐 = 2 (1 +𝑁𝑢
2000𝐴𝑔) 𝜆 √𝑓 ′𝑐 𝑏𝑑
𝑉𝑠 = 𝐴𝑣𝑓𝑦𝑡 𝑑
𝑠
Φ Vn = 464 kN
Vu = 244 kN
Vu < Φ Vn Column is adequate for shear
The calculations were done in excel. The rest of the calculations have been provided in the
Appendix. The SAP2000 analysis for Column 1 has been shown below (Figure 19). The demand
capacity ratios for axial-moment and shear were then obtained and have been summarized below
(Table 8)
Figure 19 – SAP2000 analysis of column C1
Table 8 – Demand Capacity Ratios for Axial, Moment and Shear
D/C FOR AXIAL-MOMENT
AND SHEAR
P-M V
C1 1.16 0.42
C2 1.51 0.70
21
SAP2000 analysis results for column 1 have been illustrated below (Figures 20 and 21).
Figure 20 – Axial Force Results for Column C1 in SAP2000
Figure 21 – Bending Moment and Shear Force Results for Column C1 in SAP2000
22
Exterior Joint Shear (1st Floor)
Column Shear
𝑀𝑝𝑟𝑔2 + 𝑀𝑝𝑟𝑔1 + ((𝑀𝑝𝑟𝑔1 + 𝑀𝑝𝑟𝑔1
𝐿𝑛) ∗
𝑑𝑐
2+ (
(𝑀𝑝𝑟𝑔2 + 𝑀𝑝𝑟𝑔2
𝐿𝑛) ∗
𝑑𝑐
2− 𝑉𝑐𝑜𝑙 ∗ 𝐻 = 0
Where,
Mprg1 - Probable moment of girder 1 (G1)
Mprg2 - Probable moment of girder 2 (G2)
Ln – clear span of the beam
H – Height of the column between inflection points
Vcol = 323 kN
Joint Shear
Vcol – Vjoint - –𝑀𝑝𝑟𝑔1
𝑑𝑔1+
𝑀𝑝𝑟𝑔2
𝑑𝑔2 = 0
Demand on the joint Vu = 1397 kN
ΦVn = Φ 𝛶√𝑓′𝑐 𝐴𝑗𝑜𝑖𝑛𝑡
Φ = 0.85
𝛶 = 15 (exterior joint), 12 (corner joint)
23
Capacity of the joint Vn = 2075kN
Demand < Capacity Joint is adequate
Calculations were done in excel. Please refer to the Appendix for these calculations.
The Vcol, Vjoint and D/C for exterior and corner joints have been summarized below (Table 9).
Table 9 - Joint Shear Summary
JOINT DESIGN
Vcol (kN) Vjoint (kN) ΦVn (kN) D/C
Exterior 323 1397 2075 0.67
Corner 161 699 1660 0.42
As seen from the table, the demand capacity ratios were found to be comfortably less than 1.
This suggests that the joints are adequate for resisting the loads. The location of the joint which
was considered for calculations illustrated above has been shown below (Figure 22)
Figure 22 – Location of the joint considered for illustrated calculations
24
Wall (1st Floor)
The wall cross section considered is W23 on grid B-C between levels 1 and 2. The length of the
wall is 1800 mm and its thickness is 230 mm. SAP2000 is used to find the design moment, Mu
and axial force, Pu. Xtract and Excel are used to generate a P-M Interaction Diagram (Figure 23)
for the wall. The following calculations adhere to ACI 318-11 and ASCE 7.
Reinforcement:
Boundary
Longitudinal reinforcement – (8) 19 mm dia. bars
Transverse reinforcement – 10 mm dia. Hoops @ 100 mm o.c.
Distributed
Longitudinal reinforcement – 13 mm dia. Bars @ 250 mm o.c.
Transverse reinforcement – 10 mm dia. Bars@ 150 mm o.c.
Parameters:
𝑓𝑐′ = 42 𝑀𝑃𝑎
𝑓𝑦 = 345 𝑀𝑃𝑎
𝑓𝑦𝑡 = 295 𝑀𝑃𝑎
𝐸𝑠 = 200000 𝑀𝑃𝑎
Figure 23 - P-M Interaction Diagram of the Wall
-5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
- 2 . 0 0 E + 0 5 0 . 0 0 E + 0 0 2 . 0 0 E + 0 5 4 . 0 0 E + 0 5 6 . 0 0 E + 0 5 8 . 0 0 E + 0 5 1 . 0 0 E + 0 6
AX
IAL
FOR
CE,
P (
N)
MOMENT, M (N-M)
W1: P-M INTERACTION DIAGRAM
Mn, Pn
ΦPn, ΦMn
Mu, Pu
25
As it can be seen in the graph, the demand point lies outside of the P-M interaction diagram,
suggesting that the wall does not have adequate capacity to resist applied loads.
Pu = 971.8 kN
Mu = 828 kN-m
(Pu, Mu) > Φ (Pn, Mn)
Wall Shear Calculations
Vu ≤ Φ Vn
- Φ = 0.75
- Vnmax = 8√𝑓′𝑐 𝐴𝑐𝑣
- Vn = Vc + Vs
𝑉𝑐 =α√𝑓′𝑐 𝐴𝑐𝑣
𝑉𝑠 = 𝜌𝑡 ∗ 𝑓𝑦𝑡 ∗ 𝐴𝑐𝑣
Vnmax = 1782 kN
Φ Vn = 751 kN
Vu = 710 kN
Vu < Φ Vn Wall is adequate for shear
The calculations were done in excel. The rest of the calculations have been provided in the
Appendix. The cross sectional details of the boundary zone and the wall web have been
illustrated below (Figure 24 (a) and (b)).
Figure 24 – Cross Sectional Detail of (a) Boundary Zone and (b) Wall Web
26
SERVICEABILITY CHECKS
The model is modified for calculating deflections at the story levels by changing the stiffness
modifiers for the shear walls to 0.5 in flexure and 0.4 in shear. Torsional irregularity was not a
consideration for deflection calculations. The elastic displacements were obtained under the
strength-level design earthquake forces as specified in Section 12.8.6 of ASCE 7-10. The design
story drift (Δ) was computed as the difference of the deflections at the center of mass at the top
and bottom of the story under consideration. Section 12.9.4.2 requires that the elastic drifts be
scaled up on the basis of modal base shear. Since, the applied loads were scaled to 100% of the
prescribed base shear, the drifts were not amplified again. The deflection at Level x (δx) was
computed based on the requirements of Section 12.9.2 and using Equation 12.8-15 from Section
12.8.6 of ASCE 7-10.
δx = 𝐶𝑑 X 𝛿𝑥𝑒
𝐼𝑒
Where,
δxe = elastic displacements computed at strength-level design earthquake forces
Ie = Importance factor, 1.0 for this building
Cd = Amplification factor to obtain inelastic displacements
The amplification factors Cd for the drifts were determined from Table 12.2-1 of ASCE 7-10 and
have been summarized below (Table
Table 10 - Cd values for the building
Direction R Cd
SMRF 8 5.5
Shear wall 5 5
The allowable story drifts were obtained from Table 12.12-1 of ASCE 7-10 (shown below).
The calculated elastic and inelastic displacements at the story levels have been summarized in
Table 11 (a) and 11 (b) along the Shear Wall and SMRF directions respectively. The final
inelastic story drifts were then compared with the allowable story drifts per ASCE 7-10.
27
Table 11 (a) - Drift Calculations along Shear wall direction
Story
Level
Height,
m δxe
Inelastic
Drift,δx
Interstory
Drifts,Δx
Allowable
Drift
Limits, Δax
Limit
Satisfied
2 2.8 0.002623 0.013 0.0131 0.056 OK
3 5.4 0.007562 0.038 0.0247 0.052 OK
4 8 0.013799 0.069 0.0312 0.052 OK
5 10.6 0.020455 0.102 0.0333 0.052 OK
6 13.15 0.026897 0.134 0.0322 0.051 OK
7 15.7 0.032779 0.164 0.0294 0.051 OK
8 18.25 0.037979 0.190 0.0260 0.051 OK
9 20.75 0.043688 0.218 0.0285 0.05 OK
10 23.25 0.048523 0.243 0.0242 0.05 OK
Roof 25.75 0.051818 0.259 0.0165 0.05 OK
Table 11 (b) - Drift Calculations along SMRF direction
Story
Level
Height,
m δye
Inelastic
Drift,δy
Interstory
Drifts,Δy
Allowable
Drift
Limits, Δay
Limit
Satisfied
2 2.8 0.008602 0.047 0.0473 0.056 OK
3 5.4 0.020975 0.115 0.0681 0.052 NOT OK
4 8 0.034998 0.192 0.0771 0.052 NOT OK
5 10.6 0.048547 0.267 0.0745 0.052 NOT OK
6 13.15 0.060827 0.335 0.0675 0.051 NOT OK
7 15.7 0.072208 0.397 0.0626 0.051 NOT OK
8 18.25 0.082801 0.455 0.0583 0.051 NOT OK
9 20.75 0.091359 0.502 0.0471 0.05 OK
10 23.25 0.097499 0.536 0.0338 0.05 OK
Roof 25.75 0.101277 0.557 0.0208 0.05 OK
As seen from the comparison, the drift limits are satisfied along the shear wall direction. But in
the SMRF direction, the drifts exceed the limit for story levels 3 to 8. Hence, the ASCE
provisions for serviceability are not satisfied along the SMRF direction.
28
STRUCTURAL DETAILS VS. ACI 318-11 PROVISIONS
Special Moment Resisting Frames
Reinforcement Details
Sections 21.6.3 and 21.6.4 of ACI 318-11 provide detailing requirements of columns in special
resisting moment frames. These requirements have been briefly summarized below (Figure 26
and Table 12) [8].
Figure 26 – Column detailing requirements in SMRF’s
Table 12 – Requirements for Cross Sectional Area of Hoop Reinforcement
29
The details of the three different types of columns on the first floor of the building have been
summarized below (Table 13) to determine if their detailing conforms to the ACI requirements
provided above.
Table 13 – Detailing of Columns on first floor of building
Column 1 Column 2
Cross Section
Dimensions (mm×mm) 550×550 550×550
Area of Longitudinal Reinforcement, Ast
(mm2)
7602.65 6082.12
Spacing of Transverse Reinforcement over
length lo from each joint, s1 (mm)
100 100
Cross-sectional Area of hoop reinforcement,
Ash (mm2)
314.16 314.16
Spacing of Transverse beyond length lo
from each joint, s2 (mm)
100 100
Cover (mm) 40 40
The beams listed above were checked to determine if they conformed to the ACI 318-11
requirements mentioned in Figures 25, 26. The results have been summarized below (Table 14).
Table 14 - Summary of Beams satisfying/not satisfying ACI 318-11 Code Provisions
Code Requirements Satisfied Not Satisfied
Area of Longitudinal
Reinforcement, Ast
Column 1, Column 2 None
Spacing of Transverse
Reinforcement over length lo
from each joint, s1
Column 1, Column 2 None
Cross-sectional Area of hoop
reinforcement, Ash
Column 1, Column 2 None
Spacing of Transverse beyond
length lo from each joint, s2
Column 1, Column 2 None
Cover Column 1, Column 2 None
Strong Column Weak Beam Column 1, Column 2 None
As it can be observed from Table 14, both columns satisfied all the checks and conformed to the
ACI provisions. However, it was found that the cross sectional area of hoop reinforcement was
found to be marginally higher than the minimum requirement. Hence, we can anticipate some
amount of damage in case of earthquakes which are stronger than the design level earthquake.
This is confirmed as the full scale model which was tested at the E-Defense Shake Table was
found to have sustained considerable damage after the shake table test.
30
Beams and Joints in Special Moment Resisting Frames
Section 21.7 in ACI 318-11 provides requirements for joints in SMRF’s (Figure 27 and 28).
Even though the figures below summarize the transverse and longitudinal requirements for
beams in SMRF’s, they form a major part of joint requirements for SMRF’s. Section 21.7.3.1
also states that joints in SMRF’s should satisfy the spacing, cover and hoop reinforcement
(cross-sectional area) requirements of columns. Since the column requirements have already
been checked earlier, only the rest of the requirements have been checked in this section.
Figure 27 – Beam Transverse Reinforcement Requirements in SMRF’s
Figure 28 – Beam Longitudinal Reinforcement Requirements in SMRF’s
The details of the different types of girders on the 2nd, 3rd and 5th floors of the building have been
summarized below (Table 15) to determine if their detailing conforms to the ACI requirements
provided above.
31
Table 15 – Detailing of beams on 2nd, 3rd and 5th floors of the building
Beam 1
(2nd
floor)
Beam 2
(2nd
floor)
Beam 1
(3rd
floor)
Beam 2
(3rd
floor)
Beam 1
(5th floor)
Beam 2
(5th
floor)
Cross Section
Dimensions
(mm×mm)
350×550 350×550 350×550 350×550 350×550 350×550
Column dimension
parallel to beam
reinforcement, hc
(mm)
500 500 500 500 500 500
Area of Joint
Transverse
Reinforcement (mm2)
157.08 314.16 157.08 157.08 157.08 314.16
Spacing of Joint
Transverse
Reinforcement (mm)
150 150 150 150 150 150
Distance of first hoop
from face of
supporting member
(mm)
0 0 0 0 0 0
Spacing of hoops
(mm)
100 100 100 100 125 100
Spacing for stirrups
with seismic hooks
(mm)
100 100 100 100 125 100
Spacing of hoops at
lap splice (mm)
100 100 100 100 125 100
Spacing of
transversely
supported flexural
reinforcing bars (mm)
270 270 270 270 270 270
Distance of splice
from face of
supporting member
(mm)
862.5 862.5 875 875 875 875
Area of Top Long.
Reinforcement (mm2)
1900.66 1140.39 1900.66 1140.39 1520.53 1140.39
Area of Bottom Long.
Reinforcement (mm2)
1520.53 1140.39 1520.53 1140.39 1520.53 1140.39
32
The beams listed above were checked to determine if they conformed to the ACI 318-11
requirements mentioned in Figures 8 and 9. The results have been summarized below (Table 6).
As we observe, most of the checks were satisfied. However, the spacing of joint transverse
reinforcement was not satisfied at any of the joints. This is probably because the grade of steel
used at the joints was extremely high (785 MPa) and hence the designers probably took the
liberty of providing larger spacing. Also, the minimum distance requirement for lap splices to be
installed from the face of the supporting member was not satisfied for any of the beams. This is
probably due to difference in the Japanese and ASCE Code Provisions. The spacing of hoops at
the lap splice in the beams was satisfied by all beams except Beam 2 on the 5th floor.
Table 16 – Summary of Beams satisfying/not satisfying ACI 318-11 Code Provisions
Code Requirements Satisfied Not Satisfied
Column dimension parallel to beam
reinforcement, hc
All beams None
Area of Joint Transverse
Reinforcement
All joints None
Spacing of Joint Transverse
Reinforcement
None All joints
Distance of first hoop from face of
supporting member
All beams None
Spacing of hoops All beams None
Spacing for stirrups with seismic
hooks
All beams None
Spacing of hoops at lap splice Rest of the beams Beam 1 (5th floor)
Spacing of transversely supported
flexural reinforcing bars
All joints None
Distance of splice from face of
supporting member
None All beams
Area of Top Long. Reinforcement All beams None
Area of Bottom Long. Reinforcement All beams None
33
Special Shear Walls
In our project the structure has special reinforced concrete shear walls in its shorter (X) direction
and these shear walls go up to the first seven stories of the building. A typical special shear wall
has been shown below (Figure 29).
Figure 29 – Special RC Shear Wall Details
Section 21.9 of ACI 318-11 provides detailing requirements of boundary elements of shear walls
depending on whether they are to be detailed as ordinary boundary elements (Figure 30) or
special boundary elements (Figure 31). Since Section 21.9 of ACI 318-11 is extremely
descriptive, only the major significant provisions which matter for this project have been briefly
summarized below.
Figure 30 – Detailing requirements for ordinary boundary elements
Figure 31 – Detailing requirements for special boundary elements
34
In order to determine if specially confined boundary elements are required, we first identify the
maximum axial force demand (Pu) on the shear wall. We then calculate (Pu/Agf’c) since the gross
area (Ag) and the compressive strength (f’c) are already known to us and find the corresponding
value of (c/lw) from the graph (Figure 32).
Figure 32 – Approximate Compression Flexural Depth
Once we have obtained (c/lw), we would decide to reinforce the compression zones with special
boundary elements if the following expression is satisfied.
(𝑐
𝑙𝑤) ≥
1
900 (𝜕𝑢ℎ𝑤
)
c = largest neutral axis depth calculated for the factored axial force and nominal moment strength
𝑙𝑤 = length of the wall
ℎ𝑤 = height of the wall
𝜕𝑢 = top level design displacement
The ratio 𝜕𝑢
ℎ𝑤 should not be taken less than 0.007.
ACI 318-11 states that where special boundary elements are required, the special boundary
element reinforcement shall extend vertically from the critical section a distance not less than the
larger of 𝑙𝑤 or Mu/4Vu. Structural walls not designed according to the provisions above shall
have special boundary elements at boundaries and edges around openings of structural walls
where the maximum extreme compressive fiber compressive stresses exceed 0.2f’c and are
permitted to be discontinued when the compressive stresses are less than 0.15f’c. Shear walls are
also classified based on their (ℎ𝑤/𝑙𝑤) ratio (Table 17).
Table 17 – Classification of Shear Walls based on (𝒉𝒘/𝒍𝒘) value
Shear Wall Classification (𝒉𝒘/𝒍𝒘)
Squat Wall <1
Transition Wall 1-2
Slender Wall >2
35
The requirements for web reinforcement ratios ρl and ρt depend on the shear wall classification.
For walls with (ℎ𝑤/𝑙𝑤) less than 2 (squat and transition walls), ρl should always be equal or
higher than ρt. For other cases (slender walls), ρl is 0.0025 and ρt is calculated based on shear
requirements. Reinforcement spacing each way in structural walls should not exceed 18 inches.
The details of the shear wall in the structure have been summarized below (Table 18) to
determine if they conform to the ACI requirements provided above.
Table 18 – Summary of Shear Wall satisfying/not satisfying ACI 318-11 Code Provisions
Aspect Ratio, (ℎ𝑤/𝑙𝑤) 11.083 (Slender Wall)
900 (𝜕𝑢
ℎ𝑤) (
𝑐
𝑙𝑤)
3.16 (Special
Confinement Required)
Length of boundary element, lbe (mm) 450 (Check Satisfied)
Distance between Hoop or Tie Legs, hx (mm) 123.44 (Check Satisfied)
Hoop Spacing, s (mm) 100 (Not Satisfied)
Cross Sectional Area of Hoop Reinforcement, Ash Check Satisfied
Width of boundary element, b (mm) 230 (Check Satisfied)
Extension of horizontal reinforcement into boundary element (mm) 330 (Check Satisfied)
36
PERFORMANCE OF FULL-SCALE MODEL TESTED ON SHAKE TABLE
Performance of Columns on 1st floor
The performance of columns (corner and exterior) have been illustrated below
Figure 32 – (a) Column C1 and (b) Column C2 after the shake table test
As seen above, some cracks are pretty evident in the columns. This confirms the fact the columns
are not exactly adequate to resist the loads and validates calculations which showed some of the
demands lying outside the P-M interaction curve. However, the columns have not collapsed
either. This makes sense as the columns were found to conform to all ACI 318-11 requirements.
Performance of Girders and Joints on 2nd, 3rd and 5th floors
2nd Floor
Figure 33 – (a) Corner Joint and (b) Exterior Joint on 2nd floor after shake table test
37
3rd Floor
Figure 34 – (a) Corner Joint and (b) Exterior Joint on 3rd floor after shake table test
5th Floor
Figure 35 – (a) Corner Joint and (b) Exterior Joint on 5th floor after shake table test
As seen from Figures 33 to 35, the beam column joints look damaged with considerable amount
of cracks. This is not surprising as the spacing of joint transverse reinforcement was found not to
satisfy the minimum spacing requirements according to ACI 318-11. However, apart from
sustaining cracks, the joints did not fail on any of the floors. This validates the calculations
shown earlier which suggested that the joints were adequate to resist the anticipated loading. The
minimum distance requirement for lap splices was also found to not satisfy the ACI requirements
earlier. However, there were no signs of significant damage in the girders near the lap splice
region and hence no conclusions could be made on whether the location of the splices had any
effect on the performance of the girders.
38
Performance of Shear Wall on 1st Floor
Figure 36 – 230mm thick shear wall on first floor after shake table test
The shear cracks are pretty evident in the wall. This confirms the fact the wall is not adequate to
resist the loads and validates calculations which showed the demands lying outside the P-M
interaction curve. However, the wall has clearly not collapsed. This is not surprising as it was
found to satisfy most of the ACI 318-11 requirements except for hoop spacing
CONCLUSIONS
This project on the ten story structure had four main components to it. These components
included analysis on SAP2000 through which the demands were obtained, analysis using ASCE
7-10 requirements, calculation of member capacities and their comparison with the demands and
comparison between structural detailing provided in the drawings and ACI 318-11 detailing
requirements. The demands obtained from SAP2000 analysis were compared to the member
capacities to check if the members were adequate to resist loads. It was found that most of the
columns and walls were not satisfying the P-M Interaction curve as the demand points were
found to lie outside the P-M curve. Some of the girders were also found to not have adequate
moment capacities. The joint strength was found to be adequate to resist loads. However, the
spacing of joint transverse reinforcement did not satisfy the minimum spacing requirements of
ACI 318-11. The columns satisfied all the detailing requirements while most of the beams and
joints satisfied majority of the detailing requirements too with a few exceptions mentioned
earlier in the paper. Hence, it is observed that while most of the detailing requirements were
satisfied for the members, they were found to be inadequate in strength. Our calculations and
comparisons of structural detailing were validated by observing the performance of the full-scale
model after the shake table test in Hyogo, Japan.
39
REFERENCES
1. http://www.norsar.no/norsar/about-us/News/2011/Historical-Development-of-
Earthquake-Resistant-Design
2. http://www.cement.org/think-harder-concrete-/buildings-structures/design-aids/seismic-
design
3. Blume, J.A., Newmark, N.M., and Corning, L.H. (1961). Design of multistory reinforced
concrete buildings for earthquake motions, Portland Cement Association, Chicago, IL.
4. Moehle, J.P., Hooper, J.D., and Lubke, C.D. (2008). Seismic Design of Reinforced
Concrete Special Moment Frames: A Guide for Practicing Engineers, NEHRP Seismic
Design Technical Brief 1, Gaithersburg, MD.
5. Bagheri, B., Firoozabad, E.S., and Yahyaei, M. (2012). Comparative Study of the Static
and Dynamic Analysis of Multi-Storey Irregular Building, International Journal of Civil,
Environmental, Structural, Construction and Architectural Engineering Vol: 6, No: 11,
2012
6. ASCE (2010). Minimum design loads for buildings and other structures (ASCE 7-10),
American Society of Civil Engineers, Reston, VA.
7. ACI (2011). Building code requirements for structural concrete (ACI 318-11) and
commentary, American Concrete Institute, Farmington Hills, MI.
8. Moehle, J.P. (2014). Seismic Design of Reinforced Concrete Buildings, University of
California Berkeley, CA.
ACKNOWLEDGMENTS
The authors would like to express their very great appreciation to the following people:
1) Professor Jack P. Moehle, for allowing them to be a part of this project and providing
insight and expertise that greatly assisted the project and this paper.
2) Professor Stephen A. Mahin for expanding their knowledge of Earthquake Engineering.
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-1st Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G1
350 x 550
f'c = 42 MPa
(4) 22 mm bars at the bottom
(5) 22 mm bars at the top
Length, L (mm) 3450 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fyt (Mpa) 295 fyt (Mpa) 295
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 5 # of top bars 4
# of bottom bars 4 # of bottom bars 5
As 1520.530844 As 1900.663555
A's 1900.663555 A's 1520.530844
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 42 f'c (MPa) 42
f'c (ksi) 6.091596 f'c (ksi) 6.091596
β 0.7454202 β 0.7454202
A 9314.025399 A 9314.025399
B 615814.992 B 256589.58
C -69564286.13 C -55651428.9
c (N.A. depth), mm 59.47059922 c (N.A. depth), mm 64.74153606
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 246.0325431 Mn (kN-m) 302.8848554
εs 0.021667651 εs 0.019659333
ɸMn (kN-m) 221 kN-m ɸMn (kN-m) 273 kN-m
Mu+ (kN-m) 290 kN-m Mu- (kN-m) 304 kN-m
Mu< ɸMn NOT OK Mu< ɸMn NOT OK
D/C 1.31 D/C 1.12
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-1st Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 42 MPa 6.0915834 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 100 mm 3.93701 in
Vc 207.2 kN 46.58 K
Vs 254.9 kN 57.30 K
Vn 462.0 kN 103.87 K
ɸVn 346.5 kN 77.90 K
Demands
Mpr1 307.5407 kN-m
Mpr2 378.6061 kN-m
Vu, gravity 8.02125 kN
Vu, lateral 219 KN
Vu, total 228 kN
Vu < ɸVn OK
D/C 0.66
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-2nd Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G2
350 x 550
f'c = 42 MPa
(5) 22 mm bars at the bottom
(5) 22 mm bars at the top
Length, L (mm) 3450 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fyt (Mpa) 295 fyt (Mpa) 295
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 5 # of top bars 5
# of bottom bars 5 # of bottom bars 5
As 1900.663555 As 1900.663555
A's 1900.663555 A's 1900.663555
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 42 f'c (MPa) 42
f'c (ksi) 6.091596 f'c (ksi) 6.091596
β 0.7454202 β 0.7454202
A 9314.025399 A 9314.025399
B 484669.2066 B 484669.2066
C -69564286.13 C -69564286.13
c (N.A. depth), mm 64.23537553 c (N.A. depth), mm 64.23537553
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 302.8239031 Mn (kN-m) 302.8239031
εs 0.019837883 εs 0.019837883
ɸMn (kN-m) 273 kN-m ɸMn (kN-m) 273 kN-m
Mu+ (kN-m) 270 kN-m Mu- (kN-m) 284 kN-m
Mu< ɸMn OK Mu< ɸMn NOT OK
D/C 0.99 D/C 1.04
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-2nd Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 42 MPa 6.0915834 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 100 mm 3.93701 in
Vc 207.2 kN 46.58 K
Vs 254.9 kN 57.30 K
Vn 462.0 kN 103.87 K
ɸVn 346.5 kN 77.90 K
Demands
Mpr1 378.5299 kN-m
Mpr2 378.5299 kN-m
Vu, gravity 8.02125 kN
Vu, lateral 219 KN
Vu, total 227 kN
Vu < ɸVn OK
D/C 0.66
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-3rd Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G1
350 x 550
f'c = 33 MPa
(4) 22 mm bars at the bottom
(5) 22 mm bars at the top
Length, L (mm) 3450 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 5 # of top bars 4
# of bottom bars 4 # of bottom bars 5
As 1520.530844 As 1900.663555
A's 1900.663555 A's 1520.530844
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 42 f'c (MPa) 42
f'c (ksi) 6.091596 f'c (ksi) 6.091596
β 0.7454202 β 0.7454202
A 9314.025399 A 9314.025399
B 615814.992 B 256589.58
C -69564286.13 C -55651428.9
c (N.A. depth), mm 59.47059922 c (N.A. depth), mm 64.74153606
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 246.0325431 Mn (kN-m) 302.8848554
εs 0.021667651 εs 0.019659333
ɸMn (kN-m) 221 kN-m ɸMn (kN-m) 273 kN-m
Mu+ (kN-m) 290 kN-m Mu- (kN-m) 306 kN-m
Mu< ɸMn NOT OK Mu< ɸMn NOT OK
D/C 1.31 D/C 1.12
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-3rd Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 33 MPa 4.7862441 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 100 mm 3.93701 in
Vc 183.6 kN 41.28 K
Vs 254.9 kN 57.30 K
Vn 438.5 kN 98.58 K
ɸVn 328.9 kN 73.93 K
Demands
Mpr1 307.5407 kN-m
Mpr2 378.6061 kN-m
Vu, gravity 8.02125 kN
Vu, lateral 219 KN
Vu, total 228 kN
Vu < ɸVn OK
D/C 0.69
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-3rd Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G2
350 x 550
f'c = 33 MPa
(5) 22 mm bars at the bottom
(5) 22 mm bars at the top
Length, L (mm) 3450 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 5 # of top bars 5
# of bottom bars 5 # of bottom bars 5
As 1900.663555 As 1900.663555
A's 1900.663555 A's 1900.663555
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 42 f'c (MPa) 42
f'c (ksi) 6.091596 f'c (ksi) 6.091596
β 0.7454202 β 0.7454202
A 9314.025399 A 9314.025399
B 484669.2066 B 484669.2066
C -69564286.13 C -69564286.13
c (N.A. depth), mm 64.23537553 c (N.A. depth), mm 64.23537553
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 302.8239031 Mn (kN-m) 302.8239031
εs 0.019837883 εs 0.019837883
ɸMn (kN-m) 273 kN-m ɸMn (kN-m) 273 kN-m
Mu+ (kN-m) 277 kN-m Mu- (kN-m) 291 kN-m
Mu< ɸMn NOT OK Mu< ɸMn NOT OK
D/C 1.02 D/C 1.07
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-3rd Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 33 MPa 4.7862441 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 100 mm 3.93701 in
Vc 183.6 kN 41.28 K
Vs 254.9 kN 57.30 K
Vn 438.5 kN 98.58 K
ɸVn 328.9 kN 73.93 K
Demands
Mpr1 378.5299 kN-m
Mpr2 378.5299 kN-m
Vu, gravity 8.02125 kN
Vu, lateral 219 KN
Vu, total 227 kN
Vu < ɸVn OK
D/C 0.69
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-5th Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G1
350 x 550
f'c = 27 MPa
(4) 22 mm bars at the bottom
(4) 22 mm bars at the top
Length, L (mm) 3500 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 4 # of top bars 4
# of bottom bars 4 # of bottom bars 5
As 1520.530844 As 1900.663555
A's 1520.530844 A's 1520.530844
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 27 f'c (MPa) 27
f'c (ksi) 3.916026 f'c (ksi) 3.916026
β 0.8541987 β 0.8541987
A 6861.351058 A 6861.351058
B 387735.3653 B 256589.58
C -55651428.9 C -55651428.9
c (N.A. depth), mm 66.13354161 c (N.A. depth), mm 73.28267368
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 239.3844027 Mn (kN-m) 295.5861914
εs 0.01918239 εs 0.017018374
ɸMn (kN-m) 215 kN-m ɸMn (kN-m) 266 kN-m
Mu+ (kN-m) 268 kN-m Mu- (kN-m) 286 kN-m
Mu- (kN-m) 286 kN-m
Mu< ɸMn NOT OK
D/C 1.24
D/C 1.33
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G1-5th Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 27 MPa 3.9160179 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 125 mm 4.9212625 in
Vc 166.1 kN 37.34 K
Vs 203.9 kN 45.84 K
Vn 370.0 kN 83.18 K
ɸVn 277.5 kN 62.38 K
Demands
Mpr1 299.2305 kN-m
Mpr2 369.4827 kN-m
Vu, gravity 8.1375 kN
Vu, lateral 211 KN
Vu, total 219 kN
Vu < ɸVn OK
D/C 0.79
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-5th Floor
Appendix
AD, AS, QW
MOMENT AND SHEAR CALCULATIOINSBeam Design: G2
350 x 550
f'c = 27 MPa
(4) 22 mm bars at the bottom
(5) 22 mm bars at the top
Length, L (mm) 3500 Length, L (mm) 4000
Width, b (mm) 350 Width, b (mm) 350
Depth, d (mm) 550 Depth, d (mm) 550
fy (MPa) 345 fy (MPa) 345
ᶲ factor, M 0.9 ᶲ factor, M 0.9
Rebar dia, mm 22 Rebar dia, mm 22
# of top bars 5 # of top bars 4
# of bottom bars 4 # of bottom bars 5
As 1520.530844 As 1900.663555
A's 1900.663555 A's 1520.530844
Top Cover (mm) 61 Top Cover (mm) 61
Bottom Cover (mm) 61 Bottom Cover (mm) 61
f'c (MPa) 27 f'c (MPa) 27
f'c (ksi) 3.916026 f'c (ksi) 3.916026
β 0.8541987 β 0.8541987
A 6861.351058 A 6861.351058
B 615814.992 B 256589.58
C -69564286.13 C -55651428.9
c (N.A. depth), mm 65.3622539 c (N.A. depth), mm 73.28267368
εc 0.003 εc 0.003
Es (MPa) 200000 Es (MPa) 200000
ds 489 ds 489
Mn (kN-m) 239.3587984 Mn (kN-m) 295.5861914
εs 0.019444146 εs 0.017018374
ɸMn (kN-m) 215 kN-m ɸMn (kN-m) 266 kN-m
Mu+ (kN-m) 261 kN-m Mu- (kN-m) 275 kN-m
Mu< ɸMn NOT OK Mu< ɸMn NOT OK
D/C 1.21 D/C 1.03
FLEXURE (positive) FLEXURE (negative)
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Beam Moment and Shear Calculations
G2-5th Floor
Appendix
AD, AS, QW
Conversions
1 mm 0.0393701 in
1 mm2 0.00155 in2
1 MPa 0.1450377 ksi
1 K 4.4482 kN
Metric System US Units
Width, b 350 mm 13.779535 in
Depth, h 550 mm 21.653555 in Calculating w
cover 61 mm 2.4015761 in slab depth 120 mm
fyt 295 MPa 42.7861215 ksi trib width 1.55 m
fy 345 MPa 50.0380065 ksi SW 3 kN/m2
ᶲ 0.75 w 4.65 kN/m
f'c 27 MPa 3.9160179 ksi
ds 489 mm 19.2519789 in
stirrup dia 10 mm 0.393701 in
number of legs 2
Av 157.0796 mm2 0.24347343 in2
s 100 mm 3.93701 in
Vc 166.1 kN 37.34 K
Vs 254.9 kN 57.30 K
Vn 421.0 kN 94.64 K
ɸVn 315.7 kN 70.98 K
Demands
Mpr1 299.1985 kN-m
Mpr2 369.4827 kN-m
Vu, gravity 8.1375 kN
Vu, lateral 211 KN
Vu, total 219 kN
Vu < ɸVn OK
D/C 0.69
SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Exterior Joint Appendix
AD, AS, QW
Demand
Lc 3450 mm Lc 3450 mm 1 mm 0.0393701 in
Mpr+ 307.5406789 kN-m Mpr+ 378.5299 kN-m 1 K 4.4482 kN
Mpr- 378.6060693 kN-m Mpr- 378.5299 kN-m
h 550 mm h 550 mm
cover 61 mm cover 61 mm
d 489 mm d 489 mm
de 440.1 mm de 440.1 mm
h1 2800 mm
h2 2600 mm
dc 550 mm 21.65356
bc 550 mm 21.65356
Vcol 296.7069472 kN Vcol 323.0275 kN
Vjoint 1262.190026 kN Vjoint 1397.345 kN
Vjoint 1397 kN
Capacity
f'c (MPa) 42 Mpa
f'c (ksi) 6.091596 ksi
ϒ 15
Vn 549 K 2441.752 kN
ΦVn 2075 kN
Vjoint < Vn OK
D/C 0.67
ConversionsG1 G2
Column (C2)
JOINT SHEAR
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Interior Joint Appendix
AD, AS, QW
Demand
Lc 3450 mm
Mpr+ 307.5406789 kN-m 1 mm 0.0393701 in
Mpr- 378.6060693 kN-m 1 K 4.4482 kN
h 550 mm
cover 61 mm
d 489 mm
de 440.1 mm
h1 2800 mm
h2 2600 mm
dc 550 mm 21.65356 in
bc 550 mm 21.65356 in
Gravity Loading
Calculating w
slab depth 120 mm
trib width 1.55 m
SW 3 kN/m2
w 4.65 kN/m
Vcol 161.2980629 kN Vcol 133.3435896 kN
Vjoint 698.9747599 kN Vjoint 565.4536813 kN
Vjoint 699 kN
Capacity
f'c (MPa) 42 Mpa
f'c (ksi) 6.091596 ksi
ϒ 12
Vn 439 K 1953.402 kN
ΦVn 1660 kN
Vjoint < Vn OK
D/C 0.42
JOINT SHEAR
G1
Column (C1)
Conversions
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Column (C1) PM Interaction Diagram APPENDIX
AD, AS, QW
Demands for Envelope Case
P M3 0 6838.39 0 10520.61
KN KN-m 250.7993 6838.39 385.8451 10520.61
-1739.93 425.8359 420.4033 6474.33 646.7743 9960.51
-1729.13 176.5983 543.8825 5466.9 836.7424 8410.61
-1718.33 71.193 628.5217 4341.2 966.9564 6678.78
686.432 3031.961 1056.049 4664.56
746.52 2463.428 1008.548 3328.09
793.4803 1749.4 891.0365 1964.483
623.4142 514.8855 692.6824 572.095
329.6961 -960.9879 366.329 -1067.764
0 -2360.63 0 -2622.922
P-M3 interactionφP-φM3 interaction
-4000
-2000
0
2000
4000
6000
8000
10000
12000
0 200 400 600 800 1000 1200
Axi
al F
orc
e, P
(kN
)
Moment, M (kN-m)
C1: P-M Interaction Diagram
Pu,Mu
ΦMn, ΦPn
Pn,Mn
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Column (C1) Shear Appendix
AD, AS, QW
Section Properties in metric system Conversions
b 550 mm 1 mm 0.0393701 in
h 550 mm 1 mm2 0.00155 in2
d 440 mm 1 MPa 0.1450377 ksi
Ag 302500 mm2 1 K 4.4482 kN
f'c 42 MPa
fy 345 MPa
fyt 295 MPa
bar 10 mm
Av/leg 78.53981634 mm2
# of legs 4
s 100 mm
Vu ≤ ΦVn
Vn = Vc + Vs
Φ 0.75
Vc = 2(1 + Nu/(2000 Ag)) λ √(f'c) bd
Nu = Pu -800.5 kN -179959.6 lbs
λ 1
Ag 468.875 in2
f'c 6.091583 ksi 6091.5834 psi
b 21.65356 in
d = 0.8 b 17.32284 in
Vc 47315.73 lbs 47.32 kips 210.47 kN
Vs = Av fyt d / s
Av/leg 0.12 in2
# of legs 4
fyt 42.78612 ksi
d 17.32284 in
s 3.93701 in
Vs 91.6721 K 407.7778 kN
Vn 138.99 K 618 kN
φVn 104.2409 K 464 kN
φo 1.25
Vu,SAP 243.75 kN
Vu, Joint Shear 161.3 Kn
D/C 0.53 OK
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Column (C2) PM Interaction Diagram APPENDIX
AD, AS, QW
P M3
KN KN-m 0 6593.84 0 10144.36
-2029.32 459.1085 239.3368 6593.84 368.2105 10144.36
-2018.52 160.3768 404.0899 6264.63 621.6768 9637.9
-2007.72 140.6808 520.5027 5292.16 800.7734 8141.78
596.8154 4237.53 918.1775 6519.27
640.0707 3025.278 984.7242 4654.27
691.4461 2491.357 934.1434 3365.82
732.7736 1887.716 822.8661 2119.805
570.309 732.214 633.6767 813.5711
304.8556 -605.648 338.7285 -672.943
0 -1888.5 0 -2098.34
φP-φM3 interaction P-M3 interaction
-4000
-2000
0
2000
4000
6000
8000
10000
12000
0 200 400 600 800 1000 1200
Axi
al F
orc
e, P
(kN
)
Moment, M (kN-m)
C2: P-M Interaction Diagram
ΦMn, ΦPn
Mn,Pn
Mu,Pu
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Column (C2) Shear APPENDIX
AD, AS, QW
Section Properties in metric system Conversions
b 550 mm 1 mm 0.0393701 in
h 550 mm 1 mm2 0.00155 in2
d 440 mm 1 MPa 0.1450377 ksi
Ag 302500 mm2 1 K 4.4482 kN
f'c 42 MPa
fy 345 MPa
fyt 295 MPa
bar 10 mm
Av/leg 78.53981634 mm2
# of legs 4
s 100 mm
Vu ≤ ΦVn
Vn = Vc + Vs
Φ 0.75
Vc = 2(1 + Nu/(2000 Ag)) λ √(f'c) bd
Nu = Pu -801.5 kN -180184.4 lbs
λ 1
Ag 468.875 in2
f'c 6.091583 ksi 6091.583 psi
b 21.65356 in
d = 0.8 b 17.32284 in
Vc 47301.69 lbs 47.30 kips 210.41 kN
Vs = Av fyt d / s
Av/leg 0.12 in2
# of legs 4
fyt 42.78612 ksi
d 17.32284 in
s 3.93701 in
Vs 91.6721 K 407.7778
Vn 138.97 K 618.1862 kN
φVn 104.2303 K 463.6397 kN
φo 1.25
Vu, SAP 243.75 kN
Vu, Joint Shear 323 kN
D/C 0.70 OK
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
APPENDIX
AD, AS, QW
Wall (BC) Axial (N)
Moment1
(NS) -
about xx
(N-m)
Moment2
(EW) -
about yy
(N-m)
Moment2
(45 deg)
(N-m)
971795.3 827988.1
Mn Pn φMn φPn Axial Strain Mn Pn φMn φPn Mu Pu
N-m N N-m N N-m N N-m N N-m N
1.08E-10 1.90E+07 7.52E-11 1.07E+07 -2.00E-03 1.08E-10 1.90E+07 7.52E-11 1.07E+07 827988.1 971795.3
867.7 1.90E+07 607.4 1.07E+07 -1.94E-03 8.68E+02 1.90E+07 6.07E+02 1.07E+07
8593 1.88E+07 6015 1.07E+07 -1.88E-03 8.59E+03 1.88E+07 6.02E+03 1.07E+07
2.70E+04 1.85E+07 1.89E+04 1.07E+07 -1.82E-03 2.70E+04 1.85E+07 1.89E+04 1.07E+07
5.32E+04 1.80E+07 3.73E+04 1.07E+07 -1.76E-03 5.32E+04 1.80E+07 3.73E+04 1.07E+07
8.62E+04 1.75E+07 6.03E+04 1.07E+07 -1.70E-03 8.62E+04 1.75E+07 6.03E+04 1.07E+07
1.25E+05 1.69E+07 8.75E+04 1.07E+07 -1.64E-03 1.25E+05 1.69E+07 8.75E+04 1.07E+07
1.69E+05 1.62E+07 1.18E+05 1.07E+07 -1.58E-03 1.69E+05 1.62E+07 1.18E+05 1.07E+07
2.16E+05 1.55E+07 1.51E+05 1.07E+07 -1.52E-03 2.16E+05 1.55E+07 1.51E+05 1.07E+07
2.66E+05 1.47E+07 1.86E+05 1.03E+07 -1.46E-03 2.66E+05 1.47E+07 1.86E+05 1.03E+07
3.18E+05 1.38E+07 2.22E+05 9.67E+06 -1.40E-03 3.18E+05 1.38E+07 2.22E+05 9.67E+06
3.70E+05 1.30E+07 2.59E+05 9.07E+06 -1.34E-03 3.70E+05 1.30E+07 2.59E+05 9.07E+06
4.06E+05 1.23E+07 2.84E+05 8.58E+06 -1.28E-03 4.06E+05 1.23E+07 2.84E+05 8.58E+06
4.29E+05 1.17E+07 3.00E+05 8.18E+06 -1.22E-03 4.29E+05 1.17E+07 3.00E+05 8.18E+06
4.43E+05 1.12E+07 3.10E+05 7.85E+06 -1.16E-03 4.43E+05 1.12E+07 3.10E+05 7.85E+06
4.55E+05 1.08E+07 3.19E+05 7.53E+06 -1.10E-03 4.55E+05 1.08E+07 3.19E+05 7.53E+06
4.68E+05 1.03E+07 3.28E+05 7.20E+06 -1.04E-03 4.68E+05 1.03E+07 3.28E+05 7.20E+06
4.80E+05 9.91E+06 3.36E+05 6.93E+06 -9.80E-04 4.80E+05 9.91E+06 3.36E+05 6.93E+06
4.88E+05 9.60E+06 3.42E+05 6.72E+06 -9.20E-04 4.88E+05 9.60E+06 3.42E+05 6.72E+06
4.95E+05 9.30E+06 3.47E+05 6.51E+06 -8.60E-04 4.95E+05 9.30E+06 3.47E+05 6.51E+06
5.01E+05 9.02E+06 3.51E+05 6.31E+06 -8.00E-04 5.01E+05 9.02E+06 3.51E+05 6.31E+06
5.06E+05 8.77E+06 3.54E+05 6.14E+06 -7.40E-04 5.06E+05 8.77E+06 3.54E+05 6.14E+06
5.10E+05 8.52E+06 3.57E+05 5.97E+06 -6.80E-04 5.10E+05 8.52E+06 3.57E+05 5.97E+06
5.14E+05 8.27E+06 3.60E+05 5.79E+06 -6.20E-04 5.14E+05 8.27E+06 3.60E+05 5.79E+06
5.17E+05 8.05E+06 3.62E+05 5.63E+06 -5.60E-04 5.17E+05 8.05E+06 3.62E+05 5.63E+06
5.19E+05 7.85E+06 3.64E+05 5.49E+06 -5.00E-04 5.19E+05 7.85E+06 3.64E+05 5.49E+06
5.21E+05 7.68E+06 3.64E+05 5.38E+06 -4.40E-04 5.21E+05 7.68E+06 3.64E+05 5.38E+06
5.21E+05 7.54E+06 3.65E+05 5.28E+06 -3.80E-04 5.21E+05 7.54E+06 3.65E+05 5.28E+06
5.22E+05 7.40E+06 3.65E+05 5.18E+06 -3.20E-04 5.22E+05 7.40E+06 3.65E+05 5.18E+06
5.22E+05 7.25E+06 3.66E+05 5.07E+06 -2.60E-04 5.22E+05 7.25E+06 3.66E+05 5.07E+06
5.23E+05 7.10E+06 3.66E+05 4.97E+06 -2.00E-04 5.23E+05 7.10E+06 3.66E+05 4.97E+06
5.22E+05 6.93E+06 3.66E+05 4.85E+06 -1.40E-04 5.22E+05 6.93E+06 3.66E+05 4.85E+06
5.21E+05 6.77E+06 3.65E+05 4.74E+06 -8.00E-05 5.21E+05 6.77E+06 3.65E+05 4.74E+06
5.20E+05 6.60E+06 3.64E+05 4.62E+06 -2.00E-05 5.20E+05 6.60E+06 3.64E+05 4.62E+06
5.19E+05 6.44E+06 3.63E+05 4.51E+06 4.00E-05 5.19E+05 6.44E+06 3.63E+05 4.51E+06
5.16E+05 6.30E+06 3.61E+05 4.41E+06 1.00E-04 5.16E+05 6.30E+06 3.61E+05 4.41E+06
5.12E+05 6.17E+06 3.58E+05 4.32E+06 1.60E-04 5.12E+05 6.17E+06 3.58E+05 4.32E+06
5.08E+05 6.04E+06 3.56E+05 4.23E+06 2.20E-04 5.08E+05 6.04E+06 3.56E+05 4.23E+06
5.04E+05 5.90E+06 3.53E+05 4.13E+06 2.80E-04 5.04E+05 5.90E+06 3.53E+05 4.13E+06
5.00E+05 5.77E+06 3.50E+05 4.04E+06 3.40E-04 5.00E+05 5.77E+06 3.50E+05 4.04E+06
4.96E+05 5.63E+06 3.47E+05 3.94E+06 4.00E-04 4.96E+05 5.63E+06 3.47E+05 3.94E+06
4.92E+05 5.49E+06 3.44E+05 3.85E+06 4.60E-04 4.92E+05 5.49E+06 3.44E+05 3.85E+06
4.88E+05 5.36E+06 3.41E+05 3.75E+06 5.20E-04 4.88E+05 5.36E+06 3.41E+05 3.75E+06
4.83E+05 5.21E+06 3.38E+05 3.65E+06 5.80E-04 4.83E+05 5.21E+06 3.38E+05 3.65E+06
4.79E+05 5.07E+06 3.35E+05 3.55E+06 6.40E-04 4.79E+05 5.07E+06 3.35E+05 3.55E+06
4.74E+05 4.93E+06 3.32E+05 3.45E+06 7.00E-04 4.74E+05 4.93E+06 3.32E+05 3.45E+06
4.71E+05 4.83E+06 3.29E+05 3.38E+06 7.60E-04 4.71E+05 4.83E+06 3.29E+05 3.38E+06
4.67E+05 4.73E+06 3.27E+05 3.31E+06 8.20E-04 4.67E+05 4.73E+06 3.27E+05 3.31E+06
4.63E+05 4.62E+06 3.24E+05 3.24E+06 8.80E-04 4.63E+05 4.62E+06 3.24E+05 3.24E+06
4.59E+05 4.52E+06 3.22E+05 3.17E+06 9.40E-04 4.59E+05 4.52E+06 3.22E+05 3.17E+06
4.56E+05 4.43E+06 3.19E+05 3.10E+06 1.00E-03 4.56E+05 4.43E+06 3.19E+05 3.10E+06
4.53E+05 4.37E+06 3.17E+05 3.06E+06 1.06E-03 4.53E+05 4.37E+06 3.17E+05 3.06E+06
4.50E+05 4.30E+06 3.15E+05 3.01E+06 1.12E-03 4.50E+05 4.30E+06 3.15E+05 3.01E+06
4.47E+05 4.23E+06 3.13E+05 2.96E+06 1.18E-03 4.47E+05 4.23E+06 3.13E+05 2.96E+06
4.44E+05 4.16E+06 3.11E+05 2.91E+06 1.24E-03 4.44E+05 4.16E+06 3.11E+05 2.91E+06
4.39E+05 4.04E+06 3.08E+05 2.82E+06 1.30E-03 4.39E+05 4.04E+06 3.08E+05 2.82E+06
4.32E+05 3.84E+06 3.03E+05 2.69E+06 1.36E-03 4.32E+05 3.84E+06 3.03E+05 2.69E+06
4.24E+05 3.64E+06 2.97E+05 2.55E+06 1.42E-03 4.24E+05 3.64E+06 2.97E+05 2.55E+06
4.14E+05 3.41E+06 2.90E+05 2.38E+06 1.48E-03 4.14E+05 3.41E+06 2.90E+05 2.38E+06
4.05E+05 3.22E+06 2.83E+05 2.25E+06 1.54E-03 4.05E+05 3.22E+06 2.83E+05 2.25E+06
3.93E+05 3.00E+06 2.75E+05 2.10E+06 1.60E-03 3.93E+05 3.00E+06 2.75E+05 2.10E+06
3.77E+05 2.75E+06 2.64E+05 1.93E+06 1.66E-03 3.77E+05 2.75E+06 2.64E+05 1.93E+06
3.57E+05 2.45E+06 2.50E+05 1.72E+06 1.72E-03 3.57E+05 2.45E+06 2.50E+05 1.72E+06
3.35E+05 2.17E+06 2.35E+05 1.52E+06 1.78E-03 3.35E+05 2.17E+06 2.35E+05 1.52E+06
3.11E+05 1.88E+06 2.19E+05 1.32E+06 1.84E-03 3.11E+05 1.88E+06 2.19E+05 1.32E+06
2.83E+05 1.54E+06 2.09E+05 1.14E+06 1.90E-03 2.83E+05 1.54E+06 2.09E+05 1.14E+06
2.53E+05 1.17E+06 1.96E+05 9.10E+05 1.96E-03 2.53E+05 1.17E+06 1.96E+05 9.10E+05
2.19E+05 7.75E+05 1.79E+05 6.34E+05 2.02E-03 2.19E+05 7.75E+05 1.79E+05 6.34E+05
1.84E+05 3.56E+05 1.59E+05 3.07E+05 2.08E-03 1.84E+05 3.56E+05 1.59E+05 3.07E+05
1.47E+05 -7.62E+04 1.32E+05 -6.86E+04 2.14E-03 1.47E+05 -7.62E+04 1.32E+05 -6.86E+04
1.14E+05 -4.57E+05 1.03E+05 -4.11E+05 2.20E-03 1.14E+05 -4.57E+05 1.03E+05 -4.11E+05
8.42E+04 -7.94E+05 7.58E+04 -7.14E+05 2.26E-03 8.42E+04 -7.94E+05 7.58E+04 -7.14E+05
5.67E+04 -1.10E+06 5.10E+04 -9.94E+05 2.32E-03 5.67E+04 -1.10E+06 5.10E+04 -9.94E+05
3.33E+04 -1.37E+06 3.00E+04 -1.23E+06 2.38E-03 3.33E+04 -1.37E+06 3.00E+04 -1.23E+06
2.05E+04 -1.52E+06 1.85E+04 -1.37E+06 2.44E-03 2.05E+04 -1.52E+06 1.85E+04 -1.37E+06
1.70E+04 -1.58E+06 1.53E+04 -1.42E+06 2.50E-03 1.70E+04 -1.58E+06 1.53E+04 -1.42E+06
1.35E+04 -1.64E+06 1.22E+04 -1.48E+06 2.56E-03 1.35E+04 -1.64E+06 1.22E+04 -1.48E+06
1.00E+04 -1.70E+06 9006 -1.53E+06 2.62E-03 1.00E+04 -1.70E+06 9.01E+03 -1.53E+06
6503 -1.77E+06 5852 -1.59E+06 2.68E-03 6.50E+03 -1.77E+06 5.85E+03 -1.59E+06
2998 -1.83E+06 2698 -1.64E+06 2.74E-03 3.00E+03 -1.83E+06 2.70E+03 -1.64E+06
739.8 -1.87E+06 665.9 -1.68E+06 2.80E-03 7.40E+02 -1.87E+06 6.66E+02 -1.68E+06
34.46 -1.88E+06 31.02 -1.69E+06 2.86E-03 3.45E+01 -1.88E+06 3.10E+01 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 2.92E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 2.98E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.04E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.10E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.16E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.22E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.28E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.34E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.40E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.46E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.52E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.58E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.64E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.70E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.76E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.82E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.88E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.94E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 4.00E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.94E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.88E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.82E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.76E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.70E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.64E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.58E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.52E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.46E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.40E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.34E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.28E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.22E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.16E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.10E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 3.04E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 2.98E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
2.36E-12 -1.88E+06 2.13E-12 -1.69E+06 2.92E-03 2.36E-12 -1.88E+06 2.13E-12 -1.69E+06
-34.46 -1.88E+06 -31.02 -1.69E+06 2.86E-03
OBTAINED FROM XTRACT
-5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
-6.00E+05 -4.00E+05 -2.00E+05 0.00E+00 2.00E+05 4.00E+05 6.00E+05
W1: P-M INTERACTION DIAGRAM
-5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
- 1 . 0 0 E + 0 5 0 . 0 0 E + 0 0 1 . 0 0 E + 0 5 2 . 0 0 E + 0 5 3 . 0 0 E + 0 5 4 . 0 0 E + 0 5 5 . 0 0 E + 0 5 6 . 0 0 E + 0 5 7 . 0 0 E + 0 5 8 . 0 0 E + 0 5 9 . 0 0 E + 0 5
AX
IAL
FOR
CE,
P (
N)
MOMENT, M (N-M)
W1: P-M INTERACTION DIAGRAM
Mn, Pn
ΦPn, ΦMn
Mu, Pu
Linear Dynamic Analysis and
Seismic Evaluation of a Full Scale
RC Model
Wall Shear APPENDIX
AD, AS, QW
hw 19950 mm Conversions
lw 1800 mm 1 mm 0.0393701 in
bw 230 mm 1 mm2 0.00155 in2
f'c 42 MPa 6.091583 ksi 1 MPa 0.1450377 ksi
fyt 295 MPa 42.78612 ksi 1 K 4.4482 kN
hw/lw 11.08333 Slender
ρl 0.004617
ρt 0.00455
Acw 414000 mm2 641.7 in2
α 2
φ 0.75
Vn,max 400.6703 K 1782.27 kN
Vn 225.0917 K 1001.258 kN
φVn 168.8188 K 750.9433 kN
φo 1.25
Vu,SAP 675 kN
Vu/φVn 0.90 OK