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## Lateral Load Evaluation for Earthquake Resistant Design

Posted by Win Aung Cho on 14-Dec-2009

# Lateral Load Evaluation for Earthquake Resistant Design

Earthquake Design involves 3 analysis catagories in accordance with the UBC 97.
• Equivalent Static Load Method (Static Procedure)
• Response Spectrum Analysis (pseudo dynamic analysis)
• Time History Analysis (Real Dynamic Analysis)

## Response Spectrum Analysis

Lateral load produced by horizontal inertia forces are determined using structural dynamic properties and expected ground acceleration.
This method needs response spectrum curves relation between natural periods of building having single degree of freedom versus spactral accelerations of building. Response-spectrum analysis seeks the likely maximum response to motion equations. The earthquake ground acceleration in each direction is given as a digitized response-spectrum curve of pseudo-spectral acceleration response versus period of the structure.
Magnitude of design response spectrum can be defined by one of following three methods.

### Spectrum Curves

Magnitude of design response spectrum can be defined by one o following three methods
1. UBC 97 elastic design response spectrum containing damping of 5% Spectral Accelerations (gís) is equal to Ca at period of 0 sec, 2.5Ca between To and Ts, linearly varied between 0 sec to To and Cv/T when period is greater then Ts. Where Ts is Cv/2.5Ca and To is 0.2 times Ts. The seismic response coefficients Ca and Cv account for the potential amplification of the ground vibration generated at a specific site by an earthquake.
2. Site-specific design spectrum from the actual building location.
3. A spectrum ground motion time history analysis using accelerometer data from one or more earthquake can be performed.

### Dynamic/Modal analysis

Dynamic/Modal analysis must be active and choose one method and apply appropriate information
1. Eigen Vector Analysis
This method determines the undamped free-vibration mode shapes and frequencies of the system. These natural modes provide an excellent insight into the behaviour of the structure.
2. Ritz-Vector Analysis
This method seeks to find modes that are excited by a particular loading. Ritz vectors can provide a better basis then do eigenvectors when used for response-spectrum or time-history analyses that are based on modal superposition. Dynamic analyses based on a special set of load-dependent Ritz vectors yield more accurate results then the use of the same number of natural mode shapes. The first Ritz vector is the static displacement vector corresponding to the starting load vector.

You may specify any number of starting load vectors. Each starting load vector may be one of the following:
• An Acceleration Load in the global X, Y, or Z direction
For response-spectrum analysis, only the Acceleration Loads are needed. For modal time-history analysis, one starting load vector is needed for each Load Case or Acceleration Load that is used in any modal time-history analysis.

### The steps for response spectrum dynamic analysis

1. Construct a Lumped Mass
2. Calculate a mode-shape factors
3. Calculate the period Tm for each mode
4. Calculate the spectral acceleration (Sa,m) and seismic design coefficient for each mode from the UBC normalized response spectra
5. Calculate the base shear for each mode
6. Calculate the participating mass fraction (PM) for each mode
7. Combine the base shears into the design dynamic lateral force, Vdyne using SRSS method. (Sum of PM >= 0.9)
8. Calculate the lateral force Vstatic according to equivalent lateral force procedure
9. Determine the scaling factor for structure
• Regular structure scale = 0.9*Vstatic/Vdyne
• Irreegular structure scale = Vstatic/Vdyne
• Regular structure scale = 0.8*Vstatic/Vdyne when site-response spectrum is used
10. Distribute the scaled-up base shear to each level
11. Determine the raw deflections and element forces
12. Combine the raw results using SRSS method

### Input Steps Using ETABS 8.1.3

1. Define mass
1. Select menu item: Define > Mass Source
2. Select mass definition option: From Loads
2. Define Response Spectrum Function
1. Select menu item: Define > Response Spectrum Functions
2. Choose function type: Add UBC 97 Spectrums
3. Click: Add New Function Parameter (Ca, Cv)
4. Type function name: FUNC1
3. Define Modal Analysis Case
1. Select menu item: Set Analysis Option
2. Set Dynamic Analysis Parameter
3. Click: Eigenvectors
4. Enter maximum number of modes = 5
4. Define Response Spectra Analysis Case
1. Select menu item: Define > Response Spectrum Cases
3. Type case name = EQX
4. Set Structural and Function Damping = 0.05
7. Input Response Spectra
8. Choose FUNC1 for U1 and Enter Scale Factor = 32.2 (response values in Gís, unit dependent)
9. Click: Ok
1. Select menu item: Define > Combinations
3. Make Necessary Design Load Combination or Let Software to Define
Note: Response spectrum will automatically give +/- values
6. Perform Response Spectrum Analysis
1. Select menu item: Analyze > Run Analysis
2. Click: Run Now
7. Display output results in tabular form
8. Verification computations
2. Verify: total mass = total dead load / gravity = 1183.75 / (32.2 x 12) = 3.064
1. Mass = 3.1497 (Export joint masses to Excel and sum)
2. Weight = Mass x Gravity = 3.1497 x 12 x 32.2 = 1217 kips
3. Verify: modal participating mass ratio > 90%
1. Ratio = 92.766% for UX
2. Ratio = 93.192% for UY
4. Verify: response spectra base shear = response x weight = 246.96 kip
1. Response for period = 1.272 sec => 0.25 G => 0.25 x 1183.75 = 295.94 kip
2. Period for Response= 0.209 G => 1.53 sec
3. 0.209 x (1183.75) = 246.96 kips
5. Verify: fundamental period
1. Limiting that period by 1.4TA for zone 2,3 and 1.3 for zone 4 (Method B ñ UBC)
2. TA=Ct x Hn0.75 = 0.03 x 630.75 = 0.671 => 1.4 x 0.671 = 0.939 sec <= use this
3. Structure period T=2pi sqrt(W/(GK))
K = 1 kips / 0.025268 in = 39.576 k/in (from unit load case)
T = 2 x 3.1416 x sqrt[1183.75 / (32.2 x 39.576 x 12)] = 1.748 sec
9. Compare the Base Shear results against dynamic base shear determined from clause 1631.5.4
1. Vdynamic = (90% regular ) or (100% irregular) x statics
2. Minimum Vdynamic = V dynamic from Spectrum/R
Vdynamic = 295.94/5.5 = 53.807 kip
Vstatic = (Cv I / RT ) W = 0.23 /(5.5 x 0.939) x 1183.75 = 52.718 kip
[0.11 Ca I W < V < (2.5 Ca I /R) W ] => 28.647 < 52.718 < 118.373
Vstatic = 0.9 x 52.718 = 47.446
3. Use the greatest value to determine the scale factor => Scale Factor = Vstatic / Vdynamic
Scale = 47.446 / 53.807 < 1 <= Scale=1
4. Re-enter scale factor I/RxGxScale instead of G [32.2 ft/sec2]
10. Notes
1. It is preferred to have the same stiffness for you building in both directions in order to resist the earthquake forcesÖ
2. UBC-97 load combination is very close to ACI-318 combinations
3. Earthquake force in UBC-97 is factored by 1.4, so in service load combination you will notice that earthquake force is divided by 1.4
4. Mass participation should be at least 90% in both direction X, Y
5. For Z direction you need to perform vibration analysis
6. P-delta analysis needs to be performed as geometrical nonlinearity
7. Check if Push-Over analysis is needed because Mechanism may occur before reaching targeted displacement (calculated from linear analysis procedure).
8. Orthogonal load combinations (100%Rx + 30%RyÖ.ect) can be used if you do not use SRSS (square root of sum of squares) and CQC (completing Quadratic combinations)
9. Ductility of you structure is very important and you need to check it with acceptance percentage specified
11. UBC References
1. Seismic Zone Factor ( UBC Tbl 16 ñ I)
2. Soil profile type ( UBC Tbl 16 ñ J)
3. Seismic Dynamic Response spectrum coefficients Ca & Cv ( UBC Tbl 16 ñ R & 16 ñQ)
4. Importance Factor I (UBC Tbl 16 ñ K)
5. Measure of Over Strength R (UBC Tbl 16- N)
6. Weak Storey ( Tbl 16 ñ L)
7. Torsional Irregularity (Tbl 16-M)
8. If there is any Torsional Irregularity exists, orthogonal effects shall be considered (Ref. 1633.1)
12. Remarks
1. In order for a structure to have equal resistance to earthquake motions from all directions, the CQC3 method should be used to combine the effects of earthquake spectra applied in three dimensions. The percentage rule methods have no theoretical basis and are not invariant with respect to the reference system.
2. The CQC method has a sound theoretical basis.
3. Engineers, however, should clearly understand that the response spectrum method is an approximate method used to estimate maximum peak values of displacements and forces and that it has significant limitations.
4. It will never be accurate for nonlinear analysis of multi-degree of freedom structures.
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