p delta for semirigid diaphragms

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P-Delta for Semirigid Diaphragms 1

Implementation of P-Delta Effects for

Semirigid Diaphragms

Author: Bulent N. Alemdar

Date: July 15, 2009

IntroductionP-Delta effects are due to adverse effects of gravity loads on building responses under lateral loads(such as wind and seismic load cases). Thus, additional forces arise because of coupling between

gravity and lateral loads. RAM Frame adapted a non-iterative method to address P-Delta effects for

rigid diaphragms in which global building stiffness are modified due to diaphragm mass\weights.

Unfortunately, this method is strictly suited for only rigid diaphragm idealization where diaphragmresponse is represented with three degrees of freedom (two for lateral movements and one for rotation

around global Z-axis).

In order to capture these 2nd

order effects (P-Delta), an incremental-iterative solution strategy might

be necessary to robustly and accurately capture interaction between gravity and lateral loads (thus,analysis should be run with load combinations, not with load cases). However, such a solutionstrategy is computationally very demanding since equilibrium equations are solved at each iteration,

which nothing but simply involves assembling of global stiffness and load matrices and then solving

them at each iteration. Albeit computational expenses, this approach is necessary one to address

nonlinear effects in frame responses.

Not only computing challenges, but also running load combinations within an incremental iterative

solution scheme also leads to other challenges that are needed to be addressed, including live loadreductions, overwhelming number of load combinations, handling leaning columns and response

spectra load cases (dynamic load cases).

Even though an incremental-iterative solution with load combinations seems right method to use, anaccurately enough solution can be still pursued without using an incremental-iterative analysis to

address P-Delta effects. In the following sections, the proposed method is explained.

Proposed MethodFor semirigid diaphragms, a two-phase solution is implemented. When an analysis is kicked off,

these steps are applied as follows: if any semirigid diaphragm is detected, then

First Phase: running only for gravity load cases (dead and live) and storing axial forces

computed for columns that are located inside semirigid diaphragms. These axial forces are

also referred to as P-Delta Axial Forces for columns.

Second Phase: the P-Delta Axial Force stored for a column is utilized to compute element

geometric stiffness matrices for the column. In this phase, all remaining load cases are run

while column stiffnesses are modified with their geometric stiffness matrices.




This process is automated and transparent to the user. Hence, the program conducts two separate

analysis; one with gravity loads cases and one with remaining load cases. Note that gravity load casesare needed to be selected. Otherwise, the program shows a warning message, indicating that no

gravity load case is selected and P-Delta effects for semirigid diaphragms is ignored.

The above procedure can be regarded as a 2 iteration analysis: in the 1st

iteration, P-Delta axial forcesin columns due to gravity loads are detected and then, in the 2

nditeration, these axial forces are used

to predict P-Delta effects for all remaining load cases. On the other hand, this should not be compared

to real incremental iterative approach, where all loads are applied at the same time, and the solutionincrementally repeated until all equilibrium equations are satisfied during an iterative process (so that

there is no unbalanced loads left).

It should be noted that the current implementation also satisfies equilibrium equations but it does not

iteratively seek equilibrium conditions until full convergence is achieved, instead it stops at the end

of 2nd

iteration. The main idea imposed with this approach is that axial forces due to gravity loads do

not significantly change under lateral loads (in fact, this is the same idea assumed for the current P-Delta method implemented for rigid diaphragms).

Live load reductions are also applied to P-Delta axial forces for each column. Furthermore, Response

spectra load cases also utilize calculated P-Delta axial forces.

ExamplesTest Model 1

A simple 3D frame (45 ft bay by 60 ft bay in X and Y directions, respectively) with W12x87 and

W12x79 cross-section sizes for all columns and beams is analyzed. The story height is 15 ft, and it is

subjected to a 120 psf dead load and mass load (the total applied load is 324kips). In addition, two100 kip point loads are applied at south and north frames. No self weight and mass are considered.

Out-of-plane stiffness for semirigid diaphragm is included.

Figure 1. Test Model 1




Analysis is run once with rigid diaphragm and once with semirigid diaphragm. The results are

tabulated as shown in the following table (note that both one-way and two-way load distributionyields the same results). Below maximum deflections are reported, that are measured in X-direction

at north-west node.

Table 1. RAM Frame Analysis Results for Test Model 1

Without P-Delta (in.) With P-Delta (in). % Increase

Analysis with Rigid Diaph. 4.49 4.65 3.56

Analysis with SemirigidDiaph.

3.61 3.76 4.16

The same runs are also carried out in STAAD.Pro. For the solution with P-Delta, an iterative P-Deltamethod is utilized.

Table 2. STAAD.Pro Analysis Results for Test Model 1


Analysis with Rigid Diaph. 4.48 4.72 5.36

Analysis with SemirigidDiaph.

3.59 3.74 4.18

It is interesting to note that results for rigid diaphragm show more deflections than those with a

semirigid diaphragm. This can be attributed to out-of-plane stiffness of the diaphragm that plays an

important role in diaphragm action (its restraining effect due to its finite stiffness in the out-of-planedirection reduces the drift).




Test Model 2

The tutorial model is used to test the new implementation. It is modified from original model (allfloors are defined with 2way deck; the roof is not sloped; all floors subjected to a uniform dead load

of 75 psf and mass load of 75 psf; no line or point loads applied; no self weights and mass included

and all openings are deleted). Also note that out-of-plane deflections for the semirigid diaphragm is

included.


The following table shows the results (max. deflection) obtained from RAM Frame and STAAD.Pro

for wind load case (wind_IBC03_1_x). Again, an iterative method is used for STAAD.Pro.



Analysis with Semirigid

Diaphs.

0.540 0.565 4.62

Table 4. STAAD.Pro Analysis Results for Test Model 2



Diaphs.0.553 0.559 1.09




Test Model 3

Figure 3. Tower Model

For seismic load case in X direction, the following results (max. deflection) are obtained (note that all

diaphragms are defined as semirigid):



Max. Deflection (in.) 14.103 14.515 2.92

Structural Period (T1,s) 6.736 6.868 1.96

Table 6. STAAD.Pro (iterative analysis) Analysis Results for Test Model 3


Max. Deflection (in.) 13.764 13.757 -0.05

Structural Period (T1,s) 6.676

Also, the same model is run with SAP2000 (iterative P-Delta).

Table 7. SAP2000 Analysis Results for Test Model 3


Max. Deflection (in.) 14.293 14.80 3.55Structural Period (T1,s) 6.736 6.895 2.36




Test Model 4


For seismic load case in X direction, the following results (max. deflection) are obtained (note that all

diaphragms are defined as semirigid):

Table 8. RAM Frame Analysis Results for Test Model 4Without P-Delta (in.) With P-Delta (in). % Increase


Diaphs.0.238 0.241 1.26

Table 6. STAAD.Pro (iterative analysis) Analysis Results for Test Model 4



Diaphs.0.237 0.288 21.52

Table 7. SAP2000 Analysis Results for Test Model 4

Without P-Delta (in.) With P-Delta (in). % IncreaseAnalysis with Semirigid

Diaphs.0.246 0.2503 1.75

p delta for semirigid diaphragms

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