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AtholAthol J. CarrJ. CarrProfessorProfessor of Civil of Civil EngineeringEngineering

DepartmentDepartment of Civil of Civil andand Natural Natural ResourcesResources EngineeringEngineeringUniversity of Canterbury,University of Canterbury,

Christchurch, Christchurch, NewNew ZealandZealand..

Anlisis Ssmico de Edificios de Hormign Armado.Respuesta Dentro del Rango No Lineal

ACHISINAAsociacin Chilena de Sismologa e Ingeniera Antissmica

Santiago de Chile, 2 al 6 de junio de 2008

Section 7 P-Delta Effects

2

P-Delta EffectsIn this example when the structure deforms laterally under the horizontal force P the lateral displacement induces further moments caused by the vertical weight on the structure times the lateral displacement . The moment at the base is not Ph as would be obtained using small deflection theory but is now Ph+Mg. These extra moments will take up part of the capacity of the structure.

23

Effect of Equivalent Negative Bi-linear Slope Due to P-Delta

Mom

ent

Displacement

One of the arguments about P-Delta analyses is that then effect of the gravity loads is to reduce the effective bi-linear stiffness. This may imply a negative bi-linear factor.

With a negative bi-linear factor the structure may find it easier to go back to yielding in the initial yield direction than to yield in the other direction reducing the displacement. This will case the structure to continue to drift in one direction. This is most evident in single degree of freedom structures

4

P-Delta EffectsThese are often regarded as second order effects and are very often ignored in dynamic analyses.In terms of analysis it is always the deformed structure that is equilibrium with the applied loads.For most civil engineering structures the deformed structure has a geometry that is very close to that of the un-deformed structure and as the member orientation is almost the same as that of the original un-deformed structure we can write the equilibrium equations in theun-deformed coordinates without a significant loss of precision. This is not applicable for some classes of structures, notably suspension bridges and hanging roofs.In reality if one was to use the deformed coordinates we cannot write the equations of equilibrium until we know the deformed shape and we cannot solve for the deformed shape until we have the equations of equilibrium. The chicken and the egg problem which comes first?

35

Effects of P-Delta Requiring Extra Lateral Load Resistance

F

W1

2W

N

x 1

Fx 2

Fx 3

Some design codes (standards) require the designer to estimate the maximum inter-storey drifts in each storey and hence, from the storey gravity column forces, estimate the equivalent P-Delta lateral forces acting on the structure. These will require extra strength in the structure which already has to resist the lateral forces implied from the earthquake excitation.

6

P-Delta Effects.

If the displacements are large such for suspension structures then the non-linear equilibrium equations have to be solved, usually by incremental load methods. There are no theorems for convergence in large displacement problemsThis is in contrast to some cases involving material non-linearity where, provided we do not exceed the maximum sustainable load on the structure, iteration may give a solution.Normal structural analysis assumes that small displacement theory is valid. Here the initial geometry chosen in which the equations of equilibrium is the initial un-deformed geometry of the structure. Any displacements are assumed to be small compared with the dimensions of the structure.

47

P-Delta Effects

The alternative is to use a large displacement theory where the displacements are assumed to be significant in terms of the dimensions of the structure. This implies that columns that were vertical are no longer vertical etc. In this case only an incremental analysis is valid and at each increment of displacement the coordinates of the nodes (joints) is updated and the transformation matrices for each member have to be re-defined and new member stiffness matrices have to be computed.The total stiffness matrix for the structure will also have to be re-formed.This makes a large displacement analysis computationally more expensive

8

P-Delta Simplification

In a multi-storey frame subjected to lateral excitation the overturning moments caused by the lateral loads will increase the axial forces in the columns on one side of the structure andcorrespondingly reduce the axial forces on the other side of thestructure. This will reduce the lateral stiffness on the columns on one side with a matching ? increase of the lateral stiffness of the columns on the other side.The nett effect is no significant change in the total lateral stiffness of the frame.. This means that the lateral stiffness of the frame is reduced due to say the gravity axial column forces and then small displacement theory can be used for the rest of the analysis. The program must account for the effective lateral forces implied by the inter-storey drift acting on the vertical load carrying elements in the equilibrium equations.

59

P-Delta Simplification

With these simplifications the cost of the analysis is little different from that of a small displacement theory analysis.The limitation is that vertical earthquake components should notbe considered as these would imply a time-wise change in the total vertical load on a storey and hence change the total lateral stiffness of the storey. Sloping columns should not exist in thestructure.Such analyses are available in programs such DRAIN-2D and RuaumokoRuaumoko also offers the option of a large displacement analysis. However, for many structures the difference in the results between the full large displacement analysis and the simplified P-Delta option are not significant.In all multi-storey structural analyses, unless the gravity forces are small, at least a P-Delta option should always be used. Gravity is always acting on our structures.

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Frame Geometry 4 and 12 Storey Reinforced Concrete Frames

7.00

3.40

11.5

0

1

Floor Plan

7.00

11.5

0

Structural

2

wall

3

A

7.007.00

3.40

3.40

3.40

L2

L1

Structuralwall

L3

3.40

L4

L5

LN

B C D A

Elevation at 1 or 3

7.007.00

DB C

columnFlexible Flexible

column

Flexible column

Flexible column

Moment resisting

Moment resisting

frame

frame

The frames 1 and 3 provide the earthquake resistance in the East-West direction

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P-Delta Effects

to include p-delta Additional column

column is not deflectedForce distribution if

WN

N + 1W

W

Force distribution if column is deflected

N

yNF

XF

XF N + 1Fy

N

N + 1

WN + 1

Concept of using a pin-ended column to model P-Delta effects considering gravity loads carried by other parts of the structure but laterally supported by the part of the structure being analyzed.

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P-Delta Effects (Non-linear Geometry vs Simplified P-Delta option)

Rose12 Frame subject to El Centro 1940 N-S (x2)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 5 10 15 20 25 30

Time (s)

Hor

izon

tal d

ispl

acem

ent a

t lev

el

12 (m

)

Small displacement theory Large displacement theory P-delta included

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P-Delta Effects

Modified El Centro EW12/6/E/T

-0.600

-0.400

-0.200

0.000

0.200

0.400

0 5 10 15 20 25 30 35 40Time (sec)

Dis

p. (m

)

w/o PD w PD

The effect of P-Delta on the displacements of a twelve storey reinforced concrete frame

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Hysteretic Behaviour of Interior Column Base

Modified El Centro EW12A/6/E/T and 12A/6/E/B

With P-delta effects

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Curvature (Rad/m)

Mom

ent (

kN m

)

Takeda Bilinear

Difference in behaviour of the Elasto-plastic and Takeda hysteresis loops. The Elasto-plastic model shows creep in the lateral displacements

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Effects of 2500 Year Earthquake on Analysis

El Centro 12,500 years return period

12A/6/3101/T and 12A/6/1170/T

-0.500

0.000

0.500

1.000

1.500

2.000

0 10 20 30 40 50 60Time (sec)

Top

floor

dis

plac

emen

t (m

)

3101 w/o PD 3101 w PD 1170 wPD

The 2500 year earthquake is approximately 1.8*475 year earthquake

3101 refers to NZ Standards NZS4203:1992 and 1170 refers to NZS1170.5:2004

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P-Delta Effects

P-Delta effects are only significant if the column axial forces (P) are large (if large enough the column buckles and has no lateralstiffness) and the displacements, or inter-storey drifts (Delta) are large. If both P and Delta are small then P-Delta effects are small.The previous slide show what happens when the inter-storey drifts (column slopes) become large. The P-Delta effects become unsustainable leading to structural collapse.For tall structures always consider P-Delta effects in your analyses, gravity forces are always there.