p-delta effect on columns

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Muthu Sankari
P-Delta Effect on long columns
18-Nov-2020

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Contents
I. Introduction
II. Types of Columns
III. Long column behavior
IV. P-Delta Effects
V. Columns bent in single and double curvature
VI. Calculation of additional moment
VII. Design Example

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1. Introduction

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Introduction
Columns
A Reinforced concrete column is a compression member whose greater lateral dimension is less than or equal to
four times its lesser lateral dimensions, and in which the reinforcement is taken into account when considering
its strength.
D <= 4b

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2. Types of Column

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Types of Columns
Types of columns
• Long (Slender) column
• Short column
 Short column:
• Ratio of (le/h) <12 then Short Column
• Load carrying capacity is more
• Subjected to compressive stress
 Long column:
• Ratio of (le/h) >12 then long Column
• Load carrying capacity is less
• Subjected to buckling stress
Where
le is the effective height in the plane of buckling under consideration
h is the depth of the cross section in the plane of buckling under consideration.

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Types of Columns
Failure of columns
•“Long” columns fails structurally (buckling failure)
Slender or long columns may fail at a much lower value of the load when sudden lateral
displacement of the member takes place between the ends (Geometric failure).
• “Short” columns fails materially (yielding failure)
Short columns fail by reaching the respective stresses indicating their maximum carrying
capacities
Rebar failure
in tension
Concrete crush
in compression

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3. Long column Behavior

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Long column behaviour
End condition
The effective length Le of the column depend on the end conditions

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Types of Columns
Effective height of the column (Table 11.1 of IRC 112:2020)

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4. P-Delta Effects

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P-Delta Effects
P-Delta Effect
 P delta is a term coined from P, the load and delta, the lateral deformation
 P-delta effects are also known as second-order effects since their magnitude depends on the amount of initial
displacement or deflection.
 Generates additional shear forces and bending moments in columns because of the deformed shape.
 The moments generated will be equal to the load acting on the column times the horizontal displacement.
 Hence it is necessary to check the column capacity particularly in case of slender columns so that they do not
fail in case of these additional moments along with the axial loads

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P-Delta Effects
P-Delta Effect
 There are two distinct types of P-delta effects:
• P-∆ (sometimes referred to as “large P delta” or “P-large delta”)
• P-δ (sometimes referred to as “small P-delta” or “P-small delta”)
 P- ∆ has reference to the effects of the vertical loads acting on the laterally displaced structure.
 P- δ has reference to the effects of the axial load in an individual member subject to a deflection (curvature)
between its endpoints
• P- ∆ Effect of loads acting on the displaced location of joints or nodes in a structure
• P- δ Effect of loads acting on the deflected shape of a member b/w joints or nodes
P-∆
P-δ

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5. Columns bent in single and double curvature

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Columns bent in single and double curvature
Columns bent in single curvature • Fig-1(a ) shows column bent in single curvature under axial load
P < Pcr with constant moment Pe.
• The deflection profile marked by dotted line is due to the
constant moment.
• However, there will be additional moment of Py at a distance z
from the bottom of column which will deflect the column further,
as shown by the solid line.
• The constant moment Pe and additional moment Py are shown in
Fig-1(b) .
• Total moment M = Mo + Py = P(e + y)
• The maximum moment is P(e + Δ) at the mid-height of the
column.
Mmax = Mo + P Δ = P(e + Δ)
Fig -1(a) : Deflections Fig -1(b) : Moments
Figure-1: Column bent in single curvature (Deflection),(H=0)

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Columns bent in single and double curvature
Columns bent in Double curvature
• Fig-3: Shows a column subjected to equal end moment of
opposite signs.
• From the moment diagrams Mo and Py (Fig-3(b) and(c)), it is
clear that though Mo moments are maximum at the ends,
the Py moments are maximum at some distance from the
ends.
• The total moment can be either as shown Fig-3(d) or (e) . In
case of Fig.3d, the maximum moment remains at the ends
and in Fig.3(e) , the maximum moment is at some distance
from the ends, where Mo is comparatively smaller than Mo
max at the ends. Accordingly, the total maximum moment is
moderately higher than Mo max.

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6. Calculation of additional moment

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Calculation of additional moment
Calculation of additional moment
•The design of slender columns, in principle, is to be done
following the same procedure as those of short columns.
However, it is essential to estimate the total moment i.e.,
primary and secondary moments considering P- Δ effects.
These secondary moments and axial forces can be determined
by second-order rigorous structural analysis.
•However, Cl.39.7 of IS 456 recommends an alternative
simplified method of determining additional moments to avoid
the laborious and involved second-order analysis
M = Mo + P ea= Mo +Ma
Mo –Initial moment
Ma – Additional moment due to secondary effects

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7. Design Example

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Design example
Design Example

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Design homework
Design Homework
Try to solve the same design example using the below support condition by MIDAS, Manual calculation
compare the results.
• Both ends fixed