Diff Shortening of Tall Steel Building Columns Taranath

8/12/2019 Diff Shortening of Tall Steel Building Columns Taranath

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Differential shorteningof tall steel building columns

By Bungale S. Taranath, Ph.D., S.E.

JANUARY 2011


2/82 PDH Professional Development Advertising Section Bentley Systems, Inc.

Differential shortening of tall steel building columnsBy Bungale S. Taranath, Ph.D., S.E.

Columns in tall buildings

experience large axialdisplacements because

they are relatively long

and accumulate gravity loads from

a large number of floors. A 60-story

column of a steel building, for exam-

ple, may shorten as much as 4 to 5

inches (100 to 125 mm) at the top.

If such a shortening is not given due

consideration, problems may develop

in providing level floors and assuring

trouble-free performance of building

cladding systems. Proper awarenessof this problem is necessary on the

part of structural engineer, architect,

and curtain wall supplier to avoid

unwelcome arguments, lost time, and

money.

The maximum shortening of a

column occurs at the roof level, reduc-

ing to zero at the base. Very little can

be done to minimize the physical

phenomenon of shortening, but the

design team should be aware of the

problem particularly at the buildingexterior, so that soft joints are provided,

if need be, between the building frame

and cladding to prevent axial load

being unwittingly transferred into the

buildings faade. Before fabrication

of cladding, the in-place elevations of

the structural frame should be verified,and if required, the cladding should

be manufactured to fit the existing

field condition of the steel frame. The

design should provide for sufficient

space between the cladding panels to

allow for the movement of the struc-

ture. Insufficient space may result in

bowed cladding components or, in

extreme cases, the cladding panels

may even pop out of the building.

A similar problem occurs when

mechanical and plumbing lines areattached rigidly to the structure. Frame

shortening may force the pipes to act

as structural columns resulting in their

distress. A general remedy is to make

sure that nonstructural elements do

not bear vertical loads. This is done by

separating them from the structural

elements.

The axial loads in all columns of a

building are seldom the same, giving

rise to the problem of so-called differ-

ential shortening. The problem ismore acute in a composite structure

because steel erection columns that

are later encased in concrete are typi-

cally slender, and are therefore subject

to large axial loads during construc-

tion. Determining the magnitude of

axial shortening in a composite systemis complicated because many of the

variables that contribute to the short-

ening cannot be predicted with suffi-

cient accuracy. Consider, for example,

the lower part of the composite

column that is continually undergo-

ing creep. The steel erection column

during construction is partly enclosed

in concrete at the lower floors, with

the bare steel column projecting

several floors above the concreted

level. Another factor that is difficult topredict is the gravity load redistribu-

tion due to frame action of columns

and, if the building is founded on

compressible material, the settlement

of the foundation is another factor

that influences the relative changes

in the elevation of the columns. The

magnitude of load imbalance between

any two columns is continually chang-

ing, making an accurate assessment of

column shortening rather a challeng-

ing task.Differential rather than the abso-

lute shortening of columns is more

significant. Relative displacement

between columns occurs because

of the difference between the axial

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Learning ObjectivesAfter reading this article, you should be able to do the

following:

Explain differential shortening of columns in tall steel

buildings.

Calculate axial displacement in columns using concise

hand calculations.

Explain the concepts behind column length correc-

tions.

Explain the importance of column shortening verifica-

tion during construction.

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stresses, or P/A ratios, of columns. P

is the axial load on and A is the area

of the column under consideration.

If all columns in a building were to

have the same P/Aratio under gravity

loads, there would be no relative verti-

cal movement between the columns.In typical buildings, however, this

condition is seldom present. This is

because in buildings not all columns

are designed for the same axial loads.

For example, the design of frame

columns is governed by the combined

gravity and lateral loads while non-

frame columns are designed essen-

tially for gravity loads only. This results

in a rather large difference in the P/A

ratios between the exterior and inte-

rior columns. The differential columnshortening between perimeter and

interior columns may result in slop-

ing floors leading to unwelcome prob-

lems in setting partitions, doors, and

ceilings.

Consider, for example, a tubular

system with closely spaced exterior

columns and widely spaced interior

columns. Because of their large tribu-

tary areas, the interior columns are

more than likely to have large P/A

ratios. The exterior columns, on theother hand, usually have a small P/A

ratio for two reasons. First, their tribu-

tary areas are small because of their

close spacing. Second, they are sized

for stiffness to limit lateral displace-

ments resulting in areas much in

excess of those required from strength

consideration alone. Because of this

imbalance in the gravity stress level,

these two groups of columns undergo

different axial shortenings; the interior

columns shorten much more than theexterior columns.

The opposite happens in buildings

with interior-braced core columns and

widely spaced exterior columns; the

exterior columns experience more

axial shortening than the interior

columns. The behavior of columns in

buildings with other types of struc-

tural systems, such as interacting core

and exterior frames, tends to be some-

where in between these two limiting

cases.

Simplified method of calculating

z

, axial shortening of columnsIn a steel building, typically the

cross-sectional area of a column

increases in two-story increments from

a minimum at top to a maximum at

the base as shown in Figure 1. Theincremental steps in column areas are

due to the finite choice of available

column shapes. Similarly, the axial load

on a column increases at each floor in

a stepwise manner. In tall buildings,

the significance of these incremental

steps diminishes rapidly, allowing us

to make the following assumptions

that can be used to derive a simpli-

fied formulation for axial shortening

of columns. The first

assumption relates to

the variation of grav-

ity loads, which may be

assumed to increase linearly from top

to the bottom. The second is similar to

the first but applies to the variation ofcolumn areas. However, a linear varia-

tion using the actual column area at

the bottom appears to underestimate

the actual shortening of columns.

A slight modification in which the

column area at the bottom is taken

equal to 0.9 times the actual area (0.9

x AB) appears to work well in predict-

ing axial shortening.

Figure 1: Variation of cross-sectional area of a high-rise column.



Differential shortening of tall steel building columns

Derivation of simplifiedexpression for

z

Integral calculus in conjunction

with the simplified assumptions stated

earlier is used to derive the following

equation for z:

z= P

b/E[(-1/) ln(1 -z/A

b)]

/E[-1/2(az+ Abln(1 -z/A

b))]

where

L= height of the building

z= axial shortening at a height x

(also denoted as z), above the foun-

dation level

At= column area at top

Ab= modified column area at bottom

equal to 0.9 times actual area ofcolumn at bottom = 0.9 x A

B(where A

B

is actualcolumn area at bottom.

Ax= area of column at height x (also

denoted as z) above foundation

level

= rate of change of area of column

Pt= axial load at top

Pb= axial load at bottom

Px= axial load at height xabove foun-

dation

= rate of change of axial load

E= modulus of elasticity of steelIn = natural logarithm

The derivation of the equation is

given in Wind and Earthquake Resistant

Buildings: Structural Analysis and Design

(Bungale S. Taranath, Ph.D., S.E., CRC

Press: Taylor & Francis Group, 2005)

and Structural Analysis and Design

of Tall Buildings: Steel and Composite

Construction (Bungale S. Taranath,

Ph.D., S.E., CRC Press: Taylor & Francis

Group, 2011). Definitions of the vari-ables used in the equation are shown

schematically in Figure 2.

ExampleGiven: (See Figure 3)

Height of building: L = 682 feet =

8,184 inches (207.8 m)

Modulus of elasticity: E= 29,000 ksi

(200 x 103 MPa)

Axial of load at top: Pt = 53 kips

(237.5 kN)

Area of column at top: At= 12.48

Figure 2: Axial shortening of columns; closed-form solutions: (1) axialshortening

z; (2) variation of column area; (3) axial load variation; (4)

unit load at height z; (5) axial strain; (6) axial shortening.

Figure 3: For the example below, (1) axial load variations; (2) actual and assumed columncross-sectional areas.



square inches (8,052 square mm)

Axial load at base: Pb= 2,770 kips (12.32 x 103 kN)

Actual column area at base: AB = 147 square inches

(94.84 x 103 square mm)

Reduced column area at base: Ab = 0.9 x 147 = 133.3

square inches (86.0 x 103 square mm)

Required: Find the axial shortening of the top column

Solution: Since column shortening is calculated at top,

z= L , therefore

= (Ab A

t)/L= (133 12.48)/8,184 = 0.01476 sq. in./in.

= (Pb P

t)/L= (2,770 53)/8,184 = 0.332 kip/in.

ln (1 L/Ab) = ln (1 (0.01476 x 8,184/133.3))

= ln (0.09362)

= -2.368

Therefore,Lat top = 2,770/29,000 [(-1/0.01476) x (-2.368)]

0.332/29,000[-4,590.15 (0.01476 x 8,184 +

133.3 x -2.36847)]

= 15.32 10.20

= 5.12 in.

Similarly, the axial shortening is calculated at various

heights by substituting appropriate values for z.

Column length corrections, c

After determining axial shortening of building columns,

the next step is to assign a column length correction cforeach column. The objective is to attain as level a floor as

practical. The correction c is thus the difference between

the theoretical height of a given column and its actual

height after it has shortened. The magnitude of correction

cin a tall building of 60 stories is rather small, perhaps 1/8

inch (3.17 mm) per floor, at the most. Therefore, instead of

specifying this small correction at each level, in practice it

is common to specify lumped corrections at a few selected

floors. For example, in lieu of 1/8 inch correction at each

level, the designer would lump the correction at every

eighth floor. Thus cwould be equal to 1/8 x 8 = 1 inch

(25.4 mm). See the above referenced texts for further infor-mation on this topic.

Column shortening verification during constructionConsider Figure 4, a hypothetical building that is 48

stories tall. Identified therein are two columns: C1, an inte-

rior column with a large tributary area; and C2, an exterior

column of framed tube with a relatively small tributary area.

Under gravity loads, C1would shorten more than C

2for two

reasons: C1, designed only for gravity loads, has a P/A ratio

that is relatively high; and C2, designed as a frame column,

has its P/A ratio significantly less than that for C1because it

is lightly loaded under gravity loads.

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Assume that you

as the engineer for

the project have speci-

fied column length corrections to C1

at levels 8, 16, 24, 32, 40, and 48

with correction of 2 inches specifiedat level 24. Let us say that when steel

erection is at the level 24, the steel

erector surveys the top elevations of

columns, reports the top of column

C1is 1 inch higher than the top of C

2,

and requests that you confirm if this

is acceptable in view of the fact that

additional shortening of the column is

yet to occur.

Further calculations are needed to

verify that this 1-inch overlength of C1

will indeed come down after the appli-cation deadloads at levels 24 through

roof. This concept of verifying the over-

length of columns during construction

is shown in Figure 5. Note that Rn

shown therein corresponds to the 1

inch discussed here for the hypotheti-

cal building.

ConclusionAxial displacement in tall steel build-

ing columns is likely to be a problem if

not given proper consideration duringdesign. Fortunately, clear and concise

formulas are available for calculating

axial shortening, as are methodologies

for mitigating the adverse effects of

relative axial displacements.

Figure 5: Interpretation of column overstrength: (1) n= column shortening at nth level

due to loads on the entire height of column; (2) B= column shortening at nth level dueto loads imposed at and below nth level; (3)

Rn= column shortening yet to occur due to

loads above nth level.

Figure 4: Hypothetical framing plan: column C1, designed for gravity loads only, shortens

more than C2, designed for both gravity and lateral loads. Compensating for relative eleva-

tion difference between columns is of importance in tall buildings.

Bungale S.

Taranath, Ph.D.,

S.E., is a corpo-

rate consultant to

DeSimone ConsultingEngineers, a consult-

ing firm with offices

in New York; Miami;

San Francisco; New

Haven, Conn.; Las

Vegas; Hong Kong; and Abu Dhabi, and

has served as a principal or senior engi-

neer for several firms. He is the author

of five books on tall building design, and

has conducted seminars on tall build-

ing design around the world. He can be

reached at [email protected].



1. In a 60-story, steel-framed build-

ing, which of the following is likely

to be true?

a) Columns may not shorten at all.

b) Columns may shorten by 4 to 5

inches.

c) Columns may shorten by as much

as 1 foot.

d) Proper construction practice will

prevent significant axial shortening.

2. Which of the following statementsis most true?

a) Axial displacement is disastrous,

and should be avoided with bracing

systems and extremely rigid materi-

als.

b) Axial displacement is probably inevi-

table in large buildings, which is why

most large buildings have slightly

uneven floors.

c) Axial displacement is a minor prob-

lem for buildings that are less than

40 stories tall, and can usually beignored.

d) Large differential axial displace-

ment is inevitable in tall steel build-

ing columns, so building designers

should include mitigating meth-

ods, such as providing soft joints

between the building frame and

cladding components, based on

actual calculations and, preferably,

measurements during construction.

3. If nonstructural building elements

are not separated from verti-

cal loads, which of the following

might occur?

a) Piping and ducting may be forced to

act as structural columns, and suffer

distress.

b) Cladding system components may

bow or pop out.

c) Floors may be out of level.

d) All of the above.

4. The phrase differential shortening

refers to which of the following?

a) Building columns that shorten by

different amounts.

b) Column shortening that varies by

floor.

c) Column shortening that varies

according to temperature.

d) Building columns that shorten

during construction, as a result of

progressive loading.

5. Differential shortening may bedue to which of the following?

a) Building columns that have different

P/Aratios.

b) Building columns that support differ-

ent loads.

c) Building columns designed for grav-

ity only, and for gravity and lateral

load.

d) All of the above.

6. The P/A ratio is the ratio

between?

a) Column height and axial load.

b) Axial loads of perimeter and interior

columns.

c) Axial load and column area.

d) Axial compression measured at

the top and midpoint of building

columns.

7. In a building with closely spaced

exterior columns and widely

spaced interior columns, which ofthe following is most likely to be

true?

a) Exterior columns are likely to experi-

ence greater axial compression than

interior columns.

b) Interior columns are likely to experi-

ence greater axial compression than

exterior columns.

c) Axial compression will be roughly

equal for all columns.

d) This is a tubular system, and axial

compression will be minimal.

8. Which of the following assump-

tions is made in order to simplify

the calculation of axial shortening of

columns in steel buildings?

a) The column area at the top is

assumed to be 0.9 times the actual

column area.

b) The column area at the top is


column area floor.

c) The column area at the bottom is

assumed to be 0.9 times the actualcolumn area.

d) The column area at the bottom is


column area.

9. When determining column length

corrections, c is which of the

following?

a) The difference in shortening between

exterior and interior columns.

b) The difference in shortening between

the top and midpoint of any particu-lar column.

c) The difference in shortening due to

change in column area.

d) The difference between the theoreti-

cal height of a given column and its

actual height after it has shortened.

10. Which of the following statements

is most true?

a) Onsite verification of axial displace-

ment during construction is ineffec-tive due to progressive loading.

b) Due to a variety of factors that cant

be determined in advance, onsite

verification is the only way to deter-

mine axial shortening.

c) Onsite verification during construc-

tion is an excellent way to check

axial displacement calculations.

d) Differential shortening cant be

predicted by calculation, and can

only be measured during construc-

tion.


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Diff Shortening of Tall Steel Building Columns Taranath