Lecture # 3 بسم الله الرحمن الرحيم Design of Concrete Structure II University of Palestine Instructor: Eng. Mazen Alshorafa

Lecture # 3

الرحمن الله بسمالرحيم

Design of Concrete Structure II

University of Palestine

Instructor:

Eng. Mazen Alshorafa

Page 1



Columns

Instructor:


b

h

l

P

h

b

According to ACI Code a structural element with a ratio of

height-to least lateral dimension exceeding three used

primarily to support compressive loads is defined as column.

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Columns

Instructor:


Sec ASec A

Sec A-AMain beam

Column

Column

Beam

Loads

Columns are vertical compression members of a structural frame

intended to support the load-carrying beams. They transmit loads

from the upper floors to the lower levels and then to the soil through

the foundations.

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Instructor:


Columns

Usually columns carry bending moment as well, about one or both

axes of the cross section, and the bending action may produce tensile

forces over a part of the cross section

The main reinforcement in columns is

longitudinal, parallel to the direction of

the load and consists of bars arranged

in a square, rectangular, or circular

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Instructor:


1- Form and arrangement of reinforcement

Types of Columns

Columns are divided into three types

1- Tied Columns

It is a column in which the longitudinal reinforcement bars are tied together

with separate smaller diameter transverse bars (ties) spaced at some interval

along the column height. (Figure a)

2- Spirally-Reinforced Columns

It is a column in which the longitudinal bars are arranged in a circle surrounded

by a closely spaced continuous spiral. (Figure b)

3- Composite Columns

It is a column made of structural steel shapes or pipes surrounded by or filled

by concrete with or without longitudinal reinforcement. (Figure c)

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Types of Columns

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Instructor:


Columns may be divided into two categories

1- Short Columns, for which the strength is governed by the

strength of the materials and the geometry of the cross section

2- Slender columns, for which the strength may be significantly

reduced by lateral deflections.

2- Length of the column in relation to its lateral dimensions.

Types of Columns

3- Position of the load on the cross-section

Columns can be classified as

1-Concentrically loaded columns, are subjected to axial force only

2-Eccentrically loaded columns, are subjected to moment in addition

to the axial force.

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Instructor:


Behavior of Tied and Spirally-Reinforced Columns

Columns

Failure of a tied columnFailure of a spiral column

Deformation

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Factored Loads and Strength Reduction Factors

Columns

Factored Loads

For gravity loads only,

Pu = 1.2 PD+1.6 PL

For dead, live and wind loads,

Pu = 1.2 PD+1.0 PL+1.6 PW

For dead and wind loads,

Pu = 0.9 PD + 1.6 PW or Pu = 1.2 PD + 0.8 PW

For dead, live and earthquake loads,

Pu = 1.2 PD+1.0 PL+1.0 PE

For dead and earthquake loads,

Pu = 0.9 PD + 1.0 PE

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Instructor:


Columns

Strength Reduction Factors

Strength condition Φ

ACI Code specifies Φ values or strength reduction factors for most

situations as in the following table

Tension-controlled sections (εt ≥ 0.005) 0.90

Compression-controlled sections (εt ≤ 0.002)

Members with spiral reinforcement 0.70 Other reinforced members 0.65

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Instructor:


1- Nonsway Frames (braced)It is a structural frames whose joints are restrained against lateral

displacement by attachment to rigid elements or bracing

According to ACI Code

a column in a structure is nonsway if

Shear wallColumns

Beams

Brace X

Beams

Columns

PM

M

P

∆lc

05.0

cu

u

lv

P

tmomenPrimary

momentSecondary

Sway and Nonsway Frames

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Instructor:



05.0

cu

u

lV

PQ

1- Nonsway Frames (braced)Moreover, ACI Code assumes a story within a structure is nonsway if:

Where, Q is the stability index which is the ratio of secondary moment due to lateral displacement and primary moment,

ΣPu is the total vertical load in the story,

Vu is the story shear in the story under consideration,

Lc is length of column measured center-to center of the joints in the

frame, and Δ is the first-order relative deflection between the top and bottom of that story.

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Instructor:


2- Sway Frames (Unbraced)Structural frames, not attached to an effective bracing element, but

depend on the bending stiffness of the columns and girders to

provide resistance to lateral displacement are called “sway frames”


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Instructor:


The slenderness of columns is based on their geometry and on their

lateral bracing. As their slenderness increases, their bending stresses

increase, and thus buckling may occur.

Several items involved in the calculation of slenderness ratios, these

item unsupported column lengths, effective length factors and radii of

gyration.

Slenderness effect

Unsupported lengths (lu)

It is clear distance between floor slabs, beams, or other members

capable of providing lateral support as shown in figure.

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Instructor:


Slenderness effect

Effective length factors K

The effective length is the distance between points of zero moment in

the column Thus, the effective length factor k, is the ratio of the

effective length to the original length of column.

Typical cases illustrating the buckled shape of the column for several

end conditions and the corresponding length factor K

Points of inflection

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Instructor:


Slenderness effect


For members in a structural frame, the end restraint lies between the

hinged and fixed conditions. The actual k value can be estimated from

the Jackson and Moreland alignment charts

The effective length factor k is a function of the relative stiffness at

each end of the column. In these charts, k is determined as the

intersection of a line joining the values of ψ at the two ends of the

column. The relative stiffness of the beams and columns at each end

of the column ψ is given by the following equation

bbb

ccc

lIE

lIE

/

/

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Braced frame

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Instructor:


Unbraced frame

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Instructor:


Slenderness effect


where,

lc = length of column center-to-center of the joints

lb = length of beam center-to-center of the joints

Ec = modulus of elasticity of column concrete

Eb = modulus of elasticity of beam concrete

Ic =moment of inertia of column cross section about an axis

perpendicular to the plane of buckling being considered.

Ib =moment of inertia of beam cross section about an axis

perpendicular to the plane of buckling being considered.

Σ indicates a summation of all member stiffness connected to the joint

and lying in the plane in which buckling of the column is being

considered

bbb

ccc

lIE

lIE

/

/

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Instructor:


Slenderness effect


Consider the two-story frame shown in Figure. To determine the

effective length factor k for column EF

and

For ψ = ------ and for ψ = ------

ACI Code specifies that for columns in nonsway frames, the effective

length factor k should be taken as 1.0

A

C

B

D

F

E

G

I

H

h1

h2

L1 L2

)/()/(

)/()/(

21

21

lIElIE

hIEhIE

EHbBEb

EFcDEcE

)/()/(

)/(

21

2

lIElIE

hIE

FIbCFb

EFcF

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Instructor:


Slenderness effect


To calculate the ψ values it is necessary to use realistic moments of

inertia. Usually, the girder will be appreciably cracked on their tensile

sides, whereas the columns will probably have only a few cracks.

In the ACI code, it is stated that for determining ψ values for use in

evaluating K factors, the rigidity for beams = 0.35 Ig and for columns=

0.7 Ig as follows

Where, Ig is the gross moment of inertia

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Instructor:


Slenderness effect


ACI Code provides the following simplified equations for computing

the effective length factors for nonsway and sway frame members

For Nonsway frames,

K is the smaller of

Where, ψA and ψB are the values of ψ at the two ends of the column,

ψmin is the smaller of the two values.

0.105.07.0 BAk

0.105.085.0 min k

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Instructor:


Slenderness effect


For Sway frames,

a) Restrained at both end

For ψm > 2.0 ,

For ψm ≥ 2.0 ,

Where, ψm is the average of ψ at the two ends of the column

b) Hinged at one end

Where, ψ is the values at the restrained end of the column

3.00.2 k

mmk

1

20

20

mk 19.0

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Instructor:


According to ACI Code, columns can be classified as short when their

effective slenderness ratios satisfy the following criteria:

For Nonsway frames

For sway frames

Where,

k = effective length factor

lu = unsupported length of member

r = radius of gyration, for rectangular cross sections r = 0.30 h, and

for circular sections, r = 0.25 h

h = column dimension in the direction of bending.

4012342

1 M

M

r

lk u

22r

lk u

The ACI Procedure for Classifying Short and Slender Columns

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Instructor:


The ACI Procedure for Classifying Short and Slender Columns

02

1 M

M

Single curvature Double curvature

M1 = smaller factored end moment on column, positive if member is

bent single curvature, negative if bent in double curvature.

M2 = larger factored end moment on column, always positive.

[M1/M2] = ratio of moments at two column ends [Range -1 to 1]

02

1 M

M

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Instructor:


Chart summarizes the process of column design as per the ACI Code

Non-sway frameNon-sway frame

4012342

1 M

M

r

lukNeglectSlenderness

]Short[

.12341002

1

M

M

r

luk

Momentmagnification

]long[

100r

lukExact P ∆analysis

]long[

22r

luk

10022 r

luk

100r

luk

Column Design

Example # 1




Instructor:


Page Ex1-1

Instructor:





The frame shown in Figure consists of members with rectangular cross

sections, made of the same strength concrete. Considering buckling in

the plane of the figure.

Categorize column bc as long or short if the frame is:

a)Nonsway

b)Sway

Example # 1Example # 1

0.6x0.30.6x0.3

0.6x0.30.6x0.3

0.3

x0

.35

0.3

x0

.35

4.0

m

4.5

m

9.0 m 7.5 m

270 kN.m

400 kN.m

Page Ex1-2

Instructor:





a- Nonsway

For a column to be short,

Lu = 4-0.3-0.3=3.40 m

k is conservatively taken as 1.0

SolutionSolution

4012342

1 M

M

r

lk u

shortbeingasclassifiediscolumnei

astakenM

M

r

lk u

.,.

38.32401.4240

2712341234

38.32)35.0(3.0

)4.3(1

2

1

Page Ex1-3

Instructor:





b- Sway

For a column to be short,

Using the appropriate alignment chart, k = 1.21, and

i.e., column is classified as being slender

SolutionSolution

22r

lk u

945.0

)5.7(12)6.0)(3.0(

7.0)9(12

)6.0)(3.0(7.0

)5.4(12)4.0)(3.0(

7.0)4(12

12)35.0)(3.0(7.0

406.0

)5.7(12)6.0)(3.0(

7.0)9(12

)6.0)(3.0(7.0

)4(12)35.0)(3.0(

7.0

33

33

33

3

b

C

2218.39)35.0(3.0

)4.3(21.1

r

lk u

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Instructor:


Short axially loaded columns

For tied reinforced columns

For spirally reinforced columns

)]'85.0('85.0[52.0 fcfyfcAP ggu

)]'85.0('85.0[595.0 fcfyfcAP ggu

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Instructor:


Design Considerations

Maximum and Minimum Reinforcement Ratios

Documents

Lecture # 3 بسم الله الرحمن الرحيم Design of Concrete Structure II University of Palestine Instructor: Eng. Mazen Alshorafa