CE 412 Design of Concrete Structures

CE 412 Design of Concrete Structures

Lecture 5:

Cross-sections

Behavior in Combined Axial Load

and Flexure – Columns

November – 2020

Fawad A. NajamDepartment of Structural Engineering

NUST Institute of Civil Engineering (NICE)

National University of Sciences and Technology (NUST)

H-12 Islamabad, Pakistan

Cell: 92-334-5192533, Email: [email protected]

Naveed Anwar

Vice President, Executive Director, AIT Consulting

Director, ACECOMS

Affiliate Faculty, Structural Engineering,

Asian Institute of Technology (AIT), Thailand

2CE – 412: Design of Concrete Structures – Semester: Fall 2020

What is a Column?


What is a Column

• Generally no load between supports and are cast vertically.


Classification of Columns

• Material

• Geometry

• Stability


The Column Design Problem: Given

• Loads:

• Moments Mz, My, Px at two ends

• Geometry:

• Length, X-Section

• Adjoining Members

• Material:

• Concrete strength

• Rebar Strength

x

y

z

Upper End

Lower End

Column

Connecting

Beams in Z-Axis

Connecting

Beams in Y-Axis

Lower Column

Upper Column

z

y

x

Mz2

My2

Px

Mz1

My1

Px

a) Basic Model a) Column Loads


Column Design Problem

a) Ideal Situation

b) Practical Situation

Loads

Material

Shape & Size

Reinforcement

Solution

Loads

Trial Material

Trial Shape & Size

Trial Reinforcement

DesignAcceptable

No

Yes

Repeat


Complexity in the Column Design

• Loading

• +P, -P, Mx, My

• Slenderness

• Length (Short, Long, Very Long)

• Bracing (Sway, Non-Sway, Braced, Unbraced)

• Framing (Pin, Fixed, Free, Intermediate..)

• Section

• Geometry (Rectangular, Circular, Complex..)

• Materials (Steel, Concrete, Composite…)


Load-Shape-Slenderness

Shape

Loading

Length

V. Lon

g

Long

Short

P

P Mx

P Mx M

y

Most Simple

Problem

ShapeComplexity

Load Complexity

Slenderness


Load-Shape-Material


Load-Bracing-Length

Bracing

Length

Loading


What is Uni-axial Bending

• Uni-axial bending is induced when column bending results in only one moment stress resultants about any of

the mutually orthogonal axis.

No Bending

Mx = 0, My = 0Strain StressSection

x

y𝜺𝒄 𝒇𝒄

P

fs1

fc

fs2

P

x

y

P

ey

P

eyfs1

fs2

fc

Uni-axial Bending

Mx <> 0, My = 0


What is Bi-axial Bending

• Biaxial bending is induced when column bending results in two moment stress resultants about two mutually

orthogonal axis.

y

x

eye

ex P

x

P

ey ey

P

x

y y

Overall Design Process - ACI

Estimate Cross-section

based on “Thumb Rules”

Compute

Section Capacity Mn

Determine the

Layout of Rebars

Compute

Transverse Bars

Mn > Mu

Design

Completed

Y

Given P, Mux, Muy, fc, fy, L

Check

Slenderness

Ratio

Compute Mu

Not Slender

Compute

Design Moment

Slender

Rev

ise

Sec

tio

n/ M

ater

ialNot

OK

No


Main Design Steps

• Assume section dimensions

• Compute Design Actions

• Elastic analysis results and magnification of moments due to slenderness, minimum eccentricities etc.

• Direct determination of design actions using P-Delta or full nonlinear analysis

• Check Capacity for Design Actions

• Assume failure criteria

• Assume material layout and material models

• Compute capacity and check against actions


Reinforced Concrete Short Columns

Behaviour of Spiral Column and Tied Columns

Axially loaded column Eccentrically loaded column

Reference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.

Spiral Column Tied Column

Reference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.

Spiral Column Tied Column

Damage by 1971 San Fernando EarthquakeReference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.

19

Axially-Loaded Short

Columns


Minimum Spiral Reinforcement Ratio, ACI Section 10.9.3

0.45 1g c

s

c ys

A fP

A f

• ACI Eq. 10-5 can be solved for max spacing.

• For spirals to bind the core, they must be spaced close together. ACI sec 7.10.4.3 limits the clear spacing to not

more than 3 in.

• For concrete placing, spirals should be as far apart as possible. Clear spacing > 1-⅓ times size of aggregate

(ACI sec 3.3.3) and never less than 1 in (ACI sec 7.10.4.3) [also see ACI secs 7.9.1 & 7.10.4.6 to 7.10.4.8].

(Eq. 10-5)𝜌𝑠

21

RC Columns: Basic

Reinforcement

Configurations and Tie

Arrangements


1. Ties restrain the longitudinal bars from buckling out through the surface of the column.

• ACI sec 7.10.5.1

# 3 bars for longitudinal # 10 bars

# 4 bars for # 11, # 14, # 18 & bundled

• ACI sec 7.10.5.2

s < 16 db longitudinal bars

< 48 db tie bars

< least dimension (bmin)

• ACI sec 7.10.5.3

2. Ties hold the reinforcement cage together during the construction process.

3. Properly detailed ties confine the concrete core, providing increased ductility.

4. Ties serve as shear reinforcement for columns

• s < d/2 for effective shear reinforcement (ACI sec 11.5.4.1)

• ACI eq 11-2 and 11-14, if shear governs.

Look for ACI sec 7.10.5.4, 7.10.5.5 and 7.9.1 for further requirements.

Spacing and Construction Requirements for Ties


P-M Interaction Curve

(Capacity Curve)

Uniaxial Bending - Short Columns

Safe

Un-safe

24

Column Capacity –

Concentrically Loaded

Columns

Column Capacity

Capacity is property of the cross-

section and does not depend on the

applied actions or loads.

25

Column Capacity???

Column Capacity –

Eccentrically Loaded

Columns

26

Column Capacity –

Eccentrically Loaded

Columns


J. G. Macgregor (1997): Reinforced

Concrete Mechanics and Design, 3rd Edition.

Interaction diagram for an elastic column,

𝒇𝒄𝒖 = 𝒇𝒕𝒖

28

J. G. Macgregor (1997): Reinforced Concrete

Mechanics and Design, 3rd Edition.

Effect of Compressive or

Tensile Strength on

Interaction curve

29

Effect of Axial Load on

Moment-Curvature Curve


• The curve is generated by varying the

neutral axis depth.

P-M Interaction Curve

Safe

Un-safe

...),(1

....,1

...),(1

...,1

121

2

121

1

i

n

i

ii

x y

x

x y

n

i

iiz

yyxAydydxyxM

yxAdydxyxN

Uniaxial Bending

31

An Example P-M

Interaction Diagram used

for the Design of Short

RC Columns


What is Capacity?

• The axial-flexural capacity of the cross-section

is represented by three stress resultants.

• Capacity is property of the cross-section and

does not depend on the applied actions or

loads.

Mx

P

MyP-M Interaction Surface

Biaxial Bending


What is Capacity?

• Capacity is dependent on failure criteria, cross-section geometry and material properties.

• Maximum strain

• Stress-strain curve

• Section shape and Rebar arrangement etc.


Development of P-M Interaction Diagrams for Uniaxially Loaded RC

Short Columns – Solved Example

Courtesy: Dr. Khaliq ur Rashid Kayani (Architectural & Civil Engineering Services (ACES), Pakistan).


General Points on P-M

Interaction Diagram

Axial Load (Compressive)

1 2 2 3 3

3

1

2

0.85

0.85 .

o c si s s c

si si y c

c c

o c si

i

hP X C d C d C d C

C A f f

C f b h

P C C


Axial Load (Tensile)

3

1

3

1

t si

i

si si y

t t si i

i

P T

T A f

P X T d


1

should always be +ve

c c

si si si ci

C K f bkd

C A f f

Points below Balanced - Yield


Solved Example

Symmetric Section

112

c t

hX X

60

0.00207

4000

y

y

c

f ksi

f psi


Section dimensions

Width = 22 Height =

Steel areas and depths

Layer no. 1 area = 4 depth = 3




Concrete properties

Comp. strength (psi) beta1 Concrete Crushing Strain 0.0034000 0.85

Conc. strain at max comp. stress (Hognestad’s Model)= 0.002

Slope of stress vs. strain curve beyond peak stress (Hognestad’s Model)= z = 150

Steel properties

Yield stress (Ksi) Yield strain Elastic Modulus (ksi)

60 2.068 E-03 29000


AXIAL COMPRESSIVE LOAD ( Po)

Cc = 0.85 x 4 x 22 x 22

= 1645.6 K

Cs1 = 4 x (60 - .85 x 4) = 226.4 K

Cs2 = 2 x (60 - .85 x 4) = 113.2 K

Cs3 = 113.2, Cs4 = 226.4

Po = 2324.8 K

AXIAL TENSILE LOAD ( Pt )

Ts1 = 4 x 60 = 240 K

Ts2 = 2 x 60 = 120 K

Ts3 = 2 x 60 = 120 K

Ts4 = 4 x 60 = 240 K

Pt = 720 K


Solved Example: Complete P-M Interaction Diagram

1: Balanced Point

60

0.00207

4000

y

y

c

f ksi

f psi

11

622

633

0.00319 11.2 in.

0.003

0.003 0.00220

0.003 865 10

0.003 736 10

bal

y

s y

s

s

C

c d

c

c d

c

c d

c

1 2 360 , 25.1 , 21.3 s s sf ksi f ksi f ksi


1

1 1 1

2 2 2

3 3 3

4 4 4

0.85 715

0.85 226

0.85 43.4

42.6

240

c c

s s s c

s s s c

s s s

s s s

C f b c k

C A f f k

C A f f k

T A f k

T A f k

702 balP k

1

6

2 2 2 2

8430 k-in

0.003 268 10 rad/in

bal c si i si ii i

bal

balbal

ch h hM C C d T d

M

C

Express as Positive


2: A Point Above Balanced

4 4Pick C 19 in 0sd

1

2

2

6

3

3

19 30.003 0.00253

19

19 80.003 0.00154

19

50.4 ksi

19 140.003 790 10

19

22.9 ksi

s y

s

s

s

s

f

f

1

1 1 1

2 2 2

3 3 3

0.85 1208

0.85 226

0.85 94

0.85 39

c c

s s s c

s s s c

s s s c

C f b c k

C A f f k

C A f f k

C A f f k

1570 P C k


1

6

2 2 2

3533 1808 282 117

5510 k-in

0.003 158 10 rad/in19 in

c si ii

ch hM C C d

M


3: A Point Below Balance, Nominal Diagram

Set 2s y

1 1

6

2 2

3 3

4

0.0037.98

0.003 2

0.00187 , 54.3 ksi

6 10 , 0

0.00226 , 60 ksi

2 , 60 ksi

y

s s

s s

s s

sy y s

C d

f

f

f

f

=


1

1 1 1

2

3 3

4 4

0.85 508

0.85 204

0

120

240

c c

s s s c

s

s s y

s s y

C f b c k

C A f f k

T

T A f k

T A f k

352 P k=


1

6

2 2 2 2

7780 k-in.

0.003 376 10 rad/in.

c si i si ii i

ch h hM C C d T d

M

C

Express as Positive


4 : A Point Below Balance, Yield Diagram

1 2Set 0.002 0.667 , 0.375cm k k

1 1

1 1

6

2 2

2 2

6

3 3

9.34 in.

0.00136

39.4 ksi , 3.59 ksi

286 10

8.30 ksi , 1.06 ksi

999 10 , 29.0 ksi

cm

cm y

s c

s c

s c

s c

s s

kd d

f f

f f

f


352 P k

1

1 1 1 1

2 2 2 2

3 3 3

4 4 4

548

143

14.5

58.0

240

c c

s s s c

s s s c

s s s

s s s

C k f bkd k

C A f f k

C A f f k

T A f k

T A f k


2

6

2 2 2

7390 k-in.

0.002 214 10 rad/in.

c si i si ii i

h h hM C k kd C d T d

M

kd

Express as Positive


**** RESULTS FOR NOMINAL DIAGRAM ****

AXIAL LOAD (KIPS) NOMINAL MOMENT (K-IN.) CURAVTURE (RAD./IN.)

2324.8 0 0

1819.415 3990.68 1.36364 E-04

1724.501 4630.547 1.43541 E-04

1626.811 5219.912 1.51515 E-04

1539.457 5652.607 1.60428 E-04

1434.626 6149.517 1.70455 E-04

1325.143 6606.003 1.81818 E-04

1210.01 7027.416 1.94805 E-04

1087.924 7420.75 2.0979 E-04

963.9466 7774.937 2.27273 E-04

822.1076 8143.495 2.47934 E-04

702.1297 8436.976 2.66788 E-04

631.755 8374.65 2.854 E-04

557.8183 8286.174 3.06804 E-04

472.0286 8112.867 3.31679 E-04

377.8782 7893 3.60943 E-04

308.777 7572.274 3.95871 E-04

227.7132 7168.089 4.38283 E-04

138.3848 6692.336 4.90873 E-04

37.41104 6127.631 5.57804 E-04

-75.04004 5462.657 6.4587 E-04

**** RESULTS FOR YIELD DIAGRAM ****

AXIAL LOAD (KIPS) MOMENT (K-IN.) CURAVTURE (RAD./IN.)

702.1297 8436.976 2.667877 E-04

543.4567 8063.764 2.330024 E-04

362.9939 7160.652 2.091632 E-04

210.9273 6270.086 1.915271 E-04

95.02707 5507.163 1.780181 E-04

-3.005219 4842.844 1.67392 E-04

END OF PROGRAM


Properties of P-M Interaction Diagrams of RC Columns

56

Wight and Macgregor (2012): Reinforced

Concrete Mechanics and Design, 6th Edition.

Strain profiles

corresponding to points

at interaction diagram

58



Contributions of Steel

and Concrete to Column

Capacity

59



Contributions of Steel

and Concrete to Column

Capacity

60

Symmetrical Column

Section

Unsymmetrical Column

Section

Effect of Symmetry of RC

Column Section

61

Effect of Column Type on

Shape of Interaction

Diagram


Effect of Reinforcement Ratio Effect of Reinforcement Spacing


P-M Interaction Surface

(Capacity Surface)

Biaxial Bending - Short Columns Mx

P

My


𝑃 −𝑀𝑥 −𝑀𝑦 Interaction Surface

• The surface is generated by changing

Angle and Depth of Neutral Axis.

...),(1

....,1

...),(1

....,1

...),(1

...,1

121

3

121

2

121

1

i

n

i

ii

x y

y

i

n

i

ii

x y

x

x y

n

i

iiz

xyxAxdydxyxM

yyxAydydxyxM

yxAdydxyxN

66

𝑃 −𝑀𝑥 −𝑀𝑦 Interaction

Surface


Mx-My Interaction

-Mz

Muy

(-) Mnz

(+) Mnz

+ My

- My

+ Mz

Mx-My Interaction is the basis for many

approximate methods

P-Mx Interaction Curve

Safe

Un-safe


How to Check Capacity?

• How do we check capacity when there are three simultaneous actions and three interaction stress

resultants

• Given: Pu, Mux, Muy

• Available: Pn-Mnx-Mny Surface

• We can use the concept of Capacity Ratio, but which ratio

• Pu/Pn or Mux/Mns or Muy/Mny or …

• Three methods for computing Capacity Ratio

• Sum of Moment Ratios at Pu

• Moment Vector Ratio at Pu

• P-M vector Ratio


Sum of 𝑀𝑥 and 𝑀𝑦

• 𝑀𝑥 −𝑀𝑦 curve is plotted at applied

axial load, 𝑃𝑢

• Sum of the Ratios of Moment is each

direction gives the Capacity Ratio.


• 𝑀𝑥 −𝑀𝑦 curve is plotted at applied axial load.

• Ratio of 𝑀𝑢𝑥𝑦 vector to 𝑀𝑛𝑥𝑦 vector gives the Capacity Ratio.

Vector Moment Capacity


True 𝑃 −𝑀 Vector Capacity

• 𝑃 − 𝑀 Curve is plotted in the direction of the

resultant moment.

• Ratio of 𝑃𝑢 −𝑀𝑢𝑥𝑦 vector to 𝑃𝑛 −𝑀𝑢𝑥𝑦vector

gives the Capacity Ratio.

What is Capacity

1- Based on Sum of Moments at Pu 2- Based on Moment Vector at PU

3- Based on True Capacity Vector in 3D


Other Forms of Failure Surfaces


Design of Rectangular Columns Subjected to Biaxial Loads

1) The strain-compatibility method

2) The equivalent eccentricity method

3) Method based on 45-degree slice through interaction surface

4) Load Contour Method (Bresler and Parme)

5) Bresler’s reciprocal load method

78

Design of Rectangular

Columns Subjected to

Biaxial Loads

79

The equivalent

eccentricity method

80

Bresler’s reciprocal load

method

84

An Example P-M

Interaction Diagram used

for the Design of Short

RC Columns

CE 412 Design of Concrete Structures

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