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COLUMN DESIGN

UNIVERSITY OF WISCONSIN STOUTCOLLEGE OF SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS

LECTURE IX

Dr. Jason E. Charalambides

COLUMNS – AXIAL LOAD AND BENDING

We tend to have this image of “columns” that we envision as symmetrically shaped massive pilasters. They assume the loads applied upon them in a perfectly axial manner and transfer them straight down upon the foundations that support the structure.

However, beside axial loads, moments are also assumed by columns, either through the application of eccentric loading conditions, or by the eccentricity of the forms.

HOW BENDING IS APPLIED TO COLUMNS

Consider the effect of varying lengths of bays and the transferred moments from the beams.

Then also consider the effect of lateral forces.

HOW ARE MOMENTS PRODUCED

Gravity and lateral loads generate moments on the structure. Again, asymmetrical structural forms or variation in loading patterns are the generators of bending moments.

We will address: Behavior of elastic homogeneous

column, Behavior of RC column,

Uncracked, Cracked, Ultimate

Design of RC column. First we shall address slices of a

column and then we will extrapolate to entire columns (short vs. long).

ELASIC HOMOGENEOUS COLUMN

Now the Neutral Axis and the geometric centroid are no longer at the same location. The location of the N.A. is dependant upon the eccentricity of the resultant of loads.

óσ=(P/A)+(M*y/I)

HOW TO LOCATE THE NEUTRAL AXIS

Measure from the geometric centroid:

At N/A σ=o: cc

cc

EQUILIBRIUM

How equilibrium is established:

ÅÅ

RC COLUMN ANALYSIS

For beams, service level behavior (cracking, deflections) is very important. Structural behavior tends to be controlled more by the ultimate behavior of the column, rather than service level behavior.

So we place less emphasis on service level behavior of columns. Prior to Cracking:

Use uncracked, transformed section. Concrete cracks at fr=7.5√f`c

After Cracking: Use cracked, transformed section.

RC COLUMN ANALYSIS

fc is the.computed compression flexural fiber stress at service loadsfs is the calculated stress in reinforcement at service loads.f's is the stress in compression reinforcement.fy is the.strength of steelfr is the point where concrete cracks (modulus of rupture) 9.5.2.3Ec is the Young's modulus of Elasticity of concreten is the ratio of Elastic moduli of steel and concrete

In Class Example:

Construct the M-θ curve for the section shown. The section is subjected to a load at an eccentricity of 15 inches from its center.

In Class Example cont:

In Class Example cont:

In class example cont: After Cracking: Locate NA by trial and error:

Choose εc=>fc (reference only) Guess kd Calculate f`s, fs Calculate P & M See if calculated eccentricity (M/P) equals

given ecc, Repeat as necessary.

Note that the initial chosen fc is arbitrary. We compute P as a linear function of fc, and M as a linear function of fc. Then ecc=M/P is independent of fc.)

In Class Example cont:

In Class Example cont:

Taking Moments around the Centroid:

In Class Example cont:

In Class Example cont:

In Class Example cont:

In Class Example cont:

In Class Example cont:

In Class Example cont:

In Class Example cont:

θ/in M (k`) Comments

1.85E-05 47.56 just before cracking

4.177E-05 47.56 just after cracking

1.147E-04 130.62 fc=0.7*f`c

2.51E-04 286.26 fs=fy (assumes linear behavior)

Reading & Assignment

=Reading: Required: Furlong, Chapter 7 (7.1 through 7.5 incl.) Recommended: McCormac & Nelson, Chapter 9 and 10 (pp. 278-299) for

this week’s lectures. Practice:

Furlong: Example 7.3. McCormac & Nelson: Example 10.2.

Assignment: Assignment 8 is due one week from today.