composite course part 3

7/26/2019 Composite Course Part 3

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PART 3 - COMPLETE SECTIONBEHAVIOR


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Composite section under axial compression

cntd

Strain distribution across the section


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Composite section under pure bending



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Composite section under combined axial

loading and flexure


strccntd


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Strength of composite sections under

combined axial and flexure

Use fiber analysis approach to calculate the strength of a

composite beam-column

The fiber analysis approach consists of two subroutines

subroutine to calculate the axial load-moment-

curvature () relationship for the cross-section

subroutine to calculate the axial strength (Pcr) by finding

converged member deflections using the corresponding

relationship


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subroutine

Step 1: discretize the cross section into layers of fibers

Step 2: calculate the following properties for each fiber:

the area (Afib)

moment of inertia with respect to the centroid of the cross

section (Ifib)

centroid distance (distance from the center of the fiber to

the centroid of the cross section, yfib)

Centroid

fib


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subroutine

Step 3: apply the following procedures to obtain the

curve for each load increment (Pi):

Increase the curvature from 0 to 10y/h in increment of

0.00001, where y is the steel yield strain, and h is the

depth of the existing section.

For each increment of , perform the following sub-procedures to obtain the corresponding value of M:

1. Assume the strain value at the centroid (cntd) based on

the converged value from previous curvature increment.

2. Calculate the total strain in each fiber (strc). As shown inFig. 6, the total strain is calculated as the summation of

the centroid strain (cntd) and the bending strain (b).

Residual strains are also included if presented.


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subroutine

3. Calculate the stress in each fiber (fib) based on the

total strain (strc) and the prescribed stress-strain

curve.

4. Calculate the axial force in each fiber (ffib)as

5. Calculate the cross-section internal moment by summing

the moments from all fibers together

6. Calculate the cross-section internal axial force by

summing the forces from all fibers together


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subroutine

7. If Pcal-Pi tolerance, go to the next increment of

curvature until the limiting maximum curvature value(10y/h) is reached. The tolerance is assumed to be

0.001Pi.

8. If Pcal-Pi tolerance, change the value of cntd using

Newtons Method and restart this sub-procedureuntil converges.

The curve is obtained if the converged values of the

curvatures and the corresponding moments for each loadincrement (Pi) are calculated


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Start with the curvature

increment (m) equal to 1.

Check if PintPi tolerance,the tolerance is assumed to be

0.001Pi

Calculate the current

curvature: m = 0.00001m

Assume the strain value at

the centroid (cntd) based on

the converged value from the

previous curvature increment

(m-1)

Calculate the total strain in

each fiber (strc). The total

strain is calculated as the

summation of the centroid

strain (cntd) and the bending

strain (b). Residual strains

are also included if

presented.

Calculate the axial force in

each fiber as: ffib = fibAfib

Calculate the cross-section

internal moment (Mint) by

summing the moments from

all fibers together: Mn = ffibyfib


internal axial force by

summing the forces from all

fibers together: int = fib


internal axial force (Pint) by

summing the forces from all

fibers together: int

= fib

If yes, increase the curvature increment to

m+1 until the curvature limit (m = 10 y/h )

is reached, where y is the steel yield strain,

and h is the depth of the existing section.

If no, change the

value of cntdusing Newtons

Method, and

restart the

iteration


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subroutine

Step-1: discretize the member into segments. This

resulted in stations along the length. The number of the

segments was approximately equal to the column length-

to-depth ratio (L/h).

Step-2: use the following iterations to calculate the

member deflections for each load increment (Pi):


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subroutine

1. Assume the lateral displacement at iterationj to be the

same as the corrected displacement (Yki,j, as discussed

later) from previous iterationj-1. For the first iteration

(j=1), the lateral displacement is assumed to be the

same as the converged shape from previous load

increment (i.e., the converged shape from Pi-1). For thefirst load increment (i=1), the lateral displacement is

assumed to be the same as the sinusoidal imperfection.


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subroutine

2. Calculate the external moment at each station (Station

k) as


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subroutine

3. Obtain the curvature ki,j at each station using the

calculated curve, and calculate the rotation at

each station as

4. Calculate the displacement at each station as

5. Because of the fixity assumed at one end, all the

displacements collected at the other end. This led to thecomplication shown in Fig. 7(b). The calculated

displacement was not the same as the deflection;

therefore it was corrected as follows:


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subroutine

6. Compare the corrected displacements at each station

(Yki,j) with that from the previous iteration:

If Yki,j -Yk

i,j-1 tolerance, the converged displacements are

found;

If Yki,j -Yk

i,j-1 tolerance, use the corrected displacements

(Yki,j) as the initial lateral displacement for the next iteration(j+1) and restart the iterations. The tolerance is assumed to

be h/6000.


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subroutine

Once the converged deflections (Yki,j) were found, the

applied moment (Mk) at each station (Station k) wascalculated as the product of applied axial force (Pi) and

the deflections (Yki,j).

If the moment at any station (typically at mid-height) had

become greater than the cross-section moment capacityobtained from the corresponding subroutine,

then the column had failed due to inelastic column

buckling, and Pi was the critical buckling load (Pcr).

Otherwise, the axial load was increased from Pi to thenext load increment Pi+1 and the two subroutines (

subroutine and subroutine) presented above were

recalled until column failure due to inelastic buckling

occurred.


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Summary of the fiber analysis approach

Discretize the CFT memberinto segments, and

discretize the cross section

into layers of fibers

Check if the calculated external

moment (Mki,j) at any station

(typically at mid-height) is greater

than the section moment capacityobtained from the

subroutine.

Start the first load increment, Pi

Call the subroutine to

calculate the axial load-moment-

curvature () curve

Call the subroutine to

calculate converged member

deflections (Yki) using

corresponding curve

If yes, the axial strength (Pcr) is

reached

If no, continue to the nextload increment, Pi+1


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Fiber analysis program

Overview:

This program consists of the main code:

Main_CFT_column_buckling_Circular.m and 8

subroutines

Subroutine 1 Circular_getinput.m: discretize the CFT

cross section into fibersSubroutine 2 curvature_cal.m: calculates the P-M-

relationship

Subroutine 3 property.m: calculates the forces in each

member

Subroutine 4 strain_bending.m: calculates bending strains

Subroutine 5: thermal_strain.m: calculates thermal strains


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Overview:

Subroutine 6 residual_strain.m: calculates residual strains

Subroutine 7 steel.m: specifies steel stress-strain

relationship

Subroutine 8 concrete.m: specifies concrete stress-strain

relationship


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Input:

General analysis input

The default increment is

0.1*estimated strength.

This value specifies when

the default incrementvalue changes


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Input:

Geometric and material

properties of the CFT member


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Input check: plot to check section geometry input


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Initial output: estimated axial capacity and initial time

increment value.

The initial time increment value can be changed to

decrease the computational time.


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Output

1. The load increment to capacity ratio: This ratio should be

less than than 1%; otherwise, run the analysis with

smaller increment

2. Calculated axial capacity


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Output

3. Moment curvature response for each increment


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Output

4. P-M interaction curve


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Output

5. Stress distribution at the midspan section


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Output

6. Strain distribution at the midspan section

composite course part 3

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