Column Calculations Hypothesis

7/31/2019 Column Calculations Hypothesis

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Local axes

Y

Z Local axes convention for STAAD.

2

3 Local axes convention for ETABS.

Info about beta angle also need to be extracted and column axes / orientation should be shownaccordingly.

Procedure for calculating Muz1 (uniaxial moment capacity about Z-Zfor a given Pu)

Interaction diagram can be drawn for values or Pu and Muz for various positions of NA.

Pu = Cc + Cs

Muz = Mc + Ms

Mc and Ms denote the resultant moments due to Cc and Cs respectively, with respect to thecentroidal axis (principal axis under consideration)

DY = Dimension along Y-Y

Calculation of centroid of the section

User will be asked to draw column shape in AutoCAD for other than rectangular and circularcolumns such that the local axes of the column are as per the standard format.

Left and right edges of the column are to be identified and Dim Y of the column to be dividedinto strips of width not greater than 10 mm.


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Following calculations are made to calculate the centroidal axis of the column.

Table 1

Strip

Distance

f rom

M CE

Depth o f

section

Avg depth

o f t he

str ip

Centr e of

st r ip f r om

M CE

Area of

str ip

M o m e n t

o f s t r ip

about

M CE

S1 S2 S3 S4 S5 S6 S7

0 300

1 10 300 300 5 4500 33750

2 20 300 300 15 4500 101250

3 30 300 300 25 4500 168750

and so on . . .

Here S1, S2 . . . indicates the S No. of the data in columns

S1 = Strip No. from left edgeS2 = Distance of the right edge of the strip from MCE

(First row in the table is for the left most edge of the section)

S3 = From AutoCAD for irregular section / calculated values for rectangular or circular section

S4 = Depth of left edge + right edge/ 2

S6 = S4 x width of the strip

S7 = S6 x S5

CG of the section from MCE (in Y-Y direction) =CGy =7

6

S

S

There may be a better method to calculate centroid of the section but we have to draw thestrips in any case for calculating Cc and Mc as explained below.


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Calculating Cc and Mc

Following calculations will have to be done for various positions of NA (starting from min valueand increasing in increments (= 0.05 x DY). Strains at the centre of strip as given in S8 are for aparticular value of Xu.

Most compressed edge will be assumed to be the left edge (for calculating Muz) and bottomedge (for calculating Muy).

Table 2

Value of Xu =

Strip

Distance

f romM CE Depth o fsection

Avg

dep th o fthe s t r ip

Centr e of

st r ip f ro mM CE Area ofstr ip

Str ain at

cent re o fstr ip

Str ess in

concrete

at cent reof s t r ip

Compressive

force due toconcrete

M o m e n t

due toconcrete

S1 S2 S3 S4 S5 S6 S8 S9 S10 S11

0 300 0.0035

1 10 300 300 5 4500 0 .003413 13 .41 60345 11616413

2 20 300 300 15 4500 0 .003238 13 .41 60345 10711238

3 30 300 300 25 4500 0 .003063 13 .41 60345 9806063

and so on . . .

S1 to S6 = same as in previous table

S8 = i = Values of strain at distance yi along Y-Y for a particular value of Xu. This has beenexplained in the subsequent section. Compressive strains will be calculated as positiveand tensile strains as negative.

Value of strain the first row is the strain at MCE

S9 = 0 (for strains 0)0.447 fck (for strains 0.002)

0.447 fck [2(i/0.002) - (i/0.002)2) (for other values of strain)

S10 = S6 x S9

S11 = S10 x (CGy S5) ; CGy = Distance of the centroid from MCE as calculated in lastsection

Cc = S10 ; Mc = S11

Note: Strains at the centre of the strips are calculated from known values of Xu and strain at theMCE.

Strain at centre of ith strip =at MCE

( )Strain

Xu yi

Xu

yi = distance of the ith strip from MCE


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Calculating Cs and Ms

For calculating Cs and Ms we need to have the distance of various rows of steel from MCE(DSTi) and areas of steel in these rows (ASTi). The same can be calculated from the user inputdata for rectangular and circular columns. For other shapes, this can be taken from the

AutoCAD based on the drawing of the column given by the user.

Most compressed edge will be assumed to be the left edge (for calculating Muz) and bottomedge (for calculating Muy).

Following calculations will have to be done for various positions of NA (starting from min valueand increasing in increments (= 0.05 x DY). Strains at various distances from MCE as given inS15 are for a particular value of Xu.


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Table 3

Value of Xu =

Row

No. o fre in f

Distancef rom

M CEDSTi

Area of

re in fASTi

Str ain insteel

Str ess insteel

Compressive

force/Tensile force

M o m e n t

due tosteel

S12 S13 S14 S15 S16 S17 S18

0 0.0035

S13, S14 = calculated values from user input or picked up from AutoCAD

S15 = strain values corresponding to respective distance from MCE. This has been explainedin the subsequent section. Compressive strains will be calculated as positive and tensilestrains as negative.

Value of strain the first row is the strain at MCE

S16 = Stress in steel from stress-strain curve (refer Table 3.2 in book by Menon or SP-16).Compressive stress is taken positive and tensile stress is negative, i.e., Stress will havethe same sign as that of strain.

S17 = S14 x S16 (Compressive force = +ve ; Tensile force = -ve)

S18 = S17 x (CGy S13) ; CGy = Distance of the centroid from MCE as calculated earlier

Cs = S17 ; Ms = S18

Note: Strains at various distances from MCE are calculated from known values of Xu and strainat the MCE.

Strain at ith row =at MCE

( )Strain

Xu DSTi

Xu

DSTi = distance of the ith row from MCE

Calculations of strains (S8 / S15)

Strain at the most compressed edge and at the other edge of the section is calculated as givenbelow:

For Xu D at MCE = 0.0035


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Calculating value of Pbz

Pb is the axial load corresponding to the balanced failure. This is required to calculate the

modification factor to be applied to slenderness moments.

Xub = Distance of the NA for balanced failure from most compressed edge (MCE)

Most compressededge

0.0035

y

Xub EDY - Xub

Yield strain in steel = y =0.87

0.002y

s

f

E (for Fe 415/500 ; Fe 250 option need not be given)

Es = 2 x 105 N/mm2

EDY = DY Gross cover

Xub =0.0035

0.0035

EDY

y ; Strain at MCE = 0.0035

Transfer these values to Table 2 and 3 to calculate the value Pub.

Pub = Cc + Cs

FOLLOW THE SIMILAR PROCEDURE FOR OTHER DIRECTION.

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Column Calculations Hypothesis