5.4 steel column base platets

7/27/2019 5.4 Steel Column Base Platets

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CHAPTER 5: Structural Steel Columns - ASD Design

5.4 Steel Column Base Plates

Description

This application calculates the plan dimensions and thickness for

optimum (least weight) base plates for axially loaded wide flange

columns. The plan dimensions and thickness of the plates are

optimized for the final rounded plate dimensions and thicknesses.

The required input includes column loads, ratio of the concentric

concrete support area to the plate area, the yield strength of the

base plate steel, rounding factors for plan dimensions and

thickness, and the column widths and depths. Four base plates

are shown for illustrative purposes, however, up to 100 plates

may be designed and displayed in one run of this application.

A summary of input and calculated values is shown on pages 11 and 12.

Reference: "Design of Base Plates for Wide Flange

Columns - A Concatenation of Methods," Engineering Journal,

Volume 27 (4th quarter 1990), by W. A. Thornton


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Input

Notation

Input Variables

Column axial loads:

P 850 425 215 55( )T

kip

Column flange widths:

b 12.515 12.08 8.00 6.50( )T

in

Column depths:

d 13.88 12.38 8.12 7.75( )T

in


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In this example the column loads and column width and depth are

entered in transformed vectors with the number of vector rows

corresponding to the number of base plates to be designed.

Computed Variables

fp actual bearing pressure on support

B plate width

N plate length

tp thickness of plate

c dimension defining the assumed bearing area for the

Murray Stockwell Method (See Fig. 2)

Material Properties and Constants

Enter values for f'c, Fy, SzF, Szf, and Rif different from that shown.

Compressive strength of concrete: f'c 4 ksi

Yield strength of base plate steel: Fy 36 ksi

Factor for rounding plan dimensions of plates: SzF 1 in

Factor for rounding thickness of plates: Szf 0.125 in

Ratio of concentric area of concrete

supports to the area of the base plates between 1 and 4:

R 4 R if R 1 1 if R 4 4 R( )( ) R 4

Notes 1) The plan dimensions and plate thicknesses are rounded tomultiples of SzF and Szf, respectively.

2) The ratio R may be set at any value from 1 to 4. A value

of 1 will set theallowable bearing pressure at 0.35f'c and

require the smallest concrete bearing area and the largest

base plates. A value of 4 will set thebearing pressure at

0.70f'c and require the smallest base plates and the largest

concrete bearing areas.


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The following values are computed from the entered material strengths

and ratio R.

Ratio of the plan dimensions of the column bearing plate plate to the

minimum concentric plan dimensions of the concrete bearing area:

R 2

Allowable bending stress in plates:

Fb 0.75 Fy Fb 27 ksi

Allowable bearing pressure:

Fp if R 4( ) R 1( ) R 0.35 f'c if R 4 0.70 f'c 0.35 f'c

Fp 2.8 ksi

Calculations

Range variable i: i 0 last P( )

Minimum plate area limited to no less than the columns' plan dimensions:

Apli

ifPi

Fpbi di bi di

Pi

Fp

AplT

303.571 151.786 76.786 50.375( ) in2

All computed values are displayed in transformed vectors or matrices.

Values for each column being designed are in the corresponding

columns of the transformed vector or matrix. For example, the

minimum plate area for the first column entered is in the first column of

the transformed vector displayed above.

Minimum plate area before overhangs on both sides may be increased.(This procedure determines the area and dimensions of the smallest

plate where m equals n.):

Bi if 0.1 bi 0.025 di 0 in 0.025 di 0.1 bi

BT

0 0 0 0( ) in


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Ni if 0.025 d i 0.1 bi 0 in 0.1 bi 0.025 di

NT

0.905 0.898 0.597 0.456( ) in

A2i bi Bi di Ni A2

T185.028 160.404 69.736 53.341( ) in

2

Optimization Procedure

The optimum or least weight plate is obtained when the plate overhang

dimensions m and n are equal. Substituting the variable x for m and n,

and equating the area of the base plate in terms of the column

dimensions b and d and the plate overhang x, a quadratic equation is

obtained which is then solved for the plate overhang x. The plate size

with minimum theoretical plan dimensions is designated plate No. 0.

Since the plate dimensions obtained are usually fractional dimensions,

the next step is to round the theoretical plate width to the nearest

multiple of the sizing factor SzF, compute a new theoretical plate

length, and round that dimension up to the nearest multiple of SzF.

This procedure is then repeated starting with the theoretical minimum

plate length. The final step is to select the lighter of the two plates with

rounded dimensions. The plates with minimum rounded plan

dimensions are designated as plates No. 1 and No. 2. The optimum

plate is the plate with the lesser weight, and is designated plate No. 3.

The plate numbers correspond to the matrix column index numbers ofthe computed variables for the plate.

Plate overhang x, (with x = m = n), solved using the quadratic formula:

xi

1.6 bi 1.9 di 1.6 bi 1.9 di 2

16 0.76 bi di Apli

8

xT

2.948 0.826 0.865 0.449( ) in


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Plate No. 0, with minimum possible plate dimensions B and N:

Bi 0 if Apli

A2i 0.8 bi 2 xi if Apli

bi di= bi if Bi 0 in= biApl

i

di Ni

B0 T

15.908 12.08 8.13 6.5( ) in

Ni 0 if Apli

A2i 0.95 di 2 xi if Apli

bi di= di if Ni 0 in= diApl

i

bi Bi

N0 T

19.082 12.565 9.444 7.75( ) in

Plate No. 1, with B rounded to nearest multiple of SzF and N

rounded up to the nearest multiple of SzF:

Bi 1 if SzF floor Bi 0

SzF0.5

bi SzF floorBi 0

SzF0.5

SzF ceilbi

in

B1 T

16 13 8 7( ) in

Ni 1 if SzF ceil

Apli

Bi 1 SzF

di SzF ceilApl

i

Bi 1 SzF

SzF ceildi

in

N1 T

19 13 10 8( ) in

Plate No. 2, with N rounded to nearest multiple of SzF and B

rounded up to nearest multiple of SzF:

Ni 2 if SzF floor Ni 0

SzF0.5

di SzF floorNi 0

SzF0.5

SzF ceildi

in

N2 T

19 13 9 8( ) in

Bi 2 if SzF ceilApli

Ni 2 SzF

bi SzF ceilApli

Ni 2 SzF

SzF ceilbi

in

B2 T

16 13 9 7( ) in


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Plate overhangs m and n:

j 0 2 mi jNi j 0.95 di

2

ni jBi j 0.80 bi

2

mT

2.948

2.907

2.907

0.402

0.62

0.62

0.865

1.143

0.643

0.194

0.319

0.319

in

nT

2.948

2.994

2.994

1.208

1.668

1.668

0.865

0.8

1.3

0.65

0.9

0.9

in

There are three plates calculated for each of the columns entered. Thethree rows in the transformed m and n matrices above correspond to

the 3 plates calculated for each column. The columns in the matrices

correspond to the number of columns entered.

Plate bearing pressures:

fpi j

Pi

Bi j Ni j

fpT

2.8

2.7962.796

2.8

2.5152.515

2.8

2.6872.654

1.092

0.9820.982

ksi

Yield line method for small base plates, uniform bearing over b x d

Equivalent overhang dimension using the yield line method for small

base plates with uniform bearing over the area b x d:

n'i1

4dibi

n'T

3.295 3.057 2.015 1.774( ) in

When allowable bearing stresses are high this method will often govern

over the conventional cantilevered design method.


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Murray-Stockwell method for small lightly loaded base plates

Portion of total load acting over H shaped section (Fig. 2):

Poi j

fpi j

bi di

Po

kip

T 486.383

485.697

485.697

418.741

376.088

376.088

181.888

174.58

172.425

55

49.475

49.475

Intermediate variable for calculation of:

Xi j

4 Poi j

bi di 2

Fp

XT

0.997

0.996

0.996

1

0.898

0.898

1

0.96

0.948

0.387

0.348

0.348

Variable for determining which method governs:

i j if2 Xi j

1 1 Xi j1

2 Xi j

1 1 Xi j 1

T

1

1

1

1

1

1

1

1

1

0.698

0.653

0.653

When is equal to 1 the yield line method governs over theMurray-Stockwell method.


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Largest overhang dimension for conventional cantilever design, yield

line, or Murray-Stockwell methods:

ki j max

mi j

ni j

i j n'i

kT

3.295

3.295

3.295

3.057

3.057

3.057

2.015

2.015

2.015

1.238

1.158

1.158

in

Plate thicknesses:

tpi j

2 ki jfp

i j

Fy

tpT

1.838

1.837

1.837

1.705

1.616

1.616

1.124

1.101

1.094

0.431

0.383

0.383

in

Plate thicknesses rounded up to nearest multiple of Szf:

j1 1 2

tpi j1

Szf ceil 2 k i j1fp

i j1

Fy

1

Szf

tpT

1.838

1.875

1.875

1.705

1.625

1.625

1.124

1.125

1.125

0.431

0.5

0.5

in


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Weight of plates:

Weighti j tpi j

Bi j Ni j 3.4

12

lb

in3

WeightT

158.076

161.5

161.5

73.336

77.81

77.81

24.451

25.5

25.819

6.155

7.933

7.933

lb

Optimum (least weight) plate:

Bi 3 if Weighti 1 Weighti 2 Bi 1 Bi 2

Ni 3 if Weighti 1 Weighti 2 Ni 1 Ni 2

tpi 3

if Weighti 1 Weighti 2 tpi 1

tpi 2

Weights of the optimum base plates:

Weighti 3 if Weighti 1 Weighti 2 Weighti 1 Weighti 2

Weight3 T

161.5 77.81 25.5 7.933( ) lb

Actual bearing pressures for the optimum base plates:

fpi 3

Pi

Bi 3 Ni 3

fp3 T

2.796 2.515 2.687 0.982( ) ksi

Plan dimensions of the minimum required concentric concrete

bearing area:

Wi Bi 3 R

W

T

32 26 16 14( ) in

Li Ni 3 R

LT

38 26 20 16( ) in


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The dimensions W and L are the minimum required concrete pier size

for the bearing pressure used. In this example W and L are integers,

but they may be fractional dimensions (which should be rounded up) if

a different R value and corresponding bearing pressure is used.

Summary

Ratio of concentric area of

concrete supports to the

area of the base plates: R 4

Yield strength of

base plate steel: Fy 36 ksi

Compressive

strength of concrete: f'c 4 ksiAllowable bearing pressure: Fp 2.8 ksi

Allowable bending

stress in the plates: Fb 27 ksi

Factor for rounding plan

dimensions of plates: SzF 1 in

Factor for rounding

thickness of plates: Szf 0.125 in

Column loads: PT 850 425 215 55( ) kip

Column flange widths: bT

12.515 12.08 8 6.5( ) in

Column depths: dT

13.88 12.38 8.12 7.75( ) in

Optimum base plate width, length,

thickness weight, and bearing pressures:

Bi 3

in16

13

8

7

Ni 3

in19

13

10

8

tp

i 3

in1.875

1.625

1.125

0.5

Weighti 3

lb161.5

77.81

25.5

7.933

fp

i 3

ksi2.796

2.515

2.687

0.982


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Plan dimensions of the

minimum required concentric

concrete bearing area:

Bi 3 R

in

32

26

16

14

Ni 3 R

in

38

26

20

16

5.4 steel column base platets

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