33 Concrete Beam.xls

REINFORCED CONCRETE BEAM

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CONCRETE BEAM DESIGN fy 60 ksi f'c 3 ksi d' 2.5 in

yield strength of reinforcing compressive strength of concrete depth of compression reinforcing

bw d depth_OA

width depth of tension reinforcing overall depth

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 30

25

A's As M working Mu req'd

13 in 20.5 in 27 in Bar Qty Bar Size# 0.00 6 1.00 6 21.9 k-ft 30.7 k-ft ult

As 0.00 in2 0.44 in2 required service level moment required ultimate/factored moment

MU prov 39.80 k-ft ultimate provided OK Beam stressed to 77 % NOTE: 77% ≈ 75% see ρmin below !!! As min 0.89 > As provided 0.44 OK x balance 7.62 ≥ 0.937 required

Length t c flange

Ig I cracked Ie εt

39 4.5 78 0

21500 1365 21500 0.00507

ft in in logic

in4 in4 in4 unitless

length of joist thickness of slab spacing of joists 0 = no flanges, 1 = spandrel, 2 = T beam

20

15

10

5

0 0

As logic

0.89 in2

0 logic

15

20

25

in American nomenclature, bar size is given as a number which equates to 8 th's of an inch hence, a #6 bar is 6/8 inch = 3/4"

fy f'c

ultimate tensile strength of reinforcing, k/in 2 ultimate compression strength of concrete, k/in2

American Concrete Institute http://www.aci-int.org/general/home.asp allowable tension reinforcing strain 0.00400 minimum strain allowed

width d' -AS

200 * w * d /fy /1000 02 ACI 10.5.1 200 * 13.0 * 20.5 /60 /1000

d

MAX(0.89, 0.73) ≤ 0.44

AS

When the area of tension reinforcing steel is less than +As minimum, the amount of reinforcing provided must be increased by 1/3 beyond that needed for tensile reinforcement. 02 ACI 10.5.3 When the beam is stressed to 75% and +As = 1.33 x As provided, the minimum requirements of reinforcement have been met.

Figure 33-2 Beam cross-section.

-As, A's +As

For Seismic Resistance As min 0.89 in2

10

Bar Size #

ρmin -- Minimum Reinforcement in Flexural Members As min 1 3 √f'C /fy bw d 02 ACI 10.5.1 (10-3) 0.73 in2 3 * √ 3 *1000 * 13.0 * 20.5 /60 /1000 As min 2

5

Figure 33-1 Beam cross-section with Whitney's stress block and reinforcing.

200 * bw * d / fy ACI 21.3.2 200 * 13 * 20.5 /60 /1000

negative reinforcing - usually the top steel, in2 positive reinforcing - usually the bottom steel, in2

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CONCRETE BEAM DESIGN -- Continued

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 bE width of rectangular beam or, bE = 13" spandrel and T-beam effective d' = 3" flange, inches t = 4.5"

depth_OA = 27"

d = 21"

13"

clear = 65"

center to center = 78" Figure 33-3 T-beam framing cross section.

Spandrel beam ACI 8.10.3 bE 39 in bE 40 in bE 45.5 in bE 39 in T-beam 02 ACI 8.10.2 bE 117 in bE 85 in bE 78 in bE 78 in bE 78 in

Type of beam and overhang bE slab trib logic beam 13 1 spandrel 39 0 T-beam 78 0

NOTE: T-beam calculations are required before the doubly reinforced calculations can be done.

39' * 12 in/ft /12 4.5" * 6 + 13" beam width (78" - 13" beam width) /2 + 13" spandrel

39' * 12 in/ft /4 4.5" * 16 + 13" beam width 78'' c-c beam spacing maximum width of flange

bE 13 0 0 13

overhang OH 0 0 0 0

vLookUp Table Bar 10M 15M 20M 25M 30M 35M 45M 55M 0 3 4 5 6 7 8 9 10 11 14 18

30

25

20

15

10

5

0 0

10 15 20 Figure5 33-4 Beam cross-section.

25

30

top

Bar 6

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bottom

M

Bar 6

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STRESS-STRAIN RELATIONSHIPS I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 β1 B 1, β1 0.85 unitless reduce the .85 factor by .05 for reduction factor for higher strengths of each 1000 psi over 4000 psi concrete, unitless. ACI 10.2.7 IF( 3 ksi > 4, 0.85 - (3 ksi - 4) * 0.05, 0.85) a space is included between the B and the 1 so that the spreadsheet does not read this β1 0.85 but not less than 0.65 range name as a cell address. IF( 0.85 < 0.65, 0.65, B210) ES steel modulus of elasticity, 29000 ksi ε's limit fy /Es 0.00207 unitless fy 60 / 29000 compression strain εy = 0.00207 for reinforcing full engagement at fy = steel 60 ksi εt 0.00507 unitless allowable tension reinforcing strain 0.00400 minimum strain allowed stress, σ εt 0.00207 unitless 60 fy / 29,000 Es calculated concrete for reference only f'C εcrushing, code Note: that εt is the lower limit for tension reinforcing strain defined crushing where the minimum = 0.0040. This is conservative strain and may be increased to 0.005 where greater ductility is required 0.001 0.002 0.003 0.004 strain, εs as for tension controlled members. See 02 ACI 10.3.3, 10.3.4 and, 10.3.5. commentary for a comprehensive explanation of this new Figure 33-5 Stress and strain relationships for approach in the code. Older codes used 0.75 ρb. concrete and steel. To follow some textbook examples, you will have to set εt to 0.00207 or greater. xbalance amax

7.62 in 6.48 in

xbalance

0.003 /(0.003 + 0.00507) * 20.50" 0.85 * 7.62 in

0.003 ε'c ε's

d ε's ε's min

0.00807 unitless 0.00207 unitless

20.50" /7.62" * 0.003 MIN(0.00807, 0.00207) 0.00507 εt stress - strain

Stated another, traditional way; the basic relationship to determine x_balance * d is: ε'c εt/Es x bal * d

87 unitless 147.03 ksi ε'c ε'c + εt/Es

=

concrete strain profile Figure 33-6 The balanced strain condition.

0.003 * 29,000 ksi 29000 /0.00507 0.003 0.003 + fy /29,000

The Limit for Tension Reinforcing 0.003

0.003

=

87 87 + 147 0.003

=

0.372 * d inches = 7.62 in

εcu εy ε'S εs

concrete crushing strain, 0.003, unitless strain at first yeild, unitless strain in compression reinforcing, unitless strain in tension reinforcing, unitless

εt

allowable tension reinforcing strain, unitless

As allowable tension reinforcing strain εt increases, the corresponding Whitney's stress block decreases. The stress block is a function of β1* x balance = a. 0.004 minimum

0.005 for tension controlled members

0.0075 for elastic, tension controlled members in hinged regions

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Figure 33-7 Tensile strain versus depth of compression block.

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DEPTH of COMPRESSION BLOCK For tension and compression forces -- solve for x. Setup relationships to be solved with the quadratic equation: Flange 0.85 f'c (bE -bw) β1 x Flange 0.85 f'c (bE -bw) β1 t

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 [-b ± (b2 -4 a c)1/2 ] / [2 a] = 0

A's ((x - d')/x εcu Es - 0.85 f'c)

d' a/2

a

x d - d'

Stem 0.85 f'c bw β1 x d

d - a/2

As fy Figure 33-8 The balance of compression forces against the tension force As fy.

Figure 33-9 Calculating Whitney's stress block.

Calcualte x balance strain Tension Stem As f y = 0.85 f'c bw β1 x

Flange 0.85 f'c (bE -bw) β1 (x or t)

Compression Reinforcing +A's ((x - d')/x εcu Es - 0.85 f'c)

As f y =

0.85 f'c bw β1 x

0.85 f'c (bE -bw) β1 (x or t)

+A's (x - d')/x εcu Es

-A's 0.85 f'c

As f y

- 0.85 f'c bw β1 x

- 0.85 f'c (bE -bw) β1 (x or t)

-A's 0.85 f'c

= A's (x - d')/x εcu Es

As f y

- 0.85 f'c bw β1 x

- 0.85 f'c (bE -bw) β1 (x or t)

-A's 0.85 f'c

= A's εcu Es

As f y

- 0.85 f'c bw β1 x

- 0.85 f'c (bE -bw) β1 (x or t)

-A's 0.85 f'c

- A's εcu Es

As fy x²

- 0.85 f'c bw β1 x²

- 0.85 f'c (bE -bw) β1 (x or t) x

-A's 0.85 f'c x

- A's εcu Es x

Where the compression block a is less than flange depth t a b - 0.85 f'c bw β1 x² -0.85 f'c (bE -bw) β1 x² As fy x a x² -28.178 x

a

-A's 0.85 f'c x

- A's εcu Es x

bx 0

-28.178 k/in

0.94 in 1 logic 0.796 in 0 logic

- A's d'/x εcu Es = - A's d'/x εcu Es = - A's d' εcu Es

c + A's d' εcu Es c

0.0

0.0

26.4

[-b ± (b2 - 4 a c)1/2 ] / [2 a ]=0 [-26.4 + (26.4² - 4 * -28.178 * -0)^0.5] /[2 * -28.178] OK x balance 7.62 ≥ 0.94 0.94 * 0.85 0.796 < 4.5 flange

26.4 k

0 k-in

30 25 20 15 10

5 Where the compression block a is greater than flange depth t Figure 33-10 Beam cross-section. 0 a b c 5 10 -A's 0.85 f'c x - A's εcu Es x -0.85 fc (bE -bw) t x 0 - 0.85 f'c bw β1 x² + A's 15 d' εcu20Es 25 As fy x

a x²

bx -28.178

x

a

0.94 in 1 logic 0.796

c 0.0

0.0

26.4

[-b ± (b2 - 4 a c)1/2 ] / [2 a ]=0 [-26.4 + (26.4² - 4 * -28.178 * -0)^0.5] /[2 * -28.178] OK x balance 7.62 ≥ 0.94

0.000

26.4

0

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DEPTH of COMPRESSION BLOCK -- Continued x 0.94 in IF(0, 0.94, 0.94) a 0.796 in 0.94 * 0.85 Direct calculation of compression reinforcing stress ε's 0.00000 unitless MAX((0.94 - 2.5) /0.94 * 0.003, 0) εy 0.00207 unitless 60 /29000 f'y 0 ksi MIN(0.00000 /0.00207 * 60, 60)

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Mn Cs Mn stem Mn flange

0 k-in 531 k-in 0 k-in

(20.5 - 2.5) * 0.0 26.4 * (20.5 - 0.796 /2) 0.0 * (20.5 - MIN(4.5, 0.796) /2)

Mn Mn Mu

531 k-in 44 k-ft 40 k-ft ult

0 + 531 + 0 531 / 12 in/ft 0.9 * 531

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 x distance of the neutral axis from extreme compression fiber, in a Whitney's compression block ε's εy

compression strain compression strain limit εcu 0.003 ε's 0.00000

Calculate compression forces and moments Cs 0.0 k (0 - 0.85 * 3) * 0.00 Cc stem 26.4 k 0.85 * 3 * 13 * 0.796 Cc flange 0.0 k 0.85 * 3 * MIN(0.796, 4.5) * (13 - 13)

77 % 1 logic 1

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x 0.94

Figure 33-11 Compression reinforcing strain. Cs Cc

force in compression reinforcing, k force in compressed concrete, k

Mn Mu

ultimate moment, k-ft Mn a reduced by 0.9, k-ft ultimate

Mu required / Mu 40 provided > 31 required OK x balance 7.62 ≥ 0.937 required d - d' d - t/2

Check compression reinforcing against tension reinforcing arm 20.10 in 531 /(0 + 26 + 0) Mn comp 531 k-in Compression block and reinforcing T Mn tens

26.4 k 531 k-in

0.44 * 60 20.10 * 26.40 Tension reinforcing

d - a/2 Figure 33-12 Distances to compression forces. T

force in tension reinforcing, k

30 25 20 15 10 5 0

Figure 33-13 Beam cross-section used in 0 5 10 15 20 appropriate pages to help in understanding and verifying math.

25

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MOMENT of INERTIA Ec 3122 ksi n 9.29 unitless

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57000 * ( 3 * 1000)² /1000 29000 /3,122

For the Gross (Overall) Section d' * n * -As A*arm 0 in3 2.5 * 9.29 * 0.00 84

Area deduct

2.5 * 0.00

9

20.5 * 0.44

0

OH * t2 /2 13.0 * 4.5² /2

4813 in3

Area

0 in2

-As * n 9.29 * 0.00

4

+As * n 9.29 * 0.4

0

OH * t 13.0 * 4.5

sum CG

width * d_OA-s_area_-s_area_ 13.0 * 27.0 - 0.00 - 0.44

355 in2 13.57 in

4,813 / 354.65 from the top fiber

For Gross Section About the CG Ad2 conc (d_OA * width) * (d_OA /2 - Cg)2 2 in4 (27.0 * 13.0) * (27.0 /2 - 13.57)²

Ad2 steel

0

(OH * t) * (t/2 - Cg)2 (0.0 * 4.5) * (78.0 /2 - 13.57)²

0 in4

n * -AS * (Cg - d' )2 9.29 * 6.00 * (13.6 - 2.5)²

196

Area deduct

0 21

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w x x/2 = n A's (x - d') + n As (d' - x) This is solved with the quadratic equation.

Figure 33-14 Beam / T-beam cross section for Ad2 / A = center of gravity from the top extreme fiber. n

ratio of modulus of elasticity ES /EC, unitless

Ec d_OA

modulus of elasticity of concrete, ksi beam overal depth, inches

width * d_OA² /2 13.0 * 27.0² /2

sum

351

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d * n * +As 20.5 * 9.29 * 0.44

0

4739

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 To find the neutral axis NAx for + and - reinforcing and the stress block:

n * As * (d - Cg)2 9.29 * 0.00 * (20.5 - 13.6)² -As * (Cg - d' )2 As * (d - Cg)2

I concrete

21323 in4

width * d_OA3 /12 + OH * tf 3/12 13 * 27³ /12 + 0 * 4.5³ /12

Ig

21500 in4

2 + 0 + 0 + 196 - 0 - 21 + 21,323

NOTE: use Ad2 only. Do not rotate component areas around their own axes as in bd3/12. Most moment of inertia calculations use Ad2 + bd3/12.

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EFFECTIVE MOMENT of INERTIA For the Cracked Section Where the equilibrium equation is: Cc + Cs = T

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 (bE - bw) t x

x - t/2

(n - 1) A's

x - d'

Where n - 1 deducts the area of compression concrete

bw x

For a rectangular beam: bw x x/2 + (n - 1) A's (x - d') = n As (d - x)

d-x n As Figure 33-15 The component areas of the cracked section.

For a rectangular, spandrel, or T beam flg logic

0 logic

1 if beam has a flange

stem bw x2/2 +

flange (bE -bw) t (x - t/2) * flg logic +

compression reinforcing (n - 1) A's (x - d') =

n As (d - x)

bw x2/2 +

(bE -bw) t x * flg logic +

(bE -bw) t (- t/2) * flg logic +

(n - 1) A's (x - d')

a x2

6.5 in

bw x2/2 + 6.5

bx

4.1

(bE -bw) t x * flg logic + 0

(n - 1) A's x 0.0

-n As (-x) 4.1

(bE -bw) t (- t/2) * flg logic + 0

(n - 1) A's (- d') 0.0

- n As d

c

NAx

-83.8

3.290 in

[ -b ± √ b² - 4 a c ] /2a [ -4 + √ 4² - 4 * 6.5 * -84 ] /(2 * 6.5)

-84

tension reinforcing -n As (d - x) = 0

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EFFECTIVE MOMENT of INERTIA -- Continued I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 Two Methods of Checking NAx Take the moments of each component area about the extreme fiber in compression. Divide the sum of the moments by the area of the components. Compare this calculated NAx to the NAx calculated above. 20.500 d

A's stem flange As sum M/area

area 0.00 42.77 0.00 4.09 46.85

arm area*arm 2.500 0.000 1.645 70.341 2.250 0.000 20.500 83.786 154.126 3.290

1.645 stem 2.250 flange 2.5 d'

3.290 NAx

Take the sum of the area moments about the calculated NA x to see if they are balanced.

0.790 d' 1.040 flange

A's stem flange As

area 0.00 42.77 0.00 -4.09

arm area*arm 0.790 0.000 1.645 70.341 1.040 0.000 17.210 -70.341 0.000

1.645 stem

17.2 d

3.290 NAx Figure 33-16 Moment of inertia is calculated as a balance of forces. For the Cracked Section About NAx w * NA_x3 /3 + (bE - w) * MINA(NA_x, t)3 /12 I conc. 154 In4 13 *3.29³ /3 + ( 13 - 13) * MIN(3.29, 4.5)³ /12 Ad2

I cracked

1211 In4

1365 In4

n * As * (NA_x)2 + n * A's *(NA_x - d_)2 9.29 *0.44 *(20.5 - 3.29)² + 9.29 *0.00 *(3.29 - 2.50)² cracked section for ± reinforcing

For the Effective Moment of Inertia 7.5 f'c2 fr 411 lb/in^2 Mcr 55 k-ft fr * Ig /yt for rectangular section only +M wk'g 22 k-ft

Ie comp Ig Ie

335752 In4 -19949 In4 315803 In4 21500 In4 21500 In4

(Mcr /Ma)3 * Ig [1 - (Mcr /Ma)3] * Icr (Mcr /Ma)3 * Ig +[1 - (Mcr /Ma)3] * Icr

GOVERNS

I cracked

Icr , cracked moment of inertia

Where M wk'g is the unfactored M at the section for which the deflection is computed

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CREEP vs. TIME in MONTHS ρ' 0.00000 unitless

L_60 L_12 L_6 L_3

ξ 2.0 1.4 1.2 1.0

λ 2.000 1.400 1.200 1.000

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density of compression reinforcing

multiply computed deflection by λ

Temperature Reinforcing Minimum depth 16 in depth not to exceed 16" for temperature reinforcing calculation AS req'd 0.346 in2 @ 12" 0.432 in2 @ 15" 0.518 in2 @ 18" ρ temp ρ temp/2

0.65 in2 /ft 0.32 in2 /ft

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entire slab 0.0020 entire slab limit to 16" depth

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 ρ' density of compression reinforcing at midspan, unitless ξ

time dependent factor for sustained load, 02 ACI 9.5.2.5

λ

multiplier for additional longterm deflection

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VERIFYING GRAPHS

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I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4

This is the VBA code for the macro that Paste Value(s) the input/output values into the graphing range. Having the "New Macro" recorder input your keystrokes may not record your intentions. Sub Macro2() ' ' Macro2 Macro ' Macro recorded 9/11/2002 by Craig T. Christy ' ' Keyboard Shortcut: Ctrl+Shift+Z ' Range("r13:r18").Select If you don't want to use this range, input your own selection. Selection.Copy Selection.End(xlToRight).Select ActiveCell.Offset(0, 1).Range("A1").Select ActiveSheet.Paste Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:= _ False, Transpose:=False Selection.End(xlToLeft).Select End Sub Press [Ctrl] [Shift] z

offset to the right copy [end] [right]

labels

inputs outputs

first copy / paste by hand

data values

paste

return [end] [left] Figure 33-17 GRAPHING VALUES FLOW DIAGRAM Rectangular beam with variable +As tension reinforcing fy 60 +As 3.54 aa 0.43 0.45 0.5 0.6 0.7 0.8 0.9 f'c 3 Mu prov 39.80 41.75 43.66 48.40 57.81 67 76.38 85.53 d' 2 M working 22 71 71 71 71 71 71 71 width 12 Mu req'd 31 100 100 100 100 100 100 100 d 36 Ig /100 215 142 142 142 143 144 145 146 400 depth_OA 38 I cr /100 14 21 21 22 25 28 31 34 A's 0 Ie /100 215 44 44 45 48 51 54 57 350 As 34 a * 50 39.8190045 42.15686 44.11765 49.01961 58.8235 69 78.43 88.24 Mu prov 300 M working Mu req'd 250 Ig /100 I cr /100 200 Ie /100 150 a * 50 100 50 0 0.4

0.9

1.4

1.9

+As

2.4

2.9

3.4

3.9

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Figure 33-18 Rectangular beam +As compression reinforcing

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versus Mu Provided, Mu required, Ig, Icr, Ie.

M

Page 33 - 21

REINFORCED CONCRETE BEAM

33

EXHAUST STACK

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers A

B

C

D

351773867.xls

E

F

G

VERIFYING GRAPHS -- Continued Rectangular beam variable -As' compression reinforcing fy 60 -As 6.00 aaa f'c 3 MU prov 40 d' 2 a * 50 40 bw 12 MU req'd 31 d 22 Ig /100 215 depth_OA 24 I cr /100 14 350 A's varies Ie /100 215 As 3.5

H

I

J

K

L

M

I gross 21,500 in^4 / I cracked 1,365 in^4 / I effective 21,500 in^4 0.00 292 343 250 165 104 106

0.10 294 333 250 166 104 105

0.20 296 323 250 167 103 105

0.30 297 313 250 168 103 105

0.40 299 304 250 169 103 104

0.50 301 295 250 169 102 104

0.60 303 286 250 170 102 104

300

250

MU prov

200

a * 50 MU req'd

150

Ig /100 I cr /100 Ie /100

100

50

0 0.00

-As 1.00

2.00

3.00

4.00

5.00

6.00

7.00

Figure 33-19 A's compression reinforcing versus Mu Provided, Mu required, Ig, Icr, Ie. T-beam joist with variable spacing fy 60 joist spacing f'c 3 MU prov d' 2 MU req'd width 12 Ig /100 d 22 I cr /100 depth_OA 24 Ie /100 A's 0 a As 3 Length 40 t 4 c_350.00 varies flange 2

80 aa 39.80 250 215.00 13.65 215.00 7.964

12 257.29 250 161.13 91.38 114.21 58.824

13 260.14 250 165.79 91.48 117.03 55.490

14 262.72 250 170.33 91.77 120.05 52.157

15 265.06 250 174.76 92.20 123.24 48.824

20 25 30 273 278 281 250 250 250 195 214 230 96 101 106 141 159 178 35 28.24 23.53

300.00

MU prov 250 Ig /100 I cr /100 Ie /100

250.00 200.00 150.00 Jois t Spacing 100.00 12

22

32

42

52

62

72

82

Page 33 - 22

REINFORCED CONCRETE BEAM

33

EXHAUST STACK

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers A

B

C

D

E

351773867.xls F

G

H

I

Figure 33-20 Joist spacing versus Mu Provided, Mu required, Ig, Icr, Ie.

J

K

L

M

Page 33 - 23

REINFORCED CONCRETE BEAM

33

EXHAUST STACK

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers A

B

C

D

VERIFYING GRAPHS -- Continued T-beam variable +As tension reinforcing fy 60 +As f'c 3 Mu prov d' 2 Mu req'd width 12 Ig /100 d 22 I cr /100 depth_OA 24 Ie /100 A's 0 a * 100 As varies Length 30 t 4 c_ 30 flange 1200 2 Mu prov 1000

Mu req'd

800

Ig /100 I cr /100

351773867.xls

E

F

8.86 aa 39.80 30.7 215.001201 13.6484941 215.001201 79.638009

G

0.4 42.24 100 201.8821 23.09306 83.62034 33.72549

H

I

J

0.5 0.6 0.7 49.06 58.76 68.44 100 100 100 202.7001 203.864 205.023 26.40514 30.86259 35.0689 87.06423 91.77422 96.2889 39.21569 47.05882 54.902

K

L

M

0.8 0.9 1 ### 87.67 97.24 100 100 100 206 207 208 39 42.92 46.62 101 105 109 63 70.59 78.43

Ie /100 a * 100

600 400 200 0 0.4

1.4

2.4

3.4

4.4

5.4

6.4

7.4

8.4

Figure 33-21 T-beam with variable +As tension reinforcing versus M_provided, Mu required, Ig, Icr, Ie. T-beam variable -As' compression reinforcing fy 60 +As 10.969 aa f'c 3 Mu prov 39.80 d' 2 Mu req'd 30.7 width 12 Ig /100 215.001201 d 22 I cr /100 13.6484941 depth_OA 24 Ie /100 215.001201 A's 2 a * 100 79.638009 f'y As varies 0 Length 30 t 4 c_ 30 flange 1200 2

0.1 0.2 0.3 0.5 143.93 146.41 148.96 154.25 100 100 100 100 208.2046 209.461 210.7118 213.197 7.919034 12.73429 17.35286 26.0852 75.61118 80.81894 85.86392 95.5338 117.1421 119.2233 121.359 125.799 0 0 0 0

1 1.5 2 ### ### ### 100 100 100 219 225 231 46 62.83 78.33 118 138 157 138 152 167 0 0 0

Mu prov Mu req'd Ig /100 I cr /100 Ie /100 a * 100 f'y

1000 800 600 400 200 0 -1

+As 1

3

5

7

9

11

Page 33 - 24

REINFORCED CONCRETE BEAM

33

EXHAUST STACK

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers A

B

C

D

E

351773867.xls F

G

H

I

J

K

L

Figure 33-22 T-beam with variable -As' compression reinforcing versus M_provided, Mu required, Ig, Icr, Ie.

M

Page 33 - 25 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4

row 20

20

25

30

ith Whitney's row 30

e, bar size is given es to 8 th's of an

of reinforcing, k/in 2 ength of concrete,

row 40

row 50

row 60

ally the top steel,

ally the bottom steel,

Page 33 - 26 Christy 11:20 04/16/17

351773867.xls N

row 70

Page 33 - 27 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 ngular beam or, T-beam effective

row 80

m calculations are

culations can be

row 90

row 100 vLookUp Table Area 0.12 0.27 0.49 0.76 1.1 1.49 2.47 3.68 0 0.11 0.2 0.31 0.44 0.6 0.79 1 1.27 1.56 2.25 4 Area 0.00

Page 33 - 28 Christy 11:20 04/16/17

351773867.xls N

Area 0.44 row 130

Page 33 - 29 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4

een the B and the 1 oes not read this

εcrushing, code defined crushing

β1 xb

amax profile row 170

nforcing, unitless

ing strain, unitless

n εt increases, the

Page 33 - 30 Christy 11:20 04/16/17

351773867.xls N

row 190

Page 33 - 31 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4

row 200

Whitney's stress

row 210

row 220

=0

=0

15

20

25

30 = 0

=0

Page 33 - 32 Christy 11:20 04/16/17

351773867.xls N

row 250

Page 33 - 33 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 is from extreme

row 200

row 210

row 300 20

25

30

Page 33 - 34 Christy 11:20 04/16/17

351773867.xls N

row 310

Page 33 - 35 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 - reinforcing and

row 320

beam cross section

city E S /EC, unitless row 330

row 340

row 350

row 360

Page 33 - 36 Christy 11:20 04/16/17

351773867.xls N

row 370

Page 33 - 37 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 (bE - bw) t (n - 1) A's

acked section.

row 400

row 410

row 420

Page 33 - 38 Christy 11:20 04/16/17

351773867.xls N

row 430

Page 33 - 39 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4

row 440

row 450

row 460

moment of inertia row 470

ctored M at the ection is computed

row 480

Page 33 - 40 Christy 11:20 04/16/17

351773867.xls N

row 490

Page 33 - 41 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 mpression reinforcing

nt factor for sustained

dditional longterm row 500

row 510

row 520

row 530

row 540

Page 33 - 42 Christy 11:20 04/16/17

351773867.xls N

row 550

Page 33 - 43 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4

row 560

row 570

row 580

1 94.59 71 100 146 36 60 98.0392

3.4

3.9

Page 33 - 44 Christy 11:20 04/16/17

351773867.xls N

row 610

Page 33 - 45 Christy 11:20 04/16/17

351773867.xls N

/ I effective 21,500 in^4 0.80 307 269 250 172 101 103

MU prov a * 50 MU req'd Ig /100 I cr /100 Ie /100

6.00

7.00

35 283.39 250 245.20 110.47 196.29 20.168

MU prov 250 Ig /100 I cr /100 Ie /100 72

82

Page 33 - 46 Christy 11:20 04/16/17

351773867.xls N

row 670

Page 33 - 47 Christy 11:20 04/16/17

351773867.xls N

1.2 116.26 100 210.738 53.6796 116.919 94.1176

7.4

8.4

2.11 207.37 100 232.443 81.5564 160.861 170.925 5

9

11

Page 33 - 48 Christy 11:20 04/16/17

351773867.xls N

row 730