T20: W Brace

Example T20: W Brace with Reduced cross-section

This example shows the computation of the factored tension resistance of a W shape used as a tension member in a lateral brace in a building such as that shown in the following photo. There are 4 braces shown. Note that the upper left brace has the flange tips removed from the W-shape (to ensure ductility under seismic forces). We will compute the strength of a brace similar to that shown in the photo.

from Designer import DesignNotes, SST, Part, show

%figure "DSC6443-small.jpg"

Load and Setup the Library Modules

import pint                  # setup to use the module for computing with units
ureg = pint.UnitRegistry()
mm = ureg['mm']
inch = ureg['inch']
kN = ureg['kN']
MPa = ureg['MPa']
ureg.default_format = '~P'

notes = DesignNotes('Tr',units=kN)      # initial the note/record keeping object
RECORD = notes.record     # useful abbreviations
CHECK = notes.check
USEVARS = notes.usevars

Problem Statement

Compute the factored tension resistance, $T_r$ , of the following assembly. Steel is G40.21 350W and bolts are 3/4" ASTM A325 in 22mm punched holes.

%figure "brace1.svg"

Note that 40mm is cut from each flange tip of the W250x67.

Angles

%figure "angle.svg"

class Bolts(Part):
    "Bolts"
    grade = 'ASTM A325'
    size = '3/4"'
    d = (3/4*inch).to(mm)
    Fu = 825*MPa
    Ab = 3.14159*d**2/4.
    n = 4                    # number of bolts per end
    s = 75.*mm                  # bolt spacing
    threads_intercepted = True

Bolts.show()

Ab                  = 285   mm²
d                   = 19.05 mm
Fu                  = 825   MPa
grade               = ASTM A325 
n                   = 4     
s                   = 75    mm
size                = 3/4"  
threads_intercepted = True

class Angles(Part):
    "Angles"
    grade = 'CSA G40.21 350W'
    Fy = 350*MPa
    Fu = 450*MPa
    d,b,t,Ag,size = SST.section('L102x76x13','D,B,T,A,Dsg')
    d = d*mm
    b = b*mm
    t = t*mm
    Ag = Ag*mm*mm
    ha = (22 + 2)*mm  # hole allowance  - punched holes
    g1 = 65*mm        # gauge, longer leg
    g2 = 45*mm        # gauge, shorter leg
    s = 80*mm         # dist between innermost holes on each end

Angles.show()

Ag    = 2100 mm²
b     = 76.2 mm
d     = 102  mm
Fu    = 450  MPa
Fy    = 350  MPa
g1    = 65   mm
g2    = 45   mm
grade = CSA G40.21 350W 
ha    = 24   mm
s     = 80   mm
size  = L102x76x13 
t     = 12.7 mm

Check Details (TO BE DONE!)

Bolt spacings, edge distances
Fit within flanges (need gusset thickness)

CHECK(False,'Bolting and fitting details have not been checked.')

    Bolting and fitting details have not been checked.?  NG! *****
      ()

Net Section Fracture:

phiu = 0.75

with USEVARS(('d,b,t,ha,g1,g2,Fu,s',Angles),
             ('n',Bolts),
             locals='wg,wn1,g,wn2,wn,An,Ane', globals='phiu',
             record='Tr',label='Net section fracture, 4 angles',
             ):
    # gross width = "flattened" width of angle:
    wg = d + b - t

    # failure path 1-1: 1 hole
    wn1 = wg - 1*ha

    # failure path 2-2: 2 holes
    g = g1 + g2 - t
    wn2 = wg - 2*ha + s**2/(4*g)

    wn = min(wn1,wn2)

    An = wn*t
    Ane = 0.8*An if n >= 4 else 0.6*An   # S16-14: 12.3.3.2 (b) (i) - connected 1 leg n lines of bolts
    
    Tr = 4. * phiu*Ane*Fu    # S16-14: 13.2 a) iii)

d     = 102       mm
b     = 76.2      mm
t     = 12.7      mm
ha    = 24        mm
g1    = 65        mm
g2    = 45        mm
Fu    = 450       MPa
s     = 80        mm
n     = 4         
wg    = 165.5     mm
wn1   = 141.5     mm
g     = 97.3      mm
wn2   = 133.9     mm
wn    = 133.9     mm
An    = 1701      mm²
Ane   = 1361      mm²
Tr    = 1.837e+06 MPa·mm²
phiu  = 0.75

    Net section fracture, 4 angles: Tr = 1837 kN

Gross Section Yield:

phi = 0.9
with USEVARS(('Ag,Fy',Angles),
            label='Gross section yield, 4 angles'):
    Tr = 4. * phi*Ag*Fy    # S16-14: 13.2 a) i)

Ag    = 2100      mm²
Fy    = 350       MPa
Tr    = 2.646e+06 MPa·mm²

    Gross section yield, 4 angles: Tr = 2646 kN

Block Shear

with USEVARS(('t,d,b,g1,g2,ha,Fy,Fu',Angles),
             ('n,s',Bolts),
             locals='Agv,An,Ut',
             label='Block shear, 4 angles'):
    Agv = (40*mm + (n-1)*s)*t
    An = (min(d-g1,b-g2) - ha/2.)*t
    Ut = 0.3     # SUPER conservative
    Tr = 4. * phiu*(Ut*An*Fu + 0.6*Agv*(Fy+Fu)/2.)

t     = 12.7      mm
d     = 102       mm
b     = 76.2      mm
g1    = 65        mm
g2    = 45        mm
ha    = 24        mm
Fy    = 350       MPa
Fu    = 450       MPa
n     = 4         
s     = 75        mm
Agv   = 3366      mm²
An    = 243.8     mm²
Ut    = 0.3       
Tr    = 2.522e+06 MPa·mm²

    Block shear, 4 angles: Tr = 2522 kN

Tr for W Shape

%figure "w.svg"

class WShape(Part):
    "WShape"
    grade = 'ASTM A992'
    Fy = 345*MPa
    Fu = 450*MPa
    Ag,b,d,t,w,size = SST.section('W250x67',properties='A,B,D,T,W,Dsg')
    Ag = Ag*mm*mm
    b = b*mm
    d = d*mm
    t = t*mm
    w = w*mm
    wp = 190*mm          # width of web reinforcing PL
    tp = 8*mm            # thickness of web reinforcing PL
    wc = 40*mm           # width cut from flange tips

Net section fracture

# Path 1-1: net = gross  +  plates  -  holes
with USEVARS(('ha',Angles),
             ('Ag,tp,wp,w,Fu',WShape), 
             locals='An,Ane',
             label='Net section fracture, W shape'):
    An = Ag  +  2*tp*wp   -  2*ha*(w+tp+tp)
    Ane = 0.85*An    # S16-14: 12.3.3.2 (c) (i)
    phiu = 0.75
    Tr = phiu*Ane*Fu

ha    = 24        mm
Ag    = 8550      mm²
tp    = 8         mm
wp    = 190       mm
w     = 8.9       mm
Fu    = 450       MPa
An    = 10390     mm²
Ane   = 8836      mm²
Tr    = 2.982e+06 MPa·mm²

    Net section fracture, W shape: Tr = 2982 kN

Gross section yield

with USEVARS(('Ag,wc,t,Fy',WShape),
             locals='Agr',
             label='Gross section yield, W shape'):
    Agr = Ag - 4*wc*t   # reduced area due to flange cuts
    phi = 0.9
    Tr = phi*Fy*Agr     # S16-14: 13.2 a) i)

Ag    = 8550      mm²
wc    = 40        mm
t     = 15.7      mm
Fy    = 345       MPa
Agr   = 6038      mm²
Tr    = 1.875e+06 MPa·mm²

    Gross section yield, W shape: Tr = 1875 kN

Block Shear

with USEVARS(('ha,g2',Angles),
             ('w,tp,Fy,Fu',WShape),
             ('n,s',Bolts),
             locals='Agv,An,Ut',
             label='Block shear, W shape'):
    T = w + tp + tp          # thickness of web + reinforcing plates
    Agv = 2*(40*mm + (n-1)*s)*T
    An = (g2 + g2 + 25*mm - ha)*T   # estimate 25mm spacing between angles (gusset thickness)
    Ut = 1.0
    phiu = 0.75
    Tr = phiu*(Ut*An*Fu + 0.6*Agv*(Fy+Fu)/2.)       # S16-14: 13.11

ha    = 24        mm
g2    = 45        mm
w     = 8.9       mm
tp    = 8         mm
Fy    = 345       MPa
Fu    = 450       MPa
n     = 4         
s     = 75        mm
Agv   = 13200     mm²
An    = 2266      mm²
Ut    = 1         
Tr    = 3.125e+06 MPa·mm²

    Block shear, W shape: Tr = 3125 kN

Tearout

with USEVARS(('w,tp,Fy,Fu',WShape),
             ('n,s',Bolts),
             locals='An,Ahv,Ut',  label='Tearout, W shape'):
    Agv = 4*(40*mm + (n-1)*s)*T
    An = 0*mm*mm
    Ut = 1.0
    phiu = 0.75
    Tr = phiu*(Ut*An*Fu + 0.6*Agv*(Fy+Fu)/2.)       # S16-14: 13.11

w     = 8.9       mm
tp    = 8         mm
Fy    = 345       MPa
Fu    = 450       MPa
n     = 4         
s     = 75        mm
An    = 0         mm²
Ahv   = None      
Ut    = 1         
Tr    = 4.721e+06 MPa·mm²

    Tearout, W shape: Tr = 4721 kN

Bolt Shear

with USEVARS(('n,s,Ab,Fu,threads_intercepted',Bolts),
             locals='m,L',    label='Bolt Shear'):
    phib = 0.8
    m = 2
    Tr = 0.6*phib*n*m*Ab*Fu * 2   # S16 13.12.1.2.b)
    L = (n-1)*s   # length of connection
    if L >= 760*mm:
        Tr = Tr * (0.5/0.6)
    if threads_intercepted:
        Tr = Tr * 0.7

n                   = 4         
s                   = 75        mm
Ab                  = 285       mm²
Fu                  = 825       MPa
threads_intercepted = True      
m                   = 2         
L                   = 225       mm
Tr                  = 1.264e+06 MPa·mm²

    Bolt Shear: Tr = 1264 kN    <<<--- GOVERNS

Bolt Bearing

with USEVARS(('t,Fu',Angles),
             ('n,d',Bolts),
             record='Br', label='Bolt Bearing (on 4 angles)'):
    phibr = 0.8
    Br = 3*phibr*n*t*d*Fu * 4

t     = 12.7      mm
Fu    = 450       MPa
n     = 4         
d     = 19.05     mm
Br    = 4.181e+06 MPa·mm²

    Bolt Bearing (on 4 angles): Tr = 4181 kN

with USEVARS(('w,tp,Fu',WShape),
             ('n,d',Bolts),
             locals='t',     record='Br', label='Bolt Bearing (on web of W)'):
    t = w+tp+tp
    Br = 3*phibr*(n*2)*t*d*Fu

w     = 8.9       mm
tp    = 8         mm
Fu    = 450       MPa
n     = 4         
d     = 19.05     mm
t     = 24.9      mm
Br    = 4.098e+06 MPa·mm²

    Bolt Bearing (on web of W): Tr = 4098 kN

Summary

notes.summary()

Summary of DesignNotes for Tr
=============================

Checks:
-------
    Bolting and fitting details have not been checked.?   NG! *****
      ()

Values of Tr:
-------------
    Net section fracture, 4 angles: Tr = 1840 kN
    Gross section yield, 4 angles:  Tr = 2650 kN
    Block shear, 4 angles:          Tr = 2520 kN
    Net section fracture, W shape:  Tr = 2980 kN
    Gross section yield, W shape:   Tr = 1870 kN
    Block shear, W shape:           Tr = 3130 kN
    Tearout, W shape:               Tr = 4720 kN
    Bolt Shear:                     Tr = 1260 kN    <<<--- GOVERNS
    Bolt Bearing (on 4 angles):     Tr = 4180 kN
    Bolt Bearing (on web of W):     Tr = 4100 kN

    Governing Value:
    ----------------
       Tr = 1260 kN

To do:

check bolt detailing
check fit of angles between flanges
bolts, bearing - gusset and web
weld, reinforcing plates to web
thickness of gusset plate. Bolts allow 100-45-45 = 10mm. too little.

Notes

Note that gross section yield of the W should govern, but it does not, by a large margin.
- Obviously, more bolts are required, or detail them so that threads are not intercepted (risky for installation considerations).
- Another thing to try would be slightly larger angles. Perhaps L127x76x13 (which would not require any additional space between flanges).
- Or could shave a few more millimeters from flange tips.
- Of course, all this has to be compared with factored applied loads.