Design of a Tension Member and End Connections
This example considers the design of an HSS tension member, slotted to accept bolted plate connections. The end connections are included in the design.
This example is also available in the form of traditional hand-written design notes.
The following data are given:
| Property | Value |
|---|---|
| $T_f$ | 800 kN |
| Angles | 300W |
| Plates | 300W |
| HSS | 350W |
| W | A992 |
| Bolts | A325 3/4" |
| Holes | 22 mm punched |
| Electrodes | E49xx |
# set the default precion for display of floating point numbers
%precision 3
# import useful utilities
from sst import SST
sst = SST()
from utils import show
phi = 0.9
phiu = 0.75
phib = 0.8
phibr = 0.8
phiw = 0.67
aFy = 300 # angles
aFu = 440
pFy = 300 # plates
pFu = 440
hFy = 350 # HSS
hFu = 450
wFy = 345 # W shapes
wFu = 450
bFu = 825 # bolt
Xu = 490 # weld
bd = 25.4 * 0.75 # 3/4" bolt diameter
hd = 22 + 2 # hole diameter: 22mm punched holes
Tf = 800 # Factored applied load
1: HSS Tension Member
Tension brace will be bolted into frame. Net area fracture is often the governing case. Reasonable estimates are $A_{n} = 0.9 A_g$ and $A_{ne} = 0.85 A_n$. Therefore, a first estimate of cross-sectional area required is, without differentiating material strengths, is:
$$ \phi_u \times 0.85 \times 0.9 A_g \times Fu \ge T_f $$$$ A_g \ge \frac{T_f}{\phi_u 0.85 \times 0.9 Fu} $$Ag_reqd = Tf/(phi*0.85*0.9*450*1E-3)
%show Ag_reqd
Therefore, look for a square HSS of at least this cross-sectional area:
Gentle Reader: If you find the following 'python-pandas magic' hard to understand, don't worry about it. Its just a digital equivalent of searching through the properties and dimensions tables in Part 6 of the CISC handbook of Steel Construction.
# load a table of all HSS sections, sorted by increasing mass
hss = sst.section_tables(['HS'],'Mass')
# search for those shapes whose area meets the limit and whose B and D dimensions are equal (square)
# Then extract the first 10 rows (shapes) of the search result and choose only three columns for display.
hss.query('A>2500 & B==D').head(10)[['A','T','Mass']]
Other shapes are possible.
2: Bolting Requirements
Again, because net area considerations usually govern we should start by selecting a bolt configuration (number of bolts and layout). That will allow us to decide gusset plate thickness, which is important because the insert plate welded into the HSS must be the same thickness, and that affects the strength of the HSS.
Ab = 3.14159*bd*bd/4 # bolt shear area
vr1 = 0.60*phib*1*2*Ab*bFu * 0.7 * 1E-3 # factored shear strength of 1 bolt, double shear, threads intercepted
%show Nbolts=Tf/vr1 # number of bolts required
Therefore, use 6 bolts. But should they be in a 2x3 or a 3x2 arrangement?
3: Gusset plate design
Mostly we want the thickness so that we can determine the thickness of the insert plate in the HSS.
As the gusset plate can be made as wide as necessary, design will likely be governed by block shear and/or bearing.
Calculate resistances of a 1mm thickness of plate, then determine the thickness required.
An = (75 - (hd/2)*2)*1
Ut = 1
Agv = (2*(2*75+35))*1
vr1 = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show An,Ut,Agv,vr1
An = (75+35-(hd/2)*3)*1
Ut = 0.6
Agv = (1*(2*75+35))*1
vr2 = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show An, Ut, Agv, vr2
An = 0.
Ut = 1
Agv = (4*(2*75+35))*1
vr3 = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show Agv, vr3
# length 7-3:
l73 = 75 + 35
# length 6-7: (all lines at 45 degrees)
l67 = (75+75)/(2**0.5)
# length 5-6:
l56 = 35*(2**0.5)
# gauge and pitch for line 5-7 (they are equal):
s = g = (l56+l67)/(2**0.5)
# net area:
Ane = An = (l56+l67+l73)*1 - 2*hd*1 + (s*s/(4*g))*1 # 12.3.1
tr1 = phiu*Ane*pFu * 1E-3 # 13.2 iii)
%show l56,l67,l73,s, Ane,tr1
br1 = 3*phibr*6*1*bd*pFu * 1E-3
%show br1
# governing resistance is (the minimum of the above)
r = min(vr1,vr2,vr3,tr1,br1)
%show r
# thickness required, mm
%show Reqd T=Tf/r
The governing resistance is block shear (Path 1-3). The required thickness is thus 17.6mm, so try a 20mm gusset plate.
Dsg = 'HS127x127x6.4'
hss = sst.section(Dsg)
hssT,hssAg = hss.props('T,A')
%show hssT,hssAg
pW = 35 + 75 + 35 # width of plate based on bolt spacing and edge distance
pT = 20 # thickness of plate based on gusset thickness
Ag = hssAg
Tr = phi*Ag*pFy * 1E-3
%show Tr, Tr >= Tf
Ane = pW*pT - 2*hd*pT
Tr = phiu*Ane*pFu * 1E-3
%show Tr, Tr >= Tf
No good - make the plate wider from:
$$ \phi_u(w t - 2\times24t)F_u \ge T_f $$$$ w t \ge \frac{T_f}{\phi_u F_u} + 48 t $$$$ w \ge \frac{T_f}{\phi_u t F_u} + 48 $$%show w reqd.=Tf/(phiu * pT * pFu*1E-3) + 48
pW = 220
pT = 20
An = (75+75+35 - 24*2.5)*pT
Ut = 0.6
Agv = (35+75)*pT
Tr = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show An, Ut, Agv, Tr, Tr >= Tf
An = (75+75 - 24*2)*pT
Ut = 1.0
Agv = 2*(35+75)*pT
Tr = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show An, Ut, Agv, Tr, Tr >= Tf
An = 0
Ut = 1.0
Agv = 6*(35+75)*pT
Tr = phiu*(Ut*An*pFu + 0.6*Agv*(pFy+pFu)/2) * 1E-3
%show An, Ut, Agv, Tr, Tr >= Tf
OK, from above. (see 3.5)
# Total Required length is: (all weld parallel to axis of load)
%show 4L reqd.=Tf/(0.67*phiw*(0.707*6)*Xu*1E-3) # S16 13.13.2.2
%show L reqd.=857/4
Try 220mm of weld in 4 locations, insert plate to HSS.
# central portion is An2
w = 127.
L = 220.
pT = 20.
2*w >= L >= w
An2 = .5*w*pT + .25*L*pT
%show w,T=pT,L,An2
# edge portions are An3's:
w = (220-127)/2.
xbar = w/2.
L >= w
An3 = (1-xbar/L)*w*pT
%show w,T=pT,xbar,An3
Ane = An3 + An2 + An3
Tr = phiu * Ane * pFu * 1E-3
%show Ane, Ane/(pW*pT), Tr, Tr>=Tf
Therefore, 20mm x 220mm Insert Plate OK
# See page 7-88 CISC HB
d = 127
b = (127/2) - (20/2.)
w = t = 6.35
A = d*w + 2*(b-w)*t
xbar = b - (1/(2.*A))*((d-2*t)*w**2 + 2*t*b**2)
Lw = 220
%show d,b,t,A,xbar,Lw, xbar/Lw >= 0.1
An = 2960 - 2*20*6.35 # 20mm slot cut from both sides of HSS
Ane = max((1.1 - xbar/Lw)*An, 0.8*An) # S16 12.3.3.4
Tr = phiu*Ane*hFu * 1E-3
%show An,Ane,Ane/An,Tr,Tr>=Tf
Therefore, HSS 127x127x6.4 OK
OK, as thickness of 2 plates = thickness of insert plate, and widths are the same.
Therefore, 2-10mmx220mm Lap Plates OK
The following can be used to search for angles or W shapes to replace the HSS
##angle=sst.section_tables(['L'],'Mass')
##angle.query("Avl!='*'&A>2500/4").head(10)[['A','T','Mass']]
##angle=sst.section_tables(['W'],'Mass')
##angle.query("Avl!='*'&A>2500").head(10)[['Dsg','A','T','Mass']]
11. Comments and Observations
It may have been better to start with the design of the Insert Plate in step 3, rather than the gusset plate. Would have more quickly determined a workable bolt pattern that way.
In Step 7, it is quite possibl;e that we would have found the HSS inadequate ($T_r \lt T_f$). Had that happened, we would have the following possible remedies:
- decrease the thickness of the insert/gusset plates.
- increase the length of the HSS to insert plate weld. Neither of these would help very much.
- increase the size of the HSS (don't always have that freedom).
- add two cover plates welded to the HSS. See the photo in step 1.