T10: Lap Splice

Example T10: Tension Member Lap Splice

Compute the factored tension resistance, $T_r$, of the following plate tension member, lap splice and fasteners. Ignore the connection details at the far ends of the member (not shown). Bolts are 3/4" A325 in a bearing-type connection, in 22mm punched holes (assume threads intercepted). The plates are of CSA G40.21 350W steel.

from Designer import DesignNotes, makePart, show

%figure lap-splice-01.svg

Import and Setup 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',title='Lap Plate Splice',units=kN)
REQUIRE = notes.require      # convenient abbreviations
CHECK = notes.check
RECORD = notes.record
USEVARS = notes.usevars

Set the Problem Parameters (data)

@makePart
class Steel:
    grade = "CSA G40.21 350W"
    Fy = 350*MPa
    Fu = 450*MPa
    
@makePart
class Bolts:
    grade = "ASTM A325"
    Fu = 825*MPa 
    d = (3/4*inch).to(mm)
    Ab = 3.14159*d*d/4.
    
@makePart
class CentrePlate:
    W1 = 300*mm
    T1 = 25*mm
    C = 10*mm   # clearance between ends

@makePart
class OuterPlate:
    W2 = 220*mm
    T2 = 14*mm
    L2 = 350*mm

@makePart
class BoltGroup:    
    hd = 22*mm
    hole_type = 'punched'
    threads_intercepted = True
    HA = hd if hole_type == 'drilled' else hd+2*mm
    G = 75*mm     # gauge (transverse spacing)
    S = 75*mm     # longitudinal spacing
    NT = 3        # number of bolts across
    NL = 2        # number of lines of bolts each side
    S2 = 140*mm   # distance between inner lines of bolts

Check Bolting Details

Edge distance, end distance and spacing. If a REQuirement is not met, execution terminates with an error message.

with USEVARS((BoltGroup,'NT,NL')):
    REQUIRE( NT >= 2, "NT - Number of bolts across (on a line transverse to load)" )
    REQUIRE( NL >= 2, "NL - number of transverse lines (# of bolts per longitudinal line)" )

NT    = 3 
NL    = 2

with USEVARS((OuterPlate,'T2'),(BoltGroup,'NL'),(Bolts,'d'),globals='min_pitch,min_edge,min_end,max_edge'):
    min_edge = 32*mm      # S16 22.3.2, min edge distance, 3/4" bolt, sheared edge, Table 6
    max_edge = min(12*T2,150*mm)    # S16 22.3.3
    min_end = min_edge if NL > 2 else 1.5*d  # S16 22.3.4
    min_pitch = 2.7*d    # S16 22.3.1

T2        = 14    mm
NL        = 2     
d         = 19.05 mm
min_pitch = 51.43 mm
min_edge  = 32    mm
min_end   = 28.57 mm
max_edge  = 150   mm

Center plate:

with USEVARS((CentrePlate,'W1,C'),(BoltGroup,'S2,S,G,NT'),locals='edge,end',globals='min_pitch,min_edge,min_end,max_edge'):
    edge = (W1 - (NT-1)*G)/2.
    end = (S2 - C)/2.
    REQUIRE(edge >= min_edge)
    REQUIRE(edge <= max_edge)
    REQUIRE(end >= min_end)
    REQUIRE(S >= min_pitch)
    REQUIRE(G >= min_pitch)
    REQUIRE(S2 >= S)
    REQUIRE(S**2 + G**2 >= min_pitch**2)    # to be more precise ...

W1        = 300   mm
C         = 10    mm
S2        = 140   mm
S         = 75    mm
G         = 75    mm
NT        = 3     
edge      = 75    mm
end       = 65    mm
min_pitch = 51.43 mm
min_edge  = 32    mm
min_end   = 28.57 mm
max_edge  = 150   mm

Side plates:

with USEVARS((OuterPlate,'W2,L2'),(BoltGroup,'S2,S,G,NT,NL'),locals='edge,end',globals='min_pitch,min_edge,min_end,max_edge'):
    edge = (W2 - (NT-1)*G)/2.
    end = (L2 - ((NL-1)*G*2 + S2))/2.
    REQUIRE(edge >= min_edge)
    REQUIRE(edge <= max_edge)
    REQUIRE(end >= min_end)

W2        = 220   mm
L2        = 350   mm
S2        = 140   mm
S         = 75    mm
G         = 75    mm
NT        = 3     
NL        = 2     
edge      = 35    mm
end       = 30    mm
min_pitch = 51.43 mm
min_edge  = 32    mm
min_end   = 28.57 mm
max_edge  = 150   mm

Factored Resistance

Centre plate

%figure lap-splice-01-main.svg

Yield on Gross Area

If the gross (unreduced) cross-section reaches the yield stress, there will be considerable axial elongation as yield strains are reached over the length of the member. This is considered a failure state.

with USEVARS((CentrePlate,'W1,T1'),(Steel,'Fy'),locals='Ag,phi',
            label='Gross area yield, centre plate'):
    Ag = W1*T1    # gross x-sectional area
    phi = 0.9
    Tr = phi*Ag*Fy  # S16-14: 13.2 a) i)

W1    = 300       mm
T1    = 25        mm
Fy    = 350       MPa
Ag    = 7500      mm²
phi   = 0.9       
Tr    = 2.362e+06 MPa·mm²

    Gross area yield, centre plate: Tr = 2362 kN

Fracture on Effective Net Area

When the average stress across the net (reduced) area reaches the fracture stress, fracture will occur on that area. This is, of course, a failure mode. Normally, an account is made of how non-uniform load transfers can affect the stress distribution across the cross-section; if the stress distribution is markedly non-uniform, fracture may occur before the average stress reaches the ultimate. This account is done by computing and using an effective net area, $A_{ne}$.

Because the bolts in this connection transfer loads approximately uniformly across the entire cross-section, we can consider the effective net area to be equal to the net area in possible failure path 1 in the above figure. (S16 12.3.3.1)

with USEVARS((CentrePlate,'W1,T1'),(Steel,'Fu'),(BoltGroup,'NT,HA'), locals='wn,Ag,phiu,An,Ane',
             label='Net section fracture, centre plate'):
    wn = W1 - NT*HA    # net width: subtract total width of material removed by holes, failure path 1
    Ane = An = wn*T1
    phiu = 0.75
    Tr = phiu*Ane*Fu    # S16-14: 13.2 a) iii)

W1    = 300       mm
T1    = 25        mm
Fu    = 450       MPa
NT    = 3         
HA    = 24        mm
wn    = 228       mm
Ag    = None      
phiu  = 0.75      
An    = 5700      mm²
Ane   = 5700      mm²
Tr    = 1.924e+06 MPa·mm²

    Net section fracture, centre plate: Tr = 1924 kN

Block Shear

Other potential failure modes involve tension and shear ruptures in combination around the bolt holes. The following figure shows 4 different potential failure patterns that must be investigated.

Patterns 1 and 2 are sort of "opposites" - in Pattern 1 in the tension rupture extends across the end of the bolts, while in Pattern 2 the tension rupture extends from one line of bolts to the outside edges of the plates. Pattern 2 probably is only important when there are 2 longitudinal lines of bolts (there are three shown on the drawing, though our calculations will allow for any number, 2 or more). If there are more than 3 longitudnal lines (I.e., more than 3 bolts across), other patterns similar to Pattern 2 could be drawn but thay will not have lower strengths.

Note that Pattern 4 is often called 'tear-out' or 'pull-out'.

%figure lap-splice-01-main-blocks.svg

with USEVARS((Steel,'Fy,Fu'),locals='Fv'):
    Fv = (Fy+Fu)/2.
    if Fy > 460*MPa:      # S16-14: 13.11 footnote
        Fv = Fy
    Steel.Fv = Fv

Fy    = 350 MPa
Fu    = 450 MPa
Fv    = 400 MPa

## Pattern 1
with USEVARS((CentrePlate,'C,T1'),(BoltGroup,'S2,NL,NT,S,G'),(Steel,'Fu,Fv'),locals='Agv,An,Ut',globals='e,phiu',
             label='Block Shear Pattern 1 - centre plate'):
    e = S2/2. - C/2.     # end distance to centre of 1st bolt hole
    Agv = (e + (NL-1)*S)*T1*2.  # shear area
    An = (NT-1)*G * T1          # tension area
    Ut = 1.0
    phiu = 0.75
    Tr = phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

C     = 10        mm
T1    = 25        mm
S2    = 140       mm
NL    = 2         
NT    = 3         
S     = 75        mm
G     = 75        mm
Fu    = 450       MPa
Fv    = 400       MPa
Agv   = 7000      mm²
An    = 3750      mm²
Ut    = 1         
Tr    = 2.526e+06 MPa·mm²
e     = 65        mm
phiu  = 0.75

    Block Shear Pattern 1 - centre plate: Tr = 2526 kN

## Pattern 2
with USEVARS((CentrePlate,'C,T1,W1'),(BoltGroup,'S2,NL,NT,S,G,HA'),(Steel,'Fu,Fv'),locals='Agv,An,Ut',globals='e,phiu',
             label='Block Shear Pattern 2 - centre plate'):
    Agv = 2.*(e + (NL-1)*S)*T1  # shear area
    g1 = (W1 - (NT-1)*G)/2.  # edge distance to centre of hole
    An = (g1 + g1 - 2.*HA/2.)*T1    # to outside from edge of outside pair of holes
    if NT >= 3:
        An = An + (NT-2.)*(G - 2.*HA/2.)*T1    # additional between holes
    Ut = 0.6                 # no good guidelines in commentary - this should be conservative
    Tr = phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

C     = 10        mm
T1    = 25        mm
W1    = 300       mm
S2    = 140       mm
NL    = 2         
NT    = 3         
S     = 75        mm
G     = 75        mm
HA    = 24        mm
Fu    = 450       MPa
Fv    = 400       MPa
Agv   = 7000      mm²
An    = 4425      mm²
Ut    = 0.6       
Tr    = 2.156e+06 MPa·mm²
e     = 65        mm
phiu  = 0.75

    Block Shear Pattern 2 - centre plate: Tr = 2156 kN

## Pattern 3
with USEVARS((CentrePlate,'C,T1,W1'),(BoltGroup,'S2,NL,NT,S,G,HA'),(Steel,'Fu,Fv'),locals='Agv,An,Ut',globals='e,phiu',
             label='Block Shear Pattern 3 - centre plate'):
    Agv = (e + (NL-1)*S)*T1  # shear area
    g1 = (W1 - (NT-1)*G)/2.  # edge distance to centre of hole
    An = ((W1-g1) - (NT-0.5)*HA)*T1
    Ut = 0.6                 # no good guidelines in commentary - this should be conservative
    Tr = phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

C     = 10        mm
T1    = 25        mm
W1    = 300       mm
S2    = 140       mm
NL    = 2         
NT    = 3         
S     = 75        mm
G     = 75        mm
HA    = 24        mm
Fu    = 450       MPa
Fv    = 400       MPa
Agv   = 3500      mm²
An    = 4125      mm²
Ut    = 0.6       
Tr    = 1.465e+06 MPa·mm²
e     = 65        mm
phiu  = 0.75

    Block Shear Pattern 3 - centre plate: Tr = 1465 kN

## Pattern 4 - tear-out
with USEVARS((CentrePlate,'C,T1,W1'),(BoltGroup,'S2,NL,NT,S,G,HA'),(Steel,'Fu,Fv'),locals='Agv,An,Ut',globals='e,phiu',
             label='Block Shear Pattern 3 - centre plate'):
    Agv = (e + (NL-1)*S)*T1 * (NT*2.) # shear area
    An = 0*mm*mm
    Ut = 0.               # N.A.
    Tr = phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

C     = 10       mm
T1    = 25       mm
W1    = 300      mm
S2    = 140      mm
NL    = 2        
NT    = 3        
S     = 75       mm
G     = 75       mm
HA    = 24       mm
Fu    = 450      MPa
Fv    = 400      MPa
Agv   = 21000    mm²
An    = 0        mm²
Ut    = 0        
Tr    = 3.78e+06 MPa·mm²
e     = 65       mm
phiu  = 0.75

    Block Shear Pattern 3 - centre plate: Tr = 3780 kN

Outer Plates (Lap Plates)

%figure lap-splice-01-side.svg

The side plates have similar failure modes and so will be computed here without much additional comment. In all cases we will use the dimensions of a single plate, then will multiply the resistance by 2 to account for the two plates.

Yield on Gross Area

with USEVARS((OuterPlate,'W2,T2'),(Steel,'Fy'),locals='Ag,phi',
             label='Gross area yield, two outer plates'):
    Ag = W2*T2    # gross x-sectional area
    phi = 0.9
    Tr = 2. * phi*Ag*Fy  # S16-14: 13.2 a) i)

W2    = 220      mm
T2    = 14       mm
Fy    = 350      MPa
Ag    = 3080     mm²
phi   = 0.9      
Tr    = 1.94e+06 MPa·mm²

    Gross area yield, two outer plates: Tr = 1940 kN

Fracture on Effective Net Area

with USEVARS((OuterPlate,'W2,T2'),(BoltGroup,'NT,HA'),(Steel,'Fu'),locals='wn,An,Ane',globals='phiu',
            label='Net section fracture, two outer plates'):
    wn = W2 - NT*HA    # subtract total width of material removed by holes, failure path 2
    Ane = An = wn*T2
    phiu = 0.75
    Tr = 2. * phiu*Ane*Fu    # S1614: 13.2 a) iii)

W2    = 220       mm
T2    = 14        mm
NT    = 3         
HA    = 24        mm
Fu    = 450       MPa
wn    = 148       mm
An    = 2072      mm²
Ane   = 2072      mm²
Tr    = 1.399e+06 MPa·mm²
phiu  = 0.75

    Net section fracture, two outer plates: Tr = 1399 kN

Block Shear

%figure lap-splice-01-side-blocks.svg

## Pattern 1
with USEVARS((OuterPlate,'W2,T2,L2'),(BoltGroup,'NT,NL,G,S,HA,S2'),(Steel,'Fu,Fv'),locals='An,Agv,Ut',globals='phiu,e',
            label='Block Shear Pattern 1 - two outer plates'):
    e = (L2 - S2 - (NL-1)*S*2.)/2.   # end distance to centre of 1st bolt hole
    Agv = (e + (NL-1)*S)*T2*2.  # shear area
    An = (NT-1)*G * T2          # tension area
    Ut = 1.0
    phiu = 0.75
    Tr = 2. * phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

W2    = 220       mm
T2    = 14        mm
L2    = 350       mm
NT    = 3         
NL    = 2         
G     = 75        mm
S     = 75        mm
HA    = 24        mm
S2    = 140       mm
Fu    = 450       MPa
Fv    = 400       MPa
An    = 2100      mm²
Agv   = 2940      mm²
Ut    = 1         
Tr    = 2.476e+06 MPa·mm²
phiu  = 0.75      
e     = 30        mm

    Block Shear Pattern 1 - two outer plates: Tr = 2476 kN

## Pattern 2
with USEVARS((OuterPlate,'W2,T2,L2'),(BoltGroup,'NT,NL,G,S,HA,S2'),(Steel,'Fu,Fv'),locals='g1,An,Agv,Ut',globals='phiu,e',
            label='Block Shear Pattern 2 - two outer plates'):
    Agv = 2.*(e + (NL-1)*S)*T2  # shear area
    g1 = (W2 - (NT-1)*G)/2.  # edge distance to centre of hole
    An = (g1 + g1 - 2.*HA/2.)*T2    # to outside from edge of outside pair of holes
    if NT >= 3:
        An = An + (NT-2.)*(G - 2.*HA/2.)*T2    # additional between holes
    Ut = 0.6                 # no good guidelines in commentary - this should be conservative
    Tr = 2. * phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

W2    = 220       mm
T2    = 14        mm
L2    = 350       mm
NT    = 3         
NL    = 2         
G     = 75        mm
S     = 75        mm
HA    = 24        mm
S2    = 140       mm
Fu    = 450       MPa
Fv    = 400       MPa
g1    = 35        mm
An    = 1358      mm²
Agv   = 2940      mm²
Ut    = 0.6       
Tr    = 1.608e+06 MPa·mm²
phiu  = 0.75      
e     = 30        mm

    Block Shear Pattern 2 - two outer plates: Tr = 1608 kN

## Pattern 3
with USEVARS((OuterPlate,'W2,T2,L2'),(BoltGroup,'NT,NL,G,S,HA,S2'),(Steel,'Fu,Fv'),locals='g2,An,Agv,Ut',globals='phiu,e',
            label='Block Shear Pattern 3 - two outer plates'):
    Agv = (e + (NL-1)*S)*T2  # shear area
    g2 = (W2 - (NT-1)*G)/2.  # edge distance to centre of hole
    An = ((W2-g2) - (NT-0.5)*HA)*T2
    Ut = 0.6                 # no good guidelines in commentary - this should be conservative
    Tr = 2. * phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

W2    = 220       mm
T2    = 14        mm
L2    = 350       mm
NT    = 3         
NL    = 2         
G     = 75        mm
S     = 75        mm
HA    = 24        mm
S2    = 140       mm
Fu    = 450       MPa
Fv    = 400       MPa
g2    = 35        mm
An    = 1750      mm²
Agv   = 1470      mm²
Ut    = 0.6       
Tr    = 1.238e+06 MPa·mm²
phiu  = 0.75      
e     = 30        mm

    Block Shear Pattern 3 - two outer plates: Tr = 1238 kN

## Pattern 4 - tear-out
with USEVARS((OuterPlate,'W2,T2,L2'),(BoltGroup,'NT,NL,G,S,HA,S2'),(Steel,'Fu,Fv'),locals='g2,An,Agv,Ut',globals='phiu,e',
            label='Block Shear Pattern 4 - two outer plates'):
    Agv = (e + (NL-1)*S)*T2 * (NT*2.) # shear area
    An = 0*mm*mm
    Ut = 0.               # N.A.
    Tr = 2. * phiu*((Ut*An*Fu) + (0.6*Agv*Fv))     # S16-14: 13.11

W2    = 220       mm
T2    = 14        mm
L2    = 350       mm
NT    = 3         
NL    = 2         
G     = 75        mm
S     = 75        mm
HA    = 24        mm
S2    = 140       mm
Fu    = 450       MPa
Fv    = 400       MPa
g2    = None      
An    = 0         mm²
Agv   = 8820      mm²
Ut    = 0         
Tr    = 3.175e+06 MPa·mm²
phiu  = 0.75      
e     = 30        mm

    Block Shear Pattern 4 - two outer plates: Tr = 3175 kN

Fasteners (Bolts)

Shear

Here we compute the shear strength of one of the bolt groups.

with USEVARS((BoltGroup,'S,NT,NL,threads_intercepted'),(Bolts,'Fu,Ab'),locals='n,m,phib',
            label='Shear resistance of bolts',record='Vr'):

    n = NT*NL      # number of bolts
    m = 2          # number of faying surfaces
    phib = 0.80
    Vr = 0.60*phib*n*m*Ab*Fu      # S16-14: 13.12.1.2 c)
    if (NL-1)*S >= 760*mm:
        Vr = (0.5/0.6)*Vr
    if threads_intercepted:
        Vr = 0.7*Vr

S                   = 75     mm
NT                  = 3      
NL                  = 2      
threads_intercepted = True   
Fu                  = 825    MPa
Ab                  = 285    mm²
n                   = 6      
m                   = 2      
phib                = 0.8    
Vr                  = 948100 MPa·mm²

    Shear resistance of bolts: Tr = 948.1 kN    <<<--- GOVERNS

Bearing

with USEVARS((BoltGroup,'NT,NL,threads_intercepted'),(Bolts,'d'),(Steel,'Fu'),
            (OuterPlate,'T2'),(CentrePlate,'T1'),locals='n,t,phib', globals='phibr',
            label='Bearing resistance at bolt holes',record='Br'):
    n = NT*NL
    t = min(T1,2.*T2)
    phibr = 0.80
    Br = 3.*phibr*n*t*d*Fu

NT                  = 3         
NL                  = 2         
threads_intercepted = True      
d                   = 19.05     mm
Fu                  = 450       MPa
T2                  = 14        mm
T1                  = 25        mm
n                   = 6         
t                   = 25        mm
phib                = None      
Br                  = 3.086e+06 MPa·mm²
phibr               = 0.8

    Bearing resistance at bolt holes: Tr = 3086 kN

Summary

notes.summary()

Summary of DesignNotes for Tr: Lap Plate Splice
===============================================

Values of Tr:
-------------
    Gross area yield, centre plate:           Tr = 2360 kN
    Net section fracture, centre plate:       Tr = 1920 kN
    Block Shear Pattern 1 - centre plate:     Tr = 2530 kN
    Block Shear Pattern 2 - centre plate:     Tr = 2160 kN
    Block Shear Pattern 3 - centre plate:     Tr = 1470 kN
    Block Shear Pattern 3 - centre plate:     Tr = 3780 kN
    Gross area yield, two outer plates:       Tr = 1940 kN
    Net section fracture, two outer plates:   Tr = 1400 kN
    Block Shear Pattern 1 - two outer plates: Tr = 2480 kN
    Block Shear Pattern 2 - two outer plates: Tr = 1610 kN
    Block Shear Pattern 3 - two outer plates: Tr = 1240 kN
    Block Shear Pattern 4 - two outer plates: Tr = 3180 kN
    Shear resistance of bolts:                Tr = 948 kN    <<<--- GOVERNS
    Bearing resistance at bolt holes:         Tr = 3090 kN

    Governing Value:
    ----------------
       Tr = 948 kN

Things not done

L/r ratio
Connections at ends