Bolt Stiffness Calculation
P:=20kN=4496.18 lbf
k_s:=(A_b·E_b)/(L_s)=1.44e+09 kg/s^2
k_t:=(A_s·E_b)/(L_t)=1.31e+09 kg/s^2
k_b:=(1)/((1)/(k_s)+(1)/(k_t))=6.85e+08 kg/s^2
Joint Member Stiffness Calculation
Using frustum cone model with 30° half-angle for stress distribution.
alpha:=30deg=30 deg
L:=t_m=1.97 in
k_m:=((pi·E_m·d·tan(alpha)))/(log(((2·L·tan(alpha)+D-d)·(D+d))/((2·L·tan(alpha)+D+d)·(D-d))))=1.6e+09 kg/s^2
Member stiffness (Shigley frustum formula)
Joint Stiffness Factor
C:=(k_b)/(k_b+k_m)=0.3
Joint stiffness factor (bolt fraction)
Percentage of external load carried by bolt
C_percent:=C·100=29.93
Percentage relieved from clamped members
member_percent:=(1-C)·100=70.07
Results — Load Distribution
Additional bolt tension due to external load
Delta_Fb:=C·P=1345.61 lbf
Reduction in member clamping force
Delta_Fm:=(1-C)·P=3150.57 lbf
Total bolt tension under load
F_b:=F_i+Delta_Fb=10337.96 lbf
Remaining clamping force
F_m:=F_i-Delta_Fm=5841.78 lbf
sigma_b:=(F_b)/(A_s)=46.28 ksi
Bolt tensile stress under load
Summary
- Bolt stiffness kb and member stiffness km determine load sharing
- A lower C value means the joint is more efficient (less load goes to bolt)
- Stiffer members relative to bolt improve fatigue life
Bolted Joint Stiffness Analysis
Calculates member stiffness, bolt stiffness, and load distribution in a preloaded bolted joint.
Inputs — Bolt Properties
d_b:=16mm=0.63 in
Bolt nominal diameter
d_s:=13.546mm=0.53 in
Bolt stress area diameter (M16)
L_b:=50mm=1.97 in
Bolt grip length (clamped thickness)
L_t:=22mm=0.87 in
Threaded length within grip
L_s:=L_b-L_t=1.1 in
Unthreaded shank length within grip
E_b:=200GPa=29007.55 ksi
Bolt modulus of elasticity — steel
Inputs — Joint Member Properties
d_h:=17mm=0.67 in
Bolt hole diameter
d_w:=24mm=0.94 in
Washer or bolt head bearing diameter
E_m:=70GPa=10152.64 ksi
Member modulus of elasticity — aluminum
t_m:=L_b=1.97 in
Total clamped member thickness
Inputs — Loading
F_i:=40kN=8992.36 lbf
Bolt preload (initial tension)
External tensile load on joint
Bolt Stiffness Calculation
Bolt modeled as two springs in series: unthreaded shank + threaded section.
A_s:=(pi·d_s^2)/(4)=1.55e-03 ft^2
Bolt tensile stress area
A_b:=(pi·d_b^2)/(4)=2.16e-03 ft^2
Bolt shank cross-sectional area
Stiffness of unthreaded shank
Stiffness of threaded portion
Combined bolt stiffness (series)
Frustum cone half-angle
D:=d_w=0.94 in
Diameter at pressure cone base
d:=d_h=0.67 in
Hole diameter
Grip length for frustum