Directions
This calculator is designed to resemble a spreadsheet calculator that can be printed as a report for project recordkeeping. The term project, with regards to Precision Mindset calculators, refers to engineered designs, homework problems, tutorial problems, hobby designs or self-study.
Cells with a border are inputs and are editable. Cells without a border are outputs.
- Enter a value for the maximum external or dynamic load applied to the structure.
- Enter the number of bolts used to connect the structural components.
- Choose whether this is non-permanent or permanent connection.
- If desired, edit the value for the fraction of proof stress felt by the bolt.
- Choose the thread designation.
- Enter the length of the bolt.
- Select the bolt material. If the list does not have the material, select User Defined and enter the material properties.
- Select the bolt coating condition. If the condition is not in the list, select User Defined and enter a value for the torque coefficient, K.
- Edit the value for the frustum angle if a value other than 30 degrees is desired.
- Select the material of the washer. If the washer material is not listed, select User Defined and enter a value for the modulus of elasticity.
- Edit the washer diameter if a different diameter is used.
- Enter a value for the washer thickness.
- Repeat steps 10 through 12 for members 1 and 2.
- Select the calculate button to complete the analysis.
- Print a report for your project recordkeeping by pressing teh print button.
Assumptions
- Materials are homogeneous.
- The surfaces of the thread profiles are within the tolerance limits
Equations
- Threaded Portion of the Bolt
$$L_T = 2d+25 \:\: for \: L_b > 200 $$
$$L_T = 2d+12 \:\: for \: L_b \geq 125 \: \& \: L_b \leq 200$$
$$L_T = 2d+6 \:\: for \: L_b < 125 $$
- Effective Grip
$$L_g = t_w+t_1+\frac{t_2}{2} \:\: for \: t_2 < d$$
$$L_g = t_w+t_1+\frac{d}{2} \:\: for \: t_2 \geq d$$
- Length of the Useful Unthreaded Portion of the Bolt
$$l_d = L_b - L_T$$
- Length of the Useful Threaded Portion of the Bolt
$$l_T = L_g - l_d$$
- Top Frustum Thickness in Material 1
$$t_{1_T} = \frac{L_g}{2}-t_w$$
- Top Frustum Diameter in Material 1
$$D_{1_T} = \frac{3}{2}d+2t_w \tan \alpha $$
- Bottom Frustum Thickness in Material 1 or tmiddle
$$t_{1_B} = \frac{L_g}{2}-h$$
- Bottom Frustum Diameter in Material 1 or Dmiddle$$D_{1_B} = \frac{3}{2}d+2h \tan \alpha$$
- Effective Thickness of Member 2
$$h = \frac{t_2}{2} \:\: for \: t_2 < d $$
$$h = \frac{d}{2} \:\: for \: t_2 \geq d $$
- Bottom Frustum Thickness of Material 2 or h
$$t_{2_B} = h$$
- Bottom Frustum Diameter in Material 2
$$D_{2_B} = \frac{3}{2}d$$
- Bolt Stiffness
$$k_b = \frac{A_d A_t E_b}{A_d l_T + A_t l_d}$$
- Washer Stiffness
$$k_w = \frac{\pi E_w d \tan \alpha}{\ln \frac{\left(2t_w \tan \alpha + D_w - d\right) \left(D_w + d \right)}{\left(2t_w \tan \alpha + D_w + d \right)\left(D_w - d \right)}}$$
- Top Frustum Stiffness Due to Material 1
$$k_{1_T} = \frac{\pi E_1 d \tan \alpha}{\ln \frac{\left(2t_{1_T} \tan \alpha + D_{1_T} - d\right) \left(D_{1_T} + d \right)}{\left(2t_{1_T} \tan \alpha + D_{1_T} + d \right)\left(D_{1_T} - d \right)}}$$
- Top Frustum Stiffness
$$k_T = \frac{1}{\frac{1}{k_w} + \frac{1}{k_{1_T}}} \:\: for \: t_w > 0$$
$$k_T = k_{1_T} \:\: for \: t_w = 0$$
- Bottom Frustum Stiffness Due to Material 1 or kmiddle
$$k_{1_B} = \frac{\pi E_1 d \tan \alpha}{\ln \frac{\left(2t_{1_B} \tan \alpha + D_{1_B} - d\right) \left(D_{1_B} + d \right)}{\left(2t_{1_B} \tan \alpha + D_{1_B} + d \right)\left(D_{1_B} - d \right)}}$$
- Bottom Frustum Stiffness Due to Material 2
$$k_{2_B} = \frac{\pi E_2 d \tan \alpha}{\ln \frac{\left(2t_{2_B} \tan \alpha + D_{2_B} - d\right) \left(D_{2_B} + d \right)}{\left(2t_{2_B} \tan \alpha + D_{2_B} + d \right)\left(D_{2_B} - d \right)}}$$
- Bottom Frustum Stiffness
$$k_B = \frac{1}{\frac{1}{k_{1_B}} + \frac{1}{k_{2_B}}}$$
- Member Stiffness
$$k_M = \frac{1}{\frac{1}{k_T}+\frac{1}{k_B}}$$
- Joint Constant
$$C = \frac{k_b}{k_b + k_M}$$
- Proof Load
$$F_p = S_P A_t$$
- Preload Used
$$F_i = \zeta _1 F_p$$
- External Load Applied to a Single Screw
$$P_i = \frac{P_S}{N_b}$$
- Portion of the External Load Taken by the Bolt
$$P_b = \frac{k_b}{k_b + k_M}P_i$$
- Portion of the External Load Taken by the Members
$$P_m = \frac{k_M}{k_b + k_M}P_i$$
- External Load that will Cause Joint Seperation
$$P_o = \frac{k_b+k_M}{k_M}F_i$$
- Resultant Bolt Load
$$F_b = P_b + F_i$$
- Resultant Load on the Connected Members
$$F_m = 0 \:\: for \: P_m - F_i > 0$$
$$F_m = P_m - F_i \:\: for \: P_m - F_i \leq 0$$
- Bolt Elongation Due to the Preload
$$\delta _i = \frac{F_i}{k_b}$$
- Bolt Elongation Due to the External Load
$$\delta _p = \frac{P_b}{k_b}$$
- Total Bolt Elongation
$$\delta _T = \delta _i + \delta _p$$
- Bolt Stress Due to the Preload
$$\sigma _i = \frac{F_i}{A_t}$$
- Stress Due to Dynamic Load Amplitude, Pa
$$\sigma _a = \frac{C P_i}{2 A_t}$$
- Stress Due to the Steady Load and Dynamic Load
$$\sigma _m = \sigma _a + \sigma _i$$
- Fraction of Proof Stress Felt by the Bolt Under External Load
$$\zeta _2 = \zeta _1 + \frac{C P_i}{S_p A_t}$$
- Factor of Safety
$$FS = \frac{1- \zeta _1}{\zeta _2 - \zeta _1}$$
- Factor of Safety Against Joint Opening
$$FS_O = \frac{F_i}{\left(1-C\right)P_i}$$
- Mean Factor of Safety Guarding Against Permanent Fastener Set
$$FS_p = \frac{1}{\zeta _2}$$
- Required Torque
$$T = K \cdot F_i \cdot d$$
Background
