Calculate bolt forces with Elastic Method and Instant Center of Rotation (ICR) method.

• Introduction
• Quick Start
• Validation Examples
• Installation
• Usage
• Theoretical Background - Elastic Method
• Theoretical Background - ICR Method
• Theoretical Background - Brandt's Method for Locating ICR
• Assumptions and Limitations
• License

EZbolt is a Python program that calculates bolt forces in a bolt group subject to shear and in-plane torsion. It does so using both the Elastic Method and the Instant Center of Rotation (ICR) method as outlined in the AISC steel construction manual. The iterative algorithm for locating the center of rotation is explained in this paper by Donald Brandt: Rapid Determination of Ultimate Strength of Eccentrically Loaded Bolt Groups.. Unlike the ICR coefficient tables in the steel construction manual which is provided in 15 degree increments, EZbolt can handle any bolt arrangements, any load orientation, and any eccentricity.

Run main.py:

import ezbolt

# initialize a bolt group
bolt_group = ezbolt.BoltGroup()

# add a 3x3 bolt group with 6" width and 6" depth with lower left corner located at (0,0)
bolt_group.add_bolts(xo=0, yo=0, width=6, height=6, nx=3, ny=3)

# preview geometry
ezbolt.plotter.preview(bolt_group)

# calculate bolt demands under 50 kips horizontal shear, 50 kips vertical shear, and 200 k.in torsion
results = bolt_group.solve(Vx=50, Vy=50, torsion=200, bolt_capacity=17.9)

# plot bolt forces
ezbolt.plot_elastic(bolt_group)
ezbolt.plot_ECR(bolt_group)
ezbolt.plot_ICR(bolt_group)

# look at the bolt force tables
df1 = results["Elastic Method - Superposition"]["Bolt Force Table"]
df2 = results["Elastic Method - Center of Rotation"]["Bolt Force Table"]
df3 = results["Instant Center of Rotation Method"]["Bolt Force Table"]

ezbolt.preview() plots a bolt group preview:

ezbolt.plot_elastic() shows bolt force calculated from elastic method.

ezbolt.plot_ECR() shows bolt forces calculated from elastic center of rotation (ECR) method.

ezbolt.plot_ICR() shows bolt forces calculated from instant center of rotation (ICR) method.

BoltGroup.solve() returns a dictionary containing all relevant calculation results:

• results["Elastic Method - Superposition"]
  • ... ["Bolt Capacity"]
  • ... ["Bolt Demand"]
  • ... ["Bolt Force Table"]
  • ... ["DCR"]
• results["Elastic Method - Center of Rotation"]
  • ... ["Center of Rotation"]
  • ... ["Ce"]
  • ... ["Connection Capacity"]
  • ... ["Connection Demand"]
  • ... ["Bolt Force Table"]
  • ... ["DCR"]
• results["Instant Center of Rotation Method"]
  • ... ["ICR"]
  • ... ["Cu"]
  • ... ["Connection Capacity"]
  • ... ["Connection Demand"]
  • ... ["Bolt Force Table"]
  • ... ["DCR"]

Link to Validation Problems

Option 1: Anaconda Python

Simply run main.py using the default Anaconda base environment. The following packages are required:

• Numpy
• Matplotlib
• Pandas

Installation procedure:

1. Download Anaconda python
2. Download this package (click the green "Code" button and download zip file)
3. Open and run "main.py" in Anaconda's Spyder IDE.

Option 2: Standalone Python

1. Download this project to a folder of your choosing

   git clone https://github.com/wcfrobert/ezbolt.git
   

2. Change directory into where you downloaded ezbolt

   cd ezbolt
   

3. Create virtual environment

   py -m venv venv
   

4. Activate virtual environment

   venv\Scripts\activate
   

5. Install requirements

   pip install -r requirements.txt
   

6. run ezbolt

   py main.py
   

   Pip install is available:

pip install ezbolt

Here are all the public methods available to the user:

Adding Bolts

• ezbolt.BoltGroup.add_bolts(xo, yo, width, height, nx, ny, perimeter_only=False)
• ezbolt.BoltGroup.add_bolt_single(x, y)

Solving

• ezbolt.BoltGroup.solve(Vx, Vy, torsion, bolt_capacity=17.9, verbose=True, ecc_method="AISC")

Visualizations

• ezbolt.preview(boltgroup_object)
• ezbolt.plot_elastic(boltgroup_object, annotate_force=True)
• ezbolt.plot_ECR(boltgroup_object, annotate_force=True)
• ezbolt.plot_ICR(boltgroup_object, annotate_force=True)

For further guidance and documentation, you can access the docstring of any method using the help() command. For example, here is the output for help(ezbolt.BoltGroup.solve)

A group of bolts can be treated like any geometric section, and their geometric properties can be calculated (e.g. centroid, moment of inertia, etc):

Centroid:

$$x_{cg} = \frac{\sum x_i}{N_{bolts}}$$

$$y_{cg} = \frac{\sum y_i}{N_{bolts}}$$

Moment of inertia about x and y axis:

$$I_x = \sum (y_i - y_{cg})^2$$

$$I_y = \sum (x_i - x_{cg})^2$$

Polar moment of inertia:

$$I_z = J = I_p = I_x + I_y$$

For in-plane shear, the resulting demand on individual bolts is simply total force divided by number of bolts. We do this about the x and y components separately. Let's call this direct shear.

$$v_{dx} = \frac{-V_x}{N_{bolts}}$$

$$v_{dy} = \frac{-V_y}{N_{bolts}}$$

In-plane torsion on the bolt group is converted to shear on the individual anchors. Let's call this torsional shear. The equations below should be very familiar to most engineers. They're identical to the torsion shear stress equations for beam sections ($$\tau = Tc/J$$). Essentially, bolt force varies linearly radiating from the centroid. Bolts furthest away from the centroid naturally take more force.

$$v_{tx} = \frac{M_z (y_i - y_{cg})}{I_z}$$

$$v_{ty} = \frac{-M_z (x_i - x_{cg})}{I_z}$$

Putting it all together, we use principle of superposition to superimpose (add) the bolt demands together. In other words, we can look at each action separately, then add them together at the end.

$$v_{x} = v_{dx} + v_{tx}$$

$$v_{y} = v_{dy} + v_{ty}$$

$$v_{resultant} = \sqrt{v_{x}^2 + v_{y}^2}$$

The key assumption of elastic method is that rotational and translational actions are decoupled and do not influence each other. This is not true and produces conservative results.

A less conservative way of determining bolt forces is through the Instant Center of Rotation (ICR) method. The underlying theory is illustrated in the figure below. In short, when a bolt group is subjected to combined in-plane force and torsion, the bolt force and applied force vectors revolve around an imaginary center. The ICR method is conceptually simple, but identifying the ICR location will require iteration.

Rather than assuming elasticity and using geometric properties to determine bolt forces, we assume the bolt furthest from ICR has a deformation of 0.34", other bolts are assumed to have linearly varying deformation based on its distance from ICR (varying from 0" to 0.34"). We can then determine the corresponding force using the force-deformation relationship below.

$$\Delta_i = 0.34 (d_i / d_{max})$$

$$R_i =(1-e^{-10\Delta_i})^{0.55} \times R_{max}$$

Assuming the location of ICR has been correctly identified, equilibrium should hold.

$$\sum F_x = 0 = P_x -\sum R_{ix}$$

$$\sum F_y = 0 = P_y -\sum R_{iy}$$

$$\sum M = 0 = M_z -\sum m_i$$

• The applied load vectors (Vx, Vy, Mz) is converted into an equivalent eccentricity and load orientation

$$(V_x,V_y,M_z) \rightarrow (P, e_x,\theta)$$

• Rather than looking at DCR on the individual bolt level, ICR method provides an overall connection capacity. A coefficient (C) is determined, and we can convert bolt capacity to the overall connection capacity as follows.

$$\mbox{connection capacity} = C \times \mbox{bolt capacity}$$

• ICR method is more accurate and less conservative because it allows for plastic deformation of bolts. An appropriate analogy would be how the plastic section modulus (Zx) is larger than elastic section modulus (Sx). Technically, the elastic method also has a center about which bolt force vectors revolve. But because everything is linear, we can skip the force-deformation relationship and calculate forces from geometric properties like moment of inertia and polar moment of inertia.
• Unlike the elastic method, ICR method is not practical to do by hand as the location of ICR must be determined iteratively. There are design tables available in the steel construction manual. Generally the ICR method is more of a black-box.

The derivations will follow AISC notations which is somewhat different from what I've used above. Most notably, we will use P to denote applied force instead of V, and R to denote in-plane bolt force instead of v.

Suppose we know the exact location of ICR, first let's calculate the applied moment with respect to this new center.

$$M_p = P \times r_o$$

The maximum bolt deformation of 0.34" occurs at the bolt furthest from ICR, the other bolts have deformation varying linearly between 0 to 0.34 based on its distance to the ICR ($d_i$)

$$\Delta_{max} = 0.34$$

$$\Delta_i = 0.34 \frac{d_i}{d_{max}}$$

Next, we can calculate individual bolt forces using the following force-deformation relationship:

$$R_i = (1-e^{-10\Delta_i})^{0.55} \times R_{max}$$

Now, we can calculate the reactive moment contributions from each bolt:

$$M_i = R_i \times d_i$$

$$M_i = R_{max} (1-e^{-10\Delta_i})^{0.55} \times d_i$$

$$\sum M_i = R_{max} \times \sum (1-e^{-10\Delta_i})^{0.55} d_i$$

The applied moment, and the reactive moment are in equilibrium. Rearrange for P:

$$ M_{applied} = M_{resisting}$$

$$ M_p = \sum M_i$$

$$ P \times r_o = R_{max} \times \sum (1-e^{-10\Delta_i})^{0.55} d_i$$

$$ P = R_{max} \times \frac{\sum (1-e^{-10\Delta_i})^{0.55} d_i}{r_o}$$

Let the second term be the ICR coefficient C. You can think of C as a constant that converts an applied load (P) to the maximum bolt demand

$$ P = R_{max} \times C$$

$$ C = \frac{\sum (1-e^{-10\Delta_i})^{0.55} d_i}{r_o}$$

Set $R_{max}$ equal to the bolt capacity and back-calculate the connection capacity:

$$ R_{max} = R_{capacity}$$

$$ P_{capacity} = R_{capacity} \times C$$

Notes:

1. Despite a nonlinear bolt force-deformation, the relationship between max bolt force ($R_{max}$) and applied force ($P$) is linear. In other words, if applied force doubles, so does maximum bolt force, and vice versa. Embedded in this is the assumption that eccentricity (e = Mz / P) will remain constant
2. If we substitute 0.34 into the exponential function above, we get 0.9815 as there's a horizontal asymptote and we will never reach 1.0 exactly. We can make a simple adjustment to our "C" equation if we desire. Note that AISC does NOT make this adjustment as it is more conservative to set max bolt-force as $0.9815R_{max}$, effectively capping our utilization ratio to 98%.

$$ (1 - e^{-10(0.34)} )^{0.55} = 0.9815 $$

$$ C = \frac{\sum (1-e^{-10\Delta_i})^{0.55} d_i}{0.9815 r_o}$$

The derivation above assumes we know where ICR is. But we don't, and it is not a trivial task to find it. The original Crawford and Kulak paper (1971) is somewhat misleading. The search space for ICR is very rarely a one-dimensional line except in the very specific situation where load angle is 0 degrees. In other words, you cannot draw an orthogonal line from P to CoG, extend that line, and expect to find ICR somewhere along it. This is explained in detail by Muir and Thornton (2004); the search space for ICR is almost always two-dimensional.

Luckily for us, there exists an iterative method that converges on ICR very quickly. Brandt's method (1982) is fast and efficient, and it is what AISC uses to construct their design tables. We will implement Brandt's method here. The two key insights presented by Brandt is summarized below:

Insight #1: Elastic method also has a center of rotation and can be readily calculated

Let $$(x_{cg}, y_{cg})$$ be the coordinate of bolt group centroid, the coordinate for the elastic center of rotation (ECR) is $$(x_{cg} +a_x, y_{cg}+a_y)$$:

$$a_x = \frac{P_y}{n} \frac{J}{M_z}$$

$$a_y = \frac{P_x}{n} \frac{J}{M_z}$$

Where:

• $$P_x$$ = x component of applied load
• $$P_y$$= y component of applied load
• $$n$$ = number of bolts in bolt group
• $$J$$ = polar moment of inertia ($$I_z$$)
• $$M_z$$ = applied torsion

Insight #2: Elastic center of rotation (ECR) can be used as the initial guess of (ICR), subsequent improvements can be achieved as follows:

Let the initial guess be the ECR:

$$x_0 = x_{cg} +a_x$$

$$y_0 = y_{cg} + a_y$$

At this assumed ICR location, calculate force equilibrium. Note how moment equilibrium is enforced when we determine $$R_{max}$$ at each step.

$$\sum F_x = f_{xx} = P_x - \sum R_{x}$$

$$\sum F_y = f_{yy} = P_y - \sum R_{y}$$

The force summations will not be zero unless we are at the ICR, use the residual to determine successive guesses:

$$x_{i+1} = x_i - \frac{f_{yy}J}{nM_z}$$

$$y_{i+1} = y_i + \frac{f_{xx}J}{nM_z}$$

Repeat until the desired tolerance is achieved:

$$\mbox{residual} = \sqrt{(f_{xx})^2 + (f_{yy})^2} < tol$$

Here is the step by step procedure:

• Step 1: From applied force $(P_x, P_y, M_z)$, calculate load vector orientation ($\theta$). Note atan2 is a specialized arctan function that returns within the range between -180 to 180 degrees, rather than -90 to 90 degrees. This is to obtain a correct and unambiguous value for the angle theta.

$$P = \sqrt{P_x^2 + P_y^2}$$

$$\theta = atan2(\frac{P_y}{P_x})$$

• Step 2: Now calculate eccentricity and its x and y components. $e_x$ and $e_y$ is used to locate the point of applied load (let's call this point P). The location of P is actually ambiguous and can be anywhere along $L(x) = P_ye_x - P_xe_y$. One method is to place P such that the line P-CoG is perpendicular to L(x). Another alternative is to assume $e_y=0$ which is what AISC assumes and what we will do by default. Note both methods lead to the same result as long as P is oriented along L(x). You can change which assumption is used by ezbolt by changing the ecc_method argument in ezbolt.BoltGroup.solve()

$$e_y = 0$$

$$e_x = M_z/P_y$$

$$e = \sqrt{e_x^2 + e_y^2}$$

• Step 3: Obtain an initial guess of ICR location per Brandt's method, then calculate distance of line P-ICR ($r_o$)

$$a_x = P_y \times \frac{I_z}{M_z N_{bolt}}$$

$$a_y = P_x \times \frac{I_z}{M_z N_{bolt}}$$

$$x_{ICR} = x_{cg} - a_x$$

$$y_{ICR} = y_{cg} + a_y$$

$$r_{ox} = e_x + a_x$$

$$r_{oy} = e_y - a_y$$

$$r_{o} = \sqrt{r_{ox}^2 + r_{oy}^2}$$

• Step 4: At this point, we can already compute coefficient "C" at our assumed ICR location.

$$C = \frac{ \sum((1 - e^{-10 \Delta_i})^{0.55} d_i)}{r_o}$$

• Step 5: Next, we need to determine the maximum bolt force ($R_{max}$) at the user-specified load magnitude. This can be done by using the moment equilibrium equation; hence why we only need to check force equilibrium at the end. Moment equilibrium is established as a matter of course by enforcing a specific value of $R_{max}$

$$M_p = P_x r_{oy} - P_y r_{ox}$$

$$M_r = R_{max} \times \sum((1 - e^{-10 \Delta_i})^{0.55} d_i)$$

$$R_{max} = \frac{M_p}{\sum((1 - e^{-10 \Delta_i})^{0.55} d_i)}$$

• Step 6: Now that we have $R_{max}$, we can calculate the other bolt forces:

$$R_i = R_{max} \times \sum((1 - e^{-10 \Delta_i})^{0.55} d_i)$$

• Step 7: Now calculate the bolt forces' x and y component to check force equilibrium:

$$cos(\theta) = d_x / d = sin(\theta+90^o)$$

$$sin(\theta) = d_y / d = -cos(\theta+90^o)$$

$$R_x = R_i cos(\theta+90^o) = -R_i \frac{d_y}{d}$$

$$R_y = R_i sin(\theta+90^o) = R_i \frac{d_x}{d}$$

• Step 8: Calculate residual and repeat until a specific tolerance is achieved.

$$\sum F_x = f_{xx} = P_x - \sum R_{x}$$

$$\sum F_y = f_{yy} = P_y - \sum R_{y}$$

$$\mbox{residual} = \sqrt{(f_{xx})^2 + (f_{yy})^2} < tol$$

• Step 9: If equilibrium is not achieved, go back to step 4 with the following modifications

$$a_x = f_{yy} \times \frac{I_z}{M_z N_{bolt}}$$

$$a_y = f_{xx} \times \frac{I_z}{M_z N_{bolt}}$$

$$x_{ICR,i} = x_{ICR,i-1} - a_x$$

$$y_{ICR,i} = x_{ICR,i-1} + a_y$$

• Step 10: Once ICR has been located, calculate connection capacity:

$$P_{capacity} = C \times R_{capacity}$$

$$DCR = \frac{P}{P_{capacity}} = \frac{R_{max}}{R_{capacity}}$$

Easy enough to implement. The hard part is making sure you make an even number of sign errors.

• Sign convention follows the right-hand rule. right is +X, top is +Y, counter-clockwise is positive torsion. Note that since we are only concerned with in-plane forces, only the highlighted vectors are relevant.

• Units are in (kip, in) unless otherwise noted
• EZbolt only calculates connection capacity with respect to bolt shear. Other limit states - such as plate rupture, block shear, bearing and tearout - are not considered.
• This is not enterprise-grade software. Please do NOT use it for work. Users assume full risk and responsibility for verifying that the results are accurate.

MIT License

Copyright (c) 2023 Robert Wang