# Examples of how shearforces and bending moments are calculated¶

```
# Make the figures a bit larger
import matplotlib.pyplot as plt
plt.rcParams.update({"figure.dpi": 100})
```

```
from DAVE import *
```

```
Equilibrium-core version = 2.1
default resource folders:
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\resources
C:\Users\beneden\DAVE_models
C:\data\Dave\Book\DAVE-book\DAVE-notebooks
Blender found at: C:\Program Files\Blender Foundation\Blender 2.93\blender.exe
```

```
s = Scene()
```

# Shear forces and bending moments¶

Shear-forces and bending moments can be calculated in any frame and in any direction.

## Axis with a load¶

The simplest non-trivial example is a cantilever beam with a load at its end.

Since the bar does not need to deflect we can model this as an frame node. The force is added using a point node and a force node. Leaving all the degrees of freedom of the frame node fixed means that the frame is fixed to the world at its origin.

```
s = Scene()
s.new_frame('beam')
s.new_point('point',parent='beam',position=(10,0,0))
s.new_force('load',parent='point',force=(0,0,-20));
```

Displaying of the shear or moment line can be done in the GUI:

Programmatically a load-shear-moment diagram can be obtained from the frame. A small caveat is that the model needs to be solve before doing this.

```
s.solve_statics()
lsm = s["beam"].give_load_shear_moment_diagram()
```

```
Solved to 0.000e+00 kN
```

The obtained lsm object contains all the required data for simple and extended plotting:

```
lsm.plot_simple()
```

```
embedWindow(verbose=True): could not load ipyvtk_simple try:
> pip install -U git+https://github.com/Kitware/ipyvtk-simple.git
```

The `lsm.plot()`

method is tailored for writing to full-page PDFs. For that purpose it can be called with a `"filename"`

argument causing it to save to pdf directly.

## Off-centerline loads¶

Bending is calculated assuming that all forces pass through the X-axis of the analyzed frame (more on that later). Loads that act on a location not on the line are introduced on the nearest point on the line:

```
s = Scene()
s.new_frame('beam')
s.new_point('point',parent='beam',position=(10,0,1))
s.new_force('load',parent='point',force=(22,0,-20))
s.new_point('point2',parent='beam',position=(4,0,-1))
s.new_force('load2',parent='point2',force=(-65,0,0));
s.solve_statics()
lsm = s["beam"].give_load_shear_moment_diagram().plot_simple()
```

```
Solved to 0.000e+00 kN
```

## Distributed loads / Footprints¶

DAVE typically does not care about how loads are distributed. To be able to have distributed loads in shear and moment diagrams a new concept is introduced: **footprints**.

A footprint is a shape that defines over which length a loads is introduced on its parent. The length is the projection of the shape in direction of the X-axis of the analysis frame. The footprint is defined by a number of vertices (3d locations). For a point these define the points of the shape relative to the point.

The following example shows a load at x=10 with its footprint defined from -4 to 4.

```
s = Scene()
s.new_frame('beam')
s.new_point('point',parent='beam',position=(10,0,0))
s.new_force('load',parent='point',force=(0,0,-20));
s['point'].footprint = [(-4.0, 0.0, 0.0), (4.0, 0.0, 0.0)]
s.solve_statics()
lsm = s["beam"].give_load_shear_moment_diagram().plot_simple()
```

```
Solved to 0.000e+00 kN
```

A footprint may also be completely outide the point. In that case the resultant of the distributed load on the footprint is calculated to be equal to that of the force:

```
s = Scene()
s.new_frame('beam')
s.new_point('point',parent='beam',position=(3,0,0))
s.new_force('load',parent='point',force=(0,0,-10));
s['point'].footprint = [(0.0, 0.0, 0.0), (4.0, 0.0, 0.0)]
s.solve_statics()
lsm = s["beam"].give_load_shear_moment_diagram()
lsm.plot_simple()
```

```
Solved to 0.000e+00 kN
```

Utilizing the `plot()`

method of shows how the force and footprint are combined into a distributed force:

```
lsm.plot()
```

## Directions¶

Calculation of the shear and bending moment is not bound to the X-axis of a frame node. In fact any reference frame can be used:

Observe that the footprint is projected onto the centerline in a direction perpendicular to the centerline.

Simply provide any other frame to the `give_load_shear_moment_diagram`

method.

```
s = Scene()
s.new_frame('beam')
s.new_point('point',parent='beam',position=(5,0,3))
s.new_force('load',parent='point',force=(0,0,-10));
s['point'].footprint = [(0.0, 0.0, 0.0), (4.0, 0.0, 0.0)]
# code for some_other_frame
s.new_frame(name='some_other_frame', rotation=(0,-30,0))
s.solve_statics()
lsm = s["beam"].give_load_shear_moment_diagram(s['some_other_frame'])
lsm.plot_simple()
```

```
Solved to 0.000e+00 kN
```

The maximum shear-load in this situation is the component of the force perpendicular to the centerline: cos(30) * 10 =

```
-10*np.cos(np.radians(30))
```

```
-8.660254037844387
```

```
x,shear,moment = lsm.give_shear_and_moment()
np.min(shear)
```

```
-8.660254037844387
```

## Fluids¶

Fluids work similar but with one major difference:

Fluids, either as contents of a tank or as buoyancy, result in a distributed load that acts at the location where the vertical force meets the moment/shear axis.

A distributed load would act the the neasest points on the moment/shear axis.

The difference is illustrated in the following sketch. On the left a buoyancy shape is attached to the frame, on the right a distriubted force. The shear-force lines are shown in blue.

```
s = Scene()
beam = s.new_frame(name='beam', rotation = (0,30,0))
mesh = s.new_buoyancy(name='Buoyancy mesh',
parent='beam')
mesh.trimesh.load_file(s.get_resource_path(r'res: cube.obj'), scale = (5.0,2.0,1.0), rotation = (0.0,0.0,0.0), offset = (5,0.0,-1.0))
beam2 = s.new_frame(name='beam2', rotation = (0,30,0), position = (10,0,0))
s.new_point('point',parent='beam2',position=(5,0,-2))
s.new_force('load',parent='point',force=(0,0,20))
s['point'].footprint = [(-2.5, 0.0, 0.0), (2.5, 0.0, 0.0)]
s.solve_statics()
import matplotlib.pyplot as plt
s["beam"].give_load_shear_moment_diagram().plot_simple()
plt.title('Distributed Fluid load')
s["beam2"].give_load_shear_moment_diagram().plot_simple()
plt.title('Distributed load');
```

```
Solved to 0.000e+00 kN
```

## Fluid in partially filled tanks¶

Behaves just like any other fluid and attaches to the point on the line that is inline with the force.

```
s = Scene()
beam = s.new_frame(name='beam', rotation = (0,20,0))
tank = s.new_tank(name='Tank mesh',
parent='beam')
tank.trimesh.load_file(s.get_resource_path(r'res: cube.obj'), scale = (5.0,1.0,1.0), rotation = (0.0,0.0,0.0), offset = (-7.5,0.0,1.2))
tank.fill_pct = 30
```

## Self-weight of bodies¶

RigidBody elements can have a weight.

By default this is a point-load at the position of the cog.

```
s = Scene()
body = s.new_rigidbody('body',mass=10, cog = (5,0,0))
s.update()
s['body'].give_load_shear_moment_diagram().plot_simple()
```

Footprints are also used to define the distribution of the self-weight to a parent.

In this example the body is located on “frame” at position x=5. The cog of the body is right above its origin. The footprint of the body defines that the force is applied on “frame” from x=-3 to x=3 relative to the origin of the body. That is x=5-3 = 2 to x= 5+3 = 8 on the frame.

```
s = Scene()
frame = s.new_frame('frame')
body = s.new_rigidbody('body',parent=frame, mass=10, cog = (0,0,1), position = (5,0,0))
body.footprint = [(-3,0,0),(3,0,0)]
s.solve_statics()
s['frame'].give_load_shear_moment_diagram().plot()
```

```
Solved to 0.000e+00 kN
```

On the body itself, the footprint is used twice:

For the distribution of the connection force between the body and the frame

For the distribution of the self-weight of the body.

The result is that, in this case, we have two equal and opposite distributed forces acting on “body”, hence both the moment and shear are zero.

```
s['body'].give_load_shear_moment_diagram().plot()
```

A “nicer” example would be a rigid-body suspended from two cables:

```
s = Scene()
b = s.new_rigidbody('body',mass=10, fixed=False, cog = (0,0,0))
b.footprint = [(-3,0,0),(3,0,0)]
s.new_point('p1', parent=b, position = (-5,0,0))
s.new_point('p2', parent=b, position = ( 5,0,0))
s.new_point('s1', position = (-5,0,0))
s.new_point('s2', position = ( 5,0,0))
s.new_cable('c1','p1','s1', length=3, EA=10000)
s.new_cable('c2','p2','s2', length=3, EA=10000)
# from DAVE.jupyter import show
# show(s)
```

```
c2 <Cable>
```

```
s.solve_statics()
s['body'].give_load_shear_moment_diagram().plot()
```

```
Solved to 1.820e-06 kN
```