The following interactive animation illustrates the principle of force decomposition on the wedge.
Description
The principle is of fundamental importance in mechanics. This principle explains the functionality of a number of tools such as:
- knife blades
- saw blades
- axe blades
- drill blades
- nails
The animation is based on the following assumptions:
The tip of an ideal wedge is infinitely small. The workpiece only comes into contact with the sides of the wedge. The main force is distributed over both sides. The lateral forces (also known as cheek forces) create a parallelogram of forces.
The lines of action of the lateral forces are orthogonal to the sides. The geometric sum of the lateral forces corresponds to the amount of the main force.
The shape of a parallelogram half is identical to the shape of the wedge itself. This means that all angles of the parallelogram are known.
The angle γ (gamma) in the sketch corresponds to the angle β. As the sum of all angles in a triangle is 180 degrees, the angle α can be easily calculated.
In a symmetrical wedge, the following right-angled triangle always results on the left-hand side (in mirrored form also on the right-hand side):
The lateral force can be calculated using the following formula:
\[ F_W = \sin\left(\frac{\beta}{2}\right) \cdot F \]
- F_W: lateral force (cheek force)
- F: main force (applied force)
- \beta: wedge angle
Calculation Example
A chisel has a wedge angle of β = 60°. The applied main force is F = 200 N.
\[ F_W = \sin(30°) \cdot 200 = 100 \]
Each flank transfers a lateral force of FW = 100 N to the workpiece.
For comparison, an axe with β = 30° at F = 500 N:
\[ F_W = \sin(15°) \cdot 500 \approx 129 \]
The lateral force here is FW ≈ 129 N per flank.
Special case: Asymmetrical wedges
An asymmetrical wedge can also be shown in the animation. The halves of the force parallelogram also correspond to the wedge shape for the asymmetrical wedge (the wedge shape also appears rotated by 90 degrees here).
The following therefore applies:
\[ \gamma_1 = \gamma_2 = \beta \]
The following applies to alpha angles:
\[ \alpha_1 = 90^\circ – \beta_1 \]
and
\[ \alpha_2 = 90^\circ – \beta_2 \]
The following applies to the delta angles:
\[ \delta_1 = 180^\circ – \alpha_1 – \gamma_1 \]
and
\[ \delta_2 = 180^\circ – \alpha_2 – \gamma_2 \]
This means that all angles are known. An unknown side can be determined using the transformed sine theorem.
\[ F_S = F \times \frac{\sin(\delta)}{\sin(\gamma)} \]
- F_S: lateral force
- F: applied force
- \delta: opposing angle
- \gamma: wedge angle
Symmetric vs. Asymmetric – Applications
The choice between a symmetric and an asymmetric wedge depends on the application.
Symmetric wedges split the material evenly to both sides. Examples include axes, splitting tools, and nails.
Asymmetric wedges direct the force in one specific direction.
A plane iron has a single bevel. The flat side glides along the workpiece while the beveled side lifts the chip.
Japanese kitchen knives (e.g. Yanagiba) feature a single bevel grind. This produces a clean cut on one side.
Drill cutting edges use asymmetric wedge geometries to direct chips in a defined direction.
General information
Basically, the smaller the angle β or the point of a wedge, the wider the force parallelogram. This relationship explains the enormous cutting and splitting effect of wedges. Extremely high lateral forces can occur with a very thin wedge (e.g. a nail or a knife).
In the case shown, the wedge penetrates the workpiece from above. There are also cases where the wedge acts on a surface at an angle (e.g. with a plane or a drill cutting edge). Depending on the material of the workpiece and the orientation of the wedge, a chip is created on the workpiece.
Wedge Angle and Lateral Force
The following table shows the lateral force FW for different wedge angles at a main force of F = 100 N.
| Wedge angle β | sin(β/2) | Lateral force FW |
| 5° | 0.044 | 4.4 N |
| 15° | 0.131 | 13.1 N |
| 30° | 0.259 | 25.9 N |
| 60° | 0.500 | 50.0 N |
| 90° | 0.707 | 70.7 N |
The Wedge in Everyday Life
Wedge shaped tools appear in many areas. The choice of wedge angle determines the properties of the tool.
Knife (wedge angle 15° to 20°): The small wedge angle produces a sharp cutting edge. The blade is thin and delicate as a result.
Axe (wedge angle 25° to 35°): The larger wedge angle makes the blade robust enough to withstand repeated blows into hard wood.
Chisel (wedge angle 50° to 70°): The wedge angle is designed for controlled material removal. The wider cutting edge prevents the tool from penetrating too deeply.
Nail (wedge angle approx. 5° to 10°): The small wedge angle allows penetration into the material with minimal material displacement.
Plane (wedge angle approx. 25° to 30°): The plane iron acts as an angled wedge and produces a uniform chip.
The Wedge in History
The wedge is one of the oldest tools known to humanity. As early as the Stone Age, humans crafted hand axes from flint. These served for cutting, scraping, and splitting.
In the Bronze Age, metal chisels and axes emerged. The wedge made it possible to work stone and wood on a large scale. The pyramids of Egypt were built with the help of copper and wooden wedges.
Today, the wedge principle is found in precision tools, surgical instruments, and industrial cutting machines. The underlying force decomposition is the same in all of these applications.
Limits of the Model
The model presented is based on simplified assumptions. In practice, additional effects occur.
Friction: The model neglects friction between the wedge and the workpiece. In reality, frictional forces act along the wedge flanks and absorb part of the main force.
Material deformation: A real workpiece deforms elastically and plastically as the wedge penetrates. This deformation affects the force distribution.
Dynamic forces: When a blow occurs (e.g. a hammer striking a chisel), impact forces arise. These vary over time and exceed the static main force.
Wedge tip: The model assumes an infinitely small tip. Real tools have a finite cutting edge width that changes the force distribution.
The model provides a good approximation for the basic understanding of force decomposition. For the design of real tools, additional calculations are required.
Note on use
After starting the application, you can view the animation in full-screen mode. To do this, click on “View” and then on “Full screen”:
To exit full screen mode, press the Esc key.
Overview and download
| Title | Force distribution on the wedge |
| Target group | Teachers and lecturers |
| Platforms | Microsoft® Windows® Apple® Macintosh® (version dependent) |
| Features | Full screen mode lossless zoom Large screens and projection screens supported |
| Licence | Freeware |
| Download | Contact |
Contributors
C. Hein, S. Rikowski
Sources
Authoring tool (control elements supplied): Adobe Animate
