week 1 part 2 Introduction to Mechanical Engineering Force & Moment Systems

Introduction to Forces in Mechanical Engineering

Definition and Vector Properties of Force

• Force: In mechanical engineering, a force is understood as a fundamental interaction that, when unopposed, will change the motion of an object. It is a vector quantity, meaning it possesses several defining characteristics:

  • Magnitude: The numerical value representing the strength or intensity of the force (e.g., $8 \text{ kN}, 6 \text{ kN}$).

  • Direction: The orientation of the force in space.

  • Sense: Indicates whether the force is acting towards or away from a point along its direction.

  • Point of Application: The specific location on an object where the force is applied.

Line of Action

• The line of action of a force is an imaginary line that passes through the point of application and extends indefinitely in the same direction as the force.

• It is considered a sliding vector, meaning that for many engineering analyses (like external effects on a rigid body), the force can be considered to act anywhere along its line of action without changing its external effect.

• Examples of non-sliding forces would be internal forces or forces affecting deformable bodies where the exact point of application significantly impacts internal stress distribution.

Types of Forces

Concentrated vs. Distributed Forces

• Concentrated Force: A force considered to act at a single point on a body. This is often an idealization for practical purposes when the area over which the force acts is very small compared to the overall dimensions of the body.

• Distributed Loading: A force that is spread over a length, area, or volume of a body. Examples include pressure from fluids, weight of materials spread over a beam, or wind pressure on a surface.

  • Equivalent Concentrated Force: For distributed loadings, it is often possible to determine an equivalent concentrated force. This equivalent force has the same resultant effect (magnitude and moment) on the body as the distributed loading and is located at a specific point (often the centroid of the loading distribution). This simplification is valuable for calculating support reactions and overall equilibrium.

Contact vs. Body Forces

• Contact Force: A force that acts directly at the point of contact between two objects. These forces are transmitted through physical touch. Important considerations for contact forces include their direction and precise point of application. Examples include normal forces, friction forces, or forces from a push or pull.

• Body Force: A force that acts throughout the entire volume of a body, rather than at a single point or surface. These forces do not require direct contact. Common examples include:

  • Gravity: The force exerted by the Earth's gravitational field on every particle of a body.

  • Centrifugal Force: A fictitious force associated with rotational motion, acting outward from the center of rotation on an object.

Concurrent vs. Coplanar Forces

• Concurrent Forces: A system of forces whose lines of action all pass through a common point. Even if the forces act at different physical locations, if their extended lines of action intersect at one point, they are concurrent.

• Coplanar Forces: A force system where all the forces lie within the same two-dimensional plane. All related analysis (like resultant force determination) will occur within this single plane.

  • For coplanar forces, the resultant force will also be located in the same plane.

  • The resultant force for coplanar forces can be determined directly through vector summation.

Internal vs. External Forces

• External Forces: Forces that act on a body from its surroundings. These forces tend to move the body or hold it in place. Examples include:

  • Applied forces (e.g., a push or pull).

  • Support reactions (forces exerted by supports or foundations).

  • Friction forces.

• Internal Forces: Forces that one part of a member or system exerts on another adjacent part. These forces are generated within the body as a response to external loads and maintain the integrity of the body. They are crucial for analyzing the stress and strain within components.

  • Internal loadings (like shear forces $V$ and bending moments $M$ in beams) can be found using the method of sections, where the body is conceptually cut, and the equilibrium of the resulting free-body diagrams is analyzed.

Resultant Force

Graphical Solution for Resultant Force

• For two forces, the resultant can be found graphically by forming a parallelogram or triangle of forces. The diagonal of the parallelogram (or the closing side of the triangle) represents the resultant.

• The Cosine Law and Sine Law are trigonometric tools used to solve the triangle of forces when angles and magnitudes are known:

  • Cosine Law: $C = \sqrt{A^2 + B^2 - 2AB \cos c}$, where $C$ is the magnitude of the side opposite angle $c$, and $A, B$ are the magnitudes of the other two sides.

  • Sine Law: $\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}$, relating the ratio of a side's magnitude to the sine of its opposite angle.

• Example (Page 9): Given forces $F<em>1 = 8 \text{ kN}$ at $60^\circ$ and $F</em>B = 6 \text{ kN}$ at $-40^\circ$, these laws would be used to find the magnitude and direction of their resultant.

Force Cartesian Components

• Any force $F$ can be resolved into its orthogonal (Cartesian) components along the x and y axes:

  • $F_x$: The component of the force along the x-axis.

  • $F_y$: The component of the force along the y-axis.

• These components are crucial for algebraic (Cartesian) summation of forces.

Cartesian Solution for Resultant Force

• This method is more systematic and easily extended to multiple forces and three dimensions.

• The components of the resultant force $(F_R)$ along the x and y axes are the algebraic sums of the respective components of all individual forces:

  • $F<em>{Rx} = \sum F</em>{ix}$ (Sum of all x-components of individual forces $F_i$).

  • $F<em>{Ry} = \sum F</em>{iy}$ (Sum of all y-components of individual forces $F_i$).

• The magnitude of the resultant force $(F_R)$ is then found using the Pythagorean theorem:

  • $F<em>R = \sqrt{F</em>{Rx}^2 + F_{Ry}^2}$

• The angle $\theta$ of the resultant force with respect to the positive x-axis is determined using the arctangent function:

  • $\theta = \tan^{-1} \left(\frac{F<em>{Ry}}{F</em>{Rx}}\right)$

  • Care must be taken to determine the correct quadrant for $\theta$ based on the signs of $F<em>{Rx}$ and $F</em>{Ry}$.

• Example (Page 12): Given forces $F<em>1 = 600 \text{ N}$ at $30^\circ$ and $F</em>2 = 400 \text{ N}$ at $45^\circ$ (relative to vertical), one would resolve each force into its x and y components, sum them to get $F<em>{Rx}$ and $F</em>{Ry}$, and then calculate $F_R$ and $\theta$.

Force Effects on Materials and Structures

• When forces act on a body, they can produce various mechanical effects:

  • Axial Effects: Forces acting along the longitudinal axis of a member.

    • Tensile: Causes elongation or stretching of the member.

    • Compression: Causes shrinkage or shortening of the member.

  • Bending Effects: Forces (or moments) that cause a member to curve or deflect perpendicular to its original axis.

    • Deflection ($\delta$): The displacement of the member from its original position.

    • Induces both tensile and compressive stresses within the member (e.g., tensile on one side, compressive on the other side of a bent beam).

  • Torsion Effects: Forces (or more accurately, moments) that cause a twisting deformation about the longitudinal axis of a member.

    • Often results in shear loads and shear stress within the material.

• Practical Examples of Force Distribution on a Wing Box:

  • Engine: Thrush and weight forces.

  • LG (Landing Gear): Reaction forces during landing/ground operations.

  • Fuel: Weight distribution within the wing.

  • Payload: Weight carried in the fuselage/wing structure.

  • Lift: Aerodynamic force generated by the wing, distributed over its surface.

  • Structural Weight: Distributed weight of the wing structure itself.

  • Pressure Distribution: Aerodynamic pressures resulting from airflow over the wing surfaces.

Moment of a Force

Definition and Characteristics

• A moment of a force (often called torque) is the measure of its tendency to cause rotation or a