CHAPTER 3: FORCES IN STRUCTURES & MACHINES

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SKMJ 1902

ENGINEERING PRINCIPLES

AND PRACTICE

UTM JOHOR BAHRU

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• Overview
• Forces in Rectangular & Polar Forms
• Force Flow & Load Path

CONTENTS

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1. Overview

• Mechanical Engineers design Products, Physical Systems, and Software & Hardware.
• They must apply mathematics and physical principles to model, analyze, and predict system behavior.

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• Successful design is dependent on effective engineering analysis.
• Effective engineering analysis also require an understanding of forces in structures and machines and their tendency either to remain stationary or move.
• Mechanical engineers also evaluate position, velocity, and acceleration of machines, as well as the forces and torques that make them move.

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The fundamental principles that form the basis of mechanics are Newton’s three laws of motion:

• Every object remains in a state of rest or uniform motion of constant velocity unless an external unbalanced force acts upon it.
• An object of mass m, subject to a net force F, experiences an acceleration in the same direction as the force with a magnitude directly proportional to the magnitude of the force and inversely proportional to the mass of the object. This relationship can be expressed as F = ma.
• The forces of action and reaction between two objects are equal, opposite, and collinear.

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• Force systems and machine components fall under the Engineering sciences and analysis branch but provide support for key decisions in the design of innovative systems.

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• Seven SI base units.
• From these base units, all other SI units of measurement can be derived.
• Derived units are used for measurements such as volume, density and pressure.

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2. Forces in Rectangular & Polar Forms

• We first need to describe a force’s magnitude and direction to determine the influence of forces on a structure or machine.
• The properties of forces, equilibrium, and motion in two and three dimensions are also subjects that you will encounter later in the mechanical engineering curriculum.
• These are extensions to what you have learned in Physics and Mathematics.

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• Forces are vector quantities since their physical action involves both direction and magnitude.
• The magnitude of a force is measured by using the dimensions of Pounds (lb) or Ounces (oz) in the USCS and Newtons (N) in the SI.

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• Force vectors are denoted by using boldface notation, as in F.
• One of the common methods used to represent the influence of a force is in terms of its horizontal and vertical components.
• The force F can be broken down into its rectangular components along directions for the x- and y-axes.

2.1 Rectangular Components

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• The unit vectors i and j are used to indicate the directions in which Fx and Fy act.
• Vector i points along the positive x-direction, and j is a vector oriented in the positive y-direction

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• Rather than thinking about a force in terms of how hard it pulls rightward and upward, you could tell someone how hard the force pulls, and in which direction it does so.
• Instead of specifying Fx and Fy, we can now view the force vector F in terms of the two numbers F and θ.

2.2 Polar Components

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• Resultant of several forces
• Moment of a force
• Equilibrium of forces and moments
• Particles and rigid bodies
• Free body diagrams

2.3 Force Analysis

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3. Force Flow & Load Path

Understanding how a force or load travels through a physical system will help an engineer to quickly evaluate the system’s ability to carry the external loads and effectively devise ways to optimize its design.

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The diagrams shows how the load values differ with changing orientation of the load bearing components.

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The diagram shows how understanding Load Path in the major components of the front end of a car can help improve design of components of the main structure.

4. Dot Product

A

B

θ

Commutative law:

A ∙ B = B ∙ A

• The dot product is an algebraic operation that takes two equal-length vectors and returns a single number (a scalar)

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• In 2D we can easily visualise and use trigonometry to find the angle between two lines or two components of a force.

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• In 3D this is often difficult and vector methods should be employed for the solution.

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• Dot product which defines a particular method of “multiplying” two vectors is use to solve the problems mentioned above.

4.1. Work

4.2. Power

5. Cross Product

• A cross product is a mathematical operation between two three-dimensional vectors that results in a new vector perpendicular to both.

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• Direction: Found using the right-hand rule, where your thumb points in the direction of the new vector when your fingers curl from the first vector to the second.’

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• Magnitude: Equal to the area of the parallelogram formed by the two original vectors

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• Difference from Dot Product: While the dot product yields a single number (scalar), the cross product yields a vector.

5.1. Torque

6. Concept of Stress

• Stress - The intensity of the internal force on a specific plane (area) passing through a point.

• Two types of stress: normal and shear stress.

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• Normal stress, σ usually referred to:
1. Tensile stress: normal force “pulls” or “stretches” the area, A .

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• Compressive stress: normal force “pushes” or “compresses” the area, A.

Units of stress:

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SI: N/m² or Pascal (Pa)

– practically expressed as MPa (N/mm²)

σ = average normal stress�P = resultant normal force�A = cross sectional area of bar

Tension

Compression

• Shear stress, 𝜏

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• Due to normal stress, there will be associating shear stress felt by the body.

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• So, shear stress means intensity of force or force per unit area acting tangent to the area, A.
• Force acts along the plane (not to normal plane).

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• Symbol used for shear stress is 𝜏 (tau).

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• E.g., shear force on z axis act on the area of x and

y directions:

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5. Factor of Safety

• When designing a structural member or any mechanical element, it is necessary to restrict the stress in the material to a level that will be save.

• To ensure this safety, it important to choose an allowable load that is less than the load the member can fully support.

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• A factor of safety is needed to obtain allowable load.

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• One method used is the factor of safety (FS) → ratio of the failure load divided by the allowable load

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• In all equations, FS is chosen to be greater than 1 to avoid potential for failure. Specific values will depend on types of material used and its intended purpose

5. Factor of Safety (Cont.)

• UNCERTAINTY IN MATERIAL PROPERTIES – the composition, strength, and dimensions of the part subjected to change during fabrication (i.e. due to heating or deformation)
• UNCERTAINTY OF LOADINGS – most design loadings are engineering estimates. Changing in usage could introduce new loadings configuration and value
• UNCERTAINTY OF ANALYSES – most engineering calculations are based on assumption
• NUMBER OF LOADING CYCLES – fatigue (strength reduced due to increase of loading cycles)
• TYPES OF FAILURE – Ductile failure and Brittle type failure (no alert sign due to sudden failure/rupture).

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Summary

• Engineers often perform a force analysis to see whether a design will be feasible and safe.
• One of the skills that mechanical engineers develop is the ability to apply equations to physical problems clearly and consistently.
• Selecting the object to be included in a free body diagram, choosing the directions for coordinate axes, and picking the best point for balancing moments are some of the choices that you need to make in solving problems of this nature.

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