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Unit 6

Engineering Statics

Engineers develop new products and processes to make our lives easier, more productive, and more enjoyable. To do so, they must solve problems though scientific and mathematical techniques. One of the first engineering courses taken by engineering majors in college is a subject called Statics.

Statics is the study of Newtonian physics and how it can be used to design and analyze objects and processes with respect to movement. In statics, situations are idealized such that objects are considered rigid and in equilibrium. Rigid bodies are bodies that are simple solid objects assumed to have no ability to deform or change shape. Equilibrium refers to a condition of motion where there is no acceleration. This is an idealized and simple framework to study objects and motion in, though these conditions are not always true in real situations. After mastery of statics, engineers study Dynamics and Mechanics to understand situations where objects are accelerated, deformable, and where external and internal forces vary.

Below, we will begin our study of statics and gain a basic understanding of motion and forces. These ideas will be useful in understanding basic engineering design, product development, and testing.

Position

Position is simply an object’s place in space. Recall the Cartesian Coordinate System and how we can use x, y, and z coordinate points to locate an object. This is the mathematical definition of Position. Since coordinate points innately have direction and magnitude, it should be noted that position is a vector quantity.

Distance

Distance is how far an object moves through space, from one position to another. However, it is important to understand that distance is the total sum of all movement from one position to another. What happens to the value of distance if you move through different paths from one point to another?

We explored the distance formula during our mathematics review, but I want to point out something a little confusing at this point. In the diagram below, imagine the grid representing a map of a city and there are buildings at Point A, Point B, and Point C. The distance formula gives us the distance between Point A and Point B. That distance is a straight-line distance between the points. But what if I walk from Point A, to Point C, then to Point B? Would my distance be the same as if I walked straight from Point A to Point B? Of course not, walking straight between points is obviously a shorter distance.

It ends up that the distance formula doesn’t technically give us distance in all situations, rather it gives us the shortest path between two points. This is usually a quantity known as Displacement, which we will discuss below.

Since distance only concerns itself with total magnitude of the amount of space traveled through, this is a scalar quantity. For distance, direction doesn’t matter.

Displacement

Displacement is how far an object ends up from where it started. Here, the path to get to the new position doesn’t matter; we only consider how our new position compares to our original position. Mathematically, this is the exact same thing as the Distance Formula. Since there is a definite magnitude to displacement and since the potion change has direction, displacement is a vector quantity.

The change in position along the x-direction is Δx= x2-x1 and the change in position along the y-direction is Δy= y2-y1. So. you can see that the distance formula works well to determine displacement.

[(x2-x1) 2 + (y2-y1) 2] = c

Average Speed

Average speed is defined as the time rate of change of distance. In other words, if you take the total distance an object has traveled and divide it by the total time it took to move through that distance, you have calculated average speed. The speed an object travels at may vary over time, sometimes it may stop all together and sometimes it may be moving slowly or quickly, but none of that is measured. This is why it is considered an average speed.

The equation for average speed is

or

The units of speed are

or

Since average speed is derived from two scalar quantities, distance and time, it is also a scalar quantity. Therefore, we do not indicate direction when discussing average speed.

Average Velocity

Average velocity is defined as the time rate of change of displacement. In this case, you take the total displacement the object has traveled through and divide it by the total time it took to move through that displacement. The velocity the object travels at may vary over time, but again none of that is measured in this situation. This is why it is considered an average velocity.

The equation for average velocity is

or

The units of velocity are

or + direction

Since average velocity is derived from one vector quantity and one scalar quantity, displacement and time, it is a vector quantity. Therefore, we do indicate direction when discussing average velocity.

Average Acceleration

Average acceleration is defined as the time rate of change of velocity. In this case, you take the change in velocity the object has experienced and divide that by the total time it took to change the velocity. Generally speaking, average acceleration tells us how much, on average, an object has sped up or slowed down. The acceleration the object undergoes may vary over time, but again none of the instantaneous values are measured. This is why it is considered an average acceleration.

The equation for average acceleration is

or

The units of average acceleration are

or + direction

Since average acceleration is derived from one vector quantity and one scalar quantity, velocity and time, it is a vector quantity. Therefore, we do indicate direction when discussing average acceleration.

Mass, Inertia, Volume, & Density

Mass is often defined as the amount of matter in an object. This is a good definition because the more matter you have in an object, the heavier it feels and thus it is more “massive.” This is often confused with volume, which is the amount of space an object takes up. Don’t get confused here; just because something has more mass doesn’t mean in takes up more space. For example, a bag full of feathers has the same volume as an equal sized bag full of lead. But, the bag full of lead certainly weighs more!

Mass is a scalar quantity and thus has no direction. Units of kilograms are typically used to describe mass. Liquid or fluid volume is measured in Liters (L) and dry volume is measured in cubic meters (m3). The equation for volume depends upon the geometry of the shape being dealt with, therefore every shape has a unique equation.

There is another way to define mass, and in the study of statics this definition is usually more useful. Mass is also defined as the amount of inertia an object has. Inertia is the tendency of an object to maintain its present state of motion. This makes sense because the word “inert” is defined as “lacking the ability or strength to move.” So, something with a large mass has a large inertia and is therefore more difficult to move.

For example, a 13-pound bowling ball is harder to push than a balloon of equal radius. The bowling ball is more massive and thus has more inertia than the balloon. Since they have the same radius, they have equal volume. Therefore, the amount of mass per unit volume of the bowling ball is much greater than the amount of mass per unit volume of the balloon full of air.

The quantity that brings mass and volume together, and why many people confuse the two, is called density. Density is the mass per unit volume of an object. It is often referred to as the “thickness” of the material. This makes sense in our bowling ball and balloon example since the bowling ball is made of material that is “thicker” than the air in the balloon.

The equation for density is below.

The units of density are

or

Center of Mass and Center of Gravity

The center of mass is a reference point used for calculations in engineering statics and in physics that involves the distribution of mass in the space occupied by the object. The reference point is particularly useful for considerations of momentum and force calculations because this is the point to which forces are effectively applied to cause acceleration.

For single rigid bodies, where the bodies don’t deform, the center of mass is in a fixed location in relation to the body. If the body has a uniform density, it will be located at the centroid of the object. See the image below for examples.

Sometimes, the center of mass of an object is located outside of the physical body of the object. This occurs in open shaped objects like horseshoes. In cases where there are distributions of separate bodies, such as the planets and sun in the solar system, the center of mass may not correspond to the position of any individual member of the system. To determine the center of mass for a system up to any number (n) of objects, you use the following series summation equation:

In this equation xcm represents the center of mass of the entire system, x1 through xn represents the positions of the individual objects’ centers of mass, and m1 through mn represents the masses of each individual object.

Center of gravity refers to the average location of the weight of an object, or simply the point that gravity is actually acting on within an object. Generally, the center of mass and center of gravity exists at the same location for a given object. The only time that this isn’t true is when an object is so massive that the force of gravity is different at different points on that object. This happens because exceedingly massive objects create noticeable variations in the gravitational field around them over the span of their own volume. This is only the case for astronomically massive objects, like planets or stars.

Since center of mass and center of gravity are in the same position for objects we encounter in our daily lives, we can use this property to determine the center of gravity of irregular shapes in a simple way. If you hang an object by a string and mark a vertical line on its surface then rotate the object to a new hanging point and mark another vertical line on its surface the lines will cross each other. You can repeat this process for any number of vertical hanging lines and they will all cross the same point in the object. This point represents the center of mass of that object. This process is illustrated below.

Momentum

Momentum can be defined as “mass in motion.” This is because it is the product of an object’s mass and its velocity. Since all objects have mass, all moving objects have momentum. Objects that have no velocity have no momentum. The equation for momentum is shown below.

The units of momentum are

or + direction

Since momentum is derived from one scalar quantity and one vector quantity, mass and velocity, it is a vector quantity. Therefore, we do indicate direction when discussing momentum.

Force & Newton’s Laws of Motion

As stated above, more massive objects are harder to move than smaller masses. That is because we have to overcome larger amounts of inertia to move objects of greater mass. The thing that allows us to move a mass is called force. A force is an influence that can cause objects to change velocity. This change in velocity can come in the form of a changing speed, direction, or both. It’s interesting to note that not all accelerations result in a change in speed, rather the direction of motion is sometimes the only thing that changes. This is still considered an acceleration because changing direction of motion is still a change in velocity.

While forces “can” cause accelerations, they do not always do so when they are applied to an object. For example, a heavy box sitting on a floor might be so massive that a person can push on it as hard as they are capable and the box still not move. This is because something is pushing in the opposite direction of the person’s push, effectively canceling their effort. To understand how some forces cause acceleration and others do not, we usually look at forces in two different situations: balanced forces and unbalanced forces.

In balanced force situations, the external forces acting on a mass from all directions cancel out because they are all equal and opposite. Since each force has an equal and opposite forces acting upon it, the forces all sum to zero. In other words, the net force is zero. If the net force acting on a mass is zero, there is no acceleration. Thus, balanced forces have zero net force and zero acceleration!

F1

F2

Object

In unbalanced force situations, the external forces acting on a mass do not cancel out. Here, there is at least one force that is greater in some direction than the other forces that try to oppose it, therefore the net force is not zero. Non-zero net forces do produce accelerations. Thus, unbalanced forces have a non-zero net force and non-zero acceleration!

F2

F1

Object

Sir Isaac Newton, a famous British mathematician and physicist, worked out the basic laws of motion. These laws help us understand how objects move and serve as mathematical ways to predict motion. The laws of motion are typically referred to as “Newton’s Laws of Motion” in honor of his contributions. They are as follows:

Newton’s First Law of Motion (The Law of Inertia) – An object will maintain its present state of motion unless it is acted upon by an unbalanced external force. In other words, if an object is at rest, it will stay at rest until a force causes it to change its state of motion. Furthermore, an object in motion will continue in motion at a constant velocity until a force accelerates the object.

Newton’s Second Law of Motion (The Law of Acceleration) – An object’s acceleration is directly proportional to the force applied to the object and inversely proportional to the mass (inertia) of the object.

The equation for average acceleration based upon net force and mass is shown below.

or

The equation is often solved for net force based upon average acceleration and mass as shown below.

or

The units of net force are as follows

or + direction

In honor of Sir Isaac Newton’s contributions and to simplify our units, the units have been defined as a derived unit called the “Newton” with a symbol “N.” Therefore, we write the units of force as the symbol “N” and the direction of the net force.

Since net force is derived from one scalar quantity and one vector quantity, mass and average acceleration, it is a vector quantity. Therefore, we do indicate direction when discussing force.

Newton’s Third Law of Motion (The Law of Action/Reaction Force Pairs) – For every action force there is always an equal and opposite reaction force. This law states that there can never be an isolated force. In other words, if an object (Object A) exerts a force on another object (Object B), then Object B exerts and equal force in the opposite direction back onto Object A.

There are many types of forces, all of which push or pull upon objects and have the potential to cause acceleration. These forces are generally classified as either contact forces or field forces, which are also known as “action-at-a-distance forces.” Contact forces require that one object touches another for the force to be applied. Field forces occur because of the position an object has in a “field” of energy. The following is a list of some forces of both types.

Contact Forces Field Forces
Applied Force (FA) Gravitational Force (FG)
Normal Force (FN) Electrical Force (FE)
Frictional Force (Ff) Magnetic Force (FM)
Air Resistance Force (Fair) Nuclear Forces
Tension Force (Ftens)
Spring Force (Fspring)

Applied Forces are applied to an object by another object by direct contact. An example of this would be a person pushing a desk across the floor of the room. There person applies a force onto the desk and the desk accelerates.

Normal Forces are support forces exerted upon objects that are resting on other objects or surfaces. For example, when a book rests on a desk, the book presses downward on the desk. The desk, in return, pushes back up on the book with an equal force in the opposite direction. The latter is the normal force.

Frictional Forces are forces that typically oppose the motion of an object. There are multiple types of friction, but the three most common are static friction, kinetic friction, and rolling friction. All three of these forms of friction are dependent upon the normal force acting on the object and various coefficients of friction, with each form having a unique coefficient. Generally, the strength of the static frictional force is greater than the strength of the kinetic frictional force, which in turn is greater than that of the rolling friction.

  1. Static Friction is the friction that resists motion of an object at rest when a force is applied to it. If no friction was present, the applied force would cause the object to accelerate. But, due to interactions between the surface of the object and the surface it is resting on, a force is present that stops the object from moving.

or

  1. Kinetic Friction is the friction that resists motion of an object that is sliding across a surface. If no applied force is present, kinetic friction is what reduces the velocity of a sliding object to zero over time.

or

  1. Rolling Friction is the friction that occurs between a surface and an object that is rolling over the surface.

or

Note that the factors that influence the amount of frictional force between two surfaces are the forces pushing the surfaces together (the normal force), the texture (roughness) of the surfaces, and whether the object is moving or has a zero velocity. Note that surface area of contact has no influence on the amount of friction!

Air Resistance Forces act upon objects as they travel through the air. The force of air resistance opposes the motion of the object as it travels through air, and in this regard, it is similar to friction. However, the amount of air resistance depends upon different quantities than friction; therefore, they are not exactly the same thing. Generally, the higher the velocity an object has as it moves through air, the greater the air resistance. Also, the larger the surface area an object has, the greater the air resistance.

Tension Forces are forces transmitted through a string, cable, wire, or other material as it is pulled tight by forces acting from opposite ends of the joining material. The tension is directed along the length of the joining member and pulls equally on the opposite ends of the member.

Spring Forces are forces exerted by a compressed or stretched spring upon any object that is attached to it. The object that compresses or stretches the spring is always acted upon by a force that restores the spring to its original form, known as its equilibrium position. For most springs, the force is directly proportional to the amount of the stretch or compression of the spring and its spring constant, which basically represents how well the spring restores itself to its original shape. This is known as Hooke’s Law.

or

Gravitational Force is the force with which one body attracts another body due to the distance between them and their masses. By definition, this the weight of an object. All objects on Earth experience gravitational force in a direction “downward” toward the center of Earth. Under the influence of gravitational force near sea level on Earth, all masses accelerate downward at a rate of g = 9.81 m/s2, where “g” is known as gravitational acceleration. Gravitational force can be expressed mathematically in the two following ways:

or W=m*g

or according to Newton’s Universal Law of Gravitation,

where “m1” is the mass of object 1, “m2” is the mass of object 2, “F1” represents the force of m2 onto m1, “F2” represents the force of m1 onto m2, and “r” is the distance between their centers of mass. “G” is the universal gravitational constant, which has a value of

G = 6.67 x 10-11

Free Body Diagrams

Understanding and analyzing the forces acting on an object is a fundamental aspect of engineering. Free Body Diagrams (FBDs) serve as powerful tools to visually represent these forces and assist in the analysis of an object’s motion. A Free Body Diagram is a simplified representation of an object isolated from its surroundings. The FBD should depict all forces acting on the object without consideration for the objects internal structure or internal forces.

FBDs provide a clear and concise way to represent the forces acting on an object, making it easier to analyze and understand complex physical scenarios. They are an essential tool in problem-solving and engineering design, helping to identify and define forces acting on an object and leading to more accurate solutions. FBDs are particularly useful when studying statics, because they greatly assist in analysis of equilibrium conditions where net forces are zero.

In order to construct FBDs, you must identify the object of interest and represent it as a point (or sometimes a simple geometry, like a box or circle) with the point being located at the center of mass of the actual object. Next, you identify and include all external forces acting on the object. These may include, but are not limited to, gravitational force, normal force, frictional force, tension, or applied force. These forces are drawn as arrows originating at the center of mass point and pointing outward along the direction of the force. The length of the force arrows represents the magnitude of the force; in essence the arrows are the force vectors. The force vectors should be labeled with clear and consistent descriptive notation including symbols, values, and units of measurement when applicable. The example FBD below represents a painting being hung by two strings tied at an angle from the center of the painting. Note that the two strings have tensions in them and the weight of the painting, pointing straight down, is what is creating the tension forces.

Next, you establish a coordinate system that is convenient for mathematical analysis of the forces. Sometimes this requires you rotate the coordinate plane so that most of the forces lie along the x-axis and y-axis, instead of being tilted at various angles. This is common when objects lie on inclined surfaces, like a box on a ramp. By rotating the ramp down to a new x-axis, the vector mathematics becomes much easier because most forces will be dealt with as component forces. Note how the coordinate plane below was rotated counterclockwise so that the only vector not aligned directly with the x-axis and y-axis is the weight of the object. Also, note how the weight is simply reduced into x-components and y-components for analysis.

Impulse-Momentum Theorem

When an object experiences a collision, there are always forces involved. The greater the force of impact and the greater the period of time that the impact takes, then the greater the velocity change experienced by the object will be. This relationship between force, time, and velocity change is expressed by the Impulse-Momentum Theorem. In this theorem, impulse is defined as the product of the net force applied to the mass and the time interval over which the force is applied. Impulses lead to acceleration of the mass and thus changes in momentum. So,

or

where represents the change in momentum the object experiences. This theorem is useful for determining the amount of force an object experiences based upon the time involved in its collision and the objects change in momentum.

In application, it is interesting to note that the change in momentum the object experiences is usually constant, with the net force and timing of the impact being the quantities that vary. This is true because objects typically have constant mass during a collision and their velocity goes from some maximum value to zero. So, for a constant momentum change you can reduce the force of a collision by increasing the timing of the actual impact. This is the reason for air bags and crumple zones in automobiles.

Work

Work is a measure of the amount of energy transfer that occurs when a mass is moved through some displacement by an external force. The displacement has to be in the same direction of the force, or at least some component of the force. Assuming that the displacement and force are parallel, the following equation is used to calculate work,

The units of work are

or

Since 1 N is equivalent to 1 , the units of work can also be expressed as

or

If the force and displacement are not parallel, then we have to do a force vector analysis to determine the component of force that is parallel to the displacement. Then, we multiply the parallel component of force by the displacement to determine the work done along the axis of motion.

Since work is derived from the product of two vector quantities, force and displacement, and it lacks a direction, it is a scalar quantity. It only has a magnitude and we do not indicate direction when discussing work.

Kinetic Energy

Kinetic energy is the energy an object has because of its motion. In order to accelerate an object, an unbalanced force must be applied. This means you have to do work on the object to accelerate its mass and that takes energy. Once the energy is transferred into the object, it will then move at a new constant velocity until other forces act on the mass. The equation for kinetic energy is,

or

The units of kinetic energy are as follows

or , interestingly the same units as work!

In honor of James Prescott Joule’s contributions to the study of energy and temperature and to simplify the units of energy and work, the units N*m and/or have been defined as a derived unit called the “Joule” with a symbol “J.” Therefore, we usually write the units of work and energy as the symbol “J.”

Since kinetic energy is derived from one scalar quantity and the square of a vector quantity, mass and velocity squared, its value can only be zero or positive. Also, it has no direction, only a magnitude, and thus it is a scalar quantity.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. A gravitational field is the force field that exists in space and time around every mass or group of masses. This is often depicted as the bending of space-time due to the presence of a mass or masses. The deeper the bending of the fabric of space-time, the more gravitational force there is. So, gravitational potential energy is related to the position of an object in this bending, which is basically a potential energy well.

In a more “down to Earth” example, in order to lift something to a higher place above the ground, or whatever reference frame you are using, you must do work on the object. You apply a force to displace the mass. This transfers energy into the object. Once lifted, if you let go of the object it will fall back to the reference plane (the floor), where it originally had zero potential energy. The energy stored in the object due to its highest position is transformed back into kinetic energy as it falls.

The equation for gravitational potential energy is

or

The units of kinetic energy are as follows

or , just like those of work and kinetic energy!

Gravitational potential energy is derived from the product of one scalar and two vector quantities: mass, acceleration, and displacement (height). It lacks a direction and is therefore a scalar quantity. It only has a magnitude and we do not indicate direction when discussing gravitational potential energy.

The Work-Energy Theorem

The work-energy theorem states that the net work done by forces acting on a mass equals the change in its kinetic energy. Assuming all forces can be defined within a system, we can also state that the magnitude of net work on a system is equal to the negative of the change in potential energy. Therefore,

Again, it is worth noting that work, kinetic energy, and potential energy all have the same units. So, it should be no surprise that doing work on a system changes the energy of that system, which is exactly what the work-energy theorem tells us.

Torque

In physics and statics, torque is often defined as a rotational analogue to force. It is sometimes also called the moment of force or even abbreviated as simply “moment.” These definitions are often confusing because they make it feel like force and torque are the same thing. It is important to note that torque isn’t a force; they are two different things. To differentiate between the two concepts, I want to spend some time explaining what torque actually represents.

Torque is a rotational “twisting” that occurs because a force is acting on a lever arm and causing rotation. Torque is the result of a force being applied to a rotating body at some distance from the pivoting point. For example, a rotating wheel produces torque as it rolls forward. This torque tends to push the wheel to the left with respect to its forward motion. To begin to understand the directions of these vectors and why torque and force are not the same thing, we will examine the diagram below.

The diagram above illustrates the “right hand rule” for determining the direction of the radius, force, and torque vectors. Here, “r” represents the radius vector, or simply distance from the pivoting point to the point of application of the force. “F” represents the net force applied at a distance “r” from the pivoting point. “T” represents the torque experienced by the object. Note that all three vectors are mutually perpendicular; they all make 90° angles with respect to each other. So, torque doesn’t even point in the same direction as the force!

Another way to use your hand to predict the direction of torque is to use your right hand and curl your fingers in the direction of rotation of the object. With your thumb extended 90° out from your curled fingers, the thumb will point in the direction of the torque. The torque will tend to cause velocity changes along its direction. This can be seen if you watch a car engine as somebody raises the rpms of the motor. Though the motor is mounted to motor mounts in the engine compartment, the mounts are designed to have some “give” and you can easily see the motor rotate slightly as the engine rpms increase and decrease.

The equation for net torque is

If the force vector is not perpendicular to the radius vector, then trigonometry must be used to identify the perpendicular component and that component must be used in the torque equation.

The units of torque are

or

These are dimensionally the same as the units of work and therefore dimensionally the same as energy. However, Joules should never be used to replace Nm when labeling torque vectors because torque is not conceptually the same as work nor energy. Another difference is that torque is a vector quantity, though work and energy are not. This is because torque has direction, not just a magnitude.

Kinematic Equations

Kinematics is a branch of physics that studies the motion of objects without considering the forces that cause the motion. The kinematic equations are motion equations that play a crucial role in describing the various aspects of motion. These equations provide a mathematical framework to analyze and predict the behavior of objects in motion, and thus are of particular interest for engineering of objects or systems that have moving parts.

The kinematic equations are derived from direct manipulations of the mathematical expressions position, displacement, distance, speed, velocity, and acceleration. Please refer to the sections above to review these definitions if needed. The kinematic equations are derived in such a way as to relate the initial and final conditions of objects in motion. The equations are as follows:

In these equations, “vf” represents the final velocity of the object, “vi” represents the objects initial (or starting) velocity, is the time interval over which the velocity changed, is the displacement the object undergoes, and aave is the object’s average acceleration. The kinematic equations are only valid for motion with constant acceleration rates.

Projectile Motion

One of the most common applications of these equations is when we are trying to describe projectile motion, the motion of objects that have been launched into the air and are influenced by gravity. A great example of projectile motion is when a basketball player shoots the ball at the hoop. The curved path of the ball as it goes up in the air then falls into the hoop is a result of the ball’s inertia, gravitational force, and air resistance.

To analyze projectile motion, we need to treat the motion in both the vertical and horizontal planes as independent of each other. The only common physical quantity they share is time, which is what unifies the two dimensions. Thus, we can apply the kinematic equations independently in the x-dimension and y-dimension by utilizing the x-component and y-component vectors for the variables in each of the kinematic equations. If we assume that the projectile is in free fall, a situation with no air resistance, and that it launches from and lands in the same plane we can derive a set of equations that predict the motion of the projectile, as shown below.

The horizontal range (R) of a projectile is given by the expression

/g

where is the square of the x-component of the initial velocity of the object.

The maximum height (H) a projectile can reach is given by the expression

where is the square of the y-component of the initial velocity of the object and “g” is the acceleration due to gravity.

Finally, the time of flight for the projectile is given by

where is the y-component of the initial velocity of the object.

Please see the diagram below for a better understanding of how the velocity’s component vectors relate to the overall velocity vector and how these vectors change with time. Note that, assuming no air resistance, the x-component of velocity remains constant. The changes seen in the y-component are due to the acceleration of gravity.

V2 =

Vf

V3

V1

Vi

Special Topic – Truss Design

One of the most commonly engineered structures is a truss. Trusses are structural frameworks composed of interconnected members, designed to support loads by efficiently transferring forces through tension and compression of its members. They are widely used in engineering and construction for applications ranging from bridges and roofs to towers and cranes. In this section, we will delve deeper into the basic principes, analysis, and design considerations of trusses. We will use the topics discussed above, from statics and dynamics, as a starting place for this topic.

The basic components of a truss include members (beams), nodes (joints) and the overall geometry of the truss design. Members can be categorized as top chords, bottom chords, vertical members (posts), diagonal members (web members), and other more advanced member types. Nodes are where the members meet and are typically connected by welding, bolting, or riveting. See the picture below for an example.

The five most common truss types are as follows:

  1. Pratt Truss – diagonal members slope downwards toward the center, while vertical members support the load.
  2. Howe Truss – diagonal members slope upwards towards the center, with vertical members supporting the load.
  3. Warren Truss – diagonal members form equilateral triangles, providing a balance between materials and costs.
  4. King Post Truss – consists of a central vertical post with diagonal members extending from its top to the ends of the bottom chord.
  5. Queen Post Truss – similar to the king post truss, but has two vertical posts with diagonal members extending from each post to the bottom chord.

The analysis of trusses involves determining the internal forces, reactions, and stability of the structure. Methods such as the method of joints and the method of sections are commonly used for static analysis. Additionally, software tools like the finite element analysis (FEA) can provide more detailed insights into the behavior of complex truss structures.

In secondary schools, one of the more commonly used truss analysis software tools is the Virtual Bridge Building Software provided by the US Army Corps of Engineers. You can download the software for free from this link. This software, built by West Point Military Academy, allows students to design, build, and test bridges in a virtual setting. Of course, this simulation software could also help students visualize the basics of truss designs for applications like cranes booms and similar devices.

Designing a truss involves considering various factors to ensure structural integrity and efficiency of the design:

  1. Load Conditions – the magnitude, distribution, and type of loads that the truss will be subjected to.
  2. Material Selection – choosing appropriate materials (steel, timber, aluminum) based on factors such as strength, weight, durability, availability, and cost.
  3. Geometric Design – optimizing the geometry of the truss, including span length, member sizes, and angles, to minimize material usage while meeting design requirements.
  4. Connection Details – ensuring robust connections between members to withstand forces and prevent failures.
  5. Advanced Considerations – dynamic loads, seismic forces, temperature effects, and other complex situations are taken into consideration during the design of trusses.

Truss design is a crucial aspect of structural engineering, with applications in a wide range of construction projects. By understanding the principles, types, analysis methods, and design considerations discussed above, engineers can effectively design truss structures that meet safety, performance, and cost requirements.