What is mechanical equilibrium and why do we study it? From the blog Being Brunel: Notes From a Civil Engineer
If civil engineering was religion (and in a way it is; institutionalised by men in funny hats), the first commandment would be:
“Thou shalt always have static equilibrium”
The principle is easy: the sum of all the actions acting on a structure should come to zero.
Before Newton invented equilibrium in 1686, the world remained in check because of Aristotle‘s assertion that everything has its natural place. That is: rocks like to be on the ground, and therefore they fall, unless propelled by an external agent. About 1800 years later Gallilio chippedin; removing the sense of belonging that came from a ‘natural place’ and instead suggesting that things move at a constant speed (which includes ‘atrest’, unless you’re at the gym) until they are forced to change.
With his best seller: the Philosophiæ Naturalis Principia Mathematica Newton refined these theories with over 200 experiments (which seems overkill to me) and derived the infamous laws of motion. As far as Civil Engineers are concerned this was when it all began; hence the BN/AN (Before/After Netwon) calendar we use amongst ourselves. The logic goes as such:

Structures neither move, nor do they accelerate (this is an axiom that the public insists on…)

Therefore the ‘velocity’ of the structure, and all of the components that make it, is always zero.

That means that the net force on the structure is zero (Newton’s first law)

All forces applied to a structure are due to accelerating masses (e.g. cakemob loading; Newton’s second law)

At a global level these actions on the structure are resisted by the reactions of its supports (Newton’s third law)
Like all good articles of faith, however, this doesn’t quite stand up to scrutiny; that is Newton took the assumption that everything is a single point of mass. 50 years later, however, Euler wrote a second testament: “Laws of Motion 2: Motion Harder“, which bridges the gap. In practice, however, the distinction is rarely made by Civil Engineers.
So the next time you pickup some engineering calculations (as I’m sure you do everyday), lookout for the incantation ΣF = 0. By invoking this statement, engineers call the structure into equilibrium; and once you know that a body is in equilibrium, working out the forces at any point within the structure is as simple as balancing the equation.
Here’s the original essay: Engineering Fundamentals: Equilibrium
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Online textbook! 🙂 Hewitt Conceptual Physics Mechanical Equilibrium Chapter 2
Conceptual Physics, Paul Hewitt, Chapter 2
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Here the sum of forces = 0, so the scaffold is in equilibrium!
The “normal” force (aka the support force)
What forces act on a book, lying at rest on a table?

Weight = the force of gravity pulling the book’s mass down

The normal force, from the table, pushing back up on the book
Doesn’t seem like there would be two forces, does there? But if there was only one force, then the book would be accelerating down – it would be picking up speed, and moving through the surface of the table. Yet that obviously doesn’t happen.
Try it yourself – push your own hand down on your desktop. It won’t go through. You’re pushing down…but do you feel the table pushing back up on your hand? That’s the normal force that you are feeling!
Where does the normal force really come from?

Your hands are made of atoms, and so is the desk

Atoms have protons (+) on the inside, and electrons () on the outside

When you push your hands down on the desk, your hand’s atoms push against the desktop’s atoms.

But only the outer electrons of your hand’s atoms, come near the outer electrons of the desktop’s atoms.

And all electrons repel each other. Remember: like charges repel (just like opposite charges attract)
The upward (normal) force = +
The downward force = –
They cancel out, so the net force is 0. No net force means no acceleration!
An object at rest is one form of equilibrium. Static equilibrium.
An object moving at constant speed, in a straight line, is another example of equilibrium. Dynamic equilibrium.
So what about equilibrium for moving objects? Once in motion, if there is no net force to change the motion, then the object stays in motion!
Friction is a contact force between two objects. If the net force = 0, then that means the Push force is equal to (but opposite in direction to) the friction force.
Adding vectors
Vector addition from the PhysicsClassroom
Parallelogram Method of Vector Resolution
Finding the Resultant when adding vectors
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An equal yet opposite vector is needed for equilibrium – notice the dashed line vector.
Using the parallelogram rule we find that the tension in each rope is more than half of her weight.
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Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HSPS21. Analyze data to support the claim that Newton’s second law of motion is a
mathematical model describing change in motion (the acceleration) of objects when
acted on by a net force.
HSPS210(MA). Use freebody force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations
A FRAMEWORK FOR K12 SCIENCE EDUCATION: Practices, Crosscutting Concepts, and Core Ideas
PS2.A: FORCES AND MOTION
How can one predict an object’s continued motion, changes in motion, or stability?
Interactions of an object with another object can be explained and predicted using the concept of forces, which can cause a change in motion of one or both of the interacting objects… At the macroscale, the motion of an object subject to forces is governed by Newton’s second law of motion… An understanding of the forces between objects is important for describing how their motions change, as well as for predicting stability or instability in systems at any scale.
Massachusetts Science and Technology/Engineering Curriculum Framework (2006)
1. Motion and Forces. Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.4 Interpret and apply Newton’s three laws of motion.
1.5 Use a freebody force diagram to show forces acting on a system consisting of a pair of
interacting objects. For a diagram with only colinear forces, determine the net force acting on a system and between the objects.
1.6 Distinguish qualitatively between static and kinetic friction, and describe their effects on the motion of objects.
1.7 Describe Newton’s law of universal gravitation in terms of the attraction between two objects, their masses, and the distance between them.
1.8 Describe conceptually the forces involved in circular motion.