Forces and vectors

Statics is the study of bodies that are not accelerating. That sounds narrow, but it covers nearly everything engineers build to stand still: bridges, buildings, towers, cranes holding a load, pipe supports, machine frames, dams and retaining walls. Before you can size a beam or pick a bolt, you need to know the forces acting on it, and statics is how you find them.

Every statics problem is built from the same raw material: forces, which are vectors. This chapter sets up the tools you will use for the rest of the course. You will learn how to describe a force, add forces graphically and with trigonometry, break forces into components in two and three dimensions, write a force that acts along a cable, and use the dot product to find angles and projections.

What statics assumes

Engineering mechanics rests on a few idealizations. They are approximations, but very good ones for the problems in this course.

• A particle has mass but negligible size. We can treat a body as a particle when all the forces on it act through one point, such as a ring where several cables meet.
• A rigid body does not deform. Real steel beams bend a few millimeters under load, but that change in geometry is too small to affect the equilibrium calculation. (How much they bend is the subject of mechanics of materials, the course that follows this one.)
• A concentrated force acts at a single point. Real loads are always spread over some area, but if the area is small compared with the body, a point load is accurate.

Newton's laws are the foundation. The first law says that a body with no net force stays at rest or moves at constant velocity. Statics is the case where the net force and the net moment are both zero, so the body stays where it is. The third law says forces come in equal and opposite pairs, and you will use it every time you take a structure apart to analyze the pieces.

Units

This course uses SI units. Force is measured in newtons (N), where 1 N is the force that accelerates 1 kg at 1 m/s². Structural loads are usually quoted in kilonewtons (1 kN = 1000 N) and moments in newton-meters (N·m) or kilonewton-meters (kN·m). Distances are in meters or millimeters. Stress, which you will meet next year, is in pascals (1 Pa = 1 N/m²) and in practice megapascals (1 MPa = 1 N/mm²).

Mass and weight are different things. Mass, in kilograms, is a property of the object. Weight is the force gravity exerts on it, \(W = mg\), with \(g = 9.81\) m/s² near the Earth's surface. A 2000 kg steel beam weighs \(2000 \times 9.81 = 19\,620\) N, or 19.6 kN. When a drawing says a load is "2 tonnes", convert it to a force before using it in an equilibrium equation.

Describing a force

A force is fully described by four things: its magnitude (how large, in newtons), its direction (an angle, a slope, or a unit vector), its sense (which way along that direction it points), and its point of application. The straight line through the point of application in the direction of the force is its line of action.

For a rigid body, the principle of transmissibility says a force can slide anywhere along its line of action without changing its external effect. Pushing a stalled car from behind and pulling it with a rope from the front, along the same line, produce the same motion. This is only true for the body as a whole: the internal forces in the car are different in the two cases, and that matters when you design the bumper.

On diagrams, directions are given in one of three ways. An angle measured from a stated axis, such as "30° above the positive x-axis". A slope triangle, such as a small 3-4-5 triangle drawn beside the force: the force has the same direction as the hypotenuse, so its components are \(\tfrac{3}{5}\) and \(\tfrac{4}{5}\) of its magnitude. Or, in three dimensions, the coordinates of two points on the line of action. All three are common on engineering drawings, and you should be comfortable with each.

Adding forces: the parallelogram law

Forces add as vectors. Two forces acting at a point can be replaced by a single resultant force, found with the parallelogram law: draw the two forces tail to tail, complete the parallelogram, and the diagonal from the common tail is the resultant. The equivalent triangle rule places the forces tip to tail; the resultant runs from the first tail to the last tip.

The triangle gives you a way to calculate the resultant with two results from trigonometry. For a triangle with sides \(a, b, c\) and opposite angles \(A, B, C\):

The law of cosines gives the size of the resultant; the law of sines then gives its direction.

Graphical methods are still useful for checking. A scaled sketch on squared paper will catch a resultant that points the wrong way or is wildly too large. But for more than two forces, the component method below is faster and less error-prone.

Rectangular components in two dimensions

Any force in a plane can be replaced by two perpendicular components, one along each axis. If a force \(F\) makes an angle \(\theta\) with the positive x-axis:

With a slope triangle instead of an angle, use the ratios of its sides directly. A force of 500 N along a 3-4-5 slope pointing up and to the left has components \(F_x = -\tfrac{3}{5}(500) = -300\) N and \(F_y = +\tfrac{4}{5}(500) = 400\) N. Assign the signs from the sketch, not from the calculator.

In vector notation, using unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) along x and y, a force is written \(\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j}\). To add any number of forces, add their components:

The inverse tangent on a calculator only returns angles between −90° and +90°. If \(R_x\) is negative, the resultant is in the second or third quadrant and you must add 180°. Sketch the components and look.

Setting the work out as a table (one row per force, columns for x and y) is the habit that separates clean solutions from messy ones. It makes a sign error easy to spot and easy for a reviewer to check.

Working backwards: finding an unknown force

Often the resultant is what you need, and one of the forces is unknown. The same component equations work in reverse.

Components along non-perpendicular directions

Sometimes a force must be split into parts along two given directions that are not at right angles, for example into the two members of a bracket that meet at a joint. You can do this with the law of sines on the force triangle, or by writing the two unknown parts as components and solving two equations.

That pattern is worth remembering: the member closest to the line of the load does most of the work. The same idea explains why nearly flat cables develop enormous tension, which you will see in the next lesson.

Forces in three dimensions

Many real structures are three-dimensional: guyed towers, tripods, crane booms held by cables, pipe hangers. A force in space has three rectangular components:

Its direction can be described by the three direction angles \(\alpha, \beta, \gamma\) that the force makes with the positive x, y and z axes. Their cosines, the direction cosines, are the components of the unit vector along the force:

Because the squares must add to one, only two of the angles are independent. Given two, the third is fixed (up to the sign of its cosine, which you choose from the sketch).

A force directed along a line

The most useful 3D form comes from geometry. A cable's tension acts along the cable, so if you know the coordinates of its two ends you know its direction. The position vector from point A to point B is

Divide by its length to get a unit vector, and multiply by the tension:

Always run the position vector from the body toward the point the cable pulls it to. A cable pulls the body toward the far anchor, never away from it.

That last observation is the whole point of the calculation. A mast with two guys on one side would be pulled over. Designers place a third cable so the horizontal parts cancel, a problem you will solve in the next lesson.

The dot product

The dot (scalar) product of two vectors is

where \(\theta\) is the angle between them. It answers two questions that come up constantly in statics:

• The angle between two lines: \(\cos\theta = \mathbf{u}_1 \cdot \mathbf{u}_2\), using unit vectors along each line.
• The component of a force along a line, called its projection: \(F_\parallel = \mathbf{F}\cdot\mathbf{u}\). The perpendicular part is then \(F_\perp = \sqrt{F^2 - F_\parallel^2}\).

Projections matter whenever a member can only resist force along its own axis, like a strut, or when you need the part of a load that acts along a slope or a shaft.