← Engineering Mechanics, SOM & Structural Analysis · AAI Manager (Civil)

Chapter 1 of 13

Engineering Mechanics

In the AAI Manager (Civil) syllabus under Engineering Mechanics, SOM & Structural Analysis · 4 parts

Force Systems, Free-Body Diagrams & Equilibrium · Friction · Centroid & Centre of Gravity · Moment of Inertia

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (39 sections)

Part 1 of 4

Force Systems, Free-Body Diagrams & Equilibrium

Last reviewed 16 Sept 2026 · 7 min read

Mechanics, force and rigid bodies

Engineering mechanics studies forces and their effects on bodies. Statics deals with bodies at rest (or moving uniformly); dynamics with accelerated motion. In statics we usually treat bodies as rigid — the distance between any two points does not change — which is accurate enough for finding reactions and equilibrium.

A force is an action that changes, or tends to change, the state of rest or motion of a body. It is a vector defined by four characteristics:

Magnitude (N, kN)
Direction (line of action and angle)
Sense (which way along the line)
Point of application

DefinitionPrinciple of transmissibility

The external effect of a force on a rigid body is unchanged if the force is moved anywhere along its line of action. (It is not true for internal stresses or for deformable bodies.)

Classification of force systems

System	Description	Example
Collinear	All forces on one line	Forces in a rope in tug of war
Coplanar concurrent	In one plane, lines meet at a point	Forces at a truss joint
Coplanar parallel	In one plane, parallel lines	Loads and reactions on a beam
Coplanar non-concurrent, non-parallel	General plane system	Ladder against a wall
Non-coplanar (spatial)	In three dimensions	Forces on a tripod or tower

Resultant of forces

The resultant is the single force that produces the same external effect as the whole system.

Parallelogram and triangle laws

Two forces $P$ and $Q$ at angle $θ$ :

R = P^{2} + Q^{2} + 2 P Q cos θ, tan α = \frac{Q sin θ}{P + Q cos θ}

where $α$ is the angle of $R$ from $P$ .

$θ = 0°$ : $R = P + Q$ (maximum). $θ = 180°$ : $R = ∣ P - Q ∣$ (minimum). $θ = 90°$ : $R = P^{2} + Q^{2}$ .
If $P = Q$ : $R = 2 P cos (θ /2)$ along the bisector.

The triangle law: if two forces are drawn tip to tail, the closing side (from the start of the first to the end of the second) is the resultant. The polygon law extends this to many forces.

Resolution of forces

A force $F$ at angle $θ$ to the $x$ -axis has components $F_{x} = F cos θ$ , $F_{y} = F sin θ$ . For a coplanar concurrent system:

R = (\sum F_{x})^{2} + (\sum F_{y})^{2}, tan α = \frac{\sum F _{y}}{\sum F _{x}}

Watch the signs of $\sum F_{x}$ and $\sum F_{y}$ to place $α$ in the correct quadrant.

Moment of a force and couples

The moment of a force about a point is its turning effect: $M = F \times d$ , where $d$ is the perpendicular distance from the point to the line of action. Units N·m. Clockwise and anticlockwise moments are given opposite signs.

FormulaVarignon's theorem (principle of moments)

The moment of the resultant of a system of forces about any point equals the algebraic sum of the moments of the individual forces about the same point.

R \cdot \overset{ˉ}{d} = \sum F_{i} d_{i}

Used to locate the line of action of the resultant of parallel or general forces.

A couple is two equal, opposite, parallel, non-collinear forces. Its moment $= F \times a$ ( $a$ = arm) and is the same about every point in the plane. A couple produces rotation only; it cannot be balanced by a single force — only by another couple.

Resolution of a force into a force and a couple: a force $F$ at A is equivalent to an equal force at B plus a couple $F \times d$ ( $d$ = perpendicular distance between the lines). This is how an eccentric load on a column is replaced by an axial load plus a moment.

Part 2 of 4

Friction

Last reviewed 16 Sept 2026 · 7 min read

What friction is

When one surface slides, or tends to slide, over another, a resisting force acts tangentially at the contact and opposite to the motion or intended motion. This is friction. It arises from surface roughness and molecular adhesion.

Static friction acts while the body is at rest; it grows with the applied force up to a maximum, the limiting friction.
Kinetic (dynamic) friction acts once sliding starts; it is slightly less than limiting friction.
Rolling friction (rolling resistance) is much smaller than sliding friction.

Laws of dry (Coulomb) friction

Friction acts opposite to the direction in which the body moves or tends to move.
Limiting friction is proportional to the normal reaction: $F = μ R_{N}$ .
Limiting friction does not depend on the area of contact (for the same normal force).
Friction depends on the nature of the surfaces in contact.
Kinetic friction is somewhat less than limiting friction and is nearly independent of sliding speed (at moderate speeds).

Coefficient of friction, angle of friction and repose

FormulaKey definitions

Coefficient of friction: $μ = \frac{F _{l im}}{R _{N}}$
Angle of friction $ϕ$ : angle between the normal reaction and the resultant of normal reaction and limiting friction: $tan ϕ = μ$ .
Angle of repose $α$ : maximum inclination of a plane at which a body rests without sliding under its own weight. At this angle $W sin α = μ W cos α$ → $tan α = μ$ → angle of repose = angle of friction.
Cone of friction: the cone generated by revolving the limiting resultant reaction about the normal; semi-vertex angle $ϕ$ . A body stays at rest if the resultant reaction lies within the cone.

Body on a horizontal plane

Weight $W$ , force $P$ at angle $θ$ above the horizontal, limiting condition:

R_{N} = W - P sin θ, P cos θ = μ (W - P sin θ) \Rightarrow P = \frac{μ W}{cos θ + μ sin θ}

RememberLeast force to move a body on a horizontal plane

$P$ is least when $θ = ϕ$ (the pull is inclined at the angle of friction): $P_{min} = W sin ϕ$ . Pulling at a slight upward angle is easier than pushing horizontally, which is easier than pushing downward at an angle.

Body on an inclined plane

Plane at angle $α$ , weight $W$ , $μ = tan ϕ$ .

Case (force $P$ parallel to the plane)	Force required
To move the body up	$P = W (sin α + μ cos α) = W \frac{sin ( α + ϕ )}{cos ϕ}$
To just prevent it sliding down ( $α > ϕ$ )	$P = W (sin α - μ cos α) = W \frac{sin ( α - ϕ )}{cos ϕ}$
Horizontal force to move up	$P = W tan (α + ϕ)$
Least force to move up (pull inclined at $ϕ$ to the plane)	$P_{min} = W sin (α + ϕ)$

If $α < ϕ$ the body stays at rest without any force (self-locking).

Ladder friction

A ladder of length $L$ and weight $W$ rests against a wall at angle $θ$ with the floor. At the point of slipping, friction acts at the foot (away from the wall, preventing slip) and — if the wall is rough — upward at the top.

Equations: $\sum F_{x} = 0$ , $\sum F_{y} = 0$ , $\sum M = 0$ about the foot. With a smooth wall and floor coefficient $μ$ , a uniform ladder with no other load is about to slip when

tan θ = \frac{1}{2 μ}

Adding a person higher up the ladder makes slipping more likely.

Part 3 of 4

Centroid & Centre of Gravity

Last reviewed 16 Sept 2026 · 6 min read

Definitions

Centre of gravity (CG) — the point through which the resultant of the weights of all particles of a body acts, whatever the orientation of the body.
Centre of mass — the point at which the whole mass can be considered concentrated. In a uniform gravitational field it coincides with the CG.
Centroid — the geometric centre of a line, area or volume. For a homogeneous body the CG and the centroid coincide.

Centroids matter everywhere in structures: the neutral axis of a beam passes through the centroid; loads resolve through centroids; eccentricity of a column load is measured from it.

Centroid by the method of moments

FormulaCentroid coordinates

\overset{x}{ˉ} = \frac{\int x d A}{\int d A}, \overset{y}{ˉ} = \frac{\int y d A}{\int d A}

For composite areas made of simple parts:

\overset{x}{ˉ} = \frac{\sum A _{i} x _{i}}{\sum A _{i}}, \overset{y}{ˉ} = \frac{\sum A _{i} y _{i}}{\sum A _{i}}

For lines replace $A$ by length $L$ ; for volumes by volume $V$ ; for bodies of different materials by weight $W$ .

Axis of symmetry: the centroid lies on every axis of symmetry. If an area has two axes of symmetry, the centroid is at their intersection.

Cut-outs (holes): treat the removed part as a negative area.

Standard centroids of plane areas

Area	Area	Centroid position
Rectangle $b \times h$	$bh$	$h /2$ from base
Triangle, height $h$	$bh /2$	$h /3$ from base ( $2 h /3$ from apex) — intersection of medians
Circle, radius $r$	$π r^{2}$	Centre
Semicircle	$π r^{2} /2$	$\frac{4 r}{3 π} \approx 0.424 r$ from the diameter
Quarter circle (quadrant)	$π r^{2} /4$	$\frac{4 r}{3 π}$ from each straight edge
Circular sector, angle $2 α$	$α r^{2}$	$\frac{2 r sin α}{3 α}$ from the centre
Semi-ellipse (semi-axes $a$ , $b$ , cut along $a$ )	$π ab /2$	$\frac{4 b}{3 π}$ from the major axis
Parabolic spandrel ( $y = k x^{2}$ , under curve from vertex to $x = a$ )	$ah /3$	$\overset{x}{ˉ} = 3 a /4$ , $\overset{y}{ˉ} = 3 h /10$
Semi-parabolic area (between curve and $y$ -axis)	$2 ah /3$	$\overset{x}{ˉ} = 3 a /8$ , $\overset{y}{ˉ} = 3 h /5$
Trapezium, parallel sides $a$ (bottom), $b$ (top), height $h$	$(a + b) h /2$	$\frac{h}{3} \cdot \frac{a + 2 b}{a + b}$ from side $a$

Part 4 of 4

Moment of Inertia

Last reviewed 16 Sept 2026 · 5 min read

Second moment of area

The moment of inertia of an area (second moment of area) about an axis is the sum of each elemental area multiplied by the square of its distance from the axis:

I_{xx} = \int y^{2} d A, I_{y y} = \int x^{2} d A

Units: mm⁴ or m⁴. It measures how the area is spread away from the axis, and so governs a beam's resistance to bending ( $σ = M y / I$ , deflection $\propto 1/ E I$ ) and a column's resistance to buckling.

Mass moment of inertia $I = \int r^{2} d m$ (kg·m²) measures resistance to angular acceleration in dynamics — a different quantity with the same mathematical form.

Radius of gyration

k = \frac{I}{A} \Leftrightarrow I = A k^{2}

It is the distance from the axis at which the whole area could be concentrated to give the same moment of inertia. Columns buckle about the axis of least radius of gyration.

Theorems

FormulaParallel axis theorem

I_{A B} = I_{G} + A h^{2}

$I_{G}$ = moment of inertia about a centroidal axis; $h$ = distance to a parallel axis AB. The centroidal moment of inertia is the minimum among all parallel axes.

FormulaPerpendicular axis theorem (plane areas)

I_{z z} = I_{xx} + I_{y y} = J

The moment of inertia about an axis perpendicular to the plane (through the point where $x$ and $y$ meet) equals the sum of those about the two in-plane axes. This is the polar moment of inertia $J$ .

Standard values

Section	About	Moment of inertia
Rectangle $b \times d$	Centroidal axis parallel to $b$	$\frac{b d ^{3}}{12}$
Rectangle $b \times d$	Base (side $b$ )	$\frac{b d ^{3}}{3}$
Hollow rectangle	Centroidal	$\frac{B D ^{3} - b d ^{3}}{12}$
Triangle, base $b$ , height $h$	Centroidal axis parallel to base	$\frac{b h ^{3}}{36}$
Triangle	Base	$\frac{b h ^{3}}{12}$
Triangle	Axis through apex parallel to base	$\frac{b h ^{3}}{4}$
Circle, diameter $d$	Any diameter	$\frac{π d ^{4}}{64}$
Circle	Polar (centre)	$\frac{π d ^{4}}{32}$
Hollow circle $D$ , $d$	Diameter	$\frac{π ( D ^{4} - d ^{4} )}{64}$
Semicircle, radius $r$	Diameter (base)	$\frac{π r ^{4}}{8}$
Semicircle	Centroidal axis parallel to base	$\approx 0.11 r^{4}$ (exactly $(\frac{π}{8} - \frac{8}{9 π}) r^{4}$ )
Quadrant	Either straight edge	$\frac{π r ^{4}}{16}$
Quadrant	Centroidal axis parallel to an edge	$\approx 0.055 r^{4}$
Square of side $a$	Diagonal	$\frac{a ^{4}}{12}$ (same as about a centroidal axis parallel to a side)
Ellipse, semi-axes $a$ (along $x$ ), $b$	$x$ -axis	$\frac{π a b ^{3}}{4}$

RememberRadius of gyration shortcuts

Rectangle about centroidal axis: $k = d / 12$ . Solid circle: $k = d /4$ . Hollow circle: $k = D^{2} + d^{2} /4$ .

Composite sections

Divide the section into simple parts; locate the overall centroid.
Find each part's centroidal $I$ .
Transfer to the overall centroidal axis with $A h^{2}$ .
Add (subtract for holes).

Product of inertia and principal axes

I_{x y} = \int x y d A, I_{x y} = I_{x^{'} y^{'}} + A \overset{x}{ˉ} \overset{y}{ˉ} (parallel axis form)

$I_{x y} = 0$ if either axis is an axis of symmetry.
Principal axes: $tan 2 θ = \frac{2 I _{x y}}{I _{y y} - I _{xx}}$ ; principal values $\frac{I _{xx} + I _{y y}}{2} \pm (\frac{I _{xx} - I _{y y}}{2})^{2} + I_{x y}^{2}$ .

(See Unsymmetrical Bending & Shear Centre for applications.)

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