Unit 2 / SPPU FE Engineering Mechanics

Unit 2: Equilibrium — SPPU FE Engineering Mechanics

Topics: Free body diagrams, support reactions, beams, Lami's theorem, and 3D equilibrium.

2019 pattern: end-sem Q1 / Q22024 pattern: end-sem Q3Highest-yield unit in the course

Which exam is this in?

2019 pattern

End-sem Q1 / Q2, 18 marks, attempt 1 of the pair

2024 pattern

End-sem Q3: solve any 2 of 4 sub-questions, 12 marks

What appears and how often

Unit II — Equilibrium

2024 pattern: Q3 (12 marks). 2019 pattern: Q1/Q2 (18 marks).

Sub-topic	Frequency	Notes
Simply supported beam reactions with UDL, UVL, point load, or moment	Every single paper	Never skipped
Equilibrium of sphere / cylinder on inclined surfaces	Very common	Two-contact-point problems
3D equilibrium: plate suspended by 3 cables	Common in 2019 pattern	Square plate 1800 kg appears almost identically in multiple papers
Wall crane / bracket reactions	Occasional	Pin at wall provides two reactions; collar or cable at the free end provides one
Types of supports — theory (roller, hinge, fixed, rocker)	Often in theory part	Asked as definition with reactions

Recurring problem families

Problem	Appeared in
Simply supported beam with UDL + point load + moment — find reactions at A and B	Every paper
Square steel plate 1800 kg, mass centre G, suspended by 3 cables — find tension in each	May/Jun 2022, Nov/Dec 2023, May/Jun 2024, May/Jun 2025
Cylinder 100 kg resting between two inclined planes — find reaction at A and B	Apr 2025
Wall crane with smooth collar at B and pin at A — find reactions	Apr 2025
Beam with 50 kNm moment + two 50 kN point loads — find support reactions	Apr 2025

Sub-topics and how often they appear

Sub-topic	Priority	Frequency
Simply supported beam reactions (UDL, UVL, point load, moment)	High	Every single paper
Equilibrium of sphere / cylinder on inclined surfaces	Medium	Very common
3D equilibrium: plate suspended by 3 cables	Medium	Common in 2019 pattern
Wall crane / bracket reactions	Medium	Occasional
Types of supports — theory (roller, hinge, fixed, rocker)	Low	Theory part only, not a full question

High — do this first, appears in every paper.

Medium — do this if you have time.

Low — rare or only asked as a short theory question, skip full preparation if time is short.

This is the highest-yield unit in the entire course. Simply supported beam reactions appear in every paper without exception across both patterns and all years. If you do nothing else in Unit 2, own the beam reaction procedure completely.

High priority

Simply Supported Beam Reactions

Equilibrium conditions

ΣF_x = 0

ΣF_y = 0

ΣM about any point = 0

Distributed loads — how to convert to point loads

Load type	Total load	Where it acts
UDL (uniform distributed load, w kN/m over length L)	W = w × L	Midpoint of the loaded length
UVL (uniformly varying load, 0 to w kN/m over length L)	W = ½ × w × L	L / 3 from the larger end

Standard support reactions

Support	Reactions provided
Roller	One: perpendicular to surface
Hinge / pin	Two: horizontal + vertical
Fixed / built-in	Three: horizontal + vertical + moment

Standard problem setup

A simply supported beam of known span is given. It carries some combination of point loads, a UDL over part or all of the span, a UVL, and/or an applied moment. Find the reactions at the supports.

Step 1 — Draw the free body diagram. Mark every load, every support reaction, and all dimensions. Do not skip this.

Step 2 — Convert all distributed loads to equivalent point loads. For UDL: total = w × L acting at midpoint. For UVL: total = ½wL acting at L / 3 from the larger end.

Step 3 — Take moments about one support, usually A, to eliminate that reaction from the equation. Solve for the other reaction, R_B.

Step 4 — Apply ΣF_y = 0 to find the remaining reaction, R_A.

Step 5 — Check: R_A + R_B = total downward load. If it doesn't balance, you made a sign error.

Common mistakes

Skipping the FBD. If you don't draw it first, you will miss a load or get a moment arm wrong.

Wrong location for UVL resultant. It acts at L / 3 from the larger end, not the midpoint.

Applied couple sign error. A clockwise couple is negative when anticlockwise is positive.

Not checking the answer. R_A + R_B must equal the total downward load.

Moment arm for inclined loads. Resolve the force before writing the moment equation.

Most recycled version: a beam with a UDL over the full span, one or two point loads, and sometimes an applied couple. The procedure is identical every time.

Medium priority

Equilibrium of Sphere / Cylinder on Inclined Surfaces

Formulas that appear

F₁ / sin α₁ = F₂ / sin α₂ = F₃ / sin α₃

α₁ is the angle between F₂ and F₃, α₂ is between F₁ and F₃, and α₃ is between F₁ and F₂.

Alternative: resolve and apply ΣF_x = 0, ΣF_y = 0. Use whichever is faster given the geometry.

Standard problem setup

A sphere or cylinder rests in a groove, between two inclined planes, or against a wall and floor. Normal reactions act perpendicular to each contact surface. Weight acts downward through the centre. Find the reactions.

Step 1 — Identify all contact surfaces. Normal reactions are perpendicular to each surface.

Step 2 — Draw the FBD of the sphere or cylinder. Three forces act on it: weight W downward, reaction R₁ at first contact, reaction R₂ at second contact.

Step 3 — If all three forces pass through a single point, apply Lami's theorem directly.

Step 4 — Otherwise resolve all forces into x and y components and apply ΣF_x = 0 and ΣF_y = 0.

Common mistakes: wrong normal direction, using Lami's theorem when the three forces are not concurrent, and using the surface angle instead of the angle between the other two forces in the theorem.

Medium priority

3D Equilibrium: Plate Suspended by 3 Cables

Vector equilibrium

ΣF = 0 → ΣF_x = 0, ΣF_y = 0, ΣF_z = 0

ΣM = 0 → ΣM_x = 0, ΣM_y = 0, ΣM_z = 0

Unit vector along a cable

û = (B − A) / |B − A|

Tension vector

T⃗ = T × û

Standard problem setup

A rectangular plate of known mass is suspended horizontally by three cables attached at known points. Find the tension in each cable.

Step 1 — Set up a coordinate system. Typically x horizontal, y horizontal, z vertical.

Step 2 — Write the position vectors of each cable attachment point and the point where the cable meets the ceiling.

Step 3 — Find the unit vector along each cable: û = (top point − attachment point) / distance.

Step 4 — Write the tension in each cable as T⃗ = T × û.

Step 5 — Apply ΣF_z = 0 and take moments about two axes to get three equations in three unknowns.

Step 6 — Solve simultaneously.

Common mistakes

Wrong unit vector direction. The cable pulls upward on the plate, so the vector points from the plate toward the ceiling.

Forgetting to include weight at the centre of mass.

Sign errors in the simultaneous equations. Write each balance equation separately before solving.

Most recycled version: a square steel plate, 1800 kg, suspended by three cables attached at corners or known points. The same geometry appears repeatedly across recent papers.

Medium priority

Wall Crane / Bracket Reactions

Formulas that appear

Equilibrium equations

ΣF_x = 0

ΣF_y = 0

ΣM about pin = 0

Standard problem setup

A wall crane has a pin joint at A and a smooth collar or cable at B. A known load hangs from a known point on the horizontal member. Find the reactions at A and B.

Step 1 — Identify the type of support at each point. A smooth collar provides a reaction perpendicular to the member only. A pin provides both horizontal and vertical reactions.

Step 2 — Draw the FBD showing all reactions and the applied load.

Step 3 — Take moments about the pin A to eliminate R_A from the first equation. Solve for the reaction at B.

Step 4 — Apply ΣF_x = 0 and ΣF_y = 0 to find the components of R_A.

Step 5 — Find |R_A| = √(R_Ax² + R_Ay²).

Common mistakes: getting the smooth collar reaction direction wrong and taking moments about the wrong point instead of the pin.

Low priority

Types of Supports — Theory

This is only asked as a short theory question or as part of a definition sub-question. It is not a full numerical question. Know the table below — that is enough.

Support	Symbol	Reactions
Roller	Circle on surface	One: perpendicular to surface
Hinge / pin	Triangle on point	Two: horizontal + vertical
Fixed / built-in	Wall with hatching	Three: horizontal + vertical + moment
Rocker	Curved base on surface	One: perpendicular to surface
Smooth surface	Surface with no symbol	One: normal to surface

MCQ Sampler

3 free concept checks for Unit 2

These starter questions cover the highest-frequency ideas first. The full bank has 50 questions, and all 50 are temporarily open through June 30, 2026, including the harder hint layer.

Open full MCQ bank

Question 1

Roller Support

For a roller support on a horizontal surface in a 2D beam problem, the reaction is usually:

Question 2

UDL Conversion

A UDL of intensity w over a loaded length L is replaced by which equivalent point load?

Question 3

Applied Couple

An applied couple on a beam appears in which equilibrium equation?

The remaining 47 questions are open right now. Use the beta window to work the harder concept checks before the paid boundary returns.

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