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3.5 · Recognize stability, constraints, and static determinacy

Learn to recognize stability, constraints, and static determinacy through clear examples and targeted practice.

University of Alberta ENGG 130: Engineering Mechanics: Statics

Planar Equilibrium

How support directions and locations affect planar equilibrium

Before solving for support reactions, check whether the supports can restrain the body and whether equilibrium provides enough independent information to find the reactions. This lesson considers one planar rigid body, such as a beam. It can translate horizontally, translate vertically, or rotate in the plane. Supports impose constraints through reaction forces. The number of reactions matters, but their directions and locations matter too: three reaction unknowns do not automatically make a body stable or make the equilibrium equations independent.

What you will learn

  • Identify the planar motions that supports must restrain.
  • Represent ideal pins, rollers, and cables by their reaction components.
  • Distinguish stability from static determinacy by checking support geometry as well as reaction count.
  • Use planar equilibrium equations to solve and verify reactions for stable, statically determinate bodies.

1. Model the body and its support constraints

Define the system as the single body you isolate for analysis. On its free-body diagram, replace each ideal support by the force components it can exert. An ideal pin restrains movement of its supported point in two coordinate directions, so its reaction has horizontal and vertical components. The pin does not by itself prevent the body from rotating about that point.
An ideal roller on a horizontal surface supplies one vertical reaction. It allows horizontal movement and rotation. A cable supplies one tension force along its own direction; it pulls on the body and cannot push. An angled cable has both horizontal and vertical force components.
These support models help identify constraints and unknown reactions. To check stability, also consider where reactions act. For example, several vertical reactions cannot prevent horizontal translation, regardless of how many there are.
pin:(Ax,Ay),roller:Ry,cable:T\text{pin}: (A_x,A_y),\quad \text{roller}: R_y,\quad \text{cable}: T
  • Count reaction components, not supports.
  • A pin has two reaction components; a roller on a horizontal surface and a cable each have one.
  • Reaction directions and locations affect stability.

2. Check stability before relying on a reaction count

In planar statics, ask whether the support constraints prevent horizontal translation, vertical translation, and rotation. Imagine the body moving in each of those ways. If it could translate or turn without violating the constraints, the arrangement is unstable as a planar body.
Choose positive xx to the right and positive yy upward; take counterclockwise moments as positive. A roller on a horizontal surface contributes only a vertical reaction. Even two such rollers cannot restrain horizontal translation. By contrast, a pin and a separated roller can restrain planar motion: the pin restrains translation at its location, and the separated vertical reaction can help resist rotation.
For one planar rigid body, equilibrium provides two force equations and one moment equation. These equations can determine reactions only when the body is stable and the equations provide independent information. A count of three unknown reactions is a useful screening check, not a proof of stability.
∑Fx=0,∑Fy=0,∑MO=0\sum F_x=0,\quad \sum F_y=0,\quad \sum M_O=0
  • Check for both translations and rotation.
  • Draw all applied forces and possible support reactions on the isolated body.
  • Three equilibrium equations do not guarantee a stable support arrangement.

3. Separate stability from static determinacy

A stable body is statically determinate when its support reactions can be found from independent equilibrium equations alone. For a single planar rigid body, there are at most three independent equilibrium equations. A stable arrangement with three independent reaction components is a common determinate case.
If a stable arrangement has more unknown reaction components than the independent equilibrium equations can determine, it is statically indeterminate at this level. Equilibrium alone is not enough to find every reaction; no additional method for such arrangements is used here. If a constraint needed to prevent a planar motion is missing, the arrangement is unstable, even if it has three reactions.
Stability asks whether the body is restrained. Determinacy asks whether equilibrium suffices to find the reactions. Keep those checks separate: identify the isolated body, sketch reactions, count unknown components, inspect directions and locations, then classify the arrangement. Only after confirming stability should you rely on the equilibrium equations to solve reactions.
nR=3n_R=3
  • Stability and determinacy are different checks.
  • For one planar rigid body, equilibrium provides at most three independent equations.
  • A reaction count cannot reveal a missing restraint by itself.

4. A routine for reaction problems

First define the isolated body and draw its free-body diagram, showing applied loads and support reactions. Choose coordinate axes and a moment sign convention. Identify the reaction unknowns and count their components. Then inspect the support directions and locations: do they restrain both translations and rotation? Decide whether the body is stable before deciding whether independent equilibrium equations can determine the reactions.
For a stable, determinate body, write force balance in the two coordinate directions and moment balance about a convenient point. Taking moments about a pin removes the pin forces from that moment equation because they act at the pin. After solving, verify horizontal and vertical force balance and at least one moment balance. If a reaction is negative relative to its assumed direction, its actual direction is opposite to the one drawn.
stability check  ⟶  independent equilibrium equations  ⟶  determinacy check\text{stability check}\;\longrightarrow\;\text{independent equilibrium equations}\;\longrightarrow\;\text{determinacy check}
  • Define the system, draw reactions, choose signs, and check stability before solving.
  • Use equilibrium equations only when the arrangement is stable and the equations are independent.
  • Verify force and moment balance; interpret negative reactions as reversed assumed directions.

Worked example

Pin and roller: stable and determinate

A 4 m horizontal beam is supported by a pin at A and a roller at B. A 6 kN downward load acts 1.5 m from A. Classify the support arrangement and find its reactions.
  1. Count and assess constraints
    The beam is the isolated rigid body. The pin supplies two reaction components and the roller supplies one, for three unknowns. The pin restrains translation at A, and the separated roller can resist rotation. This support geometry restrains planar motion.
    nR=2+1=3n_R=2+1=3
  2. Choose signs and write equilibrium
    Take right and upward as positive, with counterclockwise moments positive. The arrangement is stable, and its three reaction components can be determined by the three independent planar equilibrium equations.
    ∑Fx=0,∑Fy=0,∑MA=0\sum F_x=0,\quad \sum F_y=0,\quad \sum M_A=0
  3. Find the vertical reactions
    Take moments about A so the pin reactions have zero moment arm. The load acts clockwise and the roller reaction acts counterclockwise. Then use vertical force balance to find the pin's vertical reaction.
    4By−6(1.5)=0,By=2.25 kN,Ay=6−2.25=3.75 kN4B_y-6(1.5)=0,\quad B_y=2.25\ \mathrm{kN},\quad A_y=6-2.25=3.75\ \mathrm{kN}
  4. Find the horizontal reaction and verify
    There is no applied horizontal load, so horizontal force balance gives Ax=0A_x=0. The vertical forces and moments about A also sum to zero, confirming the solution.
    Ax=0,∑Fy=3.75+2.25−6=0,∑MA=2.25(4)−6(1.5)=0A_x=0,\quad \sum F_y=3.75+2.25-6=0,\quad \sum M_A=2.25(4)-6(1.5)=0
Answer: The beam is stable and statically determinate. Its reactions are Ax=0A_x=0, Ay=3.75 kNA_y=3.75\ \mathrm{kN}, and By=2.25 kNB_y=2.25\ \mathrm{kN}.
Check: Horizontal force balance is zero. Vertical force balance is 3.75+2.25−6=0 kN3.75+2.25-6=0\ \mathrm{kN}, and moment balance about A is 2.25(4)−6(1.5)=0 kN⋅m2.25(4)-6(1.5)=0\ \mathrm{kN\cdot m}.

Worked example

Pin and angled cable: directions matter

A 3 m horizontal beam is pinned at A and held by a cable at B. The cable pulls up and left at 135∘135^\circ from the positive horizontal axis. A 4 kN downward load acts at the midpoint. Classify the arrangement and find the reactions.
  1. Count and assess constraints
    The pin contributes two reaction components and the cable contributes one tension unknown. Because the cable is angled and attached away from A, its force has a horizontal component and a vertical component with a moment arm. Together with the pin, these constraints restrain planar motion; the arrangement is stable.
    nR=2+1=3n_R=2+1=3
  2. Resolve cable tension
    Measure the angle counterclockwise from positive xx. At 135∘135^\circ, the cable force points left and upward. Resolve it into horizontal and vertical components.
    Tx=Tcos⁡135∘=−T2,Ty=Tsin⁡135∘=T2T_x=T\cos 135^\circ=-\frac{T}{\sqrt{2}},\quad T_y=T\sin 135^\circ=\frac{T}{\sqrt{2}}
  3. Use moment balance to find tension
    Take counterclockwise moments about A. The horizontal cable component acts along the beam and has no moment about A. The vertical component acts 3 m from A.
    3T2−4(1.5)=0,T=22 kN3\frac{T}{\sqrt{2}}-4(1.5)=0,\quad T=2\sqrt{2}\ \mathrm{kN}
  4. Find pin reactions and verify
    Horizontal force balance gives a 2 kN rightward pin reaction. Vertical force balance gives a 2 kN upward pin reaction. The cable components are 2 kN left and 2 kN up; substitution confirms both force sums and the moment sum are zero.
    Ax=2 kN,Ay=2 kN,∑Fx=2−2=0,∑Fy=2+2−4=0,∑MA=3(2)−4(1.5)=0A_x=2\ \mathrm{kN},\quad A_y=2\ \mathrm{kN},\quad \sum F_x=2-2=0,\quad \sum F_y=2+2-4=0,\quad \sum M_A=3(2)-4(1.5)=0
Answer: The arrangement is stable and statically determinate. The cable tension is 22 kN2\sqrt{2}\ \mathrm{kN}, approximately 2.83 kN2.83\ \mathrm{kN}. The pin reactions are 2 kN to the right and 2 kN upward.
Check: The cable components are Tx=−2 kNT_x=-2\ \mathrm{kN} and Ty=2 kNT_y=2\ \mathrm{kN}. Combined with the pin reactions and applied load, horizontal force, vertical force, and moment sums about A are all zero.

Worked example

Three vertical rollers: a count is not a stability proof

A horizontal beam is supported by three rollers, each on a horizontal surface. Each roller can exert only a vertical reaction. Decide whether this arrangement is stable and whether the reaction count establishes static determinacy.
  1. Count the reaction components
    Each roller supplies one vertical reaction, so there are three reaction unknowns. This matches the count of planar equilibrium equations, but the count alone does not prove stability or determinacy.
    nR=3n_R=3
  2. Inspect the reaction directions
    All three reactions are vertical. None can oppose a horizontal force, so the supports do not restrain horizontal translation. The arrangement is unstable as a planar body, despite having three reaction unknowns.
    ∑Fx=0contains no support reaction\sum F_x=0\quad\text{contains no support reaction}
  3. Classify the arrangement
    Because the arrangement is unstable, the matching counts do not establish a stable, statically determinate support system. The vertical force and moment equations involve the three vertical reactions, but they cannot remedy the missing horizontal restraint.
    ∑Fy=0,∑MO=0\sum F_y=0,\quad \sum M_O=0
Answer: The arrangement is unstable because it does not restrain horizontal translation. Three reaction unknowns do not make it a stable, statically determinate arrangement.
Check: The horizontal equilibrium equation has no horizontal support reaction. That missing restraint confirms the instability.

Common mistakes and how to avoid them

Calling a body stable because it has three reaction unknowns.
Correction: Check whether reaction directions and locations restrain both translations and rotation. Three vertical roller reactions still allow horizontal translation.
Counting supports instead of reaction components.
Correction: A pin has two components; an ideal roller on a horizontal surface and a cable each have one.
Assuming that more than three reaction components automatically means a stable but indeterminate arrangement.
Correction: Check stability first. A reaction count cannot make up for a missing constraint direction.
Drawing cable tension in an arbitrary direction.
Correction: Tension acts along the cable and pulls away from the body; resolve it using the chosen axes.

Lesson summary

  • A planar rigid body has two force equilibrium equations and one moment equilibrium equation.
  • Support reactions impose constraints; their directions and locations determine whether planar motions are restrained.
  • Reaction counting screens for determinacy but does not prove stability or equation independence.
  • For a stable, statically determinate body, independent equilibrium equations are sufficient to find the reactions.

Check your understanding

Question 1

A beam rests on two rollers whose surfaces are horizontal. What is the key stability concern?
  1. The rollers cannot restrain horizontal translation.
  2. Each roller provides two reaction components.
  3. The beam cannot rotate.
  4. The arrangement has too many horizontal reactions.
Show answer and explanation
The rollers cannot restrain horizontal translation.
Each roller reaction is vertical, so neither can oppose horizontal translation.

Question 2

A planar rigid body has a pin and a roller on a horizontal surface. How many reaction components are there?
  1. Two
  2. Three
  3. Four
  4. Five
Show answer and explanation
Three
The pin contributes two components and the roller contributes one, for three total.

Question 3

Why is a count of three reaction unknowns not enough to establish static determinacy?
  1. The support constraints may fail to restrain a motion or may not give independent equations.
  2. A planar body has only two equilibrium equations.
  3. A pin reaction cannot appear on a free-body diagram.
  4. Moment balance applies only to bodies with four supports.
Show answer and explanation
The support constraints may fail to restrain a motion or may not give independent equations.
Stability and equation independence depend on constraint directions and locations, not only on the count.

Key terms

Constraint
A support restriction represented in statics by a reaction force or reaction component.
Stable
For this planar statics check, an arrangement whose constraints prevent rigid-body translation and rotation.
Statically determinate
A stable arrangement whose support reactions can be found using independent equilibrium equations alone.
Statically indeterminate
An arrangement with more unknown reaction components than independent equilibrium equations can determine.

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