CIV 2109 Lecture 5: Equilibrium & Support Reactions

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Questions and Answers

What is the role of reaction forces in a structure?

They exert opposing forces to prevent motion and maintain equilibrium. (correct)

They create internal stress within the structure.

They are responsible for the movement of the structure.

They only act on the structure when external forces are applied.

What characterizes internal forces within a structure?

They are the only forces considered in equilibrium equations.

They act only when external forces are absent.

They directly influence the structure's equilibrium.

They occur in equal but opposite pairs according to Newton's third law. (correct)

Why do internal forces not appear in the equations of equilibrium for the entire structure?

They are only relevant to the structure's movement.

They are negligible compared to external forces.

They cancel each other out. (correct)

They are too complex to include in the analysis.

How do supports affect a structure?

They prevent motion by exerting reaction forces against applied loads. Signup and view all the answers

What types of reactions do supports provide?

They can provide translation and rotational reactions. Signup and view all the answers

Which statement is true regarding the types of supports for plane structures?

Supports are categorized by the number of reactions they exert. Signup and view all the answers

What determines the reaction force a support exerts on a structure?

The type of supporting device and the movement it prevents. Signup and view all the answers

What do the equations of equilibrium primarily analyze?

The overall equilibrium and motion of the entire structure. Signup and view all the answers

What characterizes an internally stable structure?

It remains a rigid body when detached from supports. Signup and view all the answers

Which condition must be satisfied for a plane internally stable structure to be in equilibrium?

It must be supported by at least three reactions. Signup and view all the answers

What does it mean for a structure to be termed internally unstable?

It cannot maintain its shape and can undergo large displacements. Signup and view all the answers

How do physical bodies generally respond to loads?

They all deform, but the effect is often negligible in engineering. Signup and view all the answers

What is the main difference between a rigid and a nonrigid structure?

Rigid structures resist changes in shape; nonrigid structures do not. Signup and view all the answers

Which statement accurately describes an externally statically determinate structure?

All its support reactions can be calculated using equilibrium equations. Signup and view all the answers

What is the primary characteristic of an internally unstable structure when not supported?

It will undergo significant movement and deformation. Signup and view all the answers

When discussing the resistance of structures to changes in shape, which option is correct?

Only rigid structures have notable resistance. Signup and view all the answers

What is the condition for a plane structure to be considered statically determinate externally?

It must have exactly three reactions. Signup and view all the answers

What happens to a structure that has more than three reactions?

All reactions cannot be determined from the equilibrium equations. Signup and view all the answers

How is the degree of external indeterminacy defined?

It is defined as the total number of reactions plus internal forces minus three times the number of rigid members. Signup and view all the answers

What are the additional equations needed to solve for unknown reactions in statically indeterminate structures called?

Compatibility equations. Signup and view all the answers

What is referred to as the external redundants in a structure?

Reactions in excess of those necessary for equilibrium. Signup and view all the answers

If a structure is supported by fewer than three reactions, what is the implication?

It will not be able to resist movements in its plane. Signup and view all the answers

What is the role of internal hinges and rollers in a structure?

They can transmit force components to internal members. Signup and view all the answers

Which of these statements about static indeterminacy is true?

An indeterminate structure requires compatibility equations to determine unknown reactions. Signup and view all the answers

What must be true for a structure to be in equilibrium under a concurrent force system?

The lines of action of all forces intersect at a single point. Signup and view all the answers

Which of the following equations represents the equilibrium conditions for a three-dimensional force system?

$ \sigma F_x = 0, \sigma F_y = 0, \sigma F_z = 0 $ Signup and view all the answers

What is the equation used to verify equilibrium in the vertical direction at support A?

$R_A + B_y - 12k imes sin 60° - 6k = 0$ Signup and view all the answers

For static determinacy, what must the relationship $r + fi = 3nr$ satisfy?

$r + fi = 3nr$ Signup and view all the answers

For a structure in equilibrium under two forces, which condition must be satisfied?

The forces must be equal, opposite, and collinear. Signup and view all the answers

What is the calculated value of $R_A$ after solving the equations of equilibrium?

4.62 k Signup and view all the answers

What is the primary difference between external and internal forces acting on a structure?

External forces are actions of other bodies, while internal forces arise within the structure itself. Signup and view all the answers

When analyzing a plane structure under a concurrent coplanar force system, which equilibrium equations are applied?

$ \sigma F_x = 0, \sigma F_y = 0 $ Signup and view all the answers

What does the abbreviation 'FBD' stand for in structural analysis?

Free Body Diagram Signup and view all the answers

What is the calculated value of $B_y$ at the supports in Example 2?

250 kN Signup and view all the answers

For a structure experiencing three concurrent forces, which of the following statements is correct?

The forces must either be concurrent or parallel. Signup and view all the answers

How can applied forces in a structure be characterized in terms of their effect?

They are known loads, such as wind or live loads, that attempt to move the structure. Signup and view all the answers

Which equation indicates the rotational equilibrium about point B?

$-M_B + 400 kN imes m - 15 kN imes 6 m - 160 kN imes 4 m = 0$ Signup and view all the answers

What does the equation $ ext{F}_x + R_A - 12k imes cos 60° = 0$ represent?

Horizontal equilibrium at support A Signup and view all the answers

Which of the following is NOT true regarding two-force and three-force structures?

Two forces must be unequal for equilibrium. Signup and view all the answers

What value of $B_x$ is found at the supports in Example 2?

0 Signup and view all the answers

What is a characteristic of a beam supported by three non-parallel reactions at point A?

It cannot prevent rotation about point A. Signup and view all the answers

How is the degree of external indeterminacy calculated?

Using $i_e = r + f_i - 3n_r$. Signup and view all the answers

If a structure is statically indeterminate to the second degree, what does this indicate about its degree of external indeterminacy?

The degree is exactly 2. Signup and view all the answers

What is the conclusion when calculating $i_e = 4 + 1 - 3(1)$?

The structure is statically indeterminate to the first degree. Signup and view all the answers

What is the result when $i_e = 3 + 6 - 3(3)$ is calculated?

The structure is statically determinate externally. Signup and view all the answers

What implication does an external indeterminacy value of $-1$ have for a structure?

The structure is statically unstable externally. Signup and view all the answers

In which scenario would a structure be classified as statically stable?

When the degree of external indeterminacy is 0. Signup and view all the answers

What could result from having too many supports with more reactions than necessary in a structure?

The beam could become statically indeterminate. Signup and view all the answers

Which equation accurately describes the relation of supports and reactions for a beam's stability?

$i_e = r + f_i - 3n_r$. Signup and view all the answers

Study Notes

Course Information

Course Title: CIV 2109: Statics & Introduction to Structures
Lecture Number: 5
Topic: Equilibrium & Support Reactions

Equilibrium of Structures

For a structure to be in equilibrium, all the forces and couples (including support reactions) acting on it must balance each other. There must be no resultant force or resultant couple acting on the structure.
For a three-dimensional structure under three-dimensional forces and couples, the equilibrium equations are: ΣFx = 0, ΣFy = 0, ΣFz = 0, ΣMx = 0, ΣMy = 0, ΣMz = 0.
For a plane structure (in the xy plane) under coplanar forces and couples, the equilibrium conditions are: ΣFx = 0, ΣFy = 0, ΣMz = 0.
The first two equations ensure there's no resultant force acting on the structure. The last equation ensures there's no resultant moment acting on the structure.
All equilibrium equations must be satisfied simultaneously for the structure to be in equilibrium.

Concurrent Force Systems

If the lines of action of forces intersect at a single point, the structure is under concurrent forces. In this case, only force equilibrium equations need to be considered. Then, ΣFx = 0 and ΣFy = 0 must be satisfied.
For a space structure, the equations of equilibrium under concurrent three-dimensional force system are ΣFx = 0, ΣFy = 0, and ΣFz = 0.
A plane structure that is subjected to a concurrent coplanar force system satisfies ΣFx = 0, and ΣFy = 0.

Two-Force and Three-Force Structures

A structure under two forces must have the forces be equal, opposite, and collinear.
A structure under three forces must either be concurrent or parallel.

External and Internal Forces

External forces: Forces and couples exerted on a structure by other bodies.
Applied forces: Forces that tend to move a structure. Often called loads. e.g., live loads and wind loads.
Reaction forces: Forces exerted by supports on a structure to prevent it from moving.
Internal forces: Forces and couples exerted on one part of a structure by another part of the same structure. These forces hold the various portions of the structure together. Internal forces always occur in equal but opposite pairs.

Support Reactions

Supports are used to attach structures to the ground or other bodies and restrict their movements.
The type of reaction a support exerts depends on the type of supporting device and the type of movement it prevents. Common types include Roller, Rocker, and Link.
Hinge and Fixed supports may exert force and moment.

Static Determinacy, Indeterminacy & Stability

A structure is considered internally stable, or rigid, if it maintains its shape when detached from the supports.
A statically determinate structure externally is one where the number of reactions can be calculated from the equilibrium equations (r + f = 3n).
A statically indeterminate structure externally is one where the number of reactions exceed those necessary for equilibrium, and additional equations are needed for calculating unknown reactions.

Examples (Support Reactions)

Several examples of support reaction calculations are shown, demonstrating the processes for solving for support reactions under a system of forces on a structure. The examples provided demonstrate statically determinate structures.

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Description

Explore the principles of equilibrium and support reactions in structures through this quiz. Understand the requirements for a structure to maintain equilibrium and learn the relevant equations for both three-dimensional and plane structures. Test your knowledge on concurrent force systems and their implications in statics.