## What is meant by equilibrium and support reactions?

Civil engineers deal with structures that are composed of several members connected or with individual members such as beams and columns. To analyze such structures, the condition of statics must be applied. Under the static condition, the members are considered to be in equilibrium under the action of forces, that is, the members are assumed to be having negligible velocity.

These members are joined with other members or are used to support them; under such conditions, support reactions are induced. The support reactions are due to Newton's third law of motion. For analysis and calculation purposes, these support reactions must be calculated as a primary task that will aid in subsequent load calculations.

## Types of supports and support reactions

There are various types of supports encountered in civil engineering applications, each of them provides different support reactions to the members. Some of the supports along with their support reactions are discussed below.

### Hinge support

Hinge support is pin-jointed support or pivoted support. A beam fixed by hinge support can undergo rotation about the point of pivot. These kinds of supports induce one degree of freedom to the beam. As the beam is pivoted, it cannot translate in any of the spatial directions along with two degrees of freedom about the two axes being constrained. These kind of supports under two-dimensional loadings produce one vertical reaction and one axial reaction.

### Fixed support

Fixed supports do not allow any degree of freedom to the members. A member under fixed support cannot translate in any of the spatial directions, and also cannot undergo rotation about any of the axes. An example of fixed support is the fixed end of a cantilever beam. Fixed support induces a vertical reaction, a horizontal reaction, and a bending moment depending upon the loading condition.

### Roller support

Roller support is provided to take the thermal expansion into consideration. A beam with roller support can undergo translation along the horizontal spatial direction and also can rotate about the point of pivot. Hence, this kind of support produces a vertical reaction only.

## Method to determine support reaction

For the estimation of loads and deformations, support reactions should be determined initially. There are a few basic steps to be followed to determine support reactions.

### Free-body diagram

The very first step in determining unknown reactions is to first draw a free-body diagram. A free-body diagram is a diagram that has the member whose analysis is the area of interest, the member is isolated from its supports and adjacent members, and is drawn separately, indicating all the forces and unknown support reactions from other members and components. The very first force that must be shown in the free-body diagram is the self-weight of the body.

### Apply equilibrium condition

The second step in determining unknown reactions is to assume the body to be static, that is, it is in equilibrium under the influence of external forces. All the forces along the X-axis and Y-axis along with the moments should be equated to zero, that is, it is Newton's second law of motion that relates force and acceleration. As the body is static there is no acceleration, so the right-hand side is equated to zero. If the member is acted by distributed forces, a resultant of a force should be considered in applying the equilibrium equations. The resultant is an equivalent force that produces the same effect as the individual forces.

### Calculation of unknowns

Having the reaction magnitudes, the unknown force values and deformations can be easily determined using standard relations and calculations.

## Statically determinate and indeterminate condition

A member is said to be statically determinate if the number of reactions of the member is less than or equal to the number of equilibrium equations. In such situations, the reactions can be easily calculated and related force and deformation values can be estimated.

In certain cases, as in the case of propped beams, the number of reactions exceeds the equilibrium equations when these beams are under the influence of both vertical and horizontal loads, such conditions are known as statically indeterminate conditions. The reactions cannot be determined using only the equilibrium equations. An additional equation known as the compatibility equation must be used. Compatibility equations consider the deformation of the members under external loads. In a compatibility equation, the principle of superposition is followed. The deformation caused by the individual forces are equated to zero.

## Context and Applications

The area is of immense importance in both civil engineering and mechanical engineering domains. Engineers dealing with structures and components need to determine reactions to estimate the critical loads that may cause the members and components to fail.

• Bachelors in Technology (Civil Engineering)
• Bachelors in Technology (Mechanical Engineering)
• Masters in Technology (Mechanical Engineering)
• Bachelors in Science (Physics)

## Practice Problems

1. Which of the following support reactions produces only a vertical reaction force?

1. Roller support
2. Fixed support
3. Hinged support
4. All of these

Correct option- a

Explanation: A roller support provides two degrees of freedom to a member and induces only one support reaction.

2. Which of the following condition is true for statically indeterminate members?

1. Number of support reactions equals the number of equilibrium reactions
2. Number of support reactions are more than number of equilibrium reactions
3. Number of support reactions are less than the number of equilibrium reactions
4. Both a and c

Correct option- d

Explanation: A statically indeterminate member is a member where the number of reactions are either less or equal to the number of equilibrium equations.

3. Which of the following beam is usually a statically indeterminate beam?

1. Cantilever beam
2. Simply supported beam
3. Propped beams
4. Over hanged beams

Correct option- c

Explanation: A propped beam is a beam that has three supports that induce more than three support reactions under the action of both vertical and axial loads.

4. Which of the following is true for roller supports?

1. Rollers are provided to account for the thermal expansion.
2. It provides one-degree of freedom to the beam.
3. A beam remains hinged to the support.
4. Both a and c

Correct option- d

Explanation: The rollers in the roller supports are provided to take into account the expansion caused by temperature rise in the material during high-temperature applications. These beams remain pin-jointed to the supports.

5. Which of the following is the first step in determining the support reactions?

1. Drawing free-body diagrams
2. Applying equilibrium equations
3. Determining the resultant in case of distributed loads
4. All of the above

Correct option- a

Explanation: To determine the support reactions, a free body diagram of the member should be drawn.

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