All questions of Determinacy & Indeterminacy of Structures for Civil Engineering (CE) Exam

C

The degree of freedom of a joint refers to the number of independent motions or directions in which the joint can move. In the case of a rigid jointed joint plane frame, the degree of freedom can be determined by considering the constraints and restrictions imposed by the joint connections.

Rigid Jointed Joint Plane Frame:

A rigid jointed joint plane frame consists of a series of members connected at their ends by joints. These joints are assumed to be perfectly rigid, meaning that they do not allow any relative movement between the connected members. The frame is typically constructed by welding or bolting the members together at the joints.

Degree of Freedom:

The degree of freedom of a joint in a rigid jointed joint plane frame can be determined using the equation:

DOF = 6 - R - 2J

Where:
DOF = Degree of Freedom
R = Number of reactions or constraints at the joint
J = Number of members meeting at the joint

Explanation:

1. Reactions (R):
The number of reactions at a joint depends on the support conditions. A joint may have both translational and rotational reactions. In a two-dimensional plane frame, each joint can have a maximum of three reactions: two translations (horizontal and vertical) and one rotation. However, the number of reactions can be less if the joint is fully restrained or partially restrained.

2. Members (J):
The number of members meeting at a joint represents the connections between the members. Each member provides two degrees of freedom at the joint, one for each end. Therefore, the total number of degrees of freedom contributed by the members meeting at a joint is equal to twice the number of members.

By subtracting the number of reactions (R) and twice the number of members (2J) from 6, we can determine the degree of freedom (DOF) at the joint.

In the case of a rigid jointed joint plane frame, the joints are assumed to be fully restrained, meaning that all three reactions (R) are present at each joint. Additionally, each joint is connected to two members (J). Substituting these values into the equation, we have:

DOF = 6 - R - 2J
DOF = 6 - 3 - 2(2)
DOF = 6 - 3 - 4
DOF = 3

Therefore, the degree of freedom of a joint for a rigid jointed joint plane frame is 3 (option C).

For a rigid joined 2D frame structure. Degree of static indeterminacy
Ds=3m+r-3j
here
Ds=3x2+3-3x3=0.
Hence it is statically determinate.
And since reactions are not parallel or concurrent it is a stable member.

Problem Analysis

We are given a single-bay, single-storeyed portal frame ABCD with fixed column ends. We need to determine the kinematic indeterminacy of the frame.

Solution

The kinematic indeterminacy of a structure refers to the number of redundant members or joints that can be removed without causing any change in the stability or equilibrium of the structure.

In the given portal frame, the column ends are fixed, which means they are fully restrained against rotation and translation. This implies that there is no relative rotation or displacement between the column ends and the foundation.

Analysis of the Frame

To determine the kinematic indeterminacy, we need to analyze the frame and identify the degrees of freedom.

1. Number of Joints (J): In the given frame, there are four joints: A, B, C, and D.

2. Number of Members (M): There are three members: AB, BC, and CD.

Degree of Freedom Analysis

Next, we analyze the degrees of freedom at each joint to determine the kinematic indeterminacy.

1. Joint A:
- Vertical Displacement (V): Since the column end is fixed, there is no vertical displacement at joint A.
- Horizontal Displacement (H): Since the column end is fixed, there is no horizontal displacement at joint A.
- Rotation (R): Since the column end is fixed, there is no rotation at joint A.
- Total Degrees of Freedom (DOF) at Joint A: 0

2. Joint B:
- Vertical Displacement (V): The vertical displacement at joint B is not restricted as it is connected to the member AB.
- Horizontal Displacement (H): The horizontal displacement at joint B is not restricted as it is connected to the member AB.
- Rotation (R): The rotation at joint B is not restricted as it is connected to the member AB.
- Total Degrees of Freedom (DOF) at Joint B: 3

3. Joint C:
- Vertical Displacement (V): The vertical displacement at joint C is not restricted as it is connected to the member BC.
- Horizontal Displacement (H): The horizontal displacement at joint C is not restricted as it is connected to the member BC.
- Rotation (R): The rotation at joint C is not restricted as it is connected to the member BC.
- Total Degrees of Freedom (DOF) at Joint C: 3

4. Joint D:
- Vertical Displacement (V): Since the column end is fixed, there is no vertical displacement at joint D.
- Horizontal Displacement (H): Since the column end is fixed, there is no horizontal displacement at joint D.
- Rotation (R): Since the column end is fixed, there is no rotation at joint D.
- Total Degrees of Freedom (DOF) at Joint D: 0

Kinematic Indeterminacy Calculation

The kinematic indeterminacy is given by the formula:

I = 3J - 2M - R

where I is the kinematic indeterminacy, J is the number of joints, M is the number of members, and R is the number of support reactions.

In this case, there are no support reactions (R = 0) as the column ends are fixed.

Substituting the values, we have:

I = 3(4)

>j+2

Explanation:
For a pin-jointed plane frame to be stable, it must satisfy the condition of static equilibrium, which is that the sum of all forces and moments acting on the frame is zero. In other words, the number of unknown forces and moments must be equal to the number of equations of static equilibrium (i.e. three equations for a 2D frame).

In a pin-jointed plane frame, each joint can provide two equilibrium equations (one for force in the x-direction and one for force in the y-direction). Therefore, the total number of equations of equilibrium is 2j.

On the other hand, each member can carry one unknown force (either tension or compression). Therefore, the total number of unknown forces is m.

Finally, the reaction components at the supports can be expressed using two equations of equilibrium (one for force in the x-direction and one for force in the y-direction). Therefore, the total number of unknown reaction components is r.

So, by applying the principle of statics, we can write the following equation for stability of a pin-jointed plane frame:

m + r = 2j

Simplifying this equation, we get:

m - 2j + r = 0

Therefore, for a pin-jointed plane frame to be stable, the expression (m - 2j + r) must be equal to zero. However, for instability, this expression must be greater than zero.

Hence, the condition for instability can be expressed as:

m - 2j + r > 0

or equivalently,

m > 2j - r

Adding 2 to both sides, we get:

m + 2 > 2j - r + 2

which can be rewritten as:

(m + r) > 2(j + 1)

or simply:

(m + r) > 2j + 2

Therefore, the condition for instability in a pin-jointed plane frame is:

(m + r) > 2j + 2.

The degree of kinematic indeterminacy of a pin-jointed plane frame can be calculated using the formula:

Degree of Kinematic Indeterminacy = 3j - 2m

Where j is the total number of joints in the frame and m is the total number of members in the frame.

Since a pin-jointed plane frame is a 2D structure, we can assume that there are 3 reactions at each support (2 vertical and 1 horizontal). Therefore, the total number of reactions (m) is equal to 3 times the number of supports.

Hence, the formula for the degree of kinematic indeterminacy of a pin-jointed plane frame is:

Degree of Kinematic Indeterminacy = 3j - 6

Therefore, the given statement "Degree of kinematic indeterminacy of a pin jointed plane frame is given by 2j" is incorrect.

The degree of freedom for a rigid jointed plane frame without axial deformation is given by 3j, where "j" represents the number of joints in the frame.

Answer by cutting the closed loop is differing at the hinged joint due to meeting with four members and reaction due to three members

Geometrically unstable structure

A truss is a structural framework composed of straight members connected at joints. It is used to support loads and distribute them evenly throughout the structure. Trusses are commonly used in bridges, roofs, and other load-bearing structures.

Classification of trusses

Trusses can be classified into several categories based on their characteristics. One classification is based on the behavior of the truss under external loads. Trusses can be categorized as determinate or indeterminate structures.

Determinate structures

Determinate structures are those in which the reactions and internal forces can be determined solely using the equations of static equilibrium. These structures have a unique solution for all the unknowns. In other words, the reactions and forces in each member can be determined by solving a set of simultaneous equations.

Statically unstable structures

Statically unstable structures are those that cannot be analyzed using the equations of static equilibrium alone. These structures have more unknowns than the number of equations available, making it impossible to determine the reactions and internal forces without additional information or assumptions.

Structurally unstable structures

Structurally unstable structures are those that fail to maintain their shape or stability when subjected to external loads. They are unable to resist the applied loads and may collapse or deform significantly. These structures generally have insufficient stiffness or strength to support the applied loads.

Geometrically unstable structures

Geometrically unstable structures are those that undergo rigid body translation under an arbitrary load. Rigid body translation refers to the movement of the entire structure as a whole, without any deformation or change in shape of individual members. In such structures, the joints are not rigid enough to maintain the shape of the truss, causing it to move as a unit.

Explanation of the correct answer

Based on the given options, the correct answer is option 'B' - Geometrically unstable structure. This is because a truss that undergoes rigid body translation for an arbitrary load does not maintain its shape and exhibits movement as a whole. The joints in the truss are not rigid enough to resist the translation, resulting in a geometrically unstable structure.