Test: Principle of Virtual Work - Civil Engineering (CE) MCQ

Virtual Work - 

• Virtual Work is the work done by a real force acting through a virtual displacement.
• A virtual displacement is any displacement consistent with the constraints of the system/structure.
• Principle of Virtual Work states that "The principle of Virtual Work states that "Total work of external forces for small virtual displacement will be zero if the rigid body is in equilibrium."

Calculation:
Given:

When θ undergoes a positive virtual displacement of δθ, the configuration given below is formed. In this figure only the spring force F_sp, the weight of table W_t and the weight of bowl W_b do work when the virtual displacement takes place. 
The magnitude of F_sp can be given using the spring force formula,

The position of the point of application of W_b, W_t and F_sp are denoted by yG_p, y_Gt and x_c respectively. 

From the Concept of Virtual Work,

Since W_b, W_t, and F_sp act towards the negative sense of their corresponding virtual displacement, their work is negative.
Put the value of  and F_sp = 0.25kcosθ N and all other values in equation (iv), then we have
−4.905(0.25cosθδθ) − 24.525(0.25cosθδθ) − 0.25kcosθ(−0.25sinθδθ) = 0
δθ(−7.3575cosθ + 0.0625ksinθcosθ) = 0
Since δθ ≠ 0 then

When θ = 45°, then

k = 166.48 ​N/m ≃ 166 N/m

Virtual work principle states that “when a body is in equilibrium then for a small change in displacement, the total work done should be zero”.
The forces which are considered for virtual work is:

• The tension in the body when displacement alters the length of the body.
• The thrust in the body when displacement alters the length of the body.
• Weight of the body.

If ‘w’ is the weight of the body and ‘h’ is the distance between the centre of gravity and the fixed plane, then the virtual work should be ‘wh’ when the centre of gravity is below the fixed plane. When the centre of gravity is above the fixed plane, the virtual work is ‘-wΔh’. If the centre of gravity is not moving in the upward or downward direction, then the weight of the body is not doing any work. Hence, the work done by the self-weight of the body is taken into consideration when the centre of gravity moves vertically.
Option ‘1’ is correct.

• Virtual work is the work done by a real force acting through a virtual displacement i.e. Virtual work done by real forces.
• The initial formulation of the principle of virtual displacement was given by Bernoulli.
• A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
• The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
• In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.
• The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.

• Virtual work is the work done by a real force acting through a virtual displacement or a virtual force acting through a real displacement.
• A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
• The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
• In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.

∴ The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.
⇒ A system of connected rigid bodies is in equilibrium, provided the virtual work done by all the external forces and couples acting on the system is zero for each independent virtual displacement of the system.

• Virtual work is the work done by a real force acting through a virtual displacement i.e. Virtual work done by real forces.
• A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
• The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
• In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.
• The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.

Virtual work load method:

• The virtual work load method is also known as unit-load method or virtual force method used to find the deflection and slope at a point of a structure with the help of law of conservation.
• When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero.

Castigliano’s method: 
It is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the energy. He is known for his two theorems.

• Castigliano’s first theorem:- The first partial derivative of total strain energy in a structure with respect to any particular deflection component at a point is equal to the force applied at that point & in the direction corresponding to the deflection component.
  
  It is applicable to linearly or non-linearly elastic structure in which the temperature is constant and the supports are unyielding.
• Castigliano’s 2nd theorem:- The first partial derivative of total strain energy in a structure with respect to the force applied at that point is equal to the deflection at the point of application of that force in the direction of its line of action.
  
  It is applicable to linearly elastic structure with constant temperature and unyielding supports.

Actual work method: 
It is the real work method which is used to find the slope and deflection of any structure with the help of force and bending moment in real conditions.

Euler’s method:
It is the first order numerical procedure for solving ordinary differential equations with a given initial value.

• Virtual work is the work done by a real force acting through a virtual displacement i.e. Virtual work done by real forces.
• A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
• The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
• In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.
• The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.

Principle of Virtual Work
It states that if a small imaginary displacement is given to the system under equilibrium. Then the algebraic sum of work done by the force and moment is always equal to zero or partial derivatives of its total potential energy with respect to each of the independent variable is zero.
If P₁, P₂,……………..P_n are force and
δ_1, δ₂…………………….δ_n are the corresponding displacement
If M₁, M₂,……………….M_n are moments and
δθ₁, δθ₂………………….. δθ_n  are corresponding angular displacement
Then according to the principle of virtual work total work is done is zero.
P₁δ₁ + P₂δ₂ …………… + M₁ δθ₁ + M­₂ δθ₂ + ……………….. = 0

Virtual work is the work done by a real force acting through a virtual displacement or a virtual force acting through a real displacement.
A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
A virtual force is any system of forces in equilibrium.
The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.
∴ The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.

Principle of virtual work: (unit-load Method)
Developed by Bernoulli:

To find Δ at point A due to loads P₁, P₂, P₃. Remove all loads, apply virtual load P’ on point A.
For simplicity P’ = 1
It creates internal load u on representative element.

Now remove this load, apply P₁, P_2, P₃ due to which pt. A will be displaced by Δ
∴ External virtual work = 1.Δ
Internal virtual work = u.dL

P' = 1 = external virtual unit load in direction of Δ.
u = internal virtual load acting on element in direction of a dL.
Δ = external displacement caused by real loads.
dL = internal deformation caused by real loads.

• Virtual work is the work done by a real force acting through a virtual displacement i.e. Virtual work done by real forces.
• The initial formulation of the principle of virtual displacement was given by Bernoulli.
• A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
• The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.
• In this method, the system is displaced through a small amount about a reference point and the work done by all the forces about the reference point is summed to zero to find the unknown reactions if any.
• The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be zero.