Calculation of Stiffness in Structural elements

Ans 1)

Case 1:

• Unit displacement u1 = 1
• Stiffness coefficient of column translation (2 similiar) 2x12EI/h^3
• Stiffness coefficient of beam rotation (2ends) EI/h^2

Case 2:

• Unit rotation u2 = 1
• Stiffness coefficient of column rotation at right end 6EI/h^2
• Stiffness coefficient of column + Beam rotation at left end 4EI/h + 4EI/L = 4EI/h + 4EI/2h = 6EI/h

Case 3:

• Unit rotation at u3 = 1
• Stiffness coefficient of column rotation at left end 6EI/h^2
• Stiffness coefficient of column + Beam rotation at right end 4EI/h + 4EI/L = 4EI/h + 4EI/2h = 6EI/h

fs = k.u

By static condensation method, deducing the 3 unknown expression into 2 unknown equation

Ans 2)

a.

• Position of the element can be described by 1 coordinate only u(t)
• The system has single degree of freedom structure.
• The column has fixed end at bottom. so the stiffness coefficient will be as follow k = 3EI/h^3
• The equation of motion shall be

• Natural frequency formula

Now put the values of k and m in the above equation

b.

• Position of the element can be described by 1 coordinate only u(t) because beam with infinite stiffness will induce uniform translation in all columns.
• The system has single degree of freedom structure.
• Two columns with both end fixed and one column has pinned-fixed . So the stiffness coefficient will be as follow

For parallel stiffness elements

• The equation of motion shall be

• Natural frequency formula

Now put the values of k and m in the above equation

Ans 3)

• Both elements have same material and section. It means they have same modulud of elasticity and second moment of inertia. But the lengths are different.
• When a mass 'M' is applied at the point, both elements will respond simultaneously. It means they are connected in parallel.

Effective stiffness of the system

• Natural frequency formula

• Time period

Ans 4)

a.

• The stiffness coefficient of the beam element is 3EI /L^3
• The stiffness coefficient of the spring is K
• The stiffness elements are in series so,

b.

• The stiffness coefficient of both the column elements is 3EI/h^3
• The stiffnesss coefficient of the spring is K, but is inclined. So the horizontal force(P) and the deformation (u) has to be resolved into inclined commponents to arrive actual coeffficient of the spring.

c.

• The stiffness coefficient of the pinned-pinned beam is
• The effective stiffness coefficient for the system has to be formed by adding the relative axial stiffness provided by the inclined strut connected at the right end.
• Force acting at the right end is mg/2 and deformation caused by the strut at the point of 'm' is half of its right end nodal deformation.