Test: Theory of Structures

For equilibrium of any body.

(i) ∑F_H = 0

(ii) ∑F_V = 0

(iii) ∑M_Z = 0

Let D_s = static Indeterminacy

D_s = R - r

R → no. of unknowns

r → No. of equilibrium equations available

If r > R

D_s < 0, system is partially constraintIf

D_s > 0, then system is over stiff.

Concrete is a brittle material and is unable to take tension. In Truss both tension and compression force is available, that’s why concrete is not used in making truss. While metal bars and channels are made of material which have both tensile as well as compressive strength.

Beam are the horizontal members which can transmit the load perpendicular to their longitudinal direction and not in axial direction while truss can transmit the load in axial direction only.

Muller Breslau Principle: The Muller-Breslau principle states that the ILD for any stress function in a structure is represented by its deflected shape obtained by removing the restrained offered by that stress function and introducing a directly related generalised unit displacement in the direction of that stress function.

The value of maximum bending moment is = 200 x 3.6 + 100 x 2.4 = 960 kN-m

D_s = m – 2j + r_e

m = 23

j = 11

∴ D_s = 23 – 22 + 5 = 6

Three-hinged arch is a determinate structure and rise in temperature do not produce any stress.

Force method is useful when D_s < D_k

Displacement method is useful when D_k < D_s­

The unit load method is extensively used in the calculation of deflection of beams, frames and trusses. Theoretically this method can be used to calculate deflections in statically determinate and indeterminate structures. However it is extensively used in evaluation of deflections of statically determinate structures only as the method requires a priori knowledge of internal stress resultants. The unit load method used in structural analysis is derived from Castigliano’s theorem.

D_k = 3j – r_e + r_r

= 3 x 4 – 5 + 1 = 8

If members are considered axially rigid

D_k = 8 – 3 = 5

According to the principle of superposition, for a linearly elastic structure, the load effects caused by two or more loadings are the sum of the load effects caused by each loading separately. Note that the principle is limited to:

• Linear material behaviour only;

• Structures undergoing small deformations only (linear geometry).

It is not applicable when:

1. The material does not obey Hooke’s law.

2. The effect of temperature changes are taken into consideration.

3. The structure is being analysed for the effect of support settlement.

• Beams that have more than one span are defined as continuous beams. Continuous beams are very common in the bridge and building structures.
• When a beam is continuous over many supports and the moment of inertia of different spans is different, the force method of analysis becomes quite cumbersome.
• However, the force method of analysis could be further simplified for this particular case (continuous beam) by choosing the unknown bending moments at the supports as unknowns.
• One compatibility equation is written at each intermediate support of a continuous beam in terms of the loads on the adjacent span and bending moment at left, center (the support where the compatibility equation is written) and rigid supports.
• Two consecutive spans of the continuous beam are considered at one time. Since the compatibility equation is written in terms of three moments, it is known as the equation of three moments.
• In this manner, each span is treated individually as a simply supported beam with external loads and two end support moments.
• For each intermediate support, one compatibility equation is written in terms of three moments. Thus, we get as many equations as there are unknowns. Each equation will have only three unknowns.
• Clapeyron first proposed this method in 1857 and it is widely known as the Clapeyron theorem.

Due to sway, the deflection of point B will be equal to that of point C.