Force Method of Analysis (Structure Indeterminate Structures)
Force Method of Analysis (Structure Indeterminate Structures)
Purpose of choosing statically indeterminate structure as compared to statically determinate structure are as follows-
- For a given loading, the maximum stress and deflection of indeterminate structures are generally smaller than that in statically determinate structures.
- Indeterminate structures have a tendency to redistribute its load to its redundant supports in the case where faulty design or overloading occurs.
As load P is increased, the plastic hinge will form at supports first and hence it will be treated as simply supported structures. The further load can also be resisted. However, in case of simple supports, the hinge will form at the center hence collapse will be early.
Although there will be cost saving in material due to lesser stress in member, the cost of construction of supports and joint may sometime offset the saving in material.
Differential settlement of supports, temperature variation, change in length due to fabrication errors in indeterminate structures will introduce internal stresses in structure.
In statically determinate structures internal stresses will not be introduced because of these factors.
In statically indeterminate structures, it is necessary to satisfy:
- Equilibrium equations
- Compatibility equations
- Force displacement requirement
There are two different ways to satisfy these requirements:
- Force Method
- Displacement Method
Difference between Force and Displacement Method
| S.No. | Force Method | Displacement Method |
| 1 | Also called- Compatibility Method, Method of Consistent Deformation, and Flexibility Method | Also called- Stiffness Method |
| 2 | Forces are unknown in this Method (Reaction, BM, SF) | Displacement is unknown in this Method (Δ, θ) |
| 3 | Force displacement equations are written and solution for unknown forces is obtained from compatibility equations. | Force displacement equations are written and solution for unknown displacement is obtained from equilibrium equations. |
| 4 | Once unknown forces are known, the reaction forces are found using equilibrium equations. | Once unknown displacements are known, internal forces are found using compatibility and load-displacement equation. |
| 5 | Example- Castigliano’s Theorem Strain energy Method Virtual Work Method/Unit Load Method Claperon’s three-moment Equations Column analogy Method Flexibility Matrix Method | Example- Slope Deflection Method Moment Distribution Method Stiffness Matrix Method Kam’s Method |
Fundamental Assumptions in Structural Analysis
Principal of Superposition
The total displacement or internal loading (stress) at a structure subjected to several external loading can be determined by adding together the displacements or internal loading (stress) caused by each of the external loads acting separately.
Two requirements for Principal of superposition are-
- Linear elastic response i.e., Hooke’s Law valid. (Load ∝ Displacement)
- Small displacement theory applies i.e., geometry of the structure must not undergo significant change when loads are applied.
The above requirements are also the requirements for Linear 1st order analysis.
Linear-1st Order Analysis
Linear method stress ∝ strain i.e., Hooke’s law is valid.
1st-order analysis means the assumption that, length of ABC = length of A’B’C’ in the following figure-
The 1st-order analysis is valid for small displacement, as otherwise, for large displacement, axial forces as shown below will also produce moments like P-Δ. Thus P-Δ effect is neglected in 1st order analysis.
However, the effect of P-Δ is considered in 2nd-order analysis.
Betti’s Theorem
The virtual work done by a P-force system is going through the deformation of the Q-force system is equal to the virtual work is done by the Q-force system is going through the deformation of the P-force system.
Hence per Betti’s Theorem–
P1ΔP1Q + P2ΔP2Q = Q2ΔQ1P + Q2ΔQ2P
Maxwell’s Reciprocal Theorem
If only two force P & Q are acting and magnitude of P & Q are unity.
ΔPQ = ΔQP
ΔPQ= Deflection at the location of P due to unit load at the location of Q
ΔQP= Deflection at the location of Q due to unit load at the location of P
i.e., deflection at the location Q due to unit load at P is equal to deflection at the location of P due to unit load at Q.
Castigliano’s Theorem
Castigliano has two theorems-
First Theorem
Second Theorem
Castigliano extended the principal of least work, later on, to the self-straining system.
If λ = small displacement in the direction of redundant force R then.
∂U/∂R= λ
Self-straining may be caused by the settlement of support of a redundant structure by an amount λ or by the initial misfit of a member by an amount λ too short/long.
Right Understanding
Right Understanding We all know that the Human Desire is to be in continuous happiness which is the need of I (self). But do you know, from where Continuous happiness will come? No, right! So continuous happiness is to be in Right Understanding, Right Feeling, and Right Thought that is Activity of I (Self). Do…
Where We Are
Where We Are (Self-Evolution) We exist as human being. We want to live a fulfilling life. We have some desires and we have some programs for the fulfilment of it. We need to understand our basic aspiration and program for its fulfillment correctly and comprehensively. Only then, we can ensure fulfillment. We should explore ourselves…
Highway Construction
Highway Construction Embankment Construction Materials and General Requirements The materials used in embankments, subgrades, earthen, shoulders, and miscellaneous backfills shall be soil, moorum, gravel, a mixture of these. Clay having liquid limit exceeding 70 and plasticity index exceeding 45; shall be considered unsuitable for embankment. Sub-grade and top 500mm portion of the embankment just below…
Special Concretes
Special Concretes Concrete is most vital material in modern construction. In addition to normal concrete, other varieties in use are, high strength and high-performance concrete, self-compacting, lightweight, high density, fiber reinforced, polymer, colored concrete, etc. The making of concrete is an art as well as a science. Special types of concrete are those with out-of-the-ordinary…
Marketing Practices
Marketing Practices Success in the world of business, no matter how you earn it, you have to rule on the marketplace. Although luck plays a role in the outcome of the market strategies. In the business decisions, there should be the understanding of market otherwise the failure will take place by the marked decisions. While…
Risk Analysis
Risk Analysis The risk that remains after the implementation of controls is called the residual risk. All systems will have residual risk because it is virtually impossible to completely eliminate risk to an IT system. In other words, we can say that there are two main parts of the security risk analysis known as Quantitative…
Recent Comments