lunchpoy.blogg.se - Flexibility method structural analysis examples

Δ nm = Deflection in the direction 'n' due to load applied in the direction 'm'. Δ mn = Deflection in the direction 'm' due to load applied in the direction 'n'. Where, P m = Load applied in the direction m. Θ = Rotation in the direction of moment M. Δ = Displacement in the direction of force P. Where, D = External dia of hollow circular shaftsĭ = Internal dia of hollow circular shaft

Strain energy stored in hollow circular shaft is,.

Where, Maximum shear stress at the surface of rod under twisting.

Strain energy stored in terms of maximum shear stress.

Strain energy stored due to shear force.

Where, M X = Bending moment at section x-x

Castigliano's theorem of minimium strain energy.

Types of Force Method/Flexibility Method/Compatibility Methods These conditions transform the basic determinate structure back to the original structure by finding the combination of redundant forces that make displacement at each of the released constraints equal to zero.

Solve for redundant forces R f ( j =1 to n ) by imposing the compatibility conditions of the original structure.

Use suitable method (given in Chapter 4) to calculate displacements at each of the released constraints in the basic determinate structure.

Apply the given loading or imposed deformation to the basic determinate structure.

For a given released constraint j, introduce an unknown redundant force R f corresponding to the type and direction of the released constraint.The system thus formed is called the basic determinate structure. This is accomplished by releasing external support conditions or by creating internal hinges. Transform the structure into a statically determinate system by releasing a number of static constraints equal to the degree of static indeterminacy, n.Determine the degree of static indeterminacy, n of the structure.The basic steps in the force method are as follows: The method is based on transforming a given structure into a statically determinate primary system and calculating the magnitude of statically redundant forces required to restore the geometric boundary conditions of the original structure. The force method is used to calculate the response of statically indeterminate structures to loads and/or imposed deformations. Another name for the method is the force method because forces are the unknown quantities in equations of compatibility. It is called the flexibility method because flexibilities appear in the equations of compatibility.

There will always be as many compatibility equations as redundants. The flexibility method is based upon the solution of "equilibrium equations and compatibility equations".Compatibility and material information are essential. However, for indeterminate structures, Statics (equilibrium) alone is not sufficient to conduct structural analysis.Material (stress-strain) relationships are needed only to calculate deflections. based on Statics) irrespective of the material information. For determinate structures, the force method allows us to find internal forces (using equilibrium i.e.