EXAMPLE 12-13 Using Castigliano's second theorem, determine the horizontal deflection for the elastic frame shown in Fig. Assume G=120 GPa. Consider only the deflection caused by bend- ing.
m is applied. Calculation Example – Determine the shear force and moment.
Calculation Example – Internal forces.
Castigliano Theorem. Calculation Example - Calculate the Axial Forces of the Truss Members. C6: Energy Methods. Pi + dPi. Consider only the deflection caused by bend- ing. EXAMPLE 12-13 Using Castigliano's second theorem, determine the horizontal deflection for the elastic frame shown in Fig. Solutions for the example problem from the topic of Castigliano’s Theorem for the Solid Mechanics II course. Castigliano's theorem may be applied directly, although the angle θ is usually a more convenient independent variable than the distance s when it comes to curved beams. Thus in the sketch, at the general free body's cut end defined by θ, the bending moment required for equilibrium is : Solid Mechanics II Course homepage. inspection. Pi P. Pi + dPi. For example, the portal frame in Fig.
Section AB My = (-20 × 1.5m) = 30 kNm. P2.
His method is simple and can be used to calculate Δ or slope θ at any point in the structure as well, similar to virtual work.
U dU dPi Pi. It is assumed that students doing this tutorial are already familiar with the following. • Calculus including partial differentiation.
The flexural rigidity El of both members is equal and constant. The flexural rigidity El of both members is equal and constant. Solution Section BC Mx = -20x. 16.1 has a degree of external statical indeterminacy of 3, while the truss of Fig. U 1 = ∫[(-20x) 2 /2EI] dx = ∫ -400x 2 /2EI = -400x 3 /6EI Knowing that the limit x = 1.5m; U 1 = 225/EI. 6.1 Elastic Strain Energy for Various Loadings - Theory - Example. Calculation Example – Calculate the moments of inertia Ix and Iy. Castigliano’s theorem.
Maxwell's Theorem of Reciprocal displacements; Betti's law Betti's Theorem For structures with multiple degree of indeterminacy Example: The displacement (rotation) at a point P in a structure due a UNIT load (moment) at point Q is equal to displacement (rotation) at a point Q in a structure due a UNIT load (moment) at point P.
Alberto Castigliano, an Italian railroad engineer, who through a stroke of brilliance derived a new method to calculate displacements and slopes. P2. •To find the displacement of the cylinder, we use Castigliano’s Theorem with the strain energy for torsion.
STRAIN ENERGY 1. U Ui = f (P1, P2,, Pn) U = U* dU = dU* Department of. • Basic torsion theory. • Basic stress and strain, • The elastic properties of materials • Basic bending theory.
The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement. 12-21(a). Chapter Eight Energy Method 8.1 Introduction 8.2 Strain energy expressions 8.3 Principal of stationary potential energy; several degrees of freedom ----- Castigliano’s first theorem ---- Examples 12-21(a). dPi. For example, for a rectangular ... a beam or frame) subjected a number of loadings. Calculation Example – Determine the magnitudes of F1,F2. Pi.
Castiglianos Theorem P1. Elastic Deflection Castigliano’s Method If deflection is not covered by simple cases in Table 5.1 (p186) Stored Elastic Energy U Complementary Energy U’ U=U' =∆⋅Q 2 Incremental: dU=dU' =∆⋅dQ Deflection: ∆=dU dQ Castiglino’s Theorem:
11 Castiglianos Second Theorem The. However, in many cases, the degree is not obvious and in other cases the internal and external indeterminacies may not be independent so that we U* (dPi)dU* U dU dPi Pi. U dPi (dPi ) Pi Pi U Pi Pi.
Castigliano's method, named after Carlo Alberto Castigliano, 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. Solid Mechanics II Course homepage. Solved Example For the frame loaded as shown below, let us find the vertical deflection at point C due to bending using Castigliano’s theorem. To develop the virtual work method in a general manner, let us consider a body or a structure of arbitrary shape (later this body will be made to represent a specific truss, beam or frame) ... Maxwell’s reciprocal theorem.
P1. C6.4 Castigliano’s Theorem. 4.7(b) has a degree of internal statical indeterminacy of 1. INTRODUCTION When an elastic body is deformed, work is done.