Structural Analysis II Course Code: CIVL322 Energy Method Dr. Aeid A. Abdulrazeg

Energy Method Widely used to obtain solutions to elasticity problems. Can be used to obtain elastic deflections of statically indeterminate structures and to determine redundant reactions. For more complicated loadings or for structures such as trusses & frames, it is suggested that energy methods be used for the computations. Most energy methods are based on the conservation of energy principal. Work done by all external forces acting on a structure, Ue is transformed into internal work or strain energy Ui Ue = Ui

Energy Method If the material’s elastic limit is not exceeded, the elastic strain energy will return the structure to its undeformed state when the loads are removed When a force F undergoes a displacement dx in the same direction as the force, the work done is d Ue = F dx If the total displacement is x, the work becomes:

Energy Method Consider the effect caused by an axial force applied to the end of a bar as shown in Figure F is gradually increased from 0 to some limiting value F = P The final elongation of the bar becomes  If the material has a linear elastic response, then F = (P/ )x Substituting into integrating from 0 to , we get:

Energy Method Strain Energy—Axial Force The normal stress is  = N/A The final strain is  = /L Consequently, N/A = E(/L) Final deflection: Substituting into with P = N,

Energy Method Strain Energy—Bending Consider the beam shown in Figure which is distorted by the gradually applied loading P and w Consequently, the strain energy or work stored in the element is determined from since the internal moment is gradually developed

Energy Method Let a statically indeterminate structure has degree of indeterminacy as (n) the selected basic determinate structure apply the unknown forces: R1, R2, …………, Rn In the direction of Ri is expressed by

Energy Method This equations will provide the (n) linear simultaneous equations with (n) unknowns (R1, R2, …………, Rn). Since the Δi is known therefore, the solution of simultaneous equations will provide the desired Ri (i = 1,2,……,n). For the structure with member subjected to the axial forces only, the equation Eq. (2) is rewritten as: Where P is the force in the member due to applied loading and unknown Ri (i = 1,2,……,n). L and AE are length and axial rigidity of the member, respectively

Energy Method For structures with member subjected to the bending moments the Eq. (2) is rewritten as

Example .1 Find the expression for the prop reaction in the propped cantilever beam shown in figure. EI is constant. A B 50 KN 6 m

Example .2 Determine the support moment for continuous beam shown in figure. EI is constant. A C 2 kN/m 3 m 5 m B

Example .3 A beam AB of span 3 m is fixed at both the ends and carries a point load of 9 kN at C distant 1 m from A. The moment inertia of the portion AC of the beam is 2I and the portion CB is I. Calculate the fixed end moments and reactions. 9KN A B C I 2I 1 m 2 m

Example .4 A portal frame ABCD is hinged at A and D and has rigid joints at B and C. the frame is loaded as shown in figure. 3 m 2 m 1 m 6 kN A B C D E