Page 1 Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Page 2 Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Let be the forces acting at from the left end on a simply supported beam of span n P P P ,...., , 2 1 n x x x ,......, , 2 1 L . Let be the displacements at the loading points respectively as shown in Fig. 3.1. Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by (vide eqn. 1.8 of lesson 1) n u u u ,..., , 2 1 n P P P ,...., , 2 1 n n u P u P u P W 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.1) Page 3 Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Let be the forces acting at from the left end on a simply supported beam of span n P P P ,...., , 2 1 n x x x ,......, , 2 1 L . Let be the displacements at the loading points respectively as shown in Fig. 3.1. Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by (vide eqn. 1.8 of lesson 1) n u u u ,..., , 2 1 n P P P ,...., , 2 1 n n u P u P u P W 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.1) Work done by the external forces is stored in the structure as strain energy in a conservative system. Hence, the strain energy of the structure is, n n u P u P u P U 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.2) Displacement below point is due to the action of acting at distances respectively from left support . Hence, may be expressed as, 1 u 1 P n P P P ,...., , 2 1 n x x x ,......, , 2 1 1 u n n P a P a P a u 1 2 12 1 11 1 .......... + + + = (3.3) In general, n i P a P a P a u n in i i i ,... 2 , 1 .......... 2 2 1 1 = + + + = (3.4) where is the flexibility coefficient at due to unit force applied at ij a i j . Substituting the values of in equation (3.2) from equation (3.4), we get, n u u u ,..., , 2 1 ...] [ 2 1 ....... ...] [ 2 1 ...] [ 2 1 2 2 1 1 2 22 1 21 2 2 12 1 11 1 + + + + + + + + + = P a P a P P a P a P P a P a P U n n n (3.5) We know from Maxwell-Betti’s reciprocal theorem ji ij a a = . Hence, equation (3.5) may be simplified as, [] 22 2 11 1 22 2 12 1 2 13 1 3 1 1 1 .... .... ... 2 nn n n n U a P a P a P a PP a PP a PP ?? =+ ++ + + ++ ?? + (3.6) Now, differentiating the strain energy with any force gives, 1 P n n P a P a P a P U 1 2 12 1 11 1 .......... + + + = ? ? (3.7) It may be observed that equation (3.7) is nothing but displacement at the loading point. 1 u In general, n n u P U = ? ? (3.8) Hence, for determinate structure within linear elastic range the partial derivative of the total strain energy with respect to any external load is equal to the Page 4 Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Let be the forces acting at from the left end on a simply supported beam of span n P P P ,...., , 2 1 n x x x ,......, , 2 1 L . Let be the displacements at the loading points respectively as shown in Fig. 3.1. Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by (vide eqn. 1.8 of lesson 1) n u u u ,..., , 2 1 n P P P ,...., , 2 1 n n u P u P u P W 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.1) Work done by the external forces is stored in the structure as strain energy in a conservative system. Hence, the strain energy of the structure is, n n u P u P u P U 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.2) Displacement below point is due to the action of acting at distances respectively from left support . Hence, may be expressed as, 1 u 1 P n P P P ,...., , 2 1 n x x x ,......, , 2 1 1 u n n P a P a P a u 1 2 12 1 11 1 .......... + + + = (3.3) In general, n i P a P a P a u n in i i i ,... 2 , 1 .......... 2 2 1 1 = + + + = (3.4) where is the flexibility coefficient at due to unit force applied at ij a i j . Substituting the values of in equation (3.2) from equation (3.4), we get, n u u u ,..., , 2 1 ...] [ 2 1 ....... ...] [ 2 1 ...] [ 2 1 2 2 1 1 2 22 1 21 2 2 12 1 11 1 + + + + + + + + + = P a P a P P a P a P P a P a P U n n n (3.5) We know from Maxwell-Betti’s reciprocal theorem ji ij a a = . Hence, equation (3.5) may be simplified as, [] 22 2 11 1 22 2 12 1 2 13 1 3 1 1 1 .... .... ... 2 nn n n n U a P a P a P a PP a PP a PP ?? =+ ++ + + ++ ?? + (3.6) Now, differentiating the strain energy with any force gives, 1 P n n P a P a P a P U 1 2 12 1 11 1 .......... + + + = ? ? (3.7) It may be observed that equation (3.7) is nothing but displacement at the loading point. 1 u In general, n n u P U = ? ? (3.8) Hence, for determinate structure within linear elastic range the partial derivative of the total strain energy with respect to any external load is equal to the displacement of the point of application of load in the direction of the applied load, provided the supports are unyielding and temperature is maintained constant. This theorem is advantageously used for calculating deflections in elastic structure. The procedure for calculating the deflection is illustrated with few examples. Example 3.1 Find the displacement and slope at the tip of a cantilever beam loaded as in Fig. 3.2. Assume the flexural rigidity of the beam EI to be constant for the beam. Moment at any section at a distance x away from the free end is given by Px M - = (1) Strain energy stored in the beam due to bending is ? = L dx EI M U 0 2 2 (2) Substituting the expression for bending moment M in equation (3.10), we get, ? = = L EI L P dx EI Px U 0 3 2 2 6 2 ) ( (3) Page 5 Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Let be the forces acting at from the left end on a simply supported beam of span n P P P ,...., , 2 1 n x x x ,......, , 2 1 L . Let be the displacements at the loading points respectively as shown in Fig. 3.1. Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by (vide eqn. 1.8 of lesson 1) n u u u ,..., , 2 1 n P P P ,...., , 2 1 n n u P u P u P W 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.1) Work done by the external forces is stored in the structure as strain energy in a conservative system. Hence, the strain energy of the structure is, n n u P u P u P U 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.2) Displacement below point is due to the action of acting at distances respectively from left support . Hence, may be expressed as, 1 u 1 P n P P P ,...., , 2 1 n x x x ,......, , 2 1 1 u n n P a P a P a u 1 2 12 1 11 1 .......... + + + = (3.3) In general, n i P a P a P a u n in i i i ,... 2 , 1 .......... 2 2 1 1 = + + + = (3.4) where is the flexibility coefficient at due to unit force applied at ij a i j . Substituting the values of in equation (3.2) from equation (3.4), we get, n u u u ,..., , 2 1 ...] [ 2 1 ....... ...] [ 2 1 ...] [ 2 1 2 2 1 1 2 22 1 21 2 2 12 1 11 1 + + + + + + + + + = P a P a P P a P a P P a P a P U n n n (3.5) We know from Maxwell-Betti’s reciprocal theorem ji ij a a = . Hence, equation (3.5) may be simplified as, [] 22 2 11 1 22 2 12 1 2 13 1 3 1 1 1 .... .... ... 2 nn n n n U a P a P a P a PP a PP a PP ?? =+ ++ + + ++ ?? + (3.6) Now, differentiating the strain energy with any force gives, 1 P n n P a P a P a P U 1 2 12 1 11 1 .......... + + + = ? ? (3.7) It may be observed that equation (3.7) is nothing but displacement at the loading point. 1 u In general, n n u P U = ? ? (3.8) Hence, for determinate structure within linear elastic range the partial derivative of the total strain energy with respect to any external load is equal to the displacement of the point of application of load in the direction of the applied load, provided the supports are unyielding and temperature is maintained constant. This theorem is advantageously used for calculating deflections in elastic structure. The procedure for calculating the deflection is illustrated with few examples. Example 3.1 Find the displacement and slope at the tip of a cantilever beam loaded as in Fig. 3.2. Assume the flexural rigidity of the beam EI to be constant for the beam. Moment at any section at a distance x away from the free end is given by Px M - = (1) Strain energy stored in the beam due to bending is ? = L dx EI M U 0 2 2 (2) Substituting the expression for bending moment M in equation (3.10), we get, ? = = L EI L P dx EI Px U 0 3 2 2 6 2 ) ( (3) Now, according to Castigliano’s theorem, the first partial derivative of strain energy with respect to external force P gives the deflection at A in the direction of applied force. Thus, A u EI PL u P U A 3 3 = = ? ? (4) To find the slope at the free end, we need to differentiate strain energy with respect to externally applied momentM atA . As there is no moment atA , apply a fictitious moment at 0 M A. Now moment at any section at a distance x away from the free end is given by 0 M Px M - - = Now, strain energy stored in the beam may be calculated as, ? + + = + = L EI L M EI PL M EI L P dx EI M Px U 0 2 0 2 0 3 2 2 0 2 2 6 2 ) ( (5) Taking partial derivative of strain energy with respect to , we get slope at 0 M A . 2 0 0 2 A M L UPL M EI EI ? ? == + ? (6) But actually there is no moment applied atA. Hence substitute in equation (3.14) we get the slope at A. 0 0 = M EI PL A 2 2 = ? (7) Example 3.2 A cantilever beam which is curved in the shape of a quadrant of a circle is loaded as shown in Fig. 3.3. The radius of curvature of curved beam isR, Young’s modulus of the material is E and second moment of the area is I about an axis perpendicular to the plane of the paper through the centroid of the cross section. Find the vertical displacement of point A on the curved beam.Read More

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