Page 1 Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,...., , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,...., , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) Page 2 Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,...., , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,...., , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) Page 3 Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,...., , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,...., , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) In the first case (4.2a), obtain deflection below A due to applied loads . This can be easily accomplished through Castigliano’s first theorem as discussed in Lesson 3. Since there is no load applied at n P P P ,...., , 2 1 A , apply a fictitious load at Q A as in Fig. 4.2. Let be the deflection below a u A . Now the strain energy s U stored in the determinate structure (i.e. the support A removed) is given by, a n n S Qu u P u P u P U 2 1 2 1 .......... 2 1 2 1 2 2 1 1 + + + + = (4.1) It is known that the displacement below point is due to action of acting at respectively and due to Q at 1 u 1 P 12 , ,...., n PP P n x x x ,......, , 2 1 A . Hence, may be expressed as, 1 u Page 4 Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,...., , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,...., , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) In the first case (4.2a), obtain deflection below A due to applied loads . This can be easily accomplished through Castigliano’s first theorem as discussed in Lesson 3. Since there is no load applied at n P P P ,...., , 2 1 A , apply a fictitious load at Q A as in Fig. 4.2. Let be the deflection below a u A . Now the strain energy s U stored in the determinate structure (i.e. the support A removed) is given by, a n n S Qu u P u P u P U 2 1 2 1 .......... 2 1 2 1 2 2 1 1 + + + + = (4.1) It is known that the displacement below point is due to action of acting at respectively and due to Q at 1 u 1 P 12 , ,...., n PP P n x x x ,......, , 2 1 A . Hence, may be expressed as, 1 u (4.2) 1111 122 1 1 .......... nn a uaP aP aP aQ =+ + + + where, is the flexibility coefficient at due to unit force applied at ij a i j . Similar equations may be written for . Substituting for in equation (4.1) from equation (4.2), we get, 23 , ,...., and n uu u u a 23 , ,...., and na uu u u 1 11 1 12 2 1 1 2 21 1 22 2 2 2 1 1 22 1 1 22 11 [ ... ] [ ... ] ....... 22 11 [ ... ] [ .... ] 22 Snna nn n n n nn n na a a an n aa U PaP a P a P aQ P a P a P a P a Q Pa P a P a P a Q Qa P a P a P a Q =+ ++ + + + + + + ++ + + + + ++ + a (4.3) Taking partial derivative of strain energy s U with respect to Q, we get deflection atA . 11 2 2 ........ s aa ann aa U aP a P a P a Q Q ? =+ + + + ? (4.4) Substitute as it is fictitious in the above equation, 0 Q = 11 2 2 ........ s aa a an U uaP aP aP Q ? == + + + ? n (4.5) Now the strain energy stored in the beam due to redundant reaction A R is, 23 6 a r R L U EI = (4.6) Now deflection at A due to a R is 3 3 a r a a R L U u R EI ? =- = ? (4.7) The deflection due to should be in the opposite direction to one caused by superposed loads , so that the net deflection at a R 12 , ,...., n PP P A is zero. From equation (4.5) and (4.7) one could write, r a a U Us u QR ? ? ==- ? ? (4.8) Since is fictitious, one could as well replace it by Q a R . Hence, Page 5 Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,...., , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,...., , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) In the first case (4.2a), obtain deflection below A due to applied loads . This can be easily accomplished through Castigliano’s first theorem as discussed in Lesson 3. Since there is no load applied at n P P P ,...., , 2 1 A , apply a fictitious load at Q A as in Fig. 4.2. Let be the deflection below a u A . Now the strain energy s U stored in the determinate structure (i.e. the support A removed) is given by, a n n S Qu u P u P u P U 2 1 2 1 .......... 2 1 2 1 2 2 1 1 + + + + = (4.1) It is known that the displacement below point is due to action of acting at respectively and due to Q at 1 u 1 P 12 , ,...., n PP P n x x x ,......, , 2 1 A . Hence, may be expressed as, 1 u (4.2) 1111 122 1 1 .......... nn a uaP aP aP aQ =+ + + + where, is the flexibility coefficient at due to unit force applied at ij a i j . Similar equations may be written for . Substituting for in equation (4.1) from equation (4.2), we get, 23 , ,...., and n uu u u a 23 , ,...., and na uu u u 1 11 1 12 2 1 1 2 21 1 22 2 2 2 1 1 22 1 1 22 11 [ ... ] [ ... ] ....... 22 11 [ ... ] [ .... ] 22 Snna nn n n n nn n na a a an n aa U PaP a P a P aQ P a P a P a P a Q Pa P a P a P a Q Qa P a P a P a Q =+ ++ + + + + + + ++ + + + + ++ + a (4.3) Taking partial derivative of strain energy s U with respect to Q, we get deflection atA . 11 2 2 ........ s aa ann aa U aP a P a P a Q Q ? =+ + + + ? (4.4) Substitute as it is fictitious in the above equation, 0 Q = 11 2 2 ........ s aa a an U uaP aP aP Q ? == + + + ? n (4.5) Now the strain energy stored in the beam due to redundant reaction A R is, 23 6 a r R L U EI = (4.6) Now deflection at A due to a R is 3 3 a r a a R L U u R EI ? =- = ? (4.7) The deflection due to should be in the opposite direction to one caused by superposed loads , so that the net deflection at a R 12 , ,...., n PP P A is zero. From equation (4.5) and (4.7) one could write, r a a U Us u QR ? ? ==- ? ? (4.8) Since is fictitious, one could as well replace it by Q a R . Hence, () sr a UU R 0 ? + = ? (4.9) or, 0 a U R ? = ? (4.10) This is the statement of theorem of least work. Where U is the total strain energy of the beam due to superimposed loads and redundant reaction . 12 , ,...., n PP P a R Example 4.1 Find the reactions of a propped cantilever beam uniformly loaded as shown in Fig. 4.3a. Assume the flexural rigidity of the beam EI to be constant throughout its length.Read More

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