Page 1 CASTIGLIANO’S FIRST THEOREM ? “When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.” 2 i i F U ? ? ? ? Where: F i = Force at i-th application point d i = Displacement at i-th point in the direction of F i U = Total strain energy Page 2 CASTIGLIANO’S FIRST THEOREM ? “When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.” 2 i i F U ? ? ? ? Where: F i = Force at i-th application point d i = Displacement at i-th point in the direction of F i U = Total strain energy CASTIGLIANO’S THEOREM ? We can also use this method to find the angle of rotation ( ?). 3 Where: M i = Moment at i-th application point ? i = Slope at i-th point resulting from M i U = Total strain energy i i M U ? ? ? ? Page 3 CASTIGLIANO’S FIRST THEOREM ? “When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.” 2 i i F U ? ? ? ? Where: F i = Force at i-th application point d i = Displacement at i-th point in the direction of F i U = Total strain energy CASTIGLIANO’S THEOREM ? We can also use this method to find the angle of rotation ( ?). 3 Where: M i = Moment at i-th application point ? i = Slope at i-th point resulting from M i U = Total strain energy i i M U ? ? ? ? CASTIGLIANO’S THEOREM 4 U stored in structure F 1 F 2 F 3 F n d 1 d 2 d 3 d n 1 1 F U ? ? ? ? 2 2 F U ? ? ? ? 3 3 F U ? ? ? ? n n F U ? ? ? ? Page 4 CASTIGLIANO’S FIRST THEOREM ? “When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.” 2 i i F U ? ? ? ? Where: F i = Force at i-th application point d i = Displacement at i-th point in the direction of F i U = Total strain energy CASTIGLIANO’S THEOREM ? We can also use this method to find the angle of rotation ( ?). 3 Where: M i = Moment at i-th application point ? i = Slope at i-th point resulting from M i U = Total strain energy i i M U ? ? ? ? CASTIGLIANO’S THEOREM 4 U stored in structure F 1 F 2 F 3 F n d 1 d 2 d 3 d n 1 1 F U ? ? ? ? 2 2 F U ? ? ? ? 3 3 F U ? ? ? ? n n F U ? ? ? ? EXAMPLE: ? CALCULATE AND FOR THE BEAM SHOWN IN FIGURE. TAKE E= AND I= B ? B ? 5 2 10 ? 8 5 10 ? 5 Page 5 CASTIGLIANO’S FIRST THEOREM ? “When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.” 2 i i F U ? ? ? ? Where: F i = Force at i-th application point d i = Displacement at i-th point in the direction of F i U = Total strain energy CASTIGLIANO’S THEOREM ? We can also use this method to find the angle of rotation ( ?). 3 Where: M i = Moment at i-th application point ? i = Slope at i-th point resulting from M i U = Total strain energy i i M U ? ? ? ? CASTIGLIANO’S THEOREM 4 U stored in structure F 1 F 2 F 3 F n d 1 d 2 d 3 d n 1 1 F U ? ? ? ? 2 2 F U ? ? ? ? 3 3 F U ? ? ? ? n n F U ? ? ? ? EXAMPLE: ? CALCULATE AND FOR THE BEAM SHOWN IN FIGURE. TAKE E= AND I= B ? B ? 5 2 10 ? 8 5 10 ? 5 For deflection at B Segment Origin Limits Mx BC B 0 to 1 -P.x -x CA C 0 to 3 -P(x+1)-40x -(x + 1) x M P ? ?? ?? ? ?? 6Read More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

30 videos|122 docs|28 tests