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a. Show that the best material for a cantilever beam of given length L and given (i.e., fixed) square cross-section (t × t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). (See Figure E.4(a).) b. Show that the best material choice for a cantilever beam of given a length L and with a given section (t x t) that will deflect least under its own self- weight is that with the largest value of M = E/p, where p is the density. (Figure E.4(b).)

a. Show that the best material for a cantilever beam of given length L and given (i.e., fixed) square cross-section (t × t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). (See Figure E.4(a).) b. Show that the best material choice for a cantilever beam of given a length L and with a given section (t x t) that will deflect least under its own self- weight is that with the largest value of M = E/p, where p is the density. (Figure E.4(b).)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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and that for the deflection of a beam under a distributed load f per unit length: 1 f Lª 8 : 8 EI where I = t/12. For a self-loaded beam f= p Ag, where p is the density of the material of the beam, A its cross-sectional area, and g the acceleration due to gravity. a. Show that the best material for a cantilever beam of given length L and given (i.e., fixed) square cross-section (t × t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). (See Figure E.4(a).) b. Show that the best material choice for a cantilever beam of given a length L and with a given section (t × t) that will deflect least under its own self- weight is that with the largest value of M = E/p, where p is the density. (Figure E.4(b).) c. Show that the material index for the lightest cantilever beam of length L and square section (not given, i.e., the area is a free variable) that will not deflect by more than ô under its own weight is M = E/p² . (See Figure E.4(c).) d. Show that the lightest cantilever beam of length L and square section (area free) that will not deflect by more than ô under an end load F is that made of the material with the largest value of M = E'/2/p (neglect self weight). (See Figure E.4(d).)
Transcribed Image Text:and that for the deflection of a beam under a distributed load f per unit length: 1 f Lª 8 : 8 EI where I = t/12. For a self-loaded beam f= p Ag, where p is the density of the material of the beam, A its cross-sectional area, and g the acceleration due to gravity. a. Show that the best material for a cantilever beam of given length L and given (i.e., fixed) square cross-section (t × t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). (See Figure E.4(a).) b. Show that the best material choice for a cantilever beam of given a length L and with a given section (t × t) that will deflect least under its own self- weight is that with the largest value of M = E/p, where p is the density. (Figure E.4(b).) c. Show that the material index for the lightest cantilever beam of length L and square section (not given, i.e., the area is a free variable) that will not deflect by more than ô under its own weight is M = E/p² . (See Figure E.4(c).) d. Show that the lightest cantilever beam of length L and square section (area free) that will not deflect by more than ô under an end load F is that made of the material with the largest value of M = E'/2/p (neglect self weight). (See Figure E.4(d).)
E7.2 Material indices for elastic beams with differing constraints (Figure E.4) Start each of the four parts of this problem by listing the function, the objective, and the constraints. You will need the equations for the deflection of a cantilever beam with a square cross-section t x t, given in Appendix B.3. The two that matter are that for the deflection & of a beam of length L under an end load F: FL³ ЗЕ Fixed (a) F, 8 Force f per unit length Fixed (b) Force f per unit length Free (c) Free (d) F, 8 FIGURE E.4
Transcribed Image Text:E7.2 Material indices for elastic beams with differing constraints (Figure E.4) Start each of the four parts of this problem by listing the function, the objective, and the constraints. You will need the equations for the deflection of a cantilever beam with a square cross-section t x t, given in Appendix B.3. The two that matter are that for the deflection & of a beam of length L under an end load F: FL³ ЗЕ Fixed (a) F, 8 Force f per unit length Fixed (b) Force f per unit length Free (c) Free (d) F, 8 FIGURE E.4
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