Tarea 2.pdfTarea 2.pdfTarea 2.pdfTarea 2.pdf, Ejercicios de Gerontología

¡Descarga Tarea 2.pdfTarea 2.pdfTarea 2.pdfTarea 2.pdf y más Ejercicios en PDF de Gerontología solo en Docsity!

PROBLEMAS • 205

*5-40. La flecha sólida de acero DF tiene un diámetro de 25 mm y está soportada por dos cojinetes lisos en D y en E. Está acoplada a un motor en C que entrega 12 kW de potencia a la flecha cuando gira a 50 rpm. Si los engranes A, B y C toman 3 kW, 4 kW y 5 kW, respectivamente, de- termine el esfuerzo cortante máximo desarrollado en la flecha en las regiones CF y BC. La flecha puede girar li- bremente en sus cojinetes de apoyo D y E.

5-41. Determine el esfuerzo cortante máximo absoluto generado en la flecha en el problema 5-40.

5-43. El motor entrega en A 50 hp cuando gira a una ve- locidad angular constante de 1350 rpm. Por medio del sis- tema de banda y polea esta carga es entregada a la flecha BC de acero del ventilador. Determine al 18 pulg más cer- cano el diámetro mínimo que puede tener esta flecha si el esfuerzo cortante permisible para el acero es (^) perm  12 klb/pulg 2.

5-42. El motor entrega 500 hp a la flecha AB de acero que es tubular y tiene un diámetro exterior de 2 pulg y un diámetro interior de 1.84 pulg. Determine la velocidad an- gular más pequeña a la que puede girar si el esfuerzo cor- tante permisible del material es (^) perm  25 klb/pulg^2.

5-39. La flecha sólida AC tiene un diámetro de 25 mm y está soportada por dos cojinetes lisos en D y E. Está aco- plada a un motor en C que suministra 3 kW de potencia a la flecha cuando gira a 50 rpm. Si los engranes A y B to- man 1 kW y 2 kW, respectivamente, determine el esfuerzo cortante máximo desarrollado en la flecha en las regiones AB y BC. La flecha puede girar libremente en sus cojine- tes de apoyo D y E.

A D B E C

1 kW

2 kW 3 kW 25 mm

Prob. 5-

A D C E F

4 kW

5 kW 12 kW 3 kW 25 mm

B

Probs. 5-40/

4 pies

A

B C

2 pies

Prob. 5-

A

B

Prob. 5-

Review Problems

3.55 The torque T is applied to the solid shaft of radius r 2. Determine the radius r 1 of the inner portion of the shaft that carries one-half of the torque.

3.56 The solid aluminium shaft ABCD carries the three torques shown. (a) Determine the smallest safe diameter of the shaft if the allowable shear stress is 15 ksi. (b) Compute the angle of rotation of end A of the shaft using G ¼ 4  106 psi.

3.57 A circular tube of outer diameter D is slipped over a 40-mm-diameter solid cylinder. The tube and cylinder are then welded together. For what value of D will the torsional strengths of the tube and cylinder be equal?

3.58 A solid steel shaft 4 m long is stressed to 70 MPa when twisted through 3. (a) Given that G ¼ 83 GPa, find the diameter of the shaft. (b) What power does this shaft transmit when running at 18 Hz?

3.59 Determine the maximum torque that can be applied to a hollow circular steel shaft of 100-mm outer diameter and 80-mm inner diameter. The shear stress is lim- ited to 70 MPa, and the angle of twist must not exceed 0.4 ^ in a length of 1.0 m. Use G ¼ 83 GPa for steel.

3.60 A 2-in.-diameter steel shaft rotates at 240 rev/min. If the shear stress is lim- ited to 12 ksi, determine the maximum horsepower that can be transmitted at that speed.

3.61 The compound shaft, consisting of steel and aluminum segments, carries the two torques shown in the figure. Determine the maximum permissible value of T subject to the following design conditions: tst a 83 MPa, tal a 55 MPa, and y a 6  (y is the angle of rotation of the free end). Use G ¼ 83 GPa for steel and G ¼ 28 GPa for aluminum.

3.62 The four gears are attached to a steel shaft that is rotating at 2 Hz. Gear B supplies 70 kW of power to the shaft. Of that power, 20 kW are used by gear A, 20 kW by gear C, and 30 kW by gear D. (a) Find the uniform shaft diameter if the shear stress in the shaft is not to exceed 60 MPa. (b) If a uniform shaft diameter of 100 mm is specified, determine the angle by which one end of the shaft lags behind the other end. Use G ¼ 83 GPa for steel.

T

r 1 r 2

FIG. P3.

4000 in.

2400 in.

1600 in.

A

B

C

D

36 in.

30 in.

15 in.

FIG. P3.

40 mm (^) D

FIG. P3.

FIG. P3.61 (^) FIG. P3.

Review Problems 103

3.63 The composite shaft consists of a copper rod that fits loosely inside an alu- minum sleeve. The two components are attached to a rigid wall at one end and joined with an end-plate at the other end. Determine the maximum shear stress in each material when the 2-kN  m torque is applied to the end-plate. Use G ¼ 26 GPa for aluminum and G ¼ 47 GPa for copper.

3.64 The torque T is applied to the solid shaft with built-in ends. (a) Show that the reactive torques at the walls are TA ¼ Tb=L and T (^) C ¼ Ta=L. (b) How would the re- sults of Part (a) change if the shaft were hollow?

3.65 A flexible shaft consists of a 0.20-in.-diameter steel rod encased in a sta- tionary tube that fits closely enough to impose a torque of intensity 0.50 lb  in./in. on the rod. (a) Determine the maximum length of the shaft if the shear stress in the rod is not to exceed 20 ksi. (b) What will be the relative angular rotation between the ends of the rod? Use G ¼ 12  10 6 psi for steel.

3.66 The shaft ABC is attached to rigid walls at A and C. The torque T 0 is dis- tributed uniformly over segment AB of the shaft. Determine the reactions at A and C.

3.67 A torque of 400 lb  ft is applied to the square tube with constant 0.10-in. wall thickness. Determine the smallest permissible dimension a if the shear stress is limited to 6500 psi.

3.68 The cross section of a brass tube is an equilateral triangle with a constant wall thickness, as shown in the figure. If the shear stress is limited to 8 ksi and the angle of twist is not to exceed 2^ per foot length, determine the largest allowable torque that can be applied to the tube. Use G ¼ 5 : 7  10 6 psi for brass.

3.69 A torsion member is made by placing a circular tube inside a square tube, as shown, and joining their ends by rigid end-plates. The tubes are made of the same material and have the same constant wall thickness t ¼ 5 mm. If a torque T is applied to the member, what fraction of T is carried by each component?

3.70 A 3-m-long aluminum tube with the cross section shown carries a 200-N  m torque. Determine (a) the maximum shear stress in the tube; and (b) the relative angle of rotation of the ends of the tube. For aluminum, use G ¼ 28 GPa.

FIG. P3.63 FIG. P3.

FIG. P3.

FIG. P3.67 FIG. P3.68 (^) FIG. P3.69 FIG. P3.

104 CHAPTER 3 Torsion

5.102 The figure shows the cross section of a fabricated wood beam. If the work- ing stress in shear for the wood is 750 psi, determine the largest allowable vertical shear force that can be carried by the beam.

5.103 The structure consists of the hollow rectangular beam ABC to which is welded the circular bent bar BDE. Calculate the maximum bending stress in the structure.

Dimensions in mm

500 500 500 50

E a b

a

D

B

A

b

C

60

70

75

15 kN

80

Section a - a

Section b - b

FIG. P5.

5.104 The beam shown in cross section is fabricated by joining two 150-mm by 150-mm wooden boards with 20-mm-thick plywood strips. Knowing that the work- ing shear stress for plywood is 2 MPa, determine the maximum allowable vertical shear force that can be carried by the beam.

5.105 The stepped beam has a rectangular cross section 2 in. wide. Determine the maximum bending stress in the beam due to the 3600-lb  ft couple.

5.106 Determine the magnitude and location of the maximum bending stress for the beam.

5.107 Determine the maximum tensile and compressive bending stresses in the beam.

FIG. P5.

5.108 The overhanging beam carries concentrated loads of magnitudes P and 2P. If the bending working stresses are 15 ksi in tension and 18 ksi in compression, de- termine the largest allowable value of P.

FIG. P5.

0.75 in. 6 in. 0.75 in.

6 in.

12 in.

2 in.

FIG. P5.

Dimensions in mm

150

150

150

75

75

20

FIG. P5.

FIG. P5.

2500 lb 12 kip · ft

4 in.

6 in.

FIG. P5.

Review Problems 189

5.109 Find the lightest S-shape for the overhanging beam if the working stress in bending is 18 ksi. What is the actual maximum bending stress in the beam selected?

5.110 The S380  74 section carries a uniformly distributed load totaling 3W and a concentrated load W. Determine the largest value of W if the working stress in bending is 120 MPa.

5.111 The cast iron beam in the figure has an overhang of length b ¼ 1 :0 m at each end. If the bending working stresses are 20 MPa in tension and 80 MPa in compression, what is the largest allowable intensity w 0 of a distributed load that can be applied to the beam? Assume that w 0 includes the weight of the beam.

FIG. P5.111, P5.

5.112 Solve Prob. 5.111 using b ¼ 3 m, all other data remaining unchanged.

5.113 A simply supported 20-ft-long beam carries a uniformly distributed load of intensity 800 lb/ft over its entire length. Find the lightest S-shape that can be used if the working stress in bending is 18 ksi. What is the actual stress in the beam selected?

5.114 The working stress in bending for the simply supported beam is 120 MPa. Find the lightest W-shape that can be used, and calculate the actual maximum bending stress in the beam selected.

5.115 The vertical shear force acting on the cross section shown in the figure is 60 kN. Determine the maximum shear stress on the section.

5.116 The cross section of a beam is formed by gluing two pieces of wood to- gether as shown. If the vertical shear force acting on the section is 60 kN, determine the shear stress (a) at the neutral axis; and (b) on the glued joint.

5000 lb 5000 lb

6 ft 10 ft 6 ft

FIG. P5.109 FIG. P5.

FIG. P5.114 (^) FIG. P5.

FIG. P5.

190 CHAPTER 5 Stresses in Beams

5.122 A simply supported beam is made of four 2-in. by 6-in. wood planks that are fastened by screws as shown in the cross section. The beam carries a concentrated load at the middle of its 12-ft span that causes a maximum bending stress of 1400 psi in the wood. (a) Determine the maximum shear stress in the wood. (b) Find the largest allowable spacing of screws if the shear force in each screw is limited to 200 lb.

5.123 A beam is fabricated by bolting together two W200  100 sections as shown. The beam supports a uniformly distributed load of intensity w 0 ¼ 30 kN/m on its 10-m simply supported span. (a) Determine the maximum bending stress in the beam. (b) If the allowable shear force in each bolt is 30 kN, calculate the largest permissible spacing of the bolts.

5.124 Two C100  10 :8 channels are joined to 120-mm by 7.5-mm plates with 10-mm rivets to form the beam shown in the figure. The beam carries a uniformly distributed loading of intensity w 0 over its 4-m simply supported span. (a) If the working bending stress is 120 MPa, find the largest allowable value of w 0. (b) Determine the largest allowable spacing of rivets using 80 MPa as the working stress for rivets in shear.

Computer Problems

C5.1 The symmetric cross section of a beam consists of three rectangles of dimen- sions bi by h (^) i (i ¼ 1 ; 2 ; 3), arranged on top of one another as shown. A bending mo- ment of magnitude M acts on the cross section about a horizontal axis. Given the values of bi, hi, and M, write an algorithm that computes the maximum bending stress acting on the cross section. Apply the algorithm to the cross sections and mo- ments shown in parts (a) and (b) of the figure.

FIG. C5.

FIG. P5.

FIG. P5.

FIG. P5.

192 CHAPTER 5 Stresses in Beams

Problems

6.1 For the simply supported beam carrying the concentrated load P at its midspan, determine (a) the equation of the elastic curve; and (b) the maximum dis- placement.

6.2 The simply supported beam carries a uniformly distributed load of intensity w 0. Determine (a) the equation of the elastic curve; and (b) the maximum displace- ment.

6.3 The intensity of the distributed load on the cantilever beam varies linearly from zero to w 0. Derive the equation of the elastic curve.

6.4 The simply supported beam carries two end couples, each of magnitude M 0 but oppositely directed. Find the location and magnitude of the maximum deflection.

6.5 Solve Prob. 6.4 if the couple M 0 acting at the left support is removed.

6.6 Compute the location and maximum value of EI d for the simply supported beam carrying the couple M 0 at the midspan. (Hint: By skew-symmetry, the de- flection at midspan is zero.)

6.7 Determine the value of EI d at midspan of the simply supported beam. Is the deflection up or down?

6.8 Determine the maximum deflection of the rectangular wood beam when it is loaded by the two end couples. Use E ¼ 12 GPa.

FIG. P6.

FIG. P6.2 (^) FIG. P6.

FIG. P6.4, P6.5 FIG. P6.

60 lb/ft

10 ft

A B 1200 lb ⋅ ft

FIG. P6.

3.6 m

9.6 kN ⋅ m 6.6 kN ⋅ m (^) 160 mm

120 mm

FIG. P6.

Problems 207

6.30 The simply supported beam carries three concentrated loads as shown in the figure. Determine (a) the equation of the elastic curve; and (b) the value of EI d at midspan.

6.31 For the overhanging beam, compute the value of EI d under the 15-kN load.

6.32 Determine the displacement midway between the supports for the overhang- ing beam.

6.33 For the overhanging beam, find the displacement at the left end.

6.34 For the overhanging beam, determine (a) the value of EI d under the 24-kN load; and (b) the maximum value of EI d between the supports.

6.35 Compute the value of EI d at the left end of the cantilever beam.

6.36 The cantilever beam carries a couple formed by two forces, each of magni- tude P ¼ 2000 lb. Determine the force R that must be applied as shown to prevent displacement of point A.

6.37 Find the maximum displacement of the cantilever beam.

6.38 Compute the value of EI d at the right end of the cantilever beam.

4 kips 8 kips

3 ft 3 ft 3 ft 3 ft

6 kips

FIG. P6.

FIG. P6.31 (^) FIG. P6.

FIG. P6.

FIG. P6.34 FIG. P6.

FIG. P6.

FIG. P6.

2 2

6 ft 3 3

FIG. P6.

218 CHAPTER 6 Deflection of Beams

Problems

6.45 Solve Sample Problem 6.10 by introducing a built-in support at A rather than at C.

6.46 For the cantilever beam ABC, compute the value of EI d at end C.

6.47 The cantilever beam ABC has the rectangular cross section shown in the figure. Using E ¼ 69 GPa, determine the maximum displacement of the beam.

6.48 The properties of the timber cantilever beam ABC are I ¼ 60 in. 4 and E ¼ 1 : 5  10 6 psi. Determine the displacement of the free end A.

6.49 For the beam described in Prob. 6.48, compute the displacement of point B.

6.50 The cantilever beam AB supporting a linearly distributed load of maximum intensity w 0 is propped at end A by the force P. (a) Find the value of P for which the deflection of A is zero. (b) Compute the corresponding value of EI y at A.

6.51 Determine the magnitude of the couple M 0 for which the slope of the beam at A is zero.

6.52 Compute the value of EI d at point B for the simply supported beam ABC.

6.53 For the simply supported beam ABCD, determine the values of EI d at (a) point B; and (b) point C.

6 2 4

4

FIG. P6.46 (^) FIG. P6.

FIG. P6.48, P6.49 FIG. P6.

12

3

M 0

3

FIG. P6.

80 lb/ft

FIG. P6.52 (^) FIG. P6.

Problems 233

6.63 The overhanging beam ABCD carries the uniformly distributed load of in- tensity 200 lb/ft over the segments AB and CD. Find the value of EI d at point B.

6.64 Determine the value of EI d at point A of the overhanging beam ABC.

6.65 The two segments of the cantilever beam ABC have di¤erent cross sections with the moments of inertia shown in the figure. Determine the expression for the maximum displacement of the beam.

6.66 The simply supported beam ABC contains two segments. The moment of inertia of the cross-sectional area for segment AB is three times larger than the mo- ment of inertia for segment BC. Find the expression for the displacement for point B.

6.67 Calculate the value of EI d at point B of the simply supported beam ABC.

6.5 Method of Superposition

The method of superposition, a popular method for finding slopes and de-

flections, is based on the principle of superposition:

If the response of a structure is linear, then the e¤ect of several loads

acting simultaneously can be obtained by superimposing (adding) the

e¤ects of the individual loads.

By ‘‘linear response’’ we mean that the relationship between the cause

(loading) and the e¤ect (deformations and internal forces) is linear. The two

requirements for linear response are (1) the material must obey Hooke’s law;

and (2) the deformations must be su‰ciently small so that their e¤ect on the

geometry is negligible.

4 ft

200 200

4 ft 4 ft

FIG. P6.

FIG. P6.64 (^) FIG. P6.

FIG. P6.66 FIG. P6.

6.5 Method of Superposition 235

The method of superposition permits us to use the known displace-

ments and slopes for simple loads to obtain the deformations for more com-

plicated loadings. To use the method e¤ectively requires access to tables that

list the formulas for slopes and deflections for various loadings, such as

Tables 6.2 and 6.3. More extensive tables can be found in most engineering

handbooks.

d ¼

Px^2

6 EI

ð 3 L  xÞ dB ¼

PL^3

3 EI

yB ¼

PL^2

2 EI

d ¼

Px^2

6 EI

ð 3 a  xÞ 0 a x a a

Pa^2

6 EI

ð 3 x  aÞ a a x a L

dB ¼

Pa^2

6 EI

ð 3 L  aÞ yB ¼

Pa^2

2 EI

d ¼

w 0 x^2

24 EI

ð 6 L^2  4 Lx þ x^2 Þ dB ¼

w 0 L^4

8 EI

yB ¼

w 0 L^3

6 EI

d ¼

w 0 x^2

24 EI

ð 6 a^2  4 ax þ x^2 Þ 0 a x a a

w 0 a^3

24 EI

ð 4 x  aÞ a a x a L

dB ¼

w 0 a^3

24 EI

ð 4 L  aÞ yB ¼

w 0 a^3

6 EI

d ¼

w 0 x^2

120 L EI

ð 10 L^3  10 L^2 x þ 5 Lx^2  x^3 Þ dB ¼

w 0 L^4

30 EI

yB ¼

w 0 L^3

24 EI

d ¼

M 0 x^2

2 EI

dB ¼

M 0 L^2

2 EI

yB ¼

M 0 L

EI

TABLE 6.2 Deflection Formulas for Cantilever Beams

236 CHAPTER 6 Deflection of Beams

Problems

7.1 Find all the support reactions for the propped cantilever beam that carries the couple M 0 at the propped end.

7.2 Determine the support reaction at A for the propped cantilever beam due to the triangular loading shown in the figure.

7.3 The beam carrying the 1200 lb  ft couple at its midpoint is built in at both ends. Find all the support reactions. (Hint: Utilize the skew-symmetry of deformation about the midpoint.)

7.4 A concentrated load is applied to the beam with built-in ends. (a) Find all the support reactions; and (b) draw the bending moment diagram. (Hint: Use symmetry.)

7.5 The beam with three supports carries a uniformly distributed load. Determine all the support reactions. (Hint: Use symmetry.)

24 kN m

6 m

FIG. P7.

FIG. P7.

1200

4 ft 4 ft

FIG. P7.

FIG. P7.

180

8 ft 8 ft FIG. P7.

Problems 255

Problems

7.9 Determine all the support reactions for the propped cantilever beam shown in the figure.

7.10 For the beam with built-in ends, determine (a) all the support reactions; and (b) the displacement at midspan.

7.11 Find the support reaction at A for the propped cantilever beam.

7.12 Determine all the support reactions for the beam with built-in ends.

7.13 For the beam with built-in ends, determine (a) all the support reactions; and (b) the displacement at the midpoint C. (Hint: Use symmetry.)

7.14 Determine all the support reactions for the beam with built-in ends.

7.15 Find all the support reactions for the beam shown in the figure.

FIG. P7.

FIG. P7.

24 kN 30 kN 3 m 3 m 2 m

FIG. P7.

FIG. P7.

FIG. P7.

2 m 4

24

FIG. P7.

3 ft 6 ft

30 kips

3 ft

FIG. P7.

Problems 259