Tema2 esfuerzos

BIBLIOGRAFIA

• Bernard J. Hamrock, Elementos de máquinas. Ed. Mc Graw Hill.

• Robert L. Norton, Diseño de máquinas. Ed. Prentice Hall.

• Shigley, Diseño en Ingeniería Mecánica, Ed. Mc Graw-Hill

Load, Stress and Strain

When I am working on a problem, I never thinkabout beauty. I only think of how to solve theproblem. But when I have finished, if the solutionis not beautiful, I know it is wrong.

Richard Buckminster Fuller

Image: A dragline lifts a large load in a mining operation.

A Simple Crane

Figure 2.1 A simple crane and forces acting on it. (a) Assembly drawing; (b) free-body diagram of forces acting on the beam.

text reference: Figure 2.1, page 30

Supports and Reactions

Table 2.1: Four types of support with their corresponding reactions.

text reference: Table 2.1, page 35

Ladder Free Body Diagram

Figure 2.5: Ladder having contact with the house and the ground while having a painter on the ladder. Used in Example 2.4. The ladder length is l.


Load Classification


Figure 2.2 Load classified as to location and method of application. (a) Normal, tensile (b) normal, compressive; (c) shear; (d) bending; (e) torsion; (f) combined

Sign Convention

Figure 2.3 Sign convention used in bending. (a) y coordinate upward; (b) y coordinate downward.


Lever Assembly

Figure 2.4 Lever assembly and results. (a) Lever assembly; (b) results showning (1) normal, tensile, (2) shear, (3) bending, (4) torsion on section B of lever assembly.


Beam Supports

Figure 2.8 Three types of beam support. (a) Simply supported; (b) cantilevered; (c) overhanging.


Simply Supported Bar

Figure 2.9 Simply supported bar with (a) midlength load and reactions; (b) free-body diagram for 0<x<l/2; (c) free body diagram for l/2<x<l; (d) shear and bending moment diagrams.


Singularity Functions (Part 1)

Table 2.2 Six singularity and load intensity functions with corresponding graphs and expressions.


Singularity Functions (Part 2)

Table 2.2 Six singularity and load intensity functions with corresponding graphs and expressions.


Shear and Moment Diagrams

Figure 2.10 (a) Shear and (b) moment diagrams for Example 2.8.


Example 2.10

Figure 2.12 Figures used in Example 2.10. (a) Load assembly drawing; (b) free-body diagram.


Ø10mm

Ø6mm

□25mm

Example

Se desea transmitir una potencia de 40 CV a través de un eje que gira a 1500 rpm mediante una chaveta de profundidad máxima= 6 mmy L= 12 mm.Datos: eje macizo de Øext=45.

Example

kgrMtFtrFtMt

cmkgn

CVMt

85025,25,1912

5,19121500

407172071720

2

2

5,11802,16,0

850

2

3,5902,12,1

850

cmkg

LhFt

Ap

Ft

cmkg

LW

Ft

Ac

Ft

d

d

40 CV a 1500 rpm H/2= 6 mmy L= 12 mm.Datos: eje macizo de Øext=45.

86,15,1180

2,2196

7,33,590

2,2196

2,21963800577,0577,0 2

d

ypd

d

Scd

YS

Sn

Sn

cmkgSS

Y

Y

Determinación del coeficiente de seguridad para un material, en este caso AISI1040 615/380

Determinación de cargas

Determinación de esfuerzos

Aplicación Criterio de Fallo

General State of Stress

Figure 2.13 Stress element showing general state of three-dimensional stress with origin placed in center of element.


2-D State of Stress

Figure 2.14 Stress element showing two-dimensional state of stress. (a) Three dimensional view; (b) plane view.


Equivalent Stresses

Figure 2.15 Illustration of equivalent stresss states; (a) Stress element oriented in the direction of applied stress. (b) stress element oriented in different (arbitrary) direction.


Stresses in Oblique Plane

Figure 2.16 Stresses in oblique plane at angle .


Stresses in Oblique Plane

22cos22

senxyyxyx

2cos22 xy

yx sen

2cos22 xy

yx sen

text reference: Shigley pag 28,29

yx

xy

2

2tan0

xy

yx

22tan0

Mohr’s Circle

Figure 2.17 Mohr’s circle diagram of Eqs. (2.13) and (2.14).


Mohr’s Circle Example

Un elemento con el siguiente estado tensional. Se desea: a) hallar los esfuerzos y las direcciones principales e indicar en el elemento su orientación correcta, con respecto al sistema xy. Se trazará otro elemento en que se muestren T1 y T2, determinando los esfuerzos normales correspondientes y marcando los signos letras.

text reference: Shigley, page 31-32

MPa

0

00

05080

Results from Example

Figure 2.18 Results from Example 2.13 (a) Mohr’s circle diagram; (b) stress element for principal normal stresses shown in x-y coordinates; (c) stress element for principal stresses shown in x-y coordinates.


Mohr’s Circle for Triaxial Stress State

Figure 2.19 Mohr’s circle for triaxial stress state. (a) Mohr’s circle representation; (b) principal stresses on two planes.


Example 3.5

Figure 2.20 Mohr’s circle diagram for Example 3.5. (a) Triaxial stress state when 1=23.43 ksi, 2=4.57 ksi, and 3=0; (b) biaxial stress state when 1=30.76 ksi and 2=-2.760 ksi; (c) triaxial stress state when 1=30.76 ksi, 2=0, and 3=-2.76 ksi.


Stresses on Octahedral Planes

Figure 2.21 Stresses acting on octahedral planes. (a) General state of stress. (b) normal stress; (c) octahedral shear stress.


Normal Strain

Figure 2.22 Normal strain of cubic element subjected to uniform tension in x direction. (a) Three dimensional view; (b) two-dimensional (or plane) view.


Shear Strain

Figure 2.23 Shear strain of cubic element subjected to shear stress. (a) Three dimensional view; (b) two-dimensional (or plane) view.


Plain Strain

Figure 2.24 Graphical depiction of plane strain element. (a) Normal strain x; (b) normal strain y; and (c) shear strain xy.


Circular Bar with Tensile Load

Figure 4.10 Circular bar with tensile load applied.


Example


Twisting due to Applied Torque

Figure 4.11 Twisting of member due to applied torque.


l

r

Hipotesis de Coulomb: secciones transversales circulares, permanecen planas.

Principio de Saint Venant: secciones transversales no circulares.

Bending of a Bar

Figure 4.12 Bar made of elastomeric material to illustrate effect of bending. (a) Undeformed bar; (b) deformed bar.


Figure 4.14 Undeformed and deformed elements in bending.


Elements in Bending

Bending Stress Distribution

Figure 4.15 Profile view of bending stress variation.


Las secciones más económicas, serán aquellas que tengan el mayor módulo resistente wz, con el menor gasto de material.

¿Calcular b´ tal que tengan el mismo valor de Wx?

Example 4.10

Figure 4.16 U-shaped cross section experiencing bending moment, used in Example 4.10.


Curved Member in Bending


r

drr n )(


0)(

dAr

rrEddA

A

n

A

r

drrEE n )(

r

drr n )(

Condición: sumatorio de esfuerzos en el rn=0

A

n

A

n

rdA

Ar

r

dArA 0


dArA

rA

1_

r

drrEE n )(

)(_

nrre

A

n

rdA

Ar

Aer

Mc

rr

EAerM

n

AeEd

ArrdAEd

r

dArArArrdA

Ed

dAr

rrrrEddA

r

rrEddArrM

A

n

A

nnn

A

nn

A

n

A

n

2

222 )2()())((

Cross Section of Curved Member

Figure 4.18 Rectangular cross section of curved member.


Example: Cross Section of Curved Member

Una sección transversal rectangular de un elemento curvo, tiene las dimensiones:

b= 1´ y h=r0-ri=3´, sometida a un momento de flexión puro de 20000lbf-pulg.

Hallar:

a) Elemento recto.

b) Elemento curvo. r=15´.

c) Elemento curvo. r=3´.


Tabla de Ganchos

Example: Cross Section of Curved Member

Una sección trapezoidal de un elemento curvo, tiene las dimensiones:

ri=10 cm

F= 125 kg

Tadm=1380 Kg/cm2

Hallar: valor de a.


01

01 2

3 bb

bbhrr in

Development of Transverse Shear

Figure 4.19 How transverse shear is developed.


Maximum Shear Stress

Table 4.3 Maximum shear stress for different beam cross sections.


Tema2 esfuerzos

Automotive

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