Transcript esfuerzos - Universidad de Salamanca

BIBLIOGRAFIA
de referencia
• 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 think
about beauty. I only think of how to solve the
problem. But when I have finished, if the solution
is 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
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
text reference: Figure 2.2, page 31
Sign Convention
Figure 2.3 Sign convention used
in bending. (a) y coordinate
upward; (b) y coordinate
downward.
text reference: Figure 2.3, page 32
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.
text reference: Figure 2.4, page 33
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.
text reference: Figure 2.5, page 36
External Rim Brake and Forces
Figure 2.6 External rim brake and forces acting on it. (a) External rim brake; (b)
external rim brake with forces acting on each part. (Linear dimensions are in
millimeters.)
text reference: Figure 2.6, page 38
Sphere and Forces
Figure 2.7 Sphere and forces acting on it. (a)
Sphere supported with wires from top and a
spring at the bottom; (b) free-body diagram of
forces acting on the sphere. Figure used in
Example 2.6.
text reference: Figure 2.7, page 38
Beam Supports
Figure 2.8 Three types of beam support. (a) Simply supported; (b) cantilevered; (c)
overhanging.
text reference: Figure 2.8, page 39
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.
text reference: Figure 2.9, page 40
Singularity Functions (Part 1)
Table 2.2 Six singularity and load intensity functions with corresponding graphs and
expressions.
text reference: Table 2.2, page 43
Singularity Functions (Part 2)
Table 2.2 Six singularity and load intensity functions with corresponding graphs and
expressions.
text reference: Table 2.2, page 43
Shear and Moment Diagrams
Figure 2.10 (a) Shear and (b) moment diagrams for Example 2.8.
text reference: Figure 2.10, page 44
Simply Supported Beam
Figure 2.11 Simply supported beam. (a) Forces acting on beam when P1=8kN, P2=5kN;
w0=4kN/m; l=12m; (b) free-body diagram showing resulting forces; (c) shear and (d)
moment diagrams of Example 2.9.
text reference: Figure 2.11, page 46
Example 2.10
Ø6mm
□25mm
Ø10mm
Figure 2.12 Figures used in Example 2.10. (a) Load assembly drawing; (b) free-body
diagram.
text reference: Figure 2.12, page 48
Example
text reference: Figure 2.12, page 48
General State of Stress
Figure 2.13 Stress element showing general state of three-dimensional stress with
origin placed in center of element.
text reference: Figure 2.13, page 49
2-D State of Stress
Figure 2.14 Stress element showing two-dimensional state of stress. (a) Three
dimensional view; (b) plane view.
text reference: Figure 2.14, page 51
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.
text reference: Figure 2.15, page 52
Stresses in Oblique Plane
Figure 2.16 Stresses in oblique plane at angle .
text reference: Figure 2.16, page 52
Mohr’s Circle
Figure 2.17 Mohr’s circle
diagram of Eqs. (2.13) and
(2.14).
text reference: Figure 2.17, page 55
Results from Example 2.13
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.
text reference: Figure 2.18, page 57
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.
text reference: Figure 2.19, page 59
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.
text reference: Figure 2.20, page 60
Stresses on Octahedral Planes
Figure 2.21 Stresses acting on octahedral planes. (a) General state of stress. (b)
normal stress; (c) octahedral shear stress.
text reference: Figure 2.21, page 61
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.
text reference: Figure 2.21, page 64
Shear Strain
Figure 2.23 Shear strain of cubic element subjected to shear stress. (a) Three
dimensional view; (b) two-dimensional (or plane) view.
text reference: Figure 2.23, page 65
Plain Strain
Figure 2.24 Graphical depiction of plane strain element. (a) Normal strain x; (b) normal
strain y; and (c) shear strain xy.
text reference: Figure 2.24, page 66
Circular Bar with
Tensile Load
Figure 4.10 Circular bar with
tensile load applied.
text reference: Figure 4.10, page 149
Twisting due to Applied Torque
Figure 4.11 Twisting of member due to
applied torque.
r
 
l
Hipotesis de Coulomb: secciones
transversales circulares, permanecen planas.
Principio de Saint Venant: secciones
transversales no circulares.
text reference: Figure 4.11, page 152
Bending of a Bar
Figure 4.12 Bar made of
elastomeric material to illustrate
effect of bending. (a)
Undeformed bar; (b) deformed
bar.
text reference: Figure 4.12, page 156
Bending in Cantilevered Bar
Figure 4.13 Bending occurring in cantilevered bar, showing neutral surface.
text reference: Figure 4.13, page 157
Elements in Bending
Figure 4.14 Undeformed and deformed
elements in bending.
text reference: Figure 4.14, page 157
Bending Stress Distribution
Figure 4.15 Profile view of bending stress variation.
text reference: Figure 4.15, page 158
Example 4.10
Figure 4.16 U-shaped cross section experiencing bending moment,
used in Example 4.10.
text reference: Figure 4.16, page 159
Curved Member in Bending
(r  rn )d

r
text reference: Figure 4.17, page 161
Curved Member in Bending
E (r  rn )d
  E 
r
(r  rn )d

r
Condición: sumatorio de esfuerzos en el rn=0
 dA 
A
Ed (r  rn )
dA  0
 A r
dA
A
 0  rn 
r
dA
A
A r
A  rn 
Curved Member in Bending
  E 
E (r  rn )d
r
Ed (r  rn ) 2
Ed (r 2  2rrn  rn )
M   (r  rn )(dA) 
dA 
dA 


 A r
 A
r
A
2
 Ed
Ed 
Ed 
2 dA

Ae
 rdA  rn A  rn A  rn   
 rdA  rn A 
  A
r 
  A


rn 
Mc
rEAe
M
 
r  rn
Aer
A
dA
A r
_
1
r   rdA
AA
_
e  (r  rn )
Cross Section of Curved Member
Figure 4.18 Rectangular cross
section of curved member.
text reference: Figure 4.18, page 162
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´.
text reference: Figure 4.18, page 162
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.
h b1  2b0
rn  ri 
3 b1  b0
text reference: Figure 4.18, page 162
Development of Transverse Shear
Figure 4.19 How transverse shear is developed.
text reference: Figure 4.19, page 165
Deformation due to
Transverse Shear
Figure 4.20 Cantilevered bar made of
highly deformable material and
marked with horizontal and vertical
grid lines to show deformation due to
transverse shear. (a) Undeformed; (b)
deformed.
text reference: Figure 4.20, page 166
Moments and Stresses on Elements
Figure 4.21 Three-dimensional and profile views of moments and stresses
associated with shaded top segment of element that has been sectioned at y’ about
neutral axis. (a) Three-dimensional view; (b) profile view.
text reference: Figure 4.21, page 166
Maximum Shear Stress
Table 4.3 Maximum shear
stress for different beam cross
sections.
text reference: Table 4.3, page 168
Strain Gage Rosette
Figure 2.25 Strain gage rosette used
in Example 2.17.
text reference: Figure 2.25, page 68