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

Load Classification

text reference: Figure 2.2, page 31

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.

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

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

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

text reference: Figure 2.12, page 48

10mm

6mm

25mm

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

Mohrs Circle

Figure 2.17 Mohrs 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) Mohrs 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

Mohrs Circle for Triaxial Stress State

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

text reference: Figure 2.19, page 59

Example 3.5

Figure 2.20 Mohrs 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.

text reference: Figure 4.11, page 152

lr

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.

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

Figure 4.14 Undeformed and deformed elements in bending.

text reference: Figure 4.14, page 157

Elements in Bending

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

text reference: Figure 4.17, page 161

rdrr n )(

Curved Member in Bending

0)( dAr rrEddA A nA

rdrrEE n )(

r

drr n )(

Condicin: sumatorio de esfuerzos en el rn=0

A

nA

n

rdAAr

rdArA 0

Curved Member in Bending

dArA

rA 1_

rdrrEE n )(

)(_

nrre

A

n

rdAAr

AerMc

rrEAerM

n

AeEdArrdAEdr

dArArArrdAEd

dAr

rrrrEddArrrEddArrM

An

Annn

A

nn

A

n

An

2

222 )2()())((

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 seccin transversal rectangular de un elemento curvo, tiene las dimensiones:

b= 1 y h=r0-ri=3, sometida a un momento de flexin 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 seccin trapezoidal de un elemento curvo, tiene las dimensiones:

ri=10 cm

F= 125 kg

Tadm=1380 Kg/cm2

Hallar: valor de a.

text reference: Figure 4.18, page 162

01

01 23 bb

bbhrr in

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

BIBLIOGRAFIAde referencia

Bernard J. Hamrock, Elementos de mquinas. Ed. Mc Graw Hill.

Robert L. Norton, Diseo de mquinas. Ed. Prentice Hall.

Shigley, Diseo en Ingeniera Mecnica, Ed. Mc Graw-Hill