Deformation

Let me tell you the secret that has led me to my goal. My strength lies in my tenacity.Louis Pateur

Image: High speed photographs of a club striking a golf ball. Note the large deformations in the golf ball.

Normal Stress

Figure 5.6 Element subjected to normal stress

text reference: Figure 5.6, page 195

Shear Stress

Figure 5.7 Element subjected to shear stress.

text reference: Figure 5.7, page 197

Twisting due to Applied Torque

Figure 4.11 Twisting of member due to applied torque.

text reference: Figure 4.11, page 152

l

r

Hipotesis de Coulomb: secciones transversales circulares, permanecen planas.

Principio de Saint Venant: secciones transversales no circulares.

Example 5.1

Figure 5.1 Cantilevered beam with concentrated force applied at free end. Used in Example 5.1.

text reference: Figure 5.1, page 184

Examen septiembre 2006

1/ Un resorte de una puerta tiene sección rectangular 50x6 (mm) y longitud 800 mm. Uno de sus extremos se haya empotrado y en el otro se producen deflexiones entre 75 y 150 mm cíclicamente. Determinar el coeficiente de seguridad en la sección más peligrosa en: a) caso citado b) si a L/2 se le practica un taladro de O 12,5mm. Condiciones de carga de (a)Material estirado en frio, E=210Gpa, Su=1034MPa y ν= 0,3 Fiabilidad 90% aT=20ºC…………3 puntos

text reference: Figure 5.1, page 184

Cantilever with Moment and Concentrated Force

Figure 5.5 Bar fixed at one end and free at other with moment applied to free end and concentrated force at any distance from free end. (a) Complete assembly; (b) free-body diagram showing effect of concentrated force; (c) free-body diagram showing effect of moment.

text reference: Figure 5.5, page 194

zEI

zM

dxdy

dxyd

)(

12

32

2

2

Example 5.8

Figure 5.8 Cantilevered bar with concentrated force acting at distance b from free end. (a) Coordinate system and important points shown; (b) fictitious force shown along with concentrated force.

text reference: Figure 5.8, page 203

Free Body Diagram of Simply Supported

Beam

Figure 5.2 Free-body diagram of force anywhere between simply supported ends. (a) Complete bar; (b) portion of bar.

text reference: Figure 5.2, page 187

Cantilevered Bar with Partial

Distributed Load

Figure 5.3 Cantilevered bar with unit step distribution over part of bar. (a) Loads and deflection acting on cantilevered bar; (b) free-body diagram of forces and moments acting on entire bar; (c) free-body diagram of forces and moments acting on portion of bar.

text reference: Figure 5.3, page 189

Cantilever with Simply Supported End

Figure 5.4 Cantilevered bar with other end simply supported and with concentrated force acting anywhere along bar. (a) Sketch of assembly; (b) free-body diagram of entire bar; (c) free-body diagram of part of bar.

text reference: Figure 5.4, page 191

Deflection of Beams

Table 5.1 Deflection for three different situations when one end is fixed and one end is free and two different situations of simply supported beams.

text reference: Table 5.1, page 193

Deflection of Beams (cont.)

Table 5.1 Deflection for three different situations when one end is fixed and one end is free and two different situations of simply supported beams.

text reference: Table 5.1, page 193

Método del momento de área

dxAEIxA

dxxMx

EIH )(

1

)(

)(1

La distancia vertical entre un punto cualquiera A en la curva de flexión y la tangente a la curva en otro punto B es igual al momento estático con respecto a A del área del diagrama del momento flexionante entre A y B, dividido entre la rigidez.

Strain EnergyAplicación: Método de Castigliano

Loading type Factors involved

Strain energy forspecial case where

al l three factorsare constant with x

Generalexpression forstrain energy

Axial P, E, A

U

P2l

2EA U

P2

2EAdx

0

l

Bending M, E, I

U

M 2

2EI

U M 2

2EIdx

0

l

Torsion T, G, J

U

T2l2GJ

U 2

2GJdx

0

l

Transverse shear(rectangular section)

V, G, A

U

3V2l

5GA U

3V2

5GAdx

0

l

Table 5.2 Strain energy for four types of loading.

text reference: Table 5.2, page 200

Example 5.9

Figure 5.9 Cable system arrangement. (a) Entire assembly; (b) free-body diagram of forces acting at point A.

text reference: Figure 5.9, page 204

Example 5.10

Figure 5.10 Cantilevered bar with 90° bend acted upon by horizontal force at free end.

text reference: Figure 5.10, page 205

Design of FOPS

Figure 5.11 Figures used in case study. (a) Forklift truck; (b) FOPS details (not to scale); (c) idealized FOPS for analysis.

text reference: Figure 5.11, page 207

FOPS Cross Brace Analysis

Figure 5.12 (a) Free-body; (b) shear; and (c) bending moment diagrams for FOPS cross brace.

text reference: Figure 5.12, page 208