chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/ch-3-review-load-and-stress-analysis.pdf ·...

Chapter 3

Load and

Stress Analysis

Shear Force and Bending

Moments in Beams

Internal shear force V & bending moment M

must ensure equilibrium

Saturday, September 26,

2015

Fig. 3−2 Mohammad Suliman Abuhaiba, Ph.D., P.E.

2

Sign Conventions for Bending

and Shear

Mohammad Suliman Abuhaiba, Ph.D., P.E.Fig. 3−3

Distributed Load on Beam

Distributed load q(x) = load intensity

Units of force per unit length

Saturday, September 26,

2015

Fig. 3−4 Mohammad Suliman Abuhaiba, Ph.D., P.E.

4

Relationships between Load, Shear,

and Bending

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

5

Plane stress occurs = stresses on one surface

are zero

Saturday, September 26,

2015

Fig. 3−8

Cartesian Stress Components

Mohammad Suliman Abuhaiba, Ph.D., P.E.

6

Plane-Stress Transformation Equations

Saturday, September 26,

2015

Fig. 3−9Mohammad Suliman Abuhaiba, Ph.D., P.E.

7

Principal Stresses for Plane Stress

principal directions

principal stresses

Zero shear stresses at principal surfaces

Third principal stress = zero for plane stress

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

8

Extreme-value Shear Stresses for

Plane Stress

Max shear stresses: on surfaces that are

±45º from principal directions

Two extreme-value shear stresses:

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

9

Mohr’s Circle Diagram

Relation between x-y stresses and principal

stresses

Relationship is a circle with center at

C = (s, t) = [(sx + sy)/2, 0]

Saturday, September 26,

2015

2

2

2

x y

xyRs s

t

Mohammad Suliman Abuhaiba, Ph.D., P.E.

10

Mohr’s

Circle

Diagram

Saturday, September 26,

2015

Fig. 3−10

Mohammad Suliman Abuhaiba, Ph.D., P.E.

11

For a stress element undergoing sx, sy, and

sz, simultaneously,

Saturday, September 26,

2015

Elastic Strain

Mohammad Suliman Abuhaiba, Ph.D., P.E.

12

Hooke’s law for shear:

Shear strain g = change in a right angle of astress element when subjected to pure

shear stress.

G = shear modulus of elasticity

For a linear, isotropic, homogeneous

material,

Saturday, September 26,

2015

Elastic Strain

Mohammad Suliman Abuhaiba, Ph.D., P.E.

13

For tension and compression,

For direct shear (no bending present),

Uniformly Distributed Stresses

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

14

Normal Stresses for Beams in Bending

Straight beam in positive bending

x axis = neutral axis

xz plane = neutral plane

Neutral axis is coincident with centroidal

axis of the cross section

Saturday, September 26,

2015

Fig. 3−13

Mohammad Suliman Abuhaiba, Ph.D., P.E.

15

Bending stress varies linearly with distance

from neutral axis, y

Saturday, September 26,

2015

Fig. 3−14

Normal Stresses for Beams in

Bending

Mohammad Suliman Abuhaiba, Ph.D., P.E.

16

Transverse Shear Stress (TSS)

TSS is always accompanied

with bending stress

Saturday, September 26,

2015

Fig. 3−18

Mohammad Suliman Abuhaiba, Ph.D., P.E.

17

Transverse Shear Stress in a

Rectangular Beam

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

18

Torsion

Angle of twist for a solid round bar

Saturday, September 26,

2015

Fig. 3−21

Mohammad Suliman Abuhaiba, Ph.D., P.E.

19

Stress Concentration

Localized increase

of stress near

discontinuities

Kt = Theoretical

(Geometric) Stress

Concentration

Factor

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

20

Theoretical Stress

Concentration Factor

A-15 and A-16

Peterson’s Stress-Concentration

Factors

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

21

Stress Concentration for Static

and Ductile Conditions

With static loads and ductile materials

Highest stressed fibers yield (cold work)

Load is shared with next fibers

Cold working is localized

Overall part does not see damage unless

ultimate strength is exceeded

Stress concentration effect is commonly

ignored for static loads on ductile

materials

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

22

Stresses in Pressurized Cylinders

Tangential and radial stresses

Saturday, September 26,

2015

Fig. 3−31

Mohammad Suliman Abuhaiba, Ph.D., P.E.

23

Special case of zero outside pressure, po = 0

Saturday, September 26,

2015

Stresses in Pressurized Cylinders

Mohammad Suliman Abuhaiba, Ph.D., P.E.

24

If ends are closed, then longitudinal stresses

also exist

Saturday, September 26,

2015

Stresses in Pressurized Cylinders

Mohammad Suliman Abuhaiba, Ph.D., P.E.

25

Thin-Walled Vessels

Cylindrical pressure vessel with wall

thickness 1/10 or less of the radius

Radial stress is quite small compared to

tangential stress

Average tangential stress

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

26

Thin-Walled Vessels

Maximum tangential stress

Longitudinal stress (if ends are closed)

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

27

Curved Beams in Bending

Saturday, September 26,

2015

Fig. 3−34

Mohammad Suliman Abuhaiba, Ph.D., P.E.

28

Location of neutral axis

Stress distribution

Saturday, September 26,

2015

Curved Beams in Bending

Mohammad Suliman Abuhaiba, Ph.D., P.E.

29

Stress at inner and outer surfaces

Saturday, September 26,

2015

Curved Beams in Bending

Mohammad Suliman Abuhaiba, Ph.D., P.E.

30

Example 3-15

Plot the distribution of stresses across

section A–A of the crane hook shown in Fig.

3–35a. The cross section is rectangular, with

b = 0.75 in and h = 4 in, and the load is F =

5000 lbf.

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

31

Example 3-15

Fig. 3−35

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

32

Formulas for Sections of

Curved Beams (Table 3-4)

Saturday, September 26,

2015

Mohammad Suliman Abuhaiba, Ph.D., P.E.

33