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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.

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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


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

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Relationships between Load, Shear,

and Bending


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

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Plane stress occurs = stresses on one surface

are zero


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Fig. 3−8

Cartesian Stress Components


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Plane-Stress Transformation Equations


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

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Principal Stresses for Plane Stress

principal directions

principal stresses

Zero shear stresses at principal surfaces

Third principal stress = zero for plane stress


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Extreme-value Shear Stresses for

Plane Stress

Max shear stresses: on surfaces that are

±45º from principal directions

Two extreme-value shear stresses:


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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]


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2

2

2

x y

xyRs s

t


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Mohr’s

Circle

Diagram


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Fig. 3−10


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For a stress element undergoing sx, sy, and

sz, simultaneously,


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Elastic Strain


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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,


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Elastic Strain


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For tension and compression,

For direct shear (no bending present),

Uniformly Distributed Stresses


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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


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Fig. 3−13


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Bending stress varies linearly with distance

from neutral axis, y


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Fig. 3−14

Normal Stresses for Beams in

Bending


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Transverse Shear Stress (TSS)

TSS is always accompanied

with bending stress


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Fig. 3−18


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Transverse Shear Stress in a

Rectangular Beam


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Torsion

Angle of twist for a solid round bar


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Fig. 3−21


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Stress Concentration

Localized increase

of stress near

discontinuities

Kt = Theoretical

(Geometric) Stress

Concentration

Factor


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Theoretical Stress

Concentration Factor

A-15 and A-16

Peterson’s Stress-Concentration

Factors


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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


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Stresses in Pressurized Cylinders

Tangential and radial stresses


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Fig. 3−31


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Special case of zero outside pressure, po = 0


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If ends are closed, then longitudinal stresses

also exist


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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


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Thin-Walled Vessels

Maximum tangential stress

Longitudinal stress (if ends are closed)


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Curved Beams in Bending


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Fig. 3−34


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Location of neutral axis

Stress distribution


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Stress at inner and outer surfaces


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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.


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Example 3-15

Fig. 3−35


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Formulas for Sections of

Curved Beams (Table 3-4)


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

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