Introduction to Stress & Bending

8/2/2019 Introduction to Stress & Bending

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Section 15: Introduction to

Stress and Bending

15-1


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Compressive stress:inner ortion

T

Tensile stress: outer

ortion C

Max stresses near theedges, less near the

ax s

neutral axis y

15-2 From: Noffal

x= by


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at neutral axis andzero at the surface

= (QV)/(I b)

Q= area moment

V= vertical shearforce

h

b

15-3 From: Noffal


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B h vi r f B n n r B n in Bending subjects bone to a combination of

ens on an compress on ens on on oneside of neutral axis, compression on the other

axis)

distance from the neutral axis (see figure)

bending fractures

15-4 From: Brown


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Compressive forceactin off-center fromlong axis

15-5 From: Noffal


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Bending Loading

15-6 From: Brown


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Muscle Activity Changing Stress

s r u on

15-7 From: Brown


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Various T es of Beam Loadin and Su ort Beam - structural member designed to support

loads applied at various points along its length.

Beam can be subjected to concentratedloads or

distributedloads or combination of both.

Beam design

is two-step process:eterm ne s ear ng orces an en ng

moments produced by applied loads

2) select cross-section best suited to resist

shearing forces and bending moments

15-8 From: Rabiei


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6.1 SHEAR AND MOMENTDIAGRAMS In order to design a beam, it is necessary to

determine the maximum shear and moment in the

beam Express V and M as functions of arbitrary position

xalong axis.

These functions can be represented by graphscalled shear and moment diagrams

Engineers need to know the variationof shear and

moment a ong t e eam to now w ere toreinforce it

15-9 From: Wang


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Diagrams

15-10 From: Hornsey


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6.1 SHEAR AND MOMENTDIAGRAMS Shear and bending-moment functions must be

determined for each regionof the beam between

any two discontinuities of loading

15-11 From: Wang


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6.1 SHEAR AND MOMENTDIAGRAMSBeam sign convention

Al h h h i f i n nv n i n i r i r r in

this course, we adopt the one often used byengineers:

15-12 From: Wang


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6.1 SHEAR AND MOMENTDIAGRAMSProcedure for analysis

Support reactions Determine all reactive forces and couple moments

acting on beam

perpendicular and parallel to beams axis

Shear and moment functions Specify separate coordinates x having an origin at

beams left end, and extending to regions of beambetween concentrated forces and/or cou le

moments, or where there is no discontinuity ofdistributed loading

15-13 From: Wang


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Shear and moment functions Section beam perpendicular to its axis at each

distance x - Make sure V and M are shown acting in positive

sense, according to sign convention Sum forces perpendicular to beams axis to get

shear

to get moment

15-14 From: Wang


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Shear and moment diagrams Plot shear diagram (V vs. x) and moment diagram

(M vs. x) ,

above axis, otherwise, negative values are plotted

below axis s conven en o s ow e s ear an momendiagrams directly below the free-body diagram

15-15 From: Wang


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.Draw the shear and moment diagrams for beamshown below.

15-16 From: Wang


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.Support reactions: Shown in free-body diagram.h r n m m n f n i n

Since there is a discontinuity of distributed loadand a concentrated load at beams center tworegions of xmust be considered.

0 x 5 m

+ Fy = 0; ... V= 5.75 N

+ M= 0; ... M= (5.75x1 + 80) kNm

15-17 From: Wang


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.Shear and moment functions 2 ,

+ F = 0 ... V= 15.75 5x kN

+ = 0 ... = 5.75x 2 + 15.75x +92.5 kNm

Check results b a l in w = dV/dx and V = dM/dx.

15-18 From: Wang


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.Shear and moment diagrams

15-19 From: Wang


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6.2 GRAPHICAL METHOD FOR

N TR TIN HEAR AND M MENTDIAGRAMSRegions of concentrated force and moment

15-20 From: Wang


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6.2 GRAPHICAL METHOD FOR

N TR TIN HEAR AND M MENTDIAGRAMSRegions of concentrated force and moment

15-21 From: Wang


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.SOLUTION:

Takin entire beam as a free-bod

calculate reactions atB andD.

Find equivalent internal force-couple

Draw the shear and bendin moment

systems or ree- o es orme y

cutting beam on either side of load

application points.

diagrams for the beam and loading

shown.

Plot results.

15-22 From: Rabiei


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Sample Problem 7.2

Taking entire beam as a free-body, calculate

reactions atB andD.

Find equivalent internal force-couple systems at

sections on either side of load application points.

= :0F 0kN20 = V kN20=V

:02 =M ( )( ) 0m0kN20 1 =+M 01 =M

mkN50kN26

mkN50kN26 33

==

==

V

MV

m ar y,

mkN50kN26

mkN50kN26

66

55

==

==

MV

MV

15-23 From: Rabiei


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Sample Problem 7.2 Plot results.

Note that shear is of constant value

etween concentrate oa s an

bending moment varies linearly.

15-24 From: Rabiei

Documents

Introduction to Stress & Bending