MDB Lecture Simple Strain

8/13/2019 MDB Lecture Simple Strain

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MECHANICS OF DEFORMABLE BODIESLecture Notes # 4

** If the unit stress is variale, the unit strain is:

HOOKES LAW: AXIAL DEFORMATION

- 0he unit stress is directl1 2ro2ortional to the unit strain u2 to the elasticlimit.

Solving for then sustituting to

0herefore, ut

So, the general formula of deformation is:

Where:e = is the deformation in mm" = is the a3ial load in 4L = is the original length in mm = is the cross-sectional area in mmE = is the &oung5s 6odulus, or 6odulus of Elasticit1 in 6"a

7hich isal7a1s constant

Retri!ti"n in #in$ t%e &"r'#(a "& )e&"r'ati"n:

Engr. erome !. !amadico "age %of $%ndSemester S.&. %'#(-%'#)

=

eL

=

dedL

* =

e = L

e=

8LE

e=

"LE

=8E

8=

"


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#. 0he load must e a3ial.

%. 0he ar must have a constant cross section and e homogenous.

(. 0he stress must not e3ceed the 2ro2ortional limit.

Shearing deformation:- n element su9ect to shear does not change the length of its sides, ut

undergoes a change in sha2e from a rectangle to 2arallelogram as sho7nin the figure.

Where: = is the shearing forces = is the shearing area; = modulus of rigidit1

Engr. erome !. !amadico "age (of $%ndSemester S.&. %'#(-%'#)

e=

L

s;

Figure 1: ShearDeformation


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TERMS RELATED TO STRAIN:

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Figure 2: Stress-strainDiagram

Figure 3: Comparative Stress-strain Diagrams fordifferent materials


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%. n luminum tue is fastened et7een a steel rod and a rone rod as sho7n.3ial loads are a22lied at the 2ositions indicated. !ind the value " that 7ill note3ceed a ma3imum overall deformation of %mm or a stress in the steel of #)'64?m%, in the luminum of D' 64?m%, or in the rone of #%' 64?m%. ssume thatthe asseml1 is suital1 raced to 2revent ucCling and that ES= %'' 3 #'

(6"a, El= B' 3 #'(6"a, and E/= D( 3 #'

(6"a.

(. 0he rigid ars sho7n are se2arated 1 a roller at and 2inned at and F. steelrod at / hel2s su22ort the load of >' C4. om2ute the vertical dis2lacement of theroller at .

Engr. erome !. !amadico "age @of $%ndSemester S.&. %'#(-%'#)

/

(> C4

F

#> C4 (' C4 #' C4

'.D m #.' m '.@ m


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). 0he rigid ar /, attached to t7o vertical rods as sho7n is horiontal efore theload " is a22lied. If the load " = >'C4, determine its vertical movement.

STATICALL INDETERMINATE MEMBERS:

- If the anal1sis could not e done 1 2lain a22lication of the three La7s ofStatic, the memer is said to e indeterminate.

- F.= /0 - F1= /0 - M = /

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- We need to a22l1 2rinci2le such as DEFORMATIONto determine theother unCno7ns.

Pra!ti!e Pr"+(e':

>. horiontal eam hinged at and carries a concentric load " = (' C4 at end F.0he eam is su22orted 1 t7o timer columns of the same material ut of differentcross-section. om2ute the value of the force at su22orts.

@. s sho7n, a rigid eam 7ith negligile mass is 2inned at one end and su22orted1 t7o rods. 0he eam as initiall1 horiontal efore load " 7as a22lied. !ind thevertical movement of " if " = #%' C4.

B. homogenous rod of constant cross-section is attached to un1ielding su22orts. Itcarries an a3ial load " as sho7n. om2ute the value of the reactions.

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MDB Lecture Simple Strain