1. The document presents mathematical equations and formulas related to stress, strain, and other mechanical properties. 2. Several equations are defined relating terms like stress (σ), strain (ε), Young's modulus (E), shear modulus (G), and Poisson's ratio (ν). 3. The relationships between different mechanical properties are explored through multiple short examples and calculations shown in equations.

1. !" #$ %" &' (" #) *" #
! + *"
, + , = -'&
− −
. / / = = =
# #
/
# &0
= + +) = = #+ &0+ ) × $ = &
$ $ $
1 ≈ #
#
# /
#
!" 2 3 %" 3 (" 0 3 *" 3
! + !"
# −
, + − 4 (0 =
) − 4 = −4 = -< 3
#
# $ )
$ + + + + +
#5 $5 )5
!" %" (" *" + #
! + %"
, +
6
4 7# 8 0 #
=09
) 4
!" %"
(" *"
! + %"
−
, + µ= =4
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

2. +
1 8
11 8
111 8
1: 8
;!< ,
!" 1 %" 11 (" 111 *" 1:
! + ("
=
, + =: ( )
= ( )
#
#
=ψ ( )
& 1
/
!" %" (" *" #
#
! + !"
, + σ = σ = σ> = 4
τ = ( )
σ −σ
/ τ = #θ − τ #θ = 0
#
#
σ −σ
τ = + τ# =
#
0 ! 1 ) &+# / < * /
!" %" + # (" *" + #
, , , ,
! + *"
, + 6
; 1 ) &+# 9
' ?
@
!" < %" <
(" < *" <
! + ("
, + < / A /
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

3. - 1 /
!" %" ( (" ; *"
! + !"
A
/
/ 8
!" %" (" # *" )
! + *"
)π#,1 π#,1
, + ; / 4( = B; / 4(# =
<# <#
4(
C / =)
4(#
/ $'
!" 0' %" & & (" )'' *" )$'
! + !"
, + τ=(+σ φ
(= →
φ= $' = 0'
τ
/ τ = σ× 0' = 0'
σ
#
? /
1
11
(DEE,(
!" D 1 E., %" D 11 E.,
(" % 1 11 E., *" F 1 11 E.,
! + %"
C
, + / D=( /
<
=
G
D=( D=(# 3
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

4. $ ! 1 /
8 ) 3 #'3
!" (C %" (1 (" (< *" (< =<
! + %"
, + << 7 ) 3 41 7 #'3
(C
(1 !
! 1 (D*,/
! / 41 = 0 (<< − #
$ )= ) & < (#' ) (<
# 3 < << = ) 3 < 3
# $
/ (1
) ! 1
A A A ;
/
= ( H
!" 4
!
%" F " ! !
(" D 8
*" D 8
! + %"
! A A
=# AFI $ " 7#
A " !
4
0/ " A
!" 0 %" (" ) *" #
! + !"
& ! > )
> $ #
(= = #) "
$
!" # %" $ (" $ *" #
! + *"
, + =)+ +)= '
C
/ != + ' × $ = )# #
#
$ $
= $# + )# = )
4 / 4 = #× + =#
! )#
C E /E = = =#
4 #
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

5. $
0 ! I
; ' "
!" ) %" (" & *" #
! + !"
, + θ= $
I
:
%= B =
; = '
: θ
= ' = ' =)
%
' A
I" J
EK
=)
4 L
: =& I : = I
E
=#
: =@
!" $ %" ) (" *" &
! + %"
, + . 8 (! ) = (! )
E
+ (! ) L
π π π
× )# × & = × ## × : + × )# × :=) I
) ) )
-
!" ) %" & - (" *" #
! + ("
# ! / /
(D*"
;! "
!" = %" 4
(" *"
! + %"
# + ! /
+
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

6. (DEE,( +
!" % " " " "
%" % " " " "
(" % " "
*" " "
! + !"
## H
J
MK J
GK M G
!" # %" $$ (" *" 0
! + !"
, + F 7& "
F 7 #) "
&
/ = = #
#)
#$ J
MK J
GK ) ( & (/
( +
1 : J
MK J
GK
11 4 J
MK J
GK
(DEE,(
!" % 1 11 E., %" 1 ;!< , 11 E.,
(" % 1 11 ;!< , *" 1 E., 11 ;!< ,
! + ("
, + < / #
#) E
!" % A %" %
(" * *" ;
! + %"
# ?
@
1 ,
11 ,
!" % 1 11 %" 1 (" 11 *" F 1 11
! + !"
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

7. !"
#& ( ) 7 $
=
( + )− ( − )/
( ) ≈
#
#× −$
=
( ) -0' /
7 #
−) −) −) −)
!" $ × %" $ × (" ) × *" - ×
! + !"
#0 1 / 8 A
# $ &
!" %" (" *"
$# $# $# $#
! + *"
, + 4 = 8=
#
)
− &
4 = (48 = ( + ( =
# # # # $#
-
#'
'
!" −# )# & '& %" $ )' $ $
(" ) 0 & '& *" & '& -
! + %"
-− λ
, +
'−λ
= ( -− λ ) ( ' − λ ) = # λ = $ )'/ $ $
#- ; D4LE A / D4 = N+ N
H DE = N+ N
H
L
!" −
E
%" O
(" O 4
*" P D
! + !"
H A
, + ! / ! = D4 × DE = Q ( − ) D4 × DE =
! = −
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

8. $ 8 +# =
/ 7 7
−# −# −# −#
!" = %" =# (" = - *" = $& -
! + *"
, + R / +# = = =
(* + # ) = * = −# ∴ =( −#
−# #
F / = =( = ( = $& -
−#
∴ 8 = $& -
$ ! H
,1
) ) ) )
!" %" (" *"
& )) ,1 0&' ,1 $') ,1 - ,1
#
! + %"
L
$# A
1 H 4 =
" E I I# 4
!" ! 8 4 =
%" ! 8 4I#
I#
(" ! 8 4 > 8 =I
*" ! 8 4I
# > 8 =I E
! + ("
$$ ! A
&# AF
) AI
F
0
)
( −
0$
( )
!
#
FI "
!" #- %" ( #-
(" ) $# *" ( ) $#
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission. #

9. ! + %"
, + ! = $0 × $ #
#
1
= >= = )&'0 × $ $
&
#
=* = . =
*< = #- 0 ×
- & &
F
'
−4 4 =
/ − =
!
− + * = −# - FI
> #
#
( )
$) ! 4LE 4L LE
S
LT H / 4 E > 4E 1
? LT L
? ? ?
!" %" (" *" U
# ) '
! + *"
$ > H /
A A #
# =4 ) =4
A (γ = # ) ! < =
1 ' +# 0/
" 4 8 #0 AF
!" %" (" *"
! + %"
$& H
# 1
/
!" $ %" $ ' (" ) *" & #
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission. $

10. ! + !"
Q!
, + ; / =
# ∂<
!
= A
α= =
# ∂<
/E = = # = A
#
E# = = A
= ( ) = (E ) = (
A
# ) =$
#
$0 ! / A H
# A4
+ 7# 0/ 7 ) 3/
7 ) / 7
A4 ! >
/ "
!" # %" - (" ) *" &
! + *"
$' A L/ #
A > 4/ L/
> E
L A
$ $
'AFI -' AFI
A "
4 L
4
#
L
E
!" ) )# %" '- (" & $' *" ' $)
! + %"
, + ! 4/ = B E/ = # &B L/ =#
/ θ = σ = σ − υ = G × C − Gω ×
= ' × # − -' × # = '-AFI #
$- )
/
# ' 0 I/ & I ) I
. /
$
I"
!" ) %" # (" *" '
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission. %

11. ! + ("
, + !
: +: &+ )
: = # '
= = I
# #
/ L = :×! = × ×)× = $
I
#
) R 1 R 11 I
& ' )
( & ' )
( )
4 !
L , # 4 T
E 1 $ C T
? ) (
(DEE,( R 1 R 11
!" 4 − / L − #/E − $/ −) %" 4 − )/ L − $/E − #/ −
(" 4 − )/ L − #/E − $/ − *" 4 − / L − $/E − #/ −)
! + ("
) ? 8 #
! & 3
"
$
I
!" $ '# %" '' (" #$ ) *" #$ )
! + !"
)# ! C -#
−-# −00
!" I< %" I<
(" $ # I
< *" $ # I
<
! + ("
$
)$ ! 8 )# I
; !
# I
$ 8
#
"
!" # %" $ (" 0#' *" #
! + %"
$
L L )# I
, + D = != = −$
=$ #
! DE & × # × I
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

12. )) !
1E(+$0
+ 7 /
7) I / 7 '3
!" $ %" # #) (" $ && *" ) )
! + %"
$& ! ( + ) −
, + != × :*;
( + ') −
× &
= $& × ) × :*; :*; = # #)
) >
+ 7 # / 7 $
70 1 *" ' /
A " 8
!" # ) %" ) (" 0 *" 0-&
! + %"
)& ! '
I . H /
;
I / I "
A
!" # %" ' (" &0 *" 0
! + ("
)0 > 4 L
4 L $
> / 4 L
> / 4/ L/
4 L F
T / "
!" & - #
0 %" # $'0 (" # ) *" #' 0)
! + %"
)( )
D
, + ( $+ )=( − D
+ D
$D
4
D
D
= ##' 0 '
C = ##' 0 ' ( $D + D
)
= # $'0
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

13. ** + ' , #- $
A
+ 7 - I "/ 7# 0
7 '
$
$
)' "
!" $$ %" $' (" )$ *" )
! + %"
)-
!" $ %" # -$ (" # & *" $-
! + ("
** + ' , %-
! A " 8
? + = $
I /
%D* 7 I
</ %D* 7 I !
< A +
7 ' / 7 #) /
$
7) / 8 7# I
<
1 ! ,
A (
E
?
;I
=" A %D* A "
A
!" %" # (" ## *" #)
! + ("
; $
I × I< −$
, + = = = -$0 × A I = ## A I
= × # I ×)
<
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

14. AI "
!" #) %" (" ' *" $$
! + !"
, + = =# × × $
AI = × '
× #) A I
<
= #) A I
. / " ' , !"
0 * 1 . / " ' , -
A !
=# ; )
#
!" ) !
( > )
0
&
!
$#
# " < = 1 ) &+#
!" # %" ##$ $ (" #$& $ *" #0 &
$
! + ("
, + $& A + ) A ; *; = '0 !
$& × # × $# × ) × ) ×# ×( − $# )× = '0× ) ×)
= #$& $
$ A "
F / <
= 1 ) &+#
!" )0 # %" 0 0 (" 0 & ) *" 0&#
! + %"
0 * 1 . / " ' , -
A #
φ I
$
I D 7 "
A
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

15. $
) A I I "
!" %" ' # (" $0 $ *" ' #
! + %"
, + ×%×C = ! ×( )
#
× × $& ×C = × &
× C= ' # $
I
#
; / A
( $
I )
!" %" '# & (" - &
# *" #&
! + ("
, + E =E −φ= − =
4 A C =4 A C ×
$
I
= ' # × = ' # $
I
%
& (
+
+ ( 2 34 4 4 4 4
4 4 4 4 4 * ' // *(
32 1
3
!" %" A (" *"
! + %"
#
0 8/ 8
A E
>
!" %" (" *" #
! + !"
, + = 8#
=E
/ = 8
4=4 = 4 − (4 = 8− 8#
> 4/ = − 8= 8=
8
' (
+
0' 5/ 6 4 4 4 4 "
4 4 4 4 ' *( /
!" %" (" *" -
! + !"
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

16. - ( ) ! +
!" A C
%" %
(" A ; < A /
*"
! + *"
& ?
@
6
!" * %" * (" * *" *
! + !"
!"
& !
8 =# − #
!" ' %" (" # *" )
! + %"
, + =# − #
=#− #
#
#
< ∴ #− # =
=
∴ =# − =
&# ? /4 ; 1
E 8 + C
/ * / , A
/
? @
!" R
%" M
(" F A
*" F
! + ("
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.

17. &$ R 8 / !* (R ;Q V4/
!" D: %" D? (" 4: *" 4?
! + !" )
# $
, +
! * ( R ; Q V 4 D:
# $ )
&) ? (DEE,( @
4+ ! 0 0
L+ ! 0 0
E+ *
+ *
!" 4/ L %" L/ E (" 4/ E *" E/
! + ("
, + 4" ( /
+#+$+)
< / #/ $/ )B / = =#
)
! 0
'/ -/ /
' + -+ +
= = =- =# + (0)
)
E" * /#/ )/ &/ '
#+)+&+'
= = = = #×#
)
& ! A M G
M & 3 G ) 3 A ! A
H 8 8
D M A /-&3 D G A /0#3
A / /
G
!" #'' %" $$) (" &&0 *" 0#
! + %"
, +
& 3 ) 3
E -&3 0#3
D 0& #''
#''
∴ 4( )= = $$)
0 + #''
&
© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the
written permission.