Download Review Formulas - Engineering Mathematics and more Study notes Engineering Mathematics in PDF only on Docsity!
g {. N cJ
s
i
H
il
s'o ag jd
P (^) Z ?1.
: FA
I U^ xBt
q 3* $ ii $l
igt Iitq';^ H
qb
sls l
Il$
{l{
sl-
ll
G
d^
rfl>
N^ C)
-l -) 3 o 2 AI UJ to 2l -A
tJ 161 q
ll
,
: ,i: a+
_so
+o: isr-ro ;-t N^3 g+aT rt;+_N
A
j a g ;ll"rr -FF; s_so-c
l=''
.ls
xl > IU l.r ll
0c lu oh EZ
9E ll o.
,s c( ll o
dlr
It t,
"f *"iL^
*i\3i
T* t'xd'.r liu;,rStl \
-c&E6.rl Ell
o ll
g
u
$l &
lrt J c,
d
<lo- rl
u
N
LIF
ilP ll Ul
o
s
q H.rrl $ \9 I-l I tct rf)
i
Efr S; :.J (^) N
i$ $ s \9< o^ o
o
lo, ^ € :.t
'l t,^ '*
[* "l^
rt *
E i ffu tslI ,o!'H
*€[
tt $9,
,\
(''
Ej o
iqE,
T * $"
ri (^3)
tru
t (^) t$ E F -'t
f
: (^) Su =s
Z
oo N
U
OC 6
a
u/, (J
z.
5 ur
=a
tJ-
?* 'l[
fJ'Ho
{rt st :rl :I dl
0l ,ll rrl
9l ot <l
Hl
Hg ll
t (^) Fll tlo
dlz
rl
ulo (^5)
uJ Xl:r^ z
)o 6ll- ul
srr (^) I
ll o- !+
c-<di
T
N r.{
El (^) "{,
El vis tltl rl tnt
:I T
a (^) dl
oi^
"
fil ti*
t
*9,IL
ll u
lro
ull
rJl =i pl E r.-l 8
as fls
f. !
9 F
5
& o
Eul
;(UE
E
$t (^) r{?*
P"b o3Sg ro 3 ,l^ ,trt^ ull n $"tf i^
rl
-( l-<
ilil
o- u-
?rll ilJ & uJ (,)
tsl h li'r lel tEJ t
,*ll
ull ell .o
a A ttl L u
d F
x
Ur o
s
ltl I 0l IL o
E t
o-ll
Fil
6
rc\
=/
o' ru o o
o
fi
J.
ffi 2l ml ul
fl
Fl ol 2l <l
nl :l al
Gl g
s
s I Fi
!o
s ltts
el lt
,
rA
lL ll
+(l
l
EI
dffiFJ dga: :l:
, AF Ab,+E
i (^) { x E
$ 3 =,
g
1?
9l lul
frl
il Fl $I url JI tllI
UI
rl
sl
Hl
n\r (^) ar+
dl"
d hl'
Hl (^) + (^) i,-t x *
"rl= (^) tlq $^
g
ll
-i, vl n
t (^) A iJ
6 7
F , --t 8* (^)! 0E 'i (^) -.{ Ur
d 1-rr^ E
+JttH
! &'5 rF a)S\
ll lt ll
sD sD (^) -
{h o(:
a- rl
Ol"t o
cz
6lu
_+r
Z ls
IN
o (^) '",
*le (^) i
illt
oD (^) 6D
6
H;s T{ fi; i (^) €f
u: ' $E 6g, * ilo n,{ Eg^ } :En a +^
'6'
.l S E^ S ll trt- -/. o^ a N-
ls.'tA=-,.,, --x
: + $ i^
'sl$'i
v (^) cP 6 (^) q -!^ t;
9l *l.o -: It ll^ T^ ll^ llll
HI ll^ tt
u tt 3 ll^
:, x
3l (^) s0 s0 (^) s0 s
N$l
e?
ut b)
Bl { I .sl oo
Sl
lr (^) ll
6l (^) u9 ui
'o
d
l
bt,
Io elL
J
*B Qz rJ gz-- ,^ .srl
el, (^) d3u
lln-c
Bsll *9-go.
uo t!
nV
aA
EF
us MD.3Q
is 5, ZE ds dd. FUO >tlj ru U)
H H^
slnr
!sEll X =& A^
l"-r
,il$ (^) -:ry
rtVArl lL lL^ l! _t,
z-z.ZA
AI u o
s{
l4j" j'
,llril
l, lv
o
ll
IU z 2 ! U 2 UT L o
!t d IL
<lo
ll € 'o
,l[
rl IL z
3 3 IL IL o
s
u
U) r.ll t-
5 ct IU F
tlu il M
,l'.J
A
r-lsr ll oa
D L ll "g
O (^) tlsr Q <ls-^ ri
It ll
M
*l
r-s
@B Its t-T^ , {-6--<
o,1'' (^) <lo-
u&
"E "p
{r
N ltl F ll,
E & A tll
E 3 ll 0-
<lu ll d
it, ll
o
s Z ,
F4!^ $o^3 IU=> srzN 8eE 92)
s
u)
o
1I :
2 o
5 IU a
t t I ( + =
8 .sr {t q-o6s .rl 3; 8tr -"d g 0 g 5
s
r il,^ fl.. il,, lltlilll €.rOts m -+[J a- -c 13 jlEl=,lx rl sl, ll -! 7S !*,^ fifi z
b+
ET Irlltllt J.y.Y-Y
7ls
."1, (^) n lt (^) s .-J (^) I sR (^) $$a I t^ '-lo O= Ncq. ti:: F.::o Hsrr? ;^ ; ldo
-ld
IN
-ld
+.
"lh
-?\
tl z u) lxls
o A- It TU E{ v)
!J E .,td 2 EtE^ AIS Hlrlr H dd ll o v)
! 8 o) !. & rl +, (,
T
a o
e
3 .l$ & rl ct, 3
a
G) ,rr (^) -ltl8 + +^ l+ ?lt E lH =l: t-ts (^) I glj jl-,, q t| #J*5$5X il 8 rE lE o lqJ
r lt
xlE
J 1,
rl r. a ;G 1l +, _-/ Sc i jlxJi ; tt (^) ,i* Ilt^ '\i ot- llllltl j-$XS f a,r lot 5lr rt ).'B l) "lil- ll . )a lo ul+ t-
el. 'l! -l! Y lt tt rt^ gr^ o 3? (^) ", ,tv el>^ Jl,^ =,i=^ J u jr, lj I"tg 5d,* 0)r 81" il
g 2 x tle
l I o NI el" I (^) lk. EI ll ) (tt -lx I lc -ls
lg
I rE IE -ls l"/)
=D
E
z. E LLI IJ z. a -z tIJ g Z L) lll (D u"l (.D lu-*=:
Yv $H
i F" I
g
's Hi;E Y)
sfi
HF ilnn& (^) i H
o'[
5+s
4 d^6
y k 5,6 6 H U
HE6g (^) E[?E $gHfi
ooud66"6'Ed
$$c a -ab
ll)A d ,0( (,F aZ
uu
uz tLo X os rev 4Z lll l-
rr ER
5tl cP rL
Ij(
L---?
t) N
ll
o-o
Z o 6 J !I! u
J .C )a
ll
oy
g
E 3
-C
,
rl
d
rst[p
lslls
|^; lli
t!lls +lr .- l.-
Hl$
,J (/) o ), ll o. Y
=ll 3
asu -rtoo )-"ll\t tl (^) - ro a , J^ -tg.
f (^) + f^ "h= t'^
dld E T dtrdtl ll^ tl d-lro
ltrtlrlfccllt,
ddd-u"eBdd
v ld*
'l
-$
o
s$ 8b aZ
F- f s
=3\t
s5J
F6P
-C-z^ "
,i*
,r (^3) u
)b g-o SE IJI J
,-) B '.o
g>*
If,IIfl
I BltS l:ll r
lslE r (^) I* ,- l,r
$lH
.J a B ll ( *Y
$l;tR
{
,t:
-l{
iF
rll i (^) rli ili
f.66ll'lrrirll
utlo: elC
J3^ 8';
slt rlg^ ulf l- I^ l- a-:f, null
+==
rlc
-s.,
,rlg t
ela Git _
ul*^ "lq
-t- ll
lr
lt^ ll*.-, Q-l (^) a zlz
-, lr
f
flz
=J
il
f
r= d G) G ,a
;€_
(n " = $ r-s { d=lza
rt €
ZIE (^) B A ,',da (i tl^6 $
}E B (^6)
j$^ -u=t
ut H *:f
_kstl dEo-
ele il
ult il -vl- cl tr, sl
llrt
aa
d Ld r. (^) s
YA n.i a I 5 slN
il
A
qr
5-l
(' (^l
5 ,!'lN
Nl$ ;l
Ittr
(d
El$
ll ,
D$
lE trN N
ll- r ll
"$ LE
El" (^) n
ilo
NP ! trrt
3o
ol r
olo
ll srl (^) - (
il s.
(,} E ID
=d
HE
E
t II
t@
I I
-lGl (^) 6li
B
-i@B ri
ff gk^ ,to
tttt
NA T,Irv
Xx
?l ql< COl c sla
{1.
tl U g (n o 6 -lcr tl -s
-ln tl
()
7
€ (^) urv
,g
gt xeE
H& ti tH Ei
9e U[.i-. S.$E"s
^"tt $t ".iS ?:^ t
gt :i=l=n2T (^) El i"^ 5l^.^
5-;:
flsi l.^ Ii^
gl ;t (^) el
=l=f
r (^) I t $l
N p m
o
ll
I
oX rl r !..)-+
( €, sltl
TA
;o
O glsr ll (^) lt
6,.
1/r : u s o st E
I d
o
4? Lua
E&
;
tt x xl$
fE CS-\o *.tr 5y .:!E 53 C a € ,rrl>
3'" (^) i3 I qq ! :l 'rd(n,
ul (^)? n 3 ol+$
El u-^ U c^
tt I
gl EiL; e
i$
gl i (^) I ,t,* "[
l-1.
T s !
o
ll
B
GI lll o 2
Sz ioo
r$30t
HE
lrlr
a-C ++ +4r 9o ++ .c
88 -a -l tE,E
Fgg
Y9Ev
E++^ u^ g^ v
I3t s $\r tl (^) n (^) v tl d l.l ro
'0D (^) aD
Itl t< IT
h, +t il) I ll x
"
s
'--'
!'
5\ b, *=.'^ /.
++;.
?",bhif A al c **lit:i^
rllTllrrlu rr^ i^
gltii
I
gl
I
a
=o<
P
L/, (-)
=I^ tl
-?
=
s d s 3 e i
{.s\
t^,1 (^) V{
tl I
!o Xs)
B 5 2 q (),
B
b tl
z
I & IL o
e
B
z lt) o
k
V\
I
I
]L
-lN
ll
? s(rp l+l
ftg
Ik
$B f"e BX
tl -o
o-
f (^) Yl
t} f^ ;^
ill {lE
. co rr I al!^ -T_
<t
gl (^) iil,i.iliT
rl -o
<so tll ooo sss
lr lr ll
l{lcs lo
.^'
N
J
N -o ?l
N _55t -+N-r tr
,o (^) F ; El'^
sJo n't*
Ill (^) I ll^ rl^ ll
Hi
jr i
j$
a
I .. t< t-
r+
N
.'lm
ll E u il J
o g a cll Rlnr
tl
il l- _o
ll
h
NI !1..r ,3r gtl
El f,
El rr
EI E
ffi
s,lp
t\
-l-
ll
;tr[
Jll$
HI zl
$l
8l
l
:l 8l II el
hl <l
HI
l- 6- IT
Hf; F
d i^?^ E fr
B (^6) $E i
HI
9l
$fil
$*rx3ru
frl
5,iitl,i, Hl
EA (^) 8l
IiEulIgs o c o
al^ ll d)
ul ll
a
{, )
6-
8rt ,x I El
f$l
$ El' $l*^ El
,lli j
g
N
a
5l
I N + N C x I
sl
lol l<
la
N
S
slm
rl t 5
J
J
t L- l-
ll-, -r e^ *l,n Ss-'
$E
il
-cl
Llo
el
url ll
gl (^) rl'
l,l il rt
l $ s N 3 I
{erp j
ts6oo
&tttttl
i-o-sx x
=. o ,6.1O oto 6to
iB
gIE
9i .!t 9s: b: t\ \l
E > llfll lo
i*
t]Jr
ers (^6) _i'.
llrt
x5)
I x
z
lol - lt .<
'-o J.
i
=ltEJ
? 1". x,
sx
f x
E
5
ll J
I x
2
s
x
{ sf J-s t (^) ll (^76) :
ZJ
fSS xll rt gx;
HT
\r
l-(, ID rsE
-pq-
3>g g^
x s^
l-o
<l
t l
xls
t-..r
4Li
3 slo *rt
$-
f x
x N lt _o
lc.l
-o rl 1
,-a
rt L r
w
lg
(a\
w
c, i'P IL
rq
M , 2 g
v,
I
x
E
u
gT rA$
E
o
gH
B f s'li$
s
a $Ii
?s €;
g E vr E^ I
ii ffi:
I AI & 2 lD
o
ru
;HN}
e ll) rl
7 t!
o
LlJ 0r 3 @ UJ
o...f IL zRt
=g.E
xls xls X
lltlll
s- -C
-z
r
2
t$
zI zx
BEAIYS
ANauvsts (conarurE^ Mu^ ,^ 6lvEN^ Ad^ c^ Ag
@
. (^) fs=$e
I flND'p
@Y Pmoo.rr*s=r.rt[ffP#ril -' OJ9^ fb
@ (^) T= A"fl
@ C= o.gik'otb
@ T=C
'Artg=o'esFab
@ (^) FIND'o'
@ (^) Mn (^) =r(d-A= c$-q
@ lulu
=QMn
vrva
@ (^) Asr= (^) fr**bA
@ Asr--As -^ As,
, (^) T,=C
' As,fg (^) = o'9s^ t"^
ob
@ Ftuo: a
O Mn= Mo, +Mn
=
't, (a-
!) +r"(a-^
a')
= As,^ f,,^ (d^ -s)*^ Asrfu^
(a -^ a)
@ (^) l'1rr (^) = QMn
@
o FIND:F
, C"=o.8Sfo'ab
, (^) Cs'- As'fs'
. (^) 6=fo
.
ts''=
GOo
@ (^) T--C*,+ Cs
. (^) A.$g --o'ss#@4b +^ A;(c*'$)
FtuP'
o
@Lr/E:
ot a=F'
Mn= lvl^otMn.
= cc(d-?)^ +^ cs(d^
-d)
o.ts fa ab^ (a -^ gt^ t^ Ar fs' (a -d')
Mrr (^) = t0 tvln
=*o{-o'sfc,
@ P--H bcl
@*^
@
@
@
Cs Cc
ll
I +
Ot
I
c
Tr +T EYACT
Q,=o.tsli ab
Q= As'fs'
T =fufg
6pepEx C=o.tsfo'ab
Tr= (^) Asr{r,
Tz= (^) Asefi
,.1,qt471V',
BEAMS
SINGLY REINrcRCED
ANALYSIS (couPure^ Mu,^ 6tveN^ Pe
7p=o.zs tL p=o.es (^) -o.o' r+
o4=#effi
@P=&
Mn (^) = fo' wbd"(L^ -o'oa ar)
,. pV
Mu=QMn
Q (^) = o'Qo^ for^
$texure
. (^) T= Asfe
, (^) C= o.lsfi ob
. fs= "*+
. qr=po
@ T--C
' Asfs^ =^
o'hsfc) ab
' As^ lr* *)=
o'ss l"' (P^ ")v
o FlNo: c
O SoLVE: a^
a=Fo
@ (^) Mn= r(a-?) (^) = "(d-?)
@ (^) Mu = Qlvln
f (^) f"'L9t'4P"
f+ (^) [i 7 lou?a
DESIGN (counrre^ As,^ GIvEN^
Mu)
@ FIND: P
@ (^) Po -- o.ts^ Po
=o.uffpffi\
@
@ (^) Mu = O^ fa'cub;12(1-o'sao)
. LISE anvEU Mu
@ Mu = @li orbd"(ro'saur)
@ FIND: (I)
@ (^) r=f
@ (^) FtxP: P
@ (^) As =Pd
DEgIGN AS^ DUBLY REINFOFCEP'/./
i l,
C=o,B9ic'ab (^) o.*
Z- Tfl
TiV
Y:_ru+r-2:+^
T=Ae$s (^) €s= (^) *
5o,ss f''
d-g
€r,
Es=2,4w
I,
l.g+eetDtttL
t!
!r
I (^4) g, -i
-m tl) d n v)lJ (^) ,(D lnl
l$
l"l
jlt
-l
(,
el*;
ll
tf
s 3 : $ v 6
$g
tr3^ =a
z ul i 3
sl sl
it
_\
E
N3D I
'
or)
ot U)l
d-In) dlrs
illl 3, 33
PEE
E9^
Br b#j
6.F.-g e! (^) - E-c I €&.r I 's
:lt \sl.o
+l
g ",1"
fls
Ss jl^
+l {r )(l'{ r'l X
zlN
+l cl sl o-l'
Hi
rl fiG 61
ie dF
sll
is
3 slI sd 6e
es ll, .O ov llr ll Es
-lalu
$- (^) :-
hu
s -o 5't s€e Fss v, vl oo
+l+
t,
"'l
6 <t< i-J (^) o uro SN dl ll 6 .t ll
.:t d
o)
3 CF o
7
sl
tl
.llol>
slS -lElt '-o.t elf, ,tl /6l
el ll
e
o^ & e
p
V U)
v €rS
tnq{
.q u o)
a =g
E S j ,b ,
-t vl U)
q
J $ E tr
u Ur E
p 2 Ul :
$ I z II lt u lu a
DS gd, F
trf ;$(n
+i (^) I
?{
o
sgag I
+*trU V
B t.^,f^ a
*siB (^) I dooov lu ll^ tl ll^ : dd s
2l
=l 3l st
6n\6- rln
r
"J
I
o
x
!, d dN d J (, i
<a
e{
{s
Be
ti"
$$*
i?r,
o oJ
z t. ,
z
lt
q
il F
gI
(6t^ $l
--l 6ol
$il
6:
EB
.j
t, (^) x
YIiI F
.'; o
e
rt
ij.;r.i !) t
:,; , "_:r.:.. (^) ,.,. 'a 2 & + $ E o t u
,
o.} u
t- 2 u u, I
oJ
f,
o
{l} rR
VA
:la sf.e u s€
{d \ r)a
:- C} r) \
..\
f e^ ii
'1,
a =t
\i (^) (/lN
*diJ
\l (^) f'J^ g (^) t<J
s \e-q*l \ llll t
zl ul
&l
3
EE oo Jrr
c l:rl^ "-.\ i'-t^ s
le
_ts
+s lt lt ct
=,E
ga.
o u z Ei
cc rrr
Bsl
aat
o
;ir-a..:':.-?..|.'.':....'..;-::."::s:...--...s.''-:a:..:.
{,
w
o, € !/< f,, , Ito 3
(l*- I sl,\ (^) S-
t 9Ps
ll (^) c
d2 E
^fiisF" S t st-g X
-"-+ d-o?
l)- o^ 3; ll rL-i^-+^ ++-S-oo
! f (^) YfiTt {s i
]d { o (^) b cl$g
Hf s^ El^ ii
5 .:i^ "P ii
l- r-+
il{ (^9) u.-' v o ul I v (^) )1,- f- -^ (,,].)
St] r>! (^) * l'--- *
\
t t:
! r{ F
e
.--- ll sold ; -DS3,
$n5 (^) i;
!:diils
-o -o x 6
It l.
-9o,a{
tJo +ll
6li +
l- s-j d+ ll ll c E=
qF^ o ro 0o 6 t \r tlu ul
$ Io
-s \t
-u^6 rlu cf+ Ul rrl-Y
:t 5
lrlrtt
d,3i
x : 3 z A I 2
rl: Z ur d , s 3 8 ul^ o lr c llt
_+
dzl
*;o6rBfil
G.. o
BE CCE
8$ Zlu 6 r-r ugl
5el o :Jl
Eit
o'
qHl '!-
i j rkr
' JE
-o6.rl ii
. (^) -€ -+\
l.r ilo^ 6!.{
I
o;[E] $t].1"Y+ l!llllll;: r-u.S 1l EE
J
{t B
r$
a gp
EE
(t61^ oo {lA
f*
:gl .',
iil" $lr
. (^) Ur l
'e \o : (^) i ;Il-
ule (^) tll
llll (^) lt il a. (^) €\
r.a 14
UT
ll .. f-ra-).'
,sl
a Z, r.al
ffi
o
t
r+t l-sl
E
G.E
a, rf) <t I
t{
9 3 l) e I { a 3 a
( .ild rrI r-E dq lltt
dJ
H
st !.(
B
1
al* (^) ds als^ el*" lt ll (^) tr rl +.t(tri
s
lu
6 J s 3 x 5
IL
& 3 d c
E /s
,i
llr +
$ i^ -.,^ G .rl^ -
P I^ ,.,i '* rcl^ AH
E (^) ;1. H*
9vl,'
Hrtrllul
