8.shear and bending moment in beam10

T.Chhay

kMlaMgkat;TTwg nigm:Um:g;Bt;kñúgFñwm
Shear and bending moment in beam
1> RbePTFñwm Types of beams
FñwmKWCaeRKOgbgÁúMepþk rWesÞIedk EdlrgnUvbnÞúkbBaÄr ¬EkgeTAnwgGkS½beNþayrbs;va¦ EdleFVIeGayva
rgnUgkarBt;.
xageRkamCaTMrg;énRbePTFñwmEdleyIgnwgCYbRbTH³
P P

w w

(a) Simple beam Deflected shape (b) Fixed beam

P
P

w w

(c) Cantilever beam (d) Propped cantilever beam

P
P P
w

(e) Overhanging beam

P P P

w

(f) Continuous beam

kMlaMgTTwg nigm:Um:g;Bt;kñúgFñwm 72

T.Chhay

FñwmEdlGacTTYlnUvbnÞúk)an luHRtaEtxøÜnvamanlMnwgCamunsin.
edIm,IdwgfaFñwmmYymanlMnwgeyIgGacrktamsmIkarxageRkam
n = r −e− f
Edl - cMnYnGBaØatielIs (redundancy)
n

r - cMnYnRbtikmμTMr (reaction of support)

e - cMnYnsmIkarlMnwgsþaTic (equilibrium equation) man 3

f - cMnYnsnøak; ¬EdltMélm:Um:g;esμIsUnü¦

kñúgkrNI n < 0 FñwmKμanlMnwg Unstable beam
kñúgkrNI n = 0 FñwmmanlMnwgkMNt;edaysmIkarsþaTic
statically determinate beam, geometrically stable
kñúgkrNI n > 0 FñwmmanlMnwgminGackMNt;edaysmIkarsþaTic
statically determinate beam, geometrically stable
]TahrN_³
n = r −e− f
r = 3 , e = 3, f = 0
⇒n=0 FñwmmanlMnwgkMNt;edaysmIkarsþaTic
n =r −e− f
r = 4 , e = 3, f = 0
⇒ n =1 FñwmmanlMnwgminGackMNt;edaysmIkarsþaTic 1dWeRk
n = r −e− f
r = 6 , e = 3, f = 1
⇒n=2 FñwmmanlMnwgminGackMNt;edaysmIkarsþaTic 2dWeRk
n = r −e− f
r = 6 , e = 3, f = 2
⇒ n =1 FñwmmanlMnwgminGackMNt;edaysmIkarsþaTic 1dWeRk
2> kMlaMgkat;TTwg nigm:Um:g;Bt; Shear force and bending moment
viFIsaRsþedIm,IedaHRsay kMlaMgkat;TTwg nigm:Umg; KWBak;Bn§½eTAnwgsmIkarlMnwgsþaTicrbs;Fñwm.
:
sMrab;FñwmEdlkMNt;edaysmIkarsþaTic CadMbUgeKRtUvkMNt;RbtikmμTMCamunsin. RbsinebIFñwmTaMgmUlman
lMnwgenaH)ann½yfa RKb;Ggát;TaMgGs;rbs;Fñwmk¾manlMnwgEdr. RbsinebIeyIgkat;FñwmCakMNat;enaH smIkar
lMnwgsþaTicGaceGayeyIgkMNt;)annUvkMlaMgkat;TTwg nigm:Um:g;EdlekIteLIgedaysarkMlaMgxageRkA.
GaRsy½edaybnÞúk niglkçxNÐTMr kMlaMgkat;TTwg nigm:Um:g;Bt;ERbRbYltamRbEvgrbs;Fñwm.

T.Chhay

k> niymn½y Definition
kMlaMgkat;TTwg V Rtg;muxkat;samBaØNamYyénr)ar KWCaplbUkBiCKNiténkMlaMgTaMgLayEdl
EkgnwgGkS½r)ar ehIysßitenAxageqVg b¤xagsþaMénmuxkat;enaH.
V = ∑ Qi = ∑Q j
left right

m:Um:g;Bt; M Rtg;muxkat;samBaØNamYyénr)ar KWCaplbUkBiCKNiténm:Um:g;TaMgGs;EdlenAxageqVg
b¤xagsþaMénmuxkat;enaH.
M = ∑ Mi = ∑ M j
left right

x> karkMNt;sBaØa Sign convention
RbsinebIr)armYyRtUv)ankat; enaHvanwgmankMlaMgkat;TTwgxagkñúgedIm,ITb;nwgkMlaMgxageRkAeGay
manlMnwg kMlaMgkat;TTwgEdleFVIeGaykMNat;r)arvilRsbTisRTnicnaLikaman sBaØaviC¢man. pÞúymkvij
ebIkMlaMgkat;enaH eFVIeGaykMNat;r)arvilRcasTisRTnicnaLikaman sBaØaGviC¢man.
sMrab;krNIm:Um:g;Bt;vij ral;m:Umg;TaMgLayNaEdleFVIeGaykMNat;r)arrgkMlaMgsgát;xagelI man
sBaØaviC¢man pÞúymkvijm:Um:g;TMagLayNa EdleFVIeGaykMNat;r)arrgkMlaMgsgát;EpñkxageRkam mansBaØa
GviC¢man.
Cutting Cutting
plane plane
V
External force

External force
V

(a) Positive shear (+) (b) Negative shear (-)

(c) Positive moment (+) (b) Negative moment (-)

K> TMnak;TMngDIepr:g;EsülrvagbnÞúkBRgay kMlaMgkat;TTwg nigm:Um:g;Bt;
eKmanbnÞúkBRgay w( x) manGMeBIelIr)ar.edayr)arenaHCar)armanlMnwgsþaTic eKkat;r)arenaH
mYykg; EdlmanRbEvg Δx mksikSa.


T.Chhay

eyIgeXIjfaRtg;muxkat; O r)armankMlaMgkñúg V nigm:Um:g;Bt; M ÉRtg;muxkat; O' r)armankMlaMg
kñúg V + ΔV nigm:Um:g;Bt; M + ΔM . edayr)ar
manlMnwgenaH kMNat; Δx k¾RtUvmanlMnwg.
⇒ ∑Y = 0
w(x)
⇒ V − w( x)Δx − V − ΔV = 0
X
⇒ − w( x)Δx − ΔV = 0 Δx

ΔV
⇒ = − w(x) w(x)
Δx
ΔV
⇒ lim = − w( x)
Δx → 0 Δx
M M+ΔM
dV
⇔ = − w(x)
dx O O'
V+ΔV
mü:ageTot ∑ M o =0 V

Δx
⇒ + ( w( x).Δx. ) + (V + ΔV )Δx − M − ΔM + M = 0 Δx

2
Δx
edaytMélénGgÁ Δx.
2
mantMéltUcq¶ayebIeRbobeFobeTAnwgGgÁdéT enaH Δx. Δ2x → 0
⇒ VΔx + ΔV .Δx − ΔM = 0
dUcKña edaytMélénGgÁ Δx.ΔV mantMéltUcq¶ayebIeRbobeFobeTAnwgGgÁdéT enaH Δx.ΔV → 0
ΔM
⇒ =V
Δx
ΔM
⇒ lim =V
Δx → 0 Δx

dM
⇒ =V
dx
]TahrN¾³ kMNt;RbtikmμTMrsMrab;FñwmdUcbgðajkñúgrUb.
20kN 30kN
2m 3m
w=12kN/m

A B
2m 8m

Beam diagram


T.Chhay

dMeNaHRsay³
W =60kN 20kN 30kN
2m 3m
w=12kN/m

A 0.5m B
2m 8m
RA RB
Load diagram

kMNt;bnÞúkcMcMnucsmmUl nwgbnÞúkBRgayesμIelIRbEvg 5m
W = w × l = 12 × 5 = 60kN
TItaMgénbnÞúkcMcMnucsmmUl
l 5
c= = = 2.5m
2 2
BIdüaRkambnÞúkkMNt;RbtikmμRtg;TMr B edayeFVIplbUkm:Um:g;Rtg;TMr A
edaysnμt; m:Um:g;bUk kalNaTisedAvilRcasTisRTnicnaLika
∑ M A = + RB × 8 − 30 × 5 − 20 × 3 − 60 × 0.5 = 0

⇒ RB = +30kN ↑
sMrab;RbtikmμRtg;TMr ARtUv)ankMNt;edayeFVIplbUkm:Um:g;Rtg;TMr B
∑ M B = − R A × 8 + 30 × 3 + 20 × 5 + 60 × 7.5 = 0

⇒ R A = +80kN ↑
epÞógpÞat;edayeFVIplbUkkMlaMgtamGkS½ ∑Y = 0
∑Y = 30 − 60 − 20 − 30 − 80 = 0 RtwmRtUv
]TahrN¾³ düaRkambnÞúkRtUv)anbgðajkñúgrUb. kMNt;kMlaMgkat;TTwg nigm:Um:g;Bt;
k> Rtg;cMnuc 5m BIcugTMrxageqVg
x> Rtg;cMnuc 12m BIcugTMrxageqVg
20kN 10kN
0.3m 0.7m 15kN
w=30kN/m

A B
15m 3m

RA RB


T.Chhay

dMeNaHRsay³
BIdüaRkambnÞúkkMNt;RbtikmμRtg;TMr B edayeFVIplbUkm:Um:g;Rtg;TMr A
edaysnμt; m:Um:g;bUk kalNaTisedAvilRcasTisRTnicnaLika
15
∑ M A = + RB × 15 − 15 × 18 − 10 × 10 − 20 × 3 − (30 × 15) × =0
2
⇒ RB = +253.7 kN ↑
sMrab;RbtikmμRtg;TMr ARtUv)ankMNt;edayeFVIplbUkm:Um:g;Rtg;TMr B
∑ M B = − R A × 15 − 15 × 3 + 10 × 5 + 20 × 12 + 450 × 7.5 = 0

⇒ R A = +241.3kN ↑
epÞógpÞat;edayeFVIplbUkkMlaMgtamGkS½ ∑Y = 0
∑Y = +253.7 + 241.7 − 15 − 10 − 20 − 450 = 0 RtwmRtUv
k> Rtg;cMnuc 5m BIcugTMrxageqVg
20kN Cutting plane
0.3m 0.2m
w=30kN/m Internal resisting shear
M
A x
5m V Internal resisting moment

RA =241.3kN

eyIgkat;FñwmenAcMnuc 5m BIcugTMrxageqVg ¬tagedayGkSr x ¦ ehIyeyIgcat;TukEpñkxageqVgénmuxkat;CaGgÁ
esrI (FBD) . kMlaMgkat;TTwgTTYl)anedayeFVIplbUkkMlaMgbBaÄrEdlmanGMeBIelIGgÁesrIesμIsUnü
∑ Y = 0 ⇒ +241.7 − 20 − 30 × 5 − V = 0

⇒ V = +71.3kN ↓
ehIym:Um:g;Bt;EdlekItBIkMlaMgxageRkARtg;muxkat; (x) )anBIeFVIplbUkm:Um:g;eFobcMnucNamYyesμIsUnü
edaysnμt;m:Umg;EdleFVIeGayFñwmrgkarsgát;EfñkxagelImansBaØabUk
5
∑ M A = 0 ⇒ + M − 20 × 3 − 30 × 5 × − 71.3 × 5 = 0
2

⇒ M = +791.5kN .m
x> Rtg;cMnuc 12m BIcugTMrxageqVg


T.Chhay

20kN 10kN Cutting plane
0.3m 0.7m 0.2m
w=30kN/m Internal resisting shear
M
A x
12m V Internal resisting moment

RA =241.3kN

eyIgkat;FñwmenAcMnuc 12m BIcugTMrxageqVg ¬tagedayGkSr x ¦ ehIyeyIgcat;TukEpñkxageqVgénmuxkat;Ca
GgÁesrI (FBD) . kMlaMgkat;TTwgTTYl)anedayeFVIplbUkkMlaMgbBaÄrEdlmanGMeBIelIGgÁesrIesμIsUnü
∑ Y = 0 ⇒ +241.7 − 20 − 30 × 12 − 10 − V = 0

⇒ V = −148.3kN ↓
rW V = +148.3kN ↑
ehIym:Um:g;Bt;EdlekItBIkMlaMgxageRkARtg;muxkat; (x) )anBIeFVIplbUkm:Um:g;eFobcMnucNamYyesμIsUnü
edaysnμt;m:Umg;EdleFVIeGayFñwmrgkarsgát;EfñkxagelImansBaØabUk
12
∑ M A = 0 ⇒ + M − 20 × 3 − 30 × 12 × − 10 × 10 + 148.3 × 12 = 0
2

⇒ M = +540.4kN .m
*** cMNaM³ kñúgkrNIEdleKcg;dwgkMlaMgkat;TTwgRtg;TItaMgbnÞúkcMcMnuc eKRtUvkat;FñwmcMnYnBIrdg mþgxagmux
bnÞúkcMcMnuc nigmþgeTotxageRkaybnÞúkcMcMnuc. ehIytMélénkMlaMgkat;TTwgTaMgBIrmantMélxusKñaesμInwgtMél
énbnÞúkcMcMnucenaH.
3> düaRkamkMlaMgkat;TTwg nigm:Um:g;Bt; Shear force and bending moment diagram
düaRkamkMlaMgkat;TTwg nigdüaRkamm:Um:g;Bt;mansar³sMxan;kñúgkardwgBIbMErbMrYlkMlaMgkat;TTwg
nigm:Um:g;Bt;tamRbEvgrbs;Fñwm nigGaceGayeyIgdwgBITItaMgEdlkMlaMgkat; nigm:Um:g;Bt;mantMélGtibrma.
edIm,IKUrdüaRkamkMlaMgkat;TTwg nigdüaRkamm:Um:g;Bt;sMrab;FñwmsþaTic eyIgGaceRbIviFImuxkat;. EtCadMbUgeyIg
RtUvkMNt;RbtikmμTMrCamunsin rYcehIyeTIbeyIgcab;epþImkat;FñwmCakMNat;². CaTUeTAeyIgRtUvkat;FñwmenAcenøaH
kMlaMgcMcMnucBIr Rtg;TItaMg x BITMreTaHxageqVgb¤xagsþaM rYceTIbGnuvtþsmIkarkMlaMgsþaTicsMrab;ral;kMNat;
nImYy².
]TahrN¾³
cUrsg;düaRkamkMlaMgkat;TTwg nigdüaRkamm:Um:g;Bt;sMrab;FñwmrgbnÞúkxageRkam


T.Chhay

15kN 10kN
2m 2m
w=4kN/m

A C D B E
6m 2m

dMeNaHRsay³
kMNt;RbtikmμRtg;TMr A edayeFVIplbUkm:Um:g;Rtg;TMr B = 0
edaysnμt;kMlaMgEdleFVIeGayvilRsbTisRTnicnaLika mansBaØaGviC¢man
2
⇒ ∑ M B = 0 ⇒ −( R A × 6) + (15 × 4) + (10 × 2) − (4 × 2 × ) = 0
2
⇒ R A = 12kN
kMNt;RbtikmμRtg;TMr B edayeFVIplbUkm:Um:g;Rtg;TMr A = 0
edaysnμt;kMlaMgEdleFVIeGayvilRsbTisRTnicnaLika mansBaØaGviC¢man
2
∑ M A = 0 ⇒ −(15 × 2) − (10 × 4) + ( RB × 6) − [4 × 2 × (6 + )] = 0
2
⇒ RB = 21kN
epÞógpÞat;edayeFVIplbUkkMlaMgtamGkS½ ∑ Y = 0
∑Y = +12 + 21 − 15 − 10 − (4 × 2) = 0 RtwmRtUv
sikSakMlaMgkat;TTwg nigm:Um:g;
- muxkat; (1 − 1) BI A → C 0 ≤ x ≤ 2m 0 1
x M
∑ Y = 0 ⇒ 12 − V1−1 = 0
⇒ V1−1 = 12kN efr CasmIkarbnÞat;efrRsbGkS½ X A
∑ M A = 0 ⇒ M 1−1 − V .x = 0
1V
⇒ M 1−1 = 12 x CasmIkarbnÞat;Edlmanrag ax + b
RA
ebI x = 0m ⇒ M = 0kN .m A
15kN
ebI x = 2m ⇒ M = 24kN.m C
2m 0 x1
M
- muxkat; (2 − 2) BI C → D 0 ≤ x ≤ 2m
A C
∑ Y = 0 ⇒ 12 − 15 − V2−2 = 0
V
⇒ V2−2 = −3kN efr CasmIkarbnÞat;efrRsbGkS½ X 1
RA
∑ M A = 0 ⇒ M 2−2 − V .( 2 + x) − 15 × 2 = 0


T.Chhay

⇒ M 2−2 = −3 x + 24 CasmIkarbnÞat;Edlmanrag ax + b
ebI x = 0m ⇒ M = 24kN.m C 15kN 10kN
0 x1
ebI x = 2m ⇒ M = 18kN .m D
2m
M

- muxkat; (3 − 3) BI D → B 0 ≤ x ≤ 2m A C D
4m V
∑ Y = 0 ⇒ 12 − 15 − 10 − V3−3 = 0 1

⇒ V3−3 = −13kN efr CasmIkarbnÞat;efrRsbGkS½ X RA

∑ M A = 0 ⇒ M 3−3 − V .(4 + x) − 15 × 2 − 10 × 4 = 0
⇒ M 1−1 = −13 x + 18 CasmIkarbnÞat;Edlmanrag ax + b
ebI x = 0m ⇒ M D = 18kN .m

ebI x = 2m ⇒ M B = −8kN .m
18
M 3−3 = 0 ⇒ x = m
13
- muxkat; (4 − 4) BI E → B 0 ≤ x ≤ 2m

∑ Y = 0 ⇒ V4−4 − 4 x = 0
4
⇒ V4−4 = 4.xkN CasmIkarbnÞat;Edlmanrag ax + b M w=4kN/m
ebI x = 0m ⇒ V E = 0kN .m

ebI x = 2m ⇒ V E = 8kN .m x 0 E
x 4
∑ M E = 0 ⇒ M 4− 4 + V . x − 4 × x × =0 V
2
⇒ M 4− 4 = 2 x 2 − 8 x CasmIkarbnÞat;Edlmanrag ax + b
ebI x = 0m ⇒ M = 0kN.m E

ebI x = 2m ⇒ M = −8kN .m B

düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;RtUv)ansg;edaytMélEdl)ansikSatammuxkat;mYy²
eyIg)an


T.Chhay

+12kN

+8kN

Shear diagram
C D B E
A V (kN)
-3kN

-13kN

-8kN

C D E Moment diagram
A
B M (kN.m)

+18kN
+24kN

4> düaRkamkMlaMgkat;TTwg edayeRbIdüaRkambnÞúk
Shear diagram by using Load diagram
CaTUeTAsMrab;FñwmsþaTickMNt; munnwgkMNt;rkkMlaMgkat;TTwgeKRtUv rkRbtikmμTMrCamunsin b¤bMElg
FñwmKMrU eTACaGgÁesrI b¤eTACadüaRkambnÞúk. kMlaMgkat;TTwgesμInwg plbUkrvagbnÞúkcMcMnuc nigRkLaépÞénbnÞúk
BRgayEpñkxageqVg b¤EpñkxagsþaM.
V = ∑ ( Pext + SQ ) = ∑ ( Pext + S Q )
left right

enAeBldüaRkambnÞúk CabnÞúkcMcMnuc enaHdüaRkamkMlaMgkat;TTwgmanragCabnÞat;efrRsbGkS½énRb
EvgFñwm. enAeBldüaRkambnÞúk CabnÞúkBRgayesμI enaHdüaRkamkMlaMgkat;TTwgmanragCabnÞat;eRTtsmIkardW
eRkTI1. ebIdüaRkambnÞúk CabnÞúkBRgayminesμI enaHdüaRkamkMlaMgkat;TTwgmanragCa)a:ra:bUlsmIkardWeRk
TI2.
]TahrN¾³ edayeRbIlT§pldüaRkambnÞúkén]TahrN¾xagelI cUrsg;düaRkamkMlaMgkat;TTwg.
dMeNaHRsay³
V A = +12kN
VC = 12 − 15 = −3kN

VD = 12 − 15 − 10 = −13kN


T.Chhay

VB = 12 − 15 − 10 + 21 = +8kN

VE = 12 − 15 − 10 + 21 − 4 × 2 = 0kN

5> düaRkamm:Um:g;Bt; edayeRbIdüaRkamkMlaMgkat;TTWg
Bending moment diagram by using Shear diagram
eKGacsg;düaRkamm:Um:g;Bt;edayeRbIdüaRkamkMlaMgkat;edaypÞal; edaymin)ac;eRbIviFImuxkat;. m:Um:g;
Bt;esμInwgplbUkrvagRklaépÞéndüaRkamkMlaMgkat;TTwg nigm:Um:g;xageRkA EpñkxageqVgb¤EpñkxagsþaM.
M = ∑ ( SV + M ext ) = ∑ ( SV + M ext )
left right

enAeBldüaRkamkMlaMgkat;TTwgmanragCabnÞat;efr enaHdüaRkamm:Um:g;manragCabnÞat;eRTtsmIkar
dWeRkTI1. enAeBldüaRkamkMlaMgkat;TTwgmanragCabnÞat;eRTtsmIkardWeRkTI1 enaHdüaRkamm:Um:g;Bt;man
ragCa)a:ra:bUlsmIkardWeRkTI2. ebIdüaRkamkMlaMgkat;TTwgmanragCa)a:ra:bUlsmIkardWeRkTI2 enaHdüaRkam
m:Um:g;Bt;manragCa)a:ra:bUlsmIkardWeRkTI3.
]TahrN¾³ edayeRbIlT§pldüaRkamkMlaMgkat;TTwgén]TahrN¾xagelI cUrsg;düaRkamm:Um:g;Bt;.
dMeNaHRsay³
edaysarFñwmenaHminrgm:Um:g;xageRkAdUcenH
m:Um:g;Bt;esμInwgplbUkRkLaépÞéndüaRkamkMlaMgkat;TTwgEpñkxageqVgb¤EpñkxagsþaM
M = 0 edayenAEpñkxageqVgcMnuc A KμankMlaMgkat;TTwg
A

M C = 12 × 2 = +24kN .m

M D = 12 × 2 − 3 × 2 = +18kN .m

M B = 12 × 2 − 3 × 2 − 13 × 2 = −8kN .m
2
M E = 12 × 2 − 3 × 2 − 13 × 2 + 8 × = 0 KN .m
2
6> viFItMrYtpl Method of superposition

viFItMrYtpl RtUv)aneRbIsMrab;sMrYlkñúgkarsikSadüaRkamkMlaMgkat;TTwg nigdüaRkamm:Um:g;Bt; sMrab;Fñwm
TaMgLayNaEdlrgkMlaMgeRkAeRcIn. edIm,IsMrYlkñúgkaredaHRsay KWeKRtUvedaHRsaykMlaMgkat; nigm:Um:g;Bt;
kñúgkrNIbnÞúknImYy² rYcehIyeTIbeKeFVIplbUkBiCKNitlT§plEdl)an.


T.Chhay

]TahrN¾³
15kN
15kN

1m
1m

w=4kN/m A B
= + 3m
A B
3m w=4kN/m

A B
3m


T.Chhay

lMhat;³
cUrsg;düaRkamm:Um:g; kMlaMgkat;TTwg nigkMlaMgtamGkS½³
1>

2>

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4>

5>

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T.Chhay

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8.shear and bending moment in beam10

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8.shear and bending moment in beam10