1. Department of Civil Engineering NPIC
cemøIysRmab;cMeNaTeRCIserIs
(Answers to selected problems)
CMBUk !
!>! DL = 18.4kN / LL = 43.2kN @>!& B y = 44.6kN / Ay = 16.8745kN /
!>@ DL = 13.1kN Ax = 22.5kN
!># P = 4.06kN / m2 @>@0 FB = 495kN / Ax = 428.7kN /
!>% FD = 90kN / FL = 92.2kN / F = 182kN A y = 22.5kN
!>^ Wc = 9.16kN / m @>@@ Ay = 73.1kN / Bx = 0 / B y = 25.9kN
!>& 5kN / m 2 @>@$ w2 = 22.85kN / m / w1 = 91.44kN
!>* DL = 6.08kN / m, LL = 10.77kN / m @>@& FA = 44.78kN / Bx = 38.78kN /
!>( p x = 0.20kN / m 2 , p y = 0.35kN / m 2 B y = 16.39kN
!>!0 Fg = 49.71kN , FR = 20.25kN @>@* wB = 36.5kN / m / w A = 178.6kN / m
!>!! Fg = 415.5kN @>#@ B y = 74.6kN , Ax = 130.5kN ,
!>!# F = 12.0kN A y = 76.6kN
!>!$ F = 1.80kN @>#$ E y = 15kN , E x = 15kN ,
!>!% sm<aFxageRkAenAelIépÞdMbUlRtUvTisxül; / Ay = 2.5kN , Ax = 0
B y = 187.5kN
p = −488 N / m 2 @>#& B y = 60kN / Ay = 20kN / E y = 40kN /
sm<aFxageRkAenAelIépÞdMbUlminRtUvTisxül; F y = 20kN
p = −236 N / m 2 @>$# B x = 124kN , B x = 111kN ,
sm<aFxagkñúg p = m134 N / m 2 Ax = 124kN / Ay = 111kN /
/
C x = 41kN C y = 159kN
CMBUk @
@>$( Ax = 2.347kN , A y = 6.629kN /
@>! sRmab; BG / w = 97.5kN / m
C x = 3.053kN , C y = 8.879kN
sRmab; ABCD / kmøaMgRbtikmμKW 21.9kN
@>@ bnÞúktambeNþay DF = 7.2kN / m
CMBUk #
enARtg; A nig E ³ F ' = 27kN
#>$ FCD = 11.3(C ), FCB = 8kN (T ),
enARtg; A, B, C, D ³ F = 54kN
FDA = 11.3kN (T ), FDE = 16kN (C ),
cemøIysRmab;cMeNaTeRCIserIs T.Chhay -550

2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
FBE = 11.3kN (C ), FBA = 24kN (T ) /
FCD = 4.08kN (T ) FDH = FDF = 0
Ax = 16kN / Ay = 8kN
#>( FAH = 25kN (C ), FAB = 20kN (T ), CMBUk $
FBC = 20kN (T ), FBH = 10kN (T ), $># N B = 0 / VB = 144kN / M B = 192kN .m
FHG = 16.7kN (C ), FHC = 8.3kN (C ), $>* N B = 0 / VB = 264kN / M B = 432kN .m
FGF = 16.7 kN (C ), FGC = 20kN (T ) $>!0 V = 2kN / M = (2 x + 16)kN .m
#>!$ FIH = 7.45kN (C ), FBC = 6.0kN (T ), $>!$ VC = 2.5kN / M C = 6kN .m / N C = 0
FBH = 1.49kN (T ) $>!( a = L
3
#>!* FKJ = 115kN (C ), FCJ = 27.0kN (T ), ⎛ 3⎞ ⎛ 4⎞
$>@^ V =
⎜ 9 ⎟
/
⎜ 6 − x ⎟kN M = ⎜ 6 x − x ⎟kN .m
⎜ 36 ⎟
FCD = 97.5kN (T ) ⎝ ⎠ ⎝ ⎠
#>@% FGF = 1.78kN (T ), FCD = 2.23kN (C ), $>#@ V = 2kN / M = (2 x + 4)kN .m
FCF = 0 $>%& Cy = 8.571kN / B y = 6.786kN
#>@( FCD = 24kN (T ), FCF = 13.4kN (C ), Ay = 13.21kN / D y = 11.78kN
FED = 13.4kN (C ), FAE = FBF = 9.49kN (C ),
$>^! FDE = 9kN / C y = 1.8kN /
FAC = FBC = 4.24kN (C ), C x = 7.2kN
FEC = FDF = 12.7kN (T )
#>#^ FDE = 10kN (T ), FDI = 3.33kN (C ), CMBUk %
FJI = 7.83kN (C ) %>@ P = 42kN
#>$! FBC = 8kN (T ), FHG = 10.1kN (C ), %>% y B = 0.867m / y D = 0.704m
FBG = 1.8kN (T ) %>!0 x B = 2.18m
#>$# /
FKJ = 15.3kN (C ) FCD = 15.3kN (T ) %>!# y = 0.0267 x 2 / Tmax = 60.0kN
/
FHD = 0.83kN (T ) FNJ = 0.83kN (C ) %>!% TB = 36.3kN / TA = 30.9kN
#>$( FAC = FAE = 0 / FAB = 4kN (T ) / %>!* B y = 0 / FH = Fmin = 100kN /
R B = FBE = 5.66kN (T ) / FBD = 2kN (C ) / Fmax = 117kN / T = 10kN
FDE = FDC = FCE = 0 %>@! M max = 3.56kN .m
#>%! FAD = 12.4kN (T ) / FAB = 5.09kN (C ) %>#! C x = Ax = 167 N / C y = 111N
FAC = 5.09kN (C ) Ay = 389 N
#>%$ FCH = 12.2kN (C ) / FCB = 4.08kN (T )
Answer to slected problems T.Chhay -551

3. Department of Civil Engineering NPIC
CMBUk ^ CMBUk *
^>!( (M C )max = 40kN .m / (VC )max = 1.5kN *>% θ max = − M 0 L / ν max = − MEIL
0
2
EI 2
^>@$ (M C )max = 30kN .m / − M 0 L2
(VC )max = 4.75kN
*>* θ max =
M 0L
2 EI
/
ν max =
8 EI
− PL 2 3
^>@* (M A )max = −106kN .m / *>!0 θB =
2 EI
/ ΔB =
PL
3EI
(VB )max = 17.5kN 2
19 Pa 3
*>!# θA =
5Pa
/ ΔC = ↓
^>#^ (VBC )max = 108.4kN / 2 EI
Pa 3
6 EI
(M C )max = 184.5kN .m
*>!% ΔD =
12 EI
↑
^>$@ (VCD )max = −30kN *>!( θA =
5M 0 a
/ M a2
ΔC = 0 ↑
12 EI 4 EI
^>%! (FHG )max(C ) = 160kN (C ) / *>@! ΔC =
84
/
↓ θA =
8
/ θB =
16
/
(FHG )max(T ) = 0 EI
40
EI EI
θC =
^>%& x = 6m / (FGF )max = 67.1kN (C ) EI
^>^0 (VC )max = 11.8kN / *>@# a= L
3
16
(M C )max = 70.9kN .m *>@% θ B = 0.00172rad / Δ C = 1.107mm ↓
^>^# (FIH )max = 38.4kN (C ) *>@* θ A = 2EI / Δ A = −3PL
PL2 3
EI
^>^& (FIH )max = 15kN (C ) *>#!> θ C = −0.00171rad / Δ C = −3.86mm
^>&0 (M C )max = 201kN .m *>#$ Δ A = 6.75mm
^>&@ Vmax = 10kN / M max = 39kN .m
*>#& θ C = − 3EI / Δ C = − EI
2 PL2 PL3
^>&% M max = 1031kN .m
CMBUk & CMBUk (
&>* M F = 8.1kN .m / M D = 14.4kN .m (># Δ B = 1.50mm ←
h
&>!! M I = 12.96kN .m / M L = 29.16kN .m (>& Δ C = 0.0375mm
v
M H = 38.88kN .m
(>!! Δ B = 0.751mm
v
&>!^ FCF = 48.08kN (>!% Δ C = 0.234mm ←
h
&>#! Ay = 11.7kN / Ax = 9kN / (>!* Δ B = 0.942mm
v
M A = 24.3kN .m
(>@! Δ E = 0.233mm ↓
v
(>@$ Δ B = 0.525mm ↓
v
cemøIysRmab;cMeNaTeRCIserIs T.Chhay -552

4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
(>@* Δ A = 12.7mm ↑
v
A y = 14.06kN/ M A = 8.44kN .m
(>#@ Δ B = 0.4mm
v
!0>@# Ay = 4.688kN / C x = 50kN /
(>#^ Δ D = 4.88mm C y = 4.688kN / M C = 81.25kN .m
v
(>$0 Δ B = 1.01mm
h
!0>@( FHG = 2.32kN (C )
(>$# Δ C = 0.9504mm →
h
!0>#@ FBE = 3.62kN (C )
3
(>$& Δ C = 56Pa ↓ !0>#& FCD = 163kN (C ) /
EI
FCA = FCB = 212kN (T )
Pa 2
(>%0 θA =
EI !0>$0 FDB = 19.2kN / FCB = 53.4kN
(>%$ θ A = 4.5(10 −3 )rad !0>$% M D = 46.9kN .m
4 PL3
(>%* Δ A = 3EI →
h
(>^@ Δ B = 10.59mm ↑ CMBUk !!
kN 3
(>^& Δ D = 59.4EI .m ↓ !!>@ M BC = 7 M 0 / M AB = − 1 M 0
2
7
2
M BA = − M 0
(>&0 Δ C = 49.1mm ↓
V
7
kN 3
(>&$ Δ C = 9.62EI .m → !!>% M AB = 5kN .m / M BA = 10kN .m /
M BC = −10kN .m / M CB = 25kN .m
H
!!>& Ay = 53.3kN / B y = 72.0kN / C x = 0 /
CMBUk !0
C y = −5.3kN
!0># B y = 153.75kN / Ay = 13.125kN /
!!>!! M BA = 9.47kN .m / M AB = 4.74kN .m /
C y = 73.125kN
M BC = −9.47 kN .m / M CB = 26.52kN .m
!0>^ Ax = 0 / B y = 5wL / C y = Ay = 3wL
4 8 !!>!% M AB = 12.4kN .m / M BA = 24.9kN .m
!0>!0 B y = −68.8kN / C y = 15.6kN / M BC = 24.9kN .m / M CB = 9kN .m
Ax = 0 / A y = 15.6kN !!>@# M AB = −3750kN .m /
!0>!# B y = 54.2kN / Bx = 0 / C y = 12.5kN / M BA = −822kN .m / M BC = 822kN .m
Ay = 13.3kN M CB = 829kN .m / M CD = −829kN .m
!0>!& M = PL / M A = M B = PL /
8 8 !!>@^ M BA = 36kN .m / M BC = −36kN .m /
Ay = B y =
P M CB = −36kN .m / M CD = 36kN .m
2
!0>@0 C y = 8.44kN / Ax = 13.5kN /
Answer to slected problems T.Chhay -553

5. Department of Civil Engineering NPIC
CMBUk !@ !$>!0 q2 = 62.5kN (C )
!@>$ M AB = −11.6kN .m / M BA = 12.8kN .m / !$>!@ q1 = 5.66kN (C ) / q5 = 4.00kN (T )
M BC = −12.8kN .m / M CB = 13.8kN .m !$>!$ D1 = −0.0624mm / q2 = 50.9 N (C )
!@>& M AB = 0 / M BA = 55.5kN .m / !$>!^ Q4 = 94.3kN / Q5 = −117kN /
M BC = −55.5kN .m / M CB = 44.25kN .m Q6 = −66.7kN
!@>( Ay = 6kN / D y = 6kN / B y = 14kN /
C y = 14kN CMBUk !%
!@>!! M B = −7.61kN .m !%>! Q5 = −7.50kN / Q6 = 5.0kN .m
!@>!^ M BA = −104kN .m / M BC = 104kN .m / !%># Q2 = −6.0kN / Q5 = 30kN .m /
M CB = 196kN .m / M CD = −196kN .m Q6 = 33.0kN
M AB = M DC = 0 !%>^ Q4 = 127.5kN / Q5 = 105kN
Q6 = 127.5kN
CMBUk !# !%>* Q3 = −204kN .m / Q5 = 38kN .m
!#>@ K A = 25.1(103 )kN .m / C AC = 0.579 2
!%>!! Q2 = − wL / Q3 = wL / Q4 = − wL
2
!#>^ K A = K B = 31.3(103 )kN .m /
6 3
C AB = C BA = 0.599
CMBUk !^
!#>( C AB = 0.707 / K A = 14.25(10 )kN .m
3
!^>$ Q5 = −36.3kN / Q6 = −46.4kN /
C BA = 0.315 / K B = 31.95( 3 )kN .m
Q7 = 36.3kN / Q8 = 46.4kN
10
!#>!@ M AB = 113kN .m / M BA = 151kN .m / Q9 = 77.1kN .m
M BC = −151kN .m / M CB = 461kN .m
!^>& Q7 = 20kN / Q8 = 0 / Q9 = 20kN
!#>!* M AB = M CD = 0 / !^>( D1 = −82.84mm / D2 = −151.1mm /
M BA = M DB = 28.3kN .m /
D3 = 0.0327rad
M BD = M DC = −28.3kN .m
!^>!# Q9 = −6.74kN / Q10 = 18.0kN /
Q11 = −17.3kN / Q12 = −18.0kN
CMBUk !$ !^>!% Q6 = 8.50kN / Q7 = 52.6kN /
!$>@ D1 = 0 / D2 = −0.694mm Q8 = 56.5kN / Q9 = 3.43kN
!$>% D4 = −0.45mm / q4 = 6.67k (T )
!$>& D1 = 2.0334(10 −6 )m / q2 = 0.508kN
cemøIysRmab;cMeNaTeRCIserIs T.Chhay -554

6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
]bsm<½n§ A
⎡6 3 ⎤
A.3 ⎢ ⎥
⎣3 14⎦
⎡ 79 ⎤
A.7 ⎢ ⎥
⎣− 5⎦
⎡ 2 48 2⎤
A.10
⎢ 0 17 2⎥
⎢ ⎥
⎢ − 6 − 5 2⎥
⎣ ⎦
A.14 − 33 , − 18
⎡0.75⎤
A.17
⎢0.25⎥
⎢ ⎥
⎢0.75⎥
⎣ ⎦
A.20 x1 = 1 , x3 = −1 , x 2 = 1
Answer to slected problems T.Chhay -555