1.3-1
Ay = 130.4 N, Bx= 135 N, By = 421.5 N,
1.3-13
N = 0 N, V = 37.5 N, M = –198.7 Nm
1.3-2
Ay = 198.3 N, Bx = –120 N, By = –38.3 N,
(b) FAC = 0.960 P
1.3-14
N = 0 N, V = –41.7 N, M = 510 Nm
1.3-3
(a) Ay = 22.7 N, By = –22.7 N, Cx = 220 N, Cy = 0
(a) MA = 0, Cy = 236 N, Dy = –75.6 N
1.3-15
1.3-5
1.3-16
1.3-17
1.3-7
1.3-8
1.3-18
L 

T   L1 + 2  = −T2 = −1100 N  m
2 

(a) Ax = 0, Ay = 4.5 kN, Ey = 22.5 kN
(b) FFE = 8.13 kN
(b) Nx = 247 N, Vx = 2229 N, Mx = –3586 Nm
(a) Fx = 0, Fy = 12.0 kN, Dy = 6.0 kN
1.3-19
1.3-20
1.3-21
Ay = –173.1 N, Dx= –236.9 N, Dy = 724.9 N,
N = 0, V = –266.1 N, M = –33.6 Nm
1.3-22
Ax = –80 N, Ay= –11.67 N, Dy = 308 N,
N = –308 N, V = –80 N, M = –53.3 Nm
1.3-23
(a) Ax = 285 N, Ay= 1330 N, Cx = –285 N,
Ax = 0, Ay = 8.67 kN, By = 71.3 kN
AD = –10.01 kN, AB = 5.01 kN, DB = 141.3 kN,
DC = 42.2 kN, CB = –181.3 kN
1.3-12
Cy = 570 N
(b) N = –222 N, V = –190 N, M = –95 Nm
Ax = 0, Ay = –86.1 kN, By = 86.1 kN
AC = 99.4 kN, BD = 102.6 kN
Ax = 0, Ay = –8.67 kN, By = 71.3 kN
AB = 5.01 kN, DC = 42.2 kN
(a) Ax = 280 N, Ay = 8.89 N, MA = –1120 Nm,
Dy = 151.1 N
(b) Συνισταμένη δύναμη στο σημείο Β: 280 Ν
Ax = 0, Ay = –86.1 kN, By = 86.1 kN
BD = 102.6 kN, DC = –94.2 kN
1.3-11
L1
= 62.5 N  m ,
2
(a) Ax = –2405 N, Ay = –247 N, MA = 5932 Nm,
Cy = 247 N
AC = 99.4 kN, AD = –49.7 kN, BC = –134 kN,
1.3-10
(a) TA = –1225 Nm
(b) T 
(b) FFE = 0
1.3-9
L1
= −1270 Ν  m ,
2
L 

T   L1 + 2  = 1130 N  m
2 

(a) R3x = 40 N, R3y = –25 N, R5x = 20 N
(b) F11 = 0, F13 = 28.3 N
(a) TA = 1270 Nm
(b) T 
(b) Συνισταμένη δύναμη στο σημείο Β: 88.2 Ν
1.3-6
(a) Az = 0, Bx = –3.75 kN
(b) FAB = 6.73 kN
(a) Ax = 57.7 N, Ay = –66.7 N, Cy = 464.3 N,
Dx = 50.3 N, Dy = –87.2 N
(c) Ax = 191.7 N, Ay = 165.5 N, MAz = 696.5 Nm,
Dx = –83.7 N, Dy = 145 N, συνισταμένη δύναμη
στο σημείο Β: 253 Ν.
(a) Ay = 4.67 P, Az = –4.0 P
(b) FAB = –8.33 P
(b) N = 0, V = –70 N, M = –36.7 Nm
(c) MA = 0, Cy = 236 N, Dy = –75.6 N, N = 0,
V = –70 N, M = –36.7 Nm
(a) Ax = –1.25 P, By = 0, Bz = –P
(b) FAB = 1.601 P
(b) N = 220 N, V = 22.7 N, M = 102 Nm
1.3-4
(a) Bx = –0.8 P, Bz =2.0 P, Oz = –1.25 P
1.3-24
(a) Ax = 10.98 kN, Ay = 29.0 kN, Ex = –8.05 kN,
Ey = –22 kN
(b) Συνισταμένη δύναμη στο σημείο C: 23.4 kΝ
1.3-25
(a) Ay = –5625 N, Ex= 0, Ey = 7875 N
(b) N = 7875 N, V = 2250 N, M = 776 Nm
1.3-26
(a) Ax = 320 N, Ay = –240 N, Cy = 192 N,
Ey = –192 N
1.4-5
(b) σcable = 185.7 MPa
1.4-6
(b) N = –312 N, V = –57.9 N, M = 289 Nm
(a) Ax = –125 N, Ay = 217 N, Bx = –281 N
(c) Pmax = 89.5 kN
1.4-7
(b) T = 310 N
1.3-28
(a) Ax = –10 kN, Ay = –2.17 kN, Cy = 9.83 kN,
Ey = 1.333 kN
1.4-8
(a) Ox = –213 N, Oy = 180 N, Oz = 56.9 N,
MOx = 37.1 Nm, MOy = 75.9 Nm, MOz = –37.8 Nm
(a) Ay = –120 N, Az = –60 N, MAx = –70 Nm,
MAy = –142.5 Nm, MAz = –180 Nm, Dx = –60 N,
Dy = 120 N, Dz = 30 N
(b) N = 120 N, V = 41.3 N, T = 142.5 Nm,
M = 180.7 Nm
1.3-31
1.4-9
1.3-32
(c) σ1 = 120.5 MPa, σ2 = 119.2 MPa, σ3 = 141.6
MPa
1.4-10
σC = 5.21 MPa
1.4-11
(a) T = 1.298 kN, σ = 118 MPa
(b) εcable = 8.4 10-4
1.4-12
Ay = 253N, Bx = 195.8 N προς τα αριστερά,
By = 504 N, Cx = 129.2 N, Cy = 26.2 N
1.3-34
(a) HB = –104.6 N, VB = 516 N, VF = 336 N
1.4-13
 18890 


28532 
(c) T = 
N
 14755 


 18890 
(a) σΑΒ = 9.95 MPa
(b) P2 = 6 kN
 245 


371 

σ=
MPa
 192 


 245 
(c) tBC = 12.62 mm
1.4-2
(a) σ1 = 23.2 MPa, σ2 = 10.61 MPa
(b) ε1 = 6.8 10-3, ε2 = 3.13 10-3
1.4-14
1.4-3
σmax = 3.9 MPa
1.4-4
(a) σ = 130.2 MPa
(b) ε = 4.652  10
-4
 25951 
(a) T =  20662  N
 31616 


 337 
(b) σ =  268  MPa
 411 


(b) N = –646 N, V = 176.8 N, M = 44.9 kNm
1.4-1
(a) T = 819 Ν, σ = 74.5 MPa
(b) εcable = 4.923 10-4
Cx = 120 N, Cy = –160 N, Cz = 506 N, Dz = 466 N,
Hy = 320 N, Hz = 499 N
1.3-33
(a) σ1 = 245 MPa, σ2 = 206 MPa
(b) d1new = 0.818 mm
(a) Ax= 26 N, Ay = 208 N, Az = –10.39 N,
MAz = 21.9 Nm
(b) TDC = 17.13 N, TEC = 30.6 N
(a) σt = 132.7 MPa
(b) αmax = 34.4°
(b) Nx = –180 N, Vres = 220 N, Tx = –75.9 Nm,
Mres = 53 Nm
1.3-30
(a) σC = 16.28 MPa
(b) xC = 489 mm, yC = 489 mm
(b) Συνισταμένη δύναμη στο σημείο D: 12.68 kΝ
1.3-29
(a) εs = 3.101  10-4
(b) δ = 0.1526 mm
(c) Συνισταμένη δύναμη στο σημείο C: 400 Ν
1.3-27
(a) RB = 400 N (αμφιέρειστη), RB = 848 N (V),
σC = 1.0 MPa (αμφιέρειστη), σC = 2.12 MPa (V)
2
2
2
(a) σ x = γω ( L − x )
2g
(b) σ max =
γω2 L2
2g
1.4-15
1.4-16
(a) TAB = 6757 N, TBC = 6358 N, TCD = 6859 N
1.7-3
P = –38.5 kN
(b) σΑΒ = 87.8 MPa, σΒC = 82.6 MPa, σCD = 89.1
MPa
1.7-4
(a) Lp = 405.25 mm
(a) TAQ = TBQ = 50.5 kN
(b) σ = 166 MPa
1.4-17
(b)
(a) TA = 5.56 kN, TB = 4.53 kN
1.5-2
1.5-3
P = 612.5 kN, gap = 5.03 mm
1.7-6
(a) P = 74.1 kN
(b) δ = ε∙L = 0.469 mm βράχυνσης,
Af − A
A
(a) Lmax = 3377 m
ΔV1 = ΔV1 f − Vol1 = −207 mm3
(a) Lmax = 7143 m
(c) d3 = 65.4 mm
(b) Lmax = 8209 m
1.7-7
Δd = –4.17  103 mm, P = 10.45 kN
Ποσοστιαία επιμήκυνση: 9.0%, 26.4%, 38.3%
1.7-8
(a) E = 104 GPa
ψαθυρό, όλκιμο, όλκιμο
(b) v = 0.34
1.7-9
11.9 103 m, 12.7 103 m, 6.1 103 m, 6.5  103 m,
23.9 103 m
1.5-5
σ = 345 MPa
1.5-6
σpl  47 MPa, Κλίση: ~2.4 GPa, σy  53 MPa,
ψαθυρό
1.5-7
σpl  486 MPa, Κλίση: ~224 GPa, σy  520 MPa,
σU  852 MPa
1.6-1
= +0.052% ,
(b) Lmax = 3881 m
Ποσοστιαία βράχυνση: 8.8%, 37.6%, 74.5%
1.5-4
= 1.003
1.7-5
(b) σΑ = 18.54 MPa, σΒ = 15.09 MPa
1.5-1
𝑉𝑖
𝑉𝑓
(a) ΔdBCinner = 0.022 mm
(b) vbrass = 0.34
(c) ΔtAB = 6.90  10-3 mm, ΔdABinner = 4.02  10-3
mm
1.7-10
(a) ΔL1 = 12.66 mm, ΔL2 = 5.06 mm, ΔL3 = 3.8
mm
(b) ΔVol1 = 21,548 mm3, ΔVol2 = 21,601 mm3,
ΔVol3 = 21,610 mm3
επιμήκυνση: 6%, βράχυνση: 32%
1.8-1
σb = 46.9 MPa, τave = 70.9 MPa
(a) Επιμήκυνση κατά 2.375 mm
1.8-2
σb = 139.86 MPa, Pult = 144.45 kN
(b) σ = 278.5 MPa
1.8-3
(a) τ = 89.1 MPa
(b) σbf = 140 MPa, σbg = 184.2 MPa
1.6-2
Επιμήκυνση κατά 4.0 mm
1.6-3
Επιμήκυνση κατά 5.5 mm
1.6-4
(a) δpset = 4.28 mm
(b) Ares = 1178 N,
(b) σΒ = 65.6 MPa
(c) τ = 5.86 MPa, σrail = 18.4 MPa
1.6-5
(a) δpset = 48.6 mm
1.8-4
1.8-5
(a) δ = 11.23 mm
1.6-7
(b) 32 mm
(c) 30 mm
(d) 328 MPa
1.7-1
Pmax = 654 kN
1.7-2
P = 27.4 kN (εφελκυσμός)
(a) τmax = 22.9 MPa
(b) σbmax = 6.75 MPa
(b) σΒ = 220 MPa
1.6-6
(a) Bx = –252.8 N, Ax = –Bx, Ay = 1150.1 N
1.8-6
T1 = 13.1176 kN, T2 = 10.772 kN, τ1ave = 25.888
MPa, τ2ave = 21.166 MPa, σb1 = 9.15 MPa, σb2 =
7.48 MPa
1.8-7
(a) Συνισταμένη δύναμη: 4882 Ν
(b) σb = 33.3 MPa
(c) τnut = 21.6 MPa, τpl = 4.28 MPa
1.8-8
G = 2.5 MPa
1.8-9
(a) γaver = 0.004
(b) τaver = 42.6 MPa
(b) V = 384 kN
0
116.7
(a) RO = { 937.5 } N, MO = {– 134.6} Nm
– 321.7
– 438.4
1.8-10
τpl = 4.28 MPa, σu = 154.7 MPa, σl = 348 MPa
(b) τ = 10 MPa
1.8-11
d = 0.213 mm, γ = 0,0164
1.8-12
τaver = 0.299 MPa
(a) Τ = 1440 N
1.8-13
(a) σb = 482 MPa, σbrg = 265 MPa, τf = 45.7 MPa
(b) τaver = 147 MPa
1.8-14
1.8-15
1.8-23
1.8-24
Για ένα ποδήλατο με L/R = 1.8:
(a) σb = 419 MPa, σbrg = 231 MPa, τf = 39.7 MPa
1.9-1
Pεπ = 2.53 kN
(a) γaver = 0.50
1.9-2
Tmax = 33.4 kNm
(b) δ = 4.92 mm
1.9-3
Pεπ = 2.67 kN
(a) Ax = 0, Ay = 765 N, MA = 520 Nm
1.9-4
(a) Pεπ = 8.74 kN
(b) Pεπ = 8.69 kN
(b) Bx = 1.349 kN, By = 0.72 kN, Bres = 1.53 kN,
Cx = –Bx
(c) Pεπ = 21.2 kN, Pεπ = 8.69 kN (καθοριστική η
διάτμηση)
(c) τΒ = 27.1 MPa, τC = 13.4 MPa
(d) σbB = 42.5 MPa, σbC = 28.1 MPa
1.9-5
Συνολικό φορτίο: 1.126 ΜΝ
1.8-16
τaver = 42.9 MPa
1.9-6
d = 56.8 mm
1.8-17
(a) τ =
1.9-7
αέρας: n = 247, νερό: n = 284
1.9-8
(a) F = 1.171 kN
P
2πrh
(b) δ = P ln  b 
2πrhG  d 
1.8-18
(b) Διάτμηση: Fa = 2.86 kN
(a) τ1 = 2.95 MPa, τ4 = 0
1.9-9
Μέγιστο φορτίο: 21.6 kN
(b) σb1 = 1.985 MPa, σb4 = 0
1.9-10
(a)
(b) Διάτμηση στο Α: Wmax = 66.5 kN
(c) σb4 = 41 MPa
1.8-19
1.8-20
1.8-21
(d) τ = 10.62 MPa
1.9-11
Pεπ = 49.1 kN
(e) σ3 = 75.1 MPa
1.9-12
Cult = 5739 N, Pmax = 445 N
(a) Ox = 55.7 N, Oy = 5.69 N, Ores = 56.0 N
1.9-13
Wmax = 1.382 kN
(b) τΟ = 3.96 MPa, σbO = 6.22 MPa
1.9-14
Pεπ = 45.8 kN
(c) τ = 2.83 MPa
1.9-15
(a) Pa = σa (0.587d 2 )
(a) Fx = 153.9 N, σ = 3.06 MPa
(b) Pa = 98.7 kN
(b) τave = 1.96 MPa
1.9-16
Pεπ = 96.5 kN
(c) σb = 1.924 MPa
1.9-17
Pmax = 557 Pa
1.9-18
(a) Pεπ = σ c
(a) P = 1736 N
(b) Cx = 1647 N, Cy = –1041 N, Cres = 1948 N
(c) τ = 137.8 MPa, σbc = 36.1 MPa
1.8-22
FA = 2T , FB = 2T, FC = T
(a) Rx = –145.4 kN, Ry = –229 kN, Rz = –51.5 kN,
Mx = –23.2 kNm, My = 0 kNm, Mz = 65.4 kNm
πd 2
R
1−  
4
L
(b) Pεπ = 9.77 kN
1.10-1
(a) dmin = 99.5 mm
2
(b) dmin = 106.2 mm
1.10-2
(a) dmin = 164.6 mm
(b) dmin = 170.9 mm
1.10-3
1.10-16 (a) dm = 24.7 mm
(b) Pmax = 49.4 kN
1.10-17 (a) Pmax = 11.56 kN
(a) dmin = 17.84 mm
(b) dA = dB = 12.1 mm, dD = 16.9 mm
(b) dmin = 18.22 mm
1.10-18 dA = dB = 8.32 mm, dD = 11.60 mm
1.10-4
dmin = 63.3 mm
1.10-5
dmin = 26 mm
 1 
1.10-19 θ = arccos 
 = 54.7
 3
1.10-6
Acable = 75 mm2, dO = 7.71 mm
A-1.1
C
dB = dD = 8.64 mm
A-1.2
C
(a) Acable = 42.9 mm2
A-1.3
D
(b) dO = 10.3 mm, dA = dB = 8.9 mm
A-1.4
A
dD = 12.6 mm
A-1.5
B
1.10-8
(b) Amin = 435 mm2
A-1.6
A
1.10-9
dmin = 9.50 mm
A-1.7
A
1.10-10 dmin = 5.96 mm
A-1.8
D
1.10-11 n = 11.8, ή 12 κοχλίες
A-1.9
A
1.10-12 (d2)min = 131 mm
A-1.10
C
1.10-13 Ac = 1043 mm2
A-1.11
D
1.10-14 (a) tmin = 18.8 mm, άρα t = 20 mm
A-1.12
D
A-1.13
D
A-1.14
A
A-1.15
B
A-1.16
C
1.10-7
(b) Dmin = 297 mm
1.10-15 (a) σDF = 65.2 MPa < σεπ, σbF = 2.72 MPa < σba
(b) Νέα τιμή της σbF: 158.9 MPa, άρα η διάμετρος
της ράβδου BC αυξάνεται στα 6 mm και η
διάμετρος στο σημείο B αυξάνεται στα 34 mm.
2.2-1
δΑ = 21.43 mm
2.2-2
δrod = 0.1471 mm, δB = 0.1912 mm
2.2-3
(a) δ =
6W
5k
(b) δ =
4W
5k
2.2-4
2.2-5
(b)
da
Es
=
= 1.646
ds
Ea
(c) La = 1.5  Ea = 0.553
Ls
(d)
(a) δ = 12.5 mm
Es
Es
= 121 GPa [χυτοσίδηρος ή
1.7
χαλκός (Παράρτημα Η)]
E1 =
(b) n = 5.8
2.2-6
h = 13.4 mm
(a) δa = Es = 2.711
2.2-7
ΔDC = 7.05 10-2 mm, δΒ = 0.264 mm
2.2-8
ΔA = 0.542 mm
δs
Ea
πρmax d 2
4k
2.2-9
h=L−
2.2-10
(a) x = 102.6 mm
(b) x = 205 mm
2.2-21
(b) θ = 54.4°, δ = 4.89 mm, RA = 60 N, RC = 40 N,
MA = 3.5 Nm
2.3-1
(c) Pmax = 12.51 N
(d) θinit = 1.325°
W = 11.76 N
2.2-12
n = 8.9
2.2-13
(a) δ4 =
2.3-2
2.3-3
26 P
3k
(c) AAB = 491 mm2
2.3-4
δΒ = 2.29 10-4 mm, δD = 4.67 10-4 mm,
δE = 6.85 10-4 mm
(a) δΒ = 1.827 mm
2.3-5
δΒ = –3.18 10-4 mm, δC = –5.31 10-4 mm, δD = –
1.38 10-3 mm
2.3-6
δΒ = 4.22 10-4 mm, δD = 7.84 10-4 mm, δE =
10.51 10-4 mm
2.3-7
δΒ = 1.85 10-5 mm, δC = 4.68 10-5 mm,
δD = –6.1 10-4 mm
2.3-8
(a) δ =
(c) δΒx = 6.71 mm
(d) Pmax = 106.1 kN
2.2-15
Pmax = 186 N
2.2-16
(a) x = 134.7 mm
(b) k1 = 0.204 kN/m
7 PL
6 Ebt
(b) δ = 0.5 mm
(c) b = 74.1 mm
(c) Lslot = 244 mm
(d) k3 = 0.638 kN/m
2.3-9
(a) δ =
(a) tc,min = 0.580 mm
7 PL
6 Ebt
(b) δr = 0.912 mm
(b) δ = 0.53 mm
(c) hmin = 1.412 mm
(c) Lslot = 299 mm
2.2-18
δΑ = 0.200 mm, δD = 0.880 mm
2.2-19
(a) δD =
P
( 28 f 2 − 9 f1 )
16
2.3-10
2.3-11
(c) δc/δa = 1.0
(c) δc/δb = 1.284
d2
365
L
236
2.3-12
(a) dmax = 23.9 mm
(b) b = 4.16 mm
(a) θ = 35.1°, δ = 44.6 mm, RA = 25 N, RC = 25 N
(b) θ = 43.3°, δ = 8.19 mm, RA = 31.5 N,
RC = 18.5 N, MA = 1.882 Nm
(a) δa = 3.1 mm
(b) δb = 2.41 mm
(c) d1 = 1.225
(d) x =
(a) δAC = 3.72 mm
(b) Ρ0 = 44.2 kN
(b) L1 = 27
L2 16
2.2-20
(a) δ = 0.296 mm (επιμήκυνση)
(b) νέα P3: 7530 N
(b) Pmax = 390 kN
2.2-17
(a) δ = 0.675 mm
(b) Pmax = 267 kN
104 P
(b) δ4 =
, λόγος: 15/4 = 3.75
45k
2.2-14
(a) δ = 0.838 mm
(b) dB = 29.4 mm
(e) P = 20.4 N
2.2-11
(a) θ = 52.7°, δ = 19.54 mm, RA = RC = 50 N
(c) x = 183.3 mm
2.3-13
(a) δ =
PL
2 EA
(b) σ ( y ) =
(c) δ =
2.3-14
P y
AL
Py
y 
PL 2
, σ ( y) =   2 − 
AL
L 
EA 3
(a) δ2-4 = 0.024 mm
2.3-28
𝛿𝐵 = −4 𝐸𝐴, 𝛿𝐹 = −
11 𝐿𝑃
3 𝐸𝐴
2.3-29
𝛿𝐵 = −
3 𝐿𝑃
,
4 𝐸𝐴
5 𝐿𝑃
4 𝐸𝐴
2.3-30
𝛿𝐵 = 𝛿𝐷 = − 2 𝐸𝐴
2.4-1
(a) P = 9.24 kN
2.3-15
3
(a) R1 = − P
2
P
3
P (εφελκυσμός), N 2 = (εφελκυσμός)
2
2
(b) N1 =
(c) x =
L
3
2 PL
3EA
(d) δ2 =
2.4-2
(a) δc =
W ( L2 − h 2 )
2 EAL
(b) δB =
WL
2EA
1 𝐿𝑃
(a) P = 104 kN
(b) Pmax=116 kN
2.4-3
Ay = –6.67 kN, By = –3.33 kN (προς τα κάτω)
δC = 6.54 10-3 mm (προς τα πάνω)
2.4-4
Α = 70.3 mm2
2.4-5
Pεπιτ = 3090 kN
2.4-6
(a) PB/P = 3/11
(b) σΒ/σΑ = 1/2
(e) β = 1/11
2.3-16
𝛿𝐹 = −
(b) Pεπ = 7.07 kN
(b) Pmax = 8.15 kN
(c) L2 = 9.16 mm
𝐿𝑃
(c) Λόγος: 1
2.4-7
(a) Αν x  L/2, RA = (−3PL)/(2(x+3L)),
RB =−P(2x+3L)/(2(x+3L)). Αν x  L/2,
RA =(−P(x+L))/(x+3L), RB=(−2PL)/(x+3L).
(b) Αν x  L/2, δ=PL(2x+3L)/[(x+3L)Eπd2].
Αν x  L/2, δ=8PL(x+L)/[3(x+3L)Επd2].
(c) β = 3
(c) x=3L/10 ή x=2L/3
WL
= 359 mm (στη θάλασσα)
2 EA
WL
δ=
= 412 mm (στον αέρα)
2 EA
2.4-8
RA = –80 kN, RC = –120 kN, δB = 0.466 mm
2.3-17
(b) δ = 0.304 mm
2.4-9
RA = –69.52 kN, RC = –20.48 kN, δB = 0.474 mm
2.3-18
δΒ = 0.937 mm, δC = 2.95 mm
2.4-10
(a) δ = 1.91 mm
2.3-19
δΒ = 2.12 mm, δC = 6.35 mm
(b) δ = 1.36 mm
2.3-20
P1 = 415 kN αριστερά, δΒ = –1.389 mm
(c) δ = 2.74 mm
2.3-21
L = 2.83 m
2.3-22
δ = 4.14 mm
2.3-23
δ=
2WL
πd 2 E
2.3-24
δ=
2 PH
3Eb 2
(d) δ =
2.3-25
(b) δ = 3.55 m
2.3-26
(a) δ = 2.18 mm
(d) RB=ρgπd2L/8, RA=3ρgπd2L/32
2.4-11
(a) 41.2%
(b) σΜ = 238 MPa, σO = 383 MPa
2.4-12
(a) RA = 10.5 kN (στα αριστερά),
RD = 2.0 kN (στα δεξιά)
(b) FBC = 15.0 kN (θλίψη)
2.4-13
(a) RA =
2
5
P , 𝑅𝐸 = − 3 𝑃
3
𝐿𝑃
(b) 𝛿𝐵 = − 6𝐸𝐴, δC =
(b) δ = 6.74 mm
2.3-27
δC = 0.653 mm προς τα κάτω
(c) δmax =
LP
5LP
, δD =
6 EA
6EA
5LP
(προς τα δεξιά), δΑ = δΕ = 0
6 EA
(d) Pmax = 53 kN
2.4-14
(c) db = 10.68 mm
(a) P = 13.73 kN, R1 = 9.07 kN, R2 = 4.66 kN,
σ2 = 7 MPa
− R2 x  L2
(b) δcap = 190.9 mm, N ( x) = 
,
x  L2
 R1
 R2
x  L2
 − EA x

2
δ ( x) = 
 − R2 L2 + R1 ( x − L2 ) x  L
2
 EA2
EA1
2.5-5
(b) σ c =
2.4-16
2.4-17
EαΔTB
 EA

+ 1
4
 kL

σΤ = –0.74 MPa, δΒ = 1.609 mm
2.5-7
σΤ = –24 MPa
2.5-8
(a) N = 51.8 kN, σcmax = 26.4 MPa, δC = –0.314 mm
(b) N = 31.2 kN, σcmax = 15.91 MPa, δC = –0.546
mm
(b) σa = 10.6 MPa (θλίψη), σs = 60.6 MPa
(εφελκυσμός)
2.5-9
RA = –RC = 5.2 kN, δΒ = 5.32  10-3 mm στα δεξιά
(a) Pεπ = 1504 N
2.5-10
RA = –RC = 117.4 kN, δΒ = 0
(b) Pεπ = 820 N
2.5-11
δ = 5 mm
(c) Pεπ = 703 N
2.5-12
άκρα σΤ = –22.7 MPa, μέση σΤ = –27.0 MPa
PE1
E1 + E2
2.5-13
τ = 67.7 MPa
2.5-14
ΔΤ = 34°C
2.5-15
(a) TA = 1760 N, TB = 880 N
(a) P1 =
b( E2 − E1 )
(b) e =
2( E1 + E2 )
(b) TA = 2008 N, TB = 383 N
(c) σ1 = E1
σ2
2.4-18
EαΔTB
4
2.5-6
(c) q = 1.522 kN/m
2.4-15
(a) σ c =
E2
(c) 177°C
(a) Ax = –41.2 kN, Ay = –71.4 kN, Bx = –329 kN,
By = 256 kN
2.5-16
Pεπ = 39.5 kN
2.5-17
(a) σ = –6.62 MPa
(b) Pmax = 233 kN
(b) Fk = 12.99 kN (C)
2.4-19
d2 = 9.28 mm, L2 = 1.10 m
(c) σ = –17.33 MPa
2.4-20
Pmax = 1800 N
2.4-21
(a) σc = 50.0 MPa, σD = 60.0 MPa
2.5-18
(b) T = 35°C
(b) δΒ = 0.320 mm
2
𝑃𝐿,
3
2.5-19
2
𝑃,
3
(a) σ = 98 MPa
4
𝑃
3
Ftube = –1036 kN, Feach-cable = 517.9 kN,
2.4-22
𝑀𝐴 =
2.4-23
σs = 58.9 MPa, σb = 28.0 MPa, σc = 33.6 MPa
2.4-24
Ax = 0, Ay = –6P, Dx = 0, Dy =
2.5-1
σ = 100.8 MPa
2.5-2
T = 40.3°C
2.5-3
ΔΤ = 90°C
(c) Για P1, τmax = 93.8 MPa, για P2, τmax = 133.33
MPa
2.5-4
(a) ΔΤ = 24°C, σράβδου = 57.6 MPa
(d) ΔΤ = 35°C, RA = 0, RB = 0
(b) Άγκιστρο: σbc = 42.4 MPa, Ροδέλα: σbw = 74.1
MPa
(e) RA = –249 kN, RB = 249 kN
Ax = 0, Dx = 0, Dy =
18
𝑃,
5
RF =
RF =
27
𝑃
5
δtube = –2.45 mm
PL
6 EA
2.5-20
s=
2.5-21
(a) P1 = 1027 kN, RA = –249 kN, RB = 249 kN
(b) P2 = 656 kN, RA = –249 kN, RB = 249 kN
2.5-22
(a) RA =
− s + αΔT ( L1 + L2 )
, RD = –RA
L1
L
L
+ 2 +
EA1 EA2 k3
(c) Στην περιστρεφόμενη πλευρά x: σx1 = 42 MPa,
τx1y1 = 42 MPa .Στην περιστρεφόμενη πλευρά y :
σy1 = 42 MPa (d) Στην περιστρεφόμενη πλευρά x:
σx1 = 71.7 MPa, τx1y1 = –29.7 MPa,
Στην περιστρεφόμενη πλευρά y: σy1 = 12.3 MPa
(b) δB = αΔTL1 − RA L1 ,
EA1
 L
L 
δC = αΔT ( L1 + L2 ) − RA  1 + 2 
 EA1 EA2 
2.6-7
2.5-23
TB = 2541 N, TC = 4623 N
2.6-8
2.5-24
Pεπ = 1.8 MN
2.5-25
(a) σp = –1.231 MPa, σr = 17.53 MPa
(b) σb = 11.63 MPa, τc = 1.328 MPa
2.5-26
σp = 25.0 MPa
2.5-27
σp = 15.0 MPa
2.5-28
(a) PB = 25.4 kN, Ps = –PB
(b) Sαπαιτd = 25.7 mm
(b) τmax = 8.4 MPa
2.6-9
2.6-10
2.6-11
2.5-31
(a) Fk = 727 N
(b) Ft = –727 N
(a) τ max AC =
σ AC
= 13.1 MPa ,
2
σ AB
= 52.3 MPa ,
2
σ
= BC = −66.3 MPa
2
τ max AB =
τ max BC
(b) Pmax = 159.3 kN
(c) Lf = 305.2 mm
σa = 500 MPa (θλίψη), σc = 10 MPa (εφελκυσμός)
σθ = –30 MPa, σθ + π/2 = –30 MPa,
τθ = 30 MPa
(b) Ft = 727 N
2.5-30
σθ = –5.81 MPa, σθ + π/2 = –2.85 MPa,
τθ = 4.07 MPa
(a) Fk = –727 N
(d) ΔΤ = 76.9°C
(a) Στοιχείο Α: σx = 105 MPa (θλίψη), Στοιχείο B:
τmax = 52.5 MPa
(b) θ = 33.1°
(c) δfinal = 0.35 mm
2.5-29
(a) σmax = 16.8 MPa
2.6-12
(a) (1) σx = –945 kPa, (2) σθ = –807 kPa και
τθ = 334 kPa, (3) σθ = –472 kPa, τθ = 472 kPa,
σmax = –945 kPa, τmax = –472 κPa
(b) σmax = –378 kPa, τmax = –189 κPa
2.6-13
(c) Lf = 304.8 mm
(a) τpq = 4.85 MPa
(d) ΔΤ = –76.8°C
(b) σpq = –8.7 MPa, σ  pq = π  = −2.7 MPa
2
2.6-1
Pmax = 312 kN
(c) Pmax = 127.7 kN
2.6-2
dmin = 6.81 mm
2.6-3
Pmax = 104 kN
2.6-4
(a) ΔΤmax = –46°C
(b) σpq = –21.0 MPa (θλίψη), τpq = 30 MPa
(ανθωρολογιακή)
(b) ΔΤ = +9.93°C
(c) β = 0.62
2.6-5
2.6-6

2.6-14

(a) ΔΤmax = 31.3°C
(a) τmax = 84.7 MPa
2.6-15
ΝΑC = 34.6 kN, dmin = 32.4 mm
(b) ΔΤmax = –17.38°C
2.6-16
(a) σθ = 0.57 MPa και τθ = –1.58 MPa
(c) ΔΤ = +42.7°C
(b) α = 33.3°
(a) σx = 84 MPa
(c) α = 26.6°
(b) τmax = 42 MPa
2.6-17
(a) θ = 30°, τ0 = –34.6 MPa
(b) σmax = 80 MPa, τmax = 40 MPa
2.6-18
σθ1 = 54.9 MPa, σθ2 = 18.3 MPa και τθ = 31.7 MPa
(b) δ = 1.321 mm
2.6-19
σmax = 64.4 MPa, τmax = 34.7 MPa
(c) U = 243 J
2.6-20
(a) θ = 30.96°
2.7-11
(b) Pmax = 1.53 kN
2.6-21
(a) τθ = 2.15 MPa, θ = 22°
(b) σx1 = –5.3 MPa, σy1 = –0.869 MPa
(c) kmax = 3962 kN/m
(b) U1 = (2k1 + k2 )s
2.7-12
2.8-1
2.7-1
23P 2 L
(a) U =
12 EA
2.8-2
(c) Συντελεστής σε κρούση = 160
2.8-3
2
(a) U = 5P L2
4πEd
(c) Συντελεστής σε κρούση = 133
2.8-4
2
P L PQL Q L
+
+
2 EA 2 EA 4 EA
2.7-4
(c) U =
2.7-5
Αλουμίνιο: 826 kPa, 31 m
2.7-6
(a) U =
P2 L
EA
(b) δB =
2 PL
EA
2.7-7
2.7-8
(b) Συντελεστής σε κρούση = 3.9
2.8-5
(a) δmax = 270 mm
(b) Συντελεστής σε κρούση = 4.2
v = 13.1 m/s
2.8-7
hmax = 0.27 m
2.8-8
Lmin = 9.25 m
(a) U1 = 0.00422 J
2.8-9
Lmin = 4.59 m
(b) U2 = 0.305 J
2.8-10
Vmax = 5.40 m/s
(c) U3 = 0.264 J
2.8-11
δmax = 200 mm
2.8-12
L = 25.5 m
2.8-13
(a) Συντελεστής σε κρούση = 1 + 1 + 2 EA
(a) U = 5kδ
2
W
10k
3W
W
3W
(c) F1 =
, F2 =
, F3 =
10
10
20
b
P2 L
U
=
ln 2
(a)
2 Et (b2 − b1 ) b1
(b) δ =
2.7-10
(a) δmax = 215 mm
2.8-6
(b) δ =
2.7-9
(a) δmax = 0.762 mm
(b) σmax = 177.7 MPa
U = 788 J
2
(a) δmax = 6.33 mm
(b) σmax = 359 MPa
(b) U = 1.036 J
2.7-3
(a) δmax = 0.869 mm
(c) Συντελεστής σε κρούση = 117
(b) U = 14.02 J
2.7-2
(a) U = 6.55 J
(b) σmax = 152.1 MPa
(e) ΔΤmax = 53.7°C
Pmax = 5.63 kN
2
(b) δC = 168.8 mm
(d) Lmax = 0.755 m
2.6-22
(a) x = 2s, P = 2(k1 + k2 )s
b
PL
ln 2
Et (b2 − b1 ) b1
(a) P1 = 270 kN
W
(b) 10
2.8-14
σmax = 33.3 MPa
2.10-1
(a) σmax  45 και 50 MPa
(b) σmax  76 και 61 MPa
2.10-2
(a) σmax  26 και 29 MPa
(b) σmax  25 και 22 MPa
σ1bt
3
3σ L
4
(b) PP = σY A , δP = Y
3
E
2.10-3
Pmax =
2.10-4
σmax  46 MPa
2.10-5
σmax  41.9 MPa
2.10-6
(a) Όχι, καθίσταται ασθενέστερο: P1 = 25.1 kN,
P2  14.4 kN
2.12-9
(b) d0  15.1 mm
2.10-7
dmax = 13 mm
2.11-2
(a) δC = 1.67 mm
(a) PY = σY A , δY =
2σ L
5
(b) PP = σY A , δP = Y
E
4
2.12-10 (a) WY = 28.8 kN, δΥ = 125 mm
(b) WP = 48 kN, δP = 225 mm
2.12-11 (a) PY = 300 kN, δΥ = 0.475 mm
(b) PP = 456 kN, δP = 0.75 mm
(b) δC = 5.13 mm
A-2.1
D
A-2.2
B
A-2.3
A
A-2.4
A
A-2.5
D
Για P=3.2 kN, δ=4.85 mm, για P=4.8 kN, δ=17.3
mm
A-2.6
A
A-2.7
B
2.12-1
PY = PP = 2σΥΑsinθ
A-2.8
C
2.12-2
PP = 201 kN
A-2.9
A
2.12-3
(a) PP = 5σΥΑ
A-2.10
D
2.12-4
PP = 2σY A(1 + sin α)
A-2.11
D
2.12-5
PP = 220 kN
A-2.12
C
2.12-6
PP = 82.5 kN
A-2.13
A
2.12-7
(a) PP = 102 kN
A-2.14
C
(b) καμία μεταβολή
A-2.15
D
A-2.16
C
(c) δC = 11.88 mm
2.11-3
(b) P = 79.9 kN
2.11-4
Για P=30 kN, δ=6.2 mm, για P=40 kN, δ=12.0 mm
2.11-5
Για P=106 kN, δ=4.5 mm, για P=180 kN, δ=17.5
mm
2.11-6
3σY L
2E
2.12-8
(a) PY = σY A , δY =
3.2-1
γmin = 4.2  10-3 rad, γmed = 4.6  10-3 rad
3.2-2
(a) Lmin = 162.9 mm
(b) r2,max = 65.1 mm
3.2-6
(b) dmax = 68.8 mm
3.2-3
3.2-4
3.2-5
σY L
E
(a) γ1 = 267  10-6 rad
(b) r2,min = 183.3 mm
(a) dmax = 10.54 mm
3.3-1
φ = 4.85  10-3 rad, τmax = 8.2 MPa, τΑ = 6.5 MPa
(b) Lmin = 545 mm
3.3-2
dmin = 35.7 mm
(a) γ1 = 393  10-6 rad
3.3-3
φ = 5.59  10-3 rad, τmax = 9.4 MPa, τΑ = 7.5 MPa
(b) r2,max = 50.9 mm
3.3-4
dmin = 64.7 mm
(a) γ1 = 1.967  10-4 rad
3.3-5
(a) τmax = 60.4 MPa
(c) ροπή: 6.25%, βάρος: 25%
(b) dmin = 15.87 mm
3.3-6
3.3-7
(a) τmax = 23.8 MPa
3.3-20
(b) Tmax = 0.402 Nm
(b) d = 88.4 mm
(c) θ = 9.12°/m
(c) Λόγος: 0.524
(a) τmax = 133 MPa
3.3-21
(b) φ = 3.65°
3.3-8
(a) kT = 2059 Nm
(c)
τ
kThollow
= 0.938 , max H = 1.067
τ max S
kTsolid
(d) d2 = 32.5 mm
(a) r2 = 35.2 mm
(b) Pmax = 6486 N
3.4-1
(b) τmax = 27.9 MPa, γmax = 997  10-6 rad
3.3-9
(a) φ = 5.19°
(a) τmax = 50.3 MPa, φC = 0.138°
(b) dBC = 50.1 mm, φC = –0.161o
3.4-2
(a) τbar = 79.6 MPa, τtube = 32.3 MPa
(b) φΑ = 9.43°
3.4-3
(a) τmax = τBC = 66 MPa, φD = 2.44°
(b) dAB = 76.5 mm, dBC = 60 mm, dCD = 39.5 mm,
φD = 2.6°
(a) Lmin = 838 mm
(b) Lmin = 982 mm
3.3-10
Tmax = 6.03 Nm, φ = 2.20°
3.4-4
Tεπ = 439 Nm
3.3-11
(a) τmax = 46 MPa, G = 22 GPa, γmax = 2.09410-3
3.4-5
d1 = 20.7 mm
3.4-6
(a) d = 77.5 mm
(b) Tmax = 548 Nm
3.3-12
(b) Tmax = 7765 Nm
3.3-13
τmax = 54.0 MPa
3.3-14
(a) dmin = 63.3 mm
(b) dmin = 66 mm (αύξηση διαμέτρου κατά 4.2%)
3.3-15
(a) τ2 = 25.7 MPa
(b) τ1 = 18.4 MPa
(c) θ = 3.67  10-3 rad/m = 0.21°/m
3.3-16
(a) τ2 = 30.1 MPa
(b) τ1 = 20.1 MPa
3.3-17
(b) d = 71.5 mm
(a) Tmax = 9164 Nm
3.4-7
dmin = 32.5 mm, φBD = –0.459°
3.4-8
τmax AB = –367 MPa, τmax BC = –466 MPa,
φmax = –5.63°
3.4-9
(a) d = 44.4 mm
(b) d = 51.5 mm
3.4-10
(b) dB/dA = 1.45
3.4-11
(dA)min = 63.7 mm
3.4-12
(dB)min = 48.6 mm
3.4-13
3
(a) R1 = − T
2
(c) θ = 0.306°/m
(b) T1 = 1.5T, T2 = 0.5T
(a) dmin = 100 mm
(c) x =
(b) kT = 648 kNm/rad
(d) φ2 =
(c) dmin = 88.9 mm
3.3-18
(a) dmin = 64.4 mm
(b) kT = 134.9 kNm/rad
(a) T1,max = 424 Nm
(b) T1,max = 398 Nm
12 TL
17 GI p
3 TL
2 πGtd A3
3.4-14
φ=
3.4-15
(a) φ = 2.48°
(c) dmin = 50 mm
3.3-19
7
L
17
(b) φ = 1.962°
3.4-16
(a) R1 = −
T
2
3.5-4
(b) σmax = 51.2 MPa
19 TL
(b) φ3 =
8 πGtd 3
(c) T = 20 kNm
3.5-5
 L1

3
 t01  d 01
4 Fd
φD =
πG
+
 (d
0
+
3.4-18
3.4-19
3.4-20
L42
− d01 x + d03 x ) ( t01 L2 − t01 x + t03 x )
3
01 L2
L3 
= 0.133
3 
t03 d03

(a) τ max =
16tL
π d3
(c) γmax = 1670  10-6 rad
dx
3.5-6
G = 30.0 GPa
3.5-7
T = 234 Nm
3.5-8
(a) dmin = 37.7 mm
(b) Tmax = 431 Nm
3.5-9
(b) φ =
16tL
πGd 4
(a) τ max
8t L
= A3
πd
(b) d1,max = 16.25
3.5-10
3.5-11
(a) Lmax = 4.42 m
3.5-12
3.5-13
(c) d3new = 65.9 mm
(d) Tmax1 = 1.881 kNm, Tmax2 = 1.498 kNm,
φmax1 = 1.492°, φmax2 = 1.29°.
2
T  x−L
TBC ( x) = − 0 
για L/2 ≤ x ≤ L
3  L 
3.5-1
(a) T1επ = 1.928 kNm, T2επ = 1.536 kNm
(b) Lmid = 597 mm
 T0  2
x για 0 ≤ x ≤ L/2 και
2 
L 
3.4-22
(a) τmax = 23.9 MPa
(b) γmax = 884 10-6 rad
T0
6
T0 L
144GI P
(d) τ max =
(a) τmax = 36.7 MPa
(b) γmax = 453 10-6 rad
(b) TAB ( x) = 
(c) φc =
(a) d2 = 79.3 mm
(b) d2 = 80.5 mm
16t A L2
(b) φ =
3πGd 4
(a) RA =
(a) d1 = 14.39 mm
2
(b) φ = 170°
3.4-21
(a) d1 = 60 mm
(b) φ = 2.30°
L2
3.4-17
(a) εmax = 320 10-6
8 T0
3
3π d AB
3.7-1
(a) τmax = 36.5 MPa
(b) d = 81.9 mm
3.7-2
(a) τmax = 50.0 MPa
(b) dmin = 32.3 mm
3.7-3
n = 820 rpm
(a) Tmax = 875 Nm
3.7-4
τmax = 63.3 MPa, φAC = 0.0535 rad
(b) τmax = 25.3 MPa
3.7-5
(a) H = 20.2 MW
τmax = 41.05 MPa, γmax = 1.0  10-3 rad,
σmax = 41.05 MPa, σmin = –41.05 MPa,
εmax = 5  10-4
3.5-2
Tmax = 2.71 kNm
3.5-3
(a) σmax = 48 MPa
(b) T = 8836 Nm
(b) η διατμητική τάση υποδιπλασιάζεται
3.7-6
(a) τmax = 16.8 MPa
(b) Pmax = 267 kW
3.7-7
d = 90.3mm
3.7-8
dmin = 110 mm
3.7-9
(d1)min = 1.221d
(b)  = 0.48°
3.7-10
Pmax = 91.0kW
(c) kT = 238 kNm
3.7-11
d = 69.1 mm
3.7-12
d = 53.4 mm
3.8-1
(a) φmax =
(b) φmax =
3.8-2
(a) x =
3.8-14
(b) d2 = 56.9 mm
3T0 L
5GI P
3.8-15
9T0 L
25GI P
3.8-16
(b) T2επ = 6.35 kNm
(d) Tmax = 6.35 kNm
TL
= 0
8GI P
2bτ επ
=
Gd
φmax
3.8-4
Pεπ = 2710 N
3.8-5
(a) T0max = 150 Nm
3.8-17
3.8-18
(c) max = 7.51°
(d) Tmax = 2.48 kNm (καθοριστική η διατμητική
τάση στους κοχλίες)
(a) T0max = 150 Nm
(a)
dA
a
=
L d A + dB
(b)
d4
a
= 4 A 4
L d A + dB
3.8-8
TA = TB = 225 Nm, φL/2 = –4.34°
3.8-9
TA = –265.4 kNm, TC = 85.4 kNm,
(e) R2 =
(a) TA =
Lt0
Lt
, TB = 0
6
3
(b) φmax
3L2t0
 L 
= φ
=
−

27GI P
 3
(a) x = 767 mm
(a) τ1 = 32.7 MPa, τ2 = 49.0 MPa
3.9-1
3.8-13
(a) τ2 = 38.1 MPa, τ1 = 25.4 MPa
L2
G2 I p 2
(a) U = 41.9 J
(b) φ = 1.29°
3.9-2
(a) U = 5.36 J
(b) φ = 1.53°
3.9-3
U = 15.2 J
3.9-4
U = 1.84 J
3.9-5
U = 81 kNm
3.9-6
(c) U 3 =
3.9-7
U=
19T02 L
32GI P
3.9-8
φ=
T0 LA LB
G( LB I PA + LA I PB )
(b)  = 1.030°
(c) kT = 22.3 kNm
R1 = − R2 με
(f) βmax = 29.1°
(b) max = –1.031° στη θέση x = 767 mm
3.8-12
β
και
fT 1 + fT 2
και fT 2 =
τΑΒ,max = 92.3 MPa, τBC,max = –77.1 MPa
3.8-11
(a) R1 = –0.77T, R2 = –0.23T
(b) Tmax = 2.79 kNm
(b) T0max = 140 Nm
3.8-10
(a) TA = 1720 Nm, TB = 2780 Nm
(b) TA = 983 Nm, TB = 3517 Nm
(b) T0max = 140 Nm
3.8-7
(a) T1επ = 7.14 kNm
(c) T3επ = 7.41 kNm
3.8-3
3.8-6
(a) Tmax = 1.521 Nm
(b) d2 = 56.9 mm
L
4
(b) φmax
(a) Tmax = 1.521 kNm
T 2 L TtL2
t 2 L3
+
+
2GI P 2GI P 6GI P
fT 1 =
L1
G1 I p1
3.9-9
3.9-10
U=
t02 L3
40GI P
(a) U =
(b) 𝜑 =
T 2 L(d A + d B )
πGtd A2 d B2
2𝑇𝐿(𝑑𝐴 +𝑑𝐵 )
2 𝑑2
𝜋𝐺𝑡𝑑𝐴
𝐵
3.11-11 tmin = 4.57 mm
3.11-12 (a) t = 6.66 mm
(b) t = 7.02 mm
3.11-13 τ =
2T (1 + β )
2
tL2m β
3.12-1
Tmax  1770 Nm
3.9-11
β 2GI PA I PB
U=
2 L( I PA + I PB )
3.12-2
Rmin  4.0 mm
3.12-3
τmax  121.5 MPa για D1 = 18 mm
3.9-12
2n 2πI m L
n 2πGJ m
φ=
, τ max =
2
G
15d
L
15d
3.12-4
D2  115 mm (κάτω όριο)
3.11-1
(a) τapprox = 42.7 MPa
3.12-5
D1  33.5 mm
(b) τexact = 47 MPa
A-3.1
D
tmin =
πd
64
A-3.2
A
3.11-2
A-3.3
C
3.11-3
(a) τ = 15.0 MPa
A-3.4
A
(b)  = 0.578°
A-3.5
D
(a) τ = 9.17 MPa
A-3.6
D
(b)  = 0.140°
A-3.7
B
3.11-5
τmax = 21.54 MPa, φ = 2.96 10-3 rad
A-3.8
B
3.11-6
U1/U2 = 2
A-3.9
C
3.11-7
τ = 46.7 MPa, θ = 0.543°/m
A-3.10
B
3.11-8
τ = 35.0 MPa, φ = 0.570°
A-3.11
D
τ=
T 3
2T
2 , θ=
9b t
9Gb3t
A-3.12
B
3.11-9
A-3.13
B
φ1
1
=1+
φ2
4  β2
A-3.14
D
A-3.15
B
3.11-4
3.11-10 (a)
4.3-1
V = 1.46 kN, M = 3.72 kNm
4.3-2
V = –0.938 kN, M = 5.06 kNm
4.3-3
(a) Vmid = 0, Mmid = 0
(c) P1 = –8 kN (προς τα δεξιά)
4.3-7
(a) b/L = 1/2
(b) b/L = 1
(b) Vmid = 0, Mmid = 0
4.3-8
M = 108 Nm
4.3-4
V = 3.0 kN, M = –6 kNm
4.3-9
N = Psinθ, V = Pcosθ, M = Ρrsinθ
4.3-5
(a) V = –786 N, M = 21.8 kNm
4.3-10
V = –3172 N, M = 5553 Nm
(b) q = 5.37 kN/m (προς τα πάνω)
4.3-11
(a) P2 = 37.5 kN
4.3-6
(a) V = –1.0 kN, M = –7 kNm
(b) P2 = 4 kN
(b) P = 4.17 kN
4.3-12
V = –4.17 kN, M = 75 kNm
4.3-13
(a) VB = 24 kN, MB = 12 kNm
4.5-5
(b) Vm = 0, Mm = 30 kNm
4.3-14
(b) q0 = 4.5q
(a) VC = –15.5 kN, MC = –44.8 kNm
4.5-6
PL
2
Vmax = − P , M max =
9
3
4.5-7
Vmax =
4.5-8
(a) Vmax =
(b) MA = –81 kNm, VC = –25 kN,
MC = –30.5 kNm
4.3-15
4.3-16
2
−5
23
(a) V23 = 36 𝑞0 𝐿, M23 = 324 𝑞0 𝐿2
5
𝑞 𝐿,
144 0
(a) V23 =
−1
𝑞 𝐿,
36 0
(b) V23 =
−5
𝑞 𝐿,
72 0
4.3-18
𝐿
V(2)
4.3-19
V( 3 ) =
4.3-20
Vmid =
−𝑞0 𝐿
,
6
Vmid =
𝑃
(√2
2
4.3-21
4.3-22
=
2𝐿
M23 =
35
𝑞 𝐿2
648 0
−19
𝑞 𝐿,
81 0
𝐿
M(2)
2𝐿
− 1), Mmid =
(b) Vmid =
𝐿
−𝑃
,
√5
𝐿
11𝑃
,
36
V(3) =
4.3-25
N=
−𝑃
,
2
𝐿
Mmid =
Mmid =
M(3) =
𝐿
M(3) =
−43
𝑃𝐿
81
4.5-10
(a) Vmax = P, Mmax = –Pa
(b) M = 3Pa (ανθωρολογιακή), Vmax = 2P, Mmax =
2Pa
4.5-11
(a) Vmax =
(b) Vmax =
4.5-12
qL
5qL2
, M max =
72
2
7𝑞𝐿
,
20
91wL2 α
229 wL3α
, M max =
30 g
75 g
4.5-1
(a) Vmax = P, Mmax = Pa
𝑎
(b) Ay = –By = P(2 𝐿 − 1)
(a) Vmax =
M0
M a
, M max = 0
L
L
(b) Vmax =
𝑀
2 0,
𝐿
Mmax = 2
𝑎
M0
𝐿
Vmax = P , M max
5𝑞𝐿2
54
q0 L
q L2
, M max = − 0
6
2
(b) Vmax = −
2q0 L
4q L2
, M max = − 0
(στο σημείο
15
3
Β)
4.5-13
Vmax = –758.3 N, Mmax = 470.7 Nm
4.5-14
(a) Vmax = 1200 N, Mmax = 960 Nm
(b) Vmax = 675 N, Mmax = 640 Nm
4.5-15
Vmax = 4 kN, Mmax = –13 kNm
4.5-16
Vmax = 4.5 kN, Mmax = –12.67 kNm
4.5-17
Vmax = –8549 N, Mmax = –3240 Nm
4.5-18
Vmax = 13.15 kN, Mmax = 11.21 kNm
4.5-19
(a) Vmax =
qL
3qL2
, M max = −
8
2
PL
=
4
Mmax =
(a) Vmax = −
V = 0, M = 0
Vmax =
4.5-4
Nmax = P, Vmax = P, Mmax = – 3 PL
(a) N = 21.6 kN (θλίψη), V = 7.2 kN,
M = 50.4 kNm
Vmax =
2
4.5-9
2√5
𝑃𝐿
15
4.3-27
4.5-3
P 3L
3LP
(ακριβώς δεξιά του
=−
5 4
20
σημείου Β)
59
𝑞 𝐿2
432 0
2
𝑞 𝐿2
27 0
P
,
5
M max = −
−𝑃𝐿
4
(b) N = 21.6 kN (θλίψη), V = –5.4 kN, M = 0
4.5-2
(b) Nmax = P (θλιπτική στο τμήμα ΑΒ), Vmax =
43
Mmid = 0
−4
𝑞 𝐿,
27 0
4.3-24
=
P
(στο τμήμα ΑΒ),
2
3L 3LP
(ακριβώς δεξιά του σημείου
= RC
=
4
8
Β)
√2+2
𝑞0 𝐿2
30
M( 3 ) = 972 𝑞0 𝐿2
(a) Vmid =
V(3) =
4.3-26
−5
𝑞 𝐿2
324 0
√2−4
𝑞0 𝐿,
6
M max
23
𝑞 𝐿2
288 0
Mmid =
M23 =
41
𝑞 𝐿,
216 0
4.3-23
2M1
7 M1
, M max =
L
3
13
Vmid = 45 𝑞0 𝐿, Mmid = 180 𝑞0 𝐿2
(b) Vmid =
4.3-17
(a) q1 = 3q
4.5-20
4𝑃
,
5
Mmax = −
17𝑃𝐿
20
(b) Vmax = −
7𝑃
,
5
Mmax =
𝑃𝐿
2
(a) Nmax = −
3𝑃
,
5
Vmax = −
12𝑃
,
5
Mmax = PL
(b) Nmax = −
3𝑃
,
5
Vmax = −
(c) Nmax = −
3𝑃
,
5
Vmax =
4𝑃
,
5
4𝑃
,
5
Mmax = −
Mmax =−
4𝑃𝐿
5
(a) x = 0, Vmax = 56.8 kN
6𝑃𝐿
5
Vmax = 8.0 kN, Mmax = –8.0 kNm
4.5-22
Vmax = –6.4 kN, Mmax = –11.5 kNm
4.5-23
Σε αμφότερες τις περιπτώσεις αντιστοιχεί η ίδια
μέγιστη ροπή, ίση με PL
4.5-24
Vmax = 31.5 kN, Mmax = –73.5 kNm,
Mmid = –72.6 kNm
4.5-36
(a) Vmax = 6.4 kN, Mmax = 11.5 kNm
4.5-37
(b) xm = 7.1 m, Mmax = 202 kNm
(b) Vmax = 6.4 kN, Mmax = –11.5 kNm
4.5-26
4.5-27
MAz = –PL (ωρολογιακή), Αx = 0, Ay = 0, Cy = P/12
(προς τα πάνω), Dy = P/6 (προς τα πάνω),
Vmax = P/12, Mmax = PL
(a) Vmax = –2.88 kN, Mmax = 2.28 kNm
4.5-28
4.5-38
4.5-39
(a) Vmax = –2.4 kN, Mmax = 1.266 kNm
𝑞0 𝐿
,
2
19𝑞0 𝐿
,
18
Mmax = 𝑞0 𝐿2
𝑞0 𝐿
,
2
Mmax = 𝑞0 𝐿2
Vmax =
𝑞0 𝐿2
,
6
𝑞0 𝐿
𝑞 𝐿2
− , MBC,max = 0
2
6
(a) NAB = 0, VAB = 0, MAB,max =
4𝑞0 𝐿
9𝑞 𝐿2
, MAB,max = − 0 ,
3
10
𝑞0 𝐿
𝑞0 𝐿2
− 2 , MBC,max = 6
(b) NAB = 0, VAB,max =
NBC = 0, VBC,max =
ΜΑ = 0, Αx = 0, Ay = –18.41 kN (προς τα κάτω), Μd
= 0, Dx = –63.0 kN (προς τα αριστερά), Dy = 62.1
kN (προς τα επάνω), Νmax = –62.1 kN,
Vmax = 63.0 kN, Mmax = 756 kNm.
A-4.1
D
A-4.2
C
Ay = 6.25 kN, By = 8.75 kN, Mmax = 8.75 kNm για
x = 3m
A-4.3
D
A-4.4
A
Ay = 2.67 kN για x = 1.2m
By = 1.774 kN για x = 6.0m
Mmax = 1.9 kNm για x = 3.9m
A-4.5
A
A-4.6
C
A-4.7
B
Vmax = –2.25 kN, Mmax = 3.5 kNm
4.5-31
(a) Vmax = −
𝑞0 𝐿
,
2
(b) Vmax = −
2𝑞0 𝐿
,
3
Mmax =
𝑞0 𝐿2
3
Mmax =
2𝑞0 𝐿2
5
wL
w L2
= − 0 , M max = − 0
12
3
4.5-34
Vmax
5.4-1
(a) εmax = 8.88 10-4
(b) Rmin = 243 mm
5.4-2
Vmax =
4.5-40
4.5-30
4.5-33
𝑞0 𝐿
,
2
NBC = 0, VBC,max =
(a) a = 0.586L, Vmax = 0.293qL,
Mmax = 0.0214qL2
(b) a = 0.404L, Vmax = 0.161qL,
Mmax = 0.00882qL2
4.5-32
(a) Nmax =
(b) Nmax = −
(b) M0 = 0.523 kNm (ωρολογιακή)
4.5-29
(a) Ax = 233 N (προς τα δεξιά), Ay = 936 N (προς
τα πάνω), Βx = –233 N (προς τα αριστερά),
Νmax = –959 N, Vmax = –219 N, Mmax = 373 Nm
(b) Ax = 0, Ay = 298 N, Βx =0, By = 638 N,
Νmax = 600 N, Vmax = –219 N, Mmax = 373 Nm
(b) a = 1.72 m, Vmax = –2.54 kN, Mmax = 2.5 kNm
(c) a = 1.41 m, Mmax = 2.53 kNm
3w0 L
(προς τα
10
3w L
wL
αριστερά), Ay = − 0 (προς τα κάτω), C y = 0
20
12
w0 L
(προς τα πάνω), Dy =
(προς τα πάνω),
6
wL
w L2
(στο σημείο Β)
Vmax = 0 , M max = − 0
24
4
(ωρολογιακή), Ax = −
𝑀𝐴 = −
4.5-21
4.5-25
𝑤0 𝐿2
30
4.5-35
(b) dmax = 4.38 mm
5.4-3
(a) εmax = 5.89 10-3
(c) dmax = 5.93 mm
(b) dmax = 124.3 mm
(a) Lmin = 5.24 m
(c) Lmin = 15.45 m
5.4-4
5.4-5
(a) ρ = 85 m, κ = 0.0118 m-1, δ = 23.5 mm
(b) σt = 360M/(73bh2)
(b) hmax = 136 mm
(c) σt = 85.24M/d3
(c) δ = 75.3 mm
5.5-16
σmax = 10.965M/d3
(a) ε = 8.4 10-4
5.5-17
(a) σmax = 147.4 MPa
(b) tmax = 6.7 mm
(b) Lreqd = 6.1 m
(c) δ = 18.27 mm
(c) d = 2.54 m
(d) Lmax = 936 mm
5.4-6
5.5-18
(a) ε = 4.57 10-4
(b) Lmax = 2 m
(a) σt = 35.4 MPa, σc = 61MPa
(b) dmax = L/2, σt = 37.1 MPa, σc = 64.1 MPa
5.5-19
(a) σc = 105.8 MPa, σt = 28.8 MPa
5.4-7
δ = 11.00 mm, εx = 8.6 10-4
(b) Pmax = 893 N
5.4-8
ρ = 600 m, κ = 1.667 10-3 m-1, δΒ = 7.5 mm
(c) 1.34 m
5.5-1
(a) σmax = 361 MPa
5.5-2
(b) σc = 1.666 MPa (+14%), σt = 1.381 MPa (–9%)
(c) Lnew = 3.07 m
(c) σc = 0.728 MPa (–50%), σt = 0.757 MPa
(–50%)
(a) σmax = 250 MPa
(c) +25%
(a) σmax =186.2 MPa
5.5-21
Pmax = 2.5 kN
5.5-22
σmax = 3pL2a0/t
5.5-23
(a) σt = 101.3 MPa, σc = 180 MPa
(b) a = 4.24 m
(b) +10%
(c) +55.8%
5.5-4
(a) σc = 1.456 MPa, σc = 1.514 MPa
(b) 33.3%
(b) –19.98%
5.5-3
5.5-20
(a) σmax = 8.63 MPa
5.5-24
σ = 25.1 MPa, 17.8 MPa, –23.5 MPa
5.5-25
(a) σt = 132.8 MPa, σc = 99.6 MPa
(b) σmax = 6.49 MPa
(b) h = 82.8 mm
5.5-5
σmax = –2.4 MPa
(c) q = 1.867 kN/m, P = 3.9 kN
5.5-6
q0,max = 351 kN/m
5.5-7
σmax = 122.3 MPa
5.5-8
σmax = 203 MPa
5.5-9
σmax = 19.1 MPa
5.5-10
σmax = 101 MPa
5.5-11
σmax = 70.6 MPa
5.5-12
σmax = 7.0 MPa
5.5-13
(a) σmax = 5264 kPa
(b) s = 0.58579L, σmin = 1884 kPa
5.5-26
(b) σt = 4659 kPa (άνω μέρος της δοκού στο
σημείο C), σc = 5506 kPa (κάτω μέρος της δοκού
στο σημείο C)
5.5-27
d = 1.0 m, σmax = 1.55 MPa, d = 2 m, σmax = 7.52
MPa
5.5-28
(a) σt = 0.335 MPa, σc = 0.288 MPa
(b) Pmax = 21.9 kN
5.5-29
σmax = 2.10 MPa
5.5-15
(a) σt = 30.93M/d3
(a) Fres = 441 N
(b) σmax = 257 MPa (η βάση θλίβεται)
(c) s = 0 ή s = L, σmax = 10.89 MPa
5.5-14
(a) c1 = 91.7 mm, c2 = 108.3 mm, Iz = 7.969 107
mm4
(c) σmax = 231 MPa (η βάση εφελκύεται)
5.5-30
Pmax = 675 N
5.6-1
100 × 300
5.6-21
6.03%
5.6-2
IPN 240
5.6-22
(a) bmin = 11.91 mm
5.6-3
dmin = 100 mm
5.6-4
(a) dmin = 12.62 mm
(b) bmin = 11.92 mm
5.6-23
(b) Pmax = 39.8 N
5.6-5
(b) d = 292 mm
(a) UPN 260
5.6-24
IPN 360
(b) IPN 180
5.6-25
(a) β = 1/9
(c) HE 240A
5.6-6
(a) HE 180B
(b) 5.35%
5.6-26
(a) IPN 280
(a) αύξηση όταν d/h > 0.6861
(b) μείωση όταν d/h < 0.6861
(b) HE 260B
5.6-7
(a) smax = 1.732 m
5.7-1
(a) x =
(b) Pmax = 17.24 kN
5.6-8
(a) bmin = 161.6 mm
L σ = 4 PL σ max = 2
, max
,
9  hA3 σ B
4
(b) x = 0.209L, σ max =
(b) bmin = 141.2 mm
(c) Λόγος επιφανειών (b/a): 1.145
5.6-9
5.7-2
5.7-3
5.6-12
hmin = 30.6 mm
5.6-13
(a) Sreqd = 249 cm3
5.6-15
(b) x = 2 m, σmax = 25.2 MPa,
σ max 4
=
σm 3
(a) x = 182 mm, σmax = 8.94 MPa,
σ max
= 1.047
σB
(b) x = 109 mm, σmax = 8.85 MPa,
σ max
= 1.220
σm
(a) q0,επ = 13.17 kN/m
(b) q0,επ = 6.9 kN/m
5.6-14
σ max 9
=
σB 8
(a) smax = 429 mm
(b) hmin = 214 mm
5.6-11
(a) x = 4 m, σmax = 37.7 MPa,
(a) 50 × 250
(b) w = 8.36 kN/m2
5.6-10
0.394 PL σ max
= 3.54
,
σB
hA3
5.7-4
(a) σΑ = 210 MPa
(b) IPN 220
(b) σΒ = 221 MPa
(a) dmin = 37.6 mm
(c) x = 0.625 m
(b) dmin = 42.4 mm
(d) σmax = 231 MPa
(c) Λόγος επιφανειών (b/a): 0.557
(e) σmax = 214 MPa
(a) 100 × 250
5.7-5
(a) 1 
(b) qmax = 332 N/m
5.6-16
b = 152 mm, h = 202 mm
5.6-17
b = 259 mm
5.6-18
t = 13.61 mm
5.6-19
W1:W2: W3:W4 = 1:1.260:1.408:0.794
5.6-20
(a) qmax = 6.61 kN/m
(b) qmax = 9.37 kN/m
dB
 1.5
dA
(b) σ max = σ B =
5.7-6
hx =
hB x
L
5.7-7
by =
2bB x
L
32PL
πd B3
x
L
(b) Lmax = 2.51 m
5.7-8
hx = hB
5.8-2
(a) τmax = 731 kPa, σmax = 4.75 MPa
(b) τmin = 31.2 MPa
(b) τmax = 1462 kPa, σmax = 19.01 MPa
(c) τaver = 40.1 MPa, λόγος: 1.045
5.8-3
σx = –1.5 MPa, τxy = –0.136 MPa
(d) Vweb = 124.3 kN, λόγος: 0.956
5.8-4
qmax = 24 kN/m
5.8-5
(a) Mmax = 41.7 kNm
(b) τmin = 21.86 MPa
(b) Mmax = 3.24 kNm
(c) τaver = 27.41 MPa
5.8-6
τmax = 500 kPa
(d) Vweb = 119.7 kN
5.8-7
τmax = 22.5 MPa
5.8-8
(a) L0 = h
(b) L0 =
5.8-9
5.8-10
5.8-12
h σ επ.
2 τεπ.
5.8-14
5.10-3
(a) τmax = 41.9 MPa
(a) τmax = 28.43 MPa
(a) τmax = 38.56 MPa
(b) τmin = 34.51 MPa
(c) τaver = 41.98 MPa, λόγος: 0.919
(d) Vweb = 39.9 kN, λόγος: 0.886
5.10-4
(a) τmax = 32.28 MPa
(a) Pmax = 8.2 kN
(b) τmin = 21.45 MPa
(b) Pmax = 8.77 kN
(c) τaver = 29.24 MPa
(a) Mmax = 72.2 Nm
(d) Vweb = 196.1 kN
5.10-5
(a) τmax = 19.016 MPa
(a) δοκός 150 mm × 300 mm
(b) τmin = 16.21 MPa
(b) δοκός 200 mm × 300 mm
(c) τaver = 19.66 MPa, λόγος: 0.967
(a) P = 38.0 kN
(d) Vweb = 82.73 kN, λόγος: 0.919
(b) P = 35.6 kN
5.8-13
5.10-2
σεπ.
τεπ.
(b) Mmax= 36.0 Nm
5.8-11
5.10-1
5.10-6
(a) τmax = 28.40 MPa
(a) w1 = 9.46 kN/m2
(b) τmin = 19.35 MPa
(b) w2 = 19.88 kN/m2
(c) τaver = 25.97 MPa
(c) wεπ = 9.46 kN/m2
(d) Vweb = 58.63 kN
(a) b = 89.3 mm
5.10-7
qmax = 131.5 kN/m
(b) b = 87.8 mm
5.10-8
(a) qmax = 184.7 kN/m
5.9-1
dmin = 158 mm
5.9-2
(a) W = 28.6 kN
5.10-9
(b) W = 38.7 kN
5.10-10 V = 273 kN
5.9-3
τmax = 7.6 MPa
5.10-11 τmax = 10.17 MPa, τmin = 7.38 MPa
5.9-4
d0 = 47.6 mm
5.10-12 τmax = 19.7 MPa
5.9-5
(a) d = 328 mm
5.10-13 τmax = 13.87 MPa
(b) d = 76.4 mm
5.11-1
Vmax = 3067 MPa
(a) qο,max = 55.7 kN/m
5.11-2
Vmax = 1.924 MN
5.9-6
(b) qmax = 247 kN/m
IPN 220
5.11-3
F = 323 kN/m
5.11-4
Vmax = 10.7 kN/m
5.11-5
(a) smax = 67.0 mm
(b) smax = 53.8 mm
5.11-6
5.12-15 (a) σt = 36.7 MPa, σc = –30.9 MPa
(b) Και οι δύο τάσεις αυξάνονται κατά μέτρο.
5.12-16 (a) σ t =
(a) smax = 67.4 mm
(b) σ t =
(b) smax = 44.9 mm
5.11-8
smax = 92.3 mm
5.11-9
Vmax = 103.0 kN
(c) σt = 3.77 MPa, σc = –19.04 MPa, y0 = –139.2
mm
5.12-18 (a) σt = 3.27 MPa, σc = –24.2 MPa
(b) y0 = –76.2 mm
(c) 1
(c) σt = 1.587 MPa, σc = –20.3 MPa, y0 = –100.8
mm
(d) 1
5.12-1
σA = 92.6 MPa, σB = –171.5 MPa
5.12-2
d = 89.4 mm
5.12-3
σt = 94.6 MPa, σc = –96.7 MPa
5.12-4
σt = 5770 kPa, σc = –6668 kPa
5.12-5
tmin = 12.38 mm
5.12-6
σt = –11.83 MPa, σc = –12.33 MPa, tmin = 12.38
mm
5.12-7
σt = 2.42 MPa, σc = –2.50 MPa
5.12-8
Tmax = 108.6 kN
5.12-9
 d 2 + d12 
α = arctan  2

 4hd 2 
5.12-10 (a) dmin = 8.46 cm
(b) dmin = 8.91 cm
5.12-19 (a) σt = 112.1 MPa
(b) σt = 21.4 MPa
5.12-20 (a) y0 = –31.2 mm
(b) P = 115.3 kN
(c) y0 = 891 mm, P = 296 kN
5.13-1
(a) d = 6, 12, 18, 24 mm ,
 max = 94.1, 96.4, 103.1, 143.5 MPa
(b) R = 1.25, 2.5, 3.75, 5.0 mm ,
 max = 500, 433, 367, 333 MPa
5.13-2
(a) d = 16 mm, σmax = 81 MPa
(b) R = 4 mm, σmax ≈ 200 MPa
5.13-3
bmin ≈ 6.39 mm
5.13-4
bmin ≈ 0.33 mm
5.13-5
(a) Rmin ≈ 0.33 mm
5.12-11 Hmax = 9.77 m
(d) dmax = 104.1 mm
5.12-12 W = 33.3 kN
5.12-13 (a) σt = 826 kPa, σc = –918 kPa
(b) dmax = 0.750 m
d
5.12-14 (a) b = π −
6
9.11P
6.36 P
, σc =
2
b2
b
(b) y0 = –100.1 mm
5.11-11 (a) 2
5.11-12 smax = 165.2 mm
8P
4P
, σc = − 2
2
b
b
5.12-17 (a) σt = 723 MPa, σc = –36.2 MPa
5.11-10 smax = 61.5 mm
(b) 3
d
3
(c) ορθογώνιο παραλληλόγραμμο
(a) sA = 78.3 mm
(b) sB = 97.9 mm
5.11-7
(b) b = π −
A-5.1
A
A-5.2
C
A-5.3
A
A-5.4
D
A-5.5
A
A-5.9
C
A-5.6
B
A-5.10
A
A-5.7
B
A-5.11
B
A-5.8
C
A-5.12
B
6.3-1
σw = 5 MPa, σs = 85 MPa
6.3-15
ts = 22.7 mm
6.3-2
σw = 4.66 MPa, σs = 91.5 MPa
6.3-16
αναλογία = 0.723
6.3-3
σface = ±21.17 MPa, σcore = ±5.29 MPa
6.3-17
άνω μέρος δοκού: σw = –5.4 MPa, σs = –97.9 MPa
6.3-4
(a) Mmax = 58.7 kNm
κάτω μέρος δοκού: σw = 6.3 MPa
(b) Mmax = 90.9 kNm
κάτω μέρος χάλυβα: σs = 86 MPa
6.3-5
(c) t = 7.08 mm
6.3-1
ομοίως με 6.2-17
(a) Mmax = 20.8 kNm
6.3-2
άνω μέρος πυρήνα: σp = –0.253 MPa
άνω μέρος δοκού: σs = –66 MPa
(b) Mmax = 11.7 mm
6.3-6
(a) 𝑀𝜀𝜋,𝑇𝑖 =
𝜎𝑇𝑖 (𝐸𝑇𝑖 𝐼𝑇𝑖 +𝐸𝐶𝑢 𝐼𝐶𝑢 )
𝑑
𝐸𝑇𝑖 ( 2)
,
6.3-3
2
𝑀𝜀𝜋,𝐶𝑢 =
(b) Mmax = 27.2 kNm
𝜎𝐶𝑢 (𝐸𝑇𝑖 𝐼𝑇𝑖 +𝐸𝐶𝑢 𝐼𝐶𝑢 )
𝑑
𝐸𝐶𝑢 ( 1)
2
(b) Mmax,Ti =4989 Nm
6.3-4
tmin = 15.0 mm
6.3-5
(a) qεπ = 11.98 kN/m
(b) σwood = 1.10 MPa, σsteel = 55.8 MPa
(c) d1 = 36.4 mm
6.3-7
(a) σw = 4.99 MPa, σs = 109.7 MPa
6.3-6
6.3-9
(c) M0,max = 2.44 kNm
6.3-7
σa = 23.0 MPa, σΡ = 0.891 MPa
(a) Mεπ = 768 Nm
6.3-8
σa = 12.14 MPa, σΡ = 0.47 MPa
(b) σsa = 47.9 MPa, Mmax = 1051 Nm
6.3-9
(a) qεπ = 4.16 kN/m
(a) σface = 26.2 MPa, σcore = 0
(b) σface = 26.7 MPa, σcore = 0
6.3-10
(a) σB = 60.3 MPa, σw = 7.09 MPa
(b) tB = 25.1 mm, Mmax = 80 kNm
(b) qmax = 3.57 kN/m
6.3-8
(a) Mmax = 103 kNm
(b) qεπ = 4.39 kN/m
6.3-10
(a) σface = 14.1 MPa, σcore = 0.214 MPa
(a) σs = 93.5 MPa
(b) hs = 5.08 mm, ha = 114.92 mm
(b) σface = 14.9 MPa, σcore = 0
6.3-11
Mmax = 10.4 kNm
6.3-11
σa = 3753 kPa, σC = 4296 kPa
6.3-12
SA = 50.6 mm3, Μέταλλο Α
6.3-12
(a) σw = 5.1 MPa (θλιπτική), σs = 37.6 MPa
(εφελκυστική)
6.3-13
σsteel = 77.8 MPa (εφελκυσμός),
(b) ts = 3.09 mm
6.3-13
σconcrete = 6.86 MPa (θλίψη)
6.3-14
(a) σplywood = 7.29 MPa, σpine = 6.45 MPa
(b) Mmax = Mc = 172.9 kNm
(b) qmax = 1.43 kN/m
6.3-14
QO,max = 15.53 kN/m
(a) σc = 8.51 MPa, σs = 118.3 MPa
(c) As = 2254mm2, Mεπ = 167.8 kNm
6.3-15
(a) σc = 4.24 MPa, σs = 85.9 MPa
6.3-16
(b) Mεπ = Ms = 349 kNm
6.5-8
β = 2.93º, σt = 6.56 MPa, σc = –6.54 MPa
(a) Mmax = Ms = 10.59 kNm
6.5-9
 t ( 0 ) = − c ( 0 ) = 2.546
(b) As = 1262 mm2, Mεπ = 15.79kNm
6.3-17
M
M
,  c ( 45 ) = −3.955 3 ,
3
r
r
M
M
 t ( 90 ) = 3.867 3 ,  c ( 90 ) = −5.244 3
r
r
 t ( 45 ) = 4.535
άνω μέρος δοκού: σw = –4.5 MPa, σs = –82.2 MPa
κάτω μέρος ξύλου: σw = 2.5 MPa
κάτω μέρος χάλυβα: σs = 87 MPa
M
,
r3
6.5-10
β = –78.9º, σt = 131.1 MPa, σc = –148.5 MPa
6.5-11
β = –11.7º, σt = 118.8 MPa, σc = –103.2 MPa
6.5-12
β = –48.6º, σt = 16.85 MPa, σc = –15.79 MPa
β = 51.8º, σmax = 17.5 MPa
6.5-13
σA = –87.7 MPa, σB = 65.1 MPa
6.4-3
β = 53.1º, σmax = 440 MPa
6.8-1
(a) τmax = 29.9 MPa
6.4-4
β = 84.5º, σA = –σE = 141 MPa, σB = –σD = –89
MPa
6.3-18
Mεπ = MW = 16.39 kNm
6.4-1
h
tan  =
b
6.4-2
6.4-5
β = 47.7º, σA = –σE = 160 MPa, σB = –σD = –12
MPa
6.4-6
β = –79.3º, σmax = 8.87 MPa
6.4-7
β = –75.9º, σmax = 12.7 MPa
6.4-8
Pmax = 22.4 kN
6.4-9
qmax = 6.7 kN/m
6.4-10
β = –83.2º, σmax = 62.3 MPa
6.4-11
β = 84.6º, σmax = 31.0 MPa
6.4-12
β = 62.6º, σmax = 21.6 MPa
6.4-13
(a) σΑ = 103.94sin a + 11.48cos a [MPa]
(b) τB = 2.7 MPa
6.8-2
(a) τmax = 21.0 MPa
(b) τB = 3.4 MPa
6.8-3
(a) τmax = 21.7 MPa
(b) τmax = 21.1 MPa
6.8-4
(a) τmax = 16.6 MPa
(b) τmax = 16.3 MPa
6.9-1
e = 31.0 mm
6.9-2
e = 19.12 mm
6.9-6
(b) e =
63 r
= 1.745r
24 + 38
6.9-8
(a) e =
b 2h + 3b
2 h + 3b
(b) e =
b 43h + 48b
2 23h + 48b
(b) tan  = 21.729 tan a
6.4-14
β = 70.3º, σmax = 11.9 MPa
6.4-15
(a) β = –76.6º, σmax = 48.5 MPa
(b) β = –80.9º, σmax = 48.8 MPa
6.4-16
σmax = 14.6 MPa
6.5-1
β = 74.5º, σt = 37.6 MPa, σc = –63.7 MPa
6.5-2
β = 78.0º, σt = 7.2 MPa, σc = –13.2 MPa
6.5-3
β = 75.6º, σt = 30.4 MPa, σc = –37.7 MPa
6.5-4
β = 75.5º, σt = 22.7 MPa, σc = –27.9 MPa
6.5-5
(a) β = –30.6º, σt = 36.9 MPa, σc = –43.6 MPa
(b) β = –41.9º, σt = 52.5 MPa, σc = –42.6 MPa
6.5-6
β = 78.1º, σt = 40.7 MPa, σc = –40.7 MPa
6.5-7
β = 73.0º, σt = 10.3 MPa, σc = –9.5 MPa
6.10-1
6.10-2
f =
2 ( 2b1 + b2 )
3b1 + b2
(a) 𝑓 =
16𝑟2 (𝑟23 −𝑟13 )
3𝜋(𝑟24 −𝑟14 )
(b) f = 4/π
6.10-3
q = 125 kN/m
6.10-4
(a) 56.7%
(b) M = 12.3 kNm
6.10-5
f = 1.10
6.10-6
f = 1.15
6.10-7
Z = 188.4 cm3, f = 1.17
6.10-8
Z = 1230 cm3, f = 1.07
6.10-9
MY = 688 kNm, MP = 814 kNm, f = 1.18
6.10-17 (a) M = 209 kNm
(b) 7.6%
6.10-10 MY = 365 kNm, MP = 431 kNm, f = 1.18
6.10-18 Z = 136 103 mm3, f = 1.79
6.10-11 MY = 154.3 kNm, MP = 205 kNm, f = 1.33
6.10-19 MP = 117.3 kNm
6.10-12 MY = 672 kNm, MP = 878 kNm, f = 1.31
6.10-20 MP = 295 kNm
6.10-13 MY = 345 kNm, MP = 453 kNm, f = 1.31
Α-6.1
B
6.10-14 MY = 122 kNm, MP = 147 kNm, f = 1.20
Α-6.2
C
6.10-15 (a) M = 378 kNm
Α-6.3
B
Α-6.4
B
Α-6.5
D
7.2-15
σw = σy1 = –912 kPa, τw = –τx1y1 = 2647 kPa
7.2-16
σw = 10.0 MPa, τw = –5.0 MPa
7.2-17
σx1 = 5 MPa, σy1 = 5 MPa, τx1y1 = 5 MPa
7.2-18
σx1 = 9 MPa, σy1 = 3 MPa, τx1y1 = 5.2 MPa
7.2-19
θ = 51.8º, σy1 = 3.7 MPa, τx1y1 = –7.63 kPa
7.2-20
θ = 36.6º, σy1 = –9 MPa, τx1y1 = –14.83 kPa
7.2-21
για θ = 52º: σx1 = –136.6 MPa, σy1 = 16.6 MPa,
τx1y1 = –84 MPa
για θ = –36º: σx1 = –81.1 MPa, σy1 = –31.9 MPa,
τx1y1 = –27.2 MPa
7.2-22
για θ = 48º: σx1 = –9.84 MPa, σy1 = –46.2 MPa,
τx1y1 = 10.16 MPa
για θ = –50º: σx1 = 51.4 MPa, σy1 = –14.4 MPa,
τx1y1 = –31.3 MPa
7.2-23
σy = 25.3 MPa, τxy = 10 MPa
για θ = –40º: σx1 = –5.5 MPa, σy1 = –27.0 MPa,
τx1y1 = –28.1 ΜPa
7.2-24
σy = –77.7 MPa, τxy = –27.5 MPa
7.2-25
σb = –32.0 MPa, τb = 18.79 MPa, θ1 = 47.9º
7.2-9
για θ = 36º: σx1 = –94.1 MPa, σy1 = –22.9 MPa,
τx1y1 = 32.1 ΜPa
7.2-26
σy = –117.2 MPa
για θ = –40º: σx1 = –66.5 MPa, σy1 = –6.52 MPa,
τx1y1 = –14.52 ΜPa
7.2-27
σx1 = –1.4 MPa, σy1 = 2.4 MPa, τx1y1 = 0
7.2-10
7.3-1
σ1 = 7 MPa, σ2 = 1 MPa, τmax = 3 MPa
7.2-11
σ = 1370 kPa (εφελκυσμός), τ = 1158 kPa
(ωρολογιακά)
7.3-2
σ1 = 8.04 MPa, σ2 = 1.463 MPa, τmax = 3.29 MPa
7.3-3
σ1 = 40.8 MPa, σ2 = 7.24 MPa, θp1 = 8.68º
7.2-12
σ = 1440 kPa (εφελκυσμός), τ = 1030 kPa
(ωρολογιακά)
7.3-4
σ1 = 119.2 MPa, σ2 = 60.8 MPa, θp1 = 29.52º
7.3-5
σ1 = –43.7 MPa, σ2 = –8.31 MPa, θp1 = –23.6º
7.2-13
σx1 = 41.25 MPa, σy1 = –54.25 MPa,
τx1y1 = 69.26 MPa
7.3-6
σ1 = 53.6 MPa, θp1 = –14.2º
7.2-14
σx1 = –14.44 MPa, σy1 = 1.44 MPa,
τx1y1 = 83.75 MPa
7.3-7
σ1 = 39.9 MPa, σ2 = 124.1 MPa, θp1 = –14.2º,
τmax = –42.1 MPa
(b) 23.3%
6.10-16 (a) M = 524 kNm
(b) 36%
7.2-1
για θ = 52º: σx1 = 30.2 MPa, σy1 = 27.8 MPa,
τx1y1 = –21.9 MPa
7.2-2
για θ = 40º: σx1 = 117.2 MPa, σy1 = 62.8 MPa,
τx1y1 = –10.43 MPa
7.2-3
σx1 = 8.2 MPa, σy1 = 2.3 MPa, τx1y1 = 0.2 MPa
7.2-4
σx1 = 185 MPa, σy1 = 35 MPa, τx1y1 = –10 MPa
7.2-5
για θ = 32º: σx1 = –23.8 MPa, σy1 = –51.2 MPa,
τx1y1 = –85.1 MPa
7.2-6
7.2-7
7.2-8
7.3-8
τmax = 24.2 MPa, σx1 = –14.25 MPa, σy1 = –14.25
MPa, θs1 = 60.53º
7.3-9
τmax = 47.3 MPa, θs1 = 61.7º
7.3-10
τmax = 26.7 MPa, θs1 = 19.08º
7.3-11
(𝜏𝑚𝑎𝑥 )z1 = 1 MPa, (𝜏𝑚𝑎𝑥 )x1 = –2 MPa
7.4-1
(b) τmax = 49 MPa, σave = 49 MPa
7.4-2
σ1 = 20 kPa, σ2 = –20 kPa, τmax = 20 kPa
7.3-13
(a) σ1 = 1 MPa, θp1 = –18.43º
(b) τmax = 5 MPa, θs1 = –63.4º
7.3-14
σ1 = 0, σ2 = –5 MPa, τmax = 2.5 MPa
7.3-15
σ1 = 0, σ2 = –7.5 MPa, τmax = 3.75 MPa
7.3-16
(a) σ1 = 25 MPa, σ2 = –130 MPa
(b) τmax = 77.5 MPa, σaver = –52.5 MPa
7.3-17
(a) σ1 = 18.45 MPa, σ2 = 4.55 MPa
7.4-3
(a) σ1 = 2262 kPa, θp1 = –13.70º
(b) τmax = 1000 kPa, θs1 = –58.7º
7.3-19
(a) σ1 = 101.8 MPa, θp1 = 7.85º
(b) τmax = 48 MPa, θs1 = –37.2º
7.3-20
(a) σ1 = 29.2 MPa, θp1 = –17.98º
(b) τmax = 66.4 MPa, θs1 = –63.0º
7.3-21
(a) σ1 = –8.72 MPa, θp1 = 24.7º
7.4-4
7.4-5
(a) για θ = 55º: σx1 = 6.09 MPa, τx1y1 = –25.8 MPa
(b) τmax = 27.5 MPa, θs1 = –45º, σaver = 15.50 MPa
7.4-6
για θ = 21.8º:
(a) σx1 = –17.1 MPa, τx1y1 = 29.7 MPa
(b) τmax = 43.0 MPa, θx1 = 45º
7.4-7
(a) για θ = 52º: σx1 = 18.44 MPa, τx1y1 = –4.60 MPa
(b) σ1 = 19.00 MPa, θp1 = 45º
7.4-8
σx1 = 1.75 MPa, σy1 = 13.25 MPa, τx1y1 = 4.82 MPa
7.4-9
(a) σx1 = –3.3 MPa, σy1 = –6.7 MPa, τx1y1 = 4.7 MPa
(b) σ1 = 0, σ2 = –10 MPa, τmax = 5 MPa
7.4-10
(a) σx1 = –60.8 MPa, σy1 = 128.8 MPa, τx1y1 = –46.7
MPa
(b) σ1 = 139.6 MPa, σ2 = –71.6 MPa, τmax = 105.6
MPa
(a) σ1 = 76.3 MPa, θp1 = 107.5º
(b) τmax = 101.3 MPa, θs1 = –62.5º
(a) για θ = 25º: σx1 = –36.0 MPa, τx1y1 = 25.7 MPa
(b) τmax = 33.5 MPa, θs1 = 45º
(b) τmax = 40.8 MPa, θs1 = –20.3º
7.3-22
(a) σx1 = –37.6 MPa, σy1 = –9.4 MPa, τx1y1 = 18.8
MPa
(b) τmax = –23.5 MPa, σave = –23.5 MPa
(b) τmax = 6.95 MPa, σaver = 11.5 MPa
7.3-18
(a) σx1 = 40.1 MPa, σy1 = 16.91 MPa, τx1y1 = 26
MPa
(b) τmax = 28.5 MPa, σave = 28.5 MPa
(𝜏𝑚𝑎𝑥 )y1 = –1 MPa
7.3-12
(a) σx1 = 75 MPa, σy1 = 23 MPa,
τx1y1 = –41.6 MPa
7.4-11
(a) για θ = 36.87º: σx1 = 25.0 MPa, τx1y1 = 7.28 MPa
(b) σ1 = 26.0 MPa, θp1 = 45º
7.3-23
20.6 MPa ≤ σy ≤ 65.4 MPa
7.3-24
18.5 MPa ≤ σy ≤ 85.5 MPa
7.4-12
για θ = 40º: σx1 = 27.5 MPa, τx1y1 = –5.36 MPa
7.3-25
(a) σy = 28.6 MPa
7.4-13
για θ = –51º: σx1 = 82.7 MPa, τx1y1 = –24.6 MPa
(b) σ1 = 44 MPa, σ2 = 17.64 MPa, θp1 = -40.24º,
θp2 = 49.76º
7.4-14
για θ = –33º: σx1 = –61.7 MPa, σy1 = –171.3 MPa,
τx1y1 = –51.7 MPa
(a) σy = 23.3 MPa
7.4-15
(b) θp1 = 65.6º, σ1 = 41 MPa, θp2 = –24.4º ,
σ2 = –62.7 MPa
για θ = 14º: σx1 = –10.42 MPa, σy1 = –6.58 MPa,
τx1y1 = 3.85 MPa
7.4-16
για θ = 35º: σx1 = 46.4 MPa, τx1y1 = –9.81 MPa
7.3-27
σ1 = 3.6 kPa, σ2 = –638 kPa, τmax = 321 kPa
7.4-17
για θ = 65º: σx1 = –12.71 MPa, τx1y1 = 27.3 MPa
7.3-28
σ1 = 12 MPa, σ2 = 0, τmax = 6 MPa
7.4-18
(a) σ1 = 10,865 kPa, θp1 = 115.2º
7.3-26
(b) τmax = 4865 kPa, θs1 = 70.2º
7.4-19
7.4-20
7.4-21
7.4-22
(a) σ1 = 17.72 MPa, θs1 = 31.4º
7.5-11
(a) ΔV = –602 mm3, U = 6.2452 J
(b) τmax = 22.47 MPa, θs1 = –13.57º
7.5-12
(a) ΔV = 2766 mm3, U = 56 J
(a) σ1 = 18.2 MPa, θp1 = 123.3º
(b) tmax = 36.1 mm
(b) τmax = 15.4 MPa, θs1 = 78.3º
(c) bmin = 640 mm
(a) σ1 = –47.67 MPa, θp1 = –32.9º
7.4-24
7.4-25
7.5-1
(b) tmax = 17.02 mm
(a) σ1 = 40.0 MPa, θp1 = 68.8º
(c) bmin = 262 mm
7.5-14
(b) Δbc = εyd = –0.074 mm (μείωση)
(b) τmax = 23.6 MPa, θs1 = –71.8º
(c) Δt = εzt = –2.86  10–3 mm (μείωση)
(a) σ1 = 3.43 MPa, θp1 = –19.68º
(d) ΔV = eV0 = 430 mm3
(b) τmax = 15.13 MPa, θs1 = –64.7º
(e) U = uV0 = 71.2 N∙m
(a) σ1 = 51.6 MPa, θp1 = 9.9º
(f) tmax = 22.0 mm
(b) τmax = 26.6 MPa, θs1 = –35.1º
(g) σmax = 63.9 MPa
σx = 175.2 MPa, σy = 135.3 MPa, Δt = –7.2  10-3
mm
σx = 102.6 MPa, σy = –11.21 MPa, Δt = –1.646 
10-3 mm
7.5-3
e = –3.17  10-4
7.5-4
(a) εx = 7.25  10-5, εy = –9  10-5, εz = 7.5  10-6,
7.5-15
γxy = 6.5  10
7.5-16
v = 0.477, E = 1.006 MPa
7.6-1
εx = –3.98  10-4, εy = 3.24  10-4, εz = 2.1  10-4,
e = 1.36  10-4
7.6-3
 +y
(a)  z = − x
1 −
(c) ΔV = 287 mm3
1 −
v = 0.24, E = 112.1 GPa
7.5-7
v = 0.3, E = 204.1 GPa
7.5-8
(a) γmax = 5.85  10-4
(b) Δt = –1.32  10-3 mm
(c) ΔV = 387 mm3
(a) τmax = 60 MPa
(b) Δa = 0.1955 mm, Δb = –0.0944 mm,
Δc = –0.034 mm
(1 − 2 ) ( x +  y )
7.5-6
7.5-10
εx = –3.33  10-4, εy = 5.79  10-4, εz = –2.95  10-4,
u = 14.09 kPa
(b) u = 1.2 kPa
(b) e =
(a) εx = 0.182, εy = –0.318, εz = 0.182
(b) e = 0.045
7.6-2
-5
7.5-9
(a) Δac = εxd = 0.1296 mm (αύξηση)
(a) σ1 = 4.4 MPa, θp1 = –26.8º
7.5-2
7.5-5
(a) ΔV = 623 mm3, U = 64.5 J
(b) τmax = 54.8 MPa, θs1 = –77.9º
(b) τmax = 40.0 MPa, θs1 = 23.8º
7.4-23
7.5-13
(d) U = 109.9 J
(e) σx,max = 88.5 MPa
(f) σx,max = 82 MPa
7.6-4
(a)  max =
1 −  3
2
= 8.5 MPa
(a) γmax = 1.921  10-3
(b) Δa = aεx = –0.0525 mm, Δb= bεy = –9.67  10–3
mm, Δc= cεz = –9.67  10–3 mm
(b) Δt = –3.55  10-3 mm (μείωση)
(c) ΔV = eV0 = –2.052  10–3 mm3
(c) ΔV = 1372 mm3 (αύξηση)
(d) U = uV0 = 56.2 J
(a) ΔVb = –49.2 mm , Ub = 3.52 J
(e) σx,max = –50 MPa
3
(a) ΔVa = –71.5 mm3, Uα = 4.82 J
(f) σx,max = –65.1 MPa
7.6-5
7.7-1
(a) σx = –28.8 MPa, σy = –14.4 MPa, σz = –14.4 MPa
(b) ε1 = 1.5  10-3, ε2 = –1.0  10-3,
γmax = 2.5  10-3
(b) τmax = 7.2 MPa
(c) ΔV = –300 mm3
7.7-2
(d) U = 3.78 J
(e) σx,max = –26.5 MPa
(f) εx,max = –242  10
7.7-3
(a) σx = –82.6 MPa, σy = –54.7 MPa, σz = –54.7 MPa
1 −  3
(b)  max =
7.7-4
(c) ΔV = eV0 = –864 mm3
(e) σx,max = –73 MPa
7.6-7
7.7-5
(a) KA1 = 74.1 GPa
(c) Δψ = α = 1.419  10-4 rad (αύξηση)
7.7-6
(a) K = 4.95 GPa
7.6-10
(a) 𝑝 =
𝜈𝐹
𝐴(1−𝜈)
(b) 𝛿 =
𝐹𝐿(1+𝜈)(1−2𝜈)
𝐸𝐴(1−𝜈)
(c) 𝑢 =
7.6-11
(c) Δψ = – α = 1.89  10-4 rad (αύξηση)
7.7-7
(c) γ = –314  10–6 (αύξηση της γωνίας ced)
p0 (1 + )(1 − 2 )
E
7.7-8
(c) γ = –634  10–6 (αύξηση της γωνίας ced)
2𝐸
(a) p = –3.1 kPa
7.6-12
P = –81.5 kN
7.6-13
(a) Δd = 0.036 mm
7.7-9
εx1 = 3.97  10-4, εy1 = 3.03  10-4, γx1y1 = 1.829 
10-4
7.7-10
εx1 = 9.53  10-5, εy1 = –1.353  10-4, γx1y1 = –3.86 
10-4
7.7-11
ε1 = 554  10-6, θp1 = –22.9º, γmax = 488  10-6
(b) ΔV = 2863 mm , U = 34.4 J
7.7-12
ε1 = 172  10-6, θp1 = 163.9º, γmax = 674  10-6
7.7-13
για θ = 75º:
3
(c) h = 1620 m
7.6-15
(a) Δd = 0.168 mm
(b) Δφ = 317  10-6 rad (μείωση)
𝑝02 (1−𝜈2 )
(b) ΔV = –0.2175 cm3
7.6-14
(a) Δd = 0.1146 mm
(b) Δφ = 150  10-6 rad (μείωση)
(a) p = vp0
(b) e = −
(a) Δd = εx1Ld = 0.062 mm
(b) Δφ = – α = 1.89  10-4 rad (μείωση)
(b) E = 1.297 GPa, v = 0.40
7.6-9
(a) Δd = 0.0476 mm
(b) Δφ = – α = 1.419  10-4 rad (μείωση)
(b) E = 42 GPa, v = 0.35
7.6-8
(a) εx1 = 1.398  10-3, εy1 = –9.76  10-5,
γx1y1 = –8.63  10-4
(b) ε1 = 1.513  10-3, ε2 = –2.13  10-4,
γmax = 1.726  10-3
(d) U = uV0 = 29.9 J
(f) εx,max = –741  10-6
(a) εx1 = 1.299  10-4, εy1 = –1.299  10-4,
γx1y1 = 1.5  10-4
(b) ε1 = 1.5  10-4, ε2 = –1.5  10-4,
γmax = 3.0  10-4
= 13.92 MPa
2
(a) ε1 = 2.0  10-3, ε2 = 1.5  10-3
(b) γmax = 5.0  10-4
-6
7.6-6
(a) εx1 = –5.53  10-4, εy1 = 1.053  10-3,
γx1y1 = 1.915  10-3
(a) p = 700 MPa
(a) εx1 = 202  10-6, γx1y1 = –596  10-6
(b) k = 175 GPa
(b) ε1 = 568  10-6, θP1 = 22.8º
(c) U = 2470 J
(c) γmax = 587  10-6
ε0 = 2.77  10-4, e = 8.3  10-4, u = 0.0344 MPa
7.7-14
για θ = 45º:
(a) εx1 = –385  10-6, γx1y1 = 690  10-6
7.7-16
7.7-25
τmax,xy = 29.5 MPa, γxy,max = 7.19  10-4,
γxz,max = 9.01  10-4, γyz,max = 1.827  10-4
θp1 = 30º, ε1 = 1550  10-6, ε2 = –250  10-6,
σ1 = 68.3 MPa, σ2 = 13.67 MPa
7.7-26
σx = 91.6 MPa
7.7-27
εx1 = 3.97  10-4, εy1 = 3.03  10-4, γx1y1 = 1.829 
10-4
7.7-28
εx1 = 9.53  10-5, εy1 = –1.353  10-4, γx1y1 = –3.86 
10-4
7.7-29
ε1 = 554  10-6, θp1 = 157.1º, γmax = 488  10-6
7.7-30
ε1 = 172  10-6, θp1 = 163.9º, γmax = 674  10-6
7.7-31
θ = 75º:
 max, xy =
 xy ,max
 x − y
2
= 33.7 MPa ,
  x −  y    xy 
−3
=2 
 +
 = 1.244 10 ,
 2   2 
2
 x −z 
2
−3
 +  xz = 1.459 10 ,
2


2
 xz ,max = 2 
 y −z 
2
−4
=2 
 +  yz = 2.15  10
 2 
2
 yz ,max
(b) ε1 = 568  10-6, θP1 = 22.8º
(c) γmax = 587  10-6
7.7-32
(c) γmax = 1.345  10-3
θ = 45º:
(a) εx1 = –385  10-6, γx1y1 = 690  10-6
(a) Για θ = 50º: εx1 = –1469  10 ,
γx1y1 = –717  10-6
-6
(b) ε1 = –254  10-6, θP1 = 65.7º
(c) γmax = 1041  10-6
(b) ε1 = –732  10-6, θp1 = 166.0º
(c) γmax = 911  10-6
A-7.1
C
7.7-19
ε1 = 551  10-6, θp1 = 12.5º, γmax = 662  10-6
A-7.2
C
7.7-20
ε1 = 332  10-6, θp1 = 12.0º, γmax = 515  10-6
A-7.3
D
7.7-21
(a) P = 23.6 kN, T = –114.9 N∙m
A-7.4
A
A-7.5
B
A-7.6
A
A-7.7
C
A-7.8
D
(b) γmax = 2.84  10-4, τmax = 23.1 MPa
7.7-22
P = 121.4 kN, α = 56.7º
7.7-23
Ρ = 44.1 kN, α = 75.2º
8.2-1
pσ = 16.82 kPa, pτ = 9.6 kPa
8.2-2
pα = 1.748 kPa
8.2-3
(a) t = 112 mm
(b) pmax = 3.14 MPa
8.2-4
8.2-5
(c) r = 334 mm
8.2-6
8.2-7
(a) σmax = 4.17 MPa, εmax = 0.655
(b) pmax = 129.8 kPa
8.2-8
(a) F = 15.81 kN, σ = 3.14 MPa
(b) db = 13.23 mm
(a) σmax = 3.12 MPa, εmax = 0.438
(b) treqd = 1.29 mm
(a) t = 98 mm
(b) pmax = 3.34 MPa
3
(a) εx1 = 202  10-6, γx1y1 = –569  10-6
(a) θ = 30º: εx1 = –7.61  10-4, γx1y1 = 8.62  10-4
(b) ε1 = 4.27  10-4, θp1 = 100.1º
7.7-18
2 ( b −  c )
(c) γmax = 1041  10-6
2
7.7-17
 xy =
εx = εa, 𝜀𝑦 =
(b) ε1 = –254  10-6, θP1 = 65.7º
7.7-15
2𝜀𝑏 +2𝜀𝑐 −𝜀𝑎
,
3
7.7-24
(a) pmax = 3.51 MPa
(b) pmax = 2.93 MPa
8.2-9
(a) f = 5100 kN/m
(b) τmax = 51 MPa
(c) εmax = 344  10-6
8.2-10
8.2-11
8.3-14
(b) τ1 = 23.2 MPa, τ2 = 46.7 MPa
(b) τmax = 57.3 MPa
(c) ε1 = 3.97  10-4, ε2 = 9.33  10-5
(c) εmax = 3.87  10-4
(d) θ = 35º, σx1 = 62.0 MPa, σy1 = 78.0 MPa,
τx1y1 = 21.9 MPa
(a) tmin = 11.67 mm
8.3-15
8.2-13
D0 = 26.6 m
8.3-1
σ1 = 152.6 MPa, σ2 = 76.3 MPa, τmax,In = 38.2 MPa,
τmax,Out = 76.3 MPa
8.3-2
σ1 = 76.2 MPa, σ2 = 38.1 MPa, τmax,In = 19.05 MPa,
τmax,Out = 38.1 MPa
(c) ε1 = 178.5  10-6, ε2 = 42  10-6
8.3-4
(a) h = 22.2 m
(b) μηδέν
8.3-5
8.3-6
n = 2.38
8.4-1
σx = 13.35 MPa, σy = 0, τxy = –0.989 ΜPa
8.4-2
σx = –2.22 MPa, σy = 0, τxy = –0.139 ΜPa
8.4-3
σx = 53.3 kPa, σy = 0, τxy = –66.7 kPa,
σ1 = 98.5 kPa, σ2 = –45.1 kPa, τmax = 71.8 kPa
8.4-4
σx = 667 kPa, σy = 0, τxy = –44.4 kPa,
σ1 = 670 kPa, σ2 = –2.95 kPa, τmax = 336 kPa
8.4-5
σx = –162.5 kPa, σy = 0, τxy = 1 MPa,
σ1 = 923 kPa, σ2 = –1085 kPa, τmax = 1004 kPa
8.4-6
σx = –208 kPa, σy = 0, τxy = 84.6 kPa,
σ1 = 30.1 kPa, σ2 = –238 kPa, τmax = 134.1 kPa
(a) F = 3πρr
8.4-7
(a) σx = –39.5 MPa, σy = 0, τxy = 7.9 MPa,
σ1 = 1.522 MPa, σ2 = –41 MPa, τmax = 21.3 MPa
2
(b) τreqd = 10.91 mm
8.3-7
(d) θ = 15º, σx1 = 22.4 MPa, σy1 = 40.6 MPa,
τx1y1 = 5.25 MPa
(a) tmin = 6.43 mm
(b) pmax = 11.2 MPa
(a) p = 350 kPa
(b) σx = –46.2 MPa, σy = 0, τxy = 7.9 MPa,
σ1 = 1.315 MPa, σ2 = –47.5 MPa, τmax = 24.4 MPa
(b) εr = 8.35  10-4
8.3-8
(a) εmax = 6.67  10-5
8.4-8
(b) εr = 2.83  10
-4
8.3-9
tmin = 2.53 mm
8.3-10
tmin = 3.71 mm
8.3-11
(a) h = 7.24 m
8.4-9
(a) σ1 = 1484 kPa, σ2 = –1484 kPa, τmax = 1484 kPa
(b) σ1 = 337 kPa, σ2 = –5163 kPa, τmax = 2750 kPa
(c) σ1 = 0, σ2 = –14.48 MPa, τmax = 7238 kPa
(a) σh = 24.9 MPa
(b) σc = 49.7 MPa
8.4-10
P = 20 kN
(c) σw = 24.9 MPa
8.4-11
P = 13.38 kN
(d) τh = 12.43 MPa
8.4-12
(a) P = 30 N, σx = σy = τxy = 0
(b) P = 167.5 N, σx = 220 kPa, σy = 0, τxy = 8.8 kPa
(e) τc = 24.9 MPa
8.3-13
(a) σx = –37.4 MPa, σy = 0, τxy = 7.49 MPa,
σ1 = 1.442 MPa, σ2 = –38.9 MPa, τmax = 20.2 MPa
(b) σx = –44.4 MPa, σy = 0, τxy = 7.49 MPa,
σ1 = 1.227 MPa, σ2 = –45.7 MPa, τmax = 23.4 MPa
(b) σ1 = 817 kPa
8.3-12
(a) σ1 = 42 MPa, σ2 = 21 MPa
(b) τ1 = 10.5 MPa, τ2 = 21 MPa
(a) tmin = 7.17 mm
(b) p = 19.25 MPa
8.3-3
(a) σ1 = 93.3 MPa, σ2 = 46.7 MPa
(a) f = 5500 kN/m
(b) p = 16.33 MPa
8.2-12
(b) tmin = 3.12 mm
(a) tmin = 6.25 mm
8.4-13
P = 22.5 kN, σx = σy = 0, τxy = –950 kPa
8.4-14
(b) σ1 = 4.5 MPa, σ2 = –76.1 MPa, τmax = 40.3 MPa
8.4-15
(b) σ1 = 75.3 MPa, σ2 = –6.5 MPa, τmax = 40.9 MPa
8.4-16
(b) σ1 = 3.52 MPa, σ2 = –66.9 MPa, τmax = 35.2
MPa
8.4-17
σx = 4.03 MPa, σy = 0, τxy = –3.19 MPa,
σ1 = 5.78 MPa, σ2 = –1.754 MPa, τmax = 3.77 MPa
(b) σ1 = 37 MPa, σ2 = –0.094 MPa, τmax = 18.57
MPa
8.5-16
τΑ = 76.0 MPa, τB = 19.94 MPa, τC = 23.7 MPa
8.5-17
σ1 = 21.3 MPa, σ2 = 0, τmax = 10.6 MPa
8.5-18
σ1 = 0, σ2 = –21.7 MPa, τmax = 10.87 MPa
8.4-18
σx = –215 kPa, σy = 0, τxy = –398 kPa,
σ1 = 305 kPa, σ2 = –519 kPa, τmax = 412 kPa
8.5-19
σt,max = 6.96 MPa, σc,max = –46.2 MPa, τmax = 24
MPa
8.4-19
(b) σ1 = 0.961 MPa, σ2 = –11.62 MPa, τmax = 6.29
MPa
8.5-20
dmin = 48.4 mm
8.4-20
σ1 = 17.86 MPa, σ2 = –0.145 MPa, τmax = 9.0 MPa
8.5-21
σt(A) = 298 MPa, σc(A) = –15.45 MPa,
τmax(A) = 156.9 MPa
8.4-21
σ1/σ2 = –58
8.5-22
(a) σx = 0, σy = 2.23 MPa, τxy = 0
8.4-22
σ1/σ2 = –663
8.5-1
(a) σ1 = 27.1 MPa, σ2 = –20.6 MPa, τmax = 23.8
MPa
(b) τmax = 1.116 MPa
8.5-23
(b) σ1 = 129.1 MPa, σ2 = –3.18 MPa, τmax = 66.1
MPa
8.5-2
(a) σ1 = 51.9 MPa, σ2 = –0.438 MPa, τmax = 26.2
MPa
(b) σt = 14.04 (qR2/d3), σc = –2.41 (qR2/d3),
τmax = 8.22 (qR2/d3)
8.5-24
σ1 = 10.9 MPa, σ2 = –16.41 kPa, τmax = 5.5 MPa
8.5-4
σ1 = 42.8 MPa, σ2 = –42.8 MPa, τmax = 42.8 MPa
8.5-5
tmin = 3 mm
8.5-6
pmax = 9.60 MPa
8.5-7
(a) σmax = σ1 = 57.2 MPa, τmax = 19.72 MPa
8.5-8
8.5-25
σt = 21.6 MPa, σc = –9.4 MPa, τmax = 15.5 MPa
8.5-26
καθαρή διάτμηση: τmax = 0.804 MPa
8.5-27
(a) dmin = 47.4 mm
(b) Pmax = 55.9 MPa
(b) Tmax = 215 kNm
(a) σ1 = 29.3 MPa, σ2 = –175.9 MPa, τmax = 102.6
MPa
(c) tmin = 14.11 mm
(b) σ1 = 156.1 MPa, σ2 = –33 MPa, τmax = 94.5 MPa
(a) Pmax = 461 kN
8.5-28
8.5-29
σt = 74.2 MPa, όχι θλιπτικές τάσεις, τmax = 37.1
MPa
8.5-10
φmax = 0.552, rad = 31.6º
8.5-11
σt = 32.0 MPa, σc = –73.7 MPa, τmax = 52.8 MPa
8.5-12
σt = 16.93 MPa, σc = –41.4 MPa, τmax = 28.9 MPa
8.5-13
P = 815 kN
8.5-14
(a) σmax = σ1 = 35.8 MPa, τmax = 18.05 MPa
(b) Pmax = 6.73 kN
8.5-15
(a) σx = 0, σy = 37 MPa, τxy = 1.863 MPa
(a) σ1 = 0, σ2 = –148.1 MPa, τmax = 74.0 MPa
(b) σ1 = 7.02 MPa, σ2 = –155.1 MPa, τmax = 81.8
MPa
(b) pmax = 8.32 MPa
8.5-9
(a) σx = 0, σy = 2.34 MPa, τxy = –0.1912 MPa
(b) σ1 = 2.35 MPa, σ2 = –0.0155 MPa, τmax = 1.185
MPa
(b) σ1 = 9.9 MPa, σ2 = –4.1 MPa, τmax = 7.0 MPa
8.5-3
(a) σt = 29.15(qR2/d3), σc = –8.78(qR2/d3),
τmax = 18.97(qR2/d3)
8.5-30
μέγιστα: σt = 18.35 MPa, σC = –18.35 MPa,
τmax = 9.42 MPa
8.5-31
στο ανώτερο μέρος της δοκού: σ1 = 64.4 MPa,
σ2 = 0, τmax = 32.2 MPa
8.5-32
(a) dA1 = 26.3 mm
(b) dT1 = 21.4 mm
8.5-33
σx = 12.14 MPa, σy = 0, τxy = –3.49 MPa, σ1 = 13.07
MPa, σ2 = –0.932 MPa, τmax = 7 MPa
8.5-34
(a) σ1 = 0, σ2 = σx = –108.4 MPa, τmax = σx/2
= –54.2 MPa
(b) σ1 = 0.703 MPa, σ2 = –1.153 MPa, τmax = 0.928
MPa
(c) Pmax = 348 N
8.5-35
σx = –136.4 MPa, σy = 0, τxy = 32.7 MPa, σ1 = 7.42
MPa, σ2 = –143.8 MPa, τmax = 75.6 MPa
A-8.5
C
A-8.6
A
A-8.7
D
A-8.8
D
A-8.9
A
A-8.10
D
A-8.1
A
A-8.11
D
A-8.2
C
A-8.12
A
A-8.3
D
A-8.13
C
A-8.4
B
A-8.14
B
A-8.15
C
9.3-9
Έστω β = a/L, ώστε
9.2-1
Η ροπή M0 ασκείται στο σημείο x = L
9.2-2
Ομοιόμορφο φορτίο
9.2-3
q = (q0/L)x, τριγωνικό φορτίο με φορά προς τα
κάτω
9.2-4
(a) q = q0sin(πx/L), ημιτονικό φορτίο
Το βέλος κάμψεως στο μέσο είναι σχεδόν ίσο με
το μέγιστο βέλος (η μέγιστη διαφορά είναι μόλις
2.6%).
9.3-10
𝛿𝐵
𝑀
9.3-11
(a) δB = 25.5 mm, δC = 51.4 mm
(b) RA = RB = q0L/π
(c) Mmax = q0L2/π2
9.2-5
 x
q = q0 1 −  , τριγωνικό φορτίο με φορά προς τα
 L
(a) q = q0
1
= 3.2 × 104 [𝑁],
𝜃𝐵
𝑀
1
= 200 [𝑁∙𝜇𝑚]
(b) δB = 25.5 mm, δC = 52.2 mm
9.3-15
(a) q0 = 1186 kN/m
(b) θΒ = 1.563  10-3 rad
κάτω
9.2-6
C
= 3 3 ( −1 + 8 − 4  2 ) .
 max
L2 − x 2
, παραβολικό φορτίο με φορά
L2
προς τα κάτω
9.3-16
IPN 380
9.3-17
 = − mx 2
9.3-18
 ( x) = −
(b) RA = 2q0L/3, MA = –q0L2/A
9.3-1
δmax = 1.04 mm, θ = 9.4  10 rad = 0.05º
9.3-2
(a) q = 1438 kN/m
-4
B =
(b) r = 14.72 mm
9.3-3
δmax = 6.5 mm, θ = 0.28º
9.3-4
h = 96 mm
9.3-5
L = 3.0 m
9.3-6
δmax = 15.4 mm
9.3-7
δ/L = 1/320
9.3-8
Eg = 80.0 GPa
mL2
mL2
3L − x
, B =
, B =
6 EI
3EI
2 EI
q
2 x 4 − 12 x 2 L2 + 11L4 ) ,
(
48 EI
qL4
48 EI
9.3-19
βλ. Πίνακα Η-2, περίπτωση 9
9.3-20
βλ. Πίνακα Η-1, περίπτωση 9
9.3-21
 ( x) =
q0 L 3
x − 2 Lx 2 ) για 0 ≤ x ≤ L/2,
(
24 EI
2 3
3 2

−q0  −160 L x + 160 L x
 ( x) =


960 LEI  +80 Lx 4 − 16 x5 − 25L4 x + 3L5 
7 q0 L4
1 q0 L4
για L/2 ≤ x ≤ L,  B =
, C =
64 EI
160 EI
9.3-22
 ( x) =
2 2
3
q0 x  200 x L − 240 Lx 

 για 0 ≤ x
5760 LEI  +96 x 4 − 53L4

9.4-10
3
2
−q0 L  40 x − 120 Lx 
≤ L/2,  ( x) =

 για L/2
5760 EI  +83L2 x − 3L3 
19q0 L4
13𝑞0 𝐿3
L/2 ≤ x ≤ L,  A =
, 𝜃𝐵 = − 192𝐸𝐼
,
480 EI
7 q L4
C = 0
240 EI
3q0 L4
≤ x ≤ L,  C =
1280 EI
9.3-23
PL
−4104 x 2 + 3565 L2 ) για 0 ≤ x
(
10368 EI
2
3
P  −648 Lx + 192 x 
≤ L/3,  ( x) = −

 για
1152 EI  +64 L2 x + 389 L3 
9.5-1
B = 86.25 mm, h = 172.5 mm
9.5-2
δmax = 0.78 mm
L/3 ≤ x ≤ L/2,
9.5-3
P = 16.88 N
9.5-4
δC = 5.9 mm
9.5-5
δB = 0.03 mm, θΒ = 4.9  10-5 rad
9.5-6
δB = 0.764 mm, θΒ = 4.02  10-4 rad
9.5-7
7 PL2
5PL3
B =
, B =
9 EI
9 EI
9.5-8
11PL3
(a) 1 =
144 EI
 ( x) = −
 ( x) = −
P
144 LEI
x ≤ L,  A =
 −72 L x + 12 Lx 

 για L/2 ≤
4
3
4
 +6 x + 5L x + 49 L 
2
2
3
3
3
3565PL
3109 PL
, C =
10368EI
10368EI
( L − x ) ,  = 2M 0 L
max
2
9.4-3
 = −M 0 x
2
27 EI
2 LEI
(προς τα
κάτω)
9.4-4
q
 ( x) = −
(2 x 4 − 12 x 2 L2 + 11L4 ) ,
48 EI
B = −
(c) δ1/δ2 = 88/75 = 1.173
βλ. Πίνακα Η-1, περίπτωση 10
9.4-6
 = −q0 x 2
9.4-9
9.5-9
45L4 − 40 L3 x + 15L2 x 2 − x 4
,
360 L2 EI
19q0 L4
q0 L3
B =

=
, B
360 EI
15EI
(a) a/L = 2/3
(b) a/L = 1/2
9.5-10
(a) δC = 6.25 mm (προς τα πάνω)
(b) δC = 18.36 mm (προς τα κάτω)
9.5-11
Px 2 ( L − x )
y=
3LEI

q0  x − 5Lx
2q0 L4

=
,


max
120 LEI  +20 L3 x 2 − 16 L5 
15 EI
9.5-12
B =
9.5-13
δC = 3.76 mm, δΒ = 12.12 mm
qL2 2 2
( x − L ) για 0 ≤ x ≤ L,
16 EI
3
2 2

q  −20 L x + 27 L x
 ( x) = −

 για L ≤ x
3
4
4
48EI  −12 Lx + 2 x + 3L 
9qL4
7 qL3
≤ 3L/2,  C =
, C =
48 EI
128EI
9.5-14
(a) M = PL/2
𝜈 = −𝑞0 𝑥
3𝐿5 −5𝐿3 𝑥 2 +3𝐿𝑥 4 −𝑥 5
,
90𝐿2 𝐸𝐼
5
9.4-8
25𝑃𝐿3
(b) 𝛿2 = 384𝐸𝐼
qL3
3EI
9.4-5
9.4-7
q0 L2
( −20 x2 + 19L2 ) για 0 ≤ x ≤ L/2,
480 EI
4
5
2 3
q0  80 Lx − 16 x − 120 L x 
 ( x) = −

 για
960 LEI  +40 L3 x 2 − 25L4 x + 41L5 
 ( x) = −
 ( x) =
 ( x) = −
 max =
61q0 L4
5760 EI
4
9.5-15
2
7q0 L3
23𝑞𝐿4
, 𝛿𝐵 =
648𝐸𝐼
162 EI
(b) M = 5PL/24,  B =
PL2
12 EI
(c) M = PL/8,  B = −
PL3
24 EI
𝑃𝐿2
θΑ = 9𝐸𝐼 ωρολογιακή, θΑ = θΒ ανθωρολογιακή
23𝑃𝐿3
𝐿
𝛿𝑚𝑎𝑥 = 648𝐸𝐼 για x = 2
9.5-16
9.5-17
9.5-18
θΑ = θΒ =
M =
𝑃𝐿2
81𝐸𝐼
ωρολογιακή, 𝛿𝑚𝑎𝑥 =
(a)  A = PL2
10 L − 9a
(προς τα πάνω)
324 EI
(β) προς τα πάνω όταν a/L < 10/9 και προς τα κάτω
όταν a/L > 10/9
9.5-19
(a)  C = PH 2
(b)  max =
9.5-21
13q0 L4
q L3
B = 0 ,  B =
10 EI
180 EI
q ( L3 − 6 La 2 + 4a 3 )
 max =
9.5-23
h =
Pc 2 ( c + 3b )
Pcb 2
,  =
3EI
2 EI
9.5-34
=
PL2 ( 2 L + 3a )
3EI
9.5-35
(a) HB = 0, VB = M/L, VC = –VB
(b) 𝜃𝐴 =
(c)
B =
ML
ML
, C = −
, θD = θC
3EI
6 EI
7 ML2
(προς τα αριστερά),
24 EI
1 ML2
D =
(προς τα δεξιά)
12 EI
14
L = 1.871L
2
(d) LCD =
9.5-36
24 EI
q ( 5L4 − 24 L2 a 2 + 16a 4 )
4 ML
5 ML
, θΒ = θΑ, C =
, θD = θC
81 EI
81 EI
L 2 L2 M
=
(προς τα δεξιά),
2 81EI
5 ML2
L
 D = C =
(προς τα αριστερά)
2 162 EI
(c)  A = − B
384 EI
P
3L + a
= 8a
Q
9 L2
P 9a
P a ( 4L + a )
=
για δΒ = 0 και
για
=
qa 8 L
qa
3L2
(a) HB = 0, VB = P/3, VC = 2P/3
(b)  A = −
,
(a) P/Q = 9a/4L
(b)
5𝑀𝐿
,
6𝐸𝐼
(c)  A =
PHL2
9 3EI
δC = 3.5 mm
A =
9.5-33
L+H
3EI
9.5-20
9.5-22
q=
2√6𝑃𝐿3
2187𝐸𝐼
19
q0 L2
180
16cEI
7 L4
9.5-32
(d) LCD =
9.5-37
δD = 0
2 5L
= 0.894 L
5
(a) b/L = 0.403
(b) δC = 0.00287qL4/EI
19WL3
31104 EI
9.5-38
α = 22.5º ή α = 112.5º ή α = –67.5º ή α = –157.5º
9.5-39
𝛿𝐷 =
9.5-40
δΒ = 113.6 mm, θΒ = 0.238 rad
9.5-41
𝑀𝐴 = 81 𝑞𝐿2 , 𝜃𝐴 = 81𝐸𝐼 ανθωρολογιακή
9.5-42
𝑀𝐴 = 81 𝑞𝐿2 , 𝜃𝐴 = 27𝐸𝐼 ανθωρολογιακή
9.5-28
47 Pb3
E =
12 EI
9.5-43
d = 0.375 L, 𝜃𝐴 = 0.0635
9.5-29
δC = 5.07 mm
9.5-44
(a) 𝛿𝐶 =
9.5-30
δC = 2.05 mm προς τα κάτω
9.5-31
δΒ = 13.84 mm, θA = 0.538º
9.5-24
=
9.5-25
k = 640 N/m
9.5-26
M1 = 7800 Nm, M2 = 4200 Nm
9.5-27
=
6Pb
EI
3
2𝑃𝐿3
9𝐸𝐼
+
𝑤𝐿4
,
8𝐸𝐼
5𝑃𝐿2
14
2𝑞𝐿3
16
𝑞𝐿3
121𝑞2 𝐿4
1920𝐸𝐼
−
13𝑞1 𝐿4
960𝐸𝐼
𝑞𝐿3
𝐸𝐼
26
19𝑞0 𝐿4
384𝐸𝐼
ανθωρολογιακή
προς τα πάνω
(b) q2 = 121 q1
(c) 𝛿𝐶 =
𝑤𝐿3
6𝐸𝐼
𝜃𝐷 = 18𝐸𝐼 +
προς τα πάνω
9.6-4
23qL4
7qL3
B =
, B =
162 EI
648 EI
9.6-5
δΒ = 10.85 mm, δC = 3.36 mm
9.6-6
δΒ = 11.8 mm, δC = 4.10 mm
9.6-8
P = 64 kN
9.6-9
M L
M L
M L
 A = 0 , B = 0 ,  = 0
6 EI
3EI
16 EI
9.7-4
 =−
9.7-5
Pa ( L − 2a)
Pa ( L − a )( L − 2a )
, 1 =
, δ2=
A =
6 LEI
6 LEI
0
2
2
9.7-6
9.6-11
9.7-1
(a)  B = PL3
9.7-2
(b) r =
9.7-3
1 + 7 ( I1 I 2 )
24 EI1
 =−
2 Px (19 L2 − 27 x 2 )
για 0 ≤ x ≤ L/3,
729 EI
3
P (13L − 175L2 x + 243Lx 2 − 81x3 )
9.7-8
=
PL3
24 EI A
1 + 15 ( I1 I 2 )
128EI1
1 + 15 ( I1 I 2 )
16
 4 L ( 2 L + 3x ) 2 x 
− ,
7 −
2
L 
( L + x)

PL3
A =
24 EI A
8PL3  L
2x 1
 2L + x 
−
− + ln 
 ,

EI A  2 L + x 9 L 9
 3L  
8PL3  3 7 
A =
 ln − 
EI A  2 18 
=
19, 683PL3  81L
3361 
 81 40 x  6440 x
+ 2 ln 
+
+
−

,
2000 EI A  81L + 40 x
 121 121L  14, 641L 14, 641 
19, 683PL3
7,320,500 EI A
𝜈(𝑥) = −
9.7-12
qL3
(a)  ( x) = −
16 EI A
(b)  A =

 11  
 −2820 + 14, 641ln  9  
 

19,683𝑃𝐿3
81𝐿
[
2000𝐸𝐼𝐴 81𝐿+40𝑥
9.7-11
40𝑥
6440𝑥
+ 2 𝑙𝑛 (1 + 81𝐿 ) − 14,641𝐿 − 1],  A =

qL4
8Lx 2 
και

(
x
)
=
−
1
−
 ( L + x )3 
2 EI A


qL4 ( 3 − 4 ln 2 )
qL3
, C =
8 EI A
16 EI A
για L/3

3x 1
L
 L + x 
−
+ + ln 

 ,
2
+
8
8
2
L
x
L
L
(
)



3
PL
A =
(8ln 2 − 5)
8EI A
PL3
EI A
9.7-9
A =
για
8PL2
PL3
8𝑃𝐿3
, 𝛿𝐵 = 729𝐸𝐼,  max = 0.01363
EI
243EI
=
(a) δΒ = –2.14 mm (προς τα πάνω)
 ( x) =
για 0 ≤ x ≤ L/4,
q (13L4 + 256 L3 x − 512 Lx3 + 256 x 4 )
9.7-7
(b) δC = 18.08 mm (προς τα κάτω)
9.7-10
768𝐸𝐼
1458EI
38PL2
34 PL2
≤ x ≤ L,  A =
, C =
,
729 EI
729 EI
32 PL3
B =
2187 EI
1 + 7 ( I1 I 2 )
8
(a)  B = qL4
A =
=
2
M L
M L
 A = 0 , θΒ = 0,  = 0 (προς τα κάτω)
6 EI
27 EI
(b) r =
𝑞𝑥(21𝐿3 −64𝐿𝑥 2 +32𝑥 3 )
12, 288EI
7 qL3
31𝑞𝐿4
L/4 ≤ x ≤ L/2,  A =
, 𝛿 4096𝐸𝐼
𝑚𝑎𝑥
256 EI
2
9.6-10
𝜈=−
19,683PL3 
 11  
−2820 + 14,641ln   

7,320,500 EI A 
 9 
 ( 9 L2 + 14 Lx + x 2 ) x
x 


−
ln
1
+

  για 0 ≤ x ≤ L
2
L 

8L ( L + x )


2
4bhL max
45E
9.8-1
U=
9.8-2
(a) και
(b) U =
(c)  =
9.8-3
9.8-4
P 2 L3
96 EI
A =
M A ( L + 3a )
𝑀 𝑎(2𝐿+3𝑎)
, 𝛿𝐴 = 𝐴
6𝐸𝐼
3EI
9.9-11
C =
Pa 2 ( L + a) P( L + a) 2
+
3EI
kL2
9.9-12
37qL4
D =
(προς τα επάνω)
6144 EI
9.10-1
 max =  st 1 + 1 +
3
PL
48EI
(a) και
9.10-2
 max =
(a) U =
32EI  2
L3
9.10-3
δmax = 4.33 mm, σmax = 102.3 MPa
9.10-4
d = 281 mm
9.10-5
HE 320A
9.10-6
h = 360 mm
9.10-7
R=
9.11-1
 =−
 EI 
4
(a) U =
2
4 L3
Pa
2
(L + a)
6 EI
Pa 2 ( L + a )
(b)  C =
3EI
(c) U = 56.2J, δC = 5.0mm
L (17 L4 q 2 + 280qL2 M 0 + 2560M 02 )
9.8-6
U=
9.8-7
2 PL3 8 2 PL
B =
+
EA
3EI
9.9-2
Pa 2b 2
D =
3LEI
9.9-6
9.9-7
9.9-8
9.9-9

2h 

 st 
q 2 L3
15 EI
2
9.9-3

(b) U =
(b) U =
9.8-5
9.9-10
15,360 EI
A =
9.11-2
31qL4
C =
4096 EI
 x(T2 − T1 )( L − x)
(θετική προς τα επάνω),
2h
 L(T2 − T1 )
(ωρολογιακή),
A =
2h
 L2 (T2 − T1 )
(προς τα κάτω)
 max =
 =−
B =
B =
9.11-3
C =
9.11-4
 x 2 (T2 − T1 )
2h
 L(T2 − T1 )
(προς τα επάνω),
(ανθωρολογιακή),
h
 L2 (T2 − T1 )
2h
 ( x) =
C =
7 qL3
48 EI
Pb 2 (b + 3h)
𝑃𝑏(𝑏+2ℎ)
C =
, 𝜃𝐶 = 2𝐸𝐼
3EI
3EI m I  2
L3
8h
Pa 2 ( L + a )
C =
3EI
L3 (2 P1 + 5P2 )
L3 (5 P1 + 16 P2 )
C =
, B =
48EI
48 EI
18WEh
AL
(προς τα πάνω)
 ( x 2 − L2 )(T2 − T1 )
2h
 ( L + a )(T2 − T1 )
h
(ανθωρολογιακή),
 (T2 − T1 )(2 La + a 2 )
(a)  max =
(b)  max =
2h
T0 L3
,
(προς τα πάνω)
(προς τα κάτω)
9 3h
(
) (προς τα κάτω)
T0 L4 2 2 − 1
48h
9.11-5
(a)  max =
(b)  max =
 T0 L3
A-9.2
C
A-9.3
B
A-9.4
C
A-9.5
B
(προς τα κάτω)
A-9.6
C
(προς τα κάτω)
A-9.7
D
10.3-5
RA = RB =
(προς τα κάτω)
6h
 T0 L
(c) 𝛿𝑚𝑎𝑥 =
(d) 𝛿𝑚𝑎𝑥 =
4
(προς τα κάτω)
12h
𝑎𝑇0 𝐿3
6ℎ
𝐿4
𝑎𝑇0
12ℎ
A-9.1
C
10.3-1
(a) RA = 1163 N, RB = 698 N, MA = 814 N·m
(b) xzero = 0.875 m
 =−
(c) δmax = 0.022 mm, θB = 2.36  10 rad
-3
10.3-2
(a) h = 257 mm
(b) δ0 = 1 mm
10.3-3
RA = − RB =
 =−
3M 0
M
, MA = 0 ,
2L
2
3EI  B
3EI  B
, MA =
,
3
L
L2
 B x 2 (3L − x)
2 L3
k R qL5
qL3
qL4
+
, B = −
6 ( k R L − EI )
8 12 ( k R L − EI )
10.3-6
B =
10.3-7
RA = V (0) =
M 0 x 2 ( L − x)
4 LEI
MA =
9q0 L
11q0 L
, RB = −V ( L) =
,
40
40
7q0 L2
120
2
10.3-4
RA = RB =
qL
qL
, MA = MB =
,
2
12
qx 2 ( L − x) 2
 =−
24 EI
10.3-8
(
q0 − x 6 + 7 L3 x3 − 6q0 L4 x 2
q0 L2
7 q0 L
13q0 L
(a) RA = V (0) =
, RB = −V ( L ) =
,MA =
, =
60
60
30
360 L2 EI
2
(b) RA = V (0) = 0.31q0 L = 

−6
)
 2 − 4 + 8 
 q0 L
4

2
  2 − 4 + 8 
2  − 12 + 24
M
2
q
L
=
−
,
,
RB = −V ( L) = 0.327q0 L =  6
q
L
A
0
 0
4
4


4
3
2
2
1 
 2 − 4 + 8 x3
 2L 
x
 2L  
2  − 12 + 24 x
=
+ 2q0 L
+ q0   x 
 −q0   sin 
 − 6q0 L
EI 
4
6
4
2
 
 2L 
   
10.3-9
(
q0 x 2 33L4 − 61L3 x + 30 L2 x 2 − 2 x 4
11q0 L2
61q0 L
19q0 L
(a) RA =
, RB =
,MA =
,  ( x) = −
120
120
120
720 L2 EI
(b) RA =
48q0 L

4
L

, RB = q ( x)dx − RA =
0
2 Lq0

−
48Lq0
4
)
L
M A =  q ( x)dx − RB L =
2 L2 q0 ( − 2)

0
2
 2 Lq0 48Lq0 
− L
−
,
 4 
 
x 
16 L4 q0 − 24 L2 q0 x 2 + 8Lq0 x3 − 16 L4 q0 cos 

 2L 
 ( x) =
 4 EI
 12 1  2
 12 1 
− 2  q0 L (ανθωρολογιακή), M B =  4 − 2  q0 L2
4
 
 




1 
x 
3
2 2
4
(ανθωρολογιακή),  ( x) = 4  −q0 L4 cos 
 + 4q0 Lx − 6q0 L x + q0 L 
 EI 
 L 

10.3-10 (a) RA = V (0) =
24q0 L
4
, RB = −V ( L ) = −
24q0 L
4
,MA = 
x 
L2 sin 
+  x 2 −  Lx

2q L
qL
 L 
(b) RA = RB = 0 , M A = M B = 03 ,  = −q0 L2


 4 EI
2
10.3-11 (a) RA = V (0) =
MB = −
48(4 −  )
32( − 3)

4
4
 2 48(4 −  ) 
 2 L  16(6 −  )
q0 L , M A = −q0 
q0 L2 ,
q0 L , RB = −V ( L) =  −
 +
4
4





 
2
1
𝜋𝑥
q0 L2 , 𝜈(𝑥) = 𝜋4 𝐸𝐼 [−16𝑞0 𝐿4 𝑐𝑜𝑠 ( 2𝐿 ) + 8(4 − 𝜋)𝑞0 𝐿𝑥 3 − 8(6 − 𝜋)𝑞0 𝐿2 𝑥 2 + 16𝑞0 𝐿4 ]
2
2
qL
qL
13
7
(b) RA = V (0) =
(ανθωρολογιακή), M B = − 0
q0 L , RB = −V ( L ) =
q0 L , M A = 0
30
30
15
20
q0
 x 6 − 15 L2 x 4 + 26 L3 x 3 − 12 L4 x 2 
(ανθωρολογιακή),  ( x) =
2
360 L EI
2
10.3-12 RA = V (0) =
qL
3
7
1
 − q0 x 5 + 3q0 Lx 3 − 2q0 L2 x 2 
q0 L , RB = −V ( L ) =
q0 L , M A = 0 ,  ( x) =
20
20
120 LEI 
30
10.3-13 RA = − RB =
3M 0
M
L − 2x
, M A = − M B = 0 ,  ( x) = − M 0 x 2
για 0 ≤ x ≤ L/2
2L
4
8 LEI
10.3-14 RB = −
1  9M 0 3 M 0 2 
9M 0
9M 0
𝑀
x −
x  για 0 ≤ x ≤ L/2,
, RA =
, 𝑀𝐴 = 80 ,  ( x) =

16 
EI  48 L
8L
8L
M 0 L2 
1  9M 0 3 9M 0 2 M 0 L
 ( x) = 
x −
x +
x−
 για L/2 ≤ x ≤ L
EI  48L
16
2
8 
10.3-15 q = 8.6 kN/m
10.3-16 q = 2.95 kN/m
10.4-1
(a) RA = RC = 261 N, RB = 870 N
(b) δAB = –0.06 mm
10.4-2
RA = 1.76 kN, MA = 2.4 kNm,
RB = 3.24 kN, MB = 3.6 kNm,
δC = 0.0674 mm
10.4-3
(a) RA = 2.02 kN, MA = 1.385 kNm,
RB = 0.68 kN, MB = 0.802 kNm
(b) δmax = 5.98 mm για x = 0.466L = 2.1 m
(c) RA = RB = 2.25 kN, MA = –MB = 1.688 kNm
10.4-4
RA = 375 N, MA = 225 Nm, RB = 225 N
10.4-5
RA = Pb
10.4-6
RA = qL , M A =
10.4-7
qL
1
17
RA = − qL , RB = qL , M A = −
8
8
8
10.4-8
(a) RA =
3L2 − b 2
L+b
3L − a
, RB = Pa 2
, M A = Pab
3
3
2 L2
2L
2L
qL2
qL2
, MB =
3
6
2
4M 0
M0
, HA = −
, RB = − RA , RC = − H A
3L
3L
(b)  A = −
M 0L
M L
,  B = 0 , C =  A
18 EI
9 EI
(c) LBC = 2L
10.4-9
(a) RA =
4M 0
2M 0
4M 0
2M 0
, HA =
, RB = −
, RC = −
3L
3L
3L
3L
(b)  A =
5M 0 L
M L
M 0L
,  B = − 0 , C =
18 EI
18 EI
36 EI
(c) LAB = 2.088L
10.4-10 (a) RA =
LM 0 k R
M0
M 0 kR
, RB = − RA , M B =
(ανθωρολογιακή)
+
6 EI + 2 Lk R
L 2 ( 3EI + LkR )
 LM 0 LM 0
 4 EI + 12 EI

LM 0
LM 0
 LM 0
(b)  A =
+
 A = 
4 EI 4 ( 3EI + Lk R )
 4 EI
 5LM 0
 18EI

10.4-11 (a) H A =
3M 0
, HB = 0, VB = 0, VC = 0, HD = –HA
2L
(b)  A = −
M 0L
M L
, θD = –θΑ,  B = 0 , θC = –θB
16 EI
8 EI
(c) HA = M0/L, HB = –2M0/L, VB = M0/L, VC = –M0/L,
M L
M L
HD = HA,  A = − 0 , θD = θΑ,  B = 0 , θC =
24 EI
12 EI
θB
10.4-12
t AB LAB
=
tCD LCD
2
10.4-13 RA =
7qL
7 qL
17 qL
, RB =
, MA =
12
12
12
kR → 0
kR → 
kR →
6 EI
L
7qL2
10.4-14 RA = 2qL, M B =
12
2
5q L
qL
10.4-15 RA = RB = 0 , M A = M B = 0
4
96
10.4-16 RA =
qL
33qL
13qL
, RB =
, RC =
8
16
16
11P
= 5.18 kN (προς τα κάτω),
17
RB = RA + P = 13.18 kN (προς τα επάνω),
5 PL
MA =
= 3.53 kN  m (ωρολογιακή)
34
10.4-17 RA =
10.4-18 RB = 6.44 kN
10.4-19 RA = 174 N, MA = 87 Nm, RB = 43.5 N
10.4-20 (a) RA = RB = 45 kN, MA = –MB = 35 kNm
(b) h = 167.1 mm
10.4-21 RA = –159.6 N, RB = –679 N,
(
48EI 6 + 5 2
10.4-34 k =
3
L3
7L
10.4-35 (a) VA = VC = 3P/32, HA = P, MA = 13PL/32
MA = –347 Nm, MB = –669 Nm
(b) Mmax = 13PL/32
10.4-22 RA = 33 kN, MA = 25.3 kNm, RB = 17 kN
10.4-23 (a) RD = 2.12 kN
10.4-36 𝐻𝐴 = −
29𝑃
,
64
𝐻𝐶 = −
2
7qL
31qL
17 qL
, RB =
, MA =
48
48
48
10.4-25 (a) RA = −
10.4-38 (a) M A = M B = qb
(b) b/L = 1, M A =
23P
20 P
3PL
, RD = RE =
, MA =
17
17
17
(b) M max =
10.4-39 (a)
10.4-27 Για την κάλυψη του διακένου απαιτείται q = 48
kN/m, που δίνει:
kN

−24kN  m q = 48 m

kN
 q
M B (q ) = −
q  48
m
 2
kN
 q
q  48
− 8 − 18
m

19
3L2 − b 2
24 L
qL2
12
=
19
𝑞 𝐿,
128 0
𝑀𝑚𝑎𝑥 = 0.0147𝑞0 𝐿2
10.4-29 RA = 109.3 N, MA = 80.3 Nm, RB = 21.2 N
6 M 0 ab
3a − L
, M A = M 0b
,
3
L2
L
3b − L
M B = −M 0a
L2
10.4-30 RA = − RB =
121qL
=
= 6.05 kN  m ,
2048
5qL2
=
= 8.0 kN  m
64
10.4-33 F = 14.47 kN, MAB = 26.0 kNm, MDE = 9.66 kNm
3− 2 2
= 0.08579qL2
2
0.01307qL4
σημείων Α και Β κατά ποσότητα
EI
19q0 L2
, σmax = 13.4 MPa,
256
19q0 L4
=
= 0.00891 mm
7680 EI
10.4-40 M max =
 max
10.4-41 MA =
2
𝑞𝐿2
27
10.4-42 MA =
4
𝑞𝐿2
81
4
10.4-43 (a) Vmax = 3 𝑃 , Mmax = −
(b) Vmax =
−17
𝑃,
27
𝑃𝐿
3
Mmax = −
10𝑃𝐿
81
243ES EW IAH T
4 AL3 ES + 243IHEW
10.5-1
S=
10.5-2
(a) RA = −
10.4-31 σ = 3.14 MPa
2
19qL2
=
648
(c) το σημείο C βρίσκεται χαμηλότερα των
19
3
𝑞 𝐿2 , 𝑉𝑚𝑎𝑥
128 0
M CD ,max
29𝑃𝐿
128
d2 4
= 48 = 1.682
d1
2
(b) M max, pos = qL
10.4-28 𝑅𝐴 = 128 𝑞0 𝐿, 𝑅𝐶 = − 128 𝑞0 𝐿,
10.4-32 M AB ,max
𝑀𝑚𝑎𝑥 =
(c) για a = b = L/3, είναι M max, pos
PL
2
2qL
11qL
10.4-26 RA = RD =
, RB = RC =
5
10
𝑀𝐴 =
35𝑃
,
64
10.4-37 RA = 12.53 kN, RB = 12.45 kN, RC = 0.026 kN
(b) RA = 4.78 kN, MA = 3.74 kNm
10.4-24 RA =
) = 89.63 EI
 (T2 − T1 ) L2
3EIk
,
2h
3EI + L3k
 (T2 − T1 ) L2 3EIk
RB =
2h
3EI + L3k
M A = RB L =
 (T2 − T1 ) L3
2h
3EIk
3EI + L3k
3EI  (T2 − T1 )
(προς τα
2hL
3EI  (T2 − T1 )
επάνω), RB =
(προς τα κάτω),
2hL
3EI (T2 − T1 )
(ανθωρολογιακή)
M A = RB L =
2h
(b) RA = − RB = −
10.5-3
 (T2 − T1 ) L2
3EIk
(προς τα
2h
3EI + L3k
 (T2 − T1 ) L2 3EIk
επάνω), RB =
(προς τα
2h
3EI + L3k
 (T2 − T1 ) L3 3EIk
κάτω), M A = RB L =
2h
3EI + L3k
RA = − RB = −
(ανθωρολογιακή)
10.5-4
10.5-5
(προς τα πάνω)
10.6-1
10.6-2
 (T1 − T2 ) L2
(προς τα πάνω),
RC = −
3RB
4
=
 (T1 − T2 ) L2
2h
9 EIk
(προς τα
36 EI + L3k
πάνω)
6 EI  (T1 − T2 )
(προς τα κάτω),
Lh
3EI  (T1 − T2 )
(προς τα πάνω),
RA =
2 Lh
9 EI (T1 − T2 )
(προς τα πάνω)
RC =
2 Lh
(b) RB = −
11.2-1
Pcr = βR/L
11.2-2
(a) Pcr =
(b) Pcr =
 2 EA 2
4 L2
, t =
 2 E 2
4 L2
2
qhL
17 2 7
q L , 320E2I2,  b =
40
16 I
17 2 6
q L , 320EI2
40
(c) λ = 0.01112 mm, σb = 117.2 MPa, σt = 0.741
MPa
A-10.1
B
A-10.2
B
A-10.3
B
A-10.4
D
A-10.5
B
A-10.6
D
A-10.7
A
 L2 + 20 R
(b) Pcr =
4L
11.2-9
 a 2 + 2 R
L
(a)  =
(b)  t =
 a2 + R
L
(a) H =
(b) 4.32 MPa, 1.08 MPa και 0.48 MPa
6 EIk
(προς τα
36 EI + L3k
h
RB  (T1 − T2 ) L2
3EIk
=
κάτω), RA = −
4
2h
36 EI + L3k
(a) RB = −
 (T1 − T2 ) L2
6 EIk
(προς τα κάτω),
36 EI + L3k
h
RB  (T1 − T2 ) L2
3EIk
RA = −
=
(προς τα
4
2h
36 EI + L3k
3RB  (T1 − T2 ) L2
9 EIk
=
πάνω), RC = −
4
2h
36 EI + L3k
RB = −
Pcr = 3βR/L
11.2-10 Pcr =
3
L
5
11.2-11 Pcr =
7
L
4
11.2-3
Pcr = 9.44 kN
11.2-4
Pcr = 10 kN
11.2-5
Pcr = 6βR/L
11.2-12 𝑃𝑐𝑟 = 8 𝛽𝐿
11.2-6
Pcr = 1.688 kN
11.2-13 𝑃𝑐𝑟 =
11.2-7
Pcr = 4.58 kN
11.2-14 Pcr1 = 30.2 kN, Pcr2 = 84.6 kN
11.2-8
(a) Pcr =
9
( L − a ) (  a2 + R )
aL
9𝛽𝑅
8𝐿
11.2-15 Pcr1 = 243 kN, Pcr2 = 932 kN
9
4
11.2-16 Pcr1 = βL, Pcr2 = 𝛽𝐿
11.3-1
Pεπ = 542 kN
11.3-23 Wcr = 203 kN
11.3-2
HE 160B
11.3-24 θ = arctan0.5 = 26.57º
11.3-3
(a) Pcr = 465 kN
11.3-25 (a) qmax = 1.045 kN/m
11.3-4
(b) Pcr = 169.5 kN
(b) Ib,min = 2411 cm4
(a) Pcr = 176.7 kN
(c) s = 70 mm, 869 mm
(b) Pcr = 10.86 kN
11.3-26 Pcr = 3.56 kN
(a) Pcr = 319 kN
11.3-27 Pcr = 70.3 kN
(b) Pcr = 120.1 kN
11.4-1
Pcr = 756 kN
11.3-6
Mεπ = 1143 kNm
11.4-2
tmin = 4.98 mm
11.3-7
Pεπ = 220 kN
11.4-3
Pcr = 831 kN, 208 kN, 1700 kN, 3330 kN
11.3-8
Pcr Y = 339 kN
11.4-4
Pcr = 62.2 kN, 15.6 kN, 127 kN, 249 kN
11.3-9
(a) Pcr = 223 kN
11.4-5
Pεπ = 323 kN, 80.7 kN, 661 kN, 1292 kN
(b) d = 118.3 mm
11.4-6
Pεπ = 6.57 kN, 1.64 kN, 13.45 kN, 26.3 kN
11.3-10 (a) Pcr = 97.5 kN
11.4-7
Pcr = 295 kN
(b) d = 86.9 mm
11.4-8
Tεπ = 18.1 kN
11.4-9
(a) Qcr = 13.41 kN
11.3-5
11.3-11 Qεπ = 109.8 kN
11.3-12 (a) Qcr =
(b) Qcr =
 2 EI
L2
11.4-10 Pcr = 447 kN, 875 kN, 54.7 kN, 219 kN
2 EI
9 L2
11.4-11 Pcr =
(b) Qcr = 35.8 kN, a = 0 mm
2
2 2 EI
11.3-13 (a) Qcr =
L2
(b) M cr =
3d EI
L2
11.4-12 Lmax = 4.2 m
11.4-13 Pmax = 242 kN
2
 2I
11.3-14 T =
 AL2
11.4-14 (a) Pcr = 17.4 kN
(b) d = 72.4 mm
11.4-15 tmin = 7.8 mm
11.4-16 tmin = 10.0 mm
11.3-15 h/b = 2
11.3-16 (a) 𝑃𝑐𝑟 =
4 2 EI

 2 x  
,  = 1 − cos 

2
L
2
 L 
3𝜋3 𝐸𝑟 4
4𝐿2
11.4-17 (b) Pcr = 413.89EI/L2
11.5-1
δ = 5.98 mm, Mmax = 231 Nm
11.5-2
δ = 8.87 mm, Mmax = 2.03 kNm
11.3-17 P1 : P2 : P3 = 1.000 : 1.047 : 1.209
11.5-3
L = 4.8 m
11.3-18 Pεπ = 710 kN
11.5-4
b = 11.07 mm, h = 16.61 mm
11.3-19 Fεπ = 164 kN
11.5-5
για P = 0.3Pcr:
(b) 𝑃𝑐𝑟 =
11𝜋3 𝐸𝑟 4
4𝐿2
11.3-20 Wmax = 124 kN
11.3-21 tmin = 4.53 mm
11.3-22 Pcr = 426 kN
M
  1.721x  
 1.721x 
= 1.162 sin 
  + cos 

Pe
 L 
  L 
11.5-6

 5
P = 583.33 arccos 
 5+

2

  , όπου το Ρ

εκφράζεται σε kN και το δ σε mm. Όταν δ = 10
mm, τότε Ρ = 884 kN.
11.5-7

 5
P = 163.61 arccos 
 5+


  , όπου το Ρ

Pεπ = 57.9 kN
11.5-9
Lmax = 3.91 m
11.6-10 (a) σmax = 94.3 MPa
(b) Lmax = 4.05 m
11.6-11 (a) σmax = 68.6 MPa
(b) Pεπ = 268 kN
11.6-12 (a) σmax = 56.6 MPa
(b) n = 2.54
11.6-13 (a) σmax = 96.1 MPa
(b) n = 3.06
11.5-10 Lmax = 3.69 m
11.5-11  = e sec ( kL ) − 1 , Mmax = Pesec(kL)
11.6-14 (a) σmax = 98.9 MPa
(b) P2 = 459 kN
11.5-12 Lmax = 2.21 m
11.5-13 Lmax = 3.86 m
11.6-15 (a) σmax = 100.2 MPa
(b) n = 2.24
11.5-14 Tmax = 8.29 kN
11.5-15 (a) q0 = 3.33 kN/m
11.6-16 (a) σmax = 120.4 MPa
(b) P2 = 113.5 kN
(b) Mmax = 4.23 kNm, λόγος: 0.47
11.6-1
(a) σmax = 116.8 MPa
(b) Lmax = 1.367 m
11.6-2
Pεπ = 37.2 kN
11.6-3
bmin = 106.3 mm
11.6-4
(a) σmax = 38.8 MPa
(b) Lmax = 5.03 m
11.6-5
(a) σmax = 63.7 MPa
(b) Pεπ = 18.98 kN
11.6-6
d2 = 131 mm
11.6-7
L = 5.2 mm
11.6-8
σmax = 212 MPa
12.2-2
x=y=
12.2-3
y = 27.5 mm
5
a
12
2
12.2-4
2c = ab
12.2-5
y = 340 mm
12.2-6
y = 52.5 mm
(a) σmax = 66.5 MPa
(b) Pεπ = 779 kN
2
εκφράζεται σε kN και το δ σε mm. Όταν δ = 10
mm, τότε Ρ = 248 kN.
11.5-8
11.6-9
A-11.1
D
A-11.2
B
A-11.3
D
A-11.4
A
A-11.5
D
A-11.6
A
A-11.7
B
A-11.8
B
A-11.9
B
A-11.10 C
A-11.11 D
12.2-7
 x = 24.5 mm

 y = 49.5 mm
12.2-8
 x = 137 mm

 y = 132 mm
12.3-6
Ix = 518  103 mm4
12.3-7
Ix = 13.55  106 mm4, Iy = 4.08  106 mm4
12.3-8
Ix = Iy = 194.6  106 mm4, rx = ry = 80.1 mm
12.3-9
I1 = 61,390 cm4, I2 = 9455 cm4, r1 = 18.9 cm,
r2 = 7.42 cm
12.4-1
Ib = 72,113 cm
Ic = 11a /192
12.4-3
Ixc = 2.82  106 mm4
12.4-4
I2 = 405  103 mm4
12.4-5
Ixc = 204,493 cm4, Iyc = 14,863 cm4
12.4-6
Ixc = 106  106 mm4
12.4-7
Ixc = 656 cm4, Iyc = 237 cm4
12.4-8
b = 250 mm
12.5-2
12.5-3
12.5-4
12.5-5
12.6-2
12.6-3
12.7-6
( I P )C
r
=
( 9
− 8sin  )
18
bh ( b 2 + h 2 )
( I P )C =
I xy =
72 ( 4 −  )
r4
24
t 2 ( 2b 2 + t 2 )
I xy =
12.6-5
I12 = –540 cm4
12.6-6
Ixy = 24.3  106 mm4
12.6-7
I xc yc = −230 cm4
12.7-1
b4
I x 1 = I y1 = , I x 1 y1 = 0
12
12.7-2
I x1 =
θp1 = –29.9º, θp2 = 60.1º, I1 = 121.5  106 mm4,
I2 = 34.7  106 mm4
12.8-4
θp1 = –8.54º, θp2 = 81.46º, I1 = 17.24  106 mm4,
I2 = 4.88  106 mm4
12.8-5
θp1 = 37.7º, θp2 = 127.7º, I1 = 5.87  106 mm4,
I2 = 0.714  106 mm4
12.8-6
θp1 = 32.63º, θp2 = 122.63º, I1 = 8.76  106 mm4,
I2 = 1.00  106 mm4
12.8-7
θp1 = 16.85º, θp2 = 106.85º, I1 = 0.2390b4,
I2 = 0.0387b4
12.8-8
θp1 = 74.08º, θp2 = –15.92º, I1 = 8.29  106 mm4,
I2 = 1.00  106 mm4
12.8-9
θp1 = 75.3º, θp2 = –14.7º, I1 = 6.28  106 mm4,
I2 = 0.66  106 mm4
4
bh b + h )
b3 h3
, Iy = (
,
2
2
6 (b + h )
12 ( b 2 + h 2 )
4
1
b2 h2 ( h2 − b2 )
12 ( b 2 + h 2 )
Id = 14,696 cm4
4
a
 5
b
12.8-3
b = 2r
I x 1 y1 =
a
= 5
b
Διαπιστώνεται ότι σε κάθε σημείο υφίστανται δύο
διαφορετικά σετ κυρίων αξόνων.
24
r 4 (176 − 84 + 9 2 )
a 2 − b2
2
12.8-2
2
IP = 32,966cm4
IP =
(a) c =
(c) 1 
12.6-4
12.7-3
12.8-1
(b)
48
2
I x 1 = 8.75 106 mm 4 , I y1 = 1.02 106 mm4 ,
I x 1 y1 = −0.356 106 mm 4
bh ( b 2 + 12h 2 )
4
I x 1 = 480.2 cm4 , I y1 = 145.8 cm4 ,
I x 1 y1 = 181.2 cm 4
4
IP =
𝐼𝑥1 = 12.44 × 106 mm4 , I y1 = 9.68 106 mm 4 ,
I x 1 y1 = 6.03 106 mm 4
12.7-5
4
12.4-2
12.5-1
12.7-4