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Beam deflection formulae www.vaxasoftware.com
Try BEAM DEFLECTION CALCULATOR at vaxasoftware.com
Symbol Physical quantity Units
E·I Flexural rigidity N·m2, Pa·m4
y Deflection or deformation m
Slope, Slope of the deflection, Angle of rotation -
x Distance from support (origin) m L Length of beam (without overhang) m M Moment, Bending moment, Couple moment applied N·m P Concentrated load, Point load, Concentrated force N w Distributed load, Load per unit length N/m R Reaction load, reaction force N V Shear force, shear N
Simple beam - Uniformly distributed load
Deflection )2(24
3230AB xLxL
EI
xwy
EI
Lwy
384
5 40
MAX
at
2
Lx
Slope )46(24
3230AB xLxL
EI
w
EI
Lw
24
30
BA
Moment )(20
AB xLxw
M
8
20
MAX
LwM at
2
Lx
Shear )2(2
0AB xL
wV
Reactions 20
BA
LwRR
Simple beam - Uniform load partially distributed at left end (I)
Deflection )16249(384
3230AC xLxL
EI
xwy
)17248(384
32230CB LxLLxx
EI
Lwy
Slope )64729(384
3230AC xLxL
EI
w
)174824(384
220CB LLxx
EI
Lw
EI
wL
128
3 3
A
EI
wL
384
7 3
B
Moment )43(8
20AC xLx
wM )(
820
CB LxLw
M
Shear )83(8
0AC xL
wV
80
CB
LwV
AA RV BB RV
Reactions 8
3 0A
LwR
80
B
LwR
Simple beam - Uniform load partially distributed at left end (II)
Deflection:
)4244(24
322222340AC LxaLxxaLaLaa
LEI
xwy
)264(24
322222
0CB xLxxaxLLa
LEI
awy
Slope:
)412644(24
322222340AC LxaLxxaLaLaa
LEI
w
)6124(24
2222
0CB xLxaL
LEI
aw
Moment:
)2(2
220AC LxaLxxa
L
wM
)(
2
20
CB xLL
awM
Shear:
)22(2
20AC LxaLa
L
wV
L
awVVV
2
20
BCCB
Reactions )2(2
0A aL
L
awR
L
awR
2
20
B
Simple beam - Uniform load partially distributed
Deflection xEI
xRy
6
3A
AC xaxEI
w
EI
xRy 40
3A
CD )(246
L
xL
EI
xLRy
)(
6
)( 3B
DB
Slope: EI
xR
2
2A
AC 302
ACD )(
62ax
EI
w
EI
xR
LEI
xLR
2
)( 2B
DB
Moment xRM AAC 20ACD )(
2ax
wxRM
)(BDB xLRM Shear ACAAC RVVV )(0ACD axwRV
BBDDB RVVV
Reactions )2(2
0A bc
L
bwR )2(
20
B baL
bwR
Where:
LEI
baLRLcREILbw
6
)(336 2A
2B
30
EI
cRLcRbaRbwabw
24
812)(834 3B
2B
3A
40
30
Simple beam - Uniform load partially distributed at each end
Moment 2
21
AAC
xwxRM
)2(21
ACD axaw
xRM 2
)()(
22
BDB
xLwxLRM
Shear: xwRV 1AAC awRV 1ACD )(2BDB xLwRV Reactions:
L
cwaLawR
2
)2( 221
A
L
awcLcwR
2
)2( 212
B
Simple beam - Load increasing uniformly to right end
Deflection )3107(360
42240AB xxLL
LEI
xwy
EI
Lwy
40
MAX 00652,0 at x = 0,5193L
Slope )15307(360
42240AB xxLL
LEI
w
EI
Lw
360
7 30
A
EI
Lw
45
30
B
Moment )(6
320AB xxL
L
wM
Shear )3(6
220AB xL
L
wV
Reactions 60
A
LwR
6
2 0B
LwR
Simple beam - Load increasing uniformly to center
Deflection 2220
AC )45(960
xLLEI
xwy
2220CB ))(45(
960
)(xLL
LEI
xLwy
EI
Lwy
120
40
MAX
at
2
Lx
Slope )4)(45(192
22220AC xLxL
LEI
w
))(4)()(45(192
22220CB xLLxLL
LEI
w
EI
Lw
192
5 30
BA
Moment )43(12
320AC xxL
L
wM
))(43(12
)( 220CB xLL
L
xLwM
Shear )4(4
220AC xL
L
wV ))(4(
4220
CB xLLL
wV
Reactions 40
BA
LwRR
Simple beam - Sinusoidal distributed load
Deflection L
x
EI
Lwy
πsin
π4
40
AB
EI
Lwy
4
40
MAX π
at
2
Lx
Slope L
x
EI
Lw πcos
π3
30
AB
EI
Lw3
30
BA π
Moment L
xLwM
πsin
π2
20
AB
Shear L
xLwV
πcos
π0
AB π0
BA
LwVV
Reactions π0
BA
LwRR
Simple beam - Concentrated load at center
Deflection )43(48
22AC xL
EI
Pxy
))(43(48
)( 22CB xLL
EI
xLPy
EI
PLyy
48
3
CMAX
at
2
Lx
Slope:
)4(16
22AC xL
EI
P
)384(
1622
CB LLxxEI
P
EI
PL
16
2
BA
Moment 2AC
PxM
2
)(CB
xLPM
Shear 2AAC
PVV
2BCB
PVV
Reactions 2BA
PRR
Simple beam - Concentrated load at any point
Deflection )(6
222AC xbL
LEI
Pbxy
222CB )(
6
)(xLaL
LEI
xLPay
Slope:
)3(6
222AC xbL
LEI
Pb
222
CB )(36
xLaLLEI
Pa
LEI
bLPb
6
)( 22
A
)(
622
B aLLEI
Pa
Moment L
PbxM AC
L
xLPaM
)(CB
Shear L
PbVV AAC
L
PaVV
BCB
Reactions L
PbR A
L
PaR B
Simple beam - Two equal concentrated loads symmetrically placed
Deflection )33(6
22AC xaaL
EI
Pxy
)33(6
22CD axLx
EI
Pay
22DB )(33
6
)(xLaaL
EI
xLPy
)43(24
22MAX aL
EI
Pay
at
2
Lx
Slope )(2
22AC xaaL
EI
P
)2(
2CD xLEI
Pa
22DB )(
2xLaaL
EI
P
EI
aaLP
2
)( 2
BA
Moment PxM AC PaM CD )(DB xLPM
Shear PV AC 0CD V PV DB
Reactions PRR BA
Simple beam - Two equal concentrated loads unsymmetrically placed
Moment xRM AAC )(ACD axPxRM
)(BDB xLRM Shear AAC RV PRV ACD BDB RV
Reactions L
baLPR
)(A
L
abLPR
)(B
Simple beam - Two unequal concentrated loads unsymmetrically placed
Moment xRM AAC )(1ACD axPxRM
)(BDB xLRM Shear AAC RV 1ACD PRV BDB RV
Reactions L
bPaLPR 21
A
)(
L
aPbLPR 12
B
)(
Simple beam - Couple moment Mo at right end
Deflection )(6
220AB xL
LEI
xMy
Slope )3(6
220AB xL
LEI
M
EI
LM
60
A
EI
LM
30
B
Moment L
xMM 0
AB
Shear L
MV 0
AB
Reactions L
MR 0
A L
MR 0
B
Simple beam - Couple moment Mo at left end (I)
Deflection )32(6
220AB xLxL
LEI
xMy
EI
LMy
39
20
MAX at Lx
3
33
Slope )362(6
220AB xLxL
LEI
M
EI
LM
30
A EI
LM
60
B
Moment )(0AB xL
L
MM
Shear L
MV 0
AB
Reactions L
MR 0
A L
MR 0
B
Simple beam - Couple moment Mo at left end (II)
Deflection )32(6
220AB xLxL
LEI
xMy
EI
LMy
39
20
MAX
at Lx
3
33
Slope )362(6
220AB xLxL
LEI
M
EI
LM
30
A
EI
LM
60
B
Moment )(0AB xL
L
MM
Shear L
MV 0
AB
Reactions L
MR 0
A
L
MR 0
B
Simple beam - Couple moment Mo at center
Deflection )4(24
220AC xL
LEI
xMy
))(4(24
)( 220CB xLL
LEI
xLMy
Slope )12(24
220AC xL
LEI
M
))(12(24
220CB LxL
LEI
M
)3(6
220A bL
LEI
M
)3(
6220
B aLLEI
M
Moment L
xMM 0
AC )(0CB xL
L
MM
Shear L
MV 0
AC L
MV 0
CB
Reactions L
MR 0
A L
MR 0
B
Simple beam - Couple moment Mo at any point
Deflection )3(6
2220AC xbL
LEI
xMy
))(3(6
)( 2220CB xLaL
LEI
xLMy
Slope )33(6
2220AC xbL
LEI
M
))(33(6
2220CB xLaL
LEI
M
)3(6
220A bL
LEI
M
)3(
6220
B aLLEI
M
Moment L
xMM 0
AC )(0CB xL
L
MM
Shear L
MV 0
AC L
MV 0
CB
Reactions L
MR 0
A L
MR 0
B
Simple beam - Couple moments M1 and M2 at each end (I)
Deflection LMMxMMLEI
xLxy )2()(
6
)(2121AB
Slope:
)2)(2()23)((6
1 221
221AB LLxMMLxxMM
LEI
Moment 121AB )(1
LMxMML
M
Shear L
MMV 21
AB
Reactions L
MMR 21
A
L
MMR 12
B
Simple beam - Couple moments M1 and M2 at each end (II)
Deflection LMMxMMLEI
xLxy )2()(
6
)(2121AB
Slope:
)2)(2()23)((6
1 221
221AB LLxMMLxxMM
LEI
Moment 121AB )(1
LMxMML
M
Shear L
MMV 21
AB
Reactions L
MMR 21
A
L
MMR 21
B
Simple beam - Two equal couple moments Mo at each end
Deflection )(2
0AB xL
EI
xMy
EI
LMy
8
20
MAX
at
2
Lx
Slope )2(2
0AB xL
EI
M
EI
LM
20
BA
Moment 0AB MM
Shear 0AB V
Reactions 0BA RR
Cantilever beam - Uniformly distributed load
Deflection )64(24
22340AB xLLxx
EI
wy
EI
Lwyy
8
40
BMAX
at x = L
Slope )33(6
2230AB xLLxx
EI
w
EI
Lw
6
30
B
Moment 20AB )(
2xL
wM
2
20
AMAX
LwMM
Shear )(0AB xLwV
Reactions LwR 0A
Cantilever beam - Uniform load partially distributed at fixed end
Deflection )46(24
43220AC xaxxa
EI
wy
)4(24
30
CB axEI
awy
)4(24
30
BMAX aLEI
awyy
Slope )33(6
3220AC xaxxa
EI
w
EI
aw
6
30
BCCB
Moment 20AC )(
2xa
wM
0BCCB MMM
2
20
AMAX
awMM
Shear )(0AC xawV 0BCCB VVV
Reactions awR 0A
Cantilever beam - Uniform load partially distributed at free end
Deflection )233(12
20
AC xaLEI
bxwy
)464(24
4322340CB axaxLLxx
EI
wy
Slope )(2
0AC xaL
EI
bxw
)33(6
32230CB axLLxx
EI
w
)(6
330B aL
EI
w
Moment )2(2
0AC xaL
bwM
20
CB )(2
xLw
M
Shear bwVVV 0CAAC )(0CB xLwV
Reactions bwR 0A
Cantilever beam - Uniform load partially distributed
Deflection )236(12
20
AC xbaEI
bxwy
)4)(6)(4(24
4322340CD axaxbaxbax
EI
wy
))(])([4(24
44330DB abaabax
EI
wy
Slope )2(2
0AC xba
EI
bxw
))(3)(3(6
32230CD axbaxbax
EI
w
))((6
330DB aba
EI
w
Moment )22(2
0AC xba
bwM
20CD )(
2xba
wM
0BDDB MMM
Shear bwVVV 0CAAC )(0CD xbawV
0BDDB VVV
Reactions bwR 0A
Cantilever beam - Load increasing uniformly to free end
Deflection )1020(120
3232
0AB xxLL
LEI
xwy
EI
Lwy
120
11 40
MAX
at x = L
Slope )68(24
3230AB xxLL
LEI
xw
EI
Lw
8
30
B
Moment )32(6
3230AB xxLL
L
wM
Shear )(2
220AB xL
L
wV
Reactions 20
A
LwR
Cantilever beam - Load increasing uniformly to fixed end
Deflection )51010(120
32232
0AB xLxxLL
LEI
xwy
EILw
y30
40
MAX at x=L
Slope )464(24
32230AB xLxxLL
LEI
xw
EI
Lw
24
30
B
Moment 30AB )(
6xL
L
wM
Shear 20AB )(
2xL
L
wV
Reactions 20
A
LwR
Cantilever beam - Cosinusoidal distributed load
Deflection
332333
40
AB ππ3482
πcos48
π3xLxL
L
xL
EI
Lwy
)24π(3π
2 34
40
MAX
EI
Lwy at x = L
Slope
L
xLxLx
EI
Lw
2
πsin8ππ2
π2222
30
AB
)8π(π
23
30
B
EI
Lw
Moment
L
xLxL
LwM
2
πcos2ππ
π
22
0AB
Shear
LxLw
V2π
sin1π
2 0AB
Reactions π
2 0A
LwR
Cantilever beam - Concentrated load P at free end
Deflection )3(6
32AB xLx
EI
Py
EI
PLyy
3
3
BMAX
Slope )2(2
2AB xLx
EI
P
EI
PL
2
2
BMAX
Moment )(AB xLPM PLMM AMAX
Shear PVVV BAAB
Reactions PR A
Cantilever beam - Concentrated load P at any point
Deflection )3(6
32AC xax
EI
Py
)3(
6
2
CB axEI
Pay
)3(6
2
BMAX aLEI
Payy
Slope )2(2
2AC xax
EI
P
EI
Pa
2
2
BCCB
Moment )(AC xaPM 0BCCB MMM
PaMM AMAX
Shear PVVV CAAC 0BCCB VVV
Reactions PR A
Cantilever beam - Couple moment Mo at free end
Deflection EI
xMy
2
20
AB
EI
LMy
2
20
MAX
at x = L
Slope EI
xM 0AB
Moment 0BAAB MMMM
Shear 0BAAB VVV
Reactions 0A R
Cantilever beam - Couple moment Mo at any point
Deflection EI
xMy
2
20
AC
)2(
20
CB axEI
aMy
)2(2
0MAX aL
EI
aMy
at x = L
Slope EI
xM 0AC
EI
aM 0BCCB
Moment 0AAC MMM 0BCB MM
Shear 0CAAC VVV 0BCCB VVV
Reactions 0A R
Fixed-fixed beam - Uniformly distributed load
Deflection 22
0AB )(
24xL
EI
xwy
Slope )23(12
220AB xLxL
EI
xw
Moment )66(12
220AB xLxL
wM
Shear )2(2
0AB xL
wV
Reactions 20
BA
LwRR
Fixed-fixed beam - Uniform load partially distributed at left end (I)
Deflection )124(24 AA
20
2
AC MxRxwEI
xy
EI
LxLRMLRML
EI
xRxLRMy
6
)2(3)3(
6
)(3
BBBB2
3B
2BB
CB
Slope )63(6 AA
20AC MxRxw
EI
x
)2()(22
1BBBB
2BCB LRMLxLRMxR
EI
Moment 2
20
AAAC
xwMxRM BBCB )( MxLRM
Shear xwRV 0AAC BCB RV
Reactions L
MMLwR BA0
A 8
3
L
MMLwR BA0
B 8
Where 192
11 20
A
LwM
192
5 20
B
LwM
Fixed-fixed beam - Uniform load partially distributed at left end (II)
Deflection )124(24 AA
20
2
AC MxRxwEI
xy
EI
LxLRMLRML
EI
xRxLRMy
6
)2(3)3(
6
)(3
BBBB2
3B
2BB
CB
Slope )63(6 AA
20AC MxRxw
EI
x
)2()(22
1BBBB
2BCB LRMLxLRMxR
EI
Moment 2
20
AAAC
xwMxRM BBCB )( MxLRM
Shear xwRV 0AAC BCB RV
Reactions L
MM
L
abLwR BA0
A 2
)(
L
MM
L
awR BA
20
B 2
Where )386(12
222
20
A aLaLL
awM
)34(12 2
30
B aLL
awM
Fixed-fixed beam - Uniform load partially distributed
Deflection )3(6 AA
2
AC xRMEI
xy
2A
3A
40CD 124)(
24
1xMxRaxw
EIy
EI
LxLRMLRML
EI
xRxLRMy
6
)2(3)3(
6
)(3
BBBB2
3B
2BB
DB
Slope )2(2 AAAC xRM
EI
x
xMxRaxwEI A
2A
30CD 63)(
6
1
)2()(22
1BBBB
2BDB LRMLxLRMxR
EI
Moment xRMM AAAC 2
)( 20
AACD
axwMxRM
)(BBDB xLRMM Shear AAC RV )(0ACD axwRV BDB RV
Reactions L
MMbbcwR
2
22)2( BA0A
L
MMbbawR
2
22)2( BA0B
Where 222
0A )2)(36()362(
24bcbabcLb
L
bwM
222
0B )2)(36()362(
24babcbaLb
L
bwM
Fixed-fixed beam - Concentrated load at center
Deflection )43(48
2
AC xLEI
Pxy
)4(48
)( 2
CB LxEI
xLPy
Slope )2(8AC xL
EI
Px
)23(
822
CB xLxLEI
P
Moment )4(8AC xLP
M
)43(8CB xLP
M
Shear 2AC
PV
2CB
PV
Reactions 2BA
PRR
Fixed-fixed beam - Concentrated load at any point
Deflection )33(6 3
22
AC bxaxaLEIL
xPby
)3(6
)(3
22
CB axaLbxEIL
xLPay
Slope )32(2 3
2
AC bxaxaLEIL
xPb
23
2
CB )3(2
)(Labx
EIL
xLPa
Moment )3(3
2
AC bxaxaLL
xPbM
)2( 23
2
CB bxLxbLLL
PaM
Shear )2(3
2
AC aLL
PbV )2(
3
2
CB bLL
PaV
Reactions )2(3
2
A aLL
PbR )2(
3
2
B bLL
PaR
Fixed-fixed beam - Two equal concentrated loads symmetrically placed
Deflection )33(6
22
AC LxaaLEIL
Pxy
)33(6
22
CD aLxLxEIL
Pay
))(33(6
)( 22
DB xLLaaLEIL
xLPy
Slope )22(2
2AC LxaaL
EIL
Px
)2(
2
2
CD xLEIL
Pa
)(222
)( 2DB xLLaaL
EIL
xLP
Moment )( 2AC aaLLx
L
PM
L
PaM
2
CD )( 22DB aLaLxL
L
PM
Shear PV AC 0CD V PV DB
Reactions PRR BA
Fixed-fixed beam - Couple moment Mo at center
Deflection )2(8
20
AC LxLEI
xMy
)425(8
32320CB LxLxLx
LEI
My
Slope )3(4
0AC Lx
LEI
xM )4610(
8220
CB LxLxLEI
M
Moment )6(4
0AC Lx
L
MM )65(
40
CB xLL
MM
Shear L
MV
2
3 0AB
Reactions L
MR
2
3 0A
L
MR
2
3 0B
Fixed-fixed beam - Couple moment Mo at any point
Deflection:
)22(2 3
20
AC bLaxaLEIL
bxMy
)2(
2
)(3
20
CB aLbxEIL
xLaMy
Slope:
)32(3
0AC bLaxaL
EIL
bxM
)3(
)( 23
0CB bxL
EIL
xLaM
Moment:
)62(3
0AC bLaxaL
L
bMM
)46(
30
CB aLbLbxL
aMM
Shear 30
AB
6
L
abMV
Reactions 30
A
6
L
abMR 3
0B
6
L
abMR
Where )2(2
0A ba
L
bMM
)2(
20
B abL
aMM
Fixed-pinned beam - Uniformly distributed load
Deflection )253(48
222
0AB xLxL
EI
xwy
Slope )8156(48
220AB xLxL
EI
xw
Moment )45(8
220AB xLxL
wM
Shear )85(8
0AB xL
wV
Reactions 8
5 0A
LwR
8
3 0B
LwR
Fixed-pinned beam - Uniform load partially distributed at fixed end
Deflection:
EIL
bLxLawxaLwxLLRy
48
)3)(()(2)(8 30
40
3B
AC
EIL
bLxLawxLLRy
48
)3)(()(8 30
3B
CB
Slope:
EIL
bLawxaLwxLLR
48
)3()(8)(24 30
30
2B
AC
EIL
bLawxLLR
48
)3()(24 30
2B
CB
Moment 2
)()(2 20B
AC
xawxLRM
)(BCB xLRM
Shear )(0BAC xawRV BCB RV
Reactions L
MabLwR
2
2)( A0A
L
MawR
2
2 A2
0B
Where 2
220
A 8
)(
L
abLwM
Fixed-pinned beam - Uniform load partially distributed at supported end
Deflection )3(6 AA
2
AC MxREI
xy
EIL
aabbLxLbw
EIL
xLLwxLLRy
48
)63)((
24
)()(4
220
40
3B
CB
Slope )2(2 AAAC MxR
EI
x
EIL
aabbLbw
EIL
xLLwxLLR
48
)63(
6
)()(3 220
30
2B
CB
Moment AAAC MxRM 2
)()(2 20B
CB
xLwxLRM
Shear AAC RV )(0BCB xLwRV
Reactions L
MbwR
2
2 A2
0A
L
MbbawR
2
2)2( A0B
Where 22
20
A ))(2(16
baLbLL
bwM
Fixed-pinned beam - Uniform load partially distributed
Deflection )3(6 AA
2
AC MxREI
xy
EIL
babcbabLbxLbw
EI
cxLwxLRy
96
)2)(2(3)2(32)(
24
)()(4
2220
40
3B
CD
EIL
babcbabLbxLbw
EI
xLRy
96
)2)(2(3)2(32)(
6
)(
2220
3B
DB
Slope )2(2 AAAC MxR
EI
x
EIL
babcbabLbbw
EI
cxLwxLR
96
)2)(2(3)2(32
6
)()(3
2220
30
2B
CD
EIL
babcbabLbbw
EI
xLR
96
)2)(2(3)2(32
2
)(
2220
2B
DB
Moment AAAC MxRM
2
)()(2 20B
CD
cxLwxLRM
)(BDB xLRM
Shear AAC RV B0CD )( RcxLwV BDB RV
Reactions L
MbcbwR
2
2)2( A0A
L
MbbawR
2
2)2( A0B
Where
2
20
A 16
)2)(22()2)(2(
L
bbabcLbbabcwM
Fixed-pinned beam - Concentrated load at center
Deflection:
)119(96
2
AC xLEI
Pxy
))(53(
96
)( 22CB xLL
EI
xLPy
Slope:
)116(32AC xL
EI
Px
)5104(
3222
CB xLxLEI
P
Moment )113(16AC xL
PM
)(
16
5CB xL
PM
Shear 16
11AC
PV
16
5CB
PV
Reactions 16
11A
PR
16
5B
PR
Fixed-pinned beam - Concentrated load at any point
Deflection )333(12
22233
2
AC xbxLLbLEIL
Pbxy
)))(2(3(12
)( 223
2
CB xLbLbLEIL
xLPay
Slope )322(4
22233AC xbxLLbL
EIL
Pbx
)2242(4
22233
2
CB bxLxbLxxLLEIL
Pa
Moment:
)3(2
22233AC xbxLLbL
L
PbM
)2)((
2 3
2
CB bLxLL
PaM
Shear )3(2
223AC bL
L
PbV )2(
2 3
2
CB bLL
PaV
Reactions )3(2
223A bL
L
PbR )2(
2 3
2
B bLL
PaR
Fixed-pinned beam - Two equal concentrated loads symmetrically placed
Deflection:
)33(2))(233(12
22222
2
AC LaLaLxLLaLaEIL
Pxy
2
222
2
223
CD
12
2))((3
12
))(6))((3(
EIL
aLxLaLLPa
EIL
xLLxLaLPay
)(3))(233(12
)( 22222DB aLaLxLLaaL
EIL
xLPy
Slope:
)33(4)32)(233(12
22222AC LaLaLxLLaLa
EIL
Px
)()(4))((34
2222CD aLLxLLxLaL
EIL
Pa
)())(233(4
22222DB aLaLxLLaaL
EIL
P
Moment )332(332
22222AC aaLLxaLLa
L
PM
22CD 2))((3
2LxLaL
L
PaM
)233(2
)( 222DB LaaL
L
xLPM
Shear )332(2
222AC aaLL
L
PV 2CD 2
)(3
L
aLPaV
)233(2
222DB LaaL
L
PV
Reactions )332(2
222A aaLL
L
PR
)323(2
222B aLLa
L
PR
Fixed-pinned beam - Couple moment Mo at any point
Deflection: ))((24
2223
20
AC bLxLLbEIL
xMy
))(3)((44
)( 2233
0CB bLLxLL
EIL
xLaMy
Slope ))(32(44
2223
0AC bLxLLb
EIL
xM
)2)((344
233
0CB LxxbLL
EIL
aM
Moment ))(3(22
2223
0AC bLxLLb
L
MM
))((2
330
CB xLbLL
aMM
Shear )(2
33
0AB bL
L
aMV
Reactions )(2
33
0A bL
L
aMR
)(
2
330
B bLL
aMR
Fixed-pinned beam - Couple moment Mo at supported end
Deflection EIL
xLxMy
4
)(20
AB
Slope EIL
xLxM
4
)32(0AB
Moment L
xLMM
2
)3(0AB
Shear L
MV
2
3 0AB
Reactions L
MR
2
3 0A
L
MR
2
3 0B
Overhanging beam - Uniformly distributed load
Deflection )222(24
222232240AB xaLaLxxLL
LEI
xwy
)464(24
31
211
23210BC xaxxaLLa
EI
xwy
Slope )6246(24
222232240AB xaLaLxxLL
LEI
w
)412124(24
31
211
2320BC xaxxaLLa
EI
w
Moment )(2
220AB aLxL
L
xwM 2
10
BC )(2
xaw
M
Shear )2(2
220AB aLxL
L
wV )( 10BC xawV
Reactions )(2
220A aL
L
wR 20
B )(2
aLL
wR
Where Lxx 1
Overhanging beam - Uniformly distributed load on overhang
Deflection )(12
222
0AB xL
LEI
xawy
)464(24
31
211
2210BC xaxxaLa
EI
xwy
Slope )3(12
222
0AB xL
LEI
aw
)33(6
31
211
220BC xaxxaLa
EI
w
Moment L
xawM
2
20
AB
2
10
BC )(2
xaw
M
Shear L
awV
2
20
AB
)( 10BC xawV
Reactions L
awR
2
20
A
L
aaLwR
2
)2(0B
Where Lxx 1
Overhanging beam - Concentrated load at end of overhang
Deflection )(6
22AB xL
LEI
Paxy
)32(6
211
1BC xaxaL
EI
Pxy
Slope )3(6
22AB xL
LEI
Pa )362(
62
11BC xaxaLEI
P
Moment L
PaxM
AB )( 1BC xaPM
Shear L
PaV
AB PV BC
Reactions L
PaR
A
L
aLPR
)(B
Where Lxx 1
Overhanging beam - Concentrated load at any point between supports
Deflection )(6
222AC xbL
LEI
Pbxy
)2(6
)( 22CB xaLx
LEI
xLPay
)(6
1BD aL
LEI
Pabxy
Slope )3(6
222AC xbL
LEI
Pb
)362(6
222CB xaLxL
LEI
Pa
LEI
aLPab
6
)(BD
Moment L
PbxM AC )(CB xL
L
PaM 0BD M
Shear L
PbV AC
L
PaV
CB 0BD V
Reactions L
PbR A
L
PaR B
Where Lxx 1
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