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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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