tablas aisc guillermo

8/18/2019 Tablas Aisc Guillermo

1/20

BEAM DIAGRAMS AND FORMULAS

Nomenclature E = modulus of elasticity of steel at 29,000 ksi I = moment of inertia of beam (in. 4)

L = total length of beam between reaction points (ft) M max = maximum moment (kip-in.) M 1 = maximum moment in left section of beam (kip-in.) M 2 = maximum moment in right section of beam (kip-in.) M 3 = maximum positive moment in beam with combined end moment conditions

(kip-in.) M x = moment at distance x from end of beam (kip-in.)P = concentrated load (kips)

P1 = concentrated load nearest left reaction (kips)P2 = concentrated load nearest right reaction, and of different magnitude than P1

(kips) R = end beam reaction for any condition of symmetrical loading (kips) R1 = left end beam reaction (kips) R2 = right end or intermediate beam reaction (kips) R3 = right end beam reaction (kips)V = maximum vertical shear for any condition of symmetrical loading (kips)

V 1 = maximum vertical shear in left section of beam (kips)V 2 = vertical shear at right reaction point, or to left of intermediate reaction point

of beam (kips)V 3 = vertical shear at right reaction point, or to right of intermediate reaction point

of beam (kips)V x = vertical shear at distance x from end of beam (kips)W = total load on beam (kips)a = measured distance along beam (in.)b = measured distance along beam which may be greater or less than a (in.)l = total length of beam between reaction points (in.)w = uniformly distributed load per unit of length (kips per in.)w1 = uniformly distributed load per unit of length nearest left reaction (kips per in.)w2 = uniformly distributed load per unit of length nearest right reaction, and of

different magnitude than w1 (kips per in.) x = any distance measured along beam from left reaction (in.) x1 = any distance measured along overhang section of beam from nearest reaction

point (in.)

∆max = maximum deflection (in.)∆a = deflection at point of load (in.)∆ x = deflection at any point x distance from left reaction (in.)∆ x1 = deflection of overhang section of beam at any distance from nearest reaction

point (in.)

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BEAM DIAGRAMS AND FORMULAS 4 - 187


2/20

BEAM DIAGRAMS AND FORMULASFrequently Used Formulas

The formulas given below are frequently required in structural designing. They areincluded herein for the convenience of those engineers who have infrequent use for suchformulas and hence may find reference necessary. Variation from the standard nomen-clature on page 4-187 is noted.BEAMS

Flexural stress at extreme fiber: f = Mc / I = M / S

Flexural stress at any fiber: f = My / I y = distance from neutral axis to fiber

Average vertical shear (for maximum see below):v = V / A = V / dt (for beams and girders)

Horizontal shearing stress at any section A-A:v = VQ / Ib Q = statical moment about the neutral axis of that portion

of the cross section lying outside of section A-Ab = width at section A-A

(Intensity of vertical shear is equal to that of horizontal shear acting normal to it at thesame point and both are usually a maximum at mid-height of beam.)Shear and deflection at any point:

EI d 2 y dx2

= M x and y are abscissa and ordinate respectively of a point on the neutralaxis, referred to axes of rectangular coordinates through a selectedpoint of support.

(First integration gives slopes; second integration gives deflections. Constants of inte-gration must be determined.)

CONTINUOUS BEAMS (the theorem of three moments)Uniform load:

M al1

I 1 + 2 M b

l1 I 1

+ l2 I 2

+ M c l2

I 2 = − 1 ⁄ 4

w1l13

I 1 + w2l2

3

I 2

Concentrated loads:

M al1

I 1 + 2 M b

l1

I 1 + l2

I 2

+ M c l2

I 2 = − P 1a 1b1

I 1

1 + a 1

l1

− p 2a 2b2 I 2

1 + b2

I 2

Considering any two consecutive spans in any continuous structure: M a , M b, M c = moments at left, center, and right supports respectively, of any pair of

adjacent spansl1 and l2 = length of left and right spans, respectively, of the pair

I 1 and I 2 = moment of inertia of left and right spans, respectivelyw1 and w2 = load per unit of length on left and right spans, respectivelyP1 and P2 = concentrated loads on left and right spans, respectivelya 1 and a 2 = distance of concentrated loads from left support, in left and right spans,

respectively

b1 and b2 = distance of concentrated loads from right support, in left and right spans,respectively

The above equations are for beam with moment of inertia constant in each span butdiffering in different spans, continuous over three or more supports. By writing such anequation for each successive pair of spans and introducing the known values (usuallyzero) of end moments, all other moments can be found.


4 - 188 BEAM AND GIRDER DESIGN


3/20

BEAM DIAGRAMS AND FORMULASTable of Concentrated Load Equivalents

n Loading Coeff.

SimpleBeam

Beam Fixed OneEnd, Supported

at OtherBeam FixedBoth Ends

∞ a 0.125 0.070 0.042b — 0.125 0.083c 0.500 0.375 —d — 0.625 0.500e 0.013 0.005 0.003f 1.000 1.000 0.667g 1.000 0.415 0.300

2 a 0.250 0.156 0.125b — 0.188 0.125

c 0.500 0.313 —d — 0.688 0.500e 0.021 0.009 0.005f 2.000 1.500 1.000g 0.800 0.477 0.400

3 a 0.333 0.222 0.111b — 0.333 0.222c 1.000 0.667 —d — 1.333 1.000e 0.036 0.015 0.008

f 2.667 2.667 1.778g 1.022 0.438 0.333

4 a 0.500 0.266 0.188b — 0.469 0.313c 1.500 1.031 —d — 1.969 1.500e 0.050 0.021 0.010f 4.000 3.750 2.500g 0.950 0.428 0.320

5 a 0.600 0.360 0.200

b — 0.600 0.400c 2.000 1.400 —d — 2.600 2.000e 0.063 0.027 0.013f 4.800 4.800 3.200g 1.008 0.424 0.312

Maximum positive moment (kip-ft): aPLMaximum negative moment (kip-ft): bPLPinned end reaction (kips): cP Fixed end reaction (kips): dP Maximum deflection (in): eP l3 / EI

Equivalent simple span uniform load (kips): f P Deflection coefficient for equivalent simple span uniform load: g Number of equal load spaces: n Span of beam (ft): LSpan of beam (in): l

P

P

P P

P P P

P P P P




4/20

BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions

For meaning of symbols, see page 4-187

1. SIMPLE BEAM—UNIFORMLY DISTRIBUTED LOADTotal Equiv. Uniform Load . . . . . . = wl

R = V . . . . . . . . . . . . . . . . . = wl2

V x . . . . . . . . . . . . . . . . . = w l2

− x

M max (at center) . . . . . . . . . . . . =wl 2

8

M x . . . . . . . . . . . . . . . . . =wx2

(l − x)

∆max (at center) . . . . . . . . . . . . =5wl 4

384 EI

∆ x . . . . . . . . . . . . . . . . . =wx

24 EI (l2 − 2 lx2 + x3)

2. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO ONE END

Total Equiv. Uniform Load . . . . . . =16 W

9√ 3 = 1.0264 W

R1 = V 1 . . . . . . . . . . . . . . . . . =W 3

R2 = V 2 max . . . . . . . . . . . . . . . =2W

3V x . . . . . . . . . . . . . . . . . =

W 3

− Wx2

l2

M max (at x =l

√ 3 = .5774 l) . . . . . . . = 2Wl

9√ 3 = .1283 Wl

M x . . . . . . . . . . . . . . . . . =Wx

3 l2 (l2 − x2)

∆max (at x = l√ 1 − √ 815 = .5193 l) . . = 0.1304 Wl3

EI

∆ x . . . . . . . . . . . . . . . . . =Wx

180 EIl 2(3 x4 − 10 l2 x2 + 7 l4)

3. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO CENTER

Total Equiv. Uniform Load . . . . . . =4W 3

R = V . . . . . . . . . . . . . . . . . = W 2

V x (when x < l2

) . . . . . . . . . . =W

2 l2 (l2 − 4 x2)

M max (at center) . . . . . . . . . . . . =Wl6

M x (when x < l2

) . . . . . . . . . . = Wx

12

− 2 x2

3 l2

∆max (at center) . . . . . . . . . . . . =Wl3

60 EI

∆ x (when x < l2

) . . . . . . . . . . =Wx

480 EIl 2 (5 l2 − 4 x2)2

Moment

Shear

l x

l

R R

2 2 l l

V

V

M max

w

Moment

Shear

l x

W

R R

2 2 l l

V

V

M max

Moment

Shear

l x

W

R R

l

V

V

M max

1 2

.5774

2

1




5/20



4. SIMPLE BEAM—UNIFORMLY LOAD PARTIALLY DISTRIBUTED

R1 = V 1 (max. when a < c) . . . . . . . =wb2 l

(2c + b)

R2 = V 2 (max. when a > c) . . . . . . . =wb2 l

(2a + b)

V x (when x > a and < (a + b)) . . . = R1 − w( x − a )

M max

at x = a + R1w

. . . . . . . . = R1

a + R12w

M x (when x < a ) . . . . . . . . . = R1 x

M x (when x > a and < (a + b)) . . . = R1 x − w2

( x − a )2

M x (when x > (a + b)) . . . . . . . = R2(l − x)

5. SIMPLE BEAM—UNIFORM LOAD PARTIALLY DISTRIBUTED AT ONE END

R1 = V 1 max . . . . . . . . . . . . . . . =wa2 l

(2l − a )

R2 = V 2 . . . . . . . . . . . . . . . . =wa 2

2l

V x (when x < a ) . . . . . . . . . = R1 − wx

M max

at x =

R1

w

. . . . . . . . . . =

R12

2w

M x (when x < a ) . . . . . . . . . = R1 x − wx2

2

M x (when x > a ) . . . . . . . . . = R2 (l − x)

∆ x (when x < a ) . . . . . . . . . =wx

24 EIl (a 2(2 l − a )2 − 2 ax 2(2 l − a ) + lx3)

∆ x (when x > a ) . . . . . . . . . =wa 2(l − x)

24 EIl (4 xl − 2 x2 − a 2)

6. SIMPLE BEAM—UNIFORM LOAD PARTIALLY DISTRIBUTED AT EACH END

R1 = V 1 . . . . . . . . . . . . . . . . = w1a (2 l − a ) + w2c2

2l

R2 = V 2 . . . . . . . . . . . . . . . . =w2c(2l − c) + w1a 2

2 l

V x (when x < a ) . . . . . . . . . = R1 − w1 xV x (when x > a and < (a + b)) = R1 − w1aV x (when x > (a + b)) . . . . . . . = R2 − w2 (l − x)

M max

at x = R1w1

when R1 < w1a

= R1

2

2w1

M max at x = l −

R1w2 when R

2 < w2c =

R22

2w2

M x (when x < a ) . . . . . . . . . = R1 x − w1 x

2

2

M x (when x > a and < (a + b)) . . . = R1 x − w1a

2 (2 x − a )

M x (when x > (a + b)) . . . . . . . = R2(l − x) − w2(l − x)2

2

Moment

Shear

l

x R R

V

V

M max

a b c w b

1 2

a+ w

R 1

2

1

Moment

Shear

l

x R R

V

V

M max

a

1 2

R 1

2

1

wa

w

Moment

Shear

l

x R R

V

V

M max

a

1 2

R 1

2

1

b c

1 2

1

w a w c

w




6/20



7. SIMPLE BEAM—CONCENTRATED LOAD AT CENTER

Total Equiv. Uniform Load . . . . . . . . . . = 2P

R = V . . . . . . . . . . . . . . . . . . . . = P2

M max (at point of load) . . . . . . . . . . . =Pl4

M x

when x < 12

. . . . . . . . . . . . . =

Px2

∆max (at point of load) . . . . . . . . . . . =Pl 3

48 EI

∆ x

when x < 12

. . . . . . . . . . . . . =

Px48 EI

(3 l2 − 4 x2)

8. SIMPLE BEAM—CONCENTRATED LOAD AT ANY POINT

Total Equiv. Uniform Load . . . . . . . . . . =8Pab

l2

R1 = V 1 (max when a < b) . . . . . . . . . . . =Pbl

R2 = V 2 (max when a > b) . . . . . . . . . . . =Pal

M max (at point of load) . . . . . . . . . . . =Pab

l

M x (when x < a ) . . . . . . . . . . . . . . =Pbx

l

∆max at x = √ a (a + 2 b)3 when a > b

. . . =

Pab (a + 2 b)√ 3a (a + 2 b)27 EIl

∆a (at point of load) . . . . . . . . . . . =Pa 2b2

3 EIl

∆ x (when x < a ) . . . . . . . . . . . . . . =Pbx6 EIl

(l2 − b2 − x2)

9. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS SYMMETRICALLY PLACED

Total Equiv. Uniform Load . . . . . . . . . . =8Pa

l

R = V . . . . . . . . . . . . . . . . . . . . = P M max (between loads) . . . . . . . . . . . . = Pa

M x (when x < a ) . . . . . . . . . . . . . . = Px∆max (at center) . . . . . . . . . . . . . . . =

Pa24 EI

(3 l2 − 4 a 2)

∆ x (when x < a ) . . . . . . . . . . . . . . =Px6 EI

(3 la − 3 a 2 − x2)

∆ x (when x > a and < (l − a )) . . . . . . . =Pa6 EI

(3 lx − 3 x2 − a 2)Moment

Shear

l

x

R R

V

M max

P

a

V

a

P

Moment

Shear

l

x

R R

V

V

M max

P

a b 1

1

2

2

Moment

Shear

l

x

R R

V

V

M max

P

l l

2 2




7/20



10. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS UNSYMMETRICALLYPLACED

R1 = V 1 (max. when a < b) . . . . . . . . . =Pl

(l − a + b)

R2 = V 2 (max. when a > b) . . . . . . . . . =Pl

(l − b + a )

V x (when x > a and < (l − b)) . . . . . =Pl

(b − a )

M 1 (max. when a > b) . . . . . . . . . = R1a

M 2 (max. when a < b) . . . . . . . . . = R2b

M x (when x < a ) . . . . . . . . . . . . = R1 x

M x (when x > a and < (l − b)) . . . . . = R1 x − P ( x − a )

11. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED LOADS UNSYMMETRICALLYPLACED

R1 = V 1 . . . . . . . . . . . . . . . . . . =P1 (l − a ) + P2 b

l

R2 = V 2 . . . . . . . . . . . . . . . . . . =P1 a + P2(l − b)

l

V x (when x > a and < (l − b)) . . . . . = R1 − P1 M 1 (max. when R1 < P1) . . . . . . . . = R1a

M 2 (max. when R2 < P2) . . . . . . . . = R2b

M x (when x < a ) . . . . . . . . . . . . = R1 x

M x (when x > a and < (l − b)) . . . . . = R1 x − P ( x − a )

12. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—UNIFORMLY DISTRIBUTEDLOAD

Total Equiv. Uniform Load . . . . . . . . . . = wl

R1 = V 1 . . . . . . . . . . . . . . . . . . =3wl

8

R2 = V 2 max . . . . . . . . . . . . . . . . . . =5wl

8

V x . . . . . . . . . . . . . . . . . . = R1 − wx

M max . . . . . . . . . . . . . . . . . . =wl 2

8

M x

at x = 3

8l

. . . . . . . . . . . . . =

9

128wl 2

M x . . . . . . . . . . . . . . . . . . = R1 x − wx 2

2

∆max

at x = l16

(1 + √ 33 ) = .4215 l . . . =

wl 4

185 EI

∆ x . . . . . . . . . . . . . . . . . .wx

48 EI (l3 − 3 lx + 2 x3)

Moment

Shear

l

x

R R

V

M M 1

P

a

V

b

P

1

1

2

2

2

Moment

Shear

l

x

R R

V

M M 1

P

a

V

b

P

1

1

2

2

2

1 2

Moment

Shear

l

x

R R

V

V

M

M

1

1

1

2

2

max

4

3 8 l

l

l w




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13. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—CONCENTRATED LOAD ATCENTER

Total Equiv. Uniform Load . . . . . . . =3P2

R1 = V 1 . . . . . . . . . . . . . . . . . . =5P15

R2 = V 2 max . . . . . . . . . . . . . . . . =11 P16

M max (at fixed end) . . . . . . . . . . . =3Pl16

M 1 (at point of load) . . . . . . . . . . =5Pl

32

M x

when x < l2

. . . . . . . . . . . . =5Px16

M x

when x > l2

. . . . . . . . . . . . = P l2

− 11 x16

∆max at x = l√ 15 = .4472 l

. . . . . . . =Pl 3

48 EI √ 5 = .009317 Pl

3

EI

∆ x (at point of load) . . . . . . . . . . =7PL 3

768 EI

∆ x

when x < l

2

. . . . . . . . . . . . =Px

96 EI (3l2 − 5 x2)

∆ x

when x > l2

. . . . . . . . . . . . =P

96 EI ( x − l)2(11 x − 2 l)

14. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—CONCENTRATED LOAD ATANY POINT

R1 = V 1 . . . . . . . . . . . . . . . . . . =Pb 2

2 l3 (a + 2 l)

R2 = V 2 . . . . . . . . . . . . . . . . . . =Pa

2 l3 (3l2 − a 2)

M (at point of load) . . . . . . . . . . = R1a

M 2 (at fixed end) . . . . . . . . . . . =Pab

2 l2 (a + l)

M x (when x < a ) . . . . . . . . . . . . = R1 x M x (when x > a ) . . . . . . . . . . . . = R1 x − P ( x − a )

∆max

when a < .414 l at x = l (l2 + a 2)

(3 l2 − a 2)

=Pa (l2 + a 2)3

3 EI (3l2 − a 2)2

∆max

when a > .414 l at x = l a2 l + a

√ .......... = Pab

2

6 EI √ a2 l + a∆a (at point of load) . . . . . . . . . . =

Pa 2b3

12 EIl 3 (3 l + a )

∆ (when x < a ) . . . . . . . . . . . . = Pb2 x

12 EIl 3 (3al 2 − 2 lx2 − ax 2)

∆ x (when x > a ) . . . . . . . . . . . . =Pa

12 EIl 2(l − x)2 (3 l2 x − a 3 x − 2 a 2l)

Moment

Shear

l

x

R R

V

V

M

max

P

l l

2 2 1

1

1

2

2

3 11

l

M

Moment

Shear

l

x

R R

V

V

M

2

P

1

1

1

2

2

Pa R

M

a b

2




9/20



15. BEAM FIXED AT BOTH ENDS—UNIFORMLY DISTRIBUTED LOADS

Total Equiv. Uniform Load . . . . . . . . =2wl

3

R = V . . . . . . . . . . . . . . . . . . = wl2

V x . . . . . . . . . . . . . . . . . . = w l2

− x

M max (at ends) . . . . . . . . . . . . . =wl 2

12

M 1 (at center) . . . . . . . . . . . . . =wl 2

24

M x . . . . . . . . . . . . . . . . . . =w

12 (6 lx − l2

− 6 x2

)∆max (at center) . . . . . . . . . . . . . =

wl 4

384 EI

∆ x . . . . . . . . . . . . . . . . . . =wx2

24 EI (l − x)2

16. BEAM FIXED AT BOTH ENDS—CONCENTRATED LOAD AT CENTER

Total Equiv. Uniform Load . . . . . . . . = P

R = V . . . . . . . . . . . . . . . . . . = P2

M max (at center and ends) . . . . . . . . =Pl8

M x

when x < l2

. . . . . . . . . . . =

P8

(4 x − l)

∆max (at center) . . . . . . . . . . . . . =Pl 3

192 EI

∆ x

when x < l2

. . . . . . . . . . . =

Px 2

48 EI (3l − 4 x)

17. BEAM FIXED AT BOTH ENDS—CONCENTRATED LOAD AT ANY POINT

R1 = V 1 (max. when a < b) . . . . . . . . = Pb2

l3 (3a + b)

R2 = V 2 (max. when a > b) . . . . . . . . =Pa 2

l3 (a + 3 b)

M 1 (max. when a < b) . . . . . . . . =Pab 2

l2

M 2 (max. when a > b) . . . . . . . . =Pa 2b

l2

M a (at point of load) . . . . . . . . . =2Pa 2b2

l3

M x (when x < a ) . . . . . . . . . . . = R1 x − Pab 2

l2

∆max

when a > b at x = 2al3a + b

. . . . =

2Pa 3b2

3 EI (3a + b)2

∆a (at point of load) . . . . . . . . . =Pa 3b3

3 EIl 3

∆ x (when x < a ) . . . . . . . . . . . =Pb 2 x2

6 EIl 2 (3al − 3 ax − bx)

Moment

Shear

l x

w

R R

V

V

M

M M

1

max

l

l

2 2

l l

.2113

max

Moment

Shear

l

x

R R

V

V

M

M M max

l

4

2

l l

max

P

2

max

Moment

Shear

l

x

R R

V

V

M

M M 2 1

P

a

2

2

1

1 a b




10/20



18. CANTILEVER BEAM—LOAD INCREASING UNIFORMLY TO FIXED END

Total Equiv. Uniform Load . . . . . . . . =83

W

R = V . . . . . . . . . . . . . . . . . . . = W

V x . . . . . . . . . . . . . . . . . . . = W x2

l2

M max (at fixed end) . . . . . . . . . . . . =Wl3

M x . . . . . . . . . . . . . . . . . . . =Wx3

3 l2

∆max (at free end) . . . . . . . . . . . . . =Wl3

15 EI

∆ x . . . . . . . . . . . . . . . . . . . =W

60 EIl 2 ( x5 − 5 l4 x + 4 l5)

19. CANTILEVER BEAM—UNIFORMLY DISTRIBUTED LOAD

Total Equiv. Uniform Load . . . . . . . . = 4wl

R = V . . . . . . . . . . . . . . . . . . . = wlV x . . . . . . . . . . . . . . . . . . . = wx

M max (at fixed end) . . . . . . . . . . . . =wl 2

2

M x . . . . . . . . . . . . . . . . . . . =wx2

2

∆max (at free end) . . . . . . . . . . . . . =wl 4

8 EI

∆ x . . . . . . . . . . . . . . . . . . . =w

24 EI ( x4 − 4 l3 x + 3 l4)

20. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATEAT OTHER—UNIFORMLY DISTRIBUTED LOAD

Total Equiv. Uniform Load . . . . . . . . =83

wl

R = V . . . . . . . . . . . . . . . . . . . = wlV x . . . . . . . . . . . . . . . . . . . = wx

M max (at fixed end) . . . . . . . . . . . . =wl 2

3

M x . . . . . . . . . . . . . . . . . . . =w6

(l2 − 3 x2)

∆max (at deflected end) . . . . . . . . . . =wl 4

24 EI

∆ x . . . . . . . . . . . . . . . . . . . =w(l2 − x2)2

24 EI

Shear

l

x

R

V

M

M

M

w

max Moment

l

.4227 l

1

Shear

l

x

R

V

M

w

max Moment

l

Shear

l

x

R

W

V

M max Moment




11/20



21. CANTILEVER BEAM—CONCENTRATED LOAD AT ANY POINT

Total Equiv. Uniform Load . . . . . . . . =8Pb

l

R = V . . . . . . . . . . . . . . . . . . . = P

M max (at fixed end) . . . . . . . . . . . . = Pb

M x (when x > a ) . . . . . . . . . . . . = P ( x − a )

∆max (at free end) . . . . . . . . . . . . . =Pb 2

6 EI (3 l − b)

∆a (at point of load) . . . . . . . . . . =Pb 3

3 EI

∆ x (when x < a ) . . . . . . . . . . . . =Pb 2

6 EI (3 l − 3 x − b)

∆ x (when x > a ) . . . . . . . . . . . . =P (l − x)2

6 EI (3b − l + x)

22. CANTILEVER BEAM—CONCENTRATED LOAD AT FREE END

Total Equiv. Uniform Load . . . . . . . . = 8P

R = V . . . . . . . . . . . . . . . . . . . = P

M max (at fixed end) . . . . . . . . . . . . = Pl

M x . . . . . . . . . . . . . . . . . . . = Px

∆max (at free end) . . . . . . . . . . . . . =Pl 3

3 EI

∆ x . . . . . . . . . . . . . . . . . . . =P

6 EI (2 l3 − 3 l2 x + x3)

23. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATEAT OTHER—CONCENTRATED LOAD AT DEFLECTED END

Total Equiv. Uniform Load . . . . . . . . = 4P

R = V . . . . . . . . . . . . . . . . . . . = P

M max (at both ends) . . . . . . . . . . . . =Pl2

M x . . . . . . . . . . . . . . . . . . . = P

l

2 − x

∆max (at deflected end) . . . . . . . . . . = pl 3

12 EI

∆ x . . . . . . . . . . . . . . . . . . . =P (l − x)2

12 EI (l + 2 x)

Shear

l

x

R

V

M max Moment

P

a b

Shear

l

x R

V

M max Moment

P

Shear

l

x R

V

M

M

M

max

Moment

P

max

l

2




12/20



24. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD

R1 = V 1 . . . . . . . . . . . . . . =w2 l

(l2 − a 2)

R2 = V 2 + V 3 . . . . . . . . . . . =w2 l

(l + a )2

V 2 . . . . . . . . . . . . . . . = wa

V 3 . . . . . . . . . . . . . . . =w2 l

(l2 + a 2)

V x (between supports) . . . . . = R1 − wxV x1 (for overhang) . . . . . . . = w(a − x1)

M 1

at x = l2

1 − a2

l2 . . . . . =

w

8 l2 (l + a )2(l − a )2

M 2 (at R2) . . . . . . . . . . . . =wa 2

2

M x (between supports) . . . . . =wx2 l

(l2 − a 2 − xl)

M x1 (for overhang) . . . . . . . =w2

(a − x1)2

∆ x (between supports) . . . . . =wx

24 EIl (l4 − 2 l2 x2 + lx3 − 2 a 2l2 + 2 a 2 x2)

∆ x1 (for overhang) . . . . . . . = wx1

24 EI (4a 2l − l3 + 6 a 2 x1 − 4 ax 12 + x13)

25. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD ONOVERHANG

R1 = V 1 . . . . . . . . . . . . . . =wa 2

2 l

R2 V 1+ V 2 . . . . . . . . . . . . =wa2 l

(2l + a )

V 2 . . . . . . . . . . . . . . . = wa

V x1 (for overhang) . . . . . . . = w(a − x1)

M max (at R2) . . . . . . . . . . . =wa 2

2

M x (between supports) . . . . . =wa 2 x

2l

M x1 (for overhang) . . . . . . . =w2

(a − x1)2

∆max

between supports at x = l√ 3

=

wa 2l2

18 √ 3 EI = 0.03208 wa

2l2

EI

∆max (for overhang at x1 = a ) . . . = wa3

24 EI (4 l + 3 a )

∆ x (between supports) . . . . . =wa 2 x12 EIl

(l2 − x2)

∆ x1 (for overhang) . . . . . . . =wx 1

24 EI (4a 2l + 6 a 2 x1 − 4 ax 12 + x13)

Shear

l

x

R

2

w( +a)

Moment

l

a x 1

R 1 2

2 1 –

1 –

( )

( )

l

l

l

l

a

a

2

2

2

2

M

M

1

2

3

1V V

V

Shear

l

x wa

R

max

Moment

a x 1

R 1 2

2

1V

V

M




13/20



26. BEAM OVERHANGING ONE SUPPORT—CONCENTRATED LOAD AT END OF OVERHANG

R1 = V 1 . . . . . . . . . . . . . . . . . . . . . =Pal

R2 = V 1+ V 2 . . . . . . . . . . . . . . . . . . . =Pl

(l + a )V 2 . . . . . . . . . . . . . . . . . . . . . = P

M max (at R2) . . . . . . . . . . . . . . . . . = Pa

M x (between supports) . . . . . . . . . . =Pax

l M x1 (for overhang) . . . . . . . . . . . . . = P (a − x1)

∆max

between supports at x = l√ 3

. . . . . =

Pal 2

9√ 3 EI = .06415 Pal

2

EI

∆max (for overhang at x1 = a ) . . . . . . . . =Pa 2

3 EI (l + a )∆ x (between supports) . . . . . . . . . . =

Pax6 EIl

(l2 − x2)

∆ x1 (for overhang) . . . . . . . . . . . . . =Px 16 EI

(2al + 3 ax 1 − x12)

27. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOADBETWEEN SUPPORTS

Total Equiv. Uniform Load . . . . . . . . . . = wl

R = V . . . . . . . . . . . . . . . . . . . . . = wl2

V x . . . . . . . . . . . . . . . . . . . . . = w

l2

− x

M max (at center) . . . . . . . . . . . . . . . =wl 2

8

M x . . . . . . . . . . . . . . . . . . . . . =wx2

(l − x)

∆max (at center) . . . . . . . . . . . . . . . =5wl 4

384 EI

∆ x . . . . . . . . . . . . . . . . . . . . . =wx

24 EI (l2 − 2 lx2 + x3)

∆ x1 . . . . . . . . . . . . . . . . . . . . . =wl 3 x124 EI

28. BEAM OVERHANGING ONE SUPPORT—CONCENTRATED LOAD AT ANY POINT

BETWEEN SUPPORTSTotal Equiv. Uniform Load . . . . . . . . . . =

8Pab

l2

R1 = V 1 (max. when a < b) . . . . . . . . . . . =Pbl

R2 = V 2 (max. when a > b) . . . . . . . . . . . =Pal

M max (at point of load) . . . . . . . . . . . . =Pab

l

M x (when x < a ) . . . . . . . . . . . . . . =Pbx

l

∆max at x = √ a (a + 2 b)3 when a > b

. . . =

Pab (a + 2 b)√ 3a (a + 2 b)27 EIl

∆a (at point of load) . . . . . . . . . . . . =Pa 2b2

3 EIl

∆ x (when x < a ) . . . . . . . . . . . . . . =Pbx6 EIl

(l2 − b2 − x2)

∆ x (when x > a ) . . . . . . . . . . . . . . =Pa (l − x)

6 EIl (2 lx − x2 − a 2)

∆ x1 . . . . . . . . . . . . . . . . . . . . . =Pabx 16 EIl

(l + a )

Shear

l

x

R

max Moment

a x 1

R 1 2

2

1V

M

V

P

Shear

l

x

R

Moment

a

w x 1

R

V

V

l

l l

2 2

M max

Shear

l

x

R

Moment

x 1

R

V

V

2 1

M max

P

b a

1

2




14/20



29. CONTINUOUS BEAM—TWO EQUAL SPANS—UNIFORM LOAD ON ONE SPAN

Total Equiv. Uniform Load =4964

wl

R1 = V 1 . . . . . . . . . . . =7

16 wl

R2 = V 2+ V 3 . . . . . . . . . =58

wl

R3 = V 3 . . . . . . . . . . . = − 1

16 wl

V 2 . . . . . . . . . . . . =9

16 wl

M max

at x = 716

l . . . . . =

49512

wl 2

M 1 (at support R2) . . . . =1

16 wl 2

M x (when x < l) . . . . . =wx16

(7 l − 8 x)

∆max (at 0.472 l from R1) . . = .0092 wl 4 / EI

30. CONTINUOUS BEAM—TWO EQUAL SPANS—CONCENTRATED LOAD AT CENTEROF ONE SPAN

Total Equiv. Uniform Load =138 P

R1 = V 1 . . . . . . . . . . . =1332

P

R2 = V 2+ V 3 . . . . . . . . . =1116

P

R3 = V 3 . . . . . . . . . . . = − 3

32 P

V 2 . . . . . . . . . . . . =1932

P

M max (at point of load) . . . =13

64

Pl

M 1 (at support R2) . . . . =3

32 Pl

∆max (at 0.480 l from R1) . . = .015 Pl 3 / EI

31. CONTINUOUS BEAM—TWO EQUAL SPANS—CONCENTRATED LOAD AT ANY POINT

R1 = V 1 . . . . . . . . . . . =Pb

4l3 (4 l2 − a (l + a ))

R2 = V 2+ V 3 . . . . . . . . . =Pa

2l3 (2 l2 + b(l + a ))

R3 = V 3 . . . . . . . . . . . = − Pab

4 l3 (l + a )

V 2 . . . . . . . . . . . . =Pa

4l3 (4 l2 + b(l + a ))

M max (at point of load) . . . =Pab

4 l3 (4 l2 − a (l + a ))

M 1 (at support R2) . . . . =Pab

4 l2 (l + a )

x w l

R R R 1 2 3

l l

1

2

3 V

7 16

l Shear

Moment

V

V

M

M 1

max

R R R 1 2 3

l l

1

2

3 V

Shear

Moment

V

V

M

M 1

max

P l l

2 2

R R R 1 2 3

l l

1

2

3

V

Shear

Moment

V

V

M

M 1

max

P a b




15/20



32. BEAM—UNIFORMLY DISTRIBUTED LOAD AND VARIABLE END MOMENTS

R1 = V 1 . . . . . . . . . . . =wl2

+ M 1 − M 2

l

R2 = V 2 . . . . . . . . . . . =wl2

− M 1 − M 2

l

V x . . . . . . . . . . . . . = w l2

− x

+ M 1 − M 2

l

M 3 at x =

l2

+ M 1 − M 2

wl

. . =

wl 2

8 −

M 1 + M 22

+ ( M 1 − M 2)2

2wl 2

M x . . . . . . . . . . . . . =wx2

(l − x) +

M 1 − M 2l

x − M 1

b (to locate inflection points) = √ l24 −

M 1 + M 2w

+

M 1 − M 2wl

2

∆ x =wx

24 EI x3 −

2 l + 4 M 1wl

− 4 M 2wl

x2 + 12 M 1

w x + l2 −

8 M 1l

w −

4 M 2l

w

33. BEAM—CONCENTRATED LOAD AT CENTER AND VARIABLE END MOMENTS

R1 = V 1 . . . . . . . . . . . =P2

+ M 1 − M 2

l

R2 = V 2 . . . . . . . . . . . =P2

− M 1 − M 2

l

M 3 (at center) . . . . . . . . =Pl4

− M 1 + M 2

2

M x

when x < l2

. . . . . . =

P2

+ M 1 − M 2

l

x − M 1

M x

when x > l2

. . . . . . =

P2

(l − x) + ( M 1 − M 2) x

l − M 1

∆ x

when x < l2

= Px48 EI

3 l2 − 4 x2 − 8(l − x)Pl

[ M 1(2 l − x) + M 2(l + x)]

Shear

l

x w

R

Moment

R

V

V

2 1

M 1

1

2

l M M 1 2

M >M 1 2

M 3

M 2

b b

Shear

l

x

R

Moment

R

V

V

2 1

M 1

1

2

M M 1 2

M >M 1 2

M 3

M 2

P

l l

2 2




16/20



34. CONTINUOUS BEAM—THREE EQUAL SPANS—ONE END SPAN UNLOADED

35. CONTINUOUS BEAM—THREE EQUAL SPANS—END SPANS LOADED

36. CONTINUOUS BEAM—THREE EQUAL SPANS—ALL SPANS LOADED

w l w l

A B C D l l l

R = 0.383 A l w R = 1.20 B w l R = 0.450 C w l

R = –0.033 D w l

Shear

Moment

0.383 w l l w 0.583 l w 0.033

l w 0.617 l w 0.417

l w 0.033

0.583 0.383 l l

+0.0735 w l 2

–0.1167 2 w l

+0.0534 2 w l –0.0333 2 w l

(0.430 from A) = 0.0059 w / El l l 4 max ∆

w l w l

A B C D l l l

R = 0.450 A l w R = 0.550 B w l R = 0.550 C w l R = 0.450 D w l

Shear

Moment

0.450 w l l w 0.550

l w 0.550

l w 0.450

0.450 l

+0.1013 w l 2

–0.050 2 w l

+0.1013 2 w l

(0.479 from A or D) = 0.0099 w / El l l 4

0.450 l

max ∆

w l w l

A B C D l l l

R = 0.400 A l w R = 1.10 B w l R = 1.10 C

w l R = 0.400 D w l

Shear

Moment

0.400 w l l w 0.600

l w 0.600

l w 0.400

0.400 l

+0.080 w l 2 +0.025 2 w l +0.080 2 w l

(0.446 from A or D) = 0.0069 w / El l l 4

0.400 l

l w

0.500 l w

0.500 l w

–0.100 l w 2 –0.100 l w 2

0.500 l 0.500 l

max ∆




17/20



37. CONTINUOUS BEAM—FOUR EQUAL SPANS—THIRD SPAN UNLOADED

38. CONTINUOUS BEAM—FOUR EQUAL SPANS—LOAD FIRST AND THIRD SPANS

39. CONTINUOUS BEAM—FOUR EQUAL SPANS—ALL SPANS LOADED

w l

A B C E l l l

R = 0.380 A l w R = 1.223 B w l R = 0.357 C

w l R = 0.442 E w

l

Shear

Moment

0.380 w l

l w 0.620

l w 0.442

0.380 l

+0.072 w l 2 +0.0611 2 w l +0.0977

2 w l

(0.475 from E) = 0.0094 w / El l l 4 0.442 l

l w

0.603 l w

0.397 l w

–0.1205 l w 2 –0.0179 l w 2

0.603 l

D

l w

l

R = 0.598 D w l

0.558 l w

0.040 l w

–0.058 2 w l

max ∆

w l

A B C E l l l

R = 0.446 A l w R = 0.572 B w l R = 0.464

C w l

R = –0.054 E w l

Shear

Moment

0.446 w l

l w 0.554

l w 0.054

0.446 l

+0.0996 w l 2 +0.0805 2 w l

(0.477 from A) = 0.0097 w / El l l 4

0.518 l

0.018 l w 0.482 l w

–0.0536 l w 2 –0.0357 l w 2

D

l w

l

R = 0.572 D w l

0.054 l w

0.518 l w

–0.0536 2 w l

max ∆

w l w l

A B C E l l l

R = 0.393 A l w R = 1.143 B w l R = 0.928 C

w l R = 0.393 E w l

Shear

Moment

0.393 w l l w 0.464

l w 0.607

l w 0.393

0.393 l

+0.0772 w l 2 +0.0364 2 w l +0.0772 2 w l

(0.440 from A and D) = 0.0065 w / El l l 4 0.393 l

l w

0.536 l w

0.464 l w

–0.1071 l w 2 –0.0714 l w 2

0.536 l 0.536 l

D

l w

l

R = 1.143 D w l

0.607 l w

0.536 l w

+0.0364 w l 2

–0.1071 2 w l

max ∆




18/20

M max



40. SIMPLE BEAM—ONE CONCENTRATED MOVING LOAD

R1 max = V 1 max (at x = 0 ) . . . . . . . . . . . = P

M max

at point of load, when x = 12

. . . . . =

Pl4

41. SIMPLE BEAM—TWO EQUAL CONCENTRATED MOVING LOADS

R1 max = V 1 max (at x = 0) . . . . . . . . . . . = P

2 − al

when a (2 − √ 2 )l . . . . . = .586 l

with one load at center of span =Pl4

(Case 40)

42. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED MOVING LOADS

R1 max = V 1 max (at x = 0 ) . . . . . . . . . . . = P1 + P2 l − a

l

under P1, at x = 12

l − P 2a

P1+ P2

= (P1 + P2) x2

l

M max may occur with larger

load at center of span and other

load off span (Case 40) . . . =P1 l

4

GENERAL RULES FOR SIMPLE BEAMS CARRYING MOVING CONCENTRATED LOADS

The maximum shear due to moving concentrated loads occurs atone support when one of the loads is at that support. With severalmoving loads, the location that will produce maximum shear must bedetermined by trial.

The maximum bending moment produced by moving concentratedloads occurs under one of the loads when that load is as far from onesupport as the center of gravity of all the moving loads on the beam isfrom the other support.

In the accompanying diagram, the maximum bending momentoccurs under load P1 when x = b . It should also be noted that thiscondition occurs when the centerline of the span is midway betweenthe center of gravity of loads and the nearest concentrated load.

M max

l

1

x

2

P

R R

l

1

x a

2

P R R

P

1 2

l

1

x a

2

P P R R

1 2

P > P 1 2

l

1 2

P R R

a

P 1 2

Moment

M

l

2

x b

C.G.




19/20

BEAM DIAGRAMS AND FORMULASDesign properties of cantilevered beams

Equal loads, equally spaced

No. Spans System

2

3

4

5

≥6(even)

≥7(odd)

n ∞∞ 2 3 4 5

Typical SpanLoading

M 1M 2M 3

M 4M 5

0.086PL0.096PL0.063PL

0.039PL0.051PL

0.167PL0.188PL0.125PL

0.083PL0.104PL

0.250PL0.278PL0.167PL

0.083PL0.139PL

0.333PL0.375PL0.250PL

0.167PL0.208PL

0.429PL0.480PL0.300PL

0.171PL0.249PL

ABCDEFGH

0.414P1.172P0.438P1.063P1.086P1.109P0.977P1.000P

0.833P2.333P0.875P2.125P2.167P2.208P1.958P2.000P

1.250P3.500P1.333P3.167P3.250P3.333P2.917P3.000P

1.667P4.667P1.750P4.250P4.333P4.417P3.917P4.000P

2.071P5.857P2.200P5.300P5.429P5.557P4.871P5.000P

abcdef

0.172L0.125L0.220L0.204L0.157L0.147L

0.250L0.200L0.333L0.308L0.273L0.250L

0.200L0.143L0.250L0.231L0.182L0.167L

0.182L0.143L0.222L0.211L0.176L0.167L

0.176L0.130L0.229L0.203L0.160L0.150L

M o m e n t s

R e a c t i o n s

C a n t i l e v e r

D i m e n s i o n s

1M 1M M 1

A B

a

A

M 3 3 M

1M M 1

2 M 3 M 2 M

1M M 4 1M

C

A

D

E

D

E

C

A

b b

c c

1M M 3 M 3 1M M 5 3

M 2

M

A F G D C

d e b

M 3 3 M 3 M 3 M

1M M 3 M 3 M 1

2 M 3 M 3 M 3 M 2 M

1M M 5 3 M

5 M M 1

C

A

D

F

H

G

H

G

D

F

C

A

b f bf

d e e d

f f

1M M 3 M 3 M 3 M 3 1M M 5 3 M 3

M M 3 2 M

A F G H H D C

d e b

M 3 3 M 3 M 3 M 3 M 3 M

1M M 3 M 3 M 3 M 3 M 1

2 M 3 M 3 M 3 M 3 M 3 M 2 M

1M M 5 3 M

3 M M 3 5 M 1M

C

A

D

F

H

G

H

H

H

H

H

G

D

F

C

A

b f

f f

f f bf

d e e d

P 2 P

2 P

P 2 P

2 P

P P 2 P

2 P

P P P 2 P

2 P

P P P P




20/20

BEAM DIAGRAMS AND FORMULASCONTINUOUS BEAMS

MOMENT AND SHEAR COEFFICIENTSEQUAL SPANS, EQUALLY LOADED

MOMENTin terms of w l2

UNIFORM LOAD SHEARin terms of w l

MOMENTin terms of P l CONCENTRATED LOADSat center SHEARin terms of P

MOMENT

in terms of P l

CONCENTRATED LOADS

at1

⁄ 3 points

SHEAR

in terms of P

MOMENTin terms of P l

CONCENTRATED LOADSat 1 ⁄ 4 points

SHEARin terms of P

+.07 –.125

+.07

+.08 –.10

+.025 –.10

+.077 –.107

+.036 –.071

+.036 –.107

+.078 –.105 –.073 –.073 –.105

+.078 –.106 –.077 –.086 –.077 –.106

+.078 –.106 –.077 –.085 –.085 –.077 –.106

+.08

+.077

+.078

+.078

+.078

142

0 36

142

86 75

142

67 70

142

72 71

142

71 72

142

70 67

142

75 86

142

36 0

51

104

63

104 104

0 41 55 43

104

53 53

104

51 49 41 63 55

104

0

104

23

38

0 15 38 38

23 20 19 38

19 18

38

18 20

38

15 0

15

28

0

28

11 17 28

13 13

28

15 17

28

0 11

10 10

0 4 5 5 10

6 10

6 4 0

0 3 5 5 3 0 8 8 8

P P

P P P

P P P P P

.31 .69 .69 .31

.35 .65 .50 .50 .65 .35

.34 .66 .54 .46 .50 .50 .46 .54 .66 .34

+.156 +.156 +.157

+.178 –.15

+.10 –.15

+.175

+.171 –.138

+.11 –.119

+.13 –.119

+.11 –.158

+.171

P P

P P P

P P P P P

.67 1 .33 1.33 .67

.73 1.27 1.0 1.0 1.27 .73

.72 1 .28 1.07 .93 1.0 1.0 .93 1 .07 1.28 .72

P P

P P P

P P P P P

+.222 +.111 +.111 +.222 –.333

+.244 +.156 –.267

+.066 +.066 –.267

+.156 +.244

+.24 +.146 –.281

+.076 +.099 –.211

+.122 +.122 +.24 +.146 –.281

+.076 +.099 –.211

P P

P P P

P P P P P

1.03 1. 97 1.97 1.03

1.13 1.87 1.50 1.50 1.87 1.13

1.11 1.89 1 .60 1.40 1 .50 1.50 1 .40 1.60 1.89 1.11

P P

P P P

P P P P P

P P

P P P

P P P P P

+ .2 58 + .0 22 +.267 +.267

+ .0 22 + .2 58 -.465

+.282 +.314

+.097 -.372

+.003 +.128

+.003 -.372

+.097 +.314

+.282

+.277 +.303

+.079 -.394

+.006 +.155

+.054 -.296

+.079 +.204

+.079 -.296

+.054 +.155

+.006 -.394

+.079 +.303

+.277


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