Diagramas de momentos lrfd

BEAM DIAGRAMS AND FORMULAS

NomenclatureE = modulus of elasticity of steel at 29,000 ksiI = moment of inertia of beam (in.4)L = total length of beam between reaction points (ft)Mmax = maximum moment (kip-in.)M1 = maximum moment in left section of beam (kip-in.)M2 = maximum moment in right section of beam (kip-in.)M3 = maximum positive moment in beam with combined end moment conditions

(kip-in.)Mx = 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)V1 = maximum vertical shear in left section of beam (kips)V2 = vertical shear at right reaction point, or to left of intermediate reaction point

of beam (kips)V3 = vertical shear at right reaction point, or to right of intermediate reaction point

of beam (kips)Vx = 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 ofdifferent 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 reactionpoint (in.)

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BEAM DIAGRAMS AND FORMULAS 4 - 187

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.

BEAMSFlexural 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-A

b = 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:

EId2y dx2 = M

x and y are abscissa and ordinate respectively of a point on the neutral axis, 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:

Mal1I1

+ 2Mb l1I1

+ l2I2

+ Mc

l2I2

= − 1⁄4 w1l1

3

I1 +

w2l23

I2

Concentrated loads:

Mal1I1

+ 2Mb l1I1

+ l2I2

+ Mc

l2I2

= − P1a1b1

I1 1 +

a1

l1

−

p2a2b2

I2 1 +

b2

I2

Considering any two consecutive spans in any continuous structure:Ma, Mb, Mc = 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 pairI1 and I2 = 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, respectivelya1 and a2 = distance of concentrated loads from left support, in left and right spans,

respectivelyb1 and b2 = distance of concentrated loads from right support, in left and right spans,

respectivelyThe 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

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 0.125c 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 0.008f 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 0.200b — 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): cPFixed end reaction (kips): dPMaximum deflection (in): ePl3 / EI

Equivalent simple span uniform load (kips): f PDeflection coefficient for equivalent simple span uniform load: gNumber of equal load spaces: nSpan of beam (ft): LSpan of beam (in): l

P

P

PP

P PP

P PPP



BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions

For meaning of symbols, see page 4-187

1. SIMPLE BEAM—UNIFORMLY DISTRIBUTED LOAD

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

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

Vx . . . . . . . . . . . . . . . . . = w

l2

− x

Mmax (at center) . . . . . . . . . . . . = wl2

8

Mx . . . . . . . . . . . . . . . . . = wx2

(l − x)

∆max (at center) . . . . . . . . . . . . = 5wl4

384EI

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

24EI (l2 − 2lx2 + x3)

2. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO ONE END

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

9√3 = 1.0264W

R1 = V1 . . . . . . . . . . . . . . . . . = W3

R2 = V2 max . . . . . . . . . . . . . . . = 2W3

Vx . . . . . . . . . . . . . . . . . = W3

− Wx2

l2

Mmax (at x = l

√3 = .5774l) . . . . . . . =

2Wl

9√3 = .1283Wl

Mx . . . . . . . . . . . . . . . . . = Wx

3l2 (l2 − x2)

∆max (at x = l√1 − √815

= .5193l) . . = 0.1304Wl3

EI

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

180EIl2(3x4 − 10l2x2 + 7l4)

3. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO CENTER

Total Equiv. Uniform Load . . . . . . = 4W3

R = V . . . . . . . . . . . . . . . . . = W2

Vx (when x < l2) . . . . . . . . . . =

W

2l2 (l2 − 4x2)

Mmax (at center) . . . . . . . . . . . . = Wl6

Mx (when x < l2) . . . . . . . . . . = Wx

12

− 2x2

3l2

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

60EI

∆x (when x < l2) . . . . . . . . . . =

Wx

480EIl2 (5l2 − 4x2)2

Moment

Shear

lx l

R R

2 2l l

V

V

Mmax

w

Moment

Shear

lx W

R R

2 2l l

V

V

Mmax

Moment

Shear

lx

W

R R

l

V

V

Mmax

1 2

.5774

2

1





4. SIMPLE BEAM—UNIFORMLY LOAD PARTIALLY DISTRIBUTED

R1 = V1 (max. when a < c) . . . . . . . = wb2l

(2c + b)

R2 = V2 (max. when a > c) . . . . . . . = wb2l

(2a + b)

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

Mmaxat x = a +

R1

w

. . . . . . . . = R1 a +

R1

2w

Mx (when x < a) . . . . . . . . . = R1x

Mx (when x > a and < (a + b)) . . . = R1x − w2

(x − a)2

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

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

R1 = V1 max . . . . . . . . . . . . . . . = wa2l

(2l − a)

R2 = V2 . . . . . . . . . . . . . . . . = wa2

2l

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

Mmax

at x =

R1

w

. . . . . . . . . . = R1

2

2w

Mx (when x < a) . . . . . . . . . = R1x − wx2

2

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

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

24EIl (a2(2l − a)2 − 2ax2(2l − a) + lx3)

∆x (when x > a) . . . . . . . . . = wa2(l − x)

24EIl (4xl − 2x2 − a2)

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

R1 = V1 . . . . . . . . . . . . . . . . = w1a(2l − a) + w2c

2

2l

R2 = V2 . . . . . . . . . . . . . . . . = w2c(2l − c) + w1a

2

2l

Vx (when x < a) . . . . . . . . . = R1 − w1x

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

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

Mmaxat x =

R1

w1 when R1 < w1a

=

R12

2w1

Mmaxat x = l −

R1

w2 when R2 < w2c

= R2

2

2w2

Mx (when x < a) . . . . . . . . . = R1x − w1x

2

2

Mx (when x > a and < (a + b)) . . . = R1x − w1a

2 (2x − a)

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

2

Moment

Shear

l

xR R

VV

Mmax

a b cwb

1 2

a+ wR1

2

1

Moment

Shear

l

xR R

VV

Mmax

a

1 2

R1

2

1

wa

w

Moment

Shear

l

xR R

V

V

Mmax

a

1 2

R1

2

1

b c

1 2

1

w a w c

w





7. SIMPLE BEAM—CONCENTRATED LOAD AT CENTER

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

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

Mmax (at point of load) . . . . . . . . . . . = Pl4

Mx

when x <

12

. . . . . . . . . . . . . = Px2

∆max (at point of load) . . . . . . . . . . . = Pl3

48EI

∆x

when x <

12

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

48EI (3l2 − 4x2)

8. SIMPLE BEAM—CONCENTRATED LOAD AT ANY POINT

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

l2

R1 = V1 (max when a < b) . . . . . . . . . . . = Pbl

R2 = V2 (max when a > b) . . . . . . . . . . . = Pal

Mmax (at point of load) . . . . . . . . . . . = Pab

l

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

l

∆max

at x = √a(a + 2b)

3 when a > b

. . . =

Pab(a + 2b)√3a(a + 2b)27EIl

∆a (at point of load) . . . . . . . . . . . = Pa2b2

3EIl

∆x (when x < a) . . . . . . . . . . . . . . = Pbx6EIl

(l2 − b2 − x2)

9. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS SYMMETRICALLY PLACED

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

l

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

Mmax (between loads) . . . . . . . . . . . . = Pa

Mx (when x < a) . . . . . . . . . . . . . . = Px

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

24EI (3l2 − 4a2)

∆x (when x < a) . . . . . . . . . . . . . . = Px6EI

(3la − 3a2 − x2)

∆x (when x > a and < (l − a)) . . . . . . . = Pa6EI

(3lx − 3x2 − a2)Moment

Shear

l

x

R R

V

Mmax

P

a

V

a

P

Moment

Shear

l

x

R R

V

V

Mmax

P

a b1

1

2

2

Moment

Shear

l

x

R R

V

V

Mmax

P

l l2 2





10. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS UNSYMMETRICALLYPLACED

R1 = V1 (max. when a < b) . . . . . . . . . = Pl (l − a + b)

R2 = V2 (max. when a > b) . . . . . . . . . = Pl (l − b + a)

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

M1 (max. when a > b) . . . . . . . . . = R1a

M2 (max. when a < b) . . . . . . . . . = R2b

Mx (when x < a) . . . . . . . . . . . . = R1x

Mx (when x > a and < (l − b)) . . . . . = R1x − P(x − a)

11. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED LOADS UNSYMMETRICALLYPLACED

R1 = V1 . . . . . . . . . . . . . . . . . . = P1 (l − a) + P2 b

l

R2 = V2 . . . . . . . . . . . . . . . . . . = P1 a + P2 (l − b)

l

Vx (when x > a and < (l − b)) . . . . . = R1 − P1

M1 (max. when R1 < P1) . . . . . . . . = R1a

M2 (max. when R2 < P2) . . . . . . . . = R2b

Mx (when x < a) . . . . . . . . . . . . = R1x

Mx (when x > a and < (l − b)) . . . . . = R1x − P(x − a)

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

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

R1 = V1 . . . . . . . . . . . . . . . . . . = 3wl8

R2 = V2 max . . . . . . . . . . . . . . . . . . = 5wl8

Vx . . . . . . . . . . . . . . . . . . = R1 − wx

Mmax . . . . . . . . . . . . . . . . . . = wl2

8

Mx

at x =

38

l

. . . . . . . . . . . . . = 9

128wl2

Mx . . . . . . . . . . . . . . . . . . = R1x − wx2

2

∆max

at x =

l16

(1 + √33) = .4215l

. . . = wl4

185EI

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

48EI(l3 − 3lx + 2x3)

Moment

Shear

l

x

R R

V

M M1

P

a

V

b

P

1

1

2

2

2

Moment

Shear

l

x

R R

V

M M1

P

a

V

b

P

1

1

2

2

2

1 2

Moment

Shear

l

xR R

V

V

M

M

1

1

1

2

2

max

4

38

l

l

lw





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

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

R1 = V1 . . . . . . . . . . . . . . . . . . = 5P15

R2 = V2 max . . . . . . . . . . . . . . . . = 11P16

Mmax (at fixed end) . . . . . . . . . . . = 3Pl16

M1 (at point of load) . . . . . . . . . . = 5Pl32

Mx

when x <

l2

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

5Px16

Mx

when x >

l2

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

l2

− 11x16

∆max

at x = l√ 1

5 = .4472l

. . . . . . . =

Pl3

48EI√5 = .009317

Pl3

EI

∆x (at point of load) . . . . . . . . . . = 7PL3

768EI

∆x

when x <

l2

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

Px96EI

(3l2 − 5x2)

∆x

when x >

l2

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

P96EI

(x − l)2(11x − 2l)

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

R1 = V1 . . . . . . . . . . . . . . . . . . = Pb2

2l3 (a + 2l)

R2 = V2 . . . . . . . . . . . . . . . . . . = Pa

2l3 (3l2 − a2)

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

M2 (at fixed end) . . . . . . . . . . . = Pab

2l2 (a + l)

Mx (when x < a) . . . . . . . . . . . . = R1x

Mx (when x > a) . . . . . . . . . . . . = R1x − P(x − a)

∆max

when a < .414l at x = l

(l2 + a2)(3l2 − a2)

= Pa(l2 + a2)3

3EI(3l2 − a2)2

∆max

when a > .414l at x = l

a

2l + a

√.......... =

Pab2

6EI √a

2l + a

∆a (at point of load) . . . . . . . . . . = Pa2b3

12EIl3 (3l + a)

∆ (when x < a) . . . . . . . . . . . . = Pb2x

12EIl3 (3al2 − 2lx2 − ax2)

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

12EIl2(l − x)2 (3l2x − a3x − 2a2l)

Moment

Shear

l

x

R R

V

V

M

max

P

l l2 2

1

1

1

2

2

311 l

M

Moment

Shear

l

x

R R

V

V

M

2

P

1

1

1

2

2

PaR

M

a b

2





15. BEAM FIXED AT BOTH ENDS—UNIFORMLY DISTRIBUTED LOADS

Total Equiv. Uniform Load . . . . . . . . = 2wl3

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

Vx . . . . . . . . . . . . . . . . . . = w

l2

− x

Mmax (at ends) . . . . . . . . . . . . . = wl2

12

M1 (at center) . . . . . . . . . . . . . = wl2

24

Mx . . . . . . . . . . . . . . . . . . = w12

(6lx − l2 − 6x2)

∆max (at center) . . . . . . . . . . . . . = wl4

384EI

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

24EI (l − x)2

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

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

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

Mmax (at center and ends) . . . . . . . . = Pl8

Mx

when x <

l2

. . . . . . . . . . . = P8

(4x − l)

∆max (at center) . . . . . . . . . . . . . = Pl3

192EI

∆x

when x <

l2

. . . . . . . . . . . = Px2

48EI (3l − 4x)

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

R1 = V1 (max. when a < b) . . . . . . . . = Pb2

l3 (3a + b)

R2 = V2 (max. when a > b) . . . . . . . . = Pa2

l3 (a + 3b)

M1 (max. when a < b) . . . . . . . . = Pab2

l2

M2 (max. when a > b) . . . . . . . . = Pa2b

l2

Ma (at point of load) . . . . . . . . . = 2Pa2b2

l3

Mx (when x < a) . . . . . . . . . . . = R1x − Pab2

l2

∆max

when a > b at x =

2al

3a + b

. . . . = 2Pa3b2

3EI(3a + b)2

∆a (at point of load) . . . . . . . . . = Pa3b3

3EIl3

∆x (when x < a) . . . . . . . . . . . = Pb2x2

6EIl2 (3al − 3ax − bx)

Moment

Shear

lx

w

R R

V

V

M

M M

1

max

l

l

2 2l l

.2113

max

Moment

Shear

l

x

R R

VV

M

MM max

l4

2l l

max

P

2

max

Moment

Shear

l

x

R R

V

V

M

MM21

P

a

2

2

1

1 a b





18. CANTILEVER BEAM—LOAD INCREASING UNIFORMLY TO FIXED END

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

W

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

Vx . . . . . . . . . . . . . . . . . . . = W x2

l2

Mmax (at fixed end) . . . . . . . . . . . . = Wl3

Mx . . . . . . . . . . . . . . . . . . . = Wx3

3l2

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

15EI

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

60EIl2 (x5 − 5l4x + 4l5)

19. CANTILEVER BEAM—UNIFORMLY DISTRIBUTED LOAD

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

R = V . . . . . . . . . . . . . . . . . . . = wl

Vx . . . . . . . . . . . . . . . . . . . = wx

Mmax (at fixed end) . . . . . . . . . . . . = wl2

2

Mx . . . . . . . . . . . . . . . . . . . = wx2

2

∆max (at free end) . . . . . . . . . . . . . = wl4

8EI

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

24EI (x4 − 4l3x + 3l4)

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 . . . . . . . . . . . . . . . . . . . = wl

Vx . . . . . . . . . . . . . . . . . . . = wx

Mmax (at fixed end) . . . . . . . . . . . . = wl2

3

Mx . . . . . . . . . . . . . . . . . . . = w6

(l2 − 3x2)

∆max (at deflected end) . . . . . . . . . . = wl4

24EI

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

24EI

Shear

l

xR

V

M

M

M

w

maxMoment

l

.4227 l

1

Shear

l

xR

V

M

w

maxMoment

l

Shear

l

xR

W

V

MmaxMoment





21. CANTILEVER BEAM—CONCENTRATED LOAD AT ANY POINT

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

l

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

Mmax (at fixed end) . . . . . . . . . . . . = Pb

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

∆max (at free end) . . . . . . . . . . . . . = Pb2

6EI (3l − b)

∆a (at point of load) . . . . . . . . . . = Pb3

3EI

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

6EI (3l − 3x − b)

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

6EI (3b − l + x)

22. CANTILEVER BEAM—CONCENTRATED LOAD AT FREE END

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

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

Mmax (at fixed end) . . . . . . . . . . . . = Pl

Mx . . . . . . . . . . . . . . . . . . . = Px

∆max (at free end) . . . . . . . . . . . . . = Pl3

3EI

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

6EI (2l3 − 3l2x + 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

Mmax (at both ends) . . . . . . . . . . . . = Pl2

Mx . . . . . . . . . . . . . . . . . . . = P

l2

− x

∆max (at deflected end) . . . . . . . . . . = pl3

12EI

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

12EI (l + 2x)

Shear

l

x

R

V

MmaxMoment

P

a b

Shear

l

xR

V

Mmax

Moment

P

Shear

l

xR

V

M

M

M

max

Moment

P

max

l2





24. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD

R1 = V1 . . . . . . . . . . . . . . = w2l

(l2 − a2)

R2 = V2 + V3 . . . . . . . . . . . = w2l

(l + a)2

V2 . . . . . . . . . . . . . . . = wa

V3 . . . . . . . . . . . . . . . = w2l

(l2 + a2)

Vx (between supports) . . . . . = R1 − wx

Vx1(for overhang) . . . . . . . = w(a − x1)

M1

at x =

l2

1 −

a2

l2

. . . . . = w

8l2 (l + a)2(l − a)2

M2 (at R2) . . . . . . . . . . . . = wa2

2

Mx (between supports) . . . . . = wx2l

(l2 − a2 − xl)

Mx1(for overhang) . . . . . . . =

w2

(a − x1)2

∆x (between supports) . . . . . = wx

24EIl (l4 − 2l2x2 + lx3 − 2a2l2 + 2a2x2)

∆x1(for overhang) . . . . . . . =

wx1

24EI (4a2l − l3 + 6a2x1 − 4ax1

2 + x13)

25. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD ONOVERHANG

R1 = V1 . . . . . . . . . . . . . . = wa2

2l

R2 V1 + V2 . . . . . . . . . . . . = wa2l

(2l + a)

V2 . . . . . . . . . . . . . . . = wa

Vx1(for overhang) . . . . . . . = w(a − x1)

Mmax (at R2) . . . . . . . . . . . = wa2

2

Mx (between supports) . . . . . = wa2x

2l

Mx1(for overhang) . . . . . . . =

w2

(a − x1)2

∆max

between supports at x =

l

√3

= wa2l2

18√3EI = 0.03208

wa2l2

EI

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

24EI (4l + 3a)

∆x (between supports) . . . . . = wa2x12EIl

(l2 − x2)

∆x1(for overhang) . . . . . . . =

wx1

24EI (4a2l + 6a2x1 − 4ax1

2 + x13)

Shear

l

x

R

2

w( +a)

Moment

l

ax1

R1 22 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

maxMoment

ax1

R1 2

2

1V

V

M





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

R1 = V1 . . . . . . . . . . . . . . . . . . . . . = Pal

R2 = V1 + V2 . . . . . . . . . . . . . . . . . . . = Pl (l + a)

V2 . . . . . . . . . . . . . . . . . . . . . = PMmax (at R2) . . . . . . . . . . . . . . . . . = Pa

Mx (between supports) . . . . . . . . . . = Pax

lMx1

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

∆max

between supports at x =

l

√3

. . . . . = Pal2

9√3EI = .06415

Pal2

EI

∆max (for overhang at x1 = a) . . . . . . . . = Pa2

3EI (l + a)

∆x (between supports) . . . . . . . . . . = Pax6EIl

(l2 − x2)

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

Px1

6EI (2al + 3ax1 − x1

2)

27. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOADBETWEEN SUPPORTS

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

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

Vx . . . . . . . . . . . . . . . . . . . . . = w

l2

− x

Mmax (at center) . . . . . . . . . . . . . . . = wl2

8

Mx . . . . . . . . . . . . . . . . . . . . . = wx2

(l − x)

∆max (at center) . . . . . . . . . . . . . . . = 5wl4

384EI

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

24EI (l2 − 2lx2 + x3)

∆x1 . . . . . . . . . . . . . . . . . . . . . =

wl3x1

24EI

28. BEAM OVERHANGING ONE SUPPORT—CONCENTRATED LOAD AT ANY POINTBETWEEN SUPPORTS

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

l2

R1 = V1 (max. when a < b) . . . . . . . . . . . = Pbl

R2 = V2 (max. when a > b) . . . . . . . . . . . = Pal

Mmax (at point of load) . . . . . . . . . . . . = Pab

l

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

l

∆max

at x = √a(a + 2b)

3 when a > b

. . . =

Pab(a + 2b)√3a(a + 2b)27EIl

∆a (at point of load) . . . . . . . . . . . . = Pa2b2

3EIl

∆x (when x < a) . . . . . . . . . . . . . . = Pbx6EIl

(l2 − b2 − x2)

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

6EIl (2lx − x2 − a2)

∆x1 . . . . . . . . . . . . . . . . . . . . . =

Pabx16EIl

(l + a)

Shear

l

x

R

maxMoment

ax1

R1 2

2

1V

M

V

P

Shear

l

x

R

Moment

a

wx1

R

V

V

l

l l22

Mmax

Shear

l

x

R

Moment

x1

R

V

V

21

Mmax

P

ba

1

2





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

Total Equiv. Uniform Load = 4964

wl

R1 = V1 . . . . . . . . . . . = 716

wl

R2 = V2 + V3 . . . . . . . . . = 58

wl

R3 = V3 . . . . . . . . . . . = − 116

wl

V2 . . . . . . . . . . . . = 916

wl

Mmax

at x =

716

l

. . . . . = 49512

wl2

M1 (at support R2) . . . . = 116

wl2

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

(7l − 8x)

∆max (at 0.472l from R1) . . = .0092wl4 / EI

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

Total Equiv. Uniform Load = 138

P

R1 = V1 . . . . . . . . . . . = 1332

P

R2 = V2 + V3 . . . . . . . . . = 1116

P

R3 = V3 . . . . . . . . . . . = − 332

P

V2 . . . . . . . . . . . . = 1932

P

Mmax (at point of load) . . . = 1364

Pl

M1 (at support R2) . . . . = 332

Pl

∆max (at 0.480l from R1) . . = .015Pl3 / EI

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

R1 = V1 . . . . . . . . . . . = Pb

4l3 (4l2 − a(l + a))

R2 = V2 + V3 . . . . . . . . . = Pa

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

R3 = V3 . . . . . . . . . . . = − Pab

4l3 (l + a)

V2 . . . . . . . . . . . . = Pa

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

Mmax (at point of load) . . . = Pab

4l3 (4l2 − a(l + a))

M1 (at support R2) . . . . = Pab

4l2 (l + a)

xw l

R R R1 2 3l l

1

2

3V

716

lShear

Moment

V

V

M

M1

max

R R R1 2 3l l

1

2

3V

Shear

Moment

V

V

M

M1

max

Pl l

2 2

R R R1 2 3l l

1

2

3V

Shear

Moment

V

V

M

M1

max

Pa b





32. BEAM—UNIFORMLY DISTRIBUTED LOAD AND VARIABLE END MOMENTS

R1 = V1 . . . . . . . . . . . = wl2

+ M1 − M2

l

R2 = V2 . . . . . . . . . . . = wl2

− M1 − M2

l

Vx . . . . . . . . . . . . . = w

l2

− x +

M1 − M2

l

M3

at x =

l2

+ M1 − M2

wl

. . = wl2

8 −

M1 + M2

2 +

(M1 − M2)2

2wl2

Mx . . . . . . . . . . . . . = wx2

(l − x) + M1 − M2

l

x − M1

b (to locate inflection points) = √ l2

4 −

M1 + M2

w

+

M1 − M2

wl

2

∆x = wx

24EI x3 −

2l +

4M1

wl −

4M2

wl

x2 +

12M1

w x + l2 −

8M1l

w −

4M2l

w

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

R1 = V1 . . . . . . . . . . . = P2

+ M1 − M2

l

R2 = V2 . . . . . . . . . . . = P2

− M1 − M2

l

M3 (at center) . . . . . . . . = Pl4

− M1 + M2

2

Mx

when x <

l2

. . . . . . = P2

+ M1 − M2

l

x − M1

Mx

when x >

l2

. . . . . . = P2

(l − x) + (M1 − M2)x

l − M1

∆x

when x <

l2

=

Px48EI

3l2 − 4x2 −

8(l − x)Pl

[M1(2l − x) + M2(l + x)]

Shear

l

x w

R

Moment

R

V

V

21

M1

1

2

lM M1 2

M >M1 2

M3

M2

b b

Shear

l

x

R

Moment

R

V

V

21

M1

1

2

M M1 2

M >M1 2

M3

M2

P

l l

2 2





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

wl wl

A B C Dl l l

R = 0.383A lw R = 1.20B wl R = 0.450C w l R = –0.033D wl

Shear

Moment

0.383w llw0.583 lw0.033

lw0.617 lw0.417lw0.033

0.5830.383 ll

+0.0735 wl 2

–0.1167 2wl

+0.0534 2wl –0.0333 2wl

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

wl wl

A B C Dl l l

R = 0.450A lw R = 0.550B wl R = 0.550C wl R = 0.450D wl

Shear

Moment

0.450 wllw0.550

lw0.550lw0.450

0.450 l

+0.1013 wl 2

–0.050 2w l

+0.1013 2wl

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

0.450 l

max∆

wl wl

A B C Dl l l

R = 0.400A lw R = 1.10B wl R = 1.10C wl R = 0.400D wl

Shear

Moment

0.400 w llw0.600

lw0.600lw0.400

0.400 l

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

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

0.400 l

lw

0.500 lw

0.500 lw

–0.100 lw 2 –0.100 lw 2

0.500 l 0.500 l

max∆





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 El l l

R = 0.380A lw R = 1.223B wl R = 0.357C wl R = 0.442E wl

Shear

Moment

0.380w l

lw0.620lw0.442

0.380 l

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

(0.475 from E) = 0.0094 w / Ell l 4

0.442 l

lw

0.603 lw

0.397 lw

–0.1205 lw 2 –0.0179 lw 2

0.603 l

D

lw

l

R = 0.598D wl

0.558 lw

0.040 lw

–0.058 2wl

max∆

w l

A B C El l l

R = 0.446A lw R = 0.572B wl R = 0.464C wl R = –0.054E wl

Shear

Moment

0.446w l

lw0.554

lw0.054

0.446 l

+0.0996 wl 2 +0.0805 2wl

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

0.518 l

0.018 lw 0.482 lw

–0.0536 lw 2 –0.0357 lw 2

D

lw

l

R = 0.572D wl

0.054 lw

0.518 lw

–0.0536 2wl

max∆

w l w l

A B C El l l

R = 0.393A lw R = 1.143B wl R = 0.928C wl R = 0.393E wl

Shear

Moment

0.393 w llw0.464

lw0.607lw0.393

0.393 l

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

(0.440 from A and D) = 0.0065 w / Ell l 40.393 l

lw

0.536 lw

0.464 lw

–0.1071 lw 2 –0.0714 lw 2

0.536 l 0.536 l

D

lw

l

R = 1.143D wl

0.607 lw

0.536 lw

+0.0364 w l 2

–0.1071 2wl

max∆



Mmax



40. SIMPLE BEAM—ONE CONCENTRATED MOVING LOAD

R1 max = V1 max (at x = 0) . . . . . . . . . . . = P

Mmax at point of load, when x =

12

. . . . . =

Pl4

41. SIMPLE BEAM—TWO EQUAL CONCENTRATED MOVING LOADS

R1 max = V1 max (at x = 0) . . . . . . . . . . . = P 2 −

al

when a <, (2 − √2) l . . . . . = .586l

under load 1 at x 12

l −

a2

. . =

P2l

l −

a2

2

when a > (2 − √2)l . . . . . = .586l

with one load at center of span = Pl4

(Case 40)

42. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED MOVING LOADS

R1 max = V1 max (at x = 0) . . . . . . . . . . . = P1 + P2 l − a

l

under P1, at x = 12

l −

P2a

P1 + P2

= (P1 + P2 )x2

l

Mmax 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 at one 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.

Mmax

l

1

x

2

P

R R

l

1

x a

2

PR R

P

1 2

l

1

x a

2

P PRR

1 2

P > P1 2

l

1 2

PR R

aP1 2

Moment

M

l2

x b

C.G.



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

M1

M2

M3

M4

M5

0.086PL0.096PL0.063PL0.039PL0.051PL

0.167PL0.188PL0.125PL0.083PL0.104PL

0.250PL0.278PL0.167PL0.083PL0.139PL

0.333PL0.375PL0.250PL0.167PL0.208PL

0.429PL0.480PL0.300PL0.171PL0.249PL

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

Mom

ents

Rea

ctio

nsC

antil

ever

Dim

ensi

ons

1M

1M M1

A B

a

A

M3 3M

1M M1

2M 3M 2M

1M M4 1M

C

A

D

E

D

E

C

A

b b

c c

1M M3 M31M M5 3M 2M

A F G D C

d e b

M3 3M 3M 3M

1M M3 M3 M1

2M 3M 3M 3M 2M

1M M5 3M 5M M1

C

A

D

F

H

G

H

G

D

F

C

A

b f bf

d e e d

f f

1M M3 M3 M3 M31M M5 3M 3M M3 2M

A F G H H D C

d e b

M3 3M 3M 3M 3M 3M

1M M3 M3 M3 M3 M1

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

1M M5 3M 3M M3 5M 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 2P

2P

P 2P

2P

PP 2P

2P

PP P 2P

2P

PP P P



BEAM DIAGRAMS AND FORMULASCONTINUOUS BEAMS

MOMENT AND SHEAR COEFFICIENTSEQUAL SPANS, EQUALLY LOADED

MOMENTin terms of wl2

UNIFORM LOAD SHEARin terms of wl

MOMENTin terms of Pl

CONCENTRATED LOADSat center

SHEARin terms of P


CONCENTRATED LOADSat 1⁄3 points

SHEARin terms of P


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

1420 36

14286 75

14267 70

14272 71

14271 72

14270 67

14275 86

14236 0

51104

63104104

0 41 55 43104

53 53104

51 49 416355104

0104

2338

0 1538 38

23 20 1938

191838

18 2038

15 0

1528

028

11 1728

13 1328

15 1728

011

10100 4 55

106

106 4 0

0355308 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

PPPPP

+.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

+.258 +.022+.267 +.267

+.022 +.258-.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



FLOOR DEFLECTIONS AND VIBRATIONS

ServiceabilityServiceability checks are necessary in design to provide for the satisfactory performanceof structures. Chapter L of the LRFD Specification and Commentary contains generalguidelines on serviceability. In contrast with the factored forces used to determine therequired strength, the (unfactored) working loads are used in serviceability calculations.

The primary concern regarding the serviceability of floor beams is the prevention ofexcessive deflections and vibrations. The use of higher strength steels and compositeconstruction has resulted in shallower and lighter beams. Serviceability has become amore important consideration than in the past, as the design of more beams is governedby deflection and vibration criteria.

Deflections and CamberCriteria for acceptable vertical deflections have traditionally been set by the designengineer, based on the intended use of the given structure. What is appropriate for anoffice building, for example, may not be satisfactory for a hospital. An illustration ofdeflection criteria is the following:

1. Live load deflections shall not exceed a specified fraction of the span (e.g., 1⁄360) nor aspecific quantity (e.g., one inch). A deeper and/or heavier beam shall be selected, ifnecessary, to meet these requirements.

2. Under dead load, plus a given portion of the design live load (say, 10 psf), the floorshall be theoretically level. Where feasible and necessary, upward camber of the beamshall be specified.

Regarding camber, the engineer is cautioned that:

1. It is unrealistic to expect precision in cambering. The limits and tolerances given inPart 1 of of this Manual for cambering of rolled beams are typical for mill camber.Kloiber (1989) states that camber tolerances are dependent on the method used (hotor cold cambering) and whether done at the mill or the fabrication shop. According tothe AISC Code of Standard Practice, Section 6.4.5: “When members are specified onthe contract documents as requiring camber, the shop fabrication tolerance shall be−0 / +1⁄2 in. for members 50 ft and less in length, or −0 / + (1⁄2 in. + 1⁄8 in. for each 10 ftor fraction thereof in excess of 50 ft in length) for members over 50 ft. Membersreceived from the rolling mill with 75 percent of the specified camber require no furthercambering. For purposes of inspection, camber must be measured in the fabricator’sshop in the unstressed condition.” Some of the camber may be lost in transportationprior to placement of the beam, due to vibration.

2. There are two methods for erection of floors: uniform slab thickness and level floor.As a consequence of possible overcamber, the latter may result in a thinner concreteslab for composite action and fire protection at midspan, and may cause the shear studsto protrude above the slab.

3. Due to end restraint at the connections, actual beam deflections are often less than thecalculated values.

4. The deflections of a composite beam (under live load for shored construction, andunder dead and live loads for unshored) cannot be determined as easily and accuratelyas the deflections of a bare steel beam. Equation C-I3-6 in Section I3.2 of the


FLOOR DEFLECTIONS AND VIBRATIONS 4 - 207

Commentary on the LRFD Specification provides an approximate effective momentof inertia for partially composite beams.

5. Cambers of less than 3⁄4-in. should not be specified, and beams less than 24 ft in lengthshould not be cambered (Kloiber, 1989).

VibrationsAnnoying floor motion may be caused by the normal activities of the occupants.Remedial action is usually very difficult and expensive and not always effective. Theprevention of excessive and objectionable floor vibration should be part of the designprocess.

Several researchers have developed procedures to enable structural engineers topredict occupant acceptability of proposed floor systems. Based on field measurementof approximately 100 floor systems, Murray (1991) developed the following accept-ability criterion:

D > 35Ao f + 2.5 (4-1)

where

D = damping in percent of criticalAo = maximum initial amplitude of the floor system due to a heel-drop excitation, in.f = first natural frequency of the floor system, hz

Damping in a completed floor system can be estimated from the following ranges:

Bare Floor: 1–3 percentLower limit for thin slab of lightweight concrete; upper limit for thick slab of normalweight concrete.

Ceiling: 1–3 percentLower limit for hung ceiling; upper limit for sheetrock on furring attached to beams orjoists.

Ductwork and Mechanical: 1–10 percentDepends on amount and attachment.

Partitions: 10–20 percentIf attached to the floor system and not spaced more than every five floor beams or theeffective joist floor width.

Note: The above values are based on observation only.

Beam or girder frequency can be estimated from

f = K

gEItWL3

1⁄2

(4-2)

where

f = first natural frequency, hzK = 1.57 for simply supported beams

= 0.56 for cantilevered beams= from Figure 4-8 for overhanging beams

g = acceleration of gravity = 386 in./sec2



E = modulus of elasticity, psiIt = transformed moment of inertia of the tee-beam model, Figure 4-9, in.4 (to be

used for both composite and noncomposite construction)W= total weight supported by the tee beam, dead load plus 10–25 percent of design

live load, lbsL = tee-beam span, in.

System frequency is estimated using

1fs

2 = 1fb

2 + 1fg

2

.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40

.2

.4

.6

.8

1.0

1.2

1.4

1.6

Cantilever - Backspan Ratio, H / L

Freq

uenc

y Co

effic

ient

, K

L H

f=K gEl

WL3t

Fig. 4-8. Frequency coefficients for overhanging beams.

Actual

Beam spacing S

SlabDeck

Beam spacing S

d e

Model

Fig. 4-9. Tee-beam model for computing transformed moment of inertia.


FLOOR DEFLECTIONS AND VIBRATIONS 4 - 209

where

fs = system frequency, hzfb = beam or joist frequency, hzfg = girder frequency, hz

Amplitude from a heel-drop impact can be estimated from

Ao = Aot

Neff(4-3)

where

Ao = initial amplitude of the floor system due to a heel-drop impact, in.Neff = number of effective tee beamsAot = initial amplitude of a single tee beam due to a heel-drop impact, in.

= (DLF)maxds (4-4)

where

(DLF)max = maximum dynamic load factor, Table 4-2ds = static deflection caused by a 600 lbs force, in.

See (Murray, 1975) for equations for (DLF)max and ds

For girders, Neff = 1.0.

For beams:

1. S < 2.5ft, usual steel joist-concrete slab floor systems.

Neff = 1 + 2Σ cos

πx2xo

for x ≤ xo (4-5)

where

x = distance from the center joist to the joist under consideration, in.xo = distance from the center joist to the edge of the effective floor, in.

= 1.06εLL = joist span, in.ε = (Dx / Dy)0.25

Dx = flexural stiffness perpendicular to the joists= Ect

3 / 12Dy = flexural stiffness parallel to the joists

= EIt / SEc = modulus of elasticity of concrete, psiE = modulus of elasticity of steel, psit = slab thickness, in.It = transformed moment of inertia of the tee beam, in.4

S = joist spacing, in.

2. S > 2.5 ft, usual steel beam-concrete slab floor systems.



Neff = 2.97 − S

17.3de +

L4

135EIT(4-6)

where E is defined above and

S = beam spacing, in.de= effective slab depth, in.L = beam span, in.

Limitations:

15 ≤ (S / de) < 40; 1 × 106 ≤ (L4 / IT) ≤ 50 × 106

The amplitude of a two-way system can be estimated from

Aos = Aob + Aog / 2

where

Aos = system amplitudeAob = Aot for beamAog = Aot for girder

Additional information on building floor vibrations can be obtained from the above-referenced paper by Murray (1991) and the references cited therein.

BEAMS: OTHER SUBJECTSOther topics related to the design of flexural members covered elsewhere in this Manualinclude:

Beam Bearing Plates, in Part 11 (Volume II);Beam Web Penetrations, in Part 12 (Volume II).


BEAMS: OTHER SUBJECTS 4 - 211

Table 4-2.Dynamic Load Factors for Heel-Drop Impact

f, hz DLF F, hz DLF F, hz DLF

1.001.101.201.301.401.501.601.701.801.902.002.102.202.302.402.502.602.702.802.903.003.103.203.303.403.503.603.703.803.904.004.104.204.304.404.504.604.704.804.905.005.105.205.305.40

0.15410.16950.18470.20000.21520.23040.24560.26070.27580.29080.30580.32070.33560.35040.36510.37980.39450.40910.42360.43800.45240.46670.48090.49500.50910.52310.53690.55070.56450.57810.59160.60500.61840.63160.64480.65780.67070.68350.69620.70880.72130.73370.74590.75800.7700

5.505.605.705.805.906.006.106.206.306.406.506.606.706.806.907.007.107.207.307.407.507.607.707.807.908.008.108.208.308.408.508.608.708.808.909.009.109.209.309.409.509.609.709.809.90

0.78190.79370.80530.81680.82820.83940.85050.86150.87230.88300.89360.90400.91430.92440.93440.94430.95400.96350.97290.98210.99121.00021.00901.01761.02611.03451.04281.05091.05881.06671.07441.08201.08951.09691.10411.11131.11831.12521.13211.13881.14341.15191.15831.16471.1709

10.0010.1010.2010.3010.4010.5010.6010.7010.8010.9011.0011.1011.2011.3011.4011.5011.6011.7011.8011.9012.0012.1012.2012.3012.4012.5012.6012.7012.8012.9013.0013.1013.2013.3013.4013.5013.6013.7013.8013.9014.0014.1014.2014.3014.40

1.17701.18311.18911.19491.20071.20651.21211.21771.22311.22851.23391.23911.24431.24941.25451.25941.26431.26921.27401.27871.28341.28791.29251.29701.30141.30581.31011.31431.31851.32271.32681.33081.33481.33881.34271.34661.35041.35411.35791.36151.36521.36881.37231.37581.3793



Diagramas de momentos lrfd

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