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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.)
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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.
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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
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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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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4 - 194 BEAM AND GIRDER DESIGN
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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 ∆
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BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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 ∆
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M max
BEAM DIAGRAMS AND FORMULASFor Various Static Loading Conditions
For meaning of symbols, see page 4-187
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.
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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
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
BEAM DIAGRAMS AND FORMULAS 4 - 205
-
8/18/2019 Tablas Aisc Guillermo
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
4 - 206 BEAM AND GIRDER DESIGN