Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 1

REINFORCED CROSS SECTION PROPERTIES

REINFORCING PLATE AT TENSION FLANGE

Fig. 1. Beam with a reinforcing plate at the tension flange.

2 2y p

p

scyb

I d tI d

yI

(1)

2

12

y tw

y

y p

I hC

I

I I

(2)

2 2

2f f f p p p f

tf f p p

d t b t d t b t th

b t b t

(3)

12 1

3

ft

w c f

f f

br

t y t

b t

(4)

where Cw = warping constant, in.6

Ip = strong-axis moment of inertia of the plate, in.4 Iy = moment of inertia of the W shape about the y-axis, in.4

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 2

Iyb = moment of inertia of the built-up sshape about the y-axis, in.4 = Iy + Ip bf = flange width of W shape, in. bp = width of plate, in. d = depth of W shape, in. ht = distance between centroid of compression flange and centroid of combined plate and tension

flange, in. rt = effective radius of gyration for lateral-torsional buckling, in. tf = thickness of flange for W shape, in. tp = thickness of plate, in. tw = thickness of web for W shape, in.

y0 = distance from the shear center to the centroid of the built-up shape, in. yc = distance from the face of the compression flange to the centroid of the built-up shape, in. ysc = distance from the face of the compression flange to the shear center of the built-up shape, in.

= yc + y0 yt = distance from the face of the tension flange to the centroid of the built-up shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 3

REINFORCING PLATE AT EACH FLANGE

Fig. 2. Beam with reinforcing plates at both flanges.

22 2pc pt

y pc pc pt pc

scyb

t tI t d I I d t

yI

(5)

2 2 2 2

2 2 2 2 2y f f pc pt

w sc pc sc pc pc sc pt pc sc

I t t t tC y t d y t I y I d t y

(6)

3 3

12 3

f f pc pct

f f pc pc w c f

t b t br

b t b t t y t

(7)

where

Ipc = strong-axis moment of inertia of compression-flange plate, in.4

Ipt = strong-axis moment of inertia of tension-flange plate, in.4 Iyb = moment of inertia of the built-up shape about the y-axis, in.4 = Iy + Ipt + Ipb bpc = width of compression-flange plate, in. bpt = width of tension-flange plate, in. tpc = thickness of compression-flange plate, in. tpt = thickness of tension-flange plate, in. yc = distance from the face of the compression flange to the centroid of the built-up shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 4

ysc = distance from the outer face of the compression flange plate to the shear center of the built-up shape, in.

= tpc + yc + y0 yt = distance from the face of the tension flange to the centroid of the built-up shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 5

TEE REINFORCEMENT AT TENSION FLANGE

Fig. 3. Beam with tee reinforcement at the tension flange.

2 2y ft

yt t

scyb

I d tI d d

yI

(8)

2 2 2

2 2 2 2y f f ft

w sc sc yt t sc

I t t tC y d y I d d y

(9)

12 1

3

ft

w c f

f f

br

t y t

b t

(10)

where

Iyb = moment of inertia of the built-up shape about the y-axis, in.4 = Iy + Iyt Iyt = moment of inertia of the Tee shape about the y-axis, in.4 bft = width of flange for Tee shape, in. dt = depth of Tee shape, in. tft = thickness of flange for Tee shape, in. twt = thickness of web for Tee shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 6

yc = distance from the face of the compression flange to the centroid of the built-up shape, in. ysc = distance from the face of the compression flange to the shear center of the built-up shape, in.

= yc + y0 yt = distance from the face of the W-shape tension flange to the centroid of the built-up shape, in. ytt = distance from the face of the Tee-shape tension flange to the centroid of the built-up shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 7

WIDE FLANGE REINFORCEMENT AT TENSION FLANGE

Fig. 4. Beam with wide flange reinforcement at the tension flange.

2 2y r

yr

scyb

I d dI d

yI

(11)

2 2 2 2

2 2 2 2 2 2y f f yr fr fr

w sc sc sc r sc

I t t I t tC y d y d y d d y

(12)

12 1

3

ft

w c f

f f

br

t y t

b t

(13)

where

Iyb = moment of inertia of the built-up shape about the y-axis, in.4 = Iy + Iyr Iyr = moment of inertia of the reinforcing W shape about the y-axis, in.4 bfr = width of flange for the reinforcing W shape, in. dr = depth of reinforcing W shape, in. tfr = thickness of flange for reinforcing W shape, in. twr = thickness of web for reinforcing W shape, in. yc = distance from the face of the compression flange to the centroid of the built-up shape, in.

Reinforced Member Properties Bo Dowswell, P.E., Ph.D. Page 8

ysc = distance from the face of the compression flange to the shear center of the built-up shape, in. = yc + y0

yt = distance from the face of the reinforced W-shape tension flange to the centroid of the built-up shape, in.

ytr = distance from the face of the reinforcing W-shape tension flange to the centroid of the built-up shape, in.