REINFORCED CROSS SECTION PROPERTIES REINFORCING PLATE AT TENSION FLANGE

Fig. 1. Beam with a reinforcing plate at the tension flange. I yd tp    Ip d   2 2  y sc  I yb

Cw  Iy ht 

(1)

I y ht2 Iy 1 2 Ip

(2)

 d  t f 2  b f t f   d  t p 2  bpt p  t f b f t f  bpt p

rt 

2

(3)

bf



 t w yc  t f 12 1  3b f t f 

 

(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 1

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 2

REINFORCING PLATE AT EACH FLANGE

Fig. 2. Beam with reinforcing plates at both flanges. t pt   t pc   I y t pc  d 2  I pc   I pt  d  t pc   2  2    ysc  I yb



Iy Cw  2



2 2 2 2  tf   tf   t pc  t pt     y sc  t pc     d  y sc  t pc     I pc  y sc   I pt  d  t pc  ysc  2  2  2  2     

3

rt 

(5)

(6)

3

t f b f  t pc b pc



12 b f t f  b pc t pc  t w yc  t f



3

(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 3

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 4

TEE REINFORCEMENT AT TENSION FLANGE

Fig. 3. Beam with tee reinforcement at the tension flange. I yd t ft    I yt  d  d t   2 2   y sc  I yb Iy Cw  2

2 2 2  tf   tf   t ft    ysc     d  ysc     I yt  d  d t  ysc   2  2  2    

rt 

(8)

(9)

bf



 t w yc  t f 12 1  3b f t f 

 

(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 5

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 6

WIDE FLANGE REINFORCEMENT AT TENSION FLANGE

Fig. 4. Beam with wide flange reinforcement at the tension flange. I yd d    I yr  d  r  2 2   y sc  I yb Iy Cw  2

2 2  tf   t f   I yr  ysc     d  y sc     2  2  2  

rt 

2 2  t fr   t fr    d  y sc     d  d r  y sc    2   2    

(11)

(12)

bf



 t w yc  t f 12 1  3b f t f 

 

(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 7

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.

Reinforced Member Properties

Bo Dowswell, P.E., Ph.D.

Page 8