Question

3 (75 mm) -3" 75" (175 mm) (125 mm) 150 mm 75 mm (100 mm) 1100 mm) -22" (ww 095) - 16" wwood) - 18- (450 mm) -12 - (300 mm) 4 #8 4 x 25 mm) 5# 8 (5 x 25 mm) -12" (300mm (50 mm) 2" (50 mm) (b) 12" (300 mm) 3.9 Determine the design moment strength of the sections shown in Fig. 3.41. Neglect the lack of symmetry in (b). Given: f = 4 ksi (30 MPa) and f = 60ksi (400 MPa).

Answer 1

Solution:

FOR FIGURE (A)

For # 8 bar, diameter = 1 inch

so, area of 4 # 8 bars = 4 * ($\dpi{100} \pi$/4) * 1² = 3.14 in²

3" 6" 3" E = 0.003 - - - - --- 4 + ((a-472) 4# 8 TL & = 0.03(d-c)/C T=ASY -17-

Now consider strain diagram as shown abovw,

By rule of similar triangle,

$\dpi{100} \frac{0.003}{C} = \frac{\varepsilon _{t}}{(d - C)}$

$\dpi{100} \therefore$  $\dpi{100} \epsilon _{t}= \frac{0.03 * (d - C)}{C}$

Let 'a' is the depth of compression zone.

Let C'₁ is compression force acting at a distance = 4 / 2 = 2" from extreme top.

Let C''₁ is compression force acting at a distance = 4 + ((a-4)/2) from extreme top.

Let T is the total tensile force acting at the depth equal to 20" from top = As * fy = 3.14 * 60 = 188.4 Kilo pounds.

Now this total T has to be resisted by total compression force ( C'₁ + C''₁ ) so as to have equilibrium

hence T = C'₁ + C''₁

240 = 0.85 * f'_c * [ 2* (3 * 4 ) + 12* (a -4) ]

240 = 0.85 * 4 * [ 2* (3 * 4 ) + 12* (a -4) ]

a= 7.88 inch

Now to calculate C, C = a / 0.85 = 9.27 inch

Hence, C'₁ = 0.85 * 4 * ( 2* (3 * 4 )) = 81.6 Kilo pound

C''₁ = 0.85 * 4 * 12* (a -4)

= 0.85 * 4 * 12* (7.88 -4) = 158.3 Kilo pound

Now taking moment of the above two forces about the tensile force, we get

M_st =( C'₁ * (d-2)) + ( C''₁ * (d-(2+a/2)))

M_st =( 81.6 * (20-2)) + ( 158.3 * (20-(2+7.88/2)))

M_st = 1468.8 + 2225.7 = 3694.5 kilo pounds inch

Now, $\dpi{100} \epsilon _{t}= \frac{0.03 * (d - C)}{C}$

$\dpi{100} \epsilon _{t}=$ 0.03472 which falls in transition zone as 0.002 < $\dpi{100} \epsilon _{t}$ < 0.005

Hence $\dpi{100} \phi$ = 0.9

Design moment = $\dpi{100} \phi$ *  M_st = 0.9 * 3694.5 = 3325 kilo pounds inch