ARCH 264/564 Structural Elements

Prelim #2 and solutions

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March 11, 2008
Show all calculations and write all answers in the exam booklet.

1. Write the following number on the front cover of your exam booklet, top-left corner, above your name:
2. [10 points] Answer the following questions:

  1. The Euler critical buckling stress curve is used for steel columns that buckle in the inelastic range (true or false).

    False
    Euler buckling

  2. Unlike steel columns, which are assumed to buckle in two ways (either elastically or inelastically), wood columns traditionally were divided into three buckling modes (true or false).

    True: Short, intermediate, and long.

  3. Which of the following reinforcement ratio corresponds to the smallest reinforced concrete column, assuming that material properties and applied loads are the same in all cases?
    a) 0.01; b) 0.04; c) 0.08

    The smallest column needs the most steel, so the answer is (c) 0.08.

  4. In reinforced concrete strength design for columns, the α safety factor accounts for unintended load eccentricity (true or false)

    True.

  5. A larger slenderness ratio implies a stronger column (true or false).

    When the column is more slender, it is weaker, so the answer is false.

3. [25 points] Wood compression analysis: A Hem-Fir No.1 4x10 "pin-ended" column supports a governing load consisting of live, dead, and snow loads. If the unbraced length (height of the column) is 9'-0", how much total load (consisting of live, dead, and snow) can the column support?
Your answer should be in pound or kip units. Include all relevant adjustments (assume CM = 1.0).

The strategy is to find (a) the adjustment for stability, CP, (b) the adjusted allowable stress, F'c, and finally (c) the capacity or total load that can be safely supported, P = F'c x (area of cross-section).

(step a) Since Fc = 1300 psi (with CF = 1.0 and CD = 1.15 for snow) , F*c = 1300(1.0)(1.15) = 1495.0 psi.

FcE = (0.3 x 1,500,000) / (9x12 / 3.5)2 = 450,000 / 952.16 = 472.6 psi.

A = [1 + (472.6/1495.0)] / 1.6 = 0.823; B = [(472.6/1495.0)] / 0.8 = 0.395; Cp = A - sq.rt.(A2 - B) = 0.292.

(step b) F'c = F*cCp = 1495.0(0.292) = 436.0 psi.

(step c) Capacity, P = F'c x (area of cross-section) = 436.0(32.38) = 14,118#.

4. [20 points] Steel compression analysis: Find the capacity of a C5x9 steel compression member (Table A-4.4) used in a truss with pin-ended connections and an unbraced length of 4 feet. Assume A-36 steel.

From Table A-4.4, Iy = 0.624 in4 and A = 2.64 in2. Therefore, the radius of gyration, ry = sq.rt.(Iy / A) = 0.486.

kL / ry = 4 x 12 / 0.486 = 98.77 or approximately 99.

From Table A-7.3, Fa = 13.10 ksi.

The capacity, P = stress x area = 13.10 x 2.64 = 34.6 k.

5. [15 points] Steel column design: Select the lightest A-992 W-shape column to support a load of 176 k with an effective length of 16 feet.

All information for the solution is given in the problem statement. From Table A-7.2, and using P = 176 k and kL = 16 feet, we find the following "candidates" for lightest section:

The "winner" is W10x39 because it is lightest (only 39 pounds per linear foot of beam).

6. [30 points] Reinforced concrete column design:
A 10"x14" tied column supports a dead load, D = 140 k and a live load, L = 120 k. Assume fy = 60 ksi, and f'c = 4 ksi.

(a) Find the required steel area (Ast);

We use the strength design equation for reinforced concrete columns:
Pu ≤ φαPn, where Pn is the failure load, or resistance; and Pu is the so-called "design" load, i.e., the loads modified by their respective safety factors.

Pu = 1.2D + 1.6L = 1.2(140) + 1.6(120) = 360 k.

360 ≤ (0.65)(0.80)(0.85f'cAconc + fyAst) = 0.52[(0.85)(4)(10x14 - Ast) + (60)(Ast)]

Solving for Ast, we get: Ast ≥ 3.82 in2.

(b) Select 4 bars from Table A-4.10;

Select 4 No. 9 bars with Ast = 4.00 in2.

(c) Check that reinforcement ratio is between 1% and 8%;

ρg = 4.0 / (10x14) = 0.029 which is between 1% and 8%, so it is OK.

(d) Check that bar spacing works within given cross-section (Table A-4.11).

With 2 bars per line, we need 8 inches. Since we have 10" (shortest dimension), the bar fit is OK.
reinforced concrete column section

© 2008 Jonathan Ochshorn. First posted 11 March 2008. Last updated: 11 March 2008