A structure is statically indeterminate when the number of unknown reactions or internal forces exceeds the available equilibrium equations.
Steps:
A timber column, 8 in. by 8 in. in cross section, is reinforced on all four sides by steel plates, each plate being 8 in. wide and t in. thick. Determine the smallest value of t for which the column can support an axial load of 300 kips if the working stresses are 1200 psi for timber and 20 ksi for steel. The moduli of elasticity are 1.5x106psi for timber and 29x106psi for steel.
As our first step, we develop the equilibrium equation by Statics. Since the steel plates are distributed symmetrically along all sides of the timber section, its resultant force, Pst, will act at the center. Therefore,$$\Sigma F_y=0$$
$$P_t+P_{st}=P$$
where:
$$P_t = \text {force in timber}$$
$$P_{st} = \text {force in steel}$$
$$P = \text {applied load}$$
Then, since $\sigma = \frac {P}{A}$, we can express the forces as $P =\sigma \cdot A$
Since we have two stress conditions that must be satisfied:
Assuming timber governs, we use the working stress of timber, which is 1200psi, and express the stress in steel in terms of the stress in timber.
By contrast, assuming steel governs, we use the working stress of steel, which is 20ksi or 20000psi, and express the stress in timber in terms of the stress in steel.
A 4-ft concrete post is reinforced with four steel bars, each with a 3/4-in diameter. Es=29x106psi and Ec=3.6x106psi. Determine the following if a 150-kip axial centric force P is applied to the post.
Equilibrium Equation:
$$\Sigma F_y=0$$
$$P=4P_{st}+P_c$$
$$P=4 \cdot \sigma_{st} (A_{st}) + \sigma_{c} (A_{c})$$
Note that the cross-sectional area of concrete should be the gross area of 8in x 8in minus the four steel areas.
$$P=4 \cdot \sigma_{st} (\frac {\pi d^2}{4}) + \sigma_{c} ((8)(8)-4\cdot \frac{ \pi d^2}{4})$$
$$150kips=4 \cdot \sigma_{st} (\frac {\pi (3/4)^2}{4}) + \sigma_{c} ((8)(8)-4\cdot \frac{ \pi (3/4)^2}{4})$$
Equation based on the relationships of the deformation:
$$
\left(\frac{PL}{AE}\right)_{st} = \left(\frac{PL}{AE}\right)_{c}
$$
$$
\left(\frac{\sigma L}{E}\right)_{st} = \left(\frac{\sigma L}{E}\right)_{c}
$$
$$
\frac{\sigma_{st}}{E_{st}} = \frac{\sigma_{c}}{E_{c}}
$$
$$
\frac{\sigma_{st}}{29x10^6} = \frac{\sigma_{c}}{3.6x10^6}
$$
$$\sigma_{st}=\frac {29}{3.6} \cdot \sigma_c$$
Substituting this expression in the equilibrium equation,
$$150kips=4 \cdot \sigma_{st} (\frac {\pi (3/4)^2}{4}) + \sigma_{c} ((8)(8)-4\cdot \frac{ \pi (3/4)^2}{4})$$
$$150kips=4 \cdot (\frac {29}{3.6} \cdot \sigma_c) (\frac {\pi (3/4)^2}{4}) + \sigma_{c} ((8)(8)-4\cdot \frac{ \pi (3/4)^2}{4})$$
Solving $\sigma_c$, we obtain:
$$\sigma_c = \boxed{1.962ksi}$$
$$\sigma_{st} = \frac {29}{3.6} \cdot 1.962ksi=\boxed{15.81ksi}$$
A 1.5-m concrete post with a diameter of 450mm is reinforced with six steel bars, each with a diameter of 28mm. Est=200GPa and Ec=25GPa. A 1550-kN axial centric force P is applied to the post.
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The composite bar, firmly attached to unyielding supports, is initially stress-free. What maximum axial load P can be applied if the allowable stresses are 10 ksi for aluminum and 18 ksi for steel?
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The steel rod is stress-free before the axial loads P1 =150 kN and P2 = 90 kN are applied to the rod. Assuming that the walls are rigid, calculate the axial force in each segment after the loads are applied. Use E = 200 GPa.
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The rigid beam of negligible weight is supported by a pin at O and two vertical rods. Find the vertical displacement of the 50-kip weight.
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Additional board-style practice items for this topic.
Given is a solid 80-mm diameter steel pole 2.5m in length. Shear Modulus = 78 GPa
Which of the following gives the axial rigidity of the pole in 1x109N?
Which of the following gives the flexural rigidity of the pole in 1x109N-mm2?
Which of the following gives the torsional rigidity of the pole in 1x109N-mm2?
Part 1.
$A = \frac{\pi d^2}{4} = \frac{\pi(80)^2}{4} = 1600\pi \approx 5026.5 \text{ mm}^2$Part 2.
$I = \frac{\pi d^4}{64} = \frac{\pi(80)^4}{64} \approx 2{,}010{,}619 \text{ mm}^4$Part 3.
$J = 2I = 4{,}021{,}238 \text{ mm}^4$; $G = 78{,}000 \text{ MPa}$A 4-ft concrete post is reinforced with four steel bars, each with a 3/4-in diameter. Es=29x106psi and Ec=3.6x106psi. Determine the following if a 150-kip axial centric force P is applied to the post.
The normal stress in steel, in ksi.
The normal stress in concrete in ksi.
Part 1.
Because the steel and concrete are bonded and loaded concentrically, they have the same strain:Part 2.
From the transformed-area equilibrium:The 1.5-m concrete post is reinforced with six steel bars, each with a diameter of 28mm. Es=200GPa and Ec=25GPa. A 1550-kN axial centric force P is applied to the post.
The normal stress in steel, in MPa.
The normal stress in concrete, in MPa.
Part 1.
For the bonded reinforced concrete post, steel and concrete have the same strain, so:Part 2.
Using the same transformed-area relation:The figure shows the cross section of a circular steel tube that is filled with concrete and topped with a rigid cap. Calculate the stresses in the steel and in the concrete caused by the 200-kip axial load. Use Est=29x106psi and Ec=3.5x106psi
Determine the stress in the steel tube.
Determine the stress in the concrete section.
Part 1.
The rigid cap makes the steel and concrete shorten equally, so $\sigma_s=n\sigma_c$, where:Part 2.
From the composite axial-load equation:The rigid slab of weight W with center of gravity at G is suspended from three identical steel wires.
Determine the force carried by wire A.
Determine the force carried by wire B.
Determine the force carried by wire C.
Part 1.
Let the rod forces be $F_A$, $F_B$, and $F_C$. Since the rods are identical, force is proportional to elongation. The slab remains plane, so the displacement at B is midway between A and C:Part 2.
Using the same equations:Part 3.
The solution of the equilibrium and compatibility equations gives:Before the 400-kN load is applied, the rigid platform rests on two steel bars, each of cross-sectional area 1400 mm2, as shown in the figure. The cross-sectional area of the aluminum bar is 2800 mm2. Compute the stress in the aluminum bar after the 400-kN load is applied. Use E=200GPa for steel and E = 70GPa for aluminum. Neglect the weight of the platform.
Solution pending in psadquestions/q632.json.
The rigid bar ABC of negligible weight is suspended from three aluminum wires, each of cross-sectional area 0.3 in.2. Before the load P is applied, the middle wire is slack, being 0.2 in. longer than the other two wires. Determine the largest safe value of P if the working stress for the wires is 12 ksi. Use E = 10x106 psi for aluminum.
Solution pending in psadquestions/q633.json.
The rigid bar AB of negligible weight is supported by a pin at O. When the two steel rods are attached to the ends of the bar, there is a gap Δ = 4 mm between the lower end of the left rod and its pin support at C. Compute the stress in the left rod after its lower end is attached to the support. The cross-sectional areas are 300 mm2 for rod AC and 250 mm2 for rod BD. Use E = 200 GPa for steel.
Solution pending in psadquestions/q634.json.
The composite bar is firmly attached to unyielding supports. An axial load P = 40 kips is applied at the junction point of the two bars.
Determine the stress in the steel member.
Determine the stress in the aluminum member.
Part 1.
The joint displacement is the same for both bars, so the axial deformations are compatible:Part 2.
From compatibility, $F_{Al}=0.1724F_s$. With $F_s=34.12$ kips:The composite bar, firmly attached to unyielding supports, is initially stress-free. What maximum axial load P can be applied if the allowable stresses are 10 ksi for aluminum and 18 ksi for steel?
Solution pending in psadquestions/q636.json.
The steel rod is stress-free before the axial loads P1 =150 kN and P2 = 90 kN are applied to the rod. Assuming that the walls are rigid, calculate the axial force in each segment after the loads are applied. Use E = 200 GPa.
Force in CD
Force in BC
Force in AB
Solution pending in psadquestions/q637.json.
The rigid beam of negligible weight is supported by a pin at O and two vertical rods. Find the vertical displacement of the 50-kip weight.
Solution pending in psadquestions/q638.json.
The two vertical rods attached to the rigid bar are identical except for length. Before the 6600-lb weight was attached, the bar was horizontal. Determine the axial force in each bar caused by the application of the weight. Neglect the weight of the bar.
Force in rod A
Force in rod B
Solution pending in psadquestions/q639.json.
The rigid bar of negligible weight is pinned at O and attached to two vertical rods. Assuming that the rods were initially stress-free, what is the largest load P that can be applied without exceeding stresses of 150 MPa in the steel rod and 70 MPa in the bronze rod?
Solution pending in psadquestions/q640.json.
The rigid, homogeneous slab weighing 600 kN is supported by three rods of identical material and cross section. Before the slab was attached, the lower ends of the rods were at the same level. Compute the axial force in each rod.
Force in rod A in kN
Force in rod B
Force in rod C
Solution pending in psadquestions/q641.json.
A timber column, 8 in. by 8 in. in cross section, is reinforced on all four sides by steel plates, each plate being 8 in. wide and t in. thick. Determine the smallest value of t for which the column can support an axial load of 300 kips if the working stresses are 1200 psi for timber and 20 ksi for steel. The moduli of elasticity are 1.5x106psi for timber and 29x106psi for steel.
Solution pending in psadquestions/q642.json.
The rigid block of mass M is supported by the three symmetrically placed rods. The ends of the rods were level before the block was attached. Determine the largest allowable value of M if the properties of the rods are as listed:
Solution pending in psadquestions/q643.json.
The compound shaft composed of steel, aluminum, and bronze segments, carries the two torques as shown. If TC=250lb-ft, determine the maximum shear stress developed in each material. The moduli of rigidity for steel, aluminum, and bronze are 12x106psi, 4x106psi, and 6x106psi, respectively.
Stress in the steel rod
Stress in the bronze rod
Stress in the aluminum rod
Solution pending in psadquestions/q746.json.