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Question 1

Superposition Method and Common Loading Formulas

The Superposition Theorem states that the total moment or deflection in a linear structure under multiple loads can be found by summing the effects of individual loads applied separately. This is valid as long as the structure behaves elastically --> follows Hooke's Law. Below is a complete list of the formulas we can use for any beam, but we will later discuss how we only need to memorize at least five formulas in total if we combine the concept of integration in formulas with point loads.

Image excerpted from Pytel, F., & Kiusalaas, J. (2012). Engineering Mechanics: Statics and Dynamics 4th ed. Cengage Learning. Used under fair use for educational and illustrative purposes.

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 1: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 2: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 3: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 4: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 5: – Diagram

$$ \frac{wL^4}{8EI}=\frac{Pa^2(3L-a)}{6EI} $$ $$ \frac{5.2(5)^4}{8}=\frac{P(2.5)^2\left(3(5)-2.5\right)}{6} $$ $$ P=31.2\text{ kN} $$

Answer

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 6: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 7: – Diagram

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Superposition Method for Slope and Deflection: Superposition Formulas | Structural Theory – Problem 8: – Diagram

Answer

Part 1.

$L=12$ m; $P=12$ kN at $a=9$ m from left ($b=3$ m from right)
$EI = 200{,}000 \text{ MPa} \times 60{\times}10^6 \text{ mm}^4 = 12{\times}10^{12} \text{ N·mm}^2 = 12{,}000 \text{ kN·m}^2$
$\theta_A = \frac{Pb(L^2-b^2)}{6EIL} = \frac{12 \times 3 \times (144-9)}{6 \times 12{,}000 \times 12}$
$= \frac{12 \times 3 \times 135}{864{,}000} = \frac{4{,}860}{864{,}000}$
$\boxed{= 0.005625 \text{ rad}}$

Part 2.

$\theta_B = \frac{Pa(L^2-a^2)}{6EIL} = \frac{12 \times 9 \times (144-81)}{6 \times 12{,}000 \times 12}$
$= \frac{12 \times 9 \times 63}{864{,}000} = \frac{6{,}804}{864{,}000}$
$\boxed{= 0.007875 \approx 0.00788 \text{ rad}}$

Question 2

Problem: Cantilever Beam with Uniform Load and Concentrated Load at the Free End

A 4-m long cantilever beam carries a concentrated load of 50kN at the free end and a uniformly distributed load of 20kN/m throughout the entire span. The beam is made of steel where E=200GPa and I=200x10⁶mm⁴

Question 3

Problem: Cantilever Beam Resting on a Simply Supported Beam | Consistent Deformation with Superposition

Beam BD is fixed at D and is supported by girder AC at B with P=14kN applied at the free end.
L₁=4m, L₂=2m, L₃=4m.
The Section Modulus of both members about the plane of bending is:
S=6.18x10⁵mm³

Question 4

Problem: Application of Formulas for Fixed-Ended Beam

Refer to the load diagrams shown:

a. If M_A = -10 kN-m, L = 6 m, calculate P in kN based on Figure A.
a. 7.5
b. 10
c. 5
d. 12.5

b. If M_A = -40 kN-m, L = 10 m, calculate w in kN/m based on Figure B.
a. 6.78
b. 8.76
c. 8.67
d. 7.68

c. If M_A = -60 kN-m, M_B = -90 kN-m, L = 10 m, calculate w in kN/m based on Figure C.
a. 15
b. 18
c. 21
d. 24

Question 5

Problem: Cantilever Beam with Uniform Load Resting on a Simply Supported Beam | Consistent Deformation

Refer to the figure shown. A cantilever beam rests on B of beam ABC. THe cantilever beam carries a uniform load W=24.6kN/m throughout its length. Prior to loading, the cantilever beam is only touching beam ABC at B. Both beams have identical cross-sectional properties. Given the following data:
L₁=4m
L₂=2m
E=30GPa
I=1333x10⁶mm⁴

Question 6

Problem: Force to Apply at Mid-Length to Eliminate Deflection at Free-End

A 5-meter cantilever beam, 300mm x 400mm in cross-section carries a total uniformly distributed load of 5.20kN/m. E=25GPa. What force (in kN) should be applied at the mid-length of the beam to eliminate the deflection at the free end?

Question 7

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Question 8

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Question 9

Problem:

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Question 10

Question Bank: q21

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, E=200GPa and I=19,430,000mm⁴
Note: Point B has an internal hinge.

Determine the deflection at B in mm.

37.74
32.65
33.67
39.82

Determine the left and right slope values at B in radians.

-0.01737, 0.0072375
-0.01373, 0.0075372
-0.01773, 0.0077325
-0.013377, 0.0073325

Determine the deflection at C in mm.

21.713
23.714
25.628
26.825

Solution pending in psadquestions/q21.json.

Question 11

Question Bank: q28

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, use E=100GPa and I=10x10⁷mm⁴.

Determine the deflection at A in mm.

1
2.4
2.8
1.6

Determine the deflection at C in mm.

4.5
6.3
5.2
3.7

Solution pending in psadquestions/q28.json.

Question 12

Question Bank: q144

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

A 12m simple beam is loaded with a uniform load w = 40kN/m over the entire span and a concentrated load P = 200kN at the midspan. EI is constant.

Which of the following most nearly gives the slope at the support due to the concentrated load only?

1800/EI
4800/EI
7200/EI
7200/EI

Which of the following most nearly gives the maximum slope due to the uniform load only?

2880/EI
9600/EI
6400/EI
10800/EI

Which of the following most nearly gives the maximum deflection due to the given loads?

4680/EI
14400/EI
10000/EI
10800/EI

Solution pending in psadquestions/q144.json.

Question 13

Question Bank: q152

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

A 12 m-simply supported beam is loaded with a concentrated load P = 12 kN at 3 meters from the right support. E = 200 GPa and I = 60 000 000 mm⁴

Compute the rotation at the left support in radians.

0.005625
0.00673
0.00265
0.00375

Compute the rotation at the right support in radians.

0.00788
0.00375
0.00265
0.00525

Question 14

Question Bank: q309

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, assume EI is constant.

Determine the slope at the free end.

288/EI
144/EI
264/EI
132/EI

Determine the deflection at the free end.

1274.4/EI
1374.6/EI
1349.2/EI
1138.8/EI

Determine the slope at the midspan.

247.5/EI
240.6/EI
270.9/EI
250.5/EI

Determine the deflection at the midspan.

441.45/EI
508.95/EI
430/EI
540/EI

Solution pending in psadquestions/q309.json.

Question 15

Question Bank: q310

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, assume EI is constant.

Determine the slope at the free end.

1800/EI
1600/EI
1700/EI
1500/EI

Determine the deflection at the free end.

7200/EI
3600/EI
6800/EI
3400/EI

Determine the slope at the midspan.

1350/EI
1250/EI
1400/EI
1375/EI

Determine the deflection at the midspan.

2250/EI
2500/EI
2480/EI
2360/EI

Solution pending in psadquestions/q310.json.

Question 16

Question Bank: q311

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, assume EI is constant.

Determine the deflection at the midspan.

4258.8/EI
4285.8/EI
4288.5/EI
4528.8/EI

Solution pending in psadquestions/q311.json.

Question 17

Question Bank: q312

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

For the beam shown, assume EI is constant.

Determine the deflection at 5.5m from the left support.

964.4625/EI
835.3125/EI
129.15/EI
1093.6125/EI

Determine the deflection at point C.

834.6/EI
735/EI
99.6/EI
934.2/EI

Determine the slope at D.

334.3/EI
297.5/EI
36.8/EI
371.1/EI

Determine the deflection at E.

802.32/EI
807.5/EI
714/EI
890.64/EI

Solution pending in psadquestions/q312.json.

Question 18

Question Bank: q326

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

Using the three moment equation,

Calculate the deflection at B in mm.

0.176mm
0.167mm
0.156mm
0.165mm

Calculate the deflection at C in mm.

0.174mm
0.267mm
0.276mm
0.147mm

Solution pending in psadquestions/q326.json.

Question 19

Question Bank: q332

PSAD - Structural Theory / Slope and Deflection / Engr. Janclyde Espinosa (Clidez)

Determine the deflection at the free end using Superposition Method.

Determine the deflection at C caused by the concentrated load alone. Remove the uniformly distributed load.

466.66/EI
333.33/EI
133.33/EI
366.66/EI

Determine the slope at B caused by the uniformly distributed load.

208.33/EI
133.33/EI
236.66/EI
254.58/EI

Determine the deflection at C

50/EI (downward)
50/EI (upward)
60/EI (upward)
60/EI (downward)

Solution pending in psadquestions/q332.json.

Question 20

Question Bank: q544

PSAD - Structural Theory / Slope and Deflection / Engr. Deguma

A 6-meter-cantilever-beam (fixed at left) is loaded with 3 concentrated loads P₁, P₂ and P₃ applied at 2 m, 4 m, and 6 m from the fixed support, respectively. If P₁ = P₂ = P₃ = 30 kN and EI is constant, compute the following:

The moment at 1m from the fixed support in kN-m.

270
180
210
360

The slope at the free end.

840/EI
480/EI
720/EI
540/EI

The deflection at the free end.

3600/E I
2400/EI
1800/EI
4800/EI

Solution pending in psadquestions/q544.json.

Question 21

Question Bank: q565

PSAD - Structural Theory / Slope and Deflection / Engr. Deguma

The beam shown is acted on by moments M₁=M₂=100kN×m. If a=3m and EI is constant,

Determine the slope at A.

75/EI
120/EI
100/EI
150/EI

Determine the deflection at B.

225/EI
125/EI
175/EI
275/EI

Determine the deflection at D.

1125/EI
1325/EI
1275/EI
1175/EI

Solution pending in psadquestions/q565.json.

Superposition Method and Common Loading Formulas

Problem: Cantilever Beam with Uniform Load and Concentrated Load at the Free End

Problem: Cantilever Beam Resting on a Simply Supported Beam | Consistent Deformation with Superposition

Problem: Application of Formulas for Fixed-Ended Beam

Problem: Cantilever Beam with Uniform Load Resting on a Simply Supported Beam | Consistent Deformation

Problem: Force to Apply at Mid-Length to Eliminate Deflection at Free-End

Problem:

Problem:

Problem:

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