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Exercise set

Structural Loads and Beam Analysis Exercises

Worked civil engineering exercises for structural loads and beam analysis covering tributary loads, reactions, shear, bending moment, bending stress, serviceability, bearing, and equilibrium checks.

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Civil and Construction Engineering
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Exercise set
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v1.1.0 ยท reviewed

These exercises practise structural load and beam-analysis calculations for civil and structural engineering. The calculations are intentionally compact, but each one includes the engineering interpretation that decides whether the number is useful.

Assume static loading, small deflection, linear elastic behaviour, and idealized supports unless an exercise states otherwise. Real designs must also check the governing code, load combinations, connection details, stability, material-specific rules, construction stages, durability, fire, vibration, fatigue, and inspection requirements.

How to Use These Exercises

For each problem:

  1. draw a free-body diagram;
  2. convert loads to consistent units;
  3. check equilibrium before calculating stress or deflection;
  4. state whether the result is a serviceability, strength, stability, or local-detail check;
  5. identify what would need a code-specific design check.

The most common mistake is jumping directly to a beam formula before proving that the load path, support model, and load basis are correct. A precise calculation based on the wrong support condition is still a wrong design check.

For each result, state whether it supports a load-takeoff check, analysis-model validation, serviceability review, strength screening, local-detail check, or request for a code-specific design calculation. The same beam result can have a different meaning depending on whether the load basis is service, factored, temporary, or accidental.

Exercise 1: Area Load to Beam Line Load

A floor beam supports a tributary width of 4.0\ \text{m}. The service floor load is 3.0\ \text{kN/m}^2. The beam self-weight is estimated as 0.8\ \text{kN/m}.

Find the total service line load on the beam.

Solution

Convert area load to line load:

w_{floor}=q b_t
w_{floor}=3.0(4.0)=12.0\ \text{kN/m}

Add self-weight:

w_{total}=12.0+0.8=12.8\ \text{kN/m}

Engineering Comment

The tributary width controls the result. If the slab spans in two directions, if load sharing is different from assumed, or if construction loads govern, this simple line-load conversion may not represent the real load path.

Exercise 2: Reactions for a Beam with Uniform and Point Loads

A simply supported beam has span L=6.0\ \text{m}. It carries a uniform load w=8.0\ \text{kN/m} over the full span and a point load P=12\ \text{kN} located 2.0\ \text{m} from the left support.

Find the support reactions.

Solution

Total uniform load:

W=wL=8.0(6.0)=48\ \text{kN}

Sum of vertical loads:

R_A+R_B=48+12=60\ \text{kN}

Take moments about the left support:

R_B(6.0)=48(3.0)+12(2.0)
R_B(6.0)=144+24=168
R_B=28\ \text{kN}

Then:

R_A=60-28=32\ \text{kN}

Engineering Comment

The reactions are not equal because the point load is closer to the left support. Before using a structural-analysis model, engineers often perform this kind of hand equilibrium check to catch load-entry or support-location errors.

Exercise 3: Maximum Shear and Moment for a Uniform Load

A simply supported beam has span L=7.0\ \text{m} and carries a full-span uniform load w=5.0\ \text{kN/m}.

Find maximum shear and maximum bending moment.

Solution

Maximum shear at each support:

\displaystyle V_{max}=\frac{wL}{2}
\displaystyle V_{max}=\frac{5.0(7.0)}{2}=17.5\ \text{kN}

Maximum bending moment at midspan:

\displaystyle M_{max}=\frac{wL^2}{8}
\displaystyle M_{max}=\frac{5.0(7.0)^2}{8}=30.6\ \text{kN m}

Engineering Comment

These values are for a simply supported member with uniform load over the full span. Partial loads, continuity, fixed ends, overhangs, moving loads, or construction-stage supports can change both the maximum value and its location.

Exercise 4: Bending Stress Check

A beam has maximum bending moment M=85\ \text{kN m}. The selected section has elastic section modulus:

S=620\times10^3\ \text{mm}^3

Find the nominal bending stress.

Solution

Use:

\displaystyle \sigma=\frac{M}{S}

Convert moment:

M=85\ \text{kN m}=85\times10^6\ \text{N mm}

Stress:

\displaystyle \sigma=\frac{85\times10^6}{620\times10^3}=137\ \text{MPa}

Engineering Comment

This is a nominal elastic bending stress. A real design must also check material resistance format, lateral-torsional buckling, local buckling, shear, bearing, connection design, load factors, and whether the section modulus applies to the correct axis.

Exercise 5: Serviceability Deflection Limit

A beam spans L=5.4\ \text{m}. A service-load calculation gives maximum deflection:

\delta_{max}=14.0\ \text{mm}

The project uses a simple serviceability limit of L/360.

Check whether the beam satisfies this limit.

Solution

Convert span:

L=5.4\ \text{m}=5400\ \text{mm}

Allowable deflection:

\displaystyle \delta_{allow}=\frac{L}{360}
\displaystyle \delta_{allow}=\frac{5400}{360}=15.0\ \text{mm}

Compare:

14.0\ \text{mm}<15.0\ \text{mm}

The beam satisfies this simple deflection limit with 1.0\ \text{mm} of margin.

Engineering Comment

The margin is small. The engineer should review creep, cracking, composite action, connection flexibility, construction tolerances, vibration, finishes, and whether the correct load combination was used for serviceability.

Exercise 6: Factored Line Load

A beam has dead load D=10\ \text{kN/m} and live load L=16\ \text{kN/m}. A simplified ultimate load combination is:

w_u=1.2D+1.6L

Find the factored line load.

Solution

Substitute:

w_u=1.2(10)+1.6(16)
w_u=12+25.6=37.6\ \text{kN/m}

Engineering Comment

The factored load is not the same as service load. Do not use factored load for ordinary deflection checks unless the design basis requires it. Strength and serviceability often use different combinations.

Exercise 7: Bearing Stress at a Support Plate

A beam reaction of 95\ \text{kN} bears on a plate area 80\ \text{mm}\times120\ \text{mm}.

Find the average bearing stress.

Solution

Bearing area:

A_b=80(120)=9600\ \text{mm}^2

Average bearing stress:

\displaystyle \sigma_b=\frac{R}{A_b}
\displaystyle \sigma_b=\frac{95{,}000}{9600}=9.90\ \text{MPa}

Engineering Comment

Average bearing stress is only a screening value. Real support design may require checking local crushing, web crippling, plate bending, welds, anchors, eccentricity, grout, masonry or concrete strength, edge distance, and inspection access.

Exercise 8: Independent Equilibrium Check

A beam model reports support reactions:

R_A=31\ \text{kN}
R_B=29\ \text{kN}

The applied loads in the same load case are a 48\ \text{kN} distributed-load resultant and a 12\ \text{kN} point load.

Check vertical equilibrium.

Solution

Total reaction:

R_A+R_B=31+29=60\ \text{kN}

Total applied load:

48+12=60\ \text{kN}

Vertical force equilibrium is satisfied:

\displaystyle \sum F_y=60-60=0

Engineering Comment

Passing vertical equilibrium is necessary, but not sufficient. The moment equilibrium check may still fail if a point load is placed at the wrong location. Independent checks should verify force balance, moment balance, units, support positions, load directions, and diagram shape.

Review Checklist

Before accepting a structural load or beam-analysis result, check:

  • whether the load path is physically possible;
  • whether area loads, line loads, and point loads were converted correctly;
  • whether reactions satisfy both force and moment equilibrium;
  • whether diagrams match the support and load pattern;
  • whether stress checks use the correct section property and axis;
  • whether serviceability uses service loads rather than factored loads where appropriate;
  • whether local bearing, connection, and stability checks could govern;
  • whether the load case, support condition, span definition, section axis, and units can be traced back to the drawings or model input;
  • whether construction stages, temporary supports, moving loads, vibration, fatigue, or robustness could control instead of the simple static case;
  • whether assumptions match the structure that will actually be built.

Good structural engineering starts with equilibrium and load path. Beam formulas are powerful only when the model, loads, supports, and limit state are the right ones.

REF

See also

  1. Structural Loads and Beam Analysis Civil and Construction Engineering
  2. Reinforced Concrete and Structural Material Design Exercises Civil and Construction Engineering
  3. Geotechnical Retaining Structures and Excavation Support Exercises Civil and Construction Engineering
  4. Structural Loads and Beam Analysis Formula Sheet Civil and Construction Engineering
  5. Beam Deflection Exercises Civil and Construction Engineering
  6. Beginner's Guide to Structural Analysis Civil and Construction Engineering
  7. How Load Paths Work Civil and Construction Engineering
  8. Reinforced Concrete and Structural Material Design Civil and Construction Engineering
  9. Civil Infrastructure Asset Management, Inspection, and Rehabilitation Civil and Construction Engineering
  10. Civil Infrastructure Asset Management, Inspection, and Rehabilitation Exercises Civil and Construction Engineering
  11. Construction Planning and Site Operations Civil and Construction Engineering
  12. Construction Planning and Site Operations Exercises Civil and Construction Engineering
  13. Mechanical Stress Analysis Mechanical Engineering
  14. Mechanical Stress Analysis Formula Sheet Mechanical Engineering
  15. Design Load Civil and Construction Engineering quantity
  16. Load Factor Civil and Construction Engineering quantity
  17. Occupancy Load Civil and Construction Engineering metric
  18. Allowable Stress Civil and Construction Engineering quantity
  19. Beam Deflection Civil and Construction Engineering quantity
  20. Deflection Civil and Construction Engineering quantity
  21. Stress Mechanical Engineering quantity
  22. Free-Body Diagram Mechanical Engineering method
  23. Moment of Inertia Mechanical Engineering quantity
  24. Elastic Modulus Materials Engineering quantity
  25. Young's Modulus Materials Engineering quantity
  26. Buckling Load Mechanical Engineering quantity
  27. Euler Buckling Civil and Construction Engineering model
  28. Thermal Stress Mechanical Engineering phenomenon
  29. Reliability Industrial and Management Engineering metric
  30. Validation Industrial and Management Engineering method
  31. Uncertainty Analysis Mathematical Engineering method
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