Structures and Strength of Materials

Every machine part, bracket and frame must carry its loads without breaking, bending too much or failing after a million repetitions. Strength of materials and structural mechanics are the tools for checking that: how a load becomes stress and strain, how beams bend, how columns buckle, how shafts twist, how bolts and welds hold, and how cyclic loads wear a part out. This guide takes those subjects in order, with the formulas, worked numbers and diagrams, and finishes with complete design checks. It continues from Gears and Power Transmission (which applies the theory to shafts and gears) and connects to Fits and Tolerances and GD&T for the geometry of the parts.

Start here: what do you need?

I need to…	Go to
understand stress, strain, elastic modulus, yield and safety factors	Stress, Strain and Safety Factors
find the stress and deflection of a beam, or draw shear and moment diagrams	Beam Bending and Deflection
choose a tube, angle, channel or I-beam, or compute section properties	Section Properties and Structural Profiles
check a strut or column for buckling	Columns and Buckling
size a shaft in torsion, or combine bending, torsion and shear	Torsion, Shear and Combined Loads
size a bolted or welded joint, or find the tightening torque	Bolted and Welded Joints
check a part for fatigue, or understand notches and stress concentration	Fatigue and Stress Concentration
pick a material or look up properties, densities, expansion and hardness	Material Properties
see complete strength checks worked through from the loads to the safety factor	Worked Design Checks

A suggested order

If you are new to the subject, read the pages in this order: the first builds the basic concepts, the next four cover the standard loading cases, joints and fatigue come after, and the last two bring in the materials and the worked checks.

1. Stress, Strain and Safety Factors: Loads and stress types, Hooke’s law, elastic modulus, yield and ultimate strength, safety factors and a method for a strength check.
2. Beam Bending and Deflection: Reactions, shear and moment diagrams, bending stress, deflection formulas for the common load cases, and how much deflection is acceptable.
3. Section Properties and Structural Profiles: Area, second moment and section modulus, why shape matters, standard tubes, angles, channels, I-beams and extrusions.
4. Columns and Buckling: Compression members, the Euler load, slenderness, end conditions, the Johnson curve for short columns and design margins.
5. Torsion, Shear and Combined Loads: Torsion of shafts and tubes, direct and bending shear, combining stresses and the von Mises criterion.
6. Bolted and Welded Joints: Bolt grades, preload and torque, joint stiffness, shear and tension in bolts, thread engagement, and weld strength.
7. Fatigue and Stress Concentration: The S-N curve, the endurance limit and its modifying factors, Goodman and Gerber lines, notches and how to design against fatigue.
8. Material Properties: Properties of metals and plastics, stress–strain curves, strength for the weight, thermal behavior, hardness and unit conversions.
9. Worked Design Checks: Complete strength checks of a bracket, a shaft, a column and a bolted joint, from loads to safety factor.

A strength check in six steps

• 1. Loads: list every force, moment and pressure the part sees, including the worst case, shock and the weight.
• 2. Free-body diagram and reactions: isolate the part, apply the loads and find the support reactions by equilibrium.
• 3. Internal forces: axial force, shear force, bending moment and torque at the critical sections (the diagrams in Beam Bending and Deflection).
• 4. Stresses: normal, shear and, where they combine, the von Mises equivalent stress (Torsion, Shear and Combined Loads), with the stress concentrations at holes, notches and fillets (Fatigue and Stress Concentration).
• 5. Compare with the strength: yield for static loads, endurance limit for cyclic loads, buckling for compression members: the safety factor n = strength ÷ stress.
• 6. Check the deflection and the details: stiffness limits (Beam Bending and Deflection), joints (Bolted and Welded Joints), and the material and manufacturing assumptions.

Quick reference: formulas

Topic	Formula	Page
Axial stress and strain	σ = F ÷ A; ε = ΔL ÷ L; σ = E ε	Stress, Strain and Safety Factors
Safety factor	n = S ÷ σ	Stress, Strain and Safety Factors
Bending stress	σ = M c ÷ I = M ÷ S	Beam Bending and Deflection
Deflection, simply supported, central load	δ = P L³ ÷ (48 E I)	Beam Bending and Deflection
Second moment of area	rectangle b h³ ÷ 12; circle π d⁴ ÷ 64	Section Properties and Structural Profiles
Euler buckling	Pcr = π² E I ÷ (K L)²	Columns and Buckling
Torsion	τ = 16 T ÷ (π d³); θ = T L ÷ (G J)	Torsion, Shear and Combined Loads
Von Mises	σ′ = √(σ² + 3 τ²)	Torsion, Shear and Combined Loads
Bolt tightening	T = K d Fi	Bolted and Welded Joints
Goodman line	σa ÷ Se + σm ÷ Su = 1 ÷ n	Fatigue and Stress Concentration

Calculators and related guides

• Bolt Torque and Thread Engagement Tool: tightening torque, preload and thread engagement.
• Gears and Power Transmission: shafts, keys, bearings and gears use these methods.
• Fits and Tolerances and Fit Calculator: interference fits and shrink fits.
• GD&T: tolerancing the parts you have checked.
• 3D Printing and CNC Machining: manufacturing methods that affect strength.

Safety

• Structural failures can be sudden and fatal. For lifting equipment, pressure vessels, vehicles, buildings and anything above people, use the applicable codes and a qualified engineer; these pages are for learning, small machines and first checks.
• Safety factors cover uncertainty: unknown loads, material variation, wear and corrosion. Do not reduce them without understanding the risk.
• Test critical parts with proof loads before relying on them, and inspect them in service.
• Stored energy: springs, tensioned cables and pressurized parts release their energy when they fail.

FAQ

Where should I start?

With Stress, Strain and Safety Factors for the concepts, then Beam Bending and Deflection since most real parts are beams.

What happened to the Beams, Structural Profiles and Material Properties pages?

They are all part of this guide: Beam Bending and Deflection, Section Properties and Structural Profiles and Material Properties.

Is there a calculator for beams?

Not yet: the formulas are given with worked examples in Beam Bending and Deflection so that you can reproduce them in a spreadsheet. A beam calculator is a good candidate for the Tools section.

Do I need finite element analysis (FEA)?

For simple shapes and loads, hand calculations are quicker and often more reliable. FEA is for complex shapes and checking, and its results are only as good as the loads, supports and mesh you give it: always compare a simple case with the hand formula.

References

• Beer, F. P., Johnston, E. R. and DeWolf, J. T., Mechanics of Materials, McGraw-Hill.
• Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill; Roark, W. C., Roark’s Formulas for Stress and Strain.
• Young, W. C. and Budynas, R. G., Roark’s Formulas for Stress and Strain; AISC Steel Construction Manual (sections and column design).
• Machinery’s Handbook (strength of materials, beam formulas, section properties).

Last reviewed: October 2026. Values are typical: use the material supplier’s data and the applicable design code for real structures.