Structural Analysis: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Beam support reactions

Equilibrium
ΣFx = 0, ΣFy = 0, ΣM = 0
Uniform load
resultant w·L at the middle of the load
Moment
M = F·d

Method

  1. Draw a free-body diagram and replace each support by its reactions.
  2. Replace every distributed load by a single force at its centroid.
  3. Take moments about one support to find the reaction at the other.
  4. Use ΣFy = 0 to find the remaining reaction, and check with a second moment equation.

Shear force & bending moment

Shear and moment
V = ΣF on one side, M = ΣF·d on one side
Bending stress
σ = M·y/I
Rectangle
I = bh³/12
Section modulus
W = I/ymax
Simply supported, uniform load
Mmax = wL²/8

Method

  1. Find the support reactions from equilibrium of the whole beam.
  2. Cut the beam at the section you need and keep one side.
  3. Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
  4. For the maximum moment, find where the shear force is zero.

Truss: method of joints

Method of joints
ΣFx = 0, ΣFy = 0 at each joint
Method of sections
ΣM = 0 about a cut point
Member force
positive = tension, negative = compression

Method

  1. Find the support reactions for the whole truss.
  2. Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
  3. Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
  4. Assume tension; a negative answer means compression.

Beam deflection

Point load at midspan
δ = PL³/(48EI)
Uniform load, simply supported
δ = 5wL⁴/(384EI)
Cantilever, end load
δ = PL³/(3EI)
Cantilever, uniform load
δ = wL⁴/(8EI)
Superposition
δ = δ1 + δ2

Column buckling

Euler load
Pcr = π²EI/(KL)²
Effective length factor
K = 1 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2 fixed–free
Critical stress
σcr = Pcr/A
Slenderness
λ = KL/r, r = √(I/A)

2Practice problems

1.Beam support reactionsMedium
P1​ = 45 kNP2​ = 55 kNABCRA​ = ?2 m6 m2 m

The beam has a pin at A and a roller at B, 8 m from A, and overhangs B to the free end C. Find the vertical reaction RA, taking upward as positive.

AB = 8 mAC = 10 mP1 = 45 kN at 2 mP2 = 55 kN at 10 m
RA = kN
2.Beam support reactionsHard
w = 4 kN/mP = 15 kNABCRA​ = ?1.5 m2.5 m1 m

The beam has a pin at A and a roller at B, 4 m from A, and overhangs B to the free end C. It carries a point load P at 1.5 m from A and a uniformly distributed load w from 1.5 m to the free end at 5 m. Find the vertical reaction RA, taking upward as positive.

AB = 4 mAC = 5 mP = 15 kN at 1.5 mw = 4 kN/m from 1.5 to 5 m
RA = kN
3.Shear force & bending momentMedium
50 kN20 kNABCDEFG1 m1 m1 m1 m1 m1 m

The marks A to G split the beam into six equal parts. At which mark is the bending moment largest in size, sagging or hogging? Click it in the diagram.

50 kN @ 4 m20 kN @ 5 m
  1. 1A (x = 0 m)
  2. 2B (x = 1 m)
  3. 3C (x = 2 m)
  4. 4D (x = 3 m)
  5. 5E (x = 4 m)
  6. 6F (x = 5 m)
  7. 7G (x = 6 m)
4.Shear force & bending momentHard
12 kN/m25 kN25 kN60 kNABCDEFG2 m2 m2 m2 m2 m2 m

The marks A to G split the beam into six equal parts. At which mark is the bending moment largest in size, sagging or hogging? Click it in the diagram.

25 kN @ 0 m25 kN @ 12 m60 kN @ 4 m12 kN/m, 2–8 m
  1. 1A (x = 0 m)
  2. 2B (x = 2 m)
  3. 3C (x = 4 m)
  4. 4D (x = 6 m)
  5. 5E (x = 8 m)
  6. 6F (x = 10 m)
  7. 7G (x = 12 m)
5.Truss: method of jointsMedium
ABCDEFGH40 kN20 kN20 kN4 × 3 m4 m

Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.

B: 40 kN ↓F: 20 kN ↓G: 20 kN ↓
  1. 1member AB
  2. 2member BC
  3. 3member CD
  4. 4member DE
  5. 5member FG
  6. 6member GH
  7. 7member AF
  8. 8member HE
  9. 9member BF
  10. 10member CG
  11. 11member DH
  12. 12member FC
  13. 13member HC
6.Truss: method of jointsHard
ABCDEFGH15 kN30 kN15 kN30 kN20 kN4 × 3 m3 m

Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.

B: 15 kN ↓D: 30 kN ↓F: 15 kN ↓F: 30 kN →H: 20 kN ↓
  1. 1member AB
  2. 2member BC
  3. 3member CD
  4. 4member DE
  5. 5member FG
  6. 6member GH
  7. 7member AF
  8. 8member HE
  9. 9member BF
  10. 10member CG
  11. 11member DH
  12. 12member FC
  13. 13member HC
7.Beam deflectionMedium
w = 15 kN/mP = 21 kNABδ1.5 m

A cantilever carries a uniformly distributed load w over its whole length and a point load P at its free end. Using superposition, find the deflection δ at the free end.

P = 21 kNw = 15 kN/mL = 1.5 mE = 11 GPa (timber)I = 160 × 10⁶ mm⁴
δ = mm
8.Beam deflectionHard
P = 10 kNP = 10 kNABδ1.667 m1.666 m1.667 m

A simply supported beam carries two equal point loads P at its third points (L/3 from each support). Using superposition, find the midspan deflection δmax.

P = 10 kNL = 5 mE = 70 GPa (aluminium)I = 120 × 10⁶ mm⁴
δmax = mm
9.Column bucklingMedium
PL = 4 m40 × 100 mmcross-section

A column of length L made of steel is fixed at the base and pinned at the top. Using Euler’s formula, find its critical buckling load Pcr.

L = 4 mE = 200 GPa (steel)b × h = 40 × 100 mm
Pcr = kN
10.Column bucklingHard
P = 90 kNL = 5 md = ?cross-section

A solid round column of length L made of steel is pinned at both ends. It must carry 90 kN with a factor of safety of 2.5 against buckling. Find the smallest diameter d.

P = 90 kNn = 2.5L = 5 mE = 200 GPa (steel)
d = mm

3Solutions

1.Beam support reactions

Answer: RA = 20 kN

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Replace each distributed load by its resultant (its area) acting at the centroid of the load diagram, and take moments about a support to remove that support’s reaction from the equation.

Moments about A

Counter-clockwise positive.

ΣMA = 0: RB·8 = 45·2 + 55·10
RB = 640/8 = 80 kN

Vertical equilibrium

RA = 45 + 55 − 80 = 20 kN

Answer

RA = 20 kN
2.Beam support reactions

Answer: RA = 12 kN

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Replace each distributed load by its resultant (its area) acting at the centroid of the load diagram, and take moments about a support to remove that support’s reaction from the equation.

Resultant of the distributed load

W = 4·3.5 = 14 kN, x̄ = (1.5 + 5)/2 = 3.25 m

Equilibrium

Moments about A, counter-clockwise positive. Then vertical force balance.

ΣMA = 0: RB·4 = 15·1.5 + 14·3.25
RB = 68/4 = 17 kN
RA = 29 − 17 = 12 kN

Answer

RA = 12 kN
3.Shear force & bending moment

Answer: E (x = 4 m)

Largest bending moment: M changes slope wherever the shear force changes, so its peaks sit where the shear force crosses zero: under a point load, inside a distributed load, or over a support of an overhang. At a fixed support of a cantilever the moment is largest at the wall. Compare the size of M at those places, sagging or hogging.

Support reactions

Take moments about one support, then use the vertical force balance.

R1 = 20 kN, R2 = 50 kN

Bending moment at every mark

Sum the moments of all forces on one side of the cut. Sagging is positive, hogging negative.

MA = 0, MB = 20, MC = 40, MD = 60, ME = 80 kN·m, MF = 50, MG = 0

Answer

Mark E: ME = 80 kN·m
4.Shear force & bending moment

Answer: E (x = 8 m)

Largest bending moment: M changes slope wherever the shear force changes, so its peaks sit where the shear force crosses zero: under a point load, inside a distributed load, or over a support of an overhang. At a fixed support of a cantilever the moment is largest at the wall. Compare the size of M at those places, sagging or hogging.

Support reactions

Take moments about one support, then use the vertical force balance.

R1 = 92.67 kN, R2 = 89.33 kN

Bending moment at every mark

Sum the moments of all forces on one side of the cut. Sagging is positive, hogging negative.

MA = 0, MB = −50, MC = 61.33, MD = 4.667, ME = −100 kN·m, MF = −50, MG = 0

Answer

Mark E: ME = −100 kN·m
5.Truss: method of joints

Answer: member DH

Zero-force members: at a joint, the forces must balance in every direction. If a joint has no load and no support, and two of its three members are in line, then the balance across that line involves only the third member, so its force is zero. If only two members meet at an unloaded joint and they are not in line, both are zero.

Joint D

CD and DE lie on one straight line and the joint carries no load. Balancing forces across that line leaves only DH.

ΣF⊥ = 0 → FDH = 0

Check with the method of joints

Solving all joints gives these member forces (T = tension, C = compression).

AB: 41.25 kN (T), BC: 41.25 kN (T), CD: 18.75 kN (T), DE: 18.75 kN (T), FG: 37.5 kN (C), GH: 37.5 kN (C), AF: 68.75 kN (C), HE: 31.25 kN (C), BF: 40 kN (T), CG: 20 kN (C), DH: 0, FC: 6.25 kN (C), HC: 31.25 kN (T)

Answer

Zero-force member: DH
6.Truss: method of joints

Answer: member CG

Zero-force members: at a joint, the forces must balance in every direction. If a joint has no load and no support, and two of its three members are in line, then the balance across that line involves only the third member, so its force is zero. If only two members meet at an unloaded joint and they are not in line, both are zero.

Joint G

FG and GH lie on one straight line and the joint carries no load. Balancing forces across that line leaves only CG.

ΣF⊥ = 0 → FCG = 0

Check with the method of joints

Solving all joints gives these member forces (T = tension, C = compression).

AB: 57.5 kN (T), BC: 57.5 kN (T), CD: 52.5 kN (T), DE: 52.5 kN (T), FG: 55 kN (C), GH: 55 kN (C), AF: 38.89 kN (C), HE: 74.25 kN (C), BF: 15 kN (T), CG: 0, DH: 30 kN (T), FC: 3.536 kN (C), HC: 3.536 kN (T)

Answer

Zero-force member: CG
7.Beam deflection

Answer: δ = 18.82 mm

Beam deflection: deflections come from integrating the curvature M/(EI) twice. Standard cases are tabulated, for example PL³/(3EI) for a cantilever with an end load and 5wL⁴/(384EI) for a simply supported beam under a full uniform load. Deflections from separate loads add (superposition).

Consistent units

Work in N and mm: P in N, w in N/mm (kN/m = N/mm), L in mm, E in N/mm² (GPa × 1000), I in mm⁴.

P = 21000 N, w = 15 N/mm, L = 1500 mm
EI = 1.76 × 10¹² N·mm²

Each load on its own

δP = PL³/(3EI) = 21000·1500³/(3·1.76 × 10¹²) = 13.42 mm
δw = wL⁴/(8EI) = 15·1500⁴/(8·1.76 × 10¹²) = 5.393 mm

Add them

δ = 13.42 + 5.393 = 18.82 mm

Answer

δ = 18.82 mm
8.Beam deflection

Answer: δmax = 5.282 mm

Beam deflection: deflections come from integrating the curvature M/(EI) twice. Standard cases are tabulated, for example PL³/(3EI) for a cantilever with an end load and 5wL⁴/(384EI) for a simply supported beam under a full uniform load. Deflections from separate loads add (superposition).

Consistent units

Work in N and mm: P in N, w in N/mm (kN/m = N/mm), L in mm, E in N/mm² (GPa × 1000), I in mm⁴.

P = 10000 N, L = 5000 mm
EI = 8.4 × 10¹² N·mm²

One load at L/3

δ₁ = Pb(3L² − 4b²)/(48EI), b = L/3 = 1667 mm
δ₁ = 2.641 mm

Add both loads

δmax = 2·δ₁ = 23PL³/(648EI) = 5.282 mm

Answer

δmax = 5.282 mm
9.Column buckling

Answer: Pcr = 134.3 kN

Euler buckling: a slender column becomes unstable at Pcr = π²EI/(KL)², where KL is the effective length: K = 1 for pinned–pinned, 2 for fixed–free, 0.7 for fixed–pinned and 0.5 for fixed–fixed. It buckles about the axis with the smallest I. The formula holds for slender columns, whose critical stress stays below the yield stress.

Smallest second moment of area

A rectangle buckles about its weak axis, using the smaller dimension cubed.

Imin = h·b³/12 = 100·40³/12 = 533333 mm⁴

Effective length

KL = 0.7·4000 = 2800 mm

Critical load

Pcr = π²EI/(KL)² = π²·200000·533333/2800² = 134280 N = 134.3 kN

Answer

Pcr = 134.3 kN
10.Column buckling

Answer: d = 87.29 mm

Euler buckling: a slender column becomes unstable at Pcr = π²EI/(KL)², where KL is the effective length: K = 1 for pinned–pinned, 2 for fixed–free, 0.7 for fixed–pinned and 0.5 for fixed–fixed. It buckles about the axis with the smallest I. The formula holds for slender columns, whose critical stress stays below the yield stress.

Required critical load

Pcr = n·P = 2.5·90000 = 225000 N
KL = 1·5000 = 5000 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·225000·5000²/(π³·200000))^(1/4) = 87.29 mm

Answer

d = 87.29 mm

Check: the slenderness KL/r is 229.1, so the column is slender and Euler applies.

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