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
w = 5 kN/mP = 15 kNABMA​ = ?4 m1 m

The cantilever beam is fixed to the wall at A. It carries a uniformly distributed load w along its whole length and a point load P at 4 m from the wall. Find the magnitude of the reaction moment MA at the wall.

L = 5 mP = 15 kN at 4 mw = 5 kN/m from 0 to 5 m
MA = kN·m
2.Beam support reactionsHard
P = 60 kN, 45°Q = 25 kNABRA​ = ?1 m3 m4 m

The simply supported beam has a pin at A and a roller at B. A load P acts at 1 m from A, inclined at 45° to the horizontal, and a vertical load Q acts at 4 m from A. Find the magnitude of the total reaction force RA at the pin A.

L = 8 mP = 60 kN at 1 m, θ = 45°Q = 25 kN at 4 m
RA = kN
3.Shear force & bending momentMedium
M0​ = 30 kN·mAB1 m3 m

The simply supported beam carries a clockwise couple M0 = 30 kN·m at x = 1 m from A, and no other load. Find the bending moment Mx just to the right of the couple. Use sagging moments positive and upward shear on the left face positive.

L = 4 mM0 = 30 kN·m clockwise, at 1 m
Mx = kN·m
4.Shear force & bending momentHard
6 kN/m10 kN35 kN25 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.

10 kN @ 0 m35 kN @ 12 m25 kN @ 4 m6 kN/m, 2–10 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
ABCDEFGH20 kN20 kN15 kN20 kN4 × 4 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.

C: 20 kN ↓D: 20 kN ↓F: 15 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
ABBCACACB30 kN10 kN5 m3 m4 m

Use the method of joints to find the force FBC in member BC. Give tension as positive and compression as negative.

span = 5 mheight = 4 mB is 3 m from Avertical load = 30 kNhorizontal load = 10 kN →
FBC = kN
7.Beam deflectionMedium
w = 2 kN/mABθ4 m

A simply supported beam carries a uniformly distributed load w over its whole span. Find the slope θA of the beam at support A, in radians.

w = 2 kN/mL = 4 mE = 11 GPa (timber)I = 36 × 10⁶ mm⁴
θA = rad
8.Beam deflectionHard
w = 15 kN/mδL = 4 m

A simply supported beam carries a uniformly distributed load w. Its deflection may not exceed span/360. Find the smallest second moment of area Ireq that meets this limit.

w = 15 kN/mL = 4 mE = 11 GPa (timber)
Ireq = × 10⁶ mm⁴
9.Column bucklingMedium
PL = 6 m40 × 50 mmcross-section

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

L = 6 mE = 70 GPa (aluminium)b × h = 40 × 50 mm
Pcr = kN
10.Column bucklingHard
P = 10 kNL = 3 md = ?cross-section

A solid round column of length L made of steel is fixed at both ends. It must carry 10 kN with a factor of safety of 2.5 against buckling. Find the smallest diameter d.

P = 10 kNn = 2.5L = 3 mE = 200 GPa (steel)
d = mm

3Solutions

1.Beam support reactions

Answer: MA = 122.5 kN·m

Cantilever equilibrium: a fixed support can push up and also resist rotation, so it gives a vertical reaction RA and a reaction moment MA. ΣFy = 0 gives RA, and moments about the wall give MA = Σ(load × distance from the wall). A distributed load acts as its resultant at the centroid of the load diagram.

Resultant of the distributed load

W = 5·5 = 25 kN, x̄ = 2.5 m

Moment about the wall

MA = P·a + W·x̄ = 15·4 + 25·2.5 = 122.5 kN·m

Answer

MA = 122.5 kN·m
2.Beam support reactions

Answer: RA = 65.29 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.

Components of P

Pv = 60·sin 45° = 42.43 kN
Ph = 60·cos 45° = 42.43 kN

Reactions

ΣMA = 0: RB·8 = 42.43·1 + 25·4 → RB = 17.8 kN
Ay = 42.43 + 25 − 17.8 = 49.62 kN
Ax = Ph = 42.43 kN

Resultant at A

RA = √(42.43² + 49.62²) = 65.29 kN

Answer

RA = 65.29 kN
3.Shear force & bending moment

Answer: Mx = 22.5 kN·m

Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut. An applied couple makes the bending moment diagram jump by the size of the couple.

Support reactions

Moments about A, counter-clockwise positive: RB·L − M0 = 0. Vertical balance: RA + RB = 0.

RB = M0/L = 30/4 = 7.5 kN (upward)
RA = −RB = −7.5 kN (downward)

Cut just right of the couple

The left part now contains RA and the couple.

Mx = RA·a + M0 = −7.5·1 + 30 = 22.5 kN·m

Answer

Mx = 22.5 kN·m
4.Shear force & bending moment

Answer: F (x = 10 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 = 46.5 kN, R2 = 71.5 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 = 41, MD = 28, ME = −9, MF = −70 kN·m, MG = 0

Answer

Mark F: MF = −70 kN·m
5.Truss: method of joints

Answer: member BF

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 B

AB and BC lie on one straight line and the joint carries no load. Balancing forces across that line leaves only BF.

ΣF⊥ = 0 → FBF = 0

Check with the method of joints

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

AB: 36.25 kN (T), BC: 36.25 kN (T), CD: 38.75 kN (T), DE: 38.75 kN (T), FG: 57.5 kN (C), GH: 57.5 kN (C), AF: 51.27 kN (C), HE: 54.8 kN (C), BF: 0, CG: 20 kN (C), DH: 20 kN (T), FC: 30.05 kN (T), HC: 26.52 kN (T)

Answer

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

Answer: FBC = −29.07 kN

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB30 kN10 kNAy​Cy​Ax​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·5 = 30·3 + 10·4 → Cy = 26 kN
ΣFy = 0: Ay = 30 − 26 = 4 kN
ΣFx = 0: Ax = −10 kN

Joint C

The roller gives only a vertical reaction, so joint C has just two unknown member forces. Assume both in tension.

sin γ = 4/4.472 = 0.8944, cos γ = 2/4.472 = 0.4472
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −26/0.8944 = −29.07 kN

Answer

FBC = −29.07 kN (compression)
7.Beam deflection

Answer: θA = 0.01347 rad

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). The slope θ = dv/dx is the first integral of M/(EI); in consistent units it comes out in radians.

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.

w = 2 N/mm, L = 4000 mm
EI = 3.96 × 10¹¹ N·mm²

End slope

θA = wL³/(24EI) = 2·4000³/(24·3.96 × 10¹¹) = 0.01347 rad

Answer

θA = 0.01347 rad

That is 0.7717°.

8.Beam deflection

Answer: Ireq = 409.1 × 10⁶ mm⁴

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).

Deflection limit

δallow = L/360 = 4000/360 = 11.11 mm

Required second moment of area

I = 5wL⁴/(384·E·δallow) = 5·15·4000⁴/(384·11000·11.11) = 4.091 × 10⁸ mm⁴
Ireq = 409.1 × 10⁶ mm⁴

Answer

Ireq = 409.1 × 10⁶ mm⁴
9.Column buckling

Answer: Pcr = 10.44 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 = 50·40³/12 = 266667 mm⁴

Effective length

KL = 0.7·6000 = 4200 mm

Critical load

Pcr = π²EI/(KL)² = π²·70000·266667/4200² = 10444 N = 10.44 kN

Answer

Pcr = 10.44 kN
10.Column buckling

Answer: d = 27.6 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 = 2.5·10000 = 25000 N
KL = 0.5·3000 = 1500 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·25000·1500²/(π³·200000))^(1/4) = 27.6 mm

Answer

d = 27.6 mm

Check: the slenderness KL/r is 217.4, so the column is slender and Euler applies.

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