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​ = 20 kNP2​ = 30 kNABCRA​ = ?3 m5 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 = 20 kN at 3 mP2 = 30 kN at 10 m
RA = kN
2.Beam support reactionsHard
w = 2 kN/mP = 40 kNABCRA​ = ?2.5 m0.5 m2 m

The beam has a pin at A and a roller at B, 3 m from A, and overhangs B to the free end C. It carries a point load P at 2.5 m from A and a uniformly distributed load w from 2.5 m to the free end at 5 m. Find the vertical reaction RA, taking upward as positive.

AB = 3 mAC = 5 mP = 40 kN at 2.5 mw = 2 kN/m from 2.5 to 5 m
RA = kN
3.Shear force & bending momentMedium
w = 10 kN/mP = 10 kNAB1 m1.5 m

The cantilever is fixed at A. It carries a uniformly distributed load w over its whole length and a point load P at 1 m from A. Find the bending moment MA at the fixed support A. Use sagging moments positive and upward shear on the left face positive.

L = 2.5 mP = 10 kN at 1 mw = 10 kN/m from 0 to 2.5 m
MA = kN·m
4.Shear force & bending momentHard
w = 10 kN/mABC6 m2 m

The beam rests on a pin at A and a roller at B and overhangs B by 2 m to the free end C. A uniformly distributed load w acts over the whole length A–C. Find the distance xc from A to the point of contraflexure between the supports (where the bending moment changes sign).

L = 8 mw = 10 kN/m from 0 to 8 msupports at x = 0 and x = 6 m
xc = m
5.Truss: method of jointsMedium
ABCDEFGH35 kN25 kN15 kN40 kN4 × 4 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.

D: 35 kN ↓F: 25 kN ↓G: 15 kN ↓H: 40 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 BG
  13. 13member DG
6.Truss: method of jointsHard
ABCDEFGH20 kN20 kN35 kN20 kN4 × 4 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: 20 kN ↓F: 20 kN ↓F: 35 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 BG
  13. 13member DG
7.Beam deflectionMedium
w = 12 kN/mABθ2 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 = 12 kN/mL = 2 mE = 11 GPa (timber)I = 160 × 10⁶ mm⁴
θA = rad
8.Beam deflectionHard
w = 4 kN/mδL = 2.5 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 = 4 kN/mL = 2.5 mE = 200 GPa (steel)
Ireq = × 10⁶ mm⁴
9.Column bucklingMedium
PL = 6 md = 30 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 = 6 mE = 200 GPa (steel)d = 30 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 the base and pinned at the top. 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: RA = 5 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 = 20·3 + 30·10
RB = 360/8 = 45 kN

Vertical equilibrium

RA = 20 + 30 − 45 = 5 kN

Answer

RA = 5 kN
2.Beam support reactions

Answer: RA = 5.417 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 = 2·2.5 = 5 kN, x̄ = (2.5 + 5)/2 = 3.75 m

Equilibrium

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

ΣMA = 0: RB·3 = 40·2.5 + 5·3.75
RB = 118.8/3 = 39.58 kN
RA = 45 − 39.58 = 5.417 kN

Answer

RA = 5.417 kN
3.Shear force & bending moment

Answer: MA = −41.25 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.

Keep the free-end part

P: −10·1 = −10 kN·m
w: −10·2.5²/2 = −31.25 kN·m
MA = −10 − 31.25 = −41.25 kN·m

Answer

MA = −41.25 kN·m

Negative means hogging: tension on the top face.

4.Shear force & bending moment

Answer: xc = 5.333 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. Maximum moment: since dM/dx = V, the bending moment has its peak where the shear force passes through zero, either at a point load or inside a distributed load.

Support reactions

Equilibrium of the whole beam, including the load on the overhang.

UDL resultant: W = w·ℓ = 10·8 = 80 kN, acting at x̄ = 4 m
ΣMA = 0: RB·6 = 10·8·4
RB = 320/6 = 53.33 kN
ΣFy = 0: RA = 80 − 53.33 = 26.67 kN

Zero moment

Between the supports M(x) = RA·x − w·x²/2. Apart from x = 0 it is zero at x = 2RA/w.

RA·xc − w·xc²/2 = 0
xc = 2·26.67/10 = 5.333 m

Answer

xc = 5.333 m
5.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 C

BC and CD 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: 60 kN (T), BC: 86.67 kN (T), CD: 86.67 kN (T), DE: 93.33 kN (T), FG: 60 kN (C), GH: 93.33 kN (C), AF: 75 kN (C), HE: 116.7 kN (C), BF: 20 kN (T), CG: 0, DH: 30 kN (T), BG: 33.33 kN (C), DG: 8.333 kN (T)

Answer

Zero-force member: CG
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 C

BC and CD 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: 79.58 kN (T), BC: 70.83 kN (T), CD: 70.83 kN (T), DE: 35.42 kN (T), FG: 79.58 kN (C), GH: 35.42 kN (C), AF: 55.73 kN (C), HE: 44.27 kN (C), BF: 13.44 kN (T), CG: 0, DH: 26.56 kN (T), BG: 10.94 kN (T), DG: 44.27 kN (C)

Answer

Zero-force member: CG
7.Beam deflection

Answer: θA = 0.002273 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 = 12 N/mm, L = 2000 mm
EI = 1.76 × 10¹² N·mm²

End slope

θA = wL³/(24EI) = 12·2000³/(24·1.76 × 10¹²) = 0.002273 rad

Answer

θA = 0.002273 rad

That is 0.1302°.

8.Beam deflection

Answer: Ireq = 1.465 × 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 = 2500/360 = 6.944 mm

Required second moment of area

I = 5wL⁴/(384·E·δallow) = 5·4·2500⁴/(384·200000·6.944) = 1.465 × 10⁶ mm⁴
Ireq = 1.465 × 10⁶ mm⁴

Answer

Ireq = 1.465 × 10⁶ mm⁴
9.Column buckling

Answer: Pcr = 4.449 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 round section has the same I about every axis.

I = πd⁴/64 = π·30⁴/64 = 39761 mm⁴

Effective length

KL = 0.7·6000 = 4200 mm

Critical load

Pcr = π²EI/(KL)² = π²·200000·39761/4200² = 4449 N = 4.449 kN

Answer

Pcr = 4.449 kN
10.Column buckling

Answer: d = 32.66 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 = 0.7·3000 = 2100 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·25000·2100²/(π³·200000))^(1/4) = 32.66 mm

Answer

d = 32.66 mm

Check: the slenderness KL/r is 257.2, so the column is slender and Euler applies.

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