Structural Analysis: cram pack

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

New problemsBack to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

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​ = 25 kNP2​ = 45 kNABRA​ = ?RB​1 m1.5 m3.5 m

The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RA.

L = 6 mP1 = 25 kN at 1 mP2 = 45 kN at 2.5 m
RA = kN
2.Beam support reactionsHard
w0​ = 12 kN/mP = 30 kNABMA​ = ?4 m1 m

The cantilever beam is fixed to the wall at A. It carries a linearly varying load that is 12 kN/m at the wall and falls to zero at the free end B, plus a point load P at 4 m from the wall. Find the magnitude of the reaction moment MA at the wall.

L = 5 mw0 = 12 kN/mP = 30 kN at 4 m
MA = kN·m
3.Shear force & bending momentMedium
w = 3 kN/mABsection3 m1 m

Find the shear force Vx at the section x = 3 m from support A. Use sagging moments positive and upward shear on the left face positive.

L = 4 mw = 3 kN/m from 0 to 4 mx = 3 m
Vx = kN
4.Shear force & bending momentHard
w = 5 kN/mABC8 m1.5 m

The beam rests on a pin at A and a roller at B and overhangs B by 1.5 m to the free end C. A uniformly distributed load w acts over the whole length A–C. Find the maximum sagging bending moment Mmax between the supports.

L = 9.5 mw = 5 kN/m from 0 to 9.5 msupports at x = 0 and x = 8 m
Mmax = kN·m
5.Truss: method of jointsMedium
ABCDEFGH20 kN40 kN10 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 ↓G: 40 kN ↓H: 10 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
ABCDEFGH15 kN10 kN40 kN20 kN35 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: 15 kN ↓F: 10 kN ↓F: 40 kN →G: 20 kN ↓H: 35 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 = 5 kN/mδL = 4 m

A cantilever carries a uniformly distributed load w over its whole length. Find the deflection δ at the free end.

w = 5 kN/mL = 4 mE = 200 GPa (steel)I = 12 × 10⁶ mm⁴
δ = mm
8.Beam deflectionHard
w = 12 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 = 12 kN/mL = 4 mE = 11 GPa (timber)
Ireq = × 10⁶ mm⁴
9.Column bucklingMedium
PL = 2.5 m40 × 60 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 = 2.5 mE = 70 GPa (aluminium)b × h = 40 × 60 mm
Pcr = kN
10.Column bucklingHard
P = 65 kNL = 2 md = ?cross-section

A solid round column of length L made of steel is fixed at the base and free at the top. It must carry 65 kN with a factor of safety of 2 against buckling. Find the smallest diameter d.

P = 65 kNn = 2L = 2 mE = 200 GPa (steel)
d = mm

3Solutions

1.Beam support reactions

Answer: RA = 47.08 kN

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.

P1​ = 25 kNP2​ = 45 kNABRA​RB​1 m1.5 m3.5 m

Free-body diagram

Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.

Equilibrium

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

ΣMA = 0: RB·6 = 25·1 + 45·2.5
RB = 137.5/6 = 22.92 kN
ΣFy = 0: RA = 70 − 22.92 = 47.08 kN

Answer

RA = 47.08 kN
2.Beam support reactions

Answer: MA = 170 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 triangular load

W = ½·12·5 = 30 kN
x̄ = L/3 = 1.667 m

Moment about the wall

MA = W·x̄ + P·a = 30·1.667 + 30·4 = 170 kN·m

Answer

MA = 170 kN·m
3.Shear force & bending moment

Answer: Vx = −3 kN

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.

Support reactions

Equilibrium of the whole beam.

UDL resultant: W = w·ℓ = 3·4 = 12 kN, acting at x̄ = 2 m
ΣMA = 0: RB·4 = 3·4·2
RB = 24/4 = 6 kN
ΣFy = 0: RA = 12 − 6 = 6 kN
w = 3 kN/mABRA​ = 6RB​ = 6section3 m1 m

Cut at x = 3 m and keep the left part

Only forces left of the cut appear. A UDL portion acts as one force at the middle of that portion.

V = RA − (all downward loads left of the cut)
Vx = 6 − 3·3 = −3 kN

Answer

Vx = −3 kN
4.Shear force & bending moment

Answer: Mmax = 37.24 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. 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 = 5·9.5 = 47.5 kN, acting at x̄ = 4.75 m
ΣMA = 0: RB·8 = 5·9.5·4.75
RB = 225.6/8 = 28.2 kN
ΣFy = 0: RA = 47.5 − 28.2 = 19.3 kN

Zero shear

RA − w·x0 = 0 → x0 = 19.3/5 = 3.859 m

Maximum sagging moment

Mmax = RA·x0 − w·x0²/2 = RA²/(2w) = 19.3²/(2·5) = 37.24 kN·m

Answer

Mmax = 37.24 kN·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: 50 kN (T), BC: 73.33 kN (T), CD: 73.33 kN (T), DE: 43.33 kN (T), FG: 50 kN (C), GH: 43.33 kN (C), AF: 62.5 kN (C), HE: 54.17 kN (C), BF: 37.5 kN (T), CG: 0, DH: 22.5 kN (T), BG: 29.17 kN (C), DG: 37.5 kN (C)

Answer

Zero-force member: CG
6.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: 70 kN (T), BC: 70 kN (T), CD: 76.67 kN (T), DE: 76.67 kN (T), FG: 86.67 kN (C), GH: 86.67 kN (C), AF: 37.5 kN (C), HE: 95.83 kN (C), BF: 0, CG: 20 kN (C), DH: 15 kN (T), FC: 20.83 kN (T), HC: 12.5 kN (T)

Answer

Zero-force member: BF
7.Beam deflection

Answer: δ = 66.67 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.

w = 5 N/mm, L = 4000 mm
EI = 2.4 × 10¹² N·mm²

Deflection

δ = wL⁴/(8EI) = 5·4000⁴/(8·2.4 × 10¹²) = 66.67 mm

Answer

δ = 66.67 mm
8.Beam deflection

Answer: Ireq = 327.3 × 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) = 5·12·4000⁴/(384·11000·11.11) = 3.273 × 10⁸ mm⁴
Ireq = 327.3 × 10⁶ mm⁴

Answer

Ireq = 327.3 × 10⁶ mm⁴
9.Column buckling

Answer: Pcr = 72.19 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 = 60·40³/12 = 320000 mm⁴

Effective length

KL = 0.7·2500 = 1750 mm

Critical load

Pcr = π²EI/(KL)² = π²·70000·320000/1750² = 72189 N = 72.19 kN

Answer

Pcr = 72.19 kN
10.Column buckling

Answer: d = 68.07 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 = 2·65000 = 130000 N
KL = 2·2000 = 4000 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·130000·4000²/(π³·200000))^(1/4) = 68.07 mm

Answer

d = 68.07 mm

Check: the slenderness KL/r is 235.1, so the column is slender and Euler applies.

ExamProblems · examproblems.com/courses/structural-analysis