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​ = 45 kNABRA​RB​ = ?1 m1 m2 m

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

L = 4 mP1 = 45 kN at 1 mP2 = 45 kN at 2 m
RB = kN
2.Beam support reactionsHard
w = 8 kN/mP = 10 kNABRA​RB​ = ?2 m1 m3 m

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

L = 6 mP = 10 kN at 3 mw = 8 kN/m from 0 to 2 m
RB = kN
3.Shear force & bending momentMedium
P1​ = 45 kNP2​ = 30 kNABC1.5 m3.5 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. It carries a point load P1 between the supports and a point load P2 at C. Find the shear force Vx just to the left of support B. Use sagging moments positive and upward shear on the left face positive.

L = 7 mP1 = 45 kN at 1.5 mP2 = 30 kN at 7 msupports at x = 0 and x = 5 m
Vx = kN
4.Shear force & bending momentHard
w = 3 kN/mP = 35 kNAB2.5 m2.5 m

The simply supported beam carries a uniformly distributed load w over its whole span and a point load P. Find the maximum bending moment Mmax in the beam.

L = 5 mP = 35 kN at 2.5 mw = 3 kN/m from 0 to 5 m
Mmax = kN·m
5.Truss: method of jointsMedium
ABCDEFGH40 kN20 kN35 kN15 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: 40 kN ↓D: 20 kN ↓F: 35 kN ↓G: 15 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
ABBCACACB45 kN5 kN8 m6 m3 m

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

span = 8 mheight = 3 mB is 6 m from Avertical load = 45 kNhorizontal load = 5 kN →
FAC = kN
7.Beam deflectionMedium
P = 3 kNABδ1 m1.5 m

A cantilever of length L is fixed at A and carries a point load P at a distance a = 1 m from A. Find the deflection δB at the free end B.

P = 3 kNa = 1 mL = 2.5 mE = 70 GPa (aluminium)I = 36 × 10⁶ mm⁴
δB = mm
8.Beam deflectionHard
w = 12 kN/mP = 23 kNδL = 5 m

A simply supported beam carries a uniformly distributed load w and a point load P at midspan. Using superposition, find the midspan deflection δmax.

P = 23 kNw = 12 kN/mL = 5 mE = 200 GPa (steel)I = 23 × 10⁶ mm⁴
δmax = mm
9.Column bucklingMedium
PL = 3 md = 50 mmcross-section

A column of length L made of aluminium is pinned at both ends. Using Euler’s formula, find its critical buckling load Pcr.

L = 3 mE = 70 GPa (aluminium)d = 50 mm
Pcr = kN
10.Column bucklingHard
PL = 5 md = 60 mmcross-section

A column of length L made of aluminium is pinned at both ends. Find its slenderness and the critical buckling stress σcr from Euler’s formula.

L = 5 mE = 70 GPa (aluminium)d = 60 mm
σcr = MPa

3Solutions

1.Beam support reactions

Answer: RB = 33.75 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​ = 45 kNP2​ = 45 kNABRA​RB​1 m1 m2 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·4 = 45·1 + 45·2
RB = 135/4 = 33.75 kN
ΣFy = 0: RA = 90 − 33.75 = 56.25 kN

Answer

RB = 33.75 kN
2.Beam support reactions

Answer: RB = 7.667 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.

w = 8 kN/mP = 10 kNABRA​RB​2 m1 m3 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.

UDL resultant: W = w·ℓ = 8·2 = 16 kN, acting at x̄ = 1 m
ΣMA = 0: RB·6 = 10·3 + 8·2·1
RB = 46/6 = 7.667 kN
ΣFy = 0: RA = 26 − 7.667 = 18.33 kN

Answer

RB = 7.667 kN
3.Shear force & bending moment

Answer: Vx = −25.5 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. The overhang load also acts about A.

ΣMA = 0: RB·5 = 45·1.5 + 30·7
RB = 277.5/5 = 55.5 kN
ΣFy = 0: RA = 75 − 55.5 = 19.5 kN

Cut just left of B and keep the left part

Vx = RA − P1 = 19.5 − 45 = −25.5 kN

Answer

Vx = −25.5 kN
4.Shear force & bending moment

Answer: Mmax = 53.13 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.

UDL resultant: W = w·ℓ = 3·5 = 15 kN, acting at x̄ = 2.5 m
ΣMA = 0: RB·5 = 35·2.5 + 3·5·2.5
RB = 125/5 = 25 kN
ΣFy = 0: RA = 50 − 25 = 25 kN

Where is V = 0?

V(2.5⁻) = RA − w·2.5 = 17.5 kN
V(2.5⁺) = 17.5 − 35 = −17.5 kN
The shear changes sign at the load, so the maximum is under the load.

Maximum moment

Mmax = RA·2.5 − w·2.5²/2 = 25·2.5 − 3·2.5²/2 = 53.13 kN·m

Answer

Mmax = 53.13 kN·m

It occurs at x = 2.5 m from A.

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: 68.75 kN (T), BC: 62.5 kN (T), CD: 62.5 kN (T), DE: 41.25 kN (T), FG: 68.75 kN (C), GH: 41.25 kN (C), AF: 97.23 kN (C), HE: 58.34 kN (C), BF: 33.75 kN (T), CG: 0, DH: 41.25 kN (T), BG: 8.839 kN (T), DG: 30.05 kN (C)

Answer

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

Answer: FAC = 23.75 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.

ABBCACACB45 kN5 kNAy​Cy​Ax​

Support reactions

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

ΣMA = 0: Cy·8 = 45·6 + 5·3 → Cy = 35.63 kN
ΣFy = 0: Ay = 45 − 35.63 = 9.375 kN
ΣFx = 0: Ax = −5 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 = 3/3.606 = 0.8321, cos γ = 2/3.606 = 0.5547
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −35.63/0.8321 = −42.82 kN
ΣFx = 0: −FAC − FBC·cos γ = 0 → FAC = −(−42.82)·0.5547 = 23.75 kN

Answer

FAC = 23.75 kN (tension)
7.Beam deflection

Answer: δB = 1.29 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 = 3000 N, a = 1000 mm, L = 2500 mm
EI = 2.52 × 10¹² N·mm²

Deflection at the free end

Deflection at the load plus the slope at the load times the remaining length.

δP = Pa³/(3EI) = 0.3968 mm
θP = Pa²/(2EI) = 0.0005952 rad
δB = δP + θP·(L − a) = Pa²(3L − a)/(6EI) = 1.29 mm

Answer

δB = 1.29 mm
8.Beam deflection

Answer: δmax = 34.25 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 = 23000 N, w = 12 N/mm, L = 5000 mm, EI = 4.6 × 10¹² N·mm²

Each load on its own

δP = PL³/(48EI) = 23000·5000³/(48·4.6 × 10¹²) = 13.02 mm
δw = 5wL⁴/(384EI) = 5·12·5000⁴/(384·4.6 × 10¹²) = 21.23 mm

Add them

δmax = 13.02 + 21.23 = 34.25 mm

Answer

δmax = 34.25 mm
9.Column buckling

Answer: Pcr = 23.55 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·50⁴/64 = 306796 mm⁴

Effective length

KL = 1·3000 = 3000 mm

Critical load

Pcr = π²EI/(KL)² = π²·70000·306796/3000² = 23551 N = 23.55 kN

Answer

Pcr = 23.55 kN
10.Column buckling

Answer: σcr = 6.218 MPa

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·60⁴/64 = 636173 mm⁴

Radius of gyration and slenderness

A = 2827 mm²
r = √(I/A) = √(636173/2827) = 15 mm
KL = 1·5000 = 5000 mm
KL/r = 5000/15 = 333.3

Critical stress

σcr = π²E/(KL/r)² = π²·70000/333.3² = 6.218 MPa

Answer

σcr = 6.218 MPa

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