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1Formulas and methods
Vectors: dot & cross product
Magnitude
|a| = √(ax² + ay² + az²)
Unit vector
û = a/|a|
Components from an angle
Fx = F·cos θ, Fy = F·sin θ
Dot product
a·b = axbx + ayby + azbz = |a||b|cos θ
Scalar projection of a onto b
a·b/|b|
Work
W = F·d
Cross product
a × b = (aybz − azby, azbx − axbz, axby − aybx)
Areas
parallelogram |a × b|, triangle ½|a × b|
Moment about O
MO = r × F, r = A − O
Volume of a parallelepiped
V = |a·(b × c)|
Method
Write every vector in components; for a force given by a magnitude and an angle, use F·cos θ and F·sin θ.
For lengths and directions use the magnitude √(x² + y² + z²) and the unit vector a/|a|.
Use the dot product for angles, projections, perpendicularity and work: a·b = |a||b|cos θ.
Use the cross product for perpendicular vectors, areas and moments: write the three components carefully, keeping the order a × b.
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
Draw a free-body diagram and replace each support by its reactions.
Replace every distributed load by a single force at its centroid.
Take moments about one support to find the reaction at the other.
Use ΣFy = 0 to find the remaining reaction, and check with a second moment equation.
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
Find the support reactions for the whole truss.
Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
Assume tension; a negative answer means compression.
Frames & machines
Equilibrium of each member
ΣFx = 0, ΣFy = 0, ΣM = 0
Two-force member
force along the line through its two pins
Pin force
equal and opposite on the two members it joins
Magnitude of a pin force
F = √(Fx² + Fy²)
Lever, pliers, bell crank
Fin·a = Fout·b
Toggle
Flink = F/(2 sin θ), Q = F/(2 tan θ)
Uniform load
resultant w·L at the middle of the load
Method
Draw the whole structure and find the support reactions if there are no more than three unknowns.
Spot the two-force members: loaded only at two pins, their force acts along the line between the pins.
Take the frame apart and draw every member on its own, with equal and opposite pin forces where members meet.
For each member, take moments about a pin with unknown forces, then use ΣFx = 0 and ΣFy = 0; combine components into a magnitude at the end.
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
Find the support reactions from equilibrium of the whole beam.
Cut the beam at the section you need and keep one side.
Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
For the maximum moment, find where the shear force is zero.
Centroid & second moment of area
Centroid
ȳ = ΣAiyi/ΣAi
Rectangle
I = bh³/12
Parallel-axis theorem
I = Ic + A·d²
Composite section
I = Σ(Ic,i + Aidi²)
2Practice problems
1.Vectors: dot & cross productMedium
Find the value of k for which the vectors a = (6, 1, −5) and b = (2, k, −1) are perpendicular.
a = (6, 1, −5)b = (2, k, −1)
k =
2.Vectors: dot & cross productHard
Find the volume V of the parallelepiped whose edges from one corner are a = (3, −2, −2), b = (−2, −2, −5) and c = (4, 2, −3).
a = (3, −2, −2)b = (−2, −2, −5)c = (4, 2, −3)
V =
3.Beam support reactionsMedium
The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RB.
L = 6 mP1 = 10 kN at 2 mP2 = 10 kN at 3 m
RB = kN
4.Beam support reactionsHard
The simply supported beam has a pin at A and a roller at B. A load P acts at 1.5 m from A, inclined at 45° to the horizontal, and a vertical load Q acts at 3 m from A. Find the magnitude of the total reaction force RA at the pin A.
L = 5 mP = 25 kN at 1.5 m, θ = 45°Q = 55 kN at 3 m
RA = kN
5.Truss: method of jointsMedium
Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.
B: 10 kN ↓D: 40 kN ↓H: 20 kN ↓
1member AB
2member BC
3member CD
4member DE
5member FG
6member GH
7member AF
8member HE
9member BF
10member CG
11member DH
12member BG
13member DG
6.Truss: method of jointsHard
Use the method of joints to find the force FAB in member AB. Give tension as positive and compression as negative.
span = 6 mheight = 3 mB is 2 m from Avertical load = 40 kNhorizontal load = 25 kN →
FAB = kN
7.Frames & machinesMedium
The frame shown has pins at A, B and C; A and C are fixed to the ground 3 m apart, and B is 4 m above the ground and 1.35 m to the right of A. A vertical load of 10 kN acts on member AB at K, 0.675 m (horizontally) from A. Find the force FBC in member BC.
P = 10 kNL = 3 mH = 4 mhorizontal distance A to B = 1.35 mhorizontal distance A to K = 0.675 m
FBC = kN
8.Frames & machinesHard
The A-frame shown stands on a pin at A and a roller at B, 4 m apart. The legs meet at the pin C, 3 m high, and are tied by the horizontal bar DE at 1.2 m. A horizontal force of 4 kN to the right acts at C. Find the tension T in the tie.
P = 4 kNAB = 4 mheight of C = 3 mheight of DE = 1.2 m
T = kN
9.Shear force & bending momentMedium
Find the bending moment Mx at the section x = 2 m from support A. Use sagging moments positive and upward shear on the left face positive.
L = 8 mw = 8 kN/m from 0 to 8 mx = 2 m
Mx = kN·m
10.Shear force & bending momentHard
The simply supported beam carries a uniformly distributed load over part of its span. Find the distance x0 from A to the section where the bending moment is largest.
L = 8 mw = 12 kN/m from 5 to 8 m
x0 = m
11.Centroid & second moment of areaMedium
A channel section lies with its open side up: a base plate 180 mm wide with two upright legs, 120 mm high overall. Base and legs are all 10 mm thick. Find its second moment of area Ix about the horizontal axis x through its centroid.
b = 180 mmh = 120 mmt = 10 mm
Ix = × 10⁶ mm⁴
12.Centroid & second moment of areaHard
The T-section is used as a beam with the flange on top and carries a sagging bending moment of 15 kN·m. Find the bending stress σbottom at the bottom edge.
Dot product:a·b = axbx + ayby + azbz = |a|·|b|·cos θ, a scalar. It gives the angle between two vectors, cos θ = a·b/(|a|·|b|), and it is zero when they are perpendicular. The scalar projection of a onto b is a·b/|b|, and a constant force F moving through a displacement d does work W = F·d.
Dot product equals zero
a·b = 6·2 + 1·k + (−5)·(−1) = 0
17 + 1·k = 0
Solve for k
k = −17/1 = −17
Answer
k = −17
2.Vectors: dot & cross product
Answer: V = 92
Cross product:a × b = (aybz − azby, azbx − axbz, axby − aybx) is perpendicular to both vectors, and |a × b| = |a|·|b|·sin θ is the area of the parallelogram they span (the triangle has half of it). Order matters: b × a = −(a × b). The moment of a force F applied at point A about point O is MO = r × F, with r = A − O.
Cross product b × c
(b × c)x = bycz − bzcy = (−2)·(−3) − (−5)·2 = 16
(b × c)y = bzcx − bxcz = (−5)·4 − (−2)·(−3) = −26
(b × c)z = bxcy − bycx = (−2)·2 − (−2)·4 = 4
Dot product with a
a·(b × c) = 3·16 + (−2)·(−26) + (−2)·4 = 92
V = |92| = 92
Answer
V = 92
3.Beam support reactions
Answer: RB = 8.333 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.
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 = 10·2 + 10·3
RB = 50/6 = 8.333 kN
ΣFy = 0: RA = 20 − 8.333 = 11.67 kN
Answer
RB = 8.333 kN
4.Beam support reactions
Answer: RA = 38.65 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 = 25·sin 45° = 17.68 kN
Ph = 25·cos 45° = 17.68 kN
Reactions
ΣMA = 0: RB·5 = 17.68·1.5 + 55·3 → RB = 38.3 kN
Ay = 17.68 + 55 − 38.3 = 34.37 kN
Ax = Ph = 17.68 kN
Resultant at A
RA = √(17.68² + 34.37²) = 38.65 kN
Answer
RA = 38.65 kN
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).
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.
Support reactions
Equilibrium of the whole truss. Moments about A, counter-clockwise positive.
ΣMA = 0: Cy·6 = 40·2 + 25·3 → Cy = 25.83 kN
ΣFy = 0: Ay = 40 − 25.83 = 14.17 kN
ΣFx = 0: Ax = −25 kN
Joint A
Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.
sin α = 3/3.606 = 0.8321
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −14.17/0.8321 = −17.03 kN
Answer
FAB = −17.03 kN (compression)
7.Frames & machines
Answer: FBC = 2.434 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Two-force member BC
BC = √(1.65² + 4²) = 4.327 m
Whole frame: moments about A
The ground pushes on BC at C along CB; its horizontal component passes through A, only the vertical one has a moment.
FBC·(4/4.327)·3 = 10·0.675
FBC = 2.434 kN (compression)
Answer
FBC = 2.434 kN
8.Frames & machines
Answer: T = 3.333 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Whole frame: moments about A
By·4 = 4·3
By = 3 kN
Leg BC: moments about C
3·2 = T·1.8
T = 3.333 kN
Answer
T = 3.333 kN
A positive T means the tie is in tension, as assumed.
9.Shear force & bending moment
Answer: Mx = 48 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.
Support reactions
Equilibrium of the whole beam.
UDL resultant: W = w·ℓ = 8·8 = 64 kN, acting at x̄ = 4 m
ΣMA = 0: RB·8 = 8·8·4
RB = 256/8 = 32 kN
ΣFy = 0: RA = 64 − 32 = 32 kN
Cut at x = 2 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.
M = RA·x − (each load left of the cut × its distance to the cut)
Mx = 32·2 − 8·2·1 = 48 kN·m
Answer
Mx = 48 kN·m
10.Shear force & bending moment
Answer: x0 = 5.563 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 = 12·3 = 36 kN, acting at x̄ = 6.5 m
ΣMA = 0: RB·8 = 12·3·6.5
RB = 234/8 = 29.25 kN
ΣFy = 0: RA = 36 − 29.25 = 6.75 kN
Zero shear
The shear force starts at RA, stays constant over any unloaded part and falls with slope w inside the load.
RA − w·(x0 − 5) = 0
x0 = 5 + 6.75/12 = 5.563 m
Answer
x0 = 5.563 m
11.Centroid & second moment of area
Answer: Ix = 5.797 × 10⁶ mm⁴
Composite areas: split the section into rectangles. The centroid is the area-weighted average ȳ = ΣAᵢyᵢ/ΣAᵢ. The second moment of area about the centroidal axis adds each part’s own bh³/12 and its parallel-axis term A·d², where d is the distance from the part’s centroid to the overall centroid. Bending stress is σ = M·y/I.
Split into rectangles
Take the base at full width and the two legs above it (height h − t), so nothing is counted twice.
base: A1 = 180·10 = 1800 mm², y1 = 5 mm
two legs: A2 = 2·10·110 = 2200 mm², y2 = 10 + 110/2 = 65 mm
Composite areas: split the section into rectangles. The centroid is the area-weighted average ȳ = ΣAᵢyᵢ/ΣAᵢ. The second moment of area about the centroidal axis adds each part’s own bh³/12 and its parallel-axis term A·d², where d is the distance from the part’s centroid to the overall centroid. Bending stress is σ = M·y/I.
Split into two rectangles
Measure every y from the bottom edge.
flange: Af = 260·25 = 6500 mm², yf = 300 + 25/2 = 312.5 mm