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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 dot product a·b of a = (2, −4, −7) and b = (−6, 2, 3).
a = (2, −4, −7)b = (−6, 2, 3)
a·b =
2.Vectors: dot & cross productHard
Find the area A of the parallelogram spanned by a = (−2, −2, 0) and b = (4, −4, −6).
a = (−2, −2, 0)b = (4, −4, −6)
A =
3.Beam support reactionsMedium
The beam has a pin at A and a roller at B, 2.5 m from A, and overhangs B to the free end C. Find the vertical reaction RA, taking upward as positive.
AB = 2.5 mAC = 4 mP1 = 75 kN at 0.5 mP2 = 50 kN at 4 m
RA = 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 m from A, inclined at 45° to the horizontal, and a vertical load Q acts at 3 m from A. Find the angle α between the total reaction force at the pin A and the horizontal.
L = 4 mP = 25 kN at 1 m, θ = 45°Q = 40 kN at 3 m
α = °
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.
D: 40 kN ↓F: 40 kN ↓G: 40 kN ↓H: 10 kN ↓
1member AB
2member BC
3member CD
4member DE
5member FG
6member GH
7member AF
8member HE
9member BF
10member CG
11member DH
12member FC
13member HC
6.Truss: method of jointsHard
Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.
D: 10 kN ↓F: 15 kN ↓G: 10 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 FC
13member HC
7.Frames & machinesMedium
A horizontal beam AB is pinned to a wall at A and propped by the strut CD, pinned to the wall at C, 0.8 m below A. A load of 8 kN hangs at B. Neglect the weights. Find the magnitude of the pin force A on the beam at A.
P = 8 kNAD = 0.75 mAB = 2 mAC = 0.8 m
A = kN
8.Frames & machinesHard
The frame shown has pins at A, B and C; A and C are fixed to the ground 5 m apart, and B is 2 m high and 1.5 m to the right of A. Member AB carries a vertical load of 8 kN at K, 1.125 m (horizontally) from A, and a horizontal load of 4 kN to the right acts at B on member AB. Find the size of the horizontal reaction Ax at A.
P = 8 kNQ = 4 kNL = 5 mH = 2 mhorizontal distance A to B = 1.5 mhorizontal distance A to K = 1.125 m
Ax = kN
9.Shear force & bending momentMedium
Find the shear force Vx at the section x = 3.5 m from support A. Use sagging moments positive and upward shear on the left face positive.
L = 8 mw = 5 kN/m from 0 to 8 mx = 3.5 m
Vx = kN
10.Shear force & bending momentHard
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.
40 kN @ 0 m45 kN @ 12 m12 kN/m, 2–8 m
1A (x = 0 m)
2B (x = 2 m)
3C (x = 4 m)
4D (x = 6 m)
5E (x = 8 m)
6F (x = 10 m)
7G (x = 12 m)
11.Centroid & second moment of areaMedium
Find the second moment of area Ix of the T-section about the horizontal axis x through its centroid.
bf = 160 mmtf = 20 mmtw = 8 mmhw = 280 mm
Ix = × 10⁶ mm⁴
12.Centroid & second moment of areaHard
A cast-iron beam has the T-section shown, with the flange on top. The allowable stresses are 60 MPa in tension and 100 MPa in compression. Find the largest sagging bending moment Mallow the beam can carry.
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.
Negative: the angle between the vectors is more than 90°.
2.Vectors: dot & cross product
Answer: A = 23.32
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
(a × b)x = aybz − azby = (−2)·(−6) − 0·(−4) = 12
(a × b)y = azbx − axbz = 0·4 − (−2)·(−6) = −12
(a × b)z = axby − aybx = (−2)·(−4) − (−2)·4 = 16
Area
A = |a × b| = √(12² + (−12)² + 16²) = 23.32
Answer
A = 23.32
3.Beam support reactions
Answer: RA = 30 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·2.5 = 75·0.5 + 50·4
RB = 237.5/2.5 = 95 kN
Vertical equilibrium
RA = 75 + 50 − 95 = 30 kN
Answer
RA = 30 kN
4.Beam support reactions
Answer: α = 52.76 °
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·4 = 17.68·1 + 40·3 → RB = 34.42 kN
Ay = 17.68 + 40 − 34.42 = 23.26 kN
Ax = Ph = 17.68 kN
Direction at A
α = atan(Ay/Ax) = atan(23.26/17.68) = 52.76°
Answer
α = 52.76°
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).
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).
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.
Strut force
FCD = 8·2/(0.7295·0.75) = 29.24 kN
Components at A
Ax = FCD·cos α = 20 kN (towards the wall)
Ay = 8 − FCD·sin α = −13.33 kN
Magnitude
A = √(20² + −13.33²) = 24.04 kN
Answer
A = 24.04 kN
A negative Ay means the wall pulls the beam down at A.
8.Frames & machines
Answer: Ax = 1.95 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.
Direction of BC
BC = 4.031 m, cos = 0.8682, sin = 0.4961
Whole frame: moments about A
FBC·0.4961·5 = 8·1.125 + 4·2
FBC = 6.853 kN
Horizontal balance
Ax + 4 − 6.853·0.8682 = 0
Ax = 1.95 kN → |Ax| = 1.95 kN
Answer
Ax = 1.95 kN
Ax points to the right.
9.Shear force & bending moment
Answer: Vx = 2.5 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.
UDL resultant: W = w·ℓ = 5·8 = 40 kN, acting at x̄ = 4 m
ΣMA = 0: RB·8 = 5·8·4
RB = 160/8 = 20 kN
ΣFy = 0: RA = 40 − 20 = 20 kN
Cut at x = 3.5 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 = 20 − 5·3.5 = 2.5 kN
Answer
Vx = 2.5 kN
10.Shear force & bending moment
Answer: E (x = 8 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 = 59.33 kN, R2 = 97.67 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 = −80, MC = −65.33, MD = −98.67, ME = −180 kN·m, MF = −90, MG = 0
Answer
Mark E: ME = −180 kN·m
11.Centroid & second moment of area
Answer: Ix = 44.39 × 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 two rectangles
Measure every y from the bottom edge.
flange: Af = 160·20 = 3200 mm², yf = 280 + 20/2 = 290 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.