Physics quiz question: concrete vs. glued laminated wood beam contraction

Physics 205A Quiz 7, fall semester 2019
Cuesta College, San Luis Obispo, CA

"Finger joints in glue-lam beams, Polkky mill, Kuusamo2"
Eli Sagor
https://flic.kr/p/9xRT62

A concrete beam (linear expansion coefficient 9.8×10^–6 K^–1) and glued laminated wood beam (linear expansion coefficient 3.6×10^–6 K^–1[*]) have the same length at room temperature (293 K). If they both decrease in temperature by the same amount, the length of the concrete beam will be __________ the length of the wood beam. (A) shorter than.
(B) equal to.
(C) longer than.
(D) (Not enough information is given.)

[*] awc.org/pdf/codes-standards/publications/archives/lrfd/AWC-LRFD1996-Glulam-0203.pdf.

Correct answer: (A)

The relation between the change in length ∆L due to a temperature change ∆T is given by:

α_concrete·∆T = ∆L_concrete/L,

and similarly for the glued laminated beam:

α_wood·∆T = ∆L_wood/L,

where the original length L and the temperature change ∆T is the same for both the concrete and glued laminated beam. Solving for this common factor for both beams:

L·∆T = ∆L_concrete/α_concrete,

L·∆T = ∆L_wood/α_wood,

we can then set this common factor for both beams equal to each other:

L·∆T = L·∆T,

∆L_concrete/α_concrete = ∆L_wood/α_wood.

From inspection, since α_concrete > α_wood, then for the equality to hold, ∆L_concrete > ∆L_wood, and so the concrete beam would contract more, and as a result be shorter than the glued laminated beam.

Sections 70854, 70855
Exam code: quiz07VlnC
(A) : 18 students
(B) : 7 students
(C) : 26 students
(D) : 0 students

Success level: 35%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.05