A steel-reinforced timber beam is analyzed as one of the composite sections using a transformed area. Given modulus of material 2 (E_2) = 13,700 ksi; modulus of material 1 (base) (E_1) = 26,800 ksi, determine the modular ratio (n).

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{E_2}{E_1}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 13,700 ksi, modulus of material 1 (base) (E_1) = 26,800 ksi$$

4. Step 4 — Substitute the given values:

   $$n = \dfrac{E_2}{E_1}$$

5. Step 5 — Evaluate:

   $$n = 0.5112$$

6. Step 6 — Check: returning n = 0.5112 to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0224 — kept a factor of two that cancels in the correct rearrangement.
• 0.2556 — dropped that same factor in the other direction.
• 0.5623 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A flitch beam with steel plates bolted to wood is treated as composite sections. Given modulus of material 1 (base) (E_1) = 5,100 ksi; modular ratio (n) = 14.4500, determine the modulus of material 2 (E_2) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_2 is given, so isolate E_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_2:

   $$E_{2} = n E_1$$

3. Step 3

   $$List the givens: modulus of material 1 (base) (E_1) = 5,100 ksi, modular ratio (n) = 14.4500$$

4. Step 4 — Substitute the given values:

   $$E_{2} = 14.4500 E_1$$

5. Step 5 — Evaluate:

   $$E_{2} = 73695\ \text{ksi}$$

6. Step 6 — Check: returning E_2 = 73,695 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 147,390 — kept a factor of two that cancels in the correct rearrangement.
• 36,848 — dropped that same factor in the other direction.
• 81,065 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A composite steel-concrete deck is transformed into composite sections for stress analysis. Given modulus of material 2 (E_2) = 5,000 ksi; modular ratio (n) = 4.6500, determine the modulus of material 1 (base) (E_1) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_1 is given, so isolate E_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_1:

   $$E_{1} = \dfrac{E_2}{n}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 5,000 ksi, modular ratio (n) = 4.6500$$

4. Step 4 — Substitute the given values:

   $$E_{1} = \dfrac{E_2}{4.6500}$$

5. Step 5 — Evaluate:

   $$E_{1} = 1075\ \text{ksi}$$

6. Step 6 — Check: returning E_1 = 1,075 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2,151 — kept a factor of two that cancels in the correct rearrangement.
• 537.6 — dropped that same factor in the other direction.
• 1,183 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A steel-reinforced timber beam is analyzed as one of the composite sections using a transformed area. Given modulus of material 2 (E_2) = 16,100 ksi; modulus of material 1 (base) (E_1) = 13,700 ksi, determine the modular ratio (n).

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{E_2}{E_1}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 16,100 ksi, modulus of material 1 (base) (E_1) = 13,700 ksi$$

4. Step 4 — Substitute the given values:

   $$n = \dfrac{E_2}{E_1}$$

5. Step 5 — Evaluate:

   $$n = 1.1752$$

6. Step 6 — Check: returning n = 1.1752 to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.3504 — kept a factor of two that cancels in the correct rearrangement.
• 0.5876 — dropped that same factor in the other direction.
• 1.2927 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A flitch beam with steel plates bolted to wood is treated as composite sections. Given modulus of material 1 (base) (E_1) = 14,100 ksi; modular ratio (n) = 4.6000, determine the modulus of material 2 (E_2) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_2 is given, so isolate E_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_2:

   $$E_{2} = n E_1$$

3. Step 3

   $$List the givens: modulus of material 1 (base) (E_1) = 14,100 ksi, modular ratio (n) = 4.6000$$

4. Step 4 — Substitute the given values:

   $$E_{2} = 4.6000 E_1$$

5. Step 5 — Evaluate:

   $$E_{2} = 64860\ \text{ksi}$$

6. Step 6 — Check: returning E_2 = 64,860 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 129,720 — kept a factor of two that cancels in the correct rearrangement.
• 32,430 — dropped that same factor in the other direction.
• 71,346 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A composite steel-concrete deck is transformed into composite sections for stress analysis. Given modulus of material 2 (E_2) = 13,700 ksi; modular ratio (n) = 1.8500, determine the modulus of material 1 (base) (E_1) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_1 is given, so isolate E_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_1:

   $$E_{1} = \dfrac{E_2}{n}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 13,700 ksi, modular ratio (n) = 1.8500$$

4. Step 4 — Substitute the given values:

   $$E_{1} = \dfrac{E_2}{1.8500}$$

5. Step 5 — Evaluate:

   $$E_{1} = 7405\ \text{ksi}$$

6. Step 6 — Check: returning E_1 = 7,405 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 14,811 — kept a factor of two that cancels in the correct rearrangement.
• 3,703 — dropped that same factor in the other direction.
• 8,146 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A steel-reinforced timber beam is analyzed as one of the composite sections using a transformed area. Given modulus of material 2 (E_2) = 25,800 ksi; modulus of material 1 (base) (E_1) = 11,100 ksi, determine the modular ratio (n).

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{E_2}{E_1}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 25,800 ksi, modulus of material 1 (base) (E_1) = 11,100 ksi$$

4. Step 4 — Substitute the given values:

   $$n = \dfrac{E_2}{E_1}$$

5. Step 5 — Evaluate:

   $$n = 2.3243$$

6. Step 6 — Check: returning n = 2.3243 to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 4.6486 — kept a factor of two that cancels in the correct rearrangement.
• 1.1622 — dropped that same factor in the other direction.
• 2.5568 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A flitch beam with steel plates bolted to wood is treated as composite sections. Given modulus of material 1 (base) (E_1) = 7,900 ksi; modular ratio (n) = 3.3500, determine the modulus of material 2 (E_2) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_2 is given, so isolate E_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_2:

   $$E_{2} = n E_1$$

3. Step 3

   $$List the givens: modulus of material 1 (base) (E_1) = 7,900 ksi, modular ratio (n) = 3.3500$$

4. Step 4 — Substitute the given values:

   $$E_{2} = 3.3500 E_1$$

5. Step 5 — Evaluate:

   $$E_{2} = 26465\ \text{ksi}$$

6. Step 6 — Check: returning E_2 = 26,465 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 52,930 — kept a factor of two that cancels in the correct rearrangement.
• 13,233 — dropped that same factor in the other direction.
• 29,112 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A composite steel-concrete deck is transformed into composite sections for stress analysis. Given modulus of material 2 (E_2) = 14,500 ksi; modular ratio (n) = 7.3000, determine the modulus of material 1 (base) (E_1) in ksi.

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except E_1 is given, so isolate E_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for E_1:

   $$E_{1} = \dfrac{E_2}{n}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 14,500 ksi, modular ratio (n) = 7.3000$$

4. Step 4 — Substitute the given values:

   $$E_{1} = \dfrac{E_2}{7.3000}$$

5. Step 5 — Evaluate:

   $$E_{1} = 1986\ \text{ksi}$$

6. Step 6 — Check: returning E_1 = 1,986 ksi to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3,973 — kept a factor of two that cancels in the correct rearrangement.
• 993.2 — dropped that same factor in the other direction.
• 2,185 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections

A steel-reinforced timber beam is analyzed as one of the composite sections using a transformed area. Given modulus of material 2 (E_2) = 8,900 ksi; modulus of material 1 (base) (E_1) = 5,400 ksi, determine the modular ratio (n).

Start with the thinking

• The governing relation printed in this handbook section is Transformed area (composite sections).
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Composite sections made of two materials are analyzed using the modular ratio to transform one material into an equivalent area of the other.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$n = \dfrac{E_2}{E_1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{E_2}{E_1}$$

3. Step 3

   $$List the givens: modulus of material 2 (E_2) = 8,900 ksi, modulus of material 1 (base) (E_1) = 5,400 ksi$$

4. Step 4 — Substitute the given values:

   $$n = \dfrac{E_2}{E_1}$$

5. Step 5 — Evaluate:

   $$n = 1.6481$$

6. Step 6 — Check: returning n = 1.6481 to

   $$n = \dfrac{E_2}{E_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.2963 — kept a factor of two that cancels in the correct rearrangement.
• 0.8241 — dropped that same factor in the other direction.
• 1.8130 — rounded an intermediate value before the final step.

Reference: FE Handbook — Mechanics of Materials: Composite Sections