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eFormulas

Beams

Cantilevers

Calculation Validation
All cell formulas checked with XLC equations against calculation reference.

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Nomenclature
Young's Modulus = E
Beam 2nd moment of area = I
Applied distributed load = W
Applied point load = P
Moment = M
Reaction force = R
Deflection = δ

Case 1) Uniformly Distributed Load on Cantilever
	Native Excel: Click in the text area below and copy. Paste into cell A1 of a worksheet. <table><tbody><tr><td>W =</td> <td>500</td> <td>N</td> <td>lbf</td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td></tr> <tr><td>b =</td> <td>500</td> <td>mm</td> <td>in</td></tr> <tr><td>c =</td> <td>=B2-B3-B6</td> <td>mm</td> <td>in</td></tr> <tr><td>MA =</td> <td>=B1(B3+(B6)/(2))</td> <td>Nmm</td> <td>lbfin</td></tr> <tr><td></td> <td></td> <td></td> <td></td></tr> <tr><td>RA =</td> <td>=B1</td> <td>N</td> <td>lbf</td></tr> <tr><td>δB =</td> <td>=(B1)/(24B4B5)(8B3^3+18B3^2B6+12B3B6^2+3B6^3+12B3^2B7+12B3B6B7+4B6^2B7)</td> <td>mm</td> <td>in</td></tr></tbody></table> XLC Enhanced Excel:* Use XLC’s ‘Web Paste’ button to paste at any position in worksheet. Includes mathematical formatting and XLC equations. <table><tbody><tr><td>W =</td> <td>500</td> <td>N</td> <td>lbf</td> <td></td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td> <td></td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td> <td></td></tr> <tr><td>b =</td> <td>500</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>c =</td> <td>=B2-B3-B6</td> <td>mm</td> <td>in</td> <td>=EQS(B7,"Units= ; EqnPrefix=Eqn. ; EqnNo= 9; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$69_0")</td></tr> <tr><td>MA =</td> <td>=B1(B3+OVD(B6)/UND(2))</td> <td>Nmm</td> <td>lbfin</td> <td>=EQS(B8,"Units= ; EqnPrefix=Eqn. ; EqnNo= 10; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$70_0")</td></tr> <tr><td></td> <td></td> <td></td> <td></td> <td></td></tr> <tr><td>RA =</td> <td>=B1</td> <td>N</td> <td>lbf</td> <td>=EQS(B10,"Units= ; EqnPrefix=Eqn. ; EqnNo= 11; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$72_0")</td></tr> <tr><td>δB =</td> <td>=OVD(B1)/UND(24B4B5)(8B3^3+18B3^2B6+12B3B6^2+3B6^3+12B3^2B7+12B3B6B7+4B6^2*B7)</td> <td>mm</td> <td>in</td> <td>=EQS(B11,"Units= ; EqnPrefix=Eqn. ; EqnNo= 12; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$73_0")</td></tr></tbody></table> Note: Both include SI(mm) & US(in) units (delete as appropriate).
Case 2) Point Load on Cantilever
	Native Excel: Click in the text area below and copy. Paste into cell A1 of a worksheet. <table><tbody><tr><td>P =</td> <td>500</td> <td>N</td> <td>lbf</td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td></tr> <tr><td>b =</td> <td>=B2-B3</td> <td>mm</td> <td>in</td></tr> <tr><td>MA =</td> <td>=B1B3</td> <td>Nmm</td> <td>lbfin</td></tr> <tr><td>RA =</td> <td>=B1</td> <td>N</td> <td>lbf</td></tr> <tr><td>δC =</td> <td>=(B1B3^3)/(3B4B5)</td> <td>mm</td> <td>in</td></tr> <tr><td></td> <td></td> <td></td> <td></td></tr> <tr><td>δB =</td> <td>=(B1B3^3)/(3B4B5)(1+(3B6)/(2B3))</td> <td>mm</td> <td>in</td></tr></tbody></table> XLC Enhanced Excel: Use XLC’s ‘Web Paste’ button to paste at any position in worksheet. Includes mathematical formatting and XLC equations. <table><tbody><tr><td>P =</td> <td>500</td> <td>N</td> <td>lbf</td> <td></td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td> <td></td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td> <td></td></tr> <tr><td>b =</td> <td>=B2-B3</td> <td>mm</td> <td>in</td> <td>=EQS(B6,"Units= ; EqnPrefix=Eqn. ; EqnNo= 13; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$86_0")</td></tr> <tr><td>MA =</td> <td>=B1B3</td> <td>Nmm</td> <td>lbfin</td> <td>=EQS(B7,"Units= ; EqnPrefix=Eqn. ; EqnNo= 14; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$87_0")</td></tr> <tr><td>RA =</td> <td>=B1</td> <td>N</td> <td>lbf</td> <td>=EQS(B8,"Units= ; EqnPrefix=Eqn. ; EqnNo= 15; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$88_0")</td></tr> <tr><td>δC =</td> <td>=OVD(B1B3^3)/UND(3B4B5)</td> <td>mm</td> <td>in</td> <td>=EQS(B9,"Units= ; EqnPrefix=Eqn. ; EqnNo= 16; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$89_0")</td></tr> <tr><td></td> <td></td> <td></td> <td></td> <td></td></tr> <tr><td>δB =</td> <td>=OVD(B1B3^3)/UND(3B4B5)(1+OVD(3B6)/UND(2B3))</td> <td>mm</td> <td>in</td> <td>=EQS(B11,"Units= ; EqnPrefix=Eqn. ; EqnNo= 17; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$91_0")</td></tr></tbody></table> Note: Both include SI(mm) & US(in) units (delete as appropriate).
Case 3) Applied Moment on Cantilever
	Native Excel: Click in the text area below and copy. Paste into cell A1 of a worksheet. <table><tbody><tr><td>M =</td> <td>500000</td> <td>Nmm</td> <td>lbfin</td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td></tr> <tr><td>b =</td> <td>=B2-B3</td> <td>mm</td> <td>in</td></tr> <tr><td>MA =</td> <td>=B1</td> <td>Nmm</td> <td>lbfin</td></tr> <tr><td>RA =</td> <td>0</td> <td>N</td> <td>lbf</td></tr> <tr><td>δC =</td> <td>=(B1B3^2)/(2B4B5)</td> <td>mm</td> <td>in</td></tr> <tr><td></td> <td></td> <td></td> <td></td></tr> <tr><td>δB =</td> <td>=(B1B3^2)/(2B4B5)(1+(2B6)/(B3))</td> <td>mm</td> <td>in</td></tr></tbody></table> XLC Enhanced Excel: Use XLC’s ‘Web Paste’ button to paste at any position in worksheet. Includes mathematical formatting and XLC equations. <table><tbody><tr><td>M =</td> <td>500000</td> <td>Nmm</td> <td>lbfin</td> <td></td></tr> <tr><td>L =</td> <td>3000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>a =</td> <td>2000</td> <td>mm</td> <td>in</td> <td></td></tr> <tr><td>E =</td> <td>210000</td> <td>MPa</td> <td>psi</td> <td></td></tr> <tr><td>I =</td> <td>1000000</td> <td>mm⁴</td> <td>in⁴</td> <td></td></tr> <tr><td>b =</td> <td>=B2-B3</td> <td>mm</td> <td>in</td> <td>=EQS(B6,"Units= ; EqnPrefix=Eqn. ; EqnNo= 18; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$104_0")</td></tr> <tr><td>MA =</td> <td>=B1</td> <td>Nmm</td> <td>lbfin</td> <td>=EQS(B7,"Units= ; EqnPrefix=Eqn. ; EqnNo= 19; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$105_0")</td></tr> <tr><td>RA =</td> <td>0</td> <td>N</td> <td>lbf</td> <td></td></tr> <tr><td>δC =</td> <td>=OVD(B1B3^2)/UND(2B4B5)</td> <td>mm</td> <td>in</td> <td>=EQS(B9,"Units= ; EqnPrefix=Eqn. ; EqnNo= 20; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$107_0")</td></tr> <tr><td></td> <td></td> <td></td> <td></td> <td></td></tr> <tr><td>δB =</td> <td>=OVD(B1B3^2)/UND(2B4B5)(1+OVD(2B6)/UND(B3))</td> <td>mm</td> <td>in</td> <td>=EQS(B11,"Units= ; EqnPrefix=Eqn. ; EqnNo= 21; Multiplication= 0; ShowWorking= 0; EqnStyle= 0; Eqp$H$109_0")</td></tr></tbody></table> Note: Both include SI(mm) & US(in) units (delete as appropriate).
Case 1) Uniformly Distributed Load on Simply Supported Beam
	Native Excel: Click in the text area below and copy. Paste into cell A1 of a worksheet. XL_code XLC Enhanced Excel: Use XLC’s ‘Web Paste’ button to paste at any position in worksheet. Includes mathematical formatting and XLC equations. XLC_code Note: Both include SI(mm) & US(in) units (delete as appropriate).

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Case 1) Uniformly Distributed Load on Cantilever

Case 2) Point Load on Cantilever

Case 3) Applied Moment on Cantilever

Case 1) Uniformly Distributed Load on Simply Supported Beam