Roark's Formulas: Table 8.1-1

Shear, Moment, Slope, and Deflection Formulas for Elastic Straight Beams.

TOC

Source: Roark's Formulas for Stress and Strain, 9th edition, Table 8.1, pp. 203-204, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

1. Concentrated intermediate load

The beam has constant $E$ and $I$. Coordinate $x$ runs from the left end $A$ to the right end $B$, with $0\leq x\leq l$. Reactions and deflections are positive upward; slope is positive upward to the right; positive moment compresses the upper fibers. A guided end can translate vertically but cannot rotate.

Macaulay brackets satisfy ${\langle x-a\rangle}^n=0$ for $x<a$ and ${(x-a)}^n$ for $x>a$; the zeroth power is a unit step. Both one-sided limits are included when finding extrema at a jump. Cases with $a=0$ or $a=l$ are interpreted as limits from within the span.

	Transverse shear: $V=R_A-W{\langle x-a\rangle}^0$
	Bending moment: $M=M_A+R_Ax-W\langle x-a\rangle$
	Slope: $\theta=\theta_A+\frac{M_Ax}{EI}+\frac{R_Ax^2}{2EI}-\frac{W}{2EI}{\langle x-a\rangle}^2$
	Deflection: $y=y_A+\theta_Ax+\frac{M_Ax^2}{2EI}+\frac{R_Ax^3}{6EI}-\frac{W}{6EI}{\langle x-a\rangle}^3$

Selected maximum formulas below assume the positive load, couple, imposed displacement, or thermal curvature shown. Signed values are retained. The current-input tables report the minimum, maximum, and greatest absolute value over the span; for negative inputs the signs reverse. A reported position is one location attaining the extremum. The formulas labeled "maximum possible" vary $a$ while holding the other inputs fixed.

1a. Left end free, right end fixed (cantilever)

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=W =$ 1000 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=-W(l-a) =$ -500000 N$\cdot$mm
$\theta_A=\frac{W{(l-a)}^2}{2EI} =$ 0.000625 rad	$\theta_B=0 =$ 0 rad
$y_A=-\frac{W}{6EI}(2l^3-3l^2a+a^3) =$ -0.520833 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-500000 N$\cdot$mm at $x=$ 1000 mm	0 N$\cdot$mm at $x=$ 0 mm	-500000 N$\cdot$mm at $x=$ 1000 mm
Slope	0 rad at $x=$ 1000 mm	0.000625 rad at $x=$ 0 mm	0.000625 rad at $x=$ 0 mm
Deflection	-0.520833 mm at $x=$ 0 mm	-5.55112e-17 mm at $x=$ 1000 mm	-0.520833 mm at $x=$ 0 mm

• $M_{\max}=M_B$; maximum possible value $-Wl =$ -1000000 N$\cdot$mm at $a=0$.
• $\theta_{\max}=\theta_A$; maximum possible value $Wl^2/(2EI) =$ 0.0025 rad at $a=0$.
• $y_{\max}=y_A$; maximum possible value $-Wl^3/(3EI) =$ -1.66667 mm at $a=0$.

1b. Left end guided, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=W =$ 1000 N
$M_A=\frac{W{(l-a)}^2}{2l} =$ 125000 N$\cdot$mm	$M_B=-\frac{W(l^2-a^2)}{2l} =$ -375000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=-\frac{W}{12EI}{(l-a)}^2(l+2a) =$ -0.208333 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-375000 N$\cdot$mm at $x=$ 1000 mm	125000 N$\cdot$mm at $x=$ 0 mm	-375000 N$\cdot$mm at $x=$ 1000 mm
Slope	0 rad at $x=$ 0 mm	0.000351562 rad at $x=$ 625 mm	0.000351562 rad at $x=$ 625 mm
Deflection	-0.208333 mm at $x=$ 0 mm	-1.38778e-17 mm at $x=$ 1000 mm	-0.208333 mm at $x=$ 0 mm

• Maximum positive moment $M_A$; maximum possible value $Wl/2$ at $a=0$.
• Maximum negative moment $M_B$; maximum possible magnitude has signed value $-Wl/2$ at $a=0$.
• $y_{\max}=y_A$; maximum possible value $-Wl^3/(12EI)$ at $a=0$.

1c. Left end simply supported, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=\frac{W}{2l^3}{(l-a)}^2(2l+a) =$ 312.5 N	$R_B=\frac{Wa}{2l^3}(3l^2-a^2) =$ 687.5 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=-\frac{Wa}{2l^2}(l^2-a^2) =$ -187500 N$\cdot$mm
$\theta_A=-\frac{Wa}{4EIl}{(l-a)}^2 =$ -0.00015625 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-187500 N$\cdot$mm at $x=$ 1000 mm	156250 N$\cdot$mm at $x=$ 500 mm	-187500 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.00015625 rad at $x=$ 0 mm	0.000127841 rad at $x=$ 727.273 mm	-0.00015625 rad at $x=$ 0 mm
Deflection	-0.0465847 mm at $x=$ 447.214 mm	0 mm at $x=$ 0 mm	-0.0465847 mm at $x=$ 447.214 mm

• Maximum positive moment $Wa{(l-a)}^2(2l+a)/(2l^3)$ at $x=a$; maximum possible value $0.174Wl$ at $a=0.366l$.
• Maximum negative moment $M_B$; maximum possible magnitude has signed value $-0.1924Wl$ at $a=0.5773l$.
• If $a>(\sqrt{2}-1)l$: $y_{\max}=-\frac{Wa}{6EI}{(l-a)}^2\sqrt{\frac{a}{2l+a}}$ at $x=l\sqrt{\frac{a}{2l+a}}$.
• If $a<(\sqrt{2}-1)l$: $y_{\max}=-\frac{Wa{(l^2-a^2)}^3}{3EI{(3l^2-a^2)}^2}$ at $x=\frac{l(l^2+a^2)}{3l^2-a^2}$. The formulas agree at the transition.
• Maximum possible deflection $-0.0098Wl^3/EI$ at $x=a\approx0.414l$.

1d. Left end fixed, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=\frac{W}{l^3}{(l-a)}^2(l+2a) =$ 500 N	$R_B=\frac{Wa^2}{l^3}(3l-2a) =$ 500 N
$M_A=-\frac{Wa}{l^2}{(l-a)}^2 =$ -125000 N$\cdot$mm	$M_B=-\frac{Wa^2}{l^2}(l-a) =$ -125000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-125000 N$\cdot$mm at $x=$ 0 mm	125000 N$\cdot$mm at $x=$ 500 mm	125000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.000078125 rad at $x=$ 250 mm	0.000078125 rad at $x=$ 750 mm	0.000078125 rad at $x=$ 750 mm
Deflection	-0.0260417 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.0260417 mm at $x=$ 500 mm

• Maximum positive moment $2Wa^2{(l-a)}^2/l^3$ at $x=a$; maximum possible value $Wl/8$ at $a=l/2$.
• Maximum negative moment $M_A$ for $a<l/2$ (use $M_B$ for $a>l/2$); maximum possible magnitude has signed value $-4Wl/27\approx-0.1481Wl$ at $a=l/3$ or $2l/3$.
• For $a>l/2$: $y_{\max}=-\frac{2W{(l-a)}^2a^3}{3EI{(l+2a)}^2}$ at $x=\frac{2al}{l+2a}$. For $a<l/2$, reflect the beam: replace $a$ by $l-a$ and $x$ by $l-x$.
• Maximum possible deflection $-Wl^3/(192EI)$ at $x=a=l/2$.

1e. Left end simply supported, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=\frac{W}{l}(l-a) =$ 500 N	$R_B=\frac{Wa}{l} =$ 500 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\frac{Wa}{6EIl}(2l-a)(l-a) =$ -0.0003125 rad	$\theta_B=\frac{Wa}{6EIl}(l^2-a^2) =$ 0.0003125 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 0 mm	250000 N$\cdot$mm at $x=$ 500 mm	250000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.0003125 rad at $x=$ 0 mm	0.0003125 rad at $x=$ 1000 mm	0.0003125 rad at $x=$ 1000 mm
Deflection	-0.104167 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.104167 mm at $x=$ 500 mm

• $M_{\max}=R_Aa$ at $x=a$; maximum possible value $Wl/4$ at $a=l/2$.
• For $a<l/2$: $y_{\max}=-\frac{Wa}{3EIl}\left(\frac{l^2-a^2}{3}\right)^{3/2}$ at $x=l-\sqrt{\frac{l^2-a^2}{3}}$. For $a>l/2$, reflect the beam.
• Maximum possible deflection $-Wl^3/(48EI)$ at $x=a=l/2$.
• Maximum slope magnitude is at $A$ for $a<l/2$ and at $B$ for $a>l/2$; the signed value at $A$ reaches $-0.0642Wl^2/EI$ at $a=0.423l$.

1f. Left end guided, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$W =$ 1000 N	Concentrated downward load

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=W =$ 1000 N
$M_A=W(l-a) =$ 500000 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=\frac{W}{2EI}(l^2-a^2) =$ 0.001875 rad
$y_A=-\frac{W(l-a)}{6EI}(2l^2+2al-a^2) =$ -1.14583 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 1000 mm	500000 N$\cdot$mm at $x=$ 0 mm	500000 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0.001875 rad at $x=$ 1000 mm	0.001875 rad at $x=$ 1000 mm
Deflection	-1.14583 mm at $x=$ 0 mm	0 mm at $x=$ 1000 mm	-1.14583 mm at $x=$ 0 mm

• $M_{\max}=M_A$ for $0<x<a$; maximum possible value $Wl$ at $a=0$.
• $\theta_{\max}=\theta_B$; maximum possible value $Wl^2/(2EI)$ at $a=0$.
• $y_{\max}=y_A$; maximum possible value $-Wl^3/(3EI)$ at $a=0$.

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