Roark's Formulas: Table 8.1-3

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. 206-208, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

3. Concentrated intermediate moment

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: $V=R_A$
	Bending moment: $M=M_A+R_Ax+M_0{\langle x-a\rangle}^0$
	Slope: $\theta=\theta_A+\frac{M_Ax}{EI}+\frac{R_Ax^2}{2EI}+\frac{M_0}{EI}\langle x-a\rangle$
	Deflection: $y=y_A+\theta_Ax+\frac{M_Ax^2}{2EI}+\frac{R_Ax^3}{6EI}+\frac{M_0}{2EI}{\langle x-a\rangle}^2$

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.

3a. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=M_0 =$ 1000000 N$\cdot$mm
$\theta_A=-\frac{M_0(l-a)}{EI} =$ -0.0025 rad	$\theta_B=0 =$ 0 rad
$y_A=\frac{M_0(l^2-a^2)}{2EI} =$ 1.875 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	1000000 N$\cdot$mm at $x=$ 500 mm	1000000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.0025 rad at $x=$ 0 mm	0 rad at $x=$ 1000 mm	-0.0025 rad at $x=$ 0 mm
Deflection	0 mm at $x=$ 1000 mm	1.875 mm at $x=$ 0 mm	1.875 mm at $x=$ 0 mm

• $M_{\max}=M_0$ on the portion to the right of the applied moment.
• $\theta_{\max}=\theta_A$; maximum possible magnitude has signed value $-M_0l/EI$ at $a=0$.
• $y_{\max}=y_A$; maximum possible value $M_0l^2/(2EI)$ at $a=0$.

3b. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=-\frac{M_0(l-a)}{l} =$ -500000 N$\cdot$mm	$M_B=\frac{M_0a}{l} =$ 500000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=\frac{M_0a(l-a)}{2EI} =$ 0.625 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 $x=$ 0 mm	500000 N$\cdot$mm at $x=$ 500 mm	500000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.00125 rad at $x=$ 500 mm	0 rad at $x=$ 0 mm	-0.00125 rad at $x=$ 500 mm
Deflection	0 mm at $x=$ 1000 mm	0.625 mm at $x=$ 0 mm	0.625 mm at $x=$ 0 mm

• Maximum positive moment $M_B$; maximum possible value $M_0$ at $a=l$.
• Maximum negative moment $M_A$; maximum possible magnitude has signed value $-M_0$ at $a=0$.
• $y_{\max}=y_A$; maximum possible value $M_0l^2/(8EI)$ at $a=l/2$.

3c. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=-\frac{3M_0}{2l^3}(l^2-a^2) =$ -1125 N	$R_B=\frac{3M_0}{2l^3}(l^2-a^2) =$ 1125 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=\frac{M_0}{2l^2}(3a^2-l^2) =$ -125000 N$\cdot$mm
$\theta_A=\frac{M_0}{4EIl}(l-a)(3a-l) =$ 0.0003125 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	-562500 N$\cdot$mm at $x=$ 500 mm	437500 N$\cdot$mm at $x=$ 500 mm	-562500 N$\cdot$mm at $x=$ 500 mm
Slope	-0.000390625 rad at $x=$ 500 mm	0.0003125 rad at $x=$ 0 mm	-0.000390625 rad at $x=$ 500 mm
Deflection	-0.00514403 mm at $x=$ 777.778 mm	0.0694444 mm at $x=$ 333.333 mm	0.0694444 mm at $x=$ 333.333 mm

• Maximum positive moment $M_0+R_Aa$ just right of $x=a$; maximum possible value $M_0$ at $a=0$ or $a=l$, and minimum possible value $0.423M_0$ at $a=0.577l$.
• For $a>0.282l$, maximum negative moment $-3M_0a(l^2-a^2)/(2l^3)$ just left of $x=a$; maximum possible magnitude has signed value $-0.577M_0$ at $a=0.577l$.
• For $a<0.282l$, maximum negative moment $-M_0(l^2-3a^2)/(2l^2)$ at $B$; its greatest magnitude on this interval has signed value $-M_0/2$ at $a=0$.
• For $a>l/3$, maximum positive deflection $\frac{M_0(l-a){(3a-l)}^{3/2}}{6EI\sqrt{3(l+a)}}$ at $x=l\sqrt{\frac{3a-l}{3l+3a}}$; maximum possible value $0.0257M_0l^2/EI$ at $x=0.474l$, $a=0.721l$. There is no positive deflection for $a<l/3$.
• A negative-deflection extremum, when it lies in $a\leq x\leq l$, occurs at $x=\frac{2l^3}{3(l^2-a^2)}\left[1-\frac12\sqrt{1-6{(a/l)}^2+9{(a/l)}^4}\right]$; maximum possible negative deflection $-M_0l^2/(27EI)$ at $x=l/3$, $a=0$. At $a=l$, use the limiting solution.

3d. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=-\frac{6M_0a}{l^3}(l-a) =$ -1500 N	$R_B=-R_A =$ 1500 N
$M_A=-\frac{M_0}{l^2}(l^2-4al+3a^2) =$ 250000 N$\cdot$mm	$M_B=\frac{M_0}{l^2}(3a^2-2al) =$ -250000 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	-500000 N$\cdot$mm at $x=$ 500 mm	500000 N$\cdot$mm at $x=$ 500 mm	500000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.0003125 rad at $x=$ 500 mm	0.000104167 rad at $x=$ 833.333 mm	-0.0003125 rad at $x=$ 500 mm
Deflection	-0.0231481 mm at $x=$ 666.667 mm	0.0231481 mm at $x=$ 333.333 mm	0.0231481 mm at $x=$ 333.333 mm

• Moment just right of $x=a$: $\frac{M_0}{l^3}(4al^2-9a^2l+6a^3)$; greatest possible value $M_0$ at $a=l$.
• Moment just left of $x=a$: $\frac{M_0}{l^3}(4al^2-9a^2l+6a^3-l^3)$; greatest possible negative magnitude has signed value $-M_0$ at $a=0$.
• Positive deflection at the stationary point before the couple: $\frac{2M_A^3}{3R_A^2EI}$ at $x=\frac{l}{3a}(3a-l)$, for $l/3<a<l$. Maximum possible value $0.01617M_0l^2/EI$ at $x=0.565l$, $a=0.767l$; no positive deflection for $a<l/3$.

The source labels the two moment values at $a$ as maxima. For some load positions an end moment has a greater magnitude; the current-input extrema above include both ends and both sides of the couple.

3e. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=-\frac{M_0}{l} =$ -1000 N	$R_B=\frac{M_0}{l} =$ 1000 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\frac{M_0}{6EIl}(2l^2-6al+3a^2) =$ 0.000208333 rad	$\theta_B=\frac{M_0}{6EIl}(l^2-3a^2) =$ 0.000208333 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	-500000 N$\cdot$mm at $x=$ 500 mm	500000 N$\cdot$mm at $x=$ 500 mm	500000 N$\cdot$mm at $x=$ 500 mm
Slope	-0.000416667 rad at $x=$ 500 mm	0.000208333 rad at $x=$ 1000 mm	-0.000416667 rad at $x=$ 500 mm
Deflection	-0.0400938 mm at $x=$ 711.325 mm	0.0400938 mm at $x=$ 288.675 mm	0.0400938 mm at $x=$ 288.675 mm

• Maximum positive moment $M_0(l-a)/l$ just right of $x=a$; maximum possible value $M_0$ at $a=0$.
• Maximum negative moment $-M_0a/l$ just left of $x=a$; maximum possible magnitude has signed value $-M_0$ at $a=l$.
• For $a>(1-1/\sqrt{3})l$, maximum positive deflection $\frac{M_0{(6al-3a^2-2l^2)}^{3/2}}{9\sqrt{3}EIl}$ at $x=\sqrt{2al-a^2-2l^2/3}$.
• Maximum possible positive deflection $0.0642M_0l^2/EI$ at $x=0.577l$, $a=l$; no positive deflection for $a<0.423l$.

3f. 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, $0\leq a\leq l$
$M_0 =$ 1000000 N$\cdot$mm	Applied couple, positive as shown

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=-M_0 =$ -1000000 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=-\frac{M_0a}{EI} =$ -0.0025 rad
$y_A=\frac{M_0a}{2EI}(2l-a) =$ 1.875 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	-1000000 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 500 mm	-1000000 N$\cdot$mm at $x=$ 0 mm
Slope	-0.0025 rad at $x=$ 500 mm	0 rad at $x=$ 0 mm	-0.0025 rad at $x=$ 500 mm
Deflection	0 mm at $x=$ 1000 mm	1.875 mm at $x=$ 0 mm	1.875 mm at $x=$ 0 mm

• $M_{\max}=-M_0$ for $0<x<a$.
• $\theta_{\max}=\theta_B$; maximum possible magnitude has signed value $-M_0l/EI$ at $a=l$.
• $y_{\max}=y_A$; maximum possible value $M_0l^2/(2EI)$ at $a=l$.

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